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# Nature-Inspired Algorithms for Wireless Sensor Networks: A Comprehensive
Survey
Abhilash Singh Sandeep Sharma Jitendra Singh Indian Institute of Science
Education and Research Bhopal, India School of ICT, Gautam Buddha University,
Greater Noida, India Department of Electrical Engineering, Indian Institute
of Technology Kanpur, India
###### Abstract
In order to solve the critical issues in Wireless Sensor Networks (WSNs), with
concern for limited sensor lifetime, nature-inspired algorithms are emerging
as a suitable method. Getting optimal network coverage is one of those
challenging issues that need to be examined critically before any network
setup. Optimal network coverage not only minimizes the consumption of limited
energy of battery-driven sensors but also reduce the sensing of redundant
information. In this paper, we focus on nature-inspired optimization
algorithms concerning the optimal coverage in WSNs. In the first half of the
paper, we have briefly discussed the taxonomy of the optimization algorithms
along with the problem domains in WSNs. In the second half of the paper, we
have compared the performance of two nature-inspired algorithms for getting
optimal coverage in WSNs. The first one is a combined Improved Genetic
Algorithm and Binary Ant Colony Algorithm (IGA-BACA), and the second one is
Lion Optimization (LO). The simulation results confirm that LO gives better
network coverage, and the convergence rate of LO is faster than that of IGA-
BACA. Further, we observed that the optimal coverage is achieved at a lesser
number of generations in LO as compared to IGA-BACA. This review will help
researchers to explore the applications in this field as well as beyond this
area.
###### keywords:
Optimal Coverage, Bio-inspired Algorithm, Lion Optimization, WSNs.
††journal: Computer Science Review
## 1 Introduction
Sensors in WSNs can sense, collect and transmit information together [1]. All
these tasks need to be done effectively in order to minimize the wastage of
limited sensor battery lifetime. We cannot increase the sensor lifetime by
supplying external or additional energy because most of the sensors are
deployed in hard-to-reach areas [2, 3, 4, 5, 6, 7, 8, 9, 10]. Much work has
been done to increase the lifetime of the sensor node. Liang et al. [11], have
proposed, Huang algorithm, an optimal energy clustering algorithm to ensure
balanced depletion of energy over the whole network which prolongs the
lifetime of the system. Cardei et al. [12], have proposed TianD algorithm, to
extend the operational time of the sensors by arranging them into several
maximal disjoint set covers that are activated successively. However, there
exist some limitations in the above-listed algorithms. Huang algorithm is
highly complex, and if data for communication is large, then it may block the
channel. In contrast, the complexity of TianD algorithm is lower. However, it
is unable to point out the redundant node, which is sensing redundant
information.
In addition to the energy constraint, accurate sensing, and non-redundant
information is a critical challenge in WSNs. In order to sense non-redundant
information, the sensors need to be placed apart at a sufficient distance from
each other so that the overlapping in the sensing region is minimum. However,
if the sensors are placed at a more considerable distance away from each
other, then it will create uncovered areas which are termed as coverage hole
or blind areas. To ensure guaranteed coverage, Wang et al. [13], proposed a
Coverage Configuration Protocol (CCP) which guaranteed coverage and
connectivity with self-configuration for a wide range of applications.
However, the CCP Algorithm gives underperformance if the numbers of sensors
are significant.
After critically analyzing the problem of energy constraint and sensor node
separation (i.e., node placement), we observed that there exist a trade-off
between these two problems. In literature, researchers have proposed
individual solutions to each problem of energy constraint and node placement
but not collectively. Keeping in view the limitations of the above-proposed
solutions and instead considering the problem individually, we have combined
these two problems as a multi-objective optimization problem. To balance this
trade-off, we need to optimize a multi-objective optimization problem. To
balance this trade-off, we need to optimize the multi-objective optimization
problem. After successful optimization, we can achieve optimal coverage with
less number of sensor nodes.
Several reviews are published in context to use of nature-inspired algorithms
in WSNs [14, 15, 16, 17, 18]. However, only a few cover the optimal coverage
aspect in WSNs [19, 20, 21]. In [19], they discussed the various issues that
are generally encountered while using a nature-inspired algorithm-based
optimization technique for sensor deployment that leads to the optimal
coverage. Whereas in [20], they compared three algorithms namely, standard
Multi-Objective Evolutionary Algorithm (MOEA), Non-dominated Sorting Genetic
Algorithm (NSGA-II) and Indicator-Based Evolutionary Algorithms for optimal
coverage in WSNs. Recently, [21] efficiently discussed the theoretical,
mathematical and practical application of nature-inspired algorithms in WSNs.
They discussed the genetic algorithm, evolutionary deferential algorithm, NSGA
and genetic programming in-depth for routing, clustering, coverage and
localization in WSNs. Nevertheless, none of them provides a critical review of
the problem domains in WSNs and in particular of the optimal coverage. In this
paper, firstly, we have briefly discussed the nature-inspired algorithms and
their application in WSNs. We have also discussed the advantages and
disadvantages of the work done by various researchers. Later, we have compared
the performance of two such algorithms for the optimization of a multi-
objective optimization problem stated above. The first algorithm is IGA-BACA
[22, 23, 24, 25]. It is a hybrid of the modified evolutionary and swarm-based
nature-inspired algorithm. In contrast, the second one is LO [26], which is a
purely swarm-based nature-inspired algorithm.
The rest of the paper is organized as follows. In Section 2, we have discussed
the WSN’s problem domains that consist of the critical issues of the WSNs by
categorizing it into four categories which are followed by a brief discussion
of the taxonomy of some of the prominent optimization algorithms. Further, in
Section 3, we have briefly discussed the theoretical and mathematical aspect
of some nature-inspired algorithms. Furthermore, in Section 4, we have
discussed the solutions to the problem domains of WSNs. Afterwards, in Section
5, we have discussed the optimal coverage aspect in detail with respect to
nature-inspired algorithms. Then, we have presented the system model in
Section 6. After that, we have presented the simulation results in Section 7.
Lastly, in Section 8, we have presented the conclusion and the future scope of
the work. For better readability, the outline of the paper is shown is Fig. 1.
Figure 1: Organization of the paper. Figure 2: Problem domains of WSNs.
## 2 WSNs and Optimizations
The critical issues in WSNs can be broadly classified into three, namely
energy efficiency, Quality of Service (QoS) and security. There exist a trade-
off between all these issues. For example, if we want good QoS, then we have
to compromise with the network lifetime. Same follows with the security
parameters. A significant amount of work has been done concerning addressing
these issues individually. However, many loopholes exist when addressing these
issues individually. So, to develop a better WSNs, we need to optimize these
issues simultaneously. One way of doing this is to develop a multi-objective
function and optimize it by using a suitable optimizer or algorithm. The
selection of suitable algorithm depends upon various factors such as the
behaviour of the algorithm, type of problem, time constraint, resource
availability, and desired accuracy. We have first discussed the problem
domains in WSNs and then review the optimizations techniques that are
available to date to solve it.
Figure 3: Coverage: (a) Hexagonal shape-based, (b) Circular shape-based
(radius = a), (c) Real time-based (irregular) and (d) Circular shape-based
(radius $=a^{{}^{\prime}}$).
### 2.1 Problem Domains in WSNs
We have reviewed the potential of optimization and focused on the different
areas in WSNs, as shown in Fig. 2.
* 1.
Optimal Coverage in WSNs
* 2.
Data Aggregation in WSNs
* 3.
Energy Efficient Clustering and Routing in WSNs
* 4.
Sensor Localization in WSNs
We have first briefly discussed each of these problem domains followed by the
work done to solve these issues using optimizations in the section 4.
#### 2.1.1 Optimal Coverage in WSNs
Coverage is necessary and hence becomes an essential topic in the study of
WSNs. The coverage in a given target area is defined as finding a set of
sensors for covering the given area or all the target points. Optimal coverage
means covering the entire area or all the targets point with a minimum number
of sensors.
One of the crucial parameters in the coverage of a sensor in WSNs is the shape
of the sensing area. In Fig. 3 (a) - (d), we present four two-dimensional
geometrical-based sensing shapes. In real life, the shape of the sensing area
is irregular and complex due to the terrain features and solid structures. The
Fig. 3 (c), represents a typical example of real-life sensing shape of a
sensor. However, for computational and conceptual ease, we often adopt either
a hexagonal shape or a circular shape. The hexagonal shape is often applied
for analysis in the WSNs because of its flexibility and no overlapping, as
illustrated in Fig. 3 (a). However, because of the low complexity, the
circular shape is more popular. The limitation associated with the circular
shape is that it creates a coverage hole, as illustrated in Fig. 3 (b). This
limitation is compensated by increasing the radius of the circle, as
illustrated in Fig. 3 (d). However, this gives birth to a new issue of
overlapping regions. These overlapping regions lead to the sensing of
redundant information and wastage of the limited sensor battery. However, if
we critically compared all the three possibilities with the real-life sensing
shape, then Fig. 3 (d), comes out to be the representative of Fig. 3 (c).
The only challenging issue in this problem domain is the reduction of these
overlapping sensing regions with no coverage hole. The more the overlapping
area, the more redundant information will be sensed by the sensors and hence
more will be the wastage of the limited battery of the sensors. One way of
minimizing this redundancy is to optimize the sensor node placement, which is
a single-objective optimization problem. We can extend the single objective to
multi-objective by considering the other network parameters.
#### 2.1.2 Data Aggregation in WSNs
The second way of minimizing the sensing of redundant information is data
aggregation. It is an energy-efficient technique in WSNs. While monitoring an
area, sensors collect local information and send it either the complete
processed data or partially processed data to the data aggregation centre.
According to the data received, the data aggregation centre takes a specific
decision to improve the lifetime of the sensors by eliminating the sensing of
overlap or common regions.
Figure 4: Types of Data aggregations: (a) Tree-based, (b) Cluster-based, (c)
Grid-based and (d) Chain based.
We can broadly classify the data aggregation techniques into four, i.e., Tree-
based, Cluster-based, Grid-based and Chain based. All four types have been
illustrated in Fig. 4 (a) - (d). The Tree-based data aggregation technique is
based on tree architecture in which the source node act as coordinator, and
the data aggregation takes place at the intermediate nodes known as the
aggregator node. The lower level nodes forward the information to the upper-
level nodes. The Cluster-based aggregation technique is based on clustering
architecture. In this type of data aggregation, the network is first divided
into several clusters followed by Cluster Head (CH) selection based on sensor
parameters such as sensor energy, etc. The CH first aggregates the data
locally within the clusters, and then the aggregated data is sent to the sink.
For each new round of data transmission, a new CH is selected to avoid excess
energy consumption from the CH. In the Grid-based aggregation technique, the
network is first divided into several areas, and every area reports the
occurrence of any new event. The data aggregation take place at the grid
aggregator node, also known as the central node. In the Chain based
aggregation technique, the sensor node transfer the data to its neighbouring
node and the data aggregation take place at the lead.
The main challenging issues in this problem domain are
* 1.
To address the problem of optimal power allocation.
* 2.
Finding minimum no. of aggregation points while routing data.
* 3.
Perform consistency for large scale and dynamic WSNs.
#### 2.1.3 Energy Efficient Clustering and Routing in WSNs
Due to the limited energy supply in sensors, the need for energy-efficient
infrastructure is of utmost importance. Most of the sensor energy is consumed
in the transmission of the sensed data. The energy required for the data
transmission increases exponentially with the transmission length. Due to
which the data transmission in sensors follows multi-hop communication.
Routing in WSNs is referred to as the path traversed by the data packets to
reach the sink from the source node. First, the sensors are clustered into
groups. Then a CHs is selected for each group which collects all the data from
the non-CH sensors. Subsequently, the collected data is transmitted to the
sink using optimal routing techniques.
The main challenging issues in this problem domain are
* 1.
Selection of high energy CHs and an optimal routing path in each round.
* 2.
Maximization of the data delivered and the network lifetime.
* 3.
Communication distance minimization.
Figure 5: Working of the localization system.
#### 2.1.4 Sensor Localization in WSNs
Sensor localization is the process of calculating the location of the sensor
present in a network. It consists of two phases. The first one is the distance
estimation, and the second one is the position calculation, as illustrated in
Fig. 5. The anchor or beacon node is the node with known location either
through Global Positioning System (GPS) or by manual pre-programming during
deployment. During the first phase, the relative distance between the anchor
and the unknown node is estimated. The coordinates of the unknown node
concerning the anchor nodes are calculated in the second phase using this
gathered information. In order to localize the other nodes in the WSNs, the
available information of distances and positions are manipulated by using
various localization algorithms. A details study of such algorithms can be
found in [27].
The main challenging issues in this problem domain are
* 1.
Minimization of the localization error.
* 2.
Increasing the accuracy of the unknown node location.
### 2.2 Optimization in WSNs
An optimization can be done by a model, or by a simulator or by an algorithm.
In this paper, we have evaluated the potential of optimization of the problem
domains in WSNs based on algorithm approach. A detailed taxonomy of the
optimization algorithms that are frequently used in WSNs is shown in Fig. 6.
However, there exist more than 100 nature-inspired algorithms since 2000.
Hence, it is not possible to list all the existing algorithms in one taxonomy.
For example, Xing and Gao [28] have listed 134 such algorithms and an online
repository Bestiary list more than 200 algorithms [29]. The most recent and
complete taxonomies or databases of the nature-inspired algorithms can be
found in [30].
The optimization algorithms are classified into deterministic (local search)
and stochastic (global search). In deterministic methods, we have a
theoretical guarantee of reaching the global minimum or at least to a local
minimum, whereas stochastic methods only provide a guarantee in terms of
probability. However, stochastic methods are faster as compared to the
deterministic one. Moreover, stochastic methods are suitable for black-box
formation and ill-behaved functions. In contrast to stochastic methods, the
deterministic method mostly relies on the theoretical assumptions about the
problem formulation and also on its analytical properties.
Figure 6: Taxonomy of the optimization techniques. Figure 7: Venn diagram for
broad classification of optimization algorithm. Figure 8: Regions of
application for various algorithm.
Further, the stochastic methods are classified into a heuristic and meta-
heuristic algorithm. Both types of algorithms are used to increase the speed
of the process of finding a global optimum for the cases where finding an
optimal solution is difficult. Heuristics algorithms are problem-dependent
algorithms. Due to its adaptiveness with the problem and greedy nature, they
are highly prone to get stuck at local optima; resulting into failure of
obtaining global optima. In contrast, meta-heuristics algorithms are problem-
independent algorithms. The non-adaptive and non-greedy nature of these
algorithms enables its use as a black box. These algorithms sometimes accept a
temporary deterioration of the solution (e.g., simulated-annealing method) in
order to get the global optima. The meta-heuristic algorithms are also known
as nature-inspired algorithms, or intelligent optimization algorithms [31, 32,
33]. These algorithms are formulated by delineating inspiration from nature.
The nature-inspired/ meta-heuristic algorithms are further classified as bio-
inspired, physics-inspired, geography inspired and human-inspired. The
majority of the nature-inspired algorithms are inspired by the biological
system. Hence, a big chunk of nature-inspired algorithms are bio-inspired
(biology-inspired) (Fig. 7). The bio-inspired algorithms are further
classified into three, namely evolutionary, swarm-based and plant-based. The
evolutionary algorithms are based on the principle of evolution, such as
Darwin’s principle of selection, heredity and variation [34]. In contrast,
swarm algorithms are based on the collective intelligence [35, 36].
For representing the present status of these algorithms in context to WSNs, we
have created a Venn diagram (Fig. 8) that illustrate the different regions of
applications or problem domains. In Fig. 8, the region R 1, R 2, R 3 and R 4
represents the application area for optimal coverage, data aggregation,
energy-efficient clustering and routing and sensor localization respectively.
Also, the overlapping regions have combined regions of application (e.g., R 3
represents the application area that includes optimal coverage as well as data
aggregation). Finally, Table 1 represents the current status of the bio-
inspired algorithms in context to WSNs.
Not all the bio-inspired algorithms are of potential use in WSNs. The
algorithms for any specific problems in WSNs arena are selected based on the
analogous parameters between the problem domain and the algorithm (e.g., Table
3) [37, 38]. According to the previous studies (Table 1), only three
algorithms (PSO, GA, and ACO) covers all the problem domains of WSNs (i.e.
lies in region R 13). Hence, PSO, GA, ACO and their modifications such as IGA,
BACA and IGA-BACA (combined meta-heuristic) are suitable for the optimizations
of the problem domains in WSNs. In this study, we have evaluated the potential
of the LO for optimal coverage in WSNs.
In the next section, we have tried to elaborate and give an insight into all
these algorithms.
Table 1: Algorithms with region of application.
Algorithm | Region of application | Main references
---|---|---
GA [39] | R 13 | [40, 41, 42, 43]
Evolutionary programming [44] | R 4 | [45]
Learning classifier system [46] | R 14 | Not addressed
Genetic programming [47] | R 10 | [48, 49]
Evolutionary strategy [50] | R 10 | [51, 52]
Estimation of distribution algorithm [53, 54] | R 12 | [55, 56]
Differential evolution [57] | R 12 | [58, 59, 60]
Multi-factorial evolutionary algorithm [61] | R 2 | [62]
Multi-tasking genetic algorithm [63] | R 14 | Not addressed
ACO [64, 65, 66, 67] | R 13 | [68, 69, 70, 71]
PSO [72, 73, 74] | R 13 | [75, 76, 77, 78]
Bacterial foraging algorithm [79, 80, 81] | R 12 | [82, 83, 84]
Artificial fish swarm optimization [85, 86] | R 12 | [87, 88, 89]
Artificial bee colony [90, 91, 92, 93] | R 12 | [94, 95, 96, 97]
Bee system [98, 99] | R 14 | Not addressed
Bees algorithm [100, 101] | R 4 | [102]
Virtual bees [103] | R 14 | Not addressed
Virtual ant algorithm [104] | R 14 | Not addressed
Cat swarm [105, 106] | R 15 | [107, 108]
Accelerated particle swarm optimization [109] | R 14 | Not addressed
Good lattice swarm optimization [110] | R 14 | Not addressed
Monkey search [111] | R 3 | [112]
Firefly algorithm [113, 114, 115] | R 12 | [116, 117, 118]
Fast bacterial swarming algorithm [119] | R 14 | Not addressed
Bee colony optimization [120] | R 2 | [121]
Bee swarm optimization [122, 123] | R 14 | Not addressed
Bumblebees algorithm [124] | R 14 | Not addressed
Cuckoo search [125, 126, 127] | R 11 | [128, 129, 130]
Hierarchical swarm model [131] | R 14 | Not addressed
Consultant guided search [132, 133, 134] | R 14 | Not addressed
Bat algorithm [135] | R 12 | [136, 137, 138]
Wolf search [139] | R 14 | Not addressed
Krill herd [140] | R 15 | [141, 142]
Weightless swarm algorithm [143] | R 14 | Not addressed
Eagle strategy [144] | R 14 | Not addressed
Gray wolf optimizer [145, 146] | R 12 | [147, 148, 149]
Ant lion optimizer [150] | R 5 | [151, 152]
Dragonfly algorithm [153] | R 6 | [154, 155]
Crow search algorithm [156] | R 3 | [157]
LO [26] | R 9 | [151, 158]
Whale optimizer algorithm [159] | R 12 | [160, 161, 162]
Sperm whale algorithm [163] | R 14 | Not addressed
Red deer algorithm [164] | R 14 | Not addressed
Grasshopper optimization algorithm [165] | R 14 | Not addressed
Spotted hyena algorithm [166] | R 14 | Not addressed
Salp swarm algorithm [167] | R 10 | [168, 169]
Artificial flora optimization algorithm [170] | R 14 | Not addressed
Squirrel search algorithm [171] | R 14 | Not addressed
Shuffled shepherd algorithm [172] | R 14 | Not addressed
Group teaching algorithm [173] | R 14 | Not addressed
## 3 Theoretical Background of the Leading Algorithms in WSNs Arena
### 3.1 Mathematical Foundation of the Nature-inspired Algorithms
In this sub-section, we have discussed the generic mathematics of nature-
inspired algorithms. In computational science, any optimization algorithm can
mathematically analyze in terms of an iterative process. According to [174,
175], any nature-inspired algorithm with $k$ parameters,
$p=(p_{1},...,p_{k})$, and $m$ random variables,
$\epsilon=(\epsilon_{1},...,\epsilon_{m})$ for a single-agent trajectory-based
system can be mathematically expressed as
$x^{t=1}=\phi(x^{t},p(t),\epsilon(t))$ (1)
where, $\phi$ represent the non-linear mapping from the current solution (at
$t$) to the better solution (at $t+1$).
For population based system with $n$ swarm solution, the equation 1 can be
extended to
$\begin{bmatrix}x_{1}\\\ x_{2}\\\ \vdots\\\
x_{n}\end{bmatrix}^{t+1}=\phi\Big{(}(x^{t}_{1},x^{t}_{2},\cdots,x^{t}_{n});(p_{1},p_{2},...,p_{k});(\epsilon_{1},\epsilon_{2},...,\epsilon_{m})\Big{)}\begin{bmatrix}x_{1}\\\
x_{2}\\\ \vdots\\\ x_{n}\end{bmatrix}^{t}$ (2)
where, $(p_{1},p_{2},...,p_{k})$ represent algorithm-dependent parameters and
$(\epsilon_{1},\epsilon_{2},...,\epsilon_{m})$ represents the random variables
used for incorporating the randomization in the algorithm. This mathematical
representation can include all the nature-inspired/meta-heuristic algorithm
listed in Fig. 6.
### 3.2 Particle Swarm Optimization (PSO)
PSO was given by Kennedy and Eberhart in 1995 [72, 74]. The basic PSO was
based on the simulation of the single directed, controlled motion of a swarm
of flying birds. Each of these birds is treated as particles which regulate
their flying information by its own and neighbour’s flying experience. In
other words, it combines the self-experience with the social experience, hence
it was a social behaviour simulator. Later on, several revised versions of PSO
emerged in which additional parameters such as confidence factors
($c_{1},c_{2}$) and inertia weight ($w$) were added [176, 177]. A recent study
on PSO and its taxonomy can be found in [178].
The initialization is random, and after that, several iterations are carried
out with the particle velocity ($v$) and position ($x$) updated at the end of
each iteration, as follows: Each particle (i.e. bird) is represented by a
particle number $i$. Each particle possesses a position which is defined by
coordinates in $n$-dimension space and velocity, which reflects their
proximity to the optimal/best position. At first, the initialization is
random, and after that, the particles are manipulated by several iterations
carried out with equation 3 and 4 for position and velocity, respectively.
$x^{i}(k+1)=x^{i}(k)+v^{i}(k+1)\makebox[49.79231pt]{}$ (3)
$v^{i}(k+1)=w^{i}v^{i}(k)+c_{1}r_{1}(x^{i}_{best}-x^{i}(k))+c_{2}r_{2}(x_{gbest}-x^{i}(k))$
(4)
where;
$i$ = 1,2,…,$N_{s}$; $N_{s}$ is the size of the swarm
$k$ =1,2,… $w^{i}$ = inertia weight for each particle i
$x^{i}_{best}$ = best location of the particle
$x_{gbest}$ = best location amongst all the particle in swarm
$c_{1}$ = confidence factor which represents the private thinking of the
particle itself; assigned to $x^{i}_{best}$
$c_{2}$ = confidence factor which represents the collaboration among the
particles; assigned to $x_{gbest}$
$r_{1},r_{2}$ = random values between [0 ,1].
### 3.3 GA and Adaptive GA(or IGA)
John H. Holland and his collaborators proposed the genetic algorithm in the
1960s and 1970s [39], and since then it has become one of the widely used
meta-heuristic algorithms. It is based on the abstraction of Darwin’s
evolution principle of biological systems that has three components or genetic
operators; reproduction-crossover-mutation. Every solution is encoded in a
string (often decimal or binary) called chromosomes. The fitness function in
every iteration calculates its value. Afterwards, these values are sorted in
descending order. Solutions that are present at the top are considered as good
solutions and selected for reproduction. It discards the solutions with low
fitness values. After completion of reproduction, the selected solutions will
go through crossover and mutation. The role of the crossover operator is to
produce crossed solutions with optimal fitness values by the interchange of
genetic material. The probability of this event is known as crossover
probability, represented by ${P_{c}}$. This event is followed by mutation;
which targets to find the unexplored genetic material with a probability known
as mutation probability, represented by ${P_{m}}$. The computational equations
for ${P_{c}}$ and ${P_{m}}$ is given by 5 and 6.
$P_{c}=\begin{cases}\frac{k_{1}(f_{max}-f^{\prime})}{f_{max}-f_{avg}}&f^{\prime}>f_{avg}\\\
k_{3}&f^{\prime}<f_{avg}\end{cases}\makebox[28.45274pt]{}$ (5)
$P_{m}=\begin{cases}\frac{k_{2}(f_{max}-f)}{f_{max}-f_{avg}}&f>f_{avg}\\\
k_{4}&f<f_{avg}\end{cases}\makebox[49.79231pt]{}$ (6)
Figure 9: BACA network.
Where ${f_{max}}$ and ${f_{avg}}$ represents the highest fitness and average
fitness of the population respectively, ${f^{\prime}}$ represents the higher
fitness amongst the two solutions that are selected for crossover and ${f}$
represents the fitness of the solution that is selected for mutation. In order
to restrict the values of ${P_{c}}$ and ${P_{m}}$ in the range [0,1], the
values of the constants ${k_{1},k_{2},k_{3}}$ and ${k_{4}}$ should be less
than ${1}$. Also, the constants $k_{1}$ and $k_{3}$ should be greater than the
$k_{2}$ and $k_{4}$.
Adaptive GA (AGA) or Improved GA (IGA) is an enhanced version of conventional
GA. In AGA, ${P_{c}}$ and ${P_{m}}$ changes adaptively based on different
individuals condition that ultimately retard the possibility of premature
convergence [179].
### 3.4 ACO and BACA
ACO is based on the food searching process by ants. In this process, the ant
emits pheromone in the path. The remaining ants follow the path with a high
intensity of pheromones [180, 181]. ACO estimates the optimal path through
continuous accumulation and pheromones release process in several iterations.
The performance of ACO depends strongly on the early stage pheromones. Lack of
sufficient pheromones may result in premature convergence (i.e., local optima)
[182] and to avoid this, we use BACA. A typical example of how BACA works is
illustrated in Fig. 9. This binary coding increases the efficiency of the
algorithm [182]. Different ants search the same routine and emit the
pheromones on each edge. Out of the two binary edges, each ant selects one.
This process can be represented in the form of a matrix with only (2 $\times$
$n$)’s space. Defining a digraph ${G=(V,R)}$ with $V$ representing the node-
set and $R$ representing the path set.
$\begin{aligned}
\big{\\{}\big{\\{}v_{0}(c_{s}),v_{1}((c^{0}_{N})),v_{2}((c^{1}_{N})),v_{3}((c^{0}_{N-1})),v_{4}((c^{1}_{N-1})),...,\\\
v_{2N-3}((c^{0}_{2}),v_{2N-2}((c^{1}_{2}),v_{2N-1}((c^{0}_{1}),v_{2N}((c^{1}_{1})\big{\\}}\big{\\}}\end{aligned}\makebox[49.79231pt]{}$
(7)
In this digraph, ${c_{s}}$ represent the staring node while ${c^{0}_{j}}$ and
${c^{1}_{j}}$ represents the value 0 and 1 of ${b_{j}}$ bit used in the binary
mapping. $N$ is the encoding length (binary). For each node present in the
node set, ${(j=1,2,3,...,N)}$, there exist only two paths (0 and 1 states)
which points towards ${c^{0}_{j-1}}$ and ${c^{1}_{j-1}}$ respectively [25].
Initially, it has been assumed that all the path have same piece information
(equal to ${\tau_{i,j}(0)=C}$; $C$ is constant and
${\Delta\tau_{i,j}(0)(i,j=1,2,...,N)}$). During the path deciding phase, the
pheromones realized by $k$ ($k$ = 1, 2, 3, …, $m$; $m$ is the number of ants)
ants and the probability of movement decides the direction. The probability of
movement, ${p^{k}_{i,j}}$, is defined as
$\begin{aligned}
p^{k}_{i,j}=\frac{(\tau^{\alpha}_{i,j}(t).\eta^{\beta}_{i,j}(t))}{(\sum_{s\in
allowedk}\tau^{\alpha}_{i,j}(t).\eta^{\beta}_{i,j}(t))}\end{aligned}\makebox[49.79231pt]{}$
(8)
$k$ ants moves from point $i$ to point $j$. ${\alpha}$ and ${\beta}$ are
constants. ${\tau_{i,j}}$ and ${\eta_{i,j}}$ represents the unutilized
information and visualness respectively in the ($i$, $j$) junction at $t$
moment. ${Allowed_{k}={(0,1)}}$ represents the upcoming status. With time, the
pheromones evaporate resulting in loss of information. $\rho$ is the
perseverance factor and (1-$\rho$) represents the information loss factor. The
${\tau_{i,j}}$ for the next moment is represented by
$\begin{aligned}
\tau_{i,j}(t+1)=\rho.\tau_{i,j}(t)+\Delta\tau_{i,j}\end{aligned}\makebox[49.79231pt]{}$
(9)
$\begin{aligned}
\Delta\tau_{i,j}=\frac{1}{f_{best}(S)}\end{aligned}\makebox[49.79231pt]{}$
(10)
Where,
${f_{best}(S)}$ is the optimal cost. In a nutshell, BACA differs with the
conventional ant colony only in the way the ant selects the path.
### 3.5 IGA with BACA
The combined meta-heuristic, IGA-BACA, searches for the optimal solution by
initializing the BACA network with the final result of the IGA. Firstly the
IGA is used to optimized the randomly generated solution. Now for the same
time been, this optimized solution is feed to initialize the pheromones
information of the BACA algorithm. The IGA-BACA algorithm terminates the loop
once it meets the termination condition; otherwise, the complete process
repeats itself to meet the termination condition (i.e., optimal updated
pheromones).
### 3.6 LO
There are two types of lions; residents and nomads. Resident lions always live
in groups called pride. In general, a pride of lion typically involves about
five female along with their cubs of both sexes and one or more than one adult
male. Young males, when getting sexually mature, get excluded from their birth
pride. Nomads move either in pair or singularly. Pairing occurs among related
males who have been excluded from their maternal pride. The lion may switch
lifestyles means nomad at any time become a resident and vice versa [26].
Unlike that of cats, lions hunt together to catch their prey, which increases
the probability of success of hunting. In case if a prey manages to escape
then the new position of prey, ${PREY^{\prime}}$ is given by
$\begin{aligned} \noindent\scalebox{1.0}{ \noindent\scalebox{0.9}{
$PREY^{\prime}=PREY+rand(0,1).PI.(PREY-Hunter)$
}}\end{aligned}\makebox[49.79231pt]{}$ (11)
Where ${PREY}$ represents the current position of prey, $PI$ is the percentage
of improvement in the fitness of hunter. The formulas are proposed to mimic
encircling prey by the hunter group. The new positions, according to the
location of prey, are generated as follows:
$\begin{aligned} \noindent\scalebox{0.6}{
$Hunter^{\prime}=\begin{cases}rand((2*PREY-Hunter),PREY)&(2*PREY-
Hunter)<PREY\\\ rand(PREY,(2*PREY-Hunter))&(2*PREY-Hunter)>PREY\end{cases}$
}\end{aligned}\makebox[49.79231pt]{}$ (12)
Where Hunter is the current position of the hunter. And the new position for
centre hunters is given as
$\begin{aligned} \noindent\scalebox{0.9}{
$Hunter^{\prime}=\begin{cases}rand(Hunter,PREY)&Hunter<PREY\\\
rand(PREY,Hunter)&Hunter>PREY\end{cases}$
}\end{aligned}\makebox[49.79231pt]{}$ (13)
In the equation 12 and 13 ${rand(a,b)}$, generates a random number between
${a}$ and ${b}$, where ${a}$ and ${b}$ are upper and lower bound respectively.
A detail of the process involve in LO is mention in the pseudo code of the
literature [26].
Figure 10: Summary of the PSO approaches/solutions to the problem domains in
WSNs.
## 4 Solution to the Problem Domains and Present Status
In this section, we have presented a summary of the most prominent solutions
to the problem domains of the WSNs based on some of the bio-inspired meta-
heuristic algorithms, namely PSO, GA, and ACO.
### 4.1 Applications of PSO in WSNs
The centralized nature of PSO enables its application in the minimization of
the coverage holes for near-optimal coverage in WSNs [78, 183, 184, 185, 186,
187, 188, 189, 190]. Data aggregation is a repetitive process, which makes it
suitable for PSO [191, 192, 193, 194, 195]. PSO is suitable for selecting CH’s
with high energy in each round [196, 197, 198, 199, 200]. It also minimizes
the sensor node localization errors [76, 201, 202]. A detailed chart,
illustrating the PSO based solution, is presented in Fig. 10.
#### 4.1.1 For Optimal Coverage using PSO
Various studies have been reported to improve sensor coverage using PSO.
Mendis et al. [189] used the conventional PSO for optimization of the mobile
sink node location in WSNs. To deal with the various complexities and
challenges in different applications, various modified or improved version of
PSO are proposed in the literatures. Ngatchou et al. [185] used a modified
version of PSO, namely sequential PSO for distributed sonar sensor placement.
Sequential PSO is generally used for high dimension optimization and found
application in underwater sensor deployment optimization. Further, Li et al.
[186] also used a modified version of PSO, namely Particle Swarm Genetic
Optimization (PSGO) for optimal sensor deployment. PSGO involve selection and
mutation operator of GA, which eliminates the premature convergence issue of
PSO. Afterwards, Wang et al. [187] proposed a virtual force directed co-
evolutionary PSO (VFC-PSO) for dynamic sensor deployment in WSNs. Ab Aziz et
al. [78] has proposed a novel optimization approach combining PSO and Voronoi
diagram for sensor coverage problem in WSNs. This algorithm works efficiently
for small Region of Interest (ROI) with a high number of sensor node or vice-
versa. Subsequently, Hong and Shiu [188] used conventional PSO for searching
the near-optimal Base Station (BS) location in WSNs. Hu et al. [184] proposed
a methodology for optimal deployment of large radius sensors. They have used
PSO for optimization of the sensor deployment in order to reduce the links in
the proposed topology. Nascimento and Bastos-Filho [190] used the conventional
PSO for BS positioning to avoid the overlapping between cells. Overall,
various modified versions, including the conventional PSO, can be used for
improving the sensor coverage.
#### 4.1.2 For Sensor Localization using PSO
For accurate node localization, Low et al. [202] used the conventional PSO.
They have reported better accuracy when the results are compared with the
Gauss-Newton algorithm. Similarly, Gopakumar et al. [76] used the same
conventional PSO and reported better accuracy as compared with the simulated
annealing approach. Later, Kulkarni et al. [201] presented a comprehensive
study on node localization. They have compared the results of PSO and
bacterial foraging algorithm. They reported that the node localization in WSNs
is faster with PSO and more accurate with bacterial foraging algorithm.
#### 4.1.3 For Energy Efficient Clustering and Routing using PSO
Several studies have reported the use of PSO for energy-efficient clustering
and routing. Tillett et al. [198] have used the conventional PSO for sensor
node clustering in WSNs. They reported that the PSO outperforms simulated
annealing and random search algorithm in terms of energy-efficient clustering.
Afterwards, [200] proposed a divided range PSO algorithm for network
clustering. They reported that the proposed algorithm is efficient when then
mobile sensors are dense. Subsequently, Guru et al. [196] proposed four
variants of PSO and applied it for energy-efficient clustering. They reported
that the PSO with the supervisor-student model outperforms the other three
algorithms. Cao et al. [197] have used a hybrid of graph theory and PSO
algorithm for energy-efficient clustering in multi-hop WSNs. Latiff et al.
[199] have used the conventional PSO for re-positioning of BS in a clustered
WSNs. Overall, the use of PSO reduces energy consumption and extend the
network lifetime.
#### 4.1.4 For Data Aggregation using PSO
Veeramachaneni and Osadciw [193] have used the conventional PSO for
optimization of the accuracy and time from data aggregation aspect. In
general, they have evaluated the potential of the PSO for multi-objective
optimization. Further, Wimalajeewa and Jayaweera [191] have used the
constrained PSO for optimal power allocation. Afterwards, Veeramachaneni and
Osadciw [192] have used the hybrid version of PSO, namely ACO and PSO for
dynamic sensor management. Guo et al. [194] proposed a multi-source temporal
data aggregation algorithm (MSTDA) for data aggregation in WSNs. Subsequently,
Jiang et al. [195] have used the constrained PSO with the penalty function
concept, which increases the accuracy.
### 4.2 Applications of GA in WSNs
GA is proven to be good for random as well as deterministic deployment [38,
25, 203, 204, 205, 206]. It is also good at finding lesser number of data
aggregation points while routing the data to the base station [40, 207, 208,
209]. It is used for pre-clustering which reduces the resultant communication
distance [210, 41, 211, 212, 213, 214, 215, 216]. Besides this, the global
searching capability of the GA results into higher accuracy in locating the
sensor nodes [43, 217, 218]. A detail chart, illustrating the GA based
solution is presented in Fig. 11.
#### 4.2.1 For Optimal Coverage using GA
Various studies have evaluated the potential of GA for network coverage
optimization. Konstantinidis et al. [204] have modeled the sensor deployment
and power assignment as a multi-objective problem and used the conventional GA
for the optimization. Poe and Schmitt [206] proposed an approach for sensor
deployment over a large WSNs. They make use of conventional GA. They have
compared and reported the pros and cons of three different types of
deployment. Bhondekar et al. [205] have used the conventional GA for node
deployment in a fixed WSNs. Jia et al. [203] proposed an energy-efficient
novel network coverage approach using conventional GA. They reported that the
proposed approach results in balanced performance with high network coverage
rate. Tian et al. [25] have used a hybrid version of GA called Improved GA and
Binary ACO Algorithm (IGA-BACA) for optimal coverage in WSNs and compared
there results with conventional GA. They reported that IGA-BACA outperforms
conventional GA. They also reported a high coverage rate. Recently, Singh et
al. [38] have used the same IGA-BACA and conventional GA for optimal coverage
in WSNs with reduced sensing of redundant information.
Figure 11: Summary of the GA approaches/solutions to the problem domains in
WSNs.
#### 4.2.2 For Sensor Localization using GA
Jegede and Ferens [217] have used the conventional GA for node localization in
WSNs. Recently, Peng and Li [43] have used DV-Hop GA based algorithm for node
localization WSNs. They reported that the DV-Hop GA based algorithm
outperforms the previously proposed algorithm. More recently, Tan et al. [218]
have proposed Distance Mapping Algorithm (DMA) and integrate this with the GA
for accurate node localization in WSNs. They reported that the proposed
algorithm outperforms previously proposed algorithms in terms of accuracy and
energy consumption.
#### 4.2.3 For Energy Efficient Clustering and Routing using GA
Jin et al. [210] proposed a sensor network optimization framework for Bari et
al. [214] have used the conventional GA algorithm for energy-efficient
clustering and routing in a two-tier sensor network. They have reported that
the proposed approach shows significant improvement compared to the earlier
proposed schemes. Seo et al. [212] proposed a hybrid GA algorithm, namely
Location-Aware 2-D GA (LA2D-GA) for efficient clustering in WSNs. They
reported that the LA2D outperform its 1-D version. Hussain and Islam [211]
proposed an energy-efficient clustering and routing scheme based on
conventional GA. Further, Luo [216] proposed the first quantum GA based QoS
routing protocol for WSNs. Also, EkbataniFard et al. [215] proposed a multi-
objective GA for energy-efficient QoS routing approach in WSNs. They reported
that the proposed approach successfully reduces the average power consumption
by efficient optimization of the network parameters. Norouzi et al. [213]
proposed a dynamic clustering algorithm for WSNs based on conventional GA.
#### 4.2.4 For Data Aggregation using GA
Islam et al. [40] have proposed an energy-efficient balanced data aggregation
tree algorithm based on GA. They reported that the spanning tree-based
proposed algorithm improves the network lifetime significantly. Al-Karaki et
al. [207] have proposed a grid-based data aggregation and routing scheme for
WSNs. They reported that the proposed scheme reduces power consumption and
improves network lifetime. Similar to the Islam et al. [40], Dabbaghian et al.
[209] proposed an energy-efficient balanced data aggregation using spanning
tree and GA. They also, reported an increase in the network lifetime. However,
an improved version of spanning tree-based data aggregation algorithm is
proposed by Norouzi et al. [208]. They use the residual energy of the nodes to
further improve the network lifetime.
Figure 12: Summary of the ACO approaches/solutions to the problem domains in
WSNs.
### 4.3 Applications of ACO in WSNs
The distributed nature of ACO results in better dynamic deployment of the
sensor node for near-optimal coverage [38, 25, 219, 220, 221, 222]. ACO
performs better in case of large and dynamic WSNs [223, 224, 225, 226, 227].
It also increases the network lifetime [228, 229, 230, 231, 232, 233]. Never
the less it also increases the accuracy of the unknown node in WSNs [69, 234,
235, 236]. A detail chart, illustrating the ACO based solution is presented in
Fig. 12.
#### 4.3.1 For Optimal Coverage using ACO
Li et al. [219] have proposed an efficient sensor deployment optimization
toolbox named as DT-ACO. Also, they have proposed a real-time hardware-based
application for WSNs called EasiNet. Later, in Li et al. [220], they have
modified the previously proposed EasiNet. This modification allows them to
eliminate redundant sensors during sensor deployment. Liao et al. [221] have
proposed an efficient approach for sensor deployment using ACO. They have
formulated the deployment problem as multiple knapsack problem (MKP). They
reported a complete network coverage with prolong network lifetime. Liu [222]
proposed a novel approach for sensor deployment in WSNs using ACO with three
ant transition concept. They report a high coverage rate. Recently, Tian et
al. [25] have used the hybrid version of ACO, namely IGA-BACA. They reported a
high network coverage rate with high network lifetime. More recently, Singh et
al. [38] have used the IGA-BACA for reducing the sensing of the redundant
information with optimal coverage.
#### 4.3.2 For Sensor Localization using ACO
Qin et al. [69] have proposed a novel node localization scheme using ACO
through beacons signals. They reported a high localization accuracy with low
power consumption. Niranchana and Dinesh [235] have proposed a node
localization approach in which the prediction of the nodes is made through
interval theory and relocation of the nodes are done through ACO. They also
reported a high localization accuracy. Further, Liang et al. [234] have used
the simple ACO for node localization in WSNs. They have optimized the
trilateration positioning error function. They reported a higher-order
localization accuracy compared to the previously proposed localization
schemes. Recently, Lu and Zhang [236] have proposed an ACO based mobile anchor
node localization scheme for WSNs.
#### 4.3.3 For Energy Efficient Clustering and Routing using ACO
Camilo et al. [228] have proposed a new routing algorithm for WSNs based on
ACO. They reported a low communication load with low energy consumption.
Further, Salehpour et al. [231] also proposed a new routing algorithm with two
routing levels based on ACO. They reported a relatively low power consumption
and more load balancing. Ziyadi et al. [232] proposed an energy-aware
clustering protocol based on ACO clustering for WSNs. They reported an
increase in the network lifetime. Later, Huang et al. [230] proposed a
Prediction routing algorithm based on ACO. It was first of its kind. They
reported various advantages such as low power consumption, increase in network
lifetime, and high load balancing. Almshreqi et al. [229] proposed a self-
optimization algorithm based on ACO for balance energy consumption in WSNs.
They reported low energy consumption with reduced packet loss. Mao et al.
[233] proposed a fuzzy-based unequal clustering algorithm. They have used ACO
for energy-aware routing. They reported that the proposed algorithm
outperforms various traditional algorithm such as LEACH.
#### 4.3.4 For Data Aggregation using ACO
Ding and Liu [223] proposed an efficient self-adaptive data aggregation
algorithm for WSNs based on ACO. They reported that the proposed algorithm
outperforms the benchmark algorithms such as LEACH and PEGASIS in terms of
prolonging the network lifetime. Further, Misra and Mandal [224] proposed an
approach for efficient data aggregation algorithm for WSNs. They reported that
the proposed approach is energy-efficient. Han and Hong-xu [225] proposed a
novel approach for multi-media data aggregation in wireless sensor and actor-
network. They have compared the performance with the traditional methods such
as MEGA and reported an improvement in the stability, accuracy and network
lifetime. Yang et al. [226] proposed an energy-efficient data aggregation
algorithm based on ACO for WSNs. They reported an improvement over network
lifetime. Similarly, Xie and Shi [227] proposed a data aggregation approach
for WSNs using ACO and reported an improvement over network lifetime.
Table 2: Summary of the present status of the bio-inspired algorithms approach to the problem domains of WSNs. | Problem
---
Domians of WSNs
| Optimization
---
Algorithms
PSO | GA | ACO | LO
| Optimal
---
Coverage
Addressed | Addressed | Addressed | Addressed (in this paper)
| Data
---
Aggregation
Addressed | Addressed | Addressed | Addressed
| Energy Efficient Clustering
---
and Routing
Addressed | Addressed | Addressed | Addressed
| Sensor
---
Localization
Addressed | Addressed | Addressed | Not Addressed
PSO, GA and ACO well address all the four problem domains of WSNs. Also, some
hybrid techniques emerge for the same. Every new attempted claimed to show
improved results over the previous approaches. In continuation of that, we
have introduced the LO to solve the issues in WSNs. Table 2, shows the current
status of all the four prominent algorithms.
## 5 Optimal Coverage using IGA-BACA and LO
Getting optimal coverage in WSN belongs to a multi-objective optimization
problem. The existing sensors, $N$, is represented by set
${S=(s_{1},s_{2},...,s_{i},...,s_{N})}$. In this optimization problem, we aim
to estimate a sensor set ${S^{\prime}}$, which covers the monitoring area to
the maximum with minimum working sensors. The function for maximum coverage
and minimum sensors is ${f_{1}(S^{\prime})}$ and ${f_{2}(S^{\prime})}$. Both
these functions are conflicting in nature; undermining them both, the new
objective function by changing it to a maximal objective function
${f(S^{\prime})}$ read as;
$\begin{aligned} \noindent\scalebox{0.9}{
${f(S^{\prime})=(f_{1}(S^{\prime}).f_{1}(S^{\prime})/f_{2}(S^{\prime}))}$
}\end{aligned}\makebox[49.79231pt]{}$ (14)
The framework for obtaining the optimal coverage using IGA-BACA and LO is
explained in the upcoming subsections.
### 5.1 Mapping from Solution to Coding Space
The binary coding represents the position of the sensors in WSNs. The
corresponding control vector is ${L=(l_{1},l_{2},...,l_{i},...,l_{N})}$.
${l_{i}}$ can either have a value of zero or one which represents the inactive
or active state of a sensor respectively. The initialization of nomad and
pride in LO and gene of the chromosome in GA has one to one correlation with
the selection of nodes. Fig. 13, shows a typical example of a control vector.
The probability of the sensor to be an active sensor depends on the adaptation
or objective function (Equation 14). Higher the value, the larger will be the
probability.
Figure 13: Illustrating a control vector with 9 active sensors out of 12 is {1
0 1 1 1 1 0 1 1 1 1 0}. The dark circles represent the inactive sensors in the
monitoring area. It is represented by a binary ‘0’ in the control vector.
### 5.2 Algorithm and Process
For optimization Equation 14, we have used IGA-BACA and LO. In IGA-BACA, we
have four processes, namely reproduction, crossover, mutation and update
pheromone process. In contrast, LO has three processes, namely, mating,
sorting, and elimination.
In IGA-BACA, the first process reproduces new offspring depending on
probability infraction to their fitness value. Afterwards, the new offspring
are sorted based on their fitness values. The only offspring with high fitness
values are retained, and others are discarded. This process ensures an
increase in the average fitness of the colony. The only limitation associated
with this process is that the number of possible varieties is lost. This
limitation is subsequently overcome by the crossover and mutation process. In
the crossover process, a pair of offspring is selected based on the
probability, ${P_{c}}$. This step further increases the probability that
crossed solutions may produce offspring with high fitness value. Afterwards,
the mutation process alters an offspring based on the probability, ${P_{m}}$.
This step explores the unexplored genetic material. Lastly, the update
pheromone process (${(T_{g})}$; pheromones update operator) mapped the updated
pheromone for an optimal offspring elected by the ant sequence. The ant
release the pheromones in the optimal path traversed by them using Max-Min
rule. We can calculate the probability of the update pheromone operator by
$\begin{aligned} \noindent\scalebox{1.0}{
$P\big{\\{}T_{g}=x_{i}\big{\\}}=\frac{f(x_{i})}{\sum^{N}_{k=1}f(x_{k})}$
}\end{aligned}\makebox[49.79231pt]{}$ (15)
Where, ${N}$ is the number of offsprings. In LO, first mating with the best
nomad, both male and female, is done followed by sorting nomad lions of both
gender-based on fitness value. After which the nomad with least fitness value
is eliminated. Analogous terms between the LO parameters and WSNs are listed
in Table 3. The complete methodology for LO and IGA-BACA is illustrated in
Fig. 14.
Figure 14: Flowchart for IGA-BACA and LO.
## 6 System Model
As stated earlier, getting optimal coverage is one of the crucial problems
associated with WSNs. The network should have maximum coverage with a certain
level of QoS [237, 238, 239]. Presence of blind area significantly affects the
QoS threshold and network coverage rate, ultimately affecting the network
reliability. In order to increase network reliability, we can deploy more
sensors in critical areas. Increasing sensors will increase the network cost.
In this paper, we used the bio-inspired algorithms to find the optimal node-
set.
In this study, we have assumed that the total monitoring area, ${A}$, is a
two-dimensional plane and It is split into $m$ $\times$ $n$ equal grids. After
that, we have randomly distributed ${N}$ no. of sensors in the study area.
Mathematically, these sensors are represented by
${S=(s_{1},s_{2},...,s_{i},...,s_{N})}$. All the sensors have effective radii
(sensing radius) of ${r}$ with a coordinate ${(x_{i},y_{i})}$ for ${s_{i}}$.
In order to ensure maximum coverage, each grid in the monitoring area is
considered as a target point. Mathematically, it is represented by
$A={(a_{1},a_{2},...,a_{j},...,a_{mxn})}$. If the target point ${a_{j}}$ lies
in the sensing region of sensor ${s_{i}}$, then the Euclidean distance between
them is given by ${d(a_{j},s_{i})\leq r}$ [25].
Table 3: Analogous mapping between LO algorithm and WSNs. LO algorithm | Optimal coverage problem
---|---
Solution of a food source | Node distribution
${N}$ dimensions in each solution | ${N}$ sensor coordinates
Fitness of the solution | Coverage rate in $A$
Maximum fitness | Optimum deployment
The probability, ${P_{cov}(x,y,s_{i})}$, that any coordinate ${(x,y)}$ in $A$
is sensed by a sensor ${s_{i}(x_{i},y_{i})}$ is given by
$\begin{aligned} \noindent\scalebox{1.0}{
$P_{cov}(x,y,s_{i})=\begin{cases}1&(x-x_{i})^{2}-(y-y_{i})^{2}\leq r^{2}\\\
0&otherwise\end{cases}$ }\end{aligned}\makebox[49.79231pt]{}$ (16)
The area covered by the sensors is given by
$\begin{aligned} \noindent\scalebox{1.0}{
$A_{area}(S)=\sum_{x=1}^{m}\sum_{y=1}^{n}P_{cov}(x,y,s_{i})\Delta x\Delta y$
}\end{aligned}\makebox[49.79231pt]{}$ (17)
If $S^{\prime}$ is the set of working or active sensors, then the fitness or
objective function for the network coverage is given by
$\begin{aligned} \noindent\scalebox{1.0}{
$f_{1}(S^{\prime})=A_{area}(S^{\prime})/A_{s}$
}\end{aligned}\makebox[49.79231pt]{}$ (18)
In contrast, the objective function for the node uses rate is given by
$\begin{aligned} \noindent\scalebox{1.0}{ $f_{2}(S^{\prime})=|S^{\prime}|/N$
}\end{aligned}\makebox[49.79231pt]{}$ (19)
Where ${N}$ is the total number of sensor nodes. Equation 18 and 19 are
combined to form a multi-objective optimization coverage problem given by.
$\begin{aligned} \noindent\scalebox{1.0}{
$maxf(S^{\prime})=max(f_{1}(S^{\prime}),1-f_{2}(S^{\prime}))$
}\end{aligned}\makebox[49.79231pt]{}$ (20)
We have to maximise the equation 20 to get maximum coverage with minimum
sensor node.
## 7 Simulation Results
The simulation parameters that we have used in this study is given in Table 4.
We selected a monitoring area ($A$) of ${100}$ m $\times$ ${100}$ m in which
sensor nodes having a perception radius of 10 m ($r$ = 10 m) are deployed. The
constants $k_{1}$, $k_{2}$, $k_{3}$, and $k_{4}$ are set to 1, 0.5, 1 and 0.5
respectively. These values restrict the range of $P_{c}$ between 0.5 and 1
(i.e., 0.5 $<$ $P_{c}$ $<$ 1) and $P_{m}$ between 0.001 and 0.05 (i.e., 0.001
$<$ $P_{m}$ $<$ 0.05). The moderately large range of $P_{c}$ and small range
of $P_{m}$ is required for extensive recombination of solutions and prevention
of the disruptions of the solutions respectively which ultimately prevents the
algorithm from getting stuck into local optimum. The constant $\alpha$
controls the pheromone importance, while $\beta$ controls the distance
priority. In general, $\beta$ should be greater than $\alpha$ for the best
results. Both these parameters are interlinked. We have to fix one and vary
(iterate) the other to find the optimal set of value. In this study, we fixed
the $\alpha$ to 1 and found $\beta$ to be 6.
Table 4: Simulation parameters. Parameter | Value
---|---
Monitoring area (${A}$) | 100 m $\times$ 100 m
Perception radius ($r$) | 10 m
$N$ | 100
$k_{1}$ = $k_{3}$ | 1
$k_{2}$ = $k_{4}$ | 0.5
$\alpha(\alpha\geq 1)$ | 1
$\beta(\beta\geq 1)$ | 6
Figure 15: Optimal network coverage. Figure 16: Random distribution of 100
nodes.
We implemented the corresponding algorithm in MATLAB® (version 2017b). We
iterated the IGA-BACA algorithm for ${300}$ iterations. In doing so, we found
that only ${42}$ (out of $100$) sensors cover the monitoring area optimally,
as shown in Fig. 15. This optimal coverage is treated as a benchmark for
further analysis. In contrast, while distributing these $100$ sensors
randomly, we found a network coverage map, as shown in Fig. 16. We randomly
distribute these ${100}$ sensors in the monitoring area, as shown in Fig. 16.
Although the monitoring area is almost covered completely, there exist a
significant amount of redundant nodes which senses redundant information. The
uncovered area in the target monitoring area is considered as a coverage hole,
and in Fig. 16, we can easily detect such coverage hole or blind areas
(highlighted in red boxes). Hence, random network coverage is not usually
adopted.
Figure 17: 50 Generation of IGA-BACA. Figure 18: 100 Generation of IGA-BACA.
Figure 19: 150 Generation of IGA-BACA. Figure 20: 200 Generation of IGA-BACA.
Figure 21: 50 Generation of LO. Figure 22: 100 Generation of LO. Figure 23:
150 Generation of LO. Figure 24: 200 Generation of LO. Figure 25: 250
Generation of LO. Figure 26: Network coverage vs sensor. Figure 27: Network
coverage vs generation. Table 5: Simulation results for IGA-BACA and LO.
| Iterations
---
IGA-BACA | LO
| | Network coverage
---
rate (%)
| Active sensor
---
node
| Time (s)
---
(GPU)
| Network coverage
---
rate (%)
| Active sensor
---
node
| Time (s)
---
(GPU)
50 | 93.5 | 72 | 5 | 95.9 | 67 | 4
100 | 95.0 | 65 | 11 | 96.9 | 61 | 9
150 | 96.4 | 59 | 17 | 98.7 | 55 | 15
200 | 97.5 | 53 | 22 | 98.9 | 48 | 19
250 | 97.9 | 48 | 26 | 99.1 | 42 | 23
Network coverage using the IGA-BACA algorithm for 50, 100, 150, 200 iterations
(or generations) is shown in Fig. 17 \- 20. As we increase the number of
iterations from 50 to 200, we found that the network coverage tends towards
the optimal coverage; hence the number of redundant nodes decreases
significantly. In comparison with the IGA-BACA derived results, the network
coverage using LO algorithm for 50, 100, 150, 200 and 250 iterations is shown
in Fig. 21 \- 25. We found a similar trend of moving towards the optimal
coverage with an increase in the number of iterations. For both the algorithms
(i.e., IGA-BACA and LO), we have tabulated the network coverage rate (in
percentage), GPU processing time (in seconds) and the number of active sensors
corresponding to 50, 100, 150, 200 and 250 iterations as shown in Table 5. We
compared the results that are obtained through combined meta-heuristic IGA-
BACA with the results obtained through LO. In doing so, we observed that the
optimal network coverage is obtained by both the approach. However, IGA-BACA
algorithm requires approximately $300$ iterations and LO algorithms require
$250$ iterations to achieve the optimal coverage. Also, in LO, the optimal
coverage is obtained at a lesser number of sensors, as shown in Fig. 26.
Further, LO has a faster rate of convergence which is primarily due to the
presence of a large number of local maxima with higher values of fitness
functions (Table 5). In addition to this, we plotted the network coverage rate
against the number of iteration (Fig. 27). In doing so, we observed that the
network coverage increases as the function of iteration. Also, the convergence
of LO is better than IGA-BACA.
## 8 Conclusion
The use of nature-inspired algorithms has created a new era in next-generation
computing. These algorithms are well suited for solving multi-objective
optimization problems. Various features of nature-inspired algorithms such as
reasonable computational time, find global optimal and applicability make them
well suited for real-world optimization problems. In contrast, traditional
algorithms generally fail to provide satisfactory results mainly because of
the complexity and size of the problem structure. In this paper, we have
presented a comprehensive review of such algorithms in context to various
issues related to the WSNs.
We have evaluated the potential of two efficient meta-heuristic approaches
that compute the optimal coverage in WSNs, namely IGA-BACA and LO. We have
compared the results of both these approaches. In doing so, we observed that
as the number of iteration is increasing the network coverage rate tend
towards optimal coverage. Also, the network coverage rate is faster in LO
approach as compared to IGA-BACA. The optimal coverage is achieved with a
lesser number of iteration in case of LO as compared with other approaches. It
is due to the presence of a large number of local maxima with higher fitness
value, and hence it is hardly any chance to miss local maxima. Although LO
gives better performance than other optimization algorithms, still there is
much scope to explore this algorithm and to apply it in multi-objective
problems. For instance, if we can use machine learning approach such as
Artificial Neural Network (ANN) that incorporates combined heuristic such as
Ant Lion Optimization (ALO), IGA-BACA, etc. as our system inputs.
## Conflict of Interest
The author states that there is no conflict of interest.
## CRediT author statement
Abhilash Singh and Sandeep Sharma: Conceptualization, Methodology, Software.
Abhilash Singh and Sandeep Sharma: Data curation, Writing- Original draft
preparation, Visualization, Investigation. Sandeep Sharma: Supervision.
Abhilash Singh, Sandeep Sharma and Jitendra Singh: Software, Validation.
Abhilash Singh, Sandeep Sharma and Jitendra Singh: Writing- Reviewing and
Editing.
## Acknowledgment
We would like to acknowledge IISER Bhopal, Gautam Buddha University Greater
Noida, and IIT Kanpur for providing institutional support. We thank to the
editor and all the anonymous reviewers for providing helpful comments and
suggestions.
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figs
# HetTree: Heterogeneous Tree Graph Neural Network
Mingyu Guan Georgia Institute of TechnologyUSA<EMAIL_ADDRESS>, Jack
W. Stokes Microsoft ResearchUSA<EMAIL_ADDRESS>, Qinlong Luo
MicrosoftUSA<EMAIL_ADDRESS>, Fuchen Liu MicrosoftUSA
<EMAIL_ADDRESS>, Purvanshi Mehta Lica World IncUSA
<EMAIL_ADDRESS>, Elnaz Nouri Microsoft ReasearchUSA<EMAIL_ADDRESS>and Taesoo Kim Georgia Institute of TechnologyUSA<EMAIL_ADDRESS>
###### Abstract.
The recent past has seen an increasing interest in Heterogeneous Graph Neural
Networks (HGNNs) since many real-world graphs are heterogeneous in nature,
from citation graphs to email graphs. However, existing methods ignore a tree
hierarchy among metapaths, which is naturally constituted by different node
types and relation types. In this paper, we present HetTree, a novel
heterogeneous tree graph neural network that models both the graph structure
and heterogeneous aspects in a scalable and effective manner. Specifically,
HetTree builds a semantic tree data structure to capture the hierarchy among
metapaths. Existing tree encoding techniques aggregate children nodes by
weighting the contribution of children nodes based on similarity to the parent
node. However, we find that this tree encoding fails to capture the entire
parent-children hierarchy by only considering the parent node. Hence, HetTree
uses a novel subtree attention mechanism to emphasize metapaths that are more
helpful in encoding parent-children relationships. Moreover, instead of
separating feature learning from label learning or treating features and
labels equally by projecting them to the same latent space, HetTree proposes
to match them carefully based on corresponding metapaths, which provides more
accurate and richer information between node features and labels. Our
evaluation of HetTree on a variety of real-world datasets demonstrates that it
outperforms all existing baselines on open benchmarks and efficiently scales
to large real-world graphs with millions of nodes and edges.
## 1\. Introduction
Graph neural networks (GNNs) have been widely explored in a variety of domains
from social networks to molecular properties (PinnerSage2020, ; Park2019, ;
sun2022does, ), where graphs are usually modeled as homogeneous graphs.
However, real-world graphs are often heterogeneous in nature (OGB, ; HGB, ).
For example, as shown in Figure 1(a), a heterogeneous email graph can involve
multiple types of nodes, including Domain, Sender, Recipient, Message and IP
Address, and the relations among them. Moreover, multiple relations can exist
between two entities in complex heterogeneous graphs. For example, a _Sender_
node can be a P1 sender and/or P2 sender of a _Message_ , where the P1 sender
denotes the entity that actually sent the message while an email application
displays the P2 sender as the “From” address. If Bob sends an email on behalf
of Alice, the email appears to come from Alice and Alice is the P2 sender
while Bob is the P1 sender in this scenario. As a result, there are two
relations between Sender and Message in Figure 1(a), p1_sends and p2_sends.
Figure 1. (a) Relational scheme of a heterogeneous email graph (b) An example
of the email graph.
To better understand real-world heterogeneous graphs, various heterogeneous
graph neural networks (HGNNs) have been proposed. The most well-known
approaches are the metapath-based methods (RGCN, ; HAN, ; MAGNN, ), which
first aggregate neighbor representations along each metapath at the node level
and then aggregate these representations across metapaths at the metapath
(semantic) level. However, metapath-based approaches often involve manual
effort to select a subset of meaningful metapaths, because the node-level
aggregation along each metapath for every node is computationally expensive
(MAGNN, ; HAN, ). Other models that do not apply the metapath method, such as
HetGNN (HetGNN, ) and HGT (HGT, ), carefully encode representations for
different node types and/or relation types in heterogeneous graphs. These
fine-grained embedding methods often utilize multi-layer message passing
approaches in traditional GNNs thus facing scalability issues. To efficiently
model real-world web-scale graphs, researchers and practitioners have explored
various ways to scale HGNNs. Sampling-based methods (HGT, ; HetGNN, ) sample
sub-graphs with different strategies, while others (NARS, ; SeHGNN, ) use
model simplification to execute feature propagation as a pre-processing stage
before training.
However, a tree hierarchy among the metapaths is ignored by existing HGNNs.
Considering the example in Figure 1, intuitively, metapath
$Sender\xrightarrow{p1\\_sends}Message$ $\xrightarrow{is\\_sent\\_from}IP$ is
more related to metapath $Sender\xrightarrow{p1\\_sends}Message$ than metapath
$Sender\xrightarrow{s\\_has\\_domain\\_of}Domain$, because of the overlap of
more node types and relations. Existing tree-based HGNNs (SHGNN, ; HetGTCN, ;
T-GNN, ) use tree structure to model the local topology of nodes, where a
semantic aggregation using attention is followed to combine the semantic
information for metapaths. However, simple semantic attention cannot fully
capture the semantic tree hierarchy among metapaths, where the parent-child
relationships between metapaths are ignored.
Moreover, to augment the data usage, label utilization is widely used in GNNs
(SLE, ; UniMP, ; GAMLP, ; LEGNN, ), which leverages ground truth in the
training set and propagates them through the graph structure as inputs to the
model. These methods either completely separate feature learning from label
learning and only combine them at the end, or simply treat the features and
label vectors the same by projecting them to the same latent space. They may
be feasible in homogeneous graphs but not in the case of heterogeneous graphs,
as the features and labels are related by the corresponding metapaths.
In this paper, we present HetTree, a scalable HGNN that extracts a unified
tree representation on metapaths, semantic tree, from the input heterogeneous
graph and proposes a novel tree aggregation with subtree attention to encode
the resulting semantic tree, in which propagated labels are carefully matched
with corresponding features based on metapaths.
To scale efficiently on web-scale graphs, HetTree follows the model
simplification approach that simplifies heterogeneous feature aggregation as a
pre-processing stage. Meanwhile, label aggregation of the target node types
for each metapath is also executed. HetTree then builds a semantic tree to
capture the hierarchy among metapaths. Instead of separating feature learning
from label learning (SLE, ; GAMLP, ) or treating features and labels equally
by projecting them to the same latent space (SeHGNN, ), HetTree proposes to
match them carefully on corresponding metapaths, which provides more accurate
and richer information between node features and labels.
To encode the resulting semantic tree, HetTree uses a novel subtree attention
mechanism to emphasize children nodes that are more helpful in encoding
parent-children relationships. Existing tree encoding techniques (SHGNN, ;
HetGTCN, ; T-GNN, ) aggregate children nodes by weighting the contribution of
children nodes based on similarity to the parent node. However, in the
semantic tree, this tree encoding fails to capture the entire parent-children
hierarchy by only considering the parent node. Hence, subtree attention in
HetTree models a broader parent-children structure while enhancing
correlations, bringing a better representation for each metapath.
In summary, we make the following contributions in this paper.
* •
We observe that existing HGNNs use simple semantic attention to aggregate
metapath representations, which ignore the hierarchy of the metapath features.
HetTree takes a radically new approach to encoding metapath hierarchy by
building a semantic tree for both pre-computed features and labels. To the
best of our knowledge, HetTree is the first to explore the tree hierarchy
among metapaths constructed from a heterogeneous graph.
* •
HetTree proposes a novel tree aggregation with subtree attention to encode the
proposed semantic tree structure, which emphasizes metapaths that are more
helpful in encoding parent-children relationships. For better label usage,
HetTree matches pre-computed features and labels correspondingly, which
constitutes a complete representation of a metapath.
* •
We conduct extensive experiments on five open graph datasets from Open Graph
Benchmark (OGB, ) and HGB Benchmark (HGB, ), as well as a real-world
commercial email dataset. The results demonstrate that HetTree is able to
outperform the state-of-the-art architectures on all datasets with low
computation and memory overhead incurred.
## 2\. Related Work
### 2.1. Graph Neural Networks
Graph Neural Networks (GNNs) are neural networks that take input structured as
graphs. The fundamental task in a GNN is to generate the representation of
graph entities, such as nodes and edges, in a $d$-dimensional space, referred
to as the embedding of the entity. To generate the embedding, GNNs usually use
a multi-layer feature propagation followed by a neural network to combine
structural information from the graph structure and the input features.
Various GNN architectures exist today, but they differ in how the information
is aggregated and transformed (Hamilton2017, ; GCN2016, ; GAT2018, ;
GraphSage2017, ). Nevertheless, a main problem with these classic GNNs is that
they are hard to scale due to the feature propagation performed at each layer
of the neural networks. A number of sampling methods (GraphSage2017, ;
FastGCN2018, ; LADIES2019, ; zou2019layer, ) have been proposed to reduce both
computation and memory complexity by using only a subset of nodes and edges.
Besides sampling, other methods (SGC, ; SIGN, ; S2GC, ) simplify models by
making feature propagation an offline pre-processing stage, so that this
computation-intensive process only needs to be executed once and is not
involved during the training process.
### 2.2. Heterogeneous Graph Neural Networks
While GNNs achieve excellent performance on homogeneous graphs, many real-
world graphs tend to be heterogeneous and applying homogeneous GNN approaches
to them is not trivial. Consequently, various excellent works have
investigated heterogeneous GNN architectures (HGNNs). The most general
approach in HGNNs is the so-called metapath-based method (MAGNN, ; HAN, ),
where the feature propagation is performed based on semantic patterns and
attention mechanism is usually applied at both the node level for each
metapath and the semantic level across metapaths. Other models (RGCN, ;
HetGNN, ; HGT, ; HGB, ) encode the heterogeneous aspect of the graph at the
fine-grained level using the multi-layer, message-passing framework common in
GNNs, where different weights are learned for distinct node and edge types.
However, HGNNs also inherit the scalability problem from traditional
homogeneous GNNs. Hence, sampling (HGT, ) and model simplification (NARS, )
have also been explored in the heterogeneous graph learning domain. NARS
(NARS, ) first applies the scaling approach proposed by SIGN (SIGN, ) on
heterogeneous graphs, which samples multiple relational subgraphs using
different sets of relations and then treats them as homogeneous graphs. Though
solving the problem of scalability, this approach can result in losing
semantic information by treating heterogeneous graphs as homogeneous ones
thereby confusing model learning by mixing different metapath semantics in
relational subgraphs. SeHGNN (SeHGNN, ) computes averaged metapath features
separately and applies Transformer-like attention to learn metapath features.
However, simply applying the Transformer ignores the hierarchy among metapath
features and thus results in sub-optimal results. Additionally, existing HGNNs
represent matapaths using node types, which fails to differentiate metapaths
that have different relation compositions but the same node types along the
metapaths.
### 2.3. Tree-based HGNNs
A few HGNNs have explored tree structure based on the local topology of nodes.
T-GNN (T-GNN, ) and SHGNN (SHGNN, ) construct hierarchical tree structures at
the node level, where each hierarchical tree represents a metapath and each
level of the tree contains nodes of a certain type. Similarly, HetGTCN
(HetGTCN, ) also constructs the tree hierarchy based on the local graph
structure of each node, where tree nodes at $k^{th}$ level are $k-hop$
neighbors of the root node. To encode tree-structured data, i.e. parent-child
relationships among tree nodes, they either use a weighted sum aggregator
(T-GNN, ; HetGTCN, ) or compute weights using attention mechanism for each
parent-child pair (SHGNN, ; HetGTCN, ). Moreover, these methods utilize the
tree structure to aggregate neighbor information for each metapath first and
then use semantic attention (HAN, ) to aggregate metapath representations to
obtain the final node representation. However, the tree hierarchy among
metapaths is ignored by existing methods. To the best of our knowledge,
HetTree is the first to explore the heterogeneous aspect of graphs based on a
tree-structured hierarchy among metapaths, which is naturally constituted by
different node types and relation types.
### 2.4. Label Utilization
Label utilization has been commonly applied in graph representation learning.
Label propagation (LabelPropagation, ) uses partially observed labels and
propagates them through the network structure. Label usage (LabelUsage, ) uses
a simple but effective method that concatenates node features and one-hot
labels. SLE (SLE, ) utilizes label propagation, separates feature and label
learning with two MLPs and then adds them together to generate the final
representations. However, label utilization naturally incurs two problems:
first, using labels in training can easily cause label leakage; second, high
training accuracy and low test accuracy can occur since only the training data
has labels. UniMP (UniMP, ) randomly masks the training nodes for each epoch.
GAMLP (GAMLP, ) modifies label propagation with residual connections to each
hop results to alleviate the label leakage issue.
However, these methods either completely separate feature learning and label
learning and only combine them at the end, or they simply add the features and
label vectors together as propagation information, which may result in good
performance in homogeneous graphs but not in the case of heterogeneous graphs,
since the features and labels are related by the corresponding metapaths.
## 3\. Preliminary
In this section, we provide formal definitions of important terminologies
related to HetTree. We also summarize notations frequently used in this paper
in Table 1.
Notations | Definitions
---|---
$\mathcal{G}$ | A heterogeneous graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{O},\mathcal{R})$.
$\mathcal{V}$ | The set of nodes.
$\mathcal{E}$ | The set of edges.
$\mathcal{O}$ | The set of node types.
$\mathcal{R}$ | The set of relation types.
$\mathcal{P}^{k}$ | The set of metapaths up to hop $k$.
$P$ | A metapath.
$\mathcal{P}_{O}$ | The set of metapaths ending with node
| type $O$ excluding $P=O^{init}$.
$\mathcal{N}^{v}_{P}$ | The set of metapath-$P$-based neighbors of node $v$.
$x^{v}$ | Initial feature of node $v$.
$y^{v}$ | Ground truth label of node $v$.
$X_{P}$ | Aggregated feature along metapath $P$.
$\hat{Y}_{P}$ | Aggregated label along metapath $P$.
$T_{O}$ | Semantic tree for node type $O$.
$C_{P}$ | A node in the semantic tree for metapath $P$.
$\mathcal{M}$ | The set of Metapath features.
$M_{P}$ | Metapath feature of metapath $P$.
$Z_{P}$ | Tree-encoded metapath feature of metapath $P$.
Table 1. Notations frequently used in this paper.
###### Definition 3.0.
Heterogeneous Graph. A heterogeneous graph is denoted as
$\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{O},\mathcal{R})$, where each
node $v\in\mathcal{V}$ and edge $e\in\mathcal{E}$ are associated with a node
mapping function $\tau(v):\mathcal{V}\rightarrow\mathcal{O}$ from node set
$\mathcal{V}$ to node type set $\mathcal{O}$, and a edge mapping function
$\phi(e):\mathcal{E}\rightarrow\mathcal{R}$ from edge set $\mathcal{E}$ to
relation set $\mathcal{R}$, respectively.
###### Example 3.0 0.
Figure 1(a) shows the relational scheme of a heterogeneous email graph while
Figure 1(b) shows an illustrative example. It is composed of five types of
nodes: $Domain$,$Sender$, $Message$, $Recipient$, $IP$, and six types of
relations: s_has_domain_of(H),
r_has_domain_of(D), p1_sends(O), p2_sends (T), receives(R),
is_sent_from(F).
Figure 2. (a) The offline process of feature aggregation. The center node is
the target node with $Sender$ node type and features are aggregated for all
metapaths $\mathcal{P}^{k}$ up to hop $k$, where $k=2$ in this example. For
instance, to compute aggregated feature $X_{HH}$, features of three
metapath-$HH$-based neighbors are aggregated, including the target node
itself. (b) The offline process of label aggregation. Unlike feature
aggregation, labels are aggregated only for metapaths
$\mathcal{P}^{k}_{Sender}$ that end with node type $Sender$ up to hop $k$. For
example, to compute aggregated label $\hat{Y}_{HH}$, features of two
metapath-$HH$-based neighbors are aggregated, excluding the target node
itself. (c) An illustration of the semantic tree with height of $k$ for Sender
nodes in heterogeneous email graph. A tree node $C_{P}$ represents metapath
$P$, where $P\in\mathcal{P}^{k}$.
###### Definition 3.0.
Metapath. A metapath $P$ is a path that describes a composite relation
$R=R_{1}\circ R_{2}\circ\cdots\circ R_{l}$ between node types $O_{1}$ and
$O_{l+1}$, where $\circ$ denotes the composition operator on relations. $P$ is
denoted as
$O_{1}\xrightarrow{\text{$R_{1}$}}O_{2}\xrightarrow{\text{$R_{2}$}}\cdots\xrightarrow{\text{$R_{l}$}}O_{l+1}$,
which is abbreviated as $R_{1}R_{2}\cdots R_{l}$. A special metapath where no
relation is present but includes only a node type $O$ is simply denoted as
$O^{init}$ (abbreviated as $init$ when $O$ is specified). The set of metapaths
ending with node type $O$ excluding $P=O^{init}$ is denoted as
$\mathcal{P}_{O}$. The set of metapaths up to hop $k$ is denoted as
$\mathcal{P}^{k}$, where $l\leq k,\forall P=R_{1}R_{2}\cdots
R_{l}\in\mathcal{P}^{k}$. Metapath-based neighbors $\mathcal{N}^{v}_{P}$ of
node $v$ along metapath $P$ are the set of nodes that are connected with node
$v$ via metapath $P$. Note that when $\tau(v)=O$ and $P$ ends with $O$,
$\mathcal{N}^{v}_{P}$ includes $v$ itself.
###### Remark 3.0 0.
Most of metapath-based HGNNs denote a metapath
$P=O_{1}\xrightarrow{\text{$R_{1}$}}O_{2}\xrightarrow{\text{$R_{2}$}}\cdots\xrightarrow{\text{$R_{l}$}}O_{l+1}$
as $O_{1}O_{2}\cdots O_{l+1}$ for short. We note that this notation fails to
differentiate metapaths that have different relation compositions but the same
node types along the metapaths, as multiple relations can be present between
two node types in a heterogeneous graph.
###### Example 3.0 0.
In Figure 1, a sender can be connected to a recipient through three 2-hop
metapaths: $Sender\xrightarrow{O}Message\xrightarrow{R}Recipient$ ($OR$),
$Sender\xrightarrow{T}Message\xrightarrow{R}Recipient$ ($TR$),
$Sender\xrightarrow{H}Domain\xrightarrow{D}Recipient$ ($HD$). Moreover, let
$S=\\{OR,TR,HD\\}$, then we have $S\subset\mathcal{P}^{k}$, for any $k\geq 2$.
###### Definition 3.0.
Semantic Tree. A semantic tree $T_{O}$ with depth of $k$, for node type $O$,
is composed of tree nodes $\mathcal{C}={C_{P},\forall P\in\mathcal{P}^{k}}$
and relation edges $\mathcal{R}$. The root node represents the metapath
$P=O^{init}$, denoted as $C_{O^{init}}$ (abbreviated as $C_{init}$ when $O$ is
specified). A non-root node $C_{R_{1}R_{2}\cdots R_{l}}$ represents the
metapath from the root node $C_{init}$ to them via relation edges
$R_{1}R_{2}\cdots R_{l}$. The root node $C_{init}$ is the parent of all nodes
$C_{R_{1}}$ at depth $1$ of the semantic tree $T_{O}$. A node
$C_{R_{1}R_{2}\cdots R_{l}}$ with depth $\geq 2$ has a parent node
$C_{R_{1}R_{2}\cdots R_{l-1}}$, and they are connected by edge $R_{l}$.
###### Example 3.0 0.
A semantic tree with depth of $2$ for Sender nodes in the heterogeneous email
graph is shown in Figure 2, where the root node $C_{init}$ represents metapath
init and other nodes represent the metapath from the root node to them. For
example, node $C_{OF}$ is connected with the root node $C_{init}$ by relation
edges $O$ and $F$ in order.
## 4\. Methodology
In this section, we describe a novel heterogeneous tree graph neural network,
HetTree, for scalable and effective heterogeneous graph learning. HetTree
consists of three major components: offline feature aggregation and semantic
tree construction (Section 4.1), metapath feature transformation (Section
4.2), and semantic tree aggregation (Section 4.3). We conduct complexity
analysis on HetTree and compare it with representative tree-based and scalable
HGNNs (Section 4.4). The overall process of HetTree is shown in Algorithm 1.
### 4.1. Offline Aggregation and Semantic Tree Construction
As a pre-processing stage, node features and labels are aggregated to prepare
for training. In node classification tasks on heterogeneous graphs, initial
node features are normally associated with every node (if not, we can use
external graph embedding algorithms like ComplEx (ComplEx, ) to generate
them), while labels are often only associated with nodes of target node types
that need to be classified. For example, in the email dataset we collect, the
task is to classify Sender nodes as a compromised email account or not, and
only Sender nodes are associated with labels. Moreover, as mentioned in
Section 2.4, only labels of training nodes can be used as part of the input
features, and the problem of label leakage in label utilization should be well
addressed. We also construct a novel semantic tree structure to organize the
aggregation results and capture the hierarchy of metapaths, which can be
leveraged in semantic tree aggregation (Section 4.3). We next describe the
offline feature and label aggregation as well as semantic tree construction in
detail.
Algorithm 1 The overall process of HetTree
0: Heterogeneous graph
$\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{O},\mathcal{R})$, Node features
$\mathcal{X}$, Node labels $\mathcal{Y}$, Metapaths $\mathcal{P}^{k}$.
0: The predicted labels $Y_{pred}$.
1: /* Offline Feature and Label Aggregation */
2: for $P\in\mathcal{P}^{k}$ do
3: for $v\in\mathcal{V}$ do
4: Feature aggr. $X_{P}^{v}=aggregate(\\{x_{u},\forall
u\in\mathcal{N}_{v}^{P}\\})$
5: if $P$ ends with target node type $O_{tgt}$ then
6: Label aggr. $\hat{Y}_{P}^{v}=aggregate(\\{y_{u},\forall
u\in\mathcal{N}_{v}^{P}\setminus\\{v\\}\\})$
7: end if
8: end for
9: end for
10: /* Metapath Feature Transformation */
11: for $P\in\mathcal{P}^{k}$ do
12: $M_{P}=\begin{array}[]{lcl}MLP(X_{P}\parallel\hat{Y}_{P}),&\mbox{if
}P\in\mathcal{P}_{O_{tgt}}^{k}\\\ MLP(X_{P}),&\mbox{otherwise}\end{array}$
13: end for
14: /* Semantic Tree Aggregation */
15: for $P\in\mathcal{P}^{k}$ do
16: Subtree reference
$S_{P}=MLP(M_{P}\parallel\sum_{Q\in\mathcal{P}^{child}_{P}}M_{Q})$
17: $\alpha_{Q}=\frac{exp(\delta(W_{P}\cdot[S_{P}\parallel
Z_{Q}]))}{\sum_{B\in\mathcal{P}^{child}_{P}}exp(\delta(W_{P}\cdot[S_{P}\parallel
Z_{B}]))}$
18: $Z_{P}=M_{P}+\delta(\sum_{Q\in\mathcal{P}^{child}_{P}}\alpha_{Q}\cdot
Z_{Q})$
19: end for
20:
$Y_{pred}=\begin{array}[]{lc}MLP(\sum_{P\in\mathcal{P}^{k}}Z_{P})+MLP(X_{init})\\\
+MLP(aggregate(\\{\hat{Y}_{P},\forall
P\in\mathcal{P}_{O_{tgt}}^{k}\\}))\end{array}$
#### 4.1.1. Feature Aggregation
Figure 2(a) shows the process of offline feature aggregation. Unlike existing
metapath-based methods (HAN, ; MAGNN, ), where feature aggregation is involved
with model learning such as projection and attention, HetTree separates
feature aggregation as a pre-processing stage. Existing methods manually
select metapaths with domain knowledge. The choice saves computation
complexity but also results in information loss. As the feature aggregation
happens offline and involves no parameter learning, which is much less
expensive than existing approaches, we use all metapaths up to hop $k$, where
$k$ is a user-defined parameter. For example, when $k=2$ as shown in Figure
2(a), feature aggregation is conducted on 14 metapaths
($init,O,T,H,OO,OT,OR,OF,TT,TO,TR,TF,HH,HD$). Possible aggregators include but
are not limited to $mean$, $sum$, $max$, and $min$, and we use the $mean$
aggregator in this paper for both feature and label aggregation. For each node
$v$, we compute a set of aggregated features $\mathcal{X}^{v}$:
(1) $\mathcal{X}^{v}=\\{X^{v}_{P}=aggregate(\\{x^{u},\forall
u\in\mathcal{N}^{v}_{P}\\}),\forall P\in\mathcal{P}^{k}\\}$
where $X_{P}^{v}$ represents the aggregated feature for node $v$ along
metapath $P$, and $aggregate$ is the aggregation function ($mean$ by default).
#### 4.1.2. Label Aggregation
Figure 2(b) shows the process of offline label aggregation. The label
aggregation process is very similar to the feature aggregation process
described in Section 4.1.1, except for two differences: first, since only
labels from nodes with target node type $O_{tgt}$ (node type to be classified)
in the training set can be used, the label aggregation is only conducted for
metapaths $\mathcal{P}_{O_{tgt}}^{k}$ where they end at node type $O_{tgt}$;
second, feature aggregation applies to all nodes in $\mathcal{N}^{v}_{P}$
including $v$ itself, while $v$ is excluded during label aggregation to avoid
label leakage. For example, when $k=2$ as shown in Figure 2(b), label
aggregation is conducted on 3 metapaths ($OT,TO,HH$) excluding the center
target node, respectively. Note that only labels from nodes in the training
set are used for label aggregation and zero vectors are used for nodes in the
non-training set. Specifically, for each node $v$, we compute a set of
aggregated features $\hat{\mathcal{Y}}^{v}$:
(2) $\begin{split}\hat{y^{v}}&=\left\\{\begin{array}[]{lll}y^{v},&\mbox{if
}v\in training\\_set\\\ \textbf{0},&\mbox{otherwise}\end{array}\right.\\\
\hat{\mathcal{Y}}^{v}&=\\{\hat{Y}^{v}_{P}=aggregate(\\{\hat{y}^{u},\forall
u\in\mathcal{N}_{v}^{P}\setminus\\{v\\}\\}),\forall
P\in\mathcal{P}_{O_{tgt}}^{k}\\}\end{split}$
where $y_{v}$ is the ground truth label for node $v$, $\hat{y}_{v}$ is the
label used in label aggregation for node $v$, and $\hat{Y}_{P}^{v}$ is the
aggregated label along metapath $P$ for node $v$.
#### 4.1.3. Semantic Tree Construction
We can construct a semantic tree $T_{O}$ for node type $O$ with tree nodes
$\\{C_{P},\forall P\in\mathcal{P}^{k}\\}$. $C_{init}$ is the root node and a
non-root node $C_{R_{1}R_{2}\cdots R_{l}}$ represents the metapath from the
root node $C_{init}$ to itself via relation edges $R_{1}R_{2}\cdots R_{l}$.
The parent of all 1-hop tree nodes $C_{R_{1}}$ is $C_{init}$, which are at
depth 1 of $T_{O}$. Starting from depth 2, $C_{R_{1}R_{2}\cdots R_{l}}$ is
connected with its parent node $C_{R_{1}R_{2}\cdots R_{l-1}}$ via edge
$R_{l}$. By constructing the semantic tree, the hierarchy among metapaths can
be captured, which provides the model the structural information of the
metapaths, compared to approaches that treat the relationships between
metapaths equally. Moreover, the semantic tree is also used as the underlying
data structure for semantic tree aggregation discussed in Section 4.3, where
the metapath features are aggregated following the tree structure in a bottom-
up way. Figure 2(c) shows an illustration of the semantic tree for the Sender
node type.
### 4.2. Metapath Feature Transformation
After obtaining aggregated features and labels for metapaths, we transform
them to the same latent space. This is due to the aggregated features for
metapaths being generated from row features of metapath-based neighbors with
different node types, which may have different initial spaces. Instead of
separating the transformation of features and labels as in existing methods
(SLE, ; GAMLP, ), HetTree automatically matches and concatenates the
aggregated features and labels of the same metapath $P$ for
$P\in\mathcal{P}_{O_{tgt}}$. This gives the model more accurate label
information of its metapath-based neighbors by designating the aggregated
labels with corresponding metapaths. Specifically, we compute the metapath
features $\mathcal{M}$ as
(3)
$\mathcal{M}=\\{M_{P}=\left\\{\begin{array}[]{lcl}MLP(X_{P}\parallel\hat{Y}_{P}),&\mbox{if
}P\in\mathcal{P}_{O_{tgt}}^{k}\\\
MLP(X_{P}),&\mbox{otherwise}\end{array}\right.,\forall
P\in\mathcal{P}^{k}\\}.$
### 4.3. Semantic Tree Aggregation
As discussed in Section 4.1.3, we construct a semantic tree $T$, and each tree
node $C_{P}$ represents a metapath $P$, where $P\in\mathcal{P}^{k}$. Since we
also obtain metapath features $\mathcal{M}$, we can associate each tree node
$C_{P}$ with $M_{P}$ correspondingly. Note that the semantic tree structure is
the same for all nodes with the same node type in a heterogeneous graph, so
the target nodes (to be classified) can easily be batched. We now have tree-
structured metapath features, and the hierarchical relationship between
metapath features needs to be well modeled when aggregating them.
The tree aggregation in HetTree is conducted in a bottom-up fashion. As it
gets closer and closer to the target node as the process proceeds, the
semantic tree aggregation can gradually emphasize those metapaths that
contribute more to the local subtree structure, i.e., the parent-children
relationship. To calculate the encoded representation $Z_{P}$ for each
metapath node in the semantic tree, HetTree applies a novel subtree attention
mechanism to aggregate the children nodes thus encoding the local subtree
structure. Unlike existing tree encoding methods (TreeLSTM, ; T-GNN, ; SHGNN,
; HetGTCN, ) that use either a simple weighted-sum aggregator or attention
mechanism to emphasize parent tree nodes, HetTree proposes the subtree
attention to encode both the parent and children representation and uses it to
emphasize the hierarchical correlation between metapaths. Specifically, let
$\mathcal{P}^{child}_{P}$ be the set of metapaths that
$\\{C_{Q},Q\in\mathcal{P}^{child}_{P}\\}$ are the set of children nodes of
$C_{P}$ in the semantic tree, HetTree computes a subtree reference as
$S_{P}=MLP(M_{P}\parallel\sum_{Q\in\mathcal{P}^{child}_{P}}M_{Q})$, for each
metapath $P$ in the semantic tree. Then, the weight coefficient $a_{Q}$ of
each children node $C_{Q}$ can be calculated as:
(4) $\alpha_{Q}=\frac{exp(\delta(W_{P}\cdot[S_{P}\parallel
Z_{Q}]))}{\sum_{B\in\mathcal{P}^{child}_{P}}exp(\delta(W_{P}\cdot[S_{P}\parallel
Z_{B}]))}.$
where $\delta$ is the activation function, $W_{P}$ is a learnable projection
vector for matapath $P$ and $\parallel$ stands for concatenation. Then we can
finally compute the encoded representation $Z_{P}$ for parent node $P$ by
aggregating encoded representation of children nodes as:
(5) $Z_{P}=M_{P}+\delta(\sum_{Q\in\mathcal{P}^{child}_{P}}\alpha_{Q}\cdot
Z_{Q}).$
| | Offline
---
Preprocessing
| Feature
---
Transformation
| Neighbor
---
Aggregation
| Semantic
---
Aggregation
| Total Training
---
Complexity
HAN | - | $\mathcal{O}(nd^{2})$ | $\mathcal{O}(nmpd)$ | $\mathcal{O}(nmd^{2})$ | $\mathcal{O}(nd(mp+md))$
SHGNN | - | $\mathcal{O}(nd^{2})$ | $\mathcal{O}(nmqd)$ | $\mathcal{O}(nmd^{2})$ | $\mathcal{O}(nd(mq+md))$
HetGTCN | - | $\mathcal{O}(nd^{2})$ | $\mathcal{O}(ed)$ | $\mathcal{O}(nmd^{2})$ | $\mathcal{O}(ed+nmd^{2})$
SeHGNN | $\mathcal{O}(hef)$ | $\mathcal{O}(nmd^{2})$ | - | $\mathcal{O}(nm^{2}d^{2})$ | $\mathcal{O}(nm^{2}d^{2})$
HetTree | $\mathcal{O}(hef)$ | $\mathcal{O}(nmd^{2})$ | - | $\mathcal{O}(nmbd^{2})$ | $\mathcal{O}(nmbd^{2})$
Table 2. Time complexity comparison among HAN, SeHGNN and HetTree. $n$, $e$,
$m$, $h$ and $N$ are the number of nodes, edges, metapaths, hops, and target
nodes respectively. $f$ and $d$ are input dimensions and hidden dimensions,
respectively. $p$ is the average number of metapath neighbor nodes and $q$ is
the average total number of nodes along a metapath. $b$ represents the average
number of children nodes in the semantic tree.
After the semantic tree aggregation has finished from bottom to top, the sum
of metapath representation will be used as the final representation of the
semantic tree. Moreover, we add a feature residual and a label residual to
further emphasize the initial features and labels aggregated from the
metapath-based neighbors. Specifically,
(6) $\begin{split}Y_{pred}\quad=&\quad
MLP(\sum_{P\in\mathcal{P}^{k}}Z_{P})+MLP(X_{init})\\\
&\quad+MLP(aggregate(\\{\hat{Y}_{P},\forall
P\in\mathcal{P}_{O_{tgt}}^{k}\\})).\end{split}$
where $aggregate$ is an aggregation function, which can be $mean$, $sum$,
$max$, $min$, etc. An illustration of the semantic tree aggregation process is
shown in Figure 3, following the same example in Figure 2.
### 4.4. Complexity Analysis
Table 2 provides a theoretical analysis of the time complexity for each
component of HetTree, compared with HAN (HAN, ), SHGNN (SHGNN, ), HetGTCN
(HetGTCN, ), SeHGNN (SeHGNN, ), where $n$, $e$, $m$, $h$, and $N$ are the
number of nodes, edges, metapaths, hops, and target nodes, respectively. $f$
and $d$ are input dimensions and hidden dimensions, respectively. Besides, $p$
is the average number of neighbor nodes for each metapath and $q$ is the
average total number of nodes along a metapath. $b$ is the average number of
children nodes for a parent node in the semantic tree.
Note that HAN, SHGNN, and HetGTCN perform multi-layer neighbor aggregation,
while SeHGNN and HetTree simplify neighbor aggregation as an offline
preprocessing process. SHGNN, HetGTCN, and HetTree utilize tree-based data
structure but the former two apply the tree structure on neighborhood
aggregation, i.e., at node level, while the semantic structure in HetTree is
for semantic aggregation, i.e., at metapath level. For each node, SHGNN
constructs a tree structure for each metapath based on the node topology, and
thus all nodes along the metapath take part in neighbor aggregation. HetGTCN
performs the neighbor aggregation for not only the target nodes but also nodes
of other types, where all edges are involved in neighbor aggregation. Since
SeHGNN is a transformer-based model, the self-attention mechanism results in
quadratic complexity ($m^{2}$) with respect to the number of metapaths.
HetTree utilizes the semantic tree structure to perform semantic aggregation,
where the proposed subtree attention mechanism gives $mb$ complexity. In
general, we have $e\geq nmq$ and $q>p\gg m>b$. Hence, HetTree has the lowest
theoretical complexity compared with the baselines.
Figure 3. The overall process of semantic tree aggregation. We aggregate
metapath representations based on our semantic tree structure in a bottom-up
fashion, which gradually emphasizes those metapaths that contribute more to
the local subtree structure, i.e., the parent-children relationship. A novel
subtree attention mechanism is used to compute weight coefficients of children
nodes, and then a weighted sum of them and the parent representation is used
to represent the encoded representation $Z_{P}$ for metapath $P$.
## 5\. Experiments
We conduct extensive experiments on six heterogeneous graphs to answer the
following questions.
Q1.:
How does HetTree compare to the state-of-the-art overall on open benchmarks?
Q2.:
How does HetTree perform in a practical compromised account detection task on
a noisy real-world email graph?
Q3.:
How does each component of HetTree contribute to the performance gain?
Q4.:
Is HetTree practical w.r.t. running time and memory usage?
Experimental Setup. All of the experiments were conducted on a computer with
dual 12-core Intel Xeon Gold 6226 CPU, 384 GB of RAM, and one NVIDIA Tesla
A100 80GB GPU. The server runs 64-bit Red Hat Enterprise Linux 7.6 with CUDA
library v11.8, PyTorch v1.12.0, and DGL v0.9.
| DBLP | IMDB | ACM | Freebase
---|---|---|---|---
| Macro-F1 | Micro-F1 | Macro-F1 | Micro-F1 | Macro-F1 | Micro-F1 | Macro-F1 | Micro-F1
RGCN | 91.52 $\pm$ 0.50 | 92.07 $\pm$ 0.50 | 58.85$\pm$0.26 | 62.05$\pm$0.15 | 91.55$\pm$0.74 | 91.41$\pm$0.75 | 46.78$\pm$0.77 | 58.33$\pm$1.57
HAN | 91.67$\pm$0.49 | 92.05$\pm$0.62 | 57.74$\pm$0.96 | 64.63$\pm$0.58 | 90.89$\pm$0.43 | 90.79$\pm$0.43 | 21.31$\pm$1.68 | 54.77$\pm$1.40
HetGNN | 91.76$\pm$0.43 | 92.33$\pm$0.41 | 48.25$\pm$0.67 | 51.16$\pm$0.65 | 85.91$\pm$0.25 | 86.05$\pm$0.25 | - | -
MAGNN | 93.28$\pm$0.51 | 93.76$\pm$0.45 | 56.49$\pm$3.20 | 64.67$\pm$1.67 | 90.88$\pm$0.64 | 90.77$\pm$0.65 | - | -
HGT | 93.01$\pm$0.23 | 93.49$\pm$0.25 | 63.00$\pm$1.19 | 67.20$\pm$0.57 | 91.12$\pm$0.76 | 91.00$\pm$0.76 | 29.28$\pm$2.52 | 60.51$\pm$1.16
HGB | 94.01$\pm$0.24 | 94.46$\pm$0.22 | 63.53$\pm$1.36 | 67.36$\pm$0.57 | 93.42$\pm$0.44 | 93.35$\pm$0.45 | 47.72$\pm$1.48 | 66.29$\pm$0.45
SeHGNN | 95.06$\pm$0.17 | 95.42$\pm$0.17 | 67.11$\pm$0.25 | 69.17$\pm$0.43 | 94.05$\pm$0.35 | 93.98$\pm$0.36 | 51.87$\pm$0.86 | 65.08$\pm$0.66
HetTree | 95.34$\pm$0.17 | 95.64$\pm$0.15 | 68.43$\pm$0.31 | 70.92$\pm$0.29 | 94.26$\pm$0.20 | 94.19$\pm$0.20 | 52.35$\pm$0.96 | 66.39$\pm$0.40
Table 3. Experimental Results of HetTree and baselines over four graphs in the
HGB benchmark. "-" means that the model runs out of memory on the
corresponding graph.
Datasets. We evaluate HetTree on four graphs from the HGB (HGB, ) benchmark:
DBLP, IMDB, ACM, and Freebase, a citation graph ogbn-mag from the OGB
benchmark (OGB, ) and a real-world email dataset collected from a commercial
email platform. We summarize the six graphs in Table 4.
Graph | #Nodes | #Edges | | #Node
---
Types
| #Relation
---
Types
#Classes
DBLP | 26,128 | 239,566 | 4 | 6 | 4
IMDB | 21,420 | 86,642 | 4 | 6 | 5
ACM | 10,942 | 547,872 | 4 | 8 | 3
Freebase | 180,098 | 1,057,688 | 8 | 36 | 7
ogbn-mag | 1,939,743 | 21,111,007 | 4 | 5 | 349
Email | 7,864,303 | 34,943,782 | 5 | 6 | 2
Table 4. Statistics of datasets.
Baselines. For the four graphs from the HGB benchmark, we compare the HetTree
results to the results reported in the HGB paper (HGB, ) as well as a state-
of-the-art work SeHGNN (SeHGNN, ). For ogbn-mag, we compare the HetTree with
top-performing methods from either the baseline paper or the leaderboard of
OGB (OGB, ). For the email dataset, we compare HetTree with the best-
performing baseline SeHGNN. All experimental results reported are averaged
over five random seeds.
Ethics and Broader Impacts. This work was reviewed and approved by independent
experts in Ethics, Privacy, and Security. For the email dataset, all users’
identities were anonymized twice, and the map from the second anonymized user
IDs to the first anonymized user IDs was deleted. Furthermore, the data was
handled according to GDPR regulations. In addition, the data was secured to
protect against data leakage and the data flow was approved by a privacy
manager. Since the goal of the model is to help protect email users from
attacks, we believe there are no potential negative, societal impacts.
### 5.1. Experiments on Open Benchmarks
To answer Q1, we compare the performance of the proposed HetTree to state-of-
the-art models on five heterogeneous graphs from two open benchmarks - HGB
(HGB, ) and OGB (OGB, ).
Performance on HGB Benchmark. Table 3 shows results that compare HetTree with
best-performing baselines on four datasets from the HGB benchmark. HetTree
outperforms the baselines on all graphs in terms of both Macro-F1 and Micro-F1
scores. We notice that HetTree benefits more from its semantic tree
aggregation, which encodes the hierarchy among metapaths, on more complex
tasks with more classes. As shown in Table 3, HetTree has more performance
gain on IMDB and Freebase with 5 and 7 classes, respectively, compared with
DBLP and ACM with 3 and 4 classes, respectively. This can be attributed to
HetTree’s semantic tree aggregation that learns more information, i.e., the
hierarchy among metapaths, which is ignored by the other baselines.
Performance on Ogbn-Mag Dataset. We also evaluate HetTree on a large-scale
citation graph, ogbn-mag (OGB, ), with millions of nodes in Table 5. We report
results without self-enhanced techniques like multi-stage training
(li2018deeper, ; sun2020multi, ; LEGNN, ), which are orthogonal to HetTree and
can be incorporated for additional benefits. The results show that HetTree
maintains its benefits with large graphs and outperforms all baselines.
Methods | Validation Accuracy | Test Accuracy
---|---|---
RGCN | 48.35 $\pm$ 0.36 | 47.37 $\pm$ 0.48
HGT | 51.24 $\pm$ 0.46 | 49.82 $\pm$ 0.13
NARS | 53.72 $\pm$ 0.09 | 52.40 $\pm$ 0.16
LEGNN | 54.43 $\pm$ 0.09 | 52.76 ± 0.14
GAMLP | 55.48 $\pm$ 0.08 | 53.96 $\pm$ 0.18
SeHGNN | 56.56 $\pm$ 0.07 | 54.78 $\pm$ 0.17
HetTree | 57.31 $\pm$ 0.15 | 55.54 $\pm$ 0.17
Table 5. Detection accuracy of the HetTree model and other baselines for the
ogbn-mag dataset.
### 5.2. Experiments on Commercial Email Graph
Besides the open benchmark datasets, we also collect a large email dataset
from a commercial email platform to answer Q2. In this experiment, a subsample
of real-world email data is used, which contains five types of entities -
senders (1,169,011 nodes), recipients (1,020,982 nodes), domains (686,787
nodes), IP addresses (296,939 nodes), and messages (4,690,584 nodes). The task
is to predict if the sender is legitimate or compromised, given its domain,
messages, recipients of the message, recipients’ domains, and message IP
addresses, where a compromised account may send several types of malicious
emails including phishing emails, malware attachments, and spam.
Since the email dataset contains binary labels (i.e., legitimate and
compromised), we can construct a Receiver Operating Characteristic (ROC) curve
for the models. The ROC curve for the detection of compromised emails is
presented in Figure 4 for the email dataset, and the accuracies are shown in
Table 6. Again, HetTree significantly outperforms the best-performing baseline
SeHGNN (SeHGNN, ).
Figure 4. ROC Curves for the email dataset. The error bars for HetTree are
tiny.
Methods | Val Accuracy | Test Accuracy
---|---|---
SeHGNN | 97.22 $\pm$0.02 | 97.26 $\pm$ 0.02
HetTree | 98.48$\pm$0.01 | 98.48 $\pm$ 0.02
Table 6. Detection accuracy for the email dataset.
### 5.3. Ablation Study
We here evaluate whether adding the subtree attention component and using
labels from the training set really help or not to answer Q3. The test F1
scores or accuracy of HetTree is evaluated on IMDB, ACM, and Ogbn-Mag compared
with its three variants: "weighted-sum", "parent-att" and "no-label". Variant
weighted-sum removes the proposed subtree attention but uses a weighted child-
sum like TreeLSTM (TreeLSTM, ). Variant parent-att removes the proposed
subtree attention but computes weights of children nodes using attention on
the parent node, which is the tree-encoding method used in SHGNN (SHGNN, ).
Variant no-label does not use labels as extra inputs.
The results in Table 7 show that each component is effective for HetTree. We
notice that datasets exhibit distinct sensitivities to individual components.
The subtree attention results in more performance gain on IMDB, which could be
attributed to the sparsity of the graph. It demonstrates that subtree
attention can capture the metapath hierarchy, compared to other tree encoding
mechanisms. Ogbn-Mag is more sensitive to label utilization, which could be
attributed to the large number of classes of labels that provide richer
information through propagation.
| IMDB | ACM | Ogbn-Mag
---|---|---|---
| Micro-F1 | Micro-F1 | Accuracy
HetTree | 70.92$\pm$0.29 | 94.19$\pm$0.20 | 55.54$\pm$0.17
weighted-sum | 69.70$\pm$0.24 | 93.54$\pm$0.23 | 55.38$\pm$0.21
parent-att | 69.69$\pm$0.41 | 93.83$\pm$0.18 | 54.87$\pm$0.26
no-label | 70.11$\pm$0.25 | 93.41$\pm$0.23 | 52.93$\pm$0.12
Table 7. Effectiveness of each component of HetTree.
0.0$\times$0.5$\times$1.0$\times$1.5$\times$2.0$\times$2.5$\times$3.0$\times$3.5$\times$4.0$\times$4.5$\times$DBLPIMDBACMEpoch
TimeDBLPIMDBACM0G2G4G6G8G10GMemory Usage
Normalized Epoch Time
Memory (GB)
HANHGBSeHGNNHetTree
Figure 5. Average epoch time and memory usage on HGB datasets.
### 5.4. Computation Cost Comparison
We next investigate the computational cost of HetTree in terms of epoch time
and memory footprint to answer Q4. We select three performant models - HAN
(HAN, ), HGB (HGB, ), and SeHGNN (SeHGNN, ) \- to compare with HetTree on four
graphs in the HGB benchmark. For fair comparison, we use a 2-layer structure
for HAN and HGN, and 2-hop feature propagation for SeHGNN and HetTree. The
result in Figure 5 shows that HetTree incurs the lowest computational cost in
terms of both running time and memory usage across three datasets from the HGB
benchmark.
## 6\. Conclusion and Discussion
In this paper, we present HetTree, a novel Graph Neural Network for
heterogeneous graphs. HetTree is based on the observation that existing HGNNs
ignore a tree hierarchy among metapaths, which is naturally constituted by
different node types and relation types. HetTree builds a semantic tree
structure to capture the hierarchy among metapaths and proposes a novel
subtree attention mechanism to encode the resulting semantic tree. Compared
with existing tree-encoding techniques that weight the contribution of
children nodes based on similarity to the parent node, subtree attention in
HetTree can model the broader local structure of parent nodes and children
nodes. The evaluation shows that HetTree is able to outperform state-of-the-
art baselines on open benchmarks and efficiently scale to large real-world
graphs with millions of nodes and edges. A future direction is to generalize
the semantic tree structure to not only scalable HGNNs but also HGNNs with
multi-layer aggregation, which performs a more fine-grained encoding on
semantics and the semantic tree structure could bring additional benefits.
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|
June-WooKim∗1,2 MiikaToikkanen2 YeraChoi3 Seoung-EunMoon3† Ho-YoungJung1†
# BTS: Bridging Text and Sound Modalities for Metadata-Aided Respiratory Sound
Classification
###### Abstract
Respiratory sound classification (RSC) is challenging due to varied acoustic
signatures, primarily influenced by patient demographics and recording
environments. To address this issue, we introduce a text-audio multimodal
model that utilizes metadata of respiratory sounds, which provides useful
complementary information for RSC. Specifically, we fine-tune a pretrained
text-audio multimodal model using free-text descriptions derived from the
sound samples' metadata which includes the gender and age of patients, type of
recording devices, and recording location on the patient's body. Our method
achieves state-of-the-art performance on the ICBHI dataset, surpassing the
previous best result by a notable margin of 1.17%. This result validates the
effectiveness of leveraging metadata and respiratory sound samples in
enhancing RSC performance. Additionally, we investigate the model performance
in the case where metadata is partially unavailable, which may occur in real-
world clinical setting.
###### keywords:
Respiratory Sound Classification, Pretrained Language-Audio Model, ICBHI,
Metadata
## 1 Introduction
Identifying abnormal respiratory sounds is pivotal for diagnosing and
providing timely interventions for respiratory conditions. Automated detection
of abnormal respiratory sounds has great potential to improve health and
quality of life for those affected by respiratory diseases by identifying
risks early and expediting first aid for potentially life-threatening
conditions, such as pneumonia or chronic obstructive pulmonary disease.
Machine learning approaches have been regarded as a promising way for
automated detection of abnormal respiratory sounds. Recently, a number of
studies [1, 2, 3, 4, 5, 6, 7, 8, 9] have tackled the respiratory sound
classification (RSC) task and notably increased the performance by utilizing
models that have been pretrained on large non-medical datasets [10, 11], and
then fine-tuned on a respiratory sound dataset [12].
Nevertheless, the inherent heterogeneity of respiratory sound data presents an
obstacle to further performance improvement in RSC. The heterogeneity arises
from differences in patient demographics, recording devices, and environmental
conditions, which can significantly impact the acoustic properties of
respiratory sounds [1]. This may lead to poor generalization on unseen data,
particularly in cases underrepresented by the training data. ICBHI [12], one
of the widely adopted respiratory sound datasets, provides metadata that
associates the recorded audio with attributes of patients and recording
environments. Such metadata may be useful for addressing difficulties caused
by heterogeneity.
Some previous work has adapted the metadata associated with respiratory sound
for RSC to mitigate the heterogeneity issue. For instance, incorporating
demographic information of patients such as age and gender into the
pretraining process provides better representations of respiratory audio
samples [5]. Moreover, metadata concerning the recording environment (i.e.,
stethoscope) also provides useful information. SG-SCL [1] employed domain-
transfer techniques to reduce the effect of heterogeneity by regarding
different types of recording devices as distinct domains. Despite the
potential benefits of leveraging the metadata, these previous works did not
fully incorporate it as text data into the model inputs.
Recent developments in multimodal models, exemplified by Contrastive Language-
Image Pretraining (CLIP) [13] for text and image data and Contrastive
Language-Audio Pretraining (CLAP) [14, 15] for text and audio data, offer a
flexible framework for integrating text data with non-textual data. Several
studies [16, 17, 18, 19] have demonstrated the effectiveness of language-EEG
multimodal models for the sentiment classification and EEG-to-text decoding
tasks. Recognizing the success of multimodal models and the demonstrated
benefits of multimodal data in healthcare tasks, it is compelling to consider
them for RSC, where such method has not yet been explored.
In this paper, we take a step into a new direction and fully make use of the
respiratory audio metadata by adapting a text-audio multimodal model, aiming
not only to leverage the metadata as an additional learning signal, but to
benefit from the further context during the inference stage. Building on the
foundation of contrastive language-audio pretrained models, our work
incorporates the respiratory audio metadata alongside the sound recordings. To
this end, we format the patient's metadata into descriptions derived from key
attributes including age, gender, recording device, and recording location on
the body, and encode them with respiratory sound data into shared feature
representation by the pretrained encoders. With these joint representations,
we train a classification head for the RSC task.
Figure 1: An overall illustration of the proposed BTS architecture. The
pretrained text and audio encoders extract feature representations of text
description derived from metadata and respiratory sound samples, respectively.
After the projection, the representations are integrated by a concatenation
operation and used for RSC.
Our approach, which we name the _BTS_ (_B_ ridging the _T_ ext and _S_ ound
modalities), a method that leverages multimodal text-audio model to fully
exploit the potential of respiratory audio metadata, achieves the state-of-
the-art (SOTA) result on the ICBHI dataset, outperforming upon the previous
best [4] by 1.17%. Our results reveal the capability of contrastive language-
audio pretraining to improve RSC both in audio-only and multimodal settings.
Moreover, we demonstrate that our method retains its performance gains in the
absence of metadata during the inference. This result suggests that our
approach can be adopted for practical clinical settings where additional
information other than audio signals may be unavailable. Our main
contributions are as follows:
* •
We show that leveraging metadata of respiratory sounds improves the RSC
performance. Our approach sets the new SOTA performance on the ICBHI dataset.
* •
We thoroughly explore ways to utilize the metadata considering a real clinical
setting where the type of metadata differs from the expectation, or the
metadata is partially or totally unavailable. We demonstrate that our method
robustly performs RSC in such scenarios.
* •
We analyze how the different types of metadata affect the performance of our
model. Our result shows that the information about the recording environment,
such as the type of recording stethoscopes and recording locations of the
human body are particularly helpful in minimizing the effect of heterogeneity
in respiratory sounds.
## 2 Method
We introduce _B_ ridging _T_ ext and _S_ ound modalities (BTS), an approach
that leverages multimodal text-audio model to fully exploit the potential of
respiratory audio metadata. To mitigate the heterogeneity of respiratory
sounds, we propose to explicitly utilize the metadata, which we expect to
capture the significant sources of acoustic variability. By integrating this
metadata, we aim to reduce the heterogeneity issue and improve the RSC
performance. Toward this goal, we propose the adoption of a multimodal text-
audio model for RSC, as depicted in Figure 1.
### 2.1 CLAP Model
While the metadata of respiratory sounds can be employed for RSC in several
different ways, a free text format is flexible and easily applicable to human-
produced data such as medical records. For instance, the metadata can be
described by a vector of numeric values where each element indicates different
metadata attributes. However, this approach is usually vulnerable to changes
in input sources, such as missing data and unseen data types. In contrast,
encoders for free text data are trained to understand the given input, which
makes approaches utilizing input data in a free text format robust to the
changes. For this reason, we use CLAP (Contrastive Language-Audio Pretraining)
[15] as our starting point. The CLAP model includes both text and audio
encoders, which are trained on the large-scale LAION-Audio-630K [15] dataset
including diverse audio data.
Given the text and audio data denoted as $X_{i}^{t}$ and $X_{i}^{a}$ where
$i\in[1,N]$ indicates the data index within a batch of size $N$, CLAP
processes the text and audio data independently through dedicated encoders
$f_{t}(\cdot)$ and $f_{a}(\cdot)$ for each modality. The embedding vectors
produced by the encoders are projected onto a $d$-dimensional shared embedding
space through projection layers $h_{t}(\cdot)$ and $h_{a}(\cdot)$.
$\displaystyle z_{t}$ $\displaystyle=h_{t}(f_{t}(X_{i}^{t})),$ $\displaystyle
z_{a}$ $\displaystyle=h_{a}(f_{a}(X_{i}^{a})).$ (1)
The CLAP model is trained to maximize the similarity between the text and
audio embeddings by contrasting them with negative samples (i.e., mismatched
text or audio embeddings obtained from $X_{j\in[1,N];j\neq i}^{t}$ or
$X_{j\in[1,N];j\neq i}^{a}$).
Table 1: Examples of generated text descriptions derived from metadata. `All'
is the case that includes all attributes: age, sex, recording location, and
recording device.
Metadata | Generated text descriptions
---|---
Age | This patient is an adult patient.
Sex | This patient is a male patient.
Loc | This sound was recorded from the left anterior chest.
Dev | This sound was recorded with a Meditron stethoscope.
Age-Loc-Dev | This sound was recorded from the left anterior chest
| of an adult patient, using a Meditron stethoscope.
……… | ………
All | This sound was recorded from the left anterior chest
| of an adult male patient, using a Meditron stethoscope.
### 2.2 Text Description Generation for Metadata
Among the metadata available in the ICBHI [12] dataset, we choose four types
of data as follows: age (adult or pediatric) and gender (male or female) of
patients, recording location on the chest of the patients (trachea, anterior
left, anterior right, posterior left, posterior right, lateral left, or
lateral right), and type of recording devices (Meditron, LittC2SE, Litt3200,
or AKGC417L). Using the attributes, we construct simple text descriptions. A
generated description can include any combination of the attributes, totaling
644 unique texts. Table 1 illustrates a few examples with different
combinations of metadata.
### 2.3 Bridging Text and Sound Modalities
As shown in Figure 1, we train the text and audio encoders of CLAP for RSC by
using the respiratory sound samples and generated text descriptions. For
classification, we concatenate text and audio representations $z_{t}$ and
$z_{a}$ from text and audio pipelines as described in Figure 1. Consequently,
we can obtain the multimodal combined representations $z=concat(z_{t},z_{a})$
where $z\in\mathbb{R}^{N\times 2d}$. We then simply add a 4-dimensional linear
layer for classifier $g(\cdot)$ followed by softmax function and train it with
the Cross-Entropy loss $\mathcal{L}_{\text{CE}}$ (division by $N$ is omitted):
$\displaystyle\mathcal{L}_{\text{CE}}=-\sum_{i=1}^{n}\\!\,y_{i}\\!\,\log\,\\!(\hat{y_{i}}),$
(2)
where $n$ is number of samples, $y$ is the respiratory sound label
$\in\\{$normal, crackle, wheeze, both$\\}$, and $\hat{y}$ is the predicted
probabilities obtained by the classifier.
## 3 Experimental Setup
### 3.1 Dataset
We utilized the ICBHI Respiratory dataset [12]. The dataset contains a total
of approximately 5.5 hours of respiratory sound recordings with pre-defined
and balanced splits for training (60%) and test (40%) without patient overlap.
There are 4,142 training and 2,756 testing respiratory cycles across four
classes. Table 2 illustrates the details of the ICBHI dataset. We binarize the
age as the adult (over 18 years old) or pediatric (18 years old or under) for
simplicity. Other than the age, we follow the metadata information of the
official ICBHI records. Body mass index (BMI) data, which was provided only
for adult patients, are solely employed for further analysis. For non-adults,
we calculated it using their weight and height data.
### 3.2 Training Details
Following the data pre-processing described in [1, 3, 4, 9], we extracted the
respiratory cycles from the waveform samples and standardized them to have a
duration of 8 seconds. We then conducted resampling to 48kHz to match the
pretraining data of CLAP. We employed the CLAP [15] model pretrained on the
LAION-Audio-630K [15] dataset for all experiments. The maximum length of the
text descriptions is limited to 64 tokens, which was sufficient for avoiding
truncation of text. We fine-tuned the models using the Adam optimizer [20]
with an initial learning rate of 5e–5. The learning rate was adjusted by
cosine scheduling through a total of 50 epochs of training with a batch size
of 8. To reduce the impact of random initialization, we conducted the
experiments with five different random seeds.
### 3.3 Metrics
We adapt the _Specificity_ ($S_{p}$), _Sensitivity_ ($S_{e}$), and their
average ($Score$) as performance metrics for RSC, following the definitions in
[12]. All reported values of $S_{p}$, $S_{e}$, and $Score$ are the mean and
variance from the five runs with different seeds.
### 3.4 Baselines
We compare the proposed method with previous studies including the current
SOTA method [4], which uses Audio Spectrogram Transformer (AST) [21] as a
backbone model. We also consider the result based solely on the audio
embedding of CLAP ($z_{a}$ in Equation (1)) as an additional baseline, which
we denote as Audio-CLAP.
Table 2: Details of the ICBHI dataset including the number of audio samples
for each class and the types of metadata. L/R stands for left or right.
| Label | Train | Test | Sum
---|---|---|---|---
Lung Sound | Normal | 2,063 | 1,579 | 3,642
Crackle | 1,215 | 649 | 1,864
Wheeze | 501 | 385 | 886
Both | 363 | 143 | 506
| Type | Metadata Label
Metadata | Age | Adult, Pediatric
Sex | Male, Female
Location | Trachea, L/R Anterior, L/R Posterior, L/R Lateral
Stethoscope | Meditron, LittC2SE, Litt3200, AKGC417L
| Others | BMI (Adult only), Weight/Height (Pediatric only)
## 4 Results
### 4.1 Main Results
Table 3: The RSC performance on the ICBHI dataset with the official 60–40%
train–test split. Here, in the Pretraining Data column, IN, AS, and LA refer
to ImageNet [10], AudioSet [11], and LAION-Audio-630K [15], respectively. $*$
denotes the previous state-of-the-art ICBHI Score. The Best and second best
results are highlighted by the bold characters and underlines.
Method | Backbone | Pretraining Data | Venue | $S_{p}$ (%) | $S_{e}$ (%) | Score (%)
---|---|---|---|---|---|---
SE+SA [22] | ResNet18 | - | INTERSPEECH`20 | 81.25 | 17.84 | 49.55
LungRN+NL [23] | ResNet-NL | - | INTERSPEECH`20 | 63.20 | 41.32 | 52.26
RespireNet [9] (CBA+BRC+FT) | ResNet34 | IN | EMBC`21 | 72.30 | 40.10 | 56.20
Chang et al. [24] | CNN8-dilated | - | INTERSPEECH`22 | 69.92 | 35.85 | 52.89
Ren et al. [25] | CNN8-Pt | - | ICASSP`22 | 72.96 | 27.78 | 50.37
Wang et al. [8] (Splice) | ResNeSt | IN | ICASSP`22 | 70.40 | 40.20 | 55.30
Late-Fusion [7] | Inc-03 + VGG14 | IN | EMBC`22 | 85.60 | 30.00 | 57.30
Nguyen et al. [6] (StochNorm) | ResNet50 | IN | TBME`22 | 78.86 | 36.40 | 57.63
Nguyen et al. [6] (CoTuning) | ResNet50 | IN | TBME`22 | 79.34 | 37.24 | 58.29
Moummad et al. [5] | CNN6 | AS | WASPAA`23 | 70.09 | 40.39 | 55.24
Moummad et al. [5] (SCL) | CNN6 | AS | WASPAA`23 | 75.95 | 39.15 | 57.55
Bae et al. [4] (Fine-tuning) | AST | IN + AS | INTERSPEECH`23 | 77.14 | 41.97 | 59.55
Bae et al. [4] (Patch-Mix CL) | AST | IN + AS | INTERSPEECH`23 | 81.66 | 43.07 | $\text{62.37}^{\textbf{*}}$
Kim et al. [3] (AFT on Mixed-500) | AST | IN + AS | NeurIPSW`23 | 80.72 | 42.86 | 61.79
Kim et al. [1] (SG-SCL) | AST | IN + AS | ICASSP`24 | 79.87 | 43.55 | 61.71
Kim et al. [2] (RepAugment) | AST | IN + AS | EMBC`24 | 82.47 | 40.55 | 61.51
Audio-CLAP [ours] | CLAP | LA | INTERSPEECH`24 | $\text{{80.85}}_{\pm 3.33}$ |<EMAIL_ADDRESS>3.77}$ |<EMAIL_ADDRESS>0.37}$
BTS [ours] | CLAP | LA | INTERSPEECH`24 | $\text{81.40}_{\pm 2.57}$ | $\text{{45.67}}_{\pm 2.66}$ | $\text{{63.54}}_{\pm 0.80}$
Table 4: A comparison of the ICBHI Scores between the Audio-CLAP baseline and
BTS. The results are shown depending on the metadata classes. Note that there
is no sample of the LittC2SE in the test set. The bold characters and
underlines indicate the best and second best Score improvement.
Metadata | Method | Score
---|---|---
Type | Class | Ratio (%) | BTS | Audio-CLAP | Difference
Age | Adult | 85.70 | 64.53 | 61.67 | 2.86
Pediatric | 14.30 | 64.53 | 61.99 | 2.54
Sex | Male | 78.74 | 64.53 | 62.00 | 2.53
Female | 21.26 | 64.46 | 61.92 | 2.54
Loc | Trachea | 11.97 | 64.46 | 61.92 | 2.54
Left Anterior | 21.99 | 64.52 | 61.66 | 2.86
Right Anterior | 9.51 | 64.78 | 61.58 | 3.20
Left Posterior | 22.57 | 64.54 | 62.00 | 2.54
Right Posterior | 15.64 | 65.31 | 61.45 | 3.86
Left Lateral | 9.43 | 64.41 | 61.83 | 2.58
Right Lateral | 8.89 | 60.21 | 54.44 | 5.77
Dev | Meditron | 16.65 | 64.54 | 62.00 | 2.54
LittC2SE | 0.0 | - | - | -
Litt3200 | 16.73 | 64.52 | 61.65 | 2.87
AKGC417L | 66.62 | 64.78 | 61.53 | 3.25
Table 3 presents comprehensive ICBHI results including our method. Our method
achieves a new SOTA by 1.17% improvement from the previous best without using
any additional training techniques that other methods rely on, such as
stethoscope-specific fine-tuning [9], co-tuning [6], Patch-Mix augmentation
[4], or domain adaptation techniques [1, 3]. Particularly, it is noteworthy
that our method has a considerably higher sensitivity ($S_{e}$) than the
previous best model while maintaining a similar specificity ($S_{p}$). We
consider that the additional context from textual descriptions enhances the
model's ability to correctly identify positive cases, without increasing the
false positive ratio. Additionally, the improvement of Audio-CLAP over the
previous SOTA highlights the effectiveness of the contrastively language-audio
pretrained encoder as a strong baseline for audio-only related tasks, where
the encoder is pretrained using text descriptions as opposed to previous
backbone models for RSC that employed categorical audio labels for
pretraining.
Table 4 compares the performance of BTS and Audio-CLAP for each metadata
category within the ICBHI test set. Although the dataset is notably imbalanced
for the metadata categories, our model consistently surpasses the Audio-CLAP
baseline across all classes. It is also noteworthy that a notable enhancement
is observed in minority classes. Specifically, the result of the test samples,
of which _Loc_ is the right lateral chest, yields the Score increase of 5.77%.
This underscores the value of the metadata in not only the overall performance
improvement but also the effectiveness of accounting for underrepresented
categories.
### 4.2 Influence of Metadata on Classification Performance
Table 5: Results of the ablation study with different combinations of the
metadata. The best result is indicated by the bold.
Method | Setting | Metadata | $S_{p}$ (%) | $S_{e}$ (%) | Score (%)
---|---|---|---|---|---
BTS | (1) | All | $\text{81.40}_{\pm 2.57}$ | $\text{45.67}_{\pm 2.66}$ | $\text{{63.54}}_{\pm 0.80}$
(2) | Age-Sex-Loc | $\text{81.71}_{\pm 4.12}$ | $\text{43.63}_{\pm 3.48}$ | $\text{62.66}_{\pm 0.35}$
(2) | Age-Sex-Dev | $\text{79.49}_{\pm 3.66}$ | $\text{46.04}_{\pm 2.39}$ | $\text{62.76}_{\pm 1.09}$
(2) | Age-Loc-Dev | $\text{82.28}_{\pm 5.27}$ | $\text{43.48}_{\pm 4.38}$ | $\text{62.88}_{\pm 0.83}$
(2) | Sex-Loc-Dev | $\text{84.66}_{\pm 3.63}$ | $\text{41.09}_{\pm 3.37}$ | $\text{62.88}_{\pm 0.54}$
Audio-CLAP | (3) | - | $\text{80.85}_{\pm 3.33}$ | $\text{44.67}_{\pm 3.77}$ | $\text{62.56}_{\pm 0.37}$
We analyze the impact of metadata on classification performance by comparing
the results of three distinct experiment settings: (1) the full set of
metadata with BTS, (2) subset with exclusion of a single attribute with BTS,
and (3) audio-encoder only in the case of Audio-CLAP. The results are
summarized in Table 5, which shows that more textual context results in higher
performance. Specifically, using all metadata in (1) yields the highest Score,
while audio-encoder only (3) scored the lowest. The results of all metadata
subsets (2) fall in between them.
Furthermore, the results demonstrate that the measurement location (_Loc_) and
recording device type (_Dev_) have a larger effect than the demographic
attributes, i.e., age and gender of patients. The absence of _Loc_ and _Dev_
leads _Score_ drop of 0.78% and 0.88%, respectively, compared to the all
metadata case. This suggests that the type of recording device and the
measurement location significantly influence the acoustic properties of
respiratory sounds. Therefore, the combined use of these metadata provides the
meaningful context to understand the respiratory sounds.
### 4.3 Unknown Metadata Scenario
To probe how well the model generalizes to unseen text descriptions, we
examined the model performance for unseen test data, which additionally
includes a new metadata attribute that is not used for training. Specifically,
we added a sentence to describe the BMI of patients to the test data. The
additional sentence is written in the same style as the training descriptions,
e.g., ``The BMI of the patient was 20.50.''. The evaluation result is denoted
as BTS[BMI] in Table 6. Adding unknown metadata to text descriptions at test
time shows only minor performance degradation, which suggests that the model
performs reliably even with the unexpected metadata.
### 4.4 Missing Metadata Scenarios
Table 6: Results with variations on the metadata. The variations include the
additional BMI attribute, partial metadata, and no metadata in text
description. The Best result.
Method | $S_{p}$ (%) | $S_{e}$ (%) | Score (%)
---|---|---|---
BTS | $\text{81.40}_{\pm 2.57}$ | $\text{45.67}_{\pm 2.66}$ | $\text{{63.54}}_{\pm 0.80}$
BTS[BMI] | $\text{81.40}_{\pm 2.57}$ | $\text{45.66}_{\pm 2.65}$ | $\text{63.53}_{\pm 0.80}$
BTS[Partial Metadata] | $\text{81.29}_{\pm 2.55}$ | $\text{45.54}_{\pm 2.61}$ | $\text{63.41}_{\pm 0.78}$
BTS[No Metadata] | $\text{80.82}_{\pm 3.54}$ | $\text{45.59}_{\pm 2.59}$ | $\text{63.21}_{\pm 1.04}$
Audio-CLAP | $\text{80.85}_{\pm 3.33}$ | $\text{44.67}_{\pm 3.77}$ | $\text{62.56}_{\pm 0.37}$
To understand how metadata that is partially or entirely missing effect the
model, we conducted two experiments. We first partially removed the metadata
(BTS[Partial Metadata]), by randomly eliminating one of the metadata
attributes from test samples and substituting ``Unknown'' with 10%
probability. Then, we entirely removed the metadata (BTS[No Metadata]) and
replace the whole description by ``No description.'' Table 6 describes the
experiment results. As expected, BTS[Partial Metadata] shows a slightly
degraded Score compared to BTS, while BTS[No Metadata] results in a relatively
large performance reduction. Nevertheless, the results with missing metadata
maintain an edge over Audio-CLAP. The results show that the BTS model is
robust to missing metadata. We conjecture that the model learns to infer
certain metadata characteristics directly from the audio, thereby preserving
its strong performance even in the absence of metadata during inference.
## 5 Conclusion
In this work, we proposed to directly utilize the metadata to improve the
performance of RSC. Our experiments demonstrated that including the metadata
as additional context for the classification leads to a considerable
performance increase, which results in the new SOTA for RSC. In particular,
the experiment results showed that our method helps minimize the performance
degradation due to the acoustic variations induced by the inherent factors
relating to the demographics and recording environment. Moreover, our method
works reliably even when the metadata is with unexpected information,
partially unavailable, or even completely unavailable. Besides, we showed that
the CLAP model provides a strong baseline for audio-only tasks.
## 6 Acknowledgement
This research was supported by the MSIT (Ministry of Science and ICT), Korea,
under the ITRC (Information Technology Research Center) support program
(IITP-2024-2020-0-01808) supervised by the IITP (Institute of Information &
Communications Technology Planning & Evaluation) and by Brian Impact
Foundation, a non-profit organization dedicated to the advancement of science
and technology for all.
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|
## 5 Other Potential Science Cases
### 5.1 Galactic Mid-Plane Survey
While the Gaia mission is creating groundbreaking advances in the exploration
of the structure and kinematics of the Galaxy as mentioned above, Gaia’s
contributions to understanding the large-scale dynamics of the youngest
populations of the disk (“Extreme Population I”) will necessarily be limited,
as even $1\deg$ of latitude is the scale height of these populations at a
distance of about 5 kpc. Yet it is within this $\lesssim 100$ pc from the
Galactic mid-plane where is the main acting fields of the primary agents of
disk dynamical evolution—e.g., Giant Molecular Clouds, the spiral arms, and
the densest parts of the Galactic bar. To gather a complete understanding of
processes such as secular orbital heating and radial migration that are driven
by these perturbations requires contiguous kinematic information spanning the
very youngest stars born in the Galactic midplane to their older siblings
(well studied by other surveys) that have long since been scattered to
dynamically hotter and/or radially migrated orbits.
Moreover, without accurate proper motions, even surveys like APOGEE, which is
providing the first global view of the detailed chemistry and radial
velocities of disk and bulge stars, are limited in their ability to place this
information within a firm dynamical context; many thousands of mid-plane stars
in the APOGEE database lack Gaia astrometry, so that 3D orbits are not
possible to be inferred. Only a NIR astrometry facility can remedy this
problem by supplying a similar global view of mid-plane stellar disk dynamics.
The importance of accessing this “optically hidden Milky Way” has motivated
discussions at ESA to follow-up Gaia with a NIR counterpart mission, GaiaNIR
(e.g., Hobbs et al., 2021), but this likely will not be realized for a few
decades.
JASMINE is well-suited to be a pathfinder for a proposed large flagship all-
sky astrometric survey mission in the NIR, like the GaiaNIR concept, and is
capable of having an immediate impact in the field of Galactic archaeology. In
addition to the GCS in section 2, to address the science objectives of the
inner disk dynamics summarised in section 2.6, tracing the dynamics and
chemistry of the stars in the Galactic disk mid-plane at the various azimuthal
angles of the disk is required. One potential targeting strategy is a small
campaigns of mosaicked JASMINE pointings, which are centered on the location
of APOGEE $b=0\deg$ fields, which are located every about $5~{}\deg$ in
longitude around the entire sky. Such a strategy will instantly pay dividends
through the value-added information on tens of thousands of stars with
spectroscopy from the APOGEE and APOGEE-2 surveys, and their SDSS-V extension,
Milky Way Mapper (see section 6.1.3), that will lack Gaia’s accurate
astrometric data, because of the extreme foreground obscuration. These data
will provide a systematic probe of the dynamics of the low latitude disk,
The required astrometric accuracy should not be as strict as the GCS. Although
APOGEE spectroscopy yields $\sim 0.1$ km s-1 accuracy radial velocities for
stars, we do not require such accuracy in the transverse velocities. This is
because we are primarily interested in measuring the velocity dispersions for
young Population I stars, which are of order 10–30 km s-1 per dimension. For
example, 200 $\mu$as yr-1 proper motions provided by JASMINE translates to a
transverse velocity error of $<5$ km s-1 for a star at 5 kpc (this is a bit
higher than the approximate median distance of the APOGEE midplane stars,
which are primarily red giants), and $<10$ km s-1 for stars at 10 kpc. This
relaxed requirement allows us to survey a large range of the Galactic
longitude, when JASMINE is not observing primary science targets.
#### 5.1.1 Star forming region
A rigid adherence to the pointing strategy of the Galactic mid-plane survey is
not necessary, and other interesting science problems–e.g., astrometric and
photometric exploration of young stellar clusters and star forming regions
(which naturally lie at low latitude) can also benefit from judicious
placement of the JASMINE Galactic mid-plane Survey “pickets”. For example, the
APOGEE project has dedicated itself to intense spectroscopic probes of several
star forming regions (e.g., the Orion Complex, the Perseus Molecular Cloud,
and the Cygnus region), which may also be matched with JASMINE targeting.
Stars are mainly formed in Giant Molecular Clouds (GMCs). Moreover, star
forming regions strongly concentrate into very compact sections within GMCs.
Using the Two Micron All Sky Survey (2MASS) point source catalog, Carpenter
(2000) estimated the fraction of young stellar populations contained within
clusters to be 50$\%-100\%$ for nearby cluster-forming GMCs, such as Perseus,
Orion A, Orion B and MonR2. NIR surveys of young stellar populations using the
Spitzer Space Telescope have confirmed that clustered star formation is the
dominant mode of star formation in the Galaxy (e.g., Poulton et al., 2008;
Román-Zúñiga et al., 2008).
This traditional picture is now challenged by the recent Gaia data releases,
which are revealing more complex reality. Anders et al. (2021) claimed that
only about 16 % of the stars in the solar neighborhood formed in bound
clusters, comparing the star formation rate in the solar neighbourhood and the
populations of the young star clusters. From the more small-scale kinematics
of the OB associations, Ward et al. (2020) showed that the velocity fields of
the OB association is highly substructured (see also Wright & Mamajek, 2018),
which is not consistent with a monolithic scenario, where stars formed in the
core of bound clouds and expanded subsequently due to the outflow of the gas
caused by feedback. They discussed that these are consistent with a
hierarchical star formation model, where stars formed in large-scale
gravitationally unbound structures in molecular clouds.
Although the Gaia data is revolutionizing our understanding of star formation,
these optical observations are unavoidably missing the information deep in the
core of star forming regions where the dust extinction is too severe. The NIR
astrometric observations with JASMINE and APOGEE spectroscopic data can unveil
the whole picture of the star forming region hidden in the dust, which will
complement what the Gaia data revealed and provide a more complete picture of
these star forming regions.
#### 5.1.2 Milky Way Neighborhood Dwarf Galaxies
The dwarf galaxy population around the Milky Way is diverse and new dwarfs are
continuously being discovered (e.g., Willman et al., 2005; Belokurov et al.,
2006; Zucker et al., 2006). The once troublesome “missing satellites problem”
that plagued the $\Lambda$CDM cosmology theoretical framework is now steadily
being refined and coming in line with observations of the dwarf galaxy
populations around more massive galaxies such as the Milky Way (Simon & Geha,
2007; Kim et al., 2018; Mao et al., 2021). The ESA F-class mission ARRAKIHS (a
planned 2030 lanuch) will further this inventory with unprecedented surface
brightness levels in the Euclid VIS (0.550-0.900 $\mu$m), Y (0.920-1.230
$\mu$m) and J (1.169-1.590 $\mu$m) bands images of nearby galaxies. One
remarkable discovery that came from the recent Gaia DR2 is that of the giant
dwarf galaxy Antlia 2 (Torrealba et al., 2019). The primary reason that this
giant dwarf galaxy lurking in the dark matter halo of the Milky Way went
undetected until now is that it lies only 11 degrees off the Galactic plane.
Where Gaia has run into the limit of visual-wavelength extinction, JASMINE can
go deeper into the Galactic mid-plane, as part of the Galactic mid-plane
survey, allowing for serendipitous dwarf galaxy and globular/stellar cluster
discoveries that cannot be made via any other method. With the anticipated
astrometric precision and stellar content, we will be able to detect over-
densities of stars that contain common proper motions indicative of low-
latitude dwarf galaxies and/or clusters that are in the optical Zone of
Avoidance, within a distance of about 10 kpc, as demonstrated for Gaia data in
Antoja et al. (2015) and Ciucǎ et al. (2018). So far, there is no dwarf galaxy
found within 20 kpc, and Draco 2 at about 22 kpc is the closest (e.g.,
McConnachie & Venn, 2020). Although the total field of view of the Galactic
mid-plane survey is very limited, finding any galaxy within 20 kpc will be a
significant discovery, and will be integral to studies of the survivability of
dwarf galaxies within the inner Galaxy, their contribution to the bulge, and
their impact on the Galactic disk (e.g., D’Onghia et al., 2010).
### 5.2 X-ray Binaries
Another interesting target for which the precise astrometry with a short
cadence observations of JASMINE can provide unique and impactful data is X-ray
binaries, including gamma-ray binaries (Aharonian et al., 2005). These are
ideal laboratories for the study of high-energy astrophysics and prime future
targets for multi-messenger astronomy, including continuous gravitational wave
observations (e.g., Middleton et al., 2020). Astrometric measurements of their
companion stars enables us to measure the physical scale of the orbital
parameters, and to unveil the mass of the compact object, whether it be a
white dwarf, neutron star or BH (Yamaguchi et al., 2018b, see also section
5.3.1). We listed below examples of X-ray binaries, which are particularly
interesting targets for JASMINE, when JASMINE cannot observe the GCS field.
This merely presents examples of potential targets, and is not an exhaustive
list.
$\gamma$ Cassiopeia ($\gamma$ Cas): $\gamma$ Cas is considered to be the first
star identified as Be star (B type star with emission lines) (Secchi, 1866).
However, now $\gamma$ Cas is known to be a rare kind of Be star, which is
characterised by a hard X-ray spectrum with a thermal X-ray emission, a high
temperature ($>10$ keV) and a lack of strong variability (e.g., Motch et al.,
2015; Nazé & Motch, 2018; Tsujimoto et al., 2018). Despite the proximity of
the object (d$\sim 168$ pc) (van Leeuwen, 2007) and several decades of
observational and theoretical studies, X-ray emission mechanism and the nature
of the lower mass secondary star are still debated (e.g., Smith, 2019; Langer
et al., 2020). While the $\sim$204 day binary period with close to the
circular orbit (Miroshnichenko et al., 2002; Nemravová et al., 2012) is known,
the mass of smaller mass secondary star is not well measured and it is still
debated if the secondary star is white dwarf or neutron star (Langer et al.,
2020). With a visual magnitude of $\sim 2$, $\gamma$ Cas is too bright for
Gaia. JASMINE can adjust the exposure time to observe such a bright object
with a short cadence. Such time-series astrometric information will provide
the precise orbital parameters and mass of the secondary star from astrometric
observations (Yamaguchi et al., 2018b), which will be a crucial to
understanding the long-debated properties of $\gamma$ Cas and the other
similar systems ($\gamma$ Cas analogs).
LSI +61 303 / HESS J0632+057: These are both gamma-ray binaries, in which the
source of gamma rays may be the impact of a relativistic pulsar wind on out-
flowing protons in the disk of a Be star, UV photons from a massive main
sequence star, or by the interaction of UV photons from such a star on the
accretion disk of an X-ray binary counterpart (Mirabel, 2012). Determining the
masses of these companions can be achieved with high astrometric precision
observations
X Per / V725 Tau: Like $\gamma$ Cas, these are X-ray binaries with rapidly
rotating B stars, and likely host neutron star companions. Neutron stars are
characterize by their equation of state, which requires knowing their masses
(e.g., Demorest et al., 2010), which can be done with precise determination of
orbital parameters (Miller, 2013).
Cygnus X-1: The tens of $\mu$as astrometric regime achievable with JASMINE
will make new studies of the orbit and jet physics of this quintessential BH
binary system possible, and the NIR astrometry of this source can be compared
to the exquisite radio positional information that exists from VLBI studies
(e.g., Miller-Jones, 2014).
### 5.3 Complementary Sciences with the Galactic Center Survey data
The JASMINE GCS data will provide the accurate astrometry and time-series
photometry for all the stars with $9.5<H_{\mathrm{w}}<14.5$ mag in the JASMINE
GCS field. The GCS data should be valuable for the wide range of scientific
studies, not just for the core science of JASMINE as shown in section 2. In
this section, we highlight some of these science cases. Note that the aim of
this section is not to provide the comprehensive list, but merely list the
potential science cases. We hope that many science cases that are not as
premeditated will be developed by the wider science community.
#### 5.3.1 Hunting Inner Disk BHs
Massive stars are expected to become BHs upon their demise (e.g., Maeder,
1992). Therefore, it is expected that there are many stellar mass BHs floating
around in the Milky Way (e.g., Brown & Bethe, 1994). Stellar mass BHs are
found in the Galaxy as X-ray binaries (e.g., Özel et al., 2010), whose masses
are around 5 to 20 $M_{\odot}$. Several gravitational wave detections of
stellar mass BH binaries since the first detection of GW150914 by LIGO/VIRGO
collaborations (LIGO Scientific Collaboration and Virgo Collaboration, 2016)
revealed that there are indeed stellar mass BHs, up to $M_{\rm BH}\sim 150$ M⊙
(LIGO Scientific Collaboration and Virgo Collaboration, 2020). Two remaining
questions are what is the mass function of this BH population and how they are
spatially distributed in the Galaxy. These questions are also related to the
origin of SMBHs, as discussed in section 2.2.
One promising method to detect a large population of these stellar mass BHs is
finding a binary system of a BH and a companion star bright enough to allow
for its kinematics to be measured with astrometry and/or spectroscopic
observations. These system does not require the companion star be interacting
to the BH and be emit X-ray, i.e. non-interacting BH (Thompson et al., 2019).
Such non-interacting BHs are expected to be observed by precise astrometry
available with Gaia (e.g., Penoyre et al., 2021). Recently Shikauchi et al.
(2022) estimated that Gaia will detect $\sim 1.1-46$ non-interacting BH
binaries (see also Kawanaka et al., 2017; Yamaguchi et al., 2018a). In fact,
so far from Gaia DR3 and follow-up observations, two non-interacting BH
binaries, Gaia BH1 with a Sun-like star (El-Badry et al., 2023b) and Gaia BH2
with a red giant star (El-Badry et al., 2023a; Tanikawa et al., 2022), have
been found. JASMINE will offer similarly precise astrometry for stars in the
inner disk from the Sun to the Galactic center, where Gaia cannot observe due
to the high extinction. Therefore, JASMINE is expected to uncover the
population of BHs in the inner Galaxy. According to a similar model of
Shikauchi et al. (2022), in the JASMINE GCS region $100-1,000$ BH$-$star
binaries are expected to exist. Further study of how many of such binaries can
be detected by JASMINE is ongoing.
Another way of hunting BHs with JASMINE is microlensing. JASMINE will offer
the time-series photometry of the Galactic center region where the stellar
density is very high. We expect that JASMINE will find about 3 microlensing
events during the nominal operation of 3 years, which is an optimistic
estimate based on the VVV microlensing survey results (Navarro et al., 2020).
As suggested by Abrams & Takada (2020), the long timescale ($>100$ days)
microlensing events are expected to be dominated by a high mass ($>\sim 30$
M⊙) BH lens. Photometric microlensing itself does not give us the lens’ mass.
However, JASMINE can also detect astrometric microlensing from the centroid
shift of the source (e.g., Dominik & Sahu, 2000; Belokurov & Evans, 2002).
Astrometric microlensing enables us to measure the source mass if the mass and
distance to the lens are optimum for the measurement. Recently, the first
astrometric microlensing measurement has been reported for a microlensing
event found by the ground-based observation and followed up by the Hubble
Space Telescope for astrometry (Lam et al., 2022; Sahu et al., 2022). However,
the lens masses for the same event, OGLE-2011-BLG-0462/MOA-2011-BLG-191,
reported by the two teams are so far quite different. Sahu et al. (2022)
reported the lens mass of $7.1\pm 1.3$ M⊙, clearly indicating a BH, at
distance of $1.58\pm 0.18$ pc, while Lam et al. (2022) reported the lens mass
of $1.6-4.2$ M⊙, which could be a neutron star or BH, at a distance of
$0.69-1.37$ kpc. This tension could be due to systematic uncertainty from the
two independent measurements of photometry and astrometry. The astrometric
displacement of this event was $>$mas and large enough to be clearly detected
by JASMINE. JASMINE can provide both photometric and astrometric information,
which may help to reduce the systematic uncertainty of such microlensing
events. The chance of having such an event in the JASMINE GCS field in the
lifetime of JASMINE could be slim, but one event of precise measurement of the
high mass BH would still be an exciting outcome, because only few of such
events are expected even in the Gaia data (Rybicki et al., 2018).
#### 5.3.2 Hunting IMBH in the Galactic center
A pressing mystery is the low number of confidently confirmed IMBHs
$M_{\mathrm{BH}}=100-10^{5}$ $M_{\odot}$ (Chilingarian et al., 2018). Many
candidates are, however, known (e.g., Graham et al., 2021b), and they may form
the low-mass ($M_{\rm bh}<10^{5}\,M_{\odot}$) extension to the quadratic
BH/bulge mass scaling relation for disk galaxies (Graham & Scott, 2015). The
Galactic center is an attractive area to explore for these long-sought after
IMBHs (see Greene et al., 2019, for a review), especially if brought in
through the capture of dwarf-mass galaxies. Furthermore, Portegies Zwart et
al. (2006) demonstrated that some massive star clusters formed in the central
100 pc undergo core collapse before the massive stars die, i.e. $\sim 3$ Myr,
which induces a runaway stellar merger and creates a IMBH. They estimated that
within 10 pc from the Galactic center about 50 IMBHs may exist. Some of them
could be still within the survived star clusters (e.g., Fujii et al., 2008),
like the star clusters near the Galactic center, the Arches (Figer et al.,
2002) and the Quintuplet (Figer et al., 1999). Detecting a star cluster in a
high stellar density region like the Galactic center only with photometric
data is difficult. However, the proper motion data from JASMINE will enable
the detection of star clusters in the Galactic center (see also section 2.3),
which can inform follow-up studies of their cluster centers with X-ray and/or
radio surveys (e.g., Oka et al., 2017; Tsuboi et al., 2017).
Interestingly, so far five IMBH candidates have been discovered as high-
velocity (velocity width $>50$ km s-1) compact ($<5$ pc) clouds in the
Galactic center (Takekawa et al., 2020). The advent of the Atacama Large
Millimeter/submillimeter Array (ALMA) enables the measurement of the detailed
velocity structure of the compact clouds less than 0.1 pc from the center,
which is consistent with Keplearian rotation around a massive object whose
inferred mass between $10^{4}$ and $10^{5}$ M⊙ (e.g., Tsuboi et al., 2019;
Takekawa et al., 2020). Although further studies are required to prove that
they are the true IMBHs, these observations may indicate that several IMBHs
exist in the Galactic center.
With JASMINE, IMBHs can be detected as a binary motion of bright stars around
an IMBH or astrometric microlensing, as discussed in the previous section, if
such systems exist or such an event occurs. For example, if a 1 M⊙ AGB star is
rotating around a 1,000 M⊙ BH with the orbital period of 3 years with zero
eccentricity at a distance of 8 kpc, the semi-major axis of the orbit
corresponds to 2.6 mas, which can be detected by JASMINE.
There will be an astrometric microlensing event if an IMBH crosses in front of
a distant star. Following Toki & Takada (2021), we can consider an event for a
source star at 8 kpc, and a lens object of an IMBH crossing at 7.5 kpc. The
Einstein time-scale of this event is 713 days, and the maximum displacement
due to the astrometric microlensing is 2.9 mas (Toki & Takada, 2021). This can
be detected by JASMINE, though such an event would be extremely rare.
#### 5.3.3 Gravitational Waves
Gravitational waves (GWs) have been successfully detected by the Laser
Interferometer Gravitational-Wave Observatory (LIGO), Virgo and the Kamioka
Gravitational Wave Detector (KAGRA) collaborations. The sources for these
events are merging compact objects such as BHs and neutron stars. It is of
importance to detect gravitational waves from SMBHs binaries to study the
growth mechanism of SMBHs. These waves have much longer wavelengths than
detectable by ground-based detectors. Astrometry could be a valuable resource
to detect or constrain such low frequency gravitational wave (e.g., Klioner,
2018).
Here, we estimate strain sensitivity of the JASMINE GCS. It is well known that
the maximal magnitude of the astrometric effect of a gravitational wave is
$h/2$ for $h$ being the strain. The astrometric accuracy of single
observations of JASMINE being $\Delta\theta=4$ mas for stars with magnitude
$H_{w}<12.5$ mag and the uncertainty grows exponentially for fainter sources.
Given that each star will be observed around $N_{\rm obs}=68,000$ times,
considering a realistically expected distribution of stars in $H_{w}$
magnitudes of the GCS and using the theoretical formulation developed for
Gaia-like astrometry (Klioner, 2018), one can conclude that the full
sensitivity of JASMINE to the effects of a gravitational wave will be
$h=3\times 10^{-13}$. Here we assume that the instrument is ideally
calibrated, so that the full accuracy scales as $N_{\rm obs}^{-1/2}$ for each
source and the sensitivity is also accordingly computed from the combination
of the contributions from individual sources.
However, the astrometric effect of a gravitationa wave is proportional to the
sine of the angular separation $\chi$ between the directions of observations
and that towards the gravitational wave source (Book & Flanagan, 2011;
Klioner, 2018). Although JASMINE makes relative astrometry only, the variation
of the astrometric effect within the observed field on the sky can be
detected. Therefore, the sensitivity quoted above should be scaled by
$|\,\sin(\chi+f/2)-\sin(\chi-f/2)\,|=2\sin(f/2)\,|\,\cos\chi\,|$, where $f$ is
the extension of the observed field, being $f\approx 2^{\circ}$ for the
JASMINE GCS. Therefore, the theoretical sensitivity of JASMINE can be
estimated $h=8.6\times 10^{-12}\,|\,\cos\chi\,|$. Interestingly, the maximal
sensitivity is reached for the gravitational sources approximately in the
direction of observations or the opposite direction where
$|\,\cos\chi\,|\approx 1$. This theoretical sensitivity is valid for the
gravitational wave periods between the typical cadence of observations and the
duration of observations by JASMINE.
#### 5.3.4 Ultra Light Dark Matter
The precise measurement of the kinematics of stars revealed by JASMINE would
enable us to reconstruct the mass distribution in the Galactic center (e.g.,
Genzel et al., 1996; Chatzopoulos et al., 2015, see also sections 2.1 and
2.3). It is believed that in the central 100 pc of the Galaxy baryons dominate
the mass profile, and the total mass measured from the dynamical model is
consistent with what is expected from the stellar density profile (e.g.,
Launhardt et al., 2002; Fritz et al., 2016). However, the Galactic center is
attracting interests in testing for the existence of a particular dark matter
candidate, namely Ultra Light Dark Matter (ULDM), including Axion-like ULDM
particles (e.g., Ferreira, 2021, for a review). Although ULDM behaves like
conventional cold dark matter on the large scales, ULDM is expected to produce
a soliton core in the galactic center in the de Broglie wavelength scale due
to Bose-Einstein condensation. Schive et al. (2014) suggested that ULDM
particle masses of $\sim 8\times 10^{-23}$ eV can explain the dynamical mass
profile of the Fornax dwarf galaxy (but see also Safarzadeh & Spergel, 2020).
Bar et al. (2018) suggested that the soliton core created from the dark matter
particle mass less than $10^{-19}$ eV can influence the gravitational
potential in the Galactic center significantly. De Martino et al. (2020)
showed that velocity dispersions observed in the Galactic center imply a
soliton core as massive as $\sim 10^{9}$ M⊙, expected from ULDM particles with
$~{}10^{-22}$ eV. Maleki et al. (2020) also demonstrated that a soliton core
corresponding to a particle mass of $\sim 2.5\times 10^{-21}$ eV explains the
rotation curve of the Milky Way in the central region. Li et al. (2020) showed
that such a massive soliton core as suggested above can influence the
kinematic properties of the nuclear gas disk on the scale of $\sim 200$ pc.
Recently, Toguz et al. (2022) demonstrated that the kinematics of stars in the
NSC can provide constraints on the particle mass range of ULDM. Toguz et al.
(2022) applied a simple isotropic dynamical model to the kinematics data of
the NSC stars in Fritz et al. (2016), and rejected the mass range of ULDM
between $10^{-20.4}$ eV and $10^{-18.5}$ eV. JASMINE will provide the precise
kinematics of the stars in the NSD (section 2.3) which is the dominant stellar
component from a few pc to $\sim 200$ pc. This size corresponds to the size of
the soliton core of $~{}10^{-19}-10^{-22}$ eV ULDM. Using the dynamical
modelling of these stellar structures, the precise astrometric information of
JASMINE may uncover indirect evidence of ULDM or provide stringent constraints
on the existence of ULDM whose particle mass between $10^{-22}$ eV and
$10^{-19}$ eV.
#### 5.3.5 Identifying disrupted globular cluster population
Recent observational studies of the Galactic bulge by APOGEE have discovered
that a significant fraction of the bulge stars have unusually high [N/Fe]
(e.g., Schiavon et al., 2017). These N-rich stars are not found in the
Galactic disk, but they are ubiquitous in globular clusters. Accordingly, one
of the possible scenarios for the formation of the N-rich stars in the
Galactic bulge is that the stars originate from globular clusters that had
been completely destroyed by the strong tidal field of the Galactic bulge.
Interestingly, these N-rich stars have been discovered in elliptical galaxies
(e.g., Schiavon, 2007; van Dokkum et al., 2017), which suggests that N-rich
populations are common in galactic bulges and elliptical galaxies, i.e. not
just in the Galactic bulge, in line with the indistinguishable properties of
classical bulges and elliptical galaxies (e.g., Renzini, 1999; Kormendy et
al., 2009; Fisher & Drory, 2010).
Globular clusters can spiral into the central region of the Galactic bulge due
to dynamical friction (e.g., Tremaine et al., 1975), and they can be more
severely influenced by the tidal field of the bulge in the inner region.
Accordingly, if such a globular cluster destruction scenario for the N-rich
stars is correct, then stars from the destroyed globular clusters can be a
major population in the central region of the Galactic halo. In fact, using
APOGEE DR16, Horta et al. (2021a) estimated that N-rich stars contribute to
about 17 % of the total halo stars at 1.5 kpc from the Galactic center (see
also Fernández-Trincado et al., 2022). JASMINE will enable us to investigate
the 3D spatial distributions and kinematics of N-rich (globular cluster
origin) and N-normal halo stars in the central region through its superb
accuracy in its proper motion measurement. Because the globular cluster origin
stars could inherit unique kinematics different from the other halo stars,
such 3D dynamics of N-rich stars will contribute to our understanding of the
formation of the inner bulge. In APOGEE DR17 (Abdurro’uf et al., 2022), there
are 436 stars with the measured [N/Fe] and [Fe/H] in the JASMINE GCS field
with good quality stars, i.e. STARFLAG$=0$, ASPCAPFLAG$=0$, SNR$>70$, 3250
K$<T_{\rm eff}<$4500 K and $\log g<3$ (Kisku et al., 2021), and 6 stars of
them are N-rich stars ([N/Fe]$>0.5$, $-1.5<$[Fe/H]$<0.0$). All these stars are
bright enough for JASMINE to observe. Therefore, it is promising that JASMINE
will provide the proper motion of good number of N-rich stars in the Galactic
center field in the combination with future high-resolution high-quality
spectroscopic surveys of the Galactic center field, which will help to
discover more N-rich stars.
#### 5.3.6 Relics of Ancient Mergers
An ancient galaxy merger of Gaia-Sausage-Enceladus discovered in the Gaia data
(section 1) leaves questions like ”where is the core of the remnant now?” and
”has the core of the progenitor galaxy reached to the Galactic center?”. To
assess the possibility of identifying such merger remnants in the JASMINE GCS,
we again use APOGEE DR17, but apply a slightly different quality cut, i.e.
STARFLAG$=0$, APPCAPFLAG$=0$, SNR$>70$, 3,500 K$<T_{\rm eff}<$5,500 K and
$\log g<3.6$, following Horta et al. (2022) who used APOGEE DR17 to chemically
characterise halo substructures of the likely accreted populations. We find
that there are 284 APOGEE high-quality star data within the JASMINE GCS field.
The Gaia-Sausage-Enceladus remnants occupies a distinct stellar abundance
distribution in the [$\alpha$/Fe]-[Fe/H] plane (e.g., Haywood et al., 2018b;
Helmi et al., 2018; Das et al., 2020). Out of this sample, we find 4 stars
within the abundances expected for the Gaia-Sausage-Enceladus remnants, i.e.
stars with [Fe/H]$<-1.1$ and [Mg/Fe]$<-0.28$. All these stars are brighter
than $H_{\mathrm{w}}=14.5$ mag.
Note, however, that the APOGEE DR17 sample is not a complete sample up to
$H_{\mathrm{w}}=14.5$ mag, but has a sample selection due to colours and/or
the specific scientific targets. The JASMINE GCS will obtain the precise
proper motion for about 1,000 times more stars than present in the APOGEE
data. Obtaining accurate proper motions and orbits of these potential remnant
stars of the Gaia-Sausage-Enceladus interaction in the inner Galactic disk
will allow studies to test the association with the already measured Gaia-
Sausage-Enceladus remnants, which have so far been found exclusively in the
solar neighbourhood.
Horta et al. (2021b) found the Inner Galactic Structure (IGS) which has a
similar chemical properties to the accreted components of the Galactic halo.
They suggest that this could be a relic of an ancient accretion of another
galaxy in the Milky Way earlier than the Gaia-Sausage-Enceladus merger and
could be a more massive progenitor than Gaia-Sausage-Enceladus. Further
studies with Gaia DR3 and ground-based spectroscopic data Belokurov & Kravtsov
(e.g., 2022); Rix et al. (e.g., 2022) argue that such centrally concentrated
metal poor stars are relics of the ancient Milky Way proto-Galaxy, which could
be mix of merger and in-situ populations from the early epoch of the Milky Way
formation. JASMINE can provide the proper motion of the stars in the Galactic
center where Gaia cannot observe, and will help to identify the inner
extension of the ancient populations.
#### 5.3.7 Origin of Hyper-velocity Stars
Hills (1988) theoretically predicted that the SMBH at the Galactic center (Sgr
A$\ast$) ejects stars with extremely large velocities as a result of close
encounter and disruption of stellar binaries near the SMBH. Yu & Tremaine
(2003) expanded upon the possible ejection mechanisms. The discoveries of
young hyper-velocity stars (HVSs) in the halo (Brown et al., 2005; Zheng et
al., 2014; Huang et al., 2017; Brown, 2015; Massey et al., 2018; Koposov et
al., 2019) confirmed this prediction. Among these discoveries, the most
intriguing one is the A-type HVS dubbed S5-HVS1 (Koposov et al., 2019). Based
on the astrometric data from Gaia and a follow-up spectroscopic observation,
it turned out that this star was ejected from the Galactic center 4.8 Myr ago
with the ejection velocity of $\sim 1,800\;{\mathrm{km\;s^{-1}}}$. Some
numerical simulations suggests that the ejection rate of HVSs is around
$10^{-5}$–$10^{-4}~{}\mathrm{yr^{-1}}$ (Brown, 2015). This ejection rate
suggests that there are 1 to 10 HVSs within a sphere of radius $0.1$ kpc
centered at the Galactic center, given their typical velocity $\sim
1000\;{\mathrm{km\;s^{-1}}}$. Of course, what we can expect to observe with
JASMINE is a tiny fraction of them, because they need to be bright enough to
be detected by JASMINE. Given that the GCS area of JASMINE includes a square
region of $\pm 0.6^{\circ}$ around the Galactic center ($0.6^{\circ}$
corresponds to about 0.09 kpc at the projected distance of 8.275 kpc), it is
an enticing prospect to look for HVS candidates with JASMINE. If JASMINE
discovers an HVS within $r<0.1$ kpc from the Galactic center, this will be
very useful to understand the detailed mechanism of HVS ejection. For example,
a HVS with a velocity of $1,000\;{\mathrm{km\;s^{-1}}}$ at $r=0.1$ kpc can be
traced back to the Galactic center by integrating the orbit backward in time
for just $0.1$ Myr. This means that we can probe the environment near the SMBH
in the immediate past (just 0.1 Myr ago), such as the binary fraction near the
SMBH or the orbital distribution near the SMBH.
#### 5.3.8 X-ray Sources and the Origin of the Galactic Ridge X-ray Emission
The apparently extended hard ($\geq$ 2 keV) X-ray emission along the Galactic
plane has been known as the Galactic Ridge X-ray Emission (GRXE) since the
early 1980s (e.g., Worrall et al., 1982). This emission extends tens of
degrees in Galactic longitude and a few degrees in Galactic latitude in
$|l|<45^{\circ}$ and $|b|<1.5^{\circ}$. The GRXE has an integrated X-ray
luminosity of $\sim$1$\times$1038 ${{\rm erg}\ {\rm s}^{-1}}$ in the $2-10$
keV range (Koyama et al., 1986; Valinia & Marshall, 1998) at a distance of the
Galactic center. The X-ray spectrum is described by a two-temperature thermal
plasma ($\sim$1 and 5–10 keV) with the K shell emission lines (e.g., Koyama et
al., 1996; Yamauchi et al., 2009); from neutral or lowly-ionized Fe at 6.4 keV
(Fe ${\rm{I}}$) as well as from highly-ionized Fe at 6.7 (Fe ${\rm{XXV}}$) and
7.0 keV (Fe ${\rm{XXVI}}$).
It has been under intensive debate whether GRXE is a truly diffuse emission of
a low surface brightness along the Galactic plane or a composition of discrete
faint unresolved X-ray sources such as cataclysmic variables (CVs) and X-ray
active stars (e.g., Yuasa et al., 2012; Hong, 2012; Revnivtsev et al., 2006).
Many X-ray observations were carried out on this topic. In a Galactic bulge
region ($l=0.^{\circ}08,b=-1.^{\circ}42$), $\sim$80% of the diffuse X-ray
emission was resolved into faint X-ray point sources using the deepest X-ray
observation with the Chandra X-ray Observatory having an excellent spatial
resolution of 0.5$\arcsec$. This indicates that the apparently diffuse
emission in the Galactic bulge is primarily made of faint discrete X-ray
sources (Revnivtsev et al., 2009). There are several candidates for such
population of faint X-ray sources, including magnetic CVs (e.g., Yuasa et al.,
2012; Hong, 2012), non-magnetic CVs (Nobukawa et al., 2016) and X-ray active
stars (e.g., Revnivtsev et al., 2006).
However, it is difficult to constrain the nature of these faint X-ray point
sources from X-ray observations alone, because most of these sources are
detected only with a limited number of X-ray photons (less than 10 photons)
even with the deepest observations. Thus, follow-up observations at longer
wavelengths are needed. Because of the large interstellar absorption toward
the Galactic plane, NIR observations are more suited than optical
observations. NIR identifications of X-ray point sources were performed along
the Galactic plane (e.g., Laycock et al., 2005; Morihana et al., 2016), which
provided clues to the nature of faint X-ray point source populations that make
up the GRXE. The distance of these sources is unknown. Thus, classification of
the sources is based on the X-ray to NIR flux ratio; high values suggest
sources containing a compact object such as CVs and low values suggest sources
otherwise such as stars. If the distance is obtained for many of these sources
with JASMINE, we can discuss their nature based on the absolute luminosity
both in X-ray and NIR bands and discriminate foreground contamination in the
line of sight. A more robust classification of X-ray sources and their 3D
distribution allow us to constrain the Galactic X-ray point source population
for the different locations and components of our Galaxy, providing a hint to
understanding the formation history of our Galaxy.
A large fraction of the observing fields of the GCS using JASMINE
($-0.6^{\circ}<b<0.6^{\circ}$ and $-1.4^{\circ}<l<0.7^{\circ}$ or
$-0.7^{\circ}<l<1.4^{\circ}$ in section 2), was observed with Chandra (the
Chandra Multiwavelength Plane Survey; $-0.4^{\circ}<b<0.4^{\circ}$ and
$-1.0^{\circ}<l<1.0^{\circ}$, Grindlay et al., 2005). A total of 9,017 X-ray
point sources were detected with a total exposure of 2.5 Ms (Muno et al.,
2009). NIR identifications for these X-ray point sources were also made
(Mauerhan et al., 2009). Based on this, we estimate that $\sim$600 X-ray
sources will be identified in NIR brighter than 12.5 mag in the
$H_{\mathrm{w}}$-band in the JASMINE GCS region. This is a significant
improvement compared to the Gaia DR3 optical identification and astrometric
distances for $\sim$100 sources Gaia Collaboration et al. (2022), which are
mostly foreground sources located within 2 kpc. This will be complemented with
JASMINE.
#### 5.3.9 Observations of small solar system bodies
As solar system bodies are moving objects, they are good targets for
astrometry and time-series photometry. Precise astrometry improves the orbital
elements of small solar system bodies such as comets and asteroids. It
provides a solid foundation in several fields; Risks of minor bodies that
threaten the Earth (potentially hazardous asteroids) can be precisely
assessed; Non-gravitational effects such as the Yarkovsky effect can be
quantitatively measured; An asteroid family, asteroids derived from the same
parent body, can be identified. Astrometry of interstellar objects such as
1I/’Oumuamua and 2I/Borisov is essential to understand their origins. The
rotation periods and shapes of minor bodies are derived from time-series
photometry, leading to their internal structure estimates (bulk density). A
binary system can be identified if it shows an eclipse or mutual event. Time-
series photometry is also useful for tracking the brightness changes of active
asteroids. Since JASMINE is in a Sun-synchronous polar orbit, JASMINE’s
photometry will be complementary to ground-based observations. Finally, non-
targeted, serendipitous surveys provide opportunities to discover new minor
bodies.
The expected number of small solar system bodies via JASMINE is estimated as
follows. Here, we focus on asteroids, the most abundant objects among minor
bodies detectable by JASMINE. The spectral energy density of an asteroid is
generally dominated by two components: reflected sunlight in optical
wavelengths and thermal emission in infrared wavelengths. JASMINE’s
$H_{\mathrm{w}}$-band is located at transitional wavelengths between the two
components. Hence, unfortunately, the minor bodies are fainter in
$H_{\mathrm{w}}$-band, and it is more challenging to detect them with JASMINE.
To evaluate the observability of asteroids, we propagate the positions of
known asteroids and check if they cross the JASMINE GCS observing region. The
orbital elements of asteroids were retrieved from Lowell Minor Planet Services
operated by Lowell Observatory on 26 August 2022. Objects with large
uncertainties were removed. The total number of objects was 1,192,756, which
includes 1,148,593 Main Belt Asteroids, 28,829 Near Earth Asteroids, 11,458
Jupiter Trojans, and 3,876 Trans-Neptunian Objects. The topocentric
(geocentric) coordinates were calculated from 1 January 2028 to 31 December
2031. For the sake of simplicity, JASMINE’s observing region was defined as a
circle with a radius of 0.7 degrees centered at $(l,b)=(359.9,0.0)$ in
Galactic coordinates. The defined region differs from the current baseline of
JASMINE GCS, but the number of observable objects is not significantly
affected. With the distances, absolute magnitudes, and slope parameters, the
apparent magnitudes of the bodies in the $V$-band can be calculated (Bowell et
al., 1989). We then assume that the $V-H_{\mathrm{w}}$ color for the objects
is the same as the Sun, i.e. $(V-H_{\mathrm{w}})_{\solar}\sim 1.21$, and
convert the $V$-band to $H_{\mathrm{w}}$-band magnitude, assuming that all the
asteroids have flat reflection spectra.
Figure 21: The brightnesses of asteroids crossing the JASMINE GCS observation
field, which is approximated with a 0.7 deg radius circle region from
$(l,b)=(359.9,0.0)$ . The asteroids brighter than $17.0$ mag are shown in red.
The gray shaded regions show the seasons when the Galactic center is not
accessible by JASMINE. Figure 22: The cumulative distribution of the
$H_{\mathrm{w}}$ magnitude for asteroids observable throughout the JASMINE
operation period.
Figure 21 shows the brightnesses of asteroids crossing the JASMINE GCS region
at different epoch. Each segment shows an individual asteroid. Due to the
operation constraint of the satellite, JASMINE does not observe the Galactic
center in the gray-shaded seasons. The length of the segment represents the
observable duration, which depends on the relative motion to JASMINE. A
handful of asteroids can be observed at apparent magnitudes brighter than
$H_{\mathrm{w}}=14.5$ mag (sufficient for astrometry) and
$H_{\mathrm{w}}=17.0$ mag (for photometry). Figure 22 illustrates the
cumulative histogram of the $H_{\mathrm{w}}$ magnitude for observable
asteroids. The numbers of objects brighter than $H_{\mathrm{w}}=14.5$ and 17.0
mag are about 10 and 100, respectively, throughout the operation period. The
expected number of the potential targets for astrometry in the JASMINE GCS is
rather small. Thus, it is unlikely that there are serendipitous astrometric
measurements with JASMINE, while targeted observations for known objects to
provide additional astrometric information are preferred.
Taking advantage of accurate photometry, JASMINE may observe occultation
events by minor bodies. Occultation provides valuable information for shape
modeling and binary search. Since the brightness of a minor body does not
matter in an occultation observation, the number of potential targets
significantly increases. Detecting an occultation event is feasible with
accurate orbital elements and dedicated observation planning. Serendipitous
observations of occultation events may detect new objects that are unreachable
even by large telescopes with apertures of $\sim 10$ m; Schlichting et al.
(2009) claimed an occultation event by a Trans-Neptunian Object with a radius
of 500 m at 45 au in archival data by the Hubble Space Telescope’s Fine
Guidance Sensors. Arimatsu et al. (2019) detected an occultation event by a
Trans-Neptunian Object with a radius of 1.3 km using a coordinated observation
system of multiple low-cost commercial off-the-shelf 0.28 m aperture
telescopes, the Organized Autotelescopes for Serendipitous Event Survey
(OASES, Arimatsu et al., 2017). Occultation events sometimes reveal additional
features of satellites and rings. Rings around (2060) Chiron and (10199)
Chariklo were identified by ground-based occultation observations (e.g., Ortiz
et al., 2015; Braga-Ribas et al., 2014). Recently, the ring beyond the Roche
limit was discovered around (50000) Quaoar (Morgado et al., 2023). High
precision photometry with JASMINE has the potential to detect minor features
in light curves (Morgado et al., 2022). As for the serendipitous survey,
careful assessment of false detection is required. OASES adopted simultaneous
observations with multiple telescopes, to minimise the false detection rate.
JASMINE may be able to detect occultation events after all anomalous signals
are suppressed by careful calibration.
## 6 Synergies with the other projects
In this section, we summarize the Galactic stellar survey projects
complementary to JASMINE and planned to be operating in late-2020s. Although
there are many projects relevant to the JASMINE science cases, here, we only
list the projects and/or new instruments more relevant to the JASMINE’s main
survey targets of the GCS and the Exoplanet Survey, targeting M dwarfs.
### 6.1 Ground-Based Surveys
#### 6.1.1 PRIME
The Prime Focus Infrared Microlensing Experiment (PRIME) is a wide field (1.56
deg2) 1.8 m telescope at the South African Astronomical Observatory, which is
planned to operate from 2023. PRIME is jointly managed by Japan, the USA and
South Africa. PRIME has $z,Y,J$ and $H$-band filters and several narrow-band
filters. PRIME will observe the Galactic center region around
$-3^{\circ}<l<3^{\circ}$ and $-2^{\circ}<b<2^{\circ}$, which covers the whole
area of the JASMINE GCS field. The prime target of PRIME is a microlensing
exoplanet search. Joint observations with JASMINE can help to constrain the
parameters of the exoplanet detection further with the additional accurate
astrometry information of the source stars that JASMINE can provide. The time-
series photometry of PRIME will also find many variable stars. As mentioned
above, the PRIME data will be used to provide the catalog of Miras observable
with JASMINE.
#### 6.1.2 Vera C. Rubin Observatory/Legacy Survey of Space and Time (LSST)
The Vera C. Rubin Observatory is located on the Cerro Panchón ridge in Chile,
and will run the ten-year Legacy Survey of Space and Time (LSST) with an 8.4 m
(6.5 m effective) Simonyi Survey Telescope (Ivezić et al., 2019). The Rubin
Observatory LSST Camera will have a 3.5-degree field of view with about 32
gigapixels with 0.2 arcsec sampling pixel size. There will be six filters
($u,g,r,i,z$ and $y$) covering 320-1,050 nm. The survey is planed to begin in
2024, and the main survey will observe 18,000 deg2 region of the sky about 800
times in the planed duration of 10 years. The co-added map will reach to
$r\sim 27.5$ mag, and it is anticipated to detect about 20 billion of stars
and a similar number of galaxies. The main science drivers of LSST are probing
the properties of dark energy and dark matter, cataloging an inventory of the
solar system, exploring the transient optical sky and mapping the Milky Way.
The survey field covers the Galactic bulge and the Galactic center. The LSST
is capable of providing the astrometric measurements for fainter stars than
possible with Gaia. With the 10 year baseline, the expected uncertainties of
parallax and proper motions are respectively $\sigma_{\rm\pi}=0.6$ mas and
$\sigma_{\rm\mu}=0.2$ mas yr-1 for the stars brighter than $r=21$ mag, and
$\sigma_{\rm\pi}=2.9$ mas and $\sigma_{\rm\mu}=1.0$ mas yr-1 for the stars
brighter than $r=24$ mag. In addition, the time-series photometry of the LSST
will help to find many variable stars and microlensing events. The majority of
them will be too faint for JASMINE to follow up. However, if they are bright
enough and in the same field, JASMINE can provide more accurate astrometric
information.
#### 6.1.3 SDSS-V
The Sloan Digital Sky Survey (SDSS)-V (Kollmeier et al., 2017) is an ambitious
project to run all-sky multi-epoch spectroscopic survey, utilising telescopes
in both Northern and Southern hemispheres. The survey will provide optical and
IR spectra covering 2,500 deg2, ultra-wide field for more than 6 million
objects in five years ($2020-2025$). SDSS-V use the telescopes at Apache Point
Observatory (APO) in USA and and Las Campanas Observatory (LCO) in Chile. At
APO, 2.5 m Sloan telescope will be continuously used for SDSS-V full-time. At
LCO, more than 300 nights per year of telescope time of 2.5 m du Pont
telescope will be dedicated to this survey. The survey will also use the
smaller (1 m to 16 cm) telescopes at APO and LCO. The NIR APOGEE spectrograph
(300 fibers, $R=22,000$, $\lambda=1.5-1.7$ $\mu$m), the eBOSS optical
spectrograph (500 fibers, $R\sim 2,000$, $\lambda=0.36-1.0$ $\mu$m) and the
MaNGA multi-object IFU ($R\sim 4,000$, $\lambda=0.36-1.0$ $\mu$m) will be used
at both APO and LCO. SDSS-V will run three surveys, the Milky Way Mapper, the
Black Hole Mapper and the Local Volume Mapper. The most relevant survey to
JASMINE is the Milky Way Mapper, which plans to observe $4-5$ million stars in
the Milky Way, with the NIR APOGEE spectrograph and/or the optical BOSS
spectrograph. The Milky Way Mapper aims to understand the evolution of the
Milky Way, the physics of the stars and the interstellar medium as well as
multiple stars and exo-planetary systems. The Galactic Genesis Survey as a
part of the Milky Way Mapper targets the stars with $H<11$ mag and $G-H>3.5$
mag, which are likely to overlap with the bright target stars of the JASMINE
GCS fields, and will provide accurate radial velocity and abundance patterns.
#### 6.1.4 Subaru/PFS
Subaru Prime Focus Spectrograph (PFS: Takada et al., 2014) is the next
generation instrument of the 8.2 m Subaru telescope at the summit of Maunakea,
Hawai’i in the US, operated by National Astronomical Observatory of Japan. PFS
is a joint instrument of the institutes in Japan, Taiwan, the USA, France,
Brazil, Germany and China. PFS has $\sim 1.38$ deg2 field of view, and about
2,400 science fibers. PFS consists of blue ($\lambda=0.38-0.65$ $\mu$m, $R\sim
2,300$), red (low resolution mode: $\lambda=0.63-0.97$ $\mu$m, $R\sim 3,000$;
medium resolution mode: $\lambda=0.71-0.885$ $\mu$m, $R\sim 5,000$) and NIR
($\lambda=0.94-1.26$ $\mu$m, $R\sim 4,300$) spectrographs. It is scheduled to
start operating in 2024. About 300 nights over 5 years of Subaru time will be
dedicated for the PFS survey for cosmology, galaxy evolution and Galactic
archaeology, though the Subaru Strategic Survey Program (SSSP). The GCS field
is not included in the PFS SSSP. However, the NIR spectrograph of PFS is
especially well-suited for spectroscopic follow up for the JASMINE GCS field
stars to obtain radial velocity and chemical abundances. We plan to apply for
a Subaru Intensive Program (5 nights per semester) to follow up the JASMINE
target stars.
#### 6.1.5 ULTIMATE-Subaru
ULTIMATE-Subaru (Minowa et al., 2020) is another next generation NIR
instrument of Subaru, which is planned to be installed around 2028. ULTIMATE-
Subaru is a wide-field (14’$\times$14’) NIR imager and multi-objects
spectrograph with Ground-Layer Adaptive Optics (GLAO), which enables a spatial
resolution of FWHM$\sim$0.2” in the $K$-band. There is a planned Galactic
Center Survey of $\sim$6 deg2 region of the Galactic center, which covers the
whole JASMINE Galactic center field, $J$, $H$, $K$ and the narrow-band $K_{\rm
NB}$ filters with a high cadence of 4 days (1 months) for the high (low)
cadence field. JASMINE can provide the astrometric reference stars in NIR,
which would improve the astrometric accuracy of ULTIMATE-Subaru Galactic
center survey data. The ULTIMATE-Subaru Galactic center survey can observe
numerous stars fainter than the JASMINE magnitude limit. The combined data of
JASMINE and ULTIMATE-Subaru will provide the accurate astrometric information
of these faint stars. They would help to identify the star clusters in the
Galactic center region from the proper motion of stars, and increase the event
rate of the astrometric microlensing, which will enable the measurement of the
masses of lensed objects with high precision, and help to identify BHs and
exoplanets.
#### 6.1.6 VLT/MOONS
The Multi Object Optical and Near-infrared Spectrograph (MOONS: Cirasuolo et
al., 2011) is a next generation instrument of the Very Large Telescope (VLT)
UT1 at the European Southern Observatory (ESO) on Cerro Paranal in the Atacama
Desert of Chile, and the planned first light is in 2023. MOONS is a multi-
object (about 1,000 fibers) NIR spectrograph with a field of view of 25 arcmin
diameter. There are three channels of spectrograph, covering $RI$, $YJ$ and
$H$-bands, with both low and high resolution modes. The low resolution mode
covers the wavelength range of $0.65-1.8$ $\mu$m with $R_{RI}>4,100$,
$R_{YJ}>4,300$ and $R_{H}>6,600$. High resolution modes cover 3 disconnected
wavelength ranges $\lambda_{RI}=0.76-0.89$ $\mu$m, $\lambda_{YJ}=0.93-1.35$
$\mu$m and $\lambda_{H}=1.52-1.64$ $\mu$m with the spectral resolution of
$R_{RI}>9,200$, $R_{YJ}>4,300$ (fixed with the low resolution mode) and
$R_{H}>18,300$, respectively. MOONS science targets cover Galactic
archaeology, the growth of galaxies and the first galaxies. Galactic
archaeology studies by MOONS plan to take spectra for the several million
stars observed by Gaia and the VISTA telescope, providing the crucial
complementary information of the accurate radial velocity and detailed
chemical abundances. The NIR coverage of MOONS is capable of recording the
spectra of stars in the heavily obscured Galactic center region. Even with the
high-resolution mode in the NIR $H$-band, a signal-to-noise ratio of more than
60 can be obtained with one hour of exposure for objects brighter than $H=15$
mag. MOONS is likely to be the most powerful instrument in 2020s, capable of
taking high-resolution spectra for all the target stars in the JASMINE GCS
field.
#### 6.1.7 VISTA/4MOST
ESO’s 4-meter Multi-Object Spectroscopic Telescope (4MOST: de Jong et al.,
2019) will be installed on the VISTA telescope in Chile in 2024. 4MOST has
2,436 fibers and about a 4.2 deg2 field of view. There are two low-resolution
and one high-resolution spectrographs. The low-resolution spectrograph covers
the wavelength range of $\lambda=0.37-0.95$ $\mu$m with $R\sim 6,500$, while
the high-resolution spectrograph covers three wavelength passbands of
$\lambda=0.3926-0.4355$, $0.5160-0.5730$ and $0.6100-0.6760$ $\mu$m. The 4MOST
consortium will run 10 surveys using $70$ % of the available time in five
years, and planned to start in 2023, taking more than 20 million low-
resolution spectra and more than 3 million high-resolution spectra. 4MOST
Milky Way Disc And BuLgE Low (4MIDABLE-LR: Chiappini et al., 2019) and High-
Resolution (4MIDABLE-HR: Bensby et al., 2019) surveys will take the spectra of
about 15 million and 2 million stars in the Milky Way, respectively. Their
target includes the inner disk and bar/bulge region. Their survey focuses on
the stars for which Gaia provides precise astrometry, but are too faint for
Gaia’s radial velocity spectrograph to provide a radial velocity. Their
optical survey will not cover many stars in the JASMINE GCS region. However,
the combination of 4MIDABLE data and Gaia data will be a powerful resource to
unveil the global nature of the bar and spiral arms, and therefore highly
complementary to the JASMINE GCS.
#### 6.1.8 Subaru/IRD
The InfraRed Doppler (IRD) spectrograph ($R\approx 70,000$,
$\lambda=0.950-1.73\,\mathrm{\mu m}$) on the Subaru telescope (Kotani et al.,
2018) is one of the most powerful instruments in the world to follow up
exoplanets around M dwarfs and young stars. Besides the “validation” of
transiting-planet candidates around mid-M dwarfs identified by JASMINE, high-
dispersion spectroscopy by IRD can play a key role in further
characterizations of those planets; precision radial velocity measurements by
IRD would enable us to constrain precise planet masses as well as orbital
eccentricities. Moreover, NIR transit spectroscopy by IRD would also allow us
to constrain the stellar obliquity (spin-orbit angle) and atmospheric
composition (e.g., He I and molecular species), which are supposed to reflect
the dynamical and chemical evolution of exoplanetary systems (e.g., Hirano et
al., 2020a). Since activity-induced radial-velocity variations of host stars
are suppressed in the NIR, IRD is also an ideal tool to confirm and
characterize planets around young active stars.
### 6.2 Space Missions
#### 6.2.1 Gaia
ESA’s Gaia (Gaia Collaboration et al., 2016) was launched in December 2013.
The nominal mission lifetime was 5 years but the mission has been extended to
the second quarter of 2025. The Gaia mission is all-sky survey to provide the
precise astrometry for more than one billion stars brighter than $G\sim 21$
mag. Gaia uses a broad passband, the $G$-band that covers the wavelength range
of $\lambda\sim 0.33-1.05$ $\mu$m (Evans et al., 2018). In the final data
release after the end of the mission, the astrometric accuracy for the bright
stars with $G\lesssim 13$ mag is expected to reach about 7 $\mu$as. The
astrometric accuracy of about $149$ $\mu$as is expected to achieve for stars
brighter than $G=19$ mag666cosmos.esa.int/web/gaia/science-performance. Gaia
has three instruments, the Astrometric instrument (ASTRO), the
Spectrophotometer (BP/RP) and Radial Velocity Spectrograph (RVS). The ASTRO
provides the five astrometric parameters, stellar position, proper motion and
parallax. The Spectrophotometer consists of the BP and RP spectrophotometer.
BP and RP respectively provides a low-resolution ($R\sim 5-25$) spectra of the
wavelength ranges of $\lambda\sim 0.33-0.68$ $\mu$m and $\sim 0.64-1.05$
$\mu$m, which are used for chromaticity calibration for astrometric
measurement, and estimates of the stellar parameters and dust extinction. RVS
is an integral-field spectrograph with $R\sim 11,500$, covering
$\lambda=0.845-0.872$ $\mu$m (Cropper et al., 2018). The main aim of the RVS
is to provide the radial velocity for about 150 million stars brighter than
$G_{\rm RVS}=16$ mag, depending on the spectral type of stars, where $G_{\rm
RVS}$ is magnitude in the RVS passband. Gaia’s fourth data release is expected
to be in 2025, and all the catalog and data will be released, including all
epoch data for all sources based on 66 months of data of the nominal mission.
The final fifth data release based on about 11 years of the extended mission
is expected to be in 2030. Hence, late-2020s and early-2030s will be the truly
golden age of the Galactic archaeology. JASMINE is expected to be launched in
this golden age, and will provide the complementary data to the Gaia data,
especially for the Galactic center stars, which the optical astrometry
mission, Gaia, cannot observe.
#### 6.2.2 Nancy Grace Roman Space Telescope
The Nancy Grace Roman Space Telescope (Roman Space Telescope) is a NASA
observatory to study dark energy, exoplanets and infrared astrophysics
(Spergel et al., 2015). The telescope has a primary mirror of 2.4 m diameter.
The nominal mission lifetime is six years. The Roman Space Telescope has the
Wide Field Instrument (WFI) and the Coronagraph Instrument (CGI). WFI is a
large area, 300 megapixel, NIR camera for imagaing and slitless spectroscopy
with a grism ($R=435-865$) and prism ($R=70-170$). The imaging mode utilises
several filters covering the wavelength range of $\lambda=0.48-2.0$ $\mu$m.
The most relevant survey of the Roman Space Telescope to JASMINE’s GCS is
their Microlensing Survey, which will repeatedly observe about 2.81 deg2 (with
10 fields) around $-0.5^{\circ}<l<1.8^{\circ}$ and
$-2.2^{\circ}<b<-1^{\circ}$. There will be six 72 day campaigns over six years
with cadence of every 15 minutes with a wide filter and 12 hours with a blue
filter. The Roman Space Telescope will detect billions of bulge stars and the
study to obtain the precise astrometry is ongoing (WFIRST Astrometry Working
Group et al., 2019). The current planned survey region of Microlensing Survey
of Roman Space Telescope does not cover the Galactic center where JASMINE’s
GCS targets, because the Microlensing Survey requires it to maximize the event
rate of microlensing and therefore target the region of less dust extinction
to obtain a higher number density of background stars. However, the Roman
Space Telescope is currently gathering the community input for the survey
strategies to maximize the science output. With strong community inputs, there
could be a possibility for the Roman Space Telescope to observe the JASMINE
GCS field. Then, JASMINE astrometry results would be valuable to calibrate
their astrometry for fainter stars observed by the Roman Space Telescope.
#### 6.2.3 James Webb Space Telescope
The James Webb Space Telescope (JWST) is a NASA’s flagship space observatory
(6.5 m aperture), launched at the end of 2021 (Gardner et al., 2006). JWST has
two spectrographs in the NIR band, the NIR Spectrograph (NIRSpec) and the NIR
Imager and Slitless Spectrograph (NIRISS). NIRSpec is a medium resolution
spectrograph ($R=100-2,700$) with a wavelength coverage of 0.6–5 $\mu$m. The
saturation limit is $H\sim 10$ mag. NIRISS is a slitless spectrograph, whose
spectral resolution is about $R=700$. The saturation limit is $J\sim 8.5$ mag.
The mission lifetime of JWST is five year as designed, but 10 years as an
optimistic goal. If the lifetime of JWST overlaps the observation period of
JASMINE, these instruments will be the most powerful instruments to follow up
exoplanets found by JASMINE Exoplanet Survey, to characterize the exoplanets
atmosphere with detailed and precise transmission spectroscopy.
#### 6.2.4 CHEOPS
The CHaraterising ExOPlanet Satellite (CHEOPS), launched in December 2019, is
an ESA space mission dedicated for precision photometry to determine the
radius of transiting exoplanets (Broeg et al., 2014). The mission lifetime is
assumed to be 3.5 years (nominal). CHEOPS is a PIT type of transiting
exoplanet exploration, similar to JASMINE. The telescope diameter (32 cm) is
similar to that of JASMINE. The passband of CHEOPS is visible (see figure 9)
while that of JASMINE is NIR. In this sense, JASMINE is complementary to
CHEOPS. However, considering the difference of the launch date, JASMINE should
be regarded as a successor of CHEOPS in terms of the space facility for
photometric follow-up of transiting planets found by the ground-based surveys.
#### 6.2.5 TESS
The Transiting Exoplanet Survey Satellite (TESS) is an MIT/NASA-led all-sky
survey mission to find planets transiting bright nearby stars (Ricker et al.,
2014). TESS has four cameras, each with 10.5 cm diameter and $24\times
24\,\mathrm{deg}^{2}$ field of view ($\sim 2,000\,\mathrm{deg}^{2}$ in total)
and each camera has four 2k$\times$2k CCDs with a pixel scale of
$21^{\prime\prime}$. The detectors are sensitive from 0.6 to 1.0 $\mu$m.
During the 2 year prime mission since the launch in 2018, TESS is monitoring
the almost entire sky in 26 overlapping segments, and observe each segment for
27.4 days with 2 min cadence for the pre-selected 15,000 stars and 30 min
cadence for the full image. Extension of the mission has been approved and
TESS will keep tiling the whole sky at least till 2024. TESS has a capability
of finding Earth-sized transiting planets near the habitable zone of early- to
mid-M dwarfs, and such an example has indeed been reported (TOI-700d and e;
Gilbert et al., 2020, 2023). The larger telescope aperture and redder passband
of JASMINE will make it sensitive to similar planets around later-type M
dwarfs. Follow-up observations by JASMINE for planets detected by TESS may
lead to finding longer-period/smaller planets that were missed by TESS, as
well as to characterizing them even better through a finer sampling of the
transit light curve.
#### 6.2.6 PLATO
PLAnetary Transits and Oscillations of stars (PLATO) is the third M-class (M3)
mission under development by ESA for a planned launch in 2026 (Rauer et al.,
2014). The primary science goal is the detection and characterization of
planets transiting bright solar-type stars, in particular terrestrial planets
in the HZ. This will be achieved by high-precision, continuous photometric
monitoring of a large number of bright stars using a collection of small and
wide-field optical telescopes. According to the PLATO definition study
report,777https://sci.esa.int/web/plato/-/59252-plato-definition-study-report-
red-book the payload is planned to consist of $>20$ cameras each with 12 cm
diameter and covering the wavelength range of 0.5-1.0 $\mu$m, which result in
a total field of view of $\sim 2,000\,\mathrm{deg}^{2}$. Although the specific
observing strategy is yet to be determined, PLATO is likely to cover a
significant fraction of the entire sky, as well as to monitor certain regions
for a duration long enough ($\sim$2 year Nascimbeni et al., 2022) to find
planets in the HZ of Sun-like stars. The duration of the nominal science
operations is 4 years and may well overlap with the operation period of
JASMINE. The main targets of PLATO are bright ($V\lesssim 13$ mag) Sun-like
stars, while JASMINE targets late-type stars fainter in the optical passband,
taking advantage of the NIR photometry. Therefore, the two missions are
complementary to each other. Similarly to TESS, PLATO observations might also
provide transiting planet target candidates around M dwarfs that can be
further characterized with NIR observations by JASMINE. PLATO also aims to
characterise the properties, including the precise age estimates in the 10 %
precision level, of the host stars from the time-series photometry using
asteroseismology. The age information of a large number of stars which PLATO
will observe will be a precious information for studies of Galactic
archaeology (Miglio et al., 2017). Hence, it will be also provide
complementary data to the JASMINE GCS and mid-plane survey.
#### 6.2.7 ARIEL
Atmospheric Remote-sensing Infrared Exoplanet Large-survey (ARIEL, Tinetti et
al., 2018) is the first space telescope dedicated to the study of exoplanet
atmospheres, adopted as an ESA’s M4 mission, whose planned launch is in 2029.
The effective size of the primary mirror of ARIEL will be $\sim$1 m, which is
much smaller than JWST (6.5 m). However, ARIEL will be able to collect fluxes
in the wavelength range of 0.5-7.8 $\mu$m at one time, using five dichroic
mirrors, three NIR spectrometers and three optical photometric detectors. This
allows one to obtain an atmospheric spectrum with a very wide wavelength
coverage from a single planetary transit or eclipse observation.
ARIEL will observe a thousand exoplanets with a wide range of mass and
temperature, from hot Jupiters to warm/temperate Earths, in order to
understand the statistical properties of exoplanetary atmospheres and
planetary formation histories. While the current target list for ARIEL already
includes a large number of Jovian planets, it still lacks Neptune- and
smaller-sized planets that are suitable for atmospheric study, i.e., hosted by
nearby M dwarfs (Edwards et al. 2019). Although TESS has been increasing the
number of such targets, it may not be enough due to its limited telescope
aperture size (10 cm) and wavelength sensitivity (because of covering only the
optical). Small transiting planets around nearby M dwarfs that will be
discovered by JASMINE can thus be good targets for atmospheric
characterization by ARIEL. Given that JASMINE is planned to be launched ahead
of ARIEL, JASMINE can provide prime targets for ARIEL in a timely manner.
## 7 Summary and Conclusions
We summarize that the unique capability of the JASMINE mission will fill the
gap of left by other planned and ongoing projects of the Galactic stellar
surveys for Galactic archaeology and the habitable exoplanet searches in
late-2020s. JASMINE will be the first mission to provide the 10 $\mu$as-level
astrometry in the NIR band with the time-series photometry. JASMINE will offer
the precise astrometric information where the dust extinction is too strong
for the optical astrometry mission, Gaia, to detect any stars, such as the
Galactic center field and the Galactic mid-plane. The astrometric data of
stars hidden behind the dust in the Galactic center and Galactic mid-plane
will shed light on the formation epoch of the Galactic bar, the nature of the
spiral arms and the mechanism underlying radial migration in the inner
Galactic disk, which are likely to be remaining questions after Gaia. The
combination of time-series photometry and precise astrometry will provide a
vast opportunities of serendipitous discovery, including the possibilities of
detecting IMBH, astrometric microlensing of inner disk BHs and studying the
nature of star forming regions and X-ray sources. JASMINE will be also the
only space observatory in the late-2020s which can follow up exoplanet
transits detected by the ground-based telescope to find the planets in the
outer and habitable orbits around late-type stars, just as Spitzer space
observatory contributed to revolutionising the field.
Finally, we note that JASMINE will be a crucial science demonstration mission
for what the future NIR astrometry mission can offer. JASMINE will be a key
mission to bridge between the successful Gaia mission and the proposed Gaia’s
successor mission, GaiaNIR (e.g., Hobbs et al., 2021) to be launched in the
2040s. GaiaNIR will provide the all-sky global astrometry in NIR band,
including the Galactic disk, bar and bulge regions. Unprecedentedly high
proper motion for the stars also observed with Gaia will be obtained, taking
advantages of $\sim 20$ years of baseline between the Gaia and GaiaNIR
missions. Also, GaiaNIR will help maintaining and improving on the absolute
astrometric quality of the celestial reference frame, which otherwise degrades
with time. JASMINE will be an pioneering mission to open up the future
$\mu$as-level NIR astrometry, and become an important milestone to demonstrate
the power of NIR astrometry. About 20 years of time difference between JASMINE
and GaiaNIR will provide superb proper motion measurements for the stars
observed by JASMINE, including the NSD stars. With careful correction of
systematic errors, the combination of JASMINE and GaiaNIR will offer endeavour
to measure the acceleration of the stars and map the gravitational field in
the Galactic center.
We thank Megan Johnson and Stephen Williams for their contribution to the
early draft of this manuscript.
This work presents results from the European Space Agency (ESA) space mission
Gaia. Gaia data are being processed by the Gaia Data Processing and Analysis
Consortium (DPAC). Funding for the DPAC is provided by national institutions,
in particular the institutions participating in the Gaia MultiLateral
Agreement (MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia.
The Gaia archive website is https://archives.esac.esa.int/gaia. This work is
also based on data products from observations made with ESO Telescopes at the
La Silla or Paranal Observatories under ESO programme ID 179.B-2002. Funding
for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan
Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions.
SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
www.sdss4.org. SDSS-IV is managed by the Astrophysical Research Consortium for
the Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science, Carnegie
Mellon University, Center for Astrophysics — Harvard & Smithsonian, the
Chilean Participation Group, the French Participation Group, Instituto de
Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the
Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the
Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz
Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie
(MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-
Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatário Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.
This work is a part of MWGaiaDN, a Horizon Europe Marie Skłodowska-Curie
Actions Doctoral Network funded under grant agreement no. 101072454 and also
funded by UK Research and Innovation (EP/X031756/1). This work was partly
supported by the UK’s Science & Technology Facilities Council (STFC grant
ST/S000216/1, ST/W001136/1), JSPS KAKENHI (23H00133, 21J00106), JSPS
Postdoctoral Research Fellowship Program, the Spanish
MICIN/AEI/10.13039/501100011033, ”ERDF A way of making Europe” by the
“European Union” through grants RTI2018-095076-B-C21 and PID2021-122842OB-C21,
the Institute of Cosmos Sciences University of Barcelona (ICCUB, Unidad de
Excelencia ’María de Maeztu’) through grant CEX2019-000918-M, NASA ADAP award
program Number (80NSSC21K063), the Swedish National Space Agency (SNSA Dnr
74/14 and SNSA Dnr 64/17), the Royal Society (URF\R1\191555) and the ERC
Consolidator Grant funding scheme (project ASTEROCHRONOMETRY
https://www.asterochronometry.eu/, G.A. n. 772293).
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11institutetext: Instituto de Astrofísica de Canarias (IAC), 38205 La Laguna,
Tenerife, Spain 22institutetext: Departamento de Astrofísica, Universidad de
La Laguna (ULL), 38206, La Laguna, Tenerife, Spain 33institutetext: Leiden
Observatory, Leiden University, Postbus 9513, 2300 RA Leiden, The Netherlands
44institutetext: INAF- Palermo Astronomical Observatory, Piazza del
Parlamento, 1, 90134 Palermo, Italy 55institutetext: University of Palermo,
Department of Physics and Chemistry “Emilio Segrè”
# The strange case of Na i in the atmosphere of HD209458 b:
Reconciling low- and high-resolution spectroscopic observations
G. Morello${}^{{\ref{inst:iac},\ref{inst:ull},\ref{inst:inaf}}}$ N. Casasayas-
Barris${}^{{\ref{inst:lo}}}$ J. Orell-
Miquel${}^{{\ref{inst:iac},\ref{inst:ull}}}$ E.
Pallé${}^{{\ref{inst:iac},\ref{inst:ull}}}$ G.
Cracchiolo${}^{{\ref{inst:inaf},\ref{inst:unipa}}}$ G.
Micela${}^{{\ref{inst:inaf}}}$
###### Abstract
Aims. We aim to investigate the origin of the discrepant results reported in
the literature about the presence of Na i in the atmosphere of HD209458 b,
based on low- and high-resolution transmission spectroscopy.
Methods. We generated synthetic planetary atmosphere models and we compared
them with the transmission light curves and spectra observed in previous
studies. Our models account for the stellar limb-darkening and Rossiter-
McLaughlin (RM) effects, and contemplate various possible scenarios for the
planetary atmosphere.
Results. We reconciled the discrepant results by identifying a range of
planetary atmospheres that are consistent with previous low- and high-
resolution spectroscopic observations. Either both datasets are interpreted as
consistent with a total absence of Na i in the planetary atmosphere (with
Hubble Space Telescope data being affected by limb darkening), or the
terminator temperature of HD209458 b has to have an upper limit of about
1000K. In particular, we find that 1D transmission spectra with lower-than-
equilibrium temperatures can also explain the previously reported detection of
absorption signal at low resolution due to differential transit depth in
adjacent bands, while the cores of the Na i D lines may be masked by the
strong RM signal seen at high resolution. We also rule out high-altitude
clouds, which would otherwise mask the absorption signal at low resolution, as
the source of the discrepancies.
Conclusions. This work highlights the synergies between different observing
techniques, specifically low- and high-resolution spectroscopy, to fully
characterise transiting exoplanet systems.
###### Key Words.:
planetary systems – planets and satellites: individual: HD209458 b – planets
and satellites: atmospheres – techniques: spectroscopic – methods:
observational
## 1 Introduction
The characterisation of exoplanet atmospheres relies on precise measurements
of the spectroscopic signal that they imprint on the observed starlight. For a
planet transiting in front of its host star, the atmosphere acts as a
wavelength-dependent annulus that affects the fraction of occulted stellar
flux (Brown, 2001). If an atom or molecule is present in the planetary
atmosphere, it causes extra absorption at specific wavelengths corresponding
to electronic transition lines. The transit depth is defined as the apparent
planet-to-star area ratio, $p^{2}=(R_{p}/R_{*})^{2}$, $R_{p}$ and $R_{*}$
being the planet and star radii. Differences in transit depths obtained on
multiple passbands or wavelength bins have led to the detection of many
chemical species in the atmospheres of exoplanets (e.g. Charbonneau et al.,
2002; Vidal-Madjar et al., 2003; Fossati et al., 2010; Deming et al., 2013;
Damiano et al., 2017). High-resolution spectroscopy has enabled us to resolve
the atmospheric spectral features and to trace their Doppler shift as it
varies with the orbital phase (e.g. Snellen et al., 2010; Casasayas-Barris et
al., 2017, 2018, 2019; Palle et al., 2020; Stangret et al., 2020). Typically,
the signal from the planet atmosphere is of the order of 10-4 or less,
relatively to the host star flux. It is therefore necessary to disentangle
this small signal from other contaminating effects with similar amplitudes,
such as stellar limb darkening (Howarth, 2011; Csizmadia et al., 2013; Morello
et al., 2017; Yan et al., 2017), magnetic activity (Ballerini et al., 2012;
Oshagh et al., 2014; Cracchiolo et al., 2021a, b), and planet self- and phase-
blend effects (Kipping & Tinetti, 2010; Martin-Lagarde et al., 2020; Morello
et al., 2021).
HD209458 b is the first planet with a reported detection of a chemical species
in its atmosphere, namely Na i (Charbonneau et al., 2002). The observations
were performed with the Hubble Space Telescope (HST)/Space Telescope Imaging
Spectrograph (STIS) using the G750M filter. In particular, Charbonneau et al.
(2002) inferred Na i absorption from the larger transit depth on a narrow band
centred on the resonance doublet at 5893 $\AA$ relative to adjacent bands.
Sing et al. (2008) reanalysed the same data, confirming the previous Na i
detection and resolving the doublet lines. Other HST/STIS datasets led to the
possible detection of H i, O i, C ii (Vidal-Madjar et al., 2004), Mg i (Vidal-
Madjar et al., 2013), and Fe ii (Cubillos et al., 2020). The near-infrared
spectrum taken with the HST/Wide Field Camera 3 (WFC3) also revealed H2O
absorption (Deming et al., 2013; Tsiaras et al., 2016). Ground-based
observations provided independent confirmation of the Na i feature with higher
spectral resolution (Snellen et al., 2008) and also revealed other features
attributed to He i (Alonso-Floriano et al., 2019), Ca i, Sc ii, H i
(Astudillo-Defru & Rojo, 2013), CO (Snellen et al., 2010), H2O (Sánchez-López
et al., 2019), HCN, CH4, C2H2 and NH3 (Giacobbe et al., 2021). However, recent
studies cast doubts on the Na i detection as well as that of other atomic and
ionic species (Casasayas-Barris et al., 2020, 2021). In particular, the
transmission spectra observed with the High Accuracy Radial velocity Planet
Searcher for the Northern hemisphere (HARPS-N), mounted on the Telescopio
Nazionale Galileo (TNG) at the Observatory of Roque de los Muchachos (ORM) in
Spain, and the Echelle Spectrograph for Rocky Exoplanet and Stable
Spectroscopic Observations (ESPRESSO), mounted on the Very Large Telescope
(VLT) at the European Southern Observatory (ESO) of Cerro Paranal in Chile,
reveal several features due to the Rossiter-McLaughlin (RM) effect and no
evidence of absorbing species in the exoplanet atmosphere.
In this paper, we try to reconcile these apparently conflicting results that
have appeared using different instruments, observing techniques, and analysis
methods. Section 2 presents a reanalysis of the HST/STIS observations that led
to the first announcement of Na i in the atmosphere of HD209458 b with updated
data detrending techniques, transit modelling tools, and system parameters.
Section 3 describes our models of the planetary atmosphere. From them, we
extract time series and spectra following the same procedures adopted by
previously published studies of low and high spectral resolution observations
aimed at exoplanet atmospheric characterisation, along with other stellar and
planetary effects. Section 4 discusses the comparison between our simulations
and the observations, providing a range of planetary atmosphere scenarios that
could explain the apparently discrepant results. Section 5 summarises the
conclusions of our study.
## 2 Reanalysis of HST data
Table 1: Summary of HST observations for identification in the online
archives. More technical information is reported in Section 2.1.
Root name | Number of spectraaa$a$For each of the four orbits. | UT start date
---|---|---
o63302030 | 36, 36, 36, 35 | 2000-04-28
o63303030 | 36, 36, 36, 35 | 2000-05-05
o63304030 | 36, 36, 36, 35 | 2000-05-12
111
Table 2: Adopted HD209458 system parameters. Stellar parameters
---
T∗,eff [K] | 6065$\pm$50 | Torres et al. (2008)
$\log{g_{*}}$ [cgs] | 4.361$\pm$0.008 | ”
$[\mathrm{Fe/H}]_{*}$ [dex] | 0.00$\pm$0.05 | ”
$M_{*}$ [$M_{\odot}$] | 1.119$\pm$0.033 | ”
$R_{*}$ [$R_{\odot}$] | 1.155$\pm$0.016 | ”
Planetary parameters
$M_{\mathrm{p}}$ [$M_{\mathrm{Jup}}$] | 0.685$\pm$0.015 | ”
$R_{\mathrm{p}}$ [$R_{\mathrm{Jup}}$] | 1.359$\pm$0.019 | ”
$a$ [au] | 0.04707$\pm$0.00047 | ”
Transit parameters
$P$ [day] | 3.52474859$\pm$0.00000038 | Knutson et al. (2007)
$E.T.$ [HJD] | 2452826.628521$\pm$0.000087 | ”
$b$ [R∗] | 0.516$\pm$0.006 | Morello (2018)
$T_{0}$ [s] | 9500$\pm$6 | ”
RM parameters
$K_{\mathrm{p}}$ [km/s] | 145.0$\pm$1.6 | Casasayas-Barris et al. (2021)
$\lambda$ [deg] | 1.58$\pm$0.08 | ”
$v\sin{i_{*}}$ [km/s] | 4.228$\pm$0.007 | ”
### 2.1 Observations
We reanalysed three transits of HD209458 b observed with HST/STIS (GO-8789,
PI: Brown) using the G750M grism, taken on 28 April, 5 May and 5 December
2000. Table 1 provides their identifiers. The spectra cover the 5808-6380
$\AA$ wavelength range and have a resolving power of 5540 at the central
wavelength of 6094 $\AA$. Each visit contains 143 spectra distributed over
four HST orbits: one before, two during, and one after the transit event. The
four HST orbits were preceded by another one to enable instrumental settling,
but it was not included in the scientific data analysis. The integration time
is 60 s per frame, followed by a reset time of about 20 s. The interval
between consecutive HST orbits is about 50 min. The spectral trace forms an
angle of less than 0.5∘ with the longest side of the 64$\times$1024 pixel
detector, and it is stable within the same pixel rows during each visit.
### 2.2 Data analysis
#### 2.2.1 Extraction
We downloaded the flat-fielded science images (extension: flt.fits) from the
Mikulski Archive for Space Telescopes (MAST)222https://archive.stsci.edu. We
considered a rectangular aperture of 17$\times$1024 pixels with the trace at
its centre and summed along the columns to extract the 1D spectra. As a
starting point, we adopted the wavelength solution from the corresponding
archive spectra (extension: x1d.fits) and then computed the cross-correlations
with a template spectrum to align the extracted spectra in the stellar rest
frame. The template spectrum was calculated as described in Section 3.1, and
purposely degraded to the same resolution as the observations.
Our analysis focused on a portion of the spectrum containing the Na i doublet.
We extracted the flux time series for three bands identical to the ‘narrow’
ones selected by Charbonneau et al. (2002) using two different sets of
apertures. The first set consisted of rectangular apertures, similar to the
ones used for the wavelength calibration, but limited in the dispersion axis
according to the bands definition. In this case, the flux is the simple sum
over the pixels contained in the rectangular aperture. The second set
consisted of tilted rectangular apertures with sides parallel and
perpendicular to the spectral trace, which was previously calculated by a
linear fit on the centroids resulting from Gaussian fits on the detector
columns. When calculating the flux from the fractions of pixels within the
tilted apertures we accounted for the non-uniform distribution inside the
pixel, that was approximated by a linear function in the cross-dispersion
direction. The resulting light curves have almost identical shapes, regardless
of the extraction method, but the tilted apertures led to higher fluxes by
$\sim$0.01$\%$, $\sim$0.04$\%$ and $\sim$0.10$\%$ for the left, central, and
right bands, respectively. This behaviour is expected as the small adjustment
for the orientation of the spectral trace reduces the flux losses from the
tails of the point spread function. For the rest of the analysis described in
this paper, both methods led to indistinguishable results.
#### 2.2.2 Detrending
The HST time series exhibit well-known systematic effects, usually referred to
as short- and long-term ramps (Brown et al., 2001). The short-term ramp is a
flux variation that follows a highly repeatable pattern for each orbit of the
same visit, excluding the first orbit. The long-term ramp approximates a
linear trend in the transit timescale. We adopted the divide-oot method to
correct for the ramp effects (Berta et al., 2012). Following this method, the
two in-transit orbits are divided by the time-weighted mean of the two out-of-
transit orbits. Since the last orbit contains one less spectrum than the
others, we duplicated the last point to enable the operations described above.
We note that the added point falls into the plateau at the end of the ramp and
out of transit, so it is expected to be similar to the previous one.
Figure 1: Left: Best-fit transit depths to HST data around the Na I lines, as
reported in Table 3. Right: divide-oot detrended and phase-folded HST light
curves (see Section 2.2.2) for the 5818-5887 $\AA$ (blue crosses), 5887-5899
$\AA$ (red dots), and 5899-5968 $\AA$ (dodger blue pluses) passbands, along
with best-fit transit models (continuous coloured lines). The detrended out-
of-transit data are not represented, as they are identical to 1 by definition
of the divide-oot method (Berta et al., 2012). The bottom panel shows the
transit model differences with respect to that of the central band. The
slightly different shape between the two side bands is due to differential
limb darkening.
#### 2.2.3 Fitting
We combined the detrended orbits from the three visits into phase-folded time
series to perform joint light-curve fits. The transit models were computed
with PYLIGHTCURVE333https://github.com/ucl-exoplanets/pylightcurve (Tsiaras et
al., 2016). Since we were primarily interested in the differences in transit
depth between the selected passbands, we kept the planet-to-star radii ratio
($p$) as the only free parameter. Table 2 reports the system parameters along
with the transit parameters that were fixed in the light-curve fits. The
stellar limb-darkening coefficients were fixed to the values obtained with
ExoTETHyS444https://github.com/ucl-exoplanets/exotethys (Morello et al.,
2020a, b)) using the four-coefficient law (Claret, 2000) and the STAGGER grid
of stellar spectra (Magic et al., 2015; Chiavassa et al., 2018). The fitting
procedure included a preliminary least-squares minimisation using
scipy.optimize.minimize with the Nelder-Mead method (Nelder & Mead, 1965).
Then we ran emcee555https://github.com/dfm/emcee (Foreman-Mackey et al., 2019)
with 80 walkers and 200 000 iterations. Each walker was initialised with a
random value close to the least-squares parameter estimate. The first 100 000
iterations were discarded as burn-in. The remaining samples were used to
determine the final best-fit parameters (median) and error bars (absolute
differences between the 16th and 84th percentiles and the median).
### 2.3 Results
Table 3 reports the measured transit depths for the three selected passbands
with the two extraction methods described in Section 2.2.1. Figure 1 (left
panel) shows the same results. The two sets of results are nearly identical
with 1-3$\%$ smaller error bars when using tilted apertures. They indicate a
greater transit depth in the central band containing the Na i doublet, and no
significant difference in transit depth between the two side bands. We
calculated the Na i absorption signal as the difference between the transit
depth in the central band and the average of the two side values, obtaining
232$\pm$62 ppm (tilted apertures) and 237$\pm$64 ppm (sum of pixel columns).
These results confirm that Na i is detected in the atmosphere of HD209458 b at
3.7$\sigma$.
We note that the differences between the best-fit light-curve models for the
side bands and for the central band are overall positive during transit, and
show a plateau at $>$200 ppm at orbital phases $|\Delta\phi|<$0.013 (see
bottom right panel of Figure 1). If stellar limb-darkening was the only
wavelength-dependent effect, the light-curve differences should have a typical
double-horned modulation with both positive and negative values (Rosenblatt,
1971; Tingley, 2004; Parviainen et al., 2019). The numerical closeness between
the plateau in the light-curve differences and the measured Na i absorption
feature also suggests that the results of the analysis are not strongly
influenced by the adopted stellar limb-darkening model. However, some authors
pointed out that stellar limb-darkening models may be biased, because of
unaccounted-for effects and lack of empirical validation (Csizmadia et al.,
2013; Espinoza & Jordán, 2015; Morello, 2018). We also attempted light-curve
fitting with free limb-darkening coefficients. The results point towards a
smaller (non-significant) Na i absorption feature and enhanced differential
limb darkening, but they are inconclusive because of the low signal-to-noise
ratio (S/N) of the data and strong parameter degeneracies. Better quality data
are needed to conclude as to the authenticity of the Na i detection at low
resolution.
Table 3: Best-fit transit depths to HST data and Na i absorption signal.
Passband | Transit Depth
---|---
($\mathrm{\AA}$) | tilted aper. ($\%$) | sums of col. ($\%$)
nl: 5818-5887 | 1.482$\pm$0.028 | 1.481$\pm$0.029
nc: 5887-5899 | 1.504$\pm$0.059 | 1.503$\pm$0.060
nr: 5899-5968 | 1.479$\pm$0.029 | 1.478$\pm$0.030
nc - (nl+nr)/2 | 232$\pm$62 ppm | 237$\pm$64 ppm
666Choice of passbands and apertures are detailed in Section 2.2.1.
## 3 Simulations
In the following sections, we describe our simulated time series of spectra
for the transit of HD209458 b, taking into account the stellar limb-darkening
and rotation, for a variety of planetary atmosphere models. Then, we processed
these synthetic data following typical procedures adopted in transmission
spectroscopy studies at low and high resolution, and in particular the average
spectra and light-curves presented as in Charbonneau et al. (2002) and
Casasayas-Barris et al. (2021). The main purpose of this exercise is to
investigate whether there are physical configurations that are compatible with
the observations both at low and high resolution. The methodology adopted to
simulate the spectra in this paper is a generalisation of that described by
Casasayas-Barris et al. (2019). The main upgrade is related to the use of a
wavelength-dependent planet radius to include the effect of its atmosphere.
### 3.1 Modelling the star
We modelled the sky-projected star disc as a grid of square cells, wherein
each cell has an associated spectrum. Each cell is identified by the Cartesian
coordinates of its centre $(x_{j},\,y_{j})$, assuming that the star disc is a
circle with unit radius centred on the origin. The cell spectrum depends on
its position because of the centre-to-limb variation (CLV, aka limb darkening)
and the local radial velocity due to the stellar rotation. We used the
Spectroscopy Made Easy (SME, Valenti & Piskunov, 1996; Piskunov & Valenti,
2017) package to generate intensity spectra for a star similar to HD209458
without rotational broadening. The SME calculation relies on a pre-computed
grid of MARCS stellar atmosphere models (Gustafsson et al., 2008). The
spectral resolving power is $\sim$8$\times$105. These intensity spectra were
calculated for 21 angles, equally spaced in $\mu$, except for $\mu=$0.005
instead of $\mu=$0 to avoid the singularity at the edge of the disc. Here,
$\mu=\cos{\theta}$, where $\theta$ is the angle between the line of sight and
the surface normal. The static cell spectrum is obtained by
$\mu$-interpolation from the pre-calculated intensity spectra, where for a
given cell
$\mu_{j}=\sqrt{1-x_{j}^{2}-y_{j}^{2}}.$ (1)
In the star rest frame, the radial velocity of a given cell is
$v_{j}=v\sin{i_{*}}\left(y_{j}\sin{\lambda}-x_{j}\cos{\lambda}\right),$ (2)
where $v\sin{i_{*}}$ is the radial component of the equatorial velocity, and
$\lambda$ is the sky-projected obliquity. We computed the Doppler shifted cell
spectra in the star rest frame over the same wavelengths of the SME models. In
this way, the disc-integrated stellar spectrum is the sum of the Doppler
shifted cell spectra multiplied by the cell area.
### 3.2 Modelling the planet
The planet is represented by a non-emitting opaque disc with a wavelength-
dependent radius to account for atmospheric absorption. We modelled the case
with constant radius equal to the value reported in Table 2, three cases with
an atmosphere including Na i absorption signature at different terminator
temperatures, and another case that also includes clouds. We used the
petitRADTRANS777https://gitlab.com/mauricemolli/petitRADTRANS/ (pRT, Mollière
et al., 2019) package to compute the apparent planetary radii at multiple
wavelengths including its atmosphere. The spectral resolving power was set to
106. Three of the simulated models assume clear atmospheres with solar
abundances and a range of isothermal temperatures ($T_{\mathrm{p,iso}}=$ 700,
1000, and 1449 K). The fourth model, with $T_{\mathrm{p,iso}}=$ 1449 K,
includes a grey cloud deck with top pressure of 1.38 mbar, as estimated by
recent retrieval studies (Tsiaras et al., 2018). The main effect of
temperature is that it changes the strength of the Na i lines, while the cloud
deck also dampens their tails (see Figure 2). We note that these are the radii
in the planet rest frame. Absorption from the planet atmosphere in the star
rest frame is Doppler shifted due to the relative radial velocity of the
planet,
$V_{\mathrm{p}}(\phi(t))=K_{\mathrm{p}}\sin{(2\pi\phi(t))},$ (3)
for a circular orbit. Here ,$K_{\mathrm{p}}$ is the orbital velocity of the
planet and $\phi(t)$ is the orbital phase,
$\phi(t)=\frac{t-E.T.}{P},$ (4)
with $E.T.$ being the reference epoch of mid-transit time, and $P$ being the
orbital period.
Figure 2: Planet-to-star radii ratios versus wavelength. Left: constant radius
(solid red line) and three pRT models with clear atmosphere and terminator
temperatures of 1449 K (dotted orange), 1000 K (dash-dotted cyan), and 700 K
(dashed blue). Right: pRT model assuming the equilibrium temperature at
terminator with a cloud deck (solid grey thick line), shifted and rescaled
model to have the continuum baseline and line peaks of the clear atmosphere
model (solid grey thin line), and cloud-free model with the same temperature
(dotted orange).
### 3.3 Modelling the transit
While transiting in front of its host star, the planet occults a time-varying
portion of the stellar disc. In our simulations, we compute the unocculted
spectrum at given instants with a cadence of 20 s. To do so, we first
calculate the sky-projected planet coordinates
$(X_{\mathrm{p}}(t),Y_{\mathrm{p}}(t))$ in units of the star radius using our
own modified version of the PYLIGHTCURVE routine. Our version takes into
account the light-travel delay (Kaplan, 2010; Bloemen et al., 2012), which has
a negligible impact in this study. Second, we compute the relative planetary
radii in the star rest frame,
$p(\lambda,t)=\frac{R_{\mathrm{p}}(\lambda,t)}{R_{*}},$ (5)
where $R_{\mathrm{p}}(\lambda,t)$ are either constant or Doppler-shifted pRT
models, interpolated over the same wavelengths of the cell spectra, and
$R_{*}$ is the star radius. Third, we apply the following mask to each cell
spectrum:
$m_{j}(\lambda,t)=\cases{0},&\text{if
}dist((X_{\mathrm{p}}(t),Y_{\mathrm{p}}(t)),\,(x_{j},y_{j}))<p(\lambda,t)\\\
1,\text{elsewhere}{},$ (6)
that is, the masked cell spectra are unchanged if the centre of the cell is
external to the planet discs at all wavelengths, partially or fully zeroed
otherwise. The stellar spectrum observed at a given instant is the sum of the
cell spectra multiplied by the mask and by the cell area.
Figure 3: Simulated transmission spectra of HD209458 b around the Na i D1
line, showing the RM+CLV effects along with atmospheric absorption. The
spectra are computed in the planet rest frame and averaged between transit
contact points as indicated in the top left or top right corner of each panel.
The different colours correspond to the same atmospheric models represented in
Figure 2. The ESPRESSO data are the same as those reported in Figure 5 of
Casasayas-Barris et al. (2021). Figure 4: Simulated transmission light curve
of HD209458 b for passbands centred on any Na i D line with
$\Delta\lambda=\mathrm{0.4\,\AA}$ and $\Delta\lambda=\mathrm{0.7\,\AA}$,
showing the RM+CLV effects along with atmospheric absorption. The light-curves
are computed in the planet rest frame. The different colours correspond to the
same atmospheric models represented in Figure 2. The ESPRESSO data are the
same as reported in Figure 6 by Casasayas-Barris et al. (2021). Figure 5:
Simulated transit light curves of HD209458 b for the same three passbands
adopted in the HST data analysis and five planetary atmosphere models. The
blue, red, and bright blue lines are obtained as described in Section 3, with
the continuous and dashed lines corresponding to clear and cloudy models,
respectively. The bottom panels show the transit model differences with
respect to that of the central band. The black and grey dotted lines are the
HST model differences from Figure 1. We note that the HST model differences
have a more trapezoidal shape than those obtained from the other set of
simulations, because of the different limb-darkening profiles of the
underlying STAGGER and MARCS stellar models.
### 3.4 Simulated transmission spectra and light curves
Starting from the time series of simulated spectra, we calculated time-
averaged spectra and band-integrated time series analogous to those that were
used for data analysis and/or visualisation in previously published papers
(Charbonneau et al., 2002; Casasayas-Barris et al., 2021). Figure 3 shows the
transmission spectra in the planet rest frame averaged between different
contact points, such as those reported in Figure 5 by Casasayas-Barris et al.
(2021). We zoomed in on the main Na i line to highlight the effects of
absorption in the planetary atmosphere, which are identical for both lines
based on our models. The Na i absorption imprints a dip near the peak of the
symmetric RM feature obtained in the T1-T4 and T2-T3 intervals, and an overall
offset due to its broad wings. The planetary Na i may also appear as a small
negative bump before or after the RM feature in the ingress or egress spectra.
We note that, depending on the S/N and resolving power of the spectroscopic
data, the double-peaked RM feature in T1-T4 or T2-T3, and the small negative
bumps in the T1-T2 and T3-T4 spectra may not be resolved. Also, the broadband
spectral offset may be altered by some data-processing steps, in particular
the continuum normalisation and the removal of instrumental systematic
effects. In these cases, the net effect of planet atmospheric absorption could
be that of reducing the amplitude of the RM feature in T1-T4 and T2-T3
spectra. However, the feature amplitudes are also sensitive to the underlying
stellar spectrum and limb-darkening profiles obtained with a different
synthesis code and/or input physics (Casasayas-Barris et al., 2021).
Consequently, the feature amplitude alone does not provide conclusive evidence
about the presence or absence of Na i absorption.
Figure 4 shows the transmission light curves for bandwidths of 0.4 and 0.7
$\mathrm{\AA}$ centred on the Na i D lines in the planet rest frame, as those
reported in Figure 6 by Casasayas-Barris et al. (2021). The Na i absorption
mostly affects the amplitude and vertical offset of the transmission time
series relative to the out-of-transit baseline, and introduces only modest
distortions during the ingress and egress phases. Even for these transmission
light curves, the effects of Na i in the planet atmosphere are degenerate with
details of the stellar spectrum template.
Moving to low resolution, Figure 5 shows the full transit light-curve models
for the three HST passbands analysed in this study (Section 2) and by
Charbonneau et al. (2002). Following the standard approach of low-resolution
transit spectroscopy studies, they consist of instantaneous band-integrated
fluxes relative to the stellar flux in the star rest frame. The decrease in
normalised flux during transit corresponds to the fraction of stellar flux
occulted by the whole planet disc, not just by its atmosphere. We subtracted
the light-curve model of the central band to highlight the wavelength-
dependent effects. If Na i is not present in the planet atmosphere, the
modulation obtained in the light-curve differences are solely due to the
stellar RM and CLV effects. In this case, we measured a peak-to-peak amplitude
of 330 ppm and mean value below 10 ppm for the modulation. If Na i is present
in the planet atmosphere, our models indicate positive light-curve differences
at all phases, with median values of $\sim$190, $\sim$300, and $\sim$520 ppm
assuming isothermal temperatures ($T_{\mathrm{p,iso}}$) of 700, 1000, and 1449
K, respectively. The cloudy model at 1449 K shows a median light-curve
difference of 300 ppm.
We note that the light-curves plotted in Figures 4 and 5 have been smoothed to
remove some fringing due to numerical errors, particularly in the models with
high-resolution signatures. We checked that fringing was significantly reduced
when decreasing the cell size adopted for the simulations, but the relevant
calculations soon became too heavy. The smoothed light curves are a good
compromise between precision of the models and computational costs.
## 4 Discussion
### 4.1 Comparison with published low-resolution spectroscopy
Charbonneau et al. (2002) announced the first detection of Na i in the
atmosphere of HD209458 b using the same HST data presented in Table 1. They
reported an absorption depth of 232$\pm$57 ppm based on the same passbands of
Table 3, but with a significantly different methodology. More specifically,
they computed the absorption light curve as a linear combination of the three
raw light curves, hence the depth is the difference between the out-of-transit
and in-transit mean values. Instrumental systematics and stellar limb-
darkening effects are ignored with their approach, although they were
mitigated by removing the first point of each HST orbit and those during the
transit ingress and egress. In this work, we instead measured the transit
depth for each passband by performing independent data detrending and light-
curve fitting, and assuming a STAGGER stellar limb-darkening model (see
Section 2). The absorption depth is a linear combination of the three transit
depths. Our best result of 232$\pm$62 ppm matches that reported by Charbonneau
et al. (2002). Turning to empirical limb-darkening coefficients in the light-
curve fits, however, we found that the HST data are consistent with the non-
detection of Na i in the atmosphere of HD209458 b.
By comparing the model light-curve differences shown in Figure 5 with the
best-fit results of Figure 1, the models that better reproduce the HST results
are those with cloud-free atmosphere and terminator temperatures of 700-1000
K, or that with partially cloudy atmosphere and terminator temperature equal
to the equilibrium temperature of 1449 K. We note that there is a degeneracy
between the terminator temperature and cloud top pressure for a given
amplitude of the Na i absorption signal at low resolution (Lecavelier Des
Etangs et al., 2008; Benneke & Seager, 2013; Heng & Kitzmann, 2017).
Sing et al. (2008) analysed the transmission spectrum of HD209458 b with
various resolving powers, obtaining similar results when considering the same
passbands as Charbonneau et al. (2002). Additionally, Sing et al. (2008)
reported a spectral signal in the opposite direction to that of Na i
absorption at the line cores, that they attributed to telluric contamination.
The removal of this effect would further increase the significance of the
measured Na i absorption. However, as pointed out by Casasayas-Barris et al.
(2020), a much more likely explanation for this signal is the combination of
RM+CLV effects instead of a telluric origin.
### 4.2 Comparison with published high-resolution spectroscopy
Snellen et al. (2008) and Albrecht et al. (2009) analysed ground-based spectra
taken with the High Dispersion Spectrograph (HDS) mounted on the Subaru
telescope at Mauna Kea in Hawaii, and with the Ultraviolet and Visual Echelle
Spectrograph (UVES) mounted on the Very Large Telescope (VLT) at Cerro Paranal
in Chile. Their results are in good agreement with those reported by Sing et
al. (2008) based on HST data, when considering similar passbands centred on
the Na i D lines.
Recently, Casasayas-Barris et al. (2020) analysed the high-resolution spectra
taken with TNG/HARPS-N, and with the Calar Alto high Resolution search for M
dwarfs with Exoearths with Near-infrared and optical Echelle Spectrographs
(CARMENES) located in Spain. Their approach differs from that of previous
studies as the transmission spectra are computed in the planet rest frame
instead of the stellar rest frame. They revealed the RM effect around the Na i
D, as well as other lines (H$\alpha$, Ca ii IRT, Mg i and K i D1), without
evidence of absorption in the exoplanet atmospheres. Casasayas-Barris et al.
(2021) confirmed these findings with higher confidence and also reported an
analogous behaviour for other lines (Fe i, Fe ii, Ca i, and V i), based on
data taken with VLT/ESPRESSO. The ESPRESSO datasets have higher S/N than any
other ground-based high-resolution dataset.
In Section 3.4, we discuss how Na i absorption may alter the RM+CLV signal in
transmission spectra. The error bars obtained by Casasayas-Barris et al.
(2021) with ESPRESSO range from 0.05$\%$ to 0.09$\%$ for bin widths of 0.03
$\AA$. Based on the models plotted in Figure 3, they should have resolved the
double peak caused by Na i absorption for both clear and cloudy atmospheric
models with equilibrium temperature of 1449 K, if present, in the ESPRESSO
data. We note that clouds tend to dampen the line tails, and hence they mostly
affect the low-resolution signal. Models with cooler temperatures pose a
greater challenge for the high-resolution signal, as the smaller dips on top
of the RM feature can be comparable with the spectral error bars. If
unresolved, the Na i dip would decrease the amplitude of the RM feature. The
ESPRESSO data produces a signal of greater amplitude than that predicted by
the models shown in Figure 3, contrary to expectations in the presence of
unresolved Na i absorption. However, the amplitude of the RM feature alone
cannot be used to validate Na i absorption in the atmosphere of HD209458 b,
given the uncertainties associated with stellar models and system parameters.
For example, Casasayas-Barris et al. (2021) obtained a larger RM feature by
assuming a MARCS model without local thermodynamic equilibrium (see their
Figure 10). We also note that the broadband spectral offset is $\sim$0.05$\%$
for the model with the strongest absorption and it can be attenuated by
ESPRESSO data processing, in particular due to continuum normalisation and
removal of observed fringe patterns (Casasayas-Barris et al., 2021).
Similar considerations apply to the transmission light curves as shown in
Figure 4. The possible Na i absorption affects the amplitude and offset of the
time signal, without significant distortions in most cases. The shape of the
transmission light curves is strongly dependent on the CLV effect, for which
the stellar models considered by Casasayas-Barris et al. (2021), which are the
same adopted in this work, do not provide a good match to the ESPRESSO data.
In conclusion, we find that atmospheric models with terminator temperature of
700-1000 K can be compatible with the non-detection of Na i at high
resolution. In Section 4.1, these models were also selected among the best
fits to the low-resolution data that led to the 3.7$\sigma$ detection of Na i.
The high-resolution observations taken with ESPRESSO added further
constraints; for example, discarding the hotter model with clouds.
### 4.3 Planet atmospheric temperature
Morello et al. (2021) calculated an equilibrium temperature of 1449 K for
HD209458 b in the case of zero reflectance and maximum circulation efficiency
(see their Table 3). Taking into account the Spitzer phase-curve measured by
Zellem et al. (2014), the terminator temperature should lie between the night-
side temperature of 970 K and the day-side temperature of 1500 K. This
temperature range can be further extended depending on the vertical profiles
(e.g. Venot et al., 2019; Drummond et al., 2020). In principle, the 3D
structure of the planetary atmosphere should be fully considered to simulate
accurate transmission spectra, but 1D models can also reproduce the same
spectra albeit with biases in physical and chemical parameters (Caldas et al.,
2019; Pluriel et al., 2020). In particular, MacDonald et al. (2020)
demonstrated that the 1D retrieval techniques can lead to significantly
underestimated temperatures for atmospheres with differing morning-evening
terminators. This effect may explain the trend that terminator temperatures
retrieved from transmission spectra are typically cooler than equilibrium
temperatures (Fisher & Heng, 2018; Tsiaras et al., 2018; Pinhas et al., 2019).
Our isothermal temperature estimate of 700-1000 K for the atmosphere of
HD209458 b aligns with this trend.
## 5 Conclusions
We investigated the puzzling question about the presence of Na i in the
atmosphere of HD209458 b, following conflicting reports from previous analyses
of transit observations with low- and high-resolution spectroscopy. By
comparing a set of models to the observations, we find that several
atmospheric scenarios are compatible with both low- and high-resolution data.
Overall, the HST and ESPRESSO datasets are consistent with a total absence of
Na i from the planetary atmosphere; otherwise, the terminator temperature of
HD209458 b has to have an upper limit of about 1000K. More precise knowledge
of the stellar intensity spectra, along with higher S/N data, is crucial to
selecting the best scenario. The lower-than-equilibrium temperature (1449 K)
on the terminator can be a consequence of heat redistribution processes or the
effect of 1D model bias. Clouds may be present, in agreement with previous
estimates, but they alone cannot help to reconcile the low- and high-
resolution observations. In some configurations, the presence of Na i is clear
from the broadband transmission spectrum, but, under certain physical
conditions, the RM effect can mask the line cores at high resolution. If the
overlapping signals at high resolution are not disentangled, the information
on the spectral continuum from low-resolution observations can be decisive in
detecting Na i absorption.
This study highlights the complementarity between low- and high-resolution
spectroscopic techniques for the characterisation of exoplanet systems. In the
literature, there are several contrasting results about exoplanet atmospheres
based on low- and high-resolution spectroscopy (Huitson et al., 2013;
Sedaghati et al., 2017; Espinoza et al., 2019; Chen et al., 2018; Allart et
al., 2020). We suggest that joint modelling efforts, such as the one proposed
in this paper, could lift the apparent discrepancies. Considering a broader
wavelength coverage should help reduce the degeneracy between stellar and
planetary signals by considering multiple lines, therefore leading to tighter
constraints on the possible exoplanet atmospheric models.
###### Acknowledgements.
The authors would like to thank the referee, Jens Hoeijmakers, for useful
comments that have improved the manuscript. G. Mo. has received funding from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No. 895525. N. C.-B. acknowledges
funding from the European Research Council under the European Union’s Horizon
2020 research and innovation program under grant agreement No. 694513. J.
O.-M. and E. P. were financed by the Spanish Ministry of Economics and
Competitiveness through grants PGC2018-098153-B-C31. G. Mi. e G. C.
acknowledge contribution from ASI-INAF agreement 2021-5-HH.0.
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|
# Optimization tools for Twin-in-the-Loop vehicle control design:
analysis and yaw-rate tracking case study
Federico Dettù<EMAIL_ADDRESS>Simone Formentin
<EMAIL_ADDRESS>Stefano Varisco<EMAIL_ADDRESS>Sergio
Matteo Savaresi<EMAIL_ADDRESS>Dipartimento di Elettronica,
Informazione e Bioingegneria, Politecnico di Milano.
Via G. Ponzio 34/5, 20133, Milano, Italy. Vehicle Dynamics, Ferrari S.p.A.
Via Abetone Inferiore 4, 41053, Maranello, Italy.
###### Abstract
Given the urgent need of simplifying the end-of-line tuning of complex vehicle
dynamics controllers, the Twin-in-the-Loop Control (TiL-C) approach was
recently proposed in the automotive field. In TiL-C, a digital twin is run on-
board to compute a nominal control action in run-time and an additional block
$C_{\delta}$ is used to compensate for the mismatch between the simulator and
the real vehicle. As the digital twin is assumed to be the best replica
available of the real plant, the key issue in TiL-C becomes the tuning of the
compensator, which must be performed relying on data only. In this paper, we
investigate the use of different black-box optimization techniques for the
calibration of $C_{\delta}$. More specifically, we compare the originally
proposed Bayesian Optimization (BO) approach with the recently developed Set
Membership Global Optimization (SMGO) and Virtual Reference Feedback Tuning
(VRFT), a one-shot direct data-driven design method. The analysis will be
carried out within a professional multibody simulation environment on a novel
TiL-C application case study – the yaw-rate tracking problem – so as to
further prove the TiL-C effictiveness on a challenging problem. Simulations
will show that the VRFT approach is capable of providing a well tuned
controller after a single iteration, while $10$ to $15$ iterations are
necessary for refining it with global optimizers. Also, SMGO is shown to
significantly reduce the computational effort required by BO.
## 1 Introduction
Figure 1: Twin-in-the-Loop vehicle control scheme.
Although control theorists have been pushing forward for decades in developing
well performing and sophisticate control algorithms, a sort of detachment
arose when looking at what control practicioners do. A few years ago, the
survey (Samad, 2017) showed that Proportional Integral Derivative control is
perceived as the most impacting technology in industry, far above _e.g._
nonlinear or adaptive controls, and somewhat more relevant than extremely
popular Model Predictive Control. In the same survey, it emerges that the lack
of competences in industry is an important perceived issue when dealing with
modern controllers. In fact, calibrating and updating a developed controller
requires adjustment of several tuning knobs, whose effect is often poorly
understood without a deep knowledge of control theory.
Concentrating our efforts toward this direction, we recently proposed an
approach for simplifying the implementation of complex controllers and easing
the end-of-line tuning phase in the vehicle domain: the Twin-in-the-Loop
Control (TiL-C) framework111The dual problem of Twin-in-the-Loop Filtering
(TiL-F) is also under investigation (Riva et al., 2022b; Delcaro et al.,
2023). (Dettù et al., 2023b), schematically represented in Fig. 1222A
discussion on robust calibration for TiL controllers has been proposed in
Dettù et al. (2023a).. TiL-C runs a digital twin (DT) –a highly sophisticated
model– inside a vehicle control unit in real-time. This is made possible by
the most recent technological advances (see _e.g._ the Autohawk platform VI-
Grade (2022)). Further, in the automotive domain, DT often come for free, as
car manufacturer already have a portfolio of well calibrated models of their
production vehicles, used for off-line testing and virtual prototyping.
The TiL-C architecture is based upon a nominal controller $C$; such a
controller is calibrated on the DT and then used on-board at run-time. This
allows for the computation of a nominal control action to be applied as a
first - open-loop - contribution to the real vehicle. It was then shown that a
simple compensator $C_{\delta}$ (_e.g._ a PID) could be sufficient to
guarantee stability and maintain the same performance attained on the DT also
on the real vehicle. By using TiL-C, the end-of-line tuning shifts from the
implementation (and sometimes redesign) of a complex controller, to the tuning
of a few gains, which is more easily done and widely understood.
An important issue in TiL-C – deeply discussed in (Dettù et al., 2023b) – is
how to use data to calibrate such parameters of the compensator $C_{\delta}$:
since this controller manages the unknown dynamics between the DT and the
vehicle, no classical model-based design techniques (_e.g._ loop shaping) can
be used. Following a recently established literature trend (Dettù et al.,
2023; Busetto et al., 2023; Coutinho et al., 2023), we first proposed Bayesian
Optimization (BO) to solve this problem. Although effectiveness of BO in
calibrating TiL-C has been shown ($100$ iterations have been considered), no
discussions concerning the selection of a specific optimization algorithm have
been considered. Also, no analyses concerning the number of iterations
necessary to get to convergence has been carried out. Being vehicle testing
very costly, assessing (and possibly reducing) the number of the necessary
experiments is extremely important. We thus concentrate our efforts to
investigate and analyse different approaches for calibrating a TiL controller.
Related works. Within the context of black-box optimization (of which BO is a
particular case), many approaches have been developed throughout the years and
successfully applied to different fields. Two classic iterative methods are
_e.g._ Extremum Seeking (ES) (Hellström et al., 2013) or Iterative Feedback
Tuning (IFT) (Hjalmarsson et al., 1998). Although being simple to apply, these
approaches lost popularity, suffering from different issues, such as the high
number of iterations required to get a solution, or the impossibility of
including safety a-priori unknown constraints during the tuning process,
potentially yielding hazardous controller attempts. Furthermore, these methods
are not global, and they might become stuck in a local minimum, requiring re-
initialization and thus increased costs.
Among global optimizers, surrogate function based ones (such as BO) gained
much popularity with respect to _e.g._ genetic algorithms (Jaen-Cuellar et
al., 2013) or particle swarm (Kim et al., 2008) ones, due to the large –often
unfeasible in real applications– number of iterations required by the latter.
As already mentioned, BO (Shahriari et al., 2015) proved to be very effective
when solving black-box optimization problems: calibrating a controller for a
partially or totally unknown plant lies within this framework. BO fits a
Gaussian Process (GP) surrogate of the unknown cost function to be minimized,
and iteratively selects new candidate points to be evaluated based on the
acquired GP knowledge. A nice BO feature is the possibility of including
a-priori unknown constraints within the optimization problems: similarly as
done for the cost function, these constraint functions are iteratively
estimated from data. This opens up the possibility of _e.g._ maintaining the
system in a safe region throughout the calibration procedure.
The most significant issue with BO is the high computational burden yield by
the GP fitting and optimization: the more data points are added, the more
cumbersome this procedure becomes, undermining application of BO in low-cost
processing units. To solve this, Sabug Jr et al. (2022) recently presented Set
Membership Global Optimization-$\Delta$ (SMGO-$\Delta$); with respect to BO,
this approach provides a computationally efficient objective and constraints
model, based on hypercones, and focusing on quantifying the uncertainty
associated to the unknown function, rather than directly trying to fit it.
SMGO-$\Delta$ also allows accounting for black-box constraints, and the
authors proven its convergence under the assumption of Lipschitz continuity of
the objective and constraints. SMGO-$\Delta$ has recently been used in
practical control tuning problems (Galbiati et al., 2022; Busetto et al.,
2023; Sabug Jr et al., 2023).
A completely different philosophy when dealing with controller tuning is the
Virtual Reference Feedback Tuning (VRFT) one: VRFT is direct data-driven
approach, thus requiring a single experiment for learning the controller. It
is based upon frequency-domain considerations, and tries to match a reference
closed-loop behaviour. VRFT has been successfully used in different control
tuning problems, _e.g._ electric motors (Busetto et al., 2023), engines
(Passenbrunner et al., 2014) or brake-by-wire systems (Radrizzani et al.,
2020); however, some critical tuning knobs (like the reference model) exist
when applying VRFT, and the resulting controller might be underperforming or
even unstable if they are not properly selected.
Finally, although well promising, TiL-C has been so far exclusively applied to
braking control (Dettù et al., 2023a, b): in the strive of widening the
portfolio of applications and further proofing its efficacy, different case
studies should be considered.
Contributions. Taking into account considerations above, we consider here
SMGO-$\Delta$ for TiL calibration, comparing it with BO, and discussing the
convergence speed of both algorithms. We include a constraint during the
optimization process, in order to guarantee that unsafe regions are avoided.
Also, we apply VRFT to the same problem, eventually showing the significant
performance obtained with just one experiment. Combination of both methods
(VRFT and BO/SMGO) is also explored. On top of this, we apply TiL-C to the
vehicle yaw-rate tracking problem: such a problem is important both in
production vehicles driving assistance systems – it is at the basics of
Electronic Stability Control (ESC) Gimondi et al. (2021) – and in autonomous
driving, where it is combined with trajectory following (Corno et al., 2023;
Falcone et al., 2007). Being an established state-of-the-art (Lucchini et al.,
2020, 2021; Spielberg et al., 2022; Beal and Gerdes, 2012; Falcone et al.,
2007), we use Model Predictive Control as a baseline controller in yaw-rate
tracking. Yaw-rate control poses critical stability problems, in that the
vehicle sideslip might utterly increase if not properly addressed, leading to
unsafe situations and undesired drifting of the vehicle (_i.e._ too large
wheel slips): it is thus an interesting and challenging test bench for TiL-C.
The remainder of the paper is as follows. In Section 2, we describe the yaw-
rate control problem, the simplified model of the vehicle and derive the
baseline controller, to be used within TiL-C. Then, in Section 3, we show how
TiL-C can be applied to the presented problem, and discuss the different
optimization tools. Section 4 compares the optimizers in terms of cost
function minimization and constraints violation, and then gives an in-depth
time-domain analysis of the obtained results, on different maneuvers.
## 2 The yaw-rate tracking problem
The yaw-rate tracking problem is tackled in this research with Twin-in-the-
Loop control. This application problem is used to evaluate and analyse
different optimization tools for TiL-C. Hence, we start off by precisely
defining a control-oriented system model and a baseline controller. The goal
is that of following a reference yaw-rate $r_{ref}$ – generated by the driver
steering action $s_{ref}$ by means of a suitable reference generation
mechanism. The controller decides a steering action $s_{cmd}$ to be passed to
a steer-by-wire system, acting onto the front wheels. Measurements used for
feedback control are: the yaw-rate $r$, the vehicle sideslip angle $\beta$
(usually retrieved by means of an estimation algorithm (Carnier et al., 2023),
not in the scopes of this work), the steering actuator position $s_{act}$, the
longitudinal component of vehicle speed $v_{x}$ and the longitudinal
acceleration $a_{x}$.
### 2.1 System modelling
In the following, we describe the model used for developing lateral dynamic
controllers. The model takes into account simplified vehicle dynamics and the
steer-by-wire actuator.
##### Vehicle model
Figure 2: Single-track vehicle model. Figure 3: Steer-by-wire actuator model,
featuring transfer function, rate limiter and saturation.
The model considered in this research for the development of benchmark lateral
dynamics controllers is a non-linear single-track one (schematically
represented in Fig. 2). Assuming that the right-left tires at front and rear
axle can be lumped together, the model reads (Beal and Gerdes, 2012)
$\begin{split}\dot{\beta}=&\dfrac{F^{f}_{y}+F^{r}_{y}}{Mv_{x}}-r\\\
\dot{r}=&\dfrac{L_{f}F^{f}_{y}-L_{r}F^{r}_{y}}{J_{zz}}\end{split},$ (1)
$J_{zz}$ and $M$ are the vehicle yaw moment of inertia and mass; $L_{f}$ and
$L_{r}$ are the distance from the vehicle center-of-mass and the front/rear
wheels, respectively. The lumped lateral forces at front and rear wheels
$F^{f}_{y}$, $F^{r}_{y}$ can be modeled in different ways; a popular one is
the Pacejka magic formula, proposed here in a simplified version (Lucchini et
al., 2020)
$F_{y}^{j}=-\dfrac{F_{z}^{j}\mathcal{C}_{j}}{\mathcal{A}_{j}\mathcal{B}_{j}}\sin\left(\mathcal{B}_{j}\textrm{atan}\left(\mathcal{A}_{j}\tan\left(\alpha_{j}\right)\right)\right),$
(2)
whereas $j=f,r$, and $\mathcal{A}_{j},\ \mathcal{B}_{j},\ \mathcal{C}_{j}$ are
suitable parameters (to be identified from data). $F^{j}_{z}$ is the wheel
vertical load, $\alpha_{j}$ is the wheel lateral sideslip angle. Under the
assumption of small vehicle sideslip and steer ($s_{act}$) angles, the lateral
slip at front and rear tires can be written as
$\alpha_{f}=\beta+\dfrac{L_{f}}{v_{x}}r-s_{act},\
\alpha_{r}=\beta-\dfrac{L_{r}}{v_{x}}r.$ (3)
On the other hand, vertical forces can be computed by taking into account
static contributions (due to vehicle mass), aerodynamic contributions (speed-
dependent) and load transfer (acceleration-dependent) (Lucchini et al., 2021),
such that
$\begin{split}F_{z}^{f}=&\dfrac{L_{r}}{L_{f}+L_{r}}Mg+k^{f}_{a}v_{x}^{2}-k_{x}a_{x}\\\
F_{z}^{r}=&\dfrac{L_{f}}{L_{f}+L_{r}}Mg+k^{r}_{a}v_{x}^{2}+k_{x}a_{x}\end{split}$
(4)
where $g$ is the gravity acceleration, $k_{a}^{f},\ k_{a}^{r}$ are aerodynamic
coefficients, and $k_{x}$ is the load transfer coefficient.
##### Steer-by-wire actuator
With regards to the steer-by-wire actuator, we consider here a model
comprising a linear part (transfer function) and a non-linear one (rate
limiter and saturation), see Fig. 3. The transfer function model from
commanded to actuated steer takes into account the low-level control, and it
has been provided by partner company Ferrari S.p.A.
$G_{act}\left(s\right)=\dfrac{58.34s^{2}+1547s+9137}{1.002s^{3}+64.55s^{2}+1549s+9137}.$
(5)
The closed-loop low-level control system modelled by $G_{act}\left(s\right)$
has a bandwidth $\omega_{act}\approx 33.8\ rad/s$. Being
$G_{act}\left(s\right)$ complex for the control-oriented modelling purposes
herein, we consider a simplified first-order transfer function
$G_{act,s}\left(s\right)=\dfrac{\omega_{act}}{s+\omega_{act}}$ (6)
The rate-limiter block implements a saturation on the time derivative of the
actuated steer angle - it models a current limit in the low-level circuits,
specifically. Then, the saturation block implements a limit on the actuated
steer angle. Mathematically, this reads
$\begin{split}\dot{s}_{rl}&=\begin{cases}-\dot{s}_{max}&\text{$\dot{s}_{tf}<-\dot{s}_{max}$}\\\
\dot{s}_{tf}&\text{$-\dot{s}_{max}\leq\dot{s}_{tf}<\dot{s}_{max}$}\\\
\dot{s}_{max}&\text{$\dot{s}_{tf}\geq\dot{s}_{max}$}\end{cases},\\\
s_{act}&=\begin{cases}-s_{max}&\text{$s_{rl}<-s_{max}$}\\\
s_{act}&\text{-$s_{max}\leq{s}_{rl}<s_{max}$}\\\
s_{max}&\text{$s_{rl}>s_{max}$}\end{cases}.\\\ \end{split}$ (7)
##### Complete model and linearization
The model in Eqs. (1)-(6) contains a non-linearity in the lateral forces
$F_{y}^{f}$, $F_{y}^{r}$. We can linearize their expression for given front
and rear sideslip angles $\bar{\alpha}_{f},\ \bar{\alpha}_{r}$
$\displaystyle F_{y}^{j}\approx$
$\displaystyle\bar{F}_{y}^{j}-\bar{C}^{j}_{\alpha}\left(\alpha_{j}-\bar{\alpha}_{j}\right),$
(8a) $\displaystyle\bar{F}^{j}_{y}\left(\bar{\alpha}_{j},F_{z}^{j}\right)=$
$\displaystyle-\dfrac{F_{z}^{j}\mathcal{C}_{j}}{\mathcal{A}_{j}\mathcal{B}_{j}}\sin\left(\mathcal{B}_{j}\arctan\left(\mathcal{A}_{j}\bar{\alpha}_{j}\right)\right),$
(8b)
$\displaystyle\bar{C}^{j}_{\alpha}\left(\bar{\alpha}_{j},F_{z}^{j}\right)=$
$\displaystyle\dfrac{F_{z}^{j}\mathcal{C}_{j}}{1+\mathcal{A}^{2}_{j}\bar{\alpha}_{j}}\cos\left(\mathcal{B}_{j}\arctan\left(\mathcal{A}_{j}\bar{\alpha}_{j}\right)\right).$
(8c)
Rewriting Eqs. (1)-(6), and considering Eq. (8), we have the following
discrete-time model
$\begin{split}x_{k+1}=&Ax_{k}+Bu_{k}+Ed_{k},\\\ y_{k}=&Cx_{k}.\end{split}$ (9)
Whereas the discretization has been carried out via forward Euler approach,
with sampling time $T_{s}=0.01s$, and the subscript $k$ denotes the time step.
The model state $x_{k}$, output $y_{k}$, disturbance $d_{k}$ and input $u_{k}$
are
$\begin{split}x_{k}=&\begin{bmatrix}\beta_{k}\\\ r_{k}\\\
s_{act,k}\end{bmatrix},\ y_{k}=r_{k},\ u_{k}=s_{cmd,k},\\\
d_{k}=&\begin{bmatrix}\dfrac{\bar{F}_{y}^{f}+\bar{C}^{f}_{\alpha}\bar{\alpha}_{f}+\bar{F}_{y}^{r}+\bar{C}^{r}_{\alpha}\bar{\alpha}_{r}}{Mv_{x}}\\\
\dfrac{L_{f}\left(\bar{F}_{y}^{f}+\bar{C}^{f}_{\alpha}\bar{\alpha}_{f}\right)-L_{r}\left(\bar{F}_{y}^{r}+\bar{C}^{r}_{\alpha}\bar{\alpha}_{r}\right)}{J_{z}}\end{bmatrix}^{t}.\end{split}$
(10)
Whereas $\left(\cdot\right)^{t}$ denotes the matrix transpose operator. And
the model matrices are given in Eq. (11).
$\boxed{A=T_{s}\begin{bmatrix}T_{s}^{-1}-\dfrac{\bar{C}^{f}_{\alpha}+\bar{C}^{r}_{\alpha}}{Mv_{x}}&\dfrac{-L_{f}\bar{C}^{f}_{\alpha}+L_{r}\bar{C}^{r}_{\alpha}-Mv^{2}_{x}}{Mv^{2}_{x}}&\dfrac{\bar{C}^{f}_{\alpha}}{Mv_{x}}\\\
\dfrac{-L_{f}\bar{C}^{f}_{\alpha}+L_{r}\bar{C}^{r}_{\alpha}}{J_{zz}}&T_{s}^{-1}-\dfrac{L_{f}^{2}\bar{C}^{f}_{\alpha}+L_{r}^{2}\bar{C}^{r}_{\alpha}}{J_{zz}v_{x}}&\dfrac{\bar{C}^{f}_{\alpha}L_{f}}{J_{zz}}\\\
0&0&T_{s}^{-1}-\omega_{act}\end{bmatrix},\ B=\begin{bmatrix}0\\\ 0\\\
T_{s}\omega_{act}\end{bmatrix},\ E=\begin{bmatrix}T_{s}&0\\\ 0&T_{s}\\\
0&0\end{bmatrix},C=\begin{bmatrix}0\\\ 1\\\ 0\end{bmatrix}^{t}.}$ (11)
###### Remark.
_Note that the model in Eq. ( 9) is linear time varying, due to the presence
of vehicle speed $v_{x}$, normal wheel forces $F^{f}_{z},\ F_{z}^{r}$ and the
linearization of $F^{f}_{y},\ F^{r}_{y}$ around the current sideslip angles.
It is common practice to disregard the vehicle speed as a state when dealing
with control-oriented modelling for lateral dynamics (Beal and Gerdes, 2012),
as the same introduces unnecessary complexity; we consider it as a slow-
varying parameter. Similar considerations can be applied to the vertical
forces (Riva et al., 2022b). Concerning the lateral forces, two possibilities
emerge: a first and simpler one consists in linearizing their expression for
zero sideslip angles, i.e. $\bar{\alpha}_{f}=\bar{\alpha}_{r}=0$ (used e.g. in
(Corno et al., 2023)), the second one consists in linearizing their expression
at each time step around the current sideslip angle values (used e.g. in (Beal
and Gerdes, 2012; Lucchini et al., 2020; Spielberg et al., 2022))._
### 2.2 Reference generator
Figure 4: Static component of $r_{ref}$ against driver steer request
$s_{ref}$. For each showed speed value, the solid line represents the digital
twin behaviour, whereas the dashed one is the static model.
The reference generation approach considered herein transforms a driver steer
request into a desired yaw-rate to be imposed onto the vehicle. This can be
realized via a static adaptive gain and a transfer function. The static gain
defines the desired steady-state vehicle behavior. Considering the available
high-fidelity model as a starting point, the reference curves are fitted so as
to mimic the nominal vehicle behaviour, saturating the yaw-rate at the maximum
obtainable value, see Fig. 4. Although the static gain guarantees that the
driver request generates a feasible and meaningful yaw-rate reference, the
same does not guarantee dynamic feasibility of said reference. Hence, we
consider a filter with unitary gain and two poles
$F_{ref,dyn}(s)=\dfrac{\left(2\pi f_{ref}\right)^{2}}{\left(s+2\pi
f_{ref}\right)^{2}}.$ (12)
The two poles are set to $f_{ref}=6.3\ Hz$ so as to assign a desired
”bandwidth” to the controller, as done in (Gimondi et al., 2021) for a similar
problem.
### 2.3 Model Predictive Control
Figure 5: High-level control system architecture, depicting the software
elements, namely the reference generator and the controller, the employed
measurements and the control variable. Figure 6: Force-slip diagram with
indicated peak force point and two examples of linearized force values before
and after the peak.
MPC can be employed to optimally solve a constrained control problem. The main
ingredients of MPC are an internal model –to be used for predicting the future
system behavior at each time step– a set of constraints on state and input
variables, and a cost function to be minimized.
##### Internal model
Considering Remark Remark, we apply here the local linearization approach at
each time step, for the lateral forces model. Hence, we use as an internal
model within the MPC the one described in Eqs. (9) and (11), which is
iteratively initialized according to the following procedure
1. 1.
At time step $k$, measure/estimate $v_{x,k},\ a_{x,k},\ s_{act,k},\
\beta_{k}$;
2. 2.
Use Eq. (3) to estimate $\alpha_{f,k},\ \alpha_{r,k}$. Use Eq. (4) to estimate
$F_{z,k}^{f},\ F_{z,k}^{r}$;
3. 3.
Linearize lateral forces via Eq. (8) to get $\bar{F}^{f}_{y,k},\
\bar{F}^{r}_{y,k},\ \bar{C}^{f}_{\alpha,k},\ \bar{C}^{r}_{\alpha,k}$;
4. 4.
Substitute Eq. (11) to get matrix $A_{k}$.
The model obtained at each time step is thus Linear Time Invariant (LTI), and
the MPC requires forward integration of this model to get predictions on the
system behavior. For the system to stay LTI, it is necessary that external or
potentially time-varying inputs are constant. This means assuming that, for
all $k$
$\begin{split}F^{f}_{z,k+i}=&F^{f}_{z,k},\ i=1,\ldots,N_{p},\\\
F^{r}_{z,k+i}=&F^{r}_{z,k},\ i=1,\ldots,N_{p},\\\ v_{x,k+i}=&v_{x,k},\
i=1,\ldots,N_{p}.\end{split}$ (13)
The assumptions above are indeed common in the literature (Lucchini et al.,
2020; Beal and Gerdes, 2012). A special note is necessary here for what
concerns the lateral force model. Consider the force-slip relation for the
considered front tires, in Fig. 6: it is evident that the maximum attainable
force is achieved for $\alpha=\alpha_{max}$, much smaller than $s_{max}$.
Going above the force peak, the slope of the curve is inverted – _i.e._
$\bar{C}_{\alpha}$ changes sign – and the predictive controller would try to
increase $\alpha$ to reduce the lateral force, given its limited knowledge of
the force model; one can verify that this yields instability. While accounting
for the complete force model is non-viable for the introduced nonlinearities,
enforcing a constraint on $\alpha$ allows for smooth operation of the tire
within its physical limits. Although a human driver might want to stay beyond
the curve for fun-to-drive reasons – _e.g._ for increased slipping of the
vehicle – no advantages in terms of steering performance exist. Let us remark
that this consideration was not carried out in previous literature (Lucchini
et al., 2020, 2021), but is non-viable when dealing with driving at the limits
of handling. Since we are considering active front steering, only the front
sideslip angle $\alpha_{f}$ is constrained.
Indeed, constraining $\alpha$ requires good knowledge of the slip value for
which force generation is maximized: imperfect knowledge might yield to worse
control performance, and an ad-hoc learning approach to estimate this
parameter in real-time should be used (_e.g._ a neural-network, as in
(Spielberg et al., 2022)). For the present research, $\alpha_{max}$ is assumed
to be known with adequate accuracy.
##### Constraints
The system in Eq. (9) is characterized by physical limitations on the actuator
(Eq. (7)). Said limitations are to be included as hard constraints within the
optimization problem.
$\begin{split}&-s_{max}\leq s_{act,k}\leq s_{max},\ \forall
k=1,\ldots,N_{p}\\\ &-\dot{s}_{max}T_{s}\leq
s_{act,k}-s_{act,k-1}\leq\dot{s}_{max}T_{s},\ \forall
k=1,\ldots,N_{p}.\end{split}$
(14)
On the other hand, as discussed above, the front sideslip angle $\alpha_{f}$
should be limited to prevent the system from going above the tire limitations.
This not being a physical limit of the system (which is always able to enter
the above-peak force region), we implement it as a soft constraint, via slack
variable $\rho$
$\begin{split}&-\alpha_{f,max}-\rho\leq\alpha_{f,k}\leq\alpha_{f,max}+\rho,\
\forall k=1,\ldots,N_{p}\end{split}$ (15)
As discussed above, the constraint based on maximum $\alpha_{f,k}$ is much
more restrictive than the actuator capabilities (constraints in Eq. (14)).
Hence, first constraint in Eq. (15) can be discarded, as it is never
activated.
##### Cost function
$\boxed{\begin{split}\mathcal{U}*=&\mathop{\mathrm{argmin}}\limits_{\epsilon,{x}_{1},\ldots,{x}_{N_{p}},u_{0},\ldots,u_{N_{p}}}\sum_{k=0}^{N_{p}-1}{{x}^{t}_{k+1}}W_{x}{{x}_{k+1}}+w_{u}u_{k}^{2}+w_{\rho}\rho^{2},\\\
\textrm{subject to}&\\\
x_{0}=&\begin{bmatrix}\beta_{0}&r_{0}&s_{0}\end{bmatrix}^{t},\\\
{x}_{k+1}=&A{x}_{k}+Bu_{k}+Ed_{k},\ \forall k=0,\ldots,N_{p}-1\\\
-\dot{s}_{max}T_{s}\leq&\delta s_{k}\leq\dot{s}_{max}T_{s},\ \forall
k=1,\ldots,N_{p}\\\
-\alpha_{f,max}-\epsilon\leq&\alpha_{f,k}\leq\alpha_{f,max}+\epsilon,\ \forall
k=1,\ldots,N_{p}.\end{split}}$
(16)
The minimized cost is given in Eq. (16), together with the constraints. More
specifically, the state vector is weighted ($W_{x}$), while a penalty on the
control effort ($w_{u}$) avoids numerical instabilities, and the slack
variable is minimized ($w_{\rho}$) to ensure that $\rho=0$ when inside the
constraint limits. Given that the prediction model is locally Linear Time-
Invariant, the Single Shooting Approach can be applied to write the states,
the constraints and the cost metric as a function of the control inputs
(_e.g._ as in (Riva et al., 2022a)). The obtained optimization problem is a
Quadratic Programming one, fastly solved via benchmark tools – qpOASES is used
in this research (Ferreau et al., 2014).
## 3 Twin-in-the-Loop Control
Figure 7: TiL control scheme for yaw-rate tracking.
The TiL scheme of Figure 7 is considered in this Section. The architecture
features the MPC controller of Section 2.3, a DT and a compensator
$C_{\delta}$.
As stated in (Dettù et al., 2023b), under the hypothesis that the simulator is
a faithful replica of the real system, $C_{\delta}$ controls a small signal
dynamic, as the open-loop steering action from the DT $\tilde{s}_{cmd}$ does
most of the job. Hence, we consider a simple PID compensator. Leveraging
findings in (Gimondi et al., 2021), we employ here a mixed yaw-rate sideslip
control, whereas the regulator $C_{\delta}$ controls a mixed signal, rather
than the yaw-rate per-se
$\epsilon=(1-\zeta)r-\zeta\beta,\ \zeta\in\left[0,1\right]$ (17)
Introducing a correction onto the yaw-rate via $\beta$ yields better results
in case of very high cornering stiffness values, and allows to account for
$\beta$ within the Single Input Single Output controller. The control law, in
continuous time ($s$ denotes the Laplace domain variable), reads
$C_{\delta}\left(s,\theta\right)=k_{p}\left(1+\dfrac{1}{sT_{I}}+\dfrac{sT_{D}}{1+sT_{D}/N_{D}}\right).$
(18)
$\theta=\left\\{k_{p},T_{I},T_{D}\right\\}$ is the vector of tuning knobs, and
the high-frequency pole is set ten times faster ($N_{D}=10$) than $T_{D}$, as
in common practice. The controller is discretized via the Tustin approach,
with sampling time $T_{s}=0.01s$.
Following the peak force limit considerations in Section 2.3, we enforce a
saturation on the actuated steer angle
$s^{lb}_{max,d}-\tilde{s}_{cmd}\leq s_{\delta}\leq
s^{ub}_{max,d}-\tilde{s}_{cmd}.$ (19)
Whereas $s^{lb}_{max,d},s^{ub}_{max,d}$ are obtained by substituting in Eq.
(3) the maximum sideslip angle $\alpha_{f,max}$, used in the MPC constraints.
### 3.1 $C_{\delta}$ tuning
To calibrate $C_{\delta}$, we proposed an optimization problem, solved via
Bayesian Optimization (BO) in (Dettù et al., 2023b): this is necessary given
that the block controls the unknown residual dynamic between the vehicle and
its replica. However, performing experiments with black-box proposed
parameters might lead to instability or unsafe configurations; we introduce
here a constraint on $\beta$, to be satisfied during the optimization process.
This guarantees that while exploring the search space, the algorithm stays
away from high sideslip values — note that using the sideslip as an indicator
of vehicle maneuverability and safety is common in the literature (Gimondi et
al., 2021; Beal and Gerdes, 2012).
$\begin{split}\theta^{*}=&\operatorname*{arg\,min}_{\theta}f_{bo}\left(\theta\right)\\\
\textrm{s.t.}\ \ \ &\theta\subseteq\Theta\in\mathbb{R}^{N_{\theta}}\\\
&g_{c}\left(\theta\right)\geq 0\end{split}$ (20)
Where
$g_{c}\left(\theta\right)=-\left|\beta\left(\theta\right)\right|+\beta_{max}$
is the constraint. The cost function $f_{bo}$ reads
$f_{bo}(\theta)=\dfrac{1}{N_{exp}}\sum_{k=1}^{N_{exp}}\left(\tilde{\epsilon}_{k}-\epsilon_{k}\left(\theta\right)\right)^{2}+\gamma_{u}\left(\dot{{s}}_{cmd,k}\right)^{2}$
(21)
Where the first term on the right hand side weights the controller tracking
performance, and the second one weights the control action derivative,
similarly to what done in Eq. (16) for the MPC. $N_{exp}$ is the number of
samples in the considered experiment. As for the cost function, also the
constraint on $\beta$ is a black-box, and needs to be adequately treated.
Constrained BO (Khosravi et al., 2022) can been used to solve the problem. CBO
estimates the black-box constraint via a Gaussian Process, and the
availability of a set of data. Analogously to cost function regression, the
optimizer iteratively adjusts its knowledge of the constraint. In practice, at
each $n$-th iteration, the black-box optimizer evaluates
$\theta^{\left(n\right)}$, and the tuple
$\left(\theta^{\left(n\right)},f_{bo}^{\left(n\right)},g^{\left(n\right)}_{c}\right)$
is collected and used to increment the dataset
$\mathcal{X}^{n}=\mathcal{X}^{n-1}\cup\left(\theta^{\left(n\right)},f_{bo}^{\left(n\right)},g^{\left(n\right)}_{c}\right)$.
The dataset contains data about the unknown cost function and constraint,
allowing estimation of the same.
Although BO proved to be effective in solving the problem, it still requires
many iterations to converge to an optimal solution; given that performing an
experiment with the vehicle might be costly, we would like to assess whether a
faster convergence can be achieved by selecting different optimization tools.
Also, the high computational burden of BO is one important issue, and has been
recently tackled (Sabug Jr et al., 2022).
#### 3.1.1 SMGO-$\Delta$
Set-Membership Global Optimization-$\Delta$ (SMGO-$\Delta$) is a novel data-
driven optimization, featuring unknown constraints (Sabug Jr et al., 2022).
Given a set of samples of the objective and constraints, SMGO-$\Delta$ uses
the existing information to build a surrogate Set Membership-based model of
the hidden functions, and iteratively selects the next point to evaluate. Such
a model is used to strategically decide between performing an exploitation
sampling in the vicinity of the current best sample, or exploration sampling
around the search space to discover the function shape. SMGO-$\Delta$ can be
used to solve a problem of the type in Eq. (20). We give here some details of
the algorithm functioning; the reader is referred to (Sabug Jr et al., 2022)
for detailed information.
By iteratively collecting data about the cost function and constraints as
detailed above, SMGO calculates Lipschitz constant
$\gamma_{f,g_{c}}^{\left(n\right)}$ and noise bound
$\varepsilon_{f,g_{c}}^{\left(n\right)}$ estimates for both $f_{bo}$ and
$g_{c}$ are computed and then used to estimate upper and lower confidence
bounds — note that BO attempts at fitting a surrogate of $f_{bo}$ and $g_{c}$,
while SMGO mostly considers the confidence bounds, thus being much lighter. As
an example, for $f_{bo}$, we have that
$\begin{split}\overline{f}_{bo}^{\left(n\right)}\left(\theta\right)=&\min_{k=1,\ldots,n}\left(f_{bo}^{\left(k\right)}+\gamma^{\left(n\right)}\left|\left|\theta-\theta^{\left(k\right)}\right|\right|\right),\\\
\underline{\smash{f}}_{bo}^{\left(n\right)}\left(\theta\right)=&\max_{k=1,\ldots,n}\left(f_{bo}^{\left(k\right)}-\gamma^{\left(n\right)}\left|\left|\theta-\theta^{\left(k\right)}\right|\right|\right).\end{split}$
(22)
The upper and lower bounds are then used to compute the mean value
$\tilde{f}_{bo}^{\left(n\right)}$ and the associated uncertainty
$\lambda^{(n)}\left(\theta\right)$
$\begin{split}\lambda^{(n)}\left(\theta\right)=&\overline{f}_{bo}^{(n)}\left(\theta\right)-\underline{\smash{f}}_{bo}^{(n)}\left(\theta\right),\\\
\tilde{f}_{bo}^{(n)}\left(\theta\right)=&\dfrac{1}{2}\left(\overline{f}_{bo}^{(n)}\left(\theta\right)+\underline{\smash{f}}_{bo}^{(n)}\left(\theta\right)\right)\end{split}$
(23)
The algorithm then selects the next point to evaluate $\tilde{\theta}^{(n)}$
based on the following problem
$\begin{split}\tilde{\theta}^{(n)}=&\operatorname*{arg\,min}_{\theta\in\mathcal{S}^{(n)}}\tilde{f}_{bo}^{(n)}\left(\theta\right)-\beta\lambda^{(n)}\left(\theta\right)\\\
\textrm{s.t.}\ \
&\Delta\tilde{g}^{(n)}\left(\theta\right)+\left(1-\delta\right)\underline{\smash{g}}^{(n)}\left(\theta\right)\geq
0\end{split}$ (24)
Whereas $\beta$ trades off between exploitation ($\beta=0$) and exploration
($\beta\geq 0$), and $\Delta$ trades off between cautiousness ($\delta=0$) and
riskiness ($\delta=1$) in constraint exploration. The candidate point
$\tilde{\theta}^{(n)}$ is then used to assess the inequality
$\underline{\smash{f}}_{bo}^{(n)}\left(\tilde{\theta}^{(n)}\right)\leq
f_{bo}^{*\left(n\right)}-\alpha_{smgo}\gamma^{(n)}_{f}$ (25)
If Eq. (25) above is satisfied, the algorithm effectively employs
$\tilde{\theta}^{(n)}$ as the next point to evaluate, otherwise, the
exploration routine is triggered, which aims at reducing the uncertainty (the
right hand side of Eq. (25)) while satisfying the constraints.
#### 3.1.2 Virtual Reference Feedback Tuning
Figure 8: Reworked TiL control scheme, for VRFT implementation.
VRFT is a direct data-driven method for the design of Single-Input Single-
Output (SISO) parametric controllers, that does not require plant knowledge
(Campi et al., 2002). It selects a controller, parametrized by $\theta$, by
minimizing the following cost (whereas the controller is linear in this setup)
$\theta^{*}_{vrft}\left(\theta\right)=\operatorname*{arg\,min}_{\theta}\left|\left|\left(\dfrac{P\left(z\right)C_{\delta}\left(\theta\right)}{1+P\left(z\right)C_{\delta}\left(\theta\right)}-M_{r}\left(z\right)\right)M_{w}\left(z\right)\right|\right|_{2}^{2}.$
(26)
The cost in Eq. (26) means that VRFT tries to find a controller matching in
the frequency domain a discrete-time reference model $M_{r}(z)$, given a
weighting function $M_{w}(z)$. Indeed, theoretical guarantees on Eq. (26) only
apply to linear plants: in our specific case, no linearity assumption can be
drawn based on the plant in Fig. 8 (the plant encompasses both the digital
twin and its interaction with the controller). However, as showed in (Dettù et
al., 2023b), if the digital twin is a good replica of the real system, we can
reasonably assume that the dynamics to be controlled is a small signal one.
Nonetheless, VRFT has been succesfully used in many real systems, where the
”linearity” concept looses its meaning (Busetto et al., 2023; Radrizzani et
al., 2020; Passenbrunner et al., 2014).
Being the plant unknown, Eq. (26) is not usable; only sequences of input
$\left(s_{\delta,1},\ldots,s_{\delta,N_{exp}}\right)$ and output
$\left(y_{\epsilon,1},\ldots,y_{\epsilon,N_{exp}}\right)$ data are available
(an open-loop experiment suffices). Hence, the cost function in Eq. (26) is
suitably rewritten as a function of available data only (Care et al., 2019)
$f_{vrft}\left(\theta\right)=\dfrac{1}{N_{exp}}\sum_{k=1}^{N_{exp}}M_{w}(z)\left(C_{\delta}\left(\theta\right)\left(M_{r}(z)^{-1}-1\right)y_{\epsilon,k}-s_{\delta,k}\right).$
(27)
In the TiL setup, regulator $C_{\delta}$ controls the error
$y_{\epsilon}=\tilde{\epsilon}-\epsilon$, and regulates it to zero.
#### 3.1.3 Comparing VRFT with global optimization
The obtained control parameters are then prone to be used as an initial guess
for a successive iterative algorithm (_e.g._ as in (Busetto et al., 2023)). As
one could note, the cost function of VRFT Eq. (27) is markedly different from
the one showed in Eq. (20). VRFT requires a reference model $M_{r}(z)$, and
cannot arbitrarily learn a controller minimizing the tracking error, as for
global optimizers. Apart from the reference model, to enhance performance is
often necessary to consider a $M_{w}(z)$, reasonably setting the frequencies
of interest. Nonetheless, we will show further on as the solution found by
VRFT is well performing if evaluated onto the cost of Eq. (21).
Also, global optimizers allow for an arbitrary cost function, _e.g._ taking
into account the control effort (see Eq. (20)): this is not possible in
classic VRFT. For a fair comparison of VRFT with BO, one could execute the
optimization routine described in Section 3.1 by using the VRFT cost in Eq.
(27), which in practice means pre-filtering the data according to $M_{r}(z)$
and $M_{w}(z)$, and removing the weight onto $\dot{s}_{act}$. VRFT is here
applied onto the tracking error $y_{\epsilon}$, and requires a virtual
reference to be passed through the desired model $M_{r}(z)$. However, the
reference for the error should be zero in a closed-loop setup like the one
considered in BO and SMGO, for this reason, the global optimizers cost
function is somewhat different in nature, in that it minimizes the $rms$ of
the error in a specific closed-loop experiment, with zero reference.
$f_{bo-
vrft}(\theta)=\dfrac{1}{N_{exp}}\sum_{k=1}^{N_{exp}}M_{w}(z)M_{r}(z)y_{\epsilon,k}^{2}$
(28)
## 4 Simulation results
In the following we conduct a series of simulation tests to assess the
performance of TiL lateral control onto the vehicle, while also analysing and
comparing the different optimization tools described above.
### 4.1 Simulation setup
#### 4.1.1 The vehicle and the digital twin
$M\ [kg]$ $J_{zz}\ [kgm^{2}]$ $L_{f,r}\ [m]$ $\mathcal{A}_{f,r}$
$\mathcal{B}_{f,r}$ $1729.1$ $2482.7$ $1.48,\ 1.16$ $10.72,\ 19.75$ $1.51,\
0.75$ $\mathcal{C}_{f,r}$ $k_{a}^{f,r}\ [kg/m]$ $k_{x}\ [kg]$ $\dot{s}_{max}\
[deg/s]$ $\alpha^{f}_{max}\ [deg]$ $20.08,\ 28.69$ $0.065,\ 0.221$ $153.63$
$100$ $9.10$
Table 1: Set of vehicle parameters for the simplified model of Section 2.1.
For testing the TiL controller, since we do not have the possibility of
performing experiments on an actual vehicle, a different instance of the
Digital Twin is generated and regarded as the "Physical Vehicle". The
considered Digital Twin is modeled in VI-CarRealTime (an off-the-shelf
commercial simulation software (VI-Grade, 2022)). It models an high-
performance two seats car, and the most relevant parameters for the problem
under analysis are given in Table 1; they have been provided by partner
company Ferrari S.p.A. Note that the displayed parameters are those of the
simplified model derived in Section 2.1 and used within the MPC model. The
true underlying model features much more parameters; _e.g._ version 5 of the
Magic Formula is used for modelling the tire-road interaction, while we used a
3-parameters approximation of it within the MPC, encompassing lateral forces
only.
We assume that the real vehicle features some different elements and non-
idealities, which are absent in the Digital Twin:
* 1.
The vehicle has real sensors, characterized by noise. Zero-mean Gaussian
distributed noise is added onto the yaw-rate measure $r$, characterized by
standard deviation $\sigma_{n,r}=0.006\ rad/s$. Similarly for the sideslip, we
mimic the presence of state-estimator, introducing a low-pass filtered noise
with standard deviation $\sigma_{n,\beta}=0.0044\ rad$. Overall, this yields
an $snr$ of $\approx 20$ in the considered signals, and the comparison between
real and noisy $\beta$ is similar to what obtained in recent works on the
topic (Carnier et al., 2023);
* 2.
Additional masses are modelled in the real vehicle. Specifically, we add a
$100\ kg$ passenger and an unbalanced load on the front trunk ($70\ kg$ on the
left, $10\ kg$ on the right). The layout is the same considered in (Dettù et
al., 2023b);
* 3.
We reduce the rear wheels cornering stiffness by the $15\ \%$, so as to mimic
a wrong tire model within the model based controller, or a modification of the
tire property due _e.g._ to low inflation pressure.
The modifications above reproduce a realistic case study, on which the
optimization is run and the controllers validated.
VI-CarRealTime is provided with a virtual driver feature, which is here used
to maintain a given speed profile (working as a longitudinal dynamics
controller): indeed, depending on the vehicle lateral dynamics, slightly
different speed profile tracking might be achieved
#### 4.1.2 Parameters selection
We here briefly describe the parameters selection for the various controllers
and optimization algorithms used. Concerning the MPC, the following weights
are chosen; they have been found through fine tuning onto the Digital Twin
$w_{u}=1,\ w_{\rho}=100,\ w_{\beta}=0.2,\ w_{r}=0.8,\ w_{s}=0.$ (29)
The same are used within the TiL control scheme of Fig. 7. In $C_{\delta}$,
the parameter $\zeta$ (see Eq. (17)) trades off yaw-rate tracking for sideslip
minimization. It is set to $0.2$ so as to focus on the first objective, while
avoiding unsafe sideslip increments.
With regards to SMGO, $\Delta=0.5$ is used, to guarantee a balanced trade-off
between cautiousness and riskiness in constraint exploring. Other tuning
parameters for SMGO are let at their default values (Sabug Jr et al., 2022).
With regards to CBO, default values, the exploration ratio is set to $0.5$,
and the acquisition function is set as the Expected Improvement one.
$\beta_{max}$ is set at $4.5\ deg$, which is $\approx 2.5\ deg$ more than what
achieved by using the nominal MPC in the optimization maneuver: we want not to
significantly exceed this limit while training the controllers onto the same
maneuver. Indeed, this is not an issue for different maneuvers, where the
sideslip might naturally exceed this limit also in nominal conditions, and
$\beta_{max}$ has to be chosen for the specific maneuver of interest.
VRFT requires tuning of reference model $M_{r}(z)$ and frequency weighting
function $M_{w}(z)$. The first one is taken from the literature (Gimondi et
al., 2021), where a controller for yaw-rate dynamics is closed at the
bandwidth of $3.5\ Hz$: we thus set it as a first-order low-pass filter, with
one pole at this frequency. The frequency range is instead taken from the
reference generation filter (see Section 4); since the reference is cut after
$f_{ref}=6.3\ Hz$, we assign the same dynamics to $M_{w}(z)$. Of course, both
reference model and weight function are suitably discretized via the Tustin
approach.
### 4.2 Optimization results comparison
Figure 9: Input and output signals for VRFT use. Figure 10: Comparison of
black-box optimization algorithms for TiL calibration. Figure 11: Comparison
of black-box optimization algorithms for TiL calibration. Highlight on the
cost function.
Controller | $rms(\tilde{r}-r)$ $\left[deg/s\right]$ | $rms(\tilde{\beta}-\beta)$ $\left[deg\right]$ | $rms(\dot{s}_{act})$ $\left[deg/s\right]$
---|---|---|---
MPC | $1.85$ | $1.81$ | $918.79$
TiL-SMGO-$\Delta$ | $0.74$ | $0.86$ | $514.41$
TiL-VRFT | $1.06$ | $0.95$ | $475.47$
TiL-SMGO-$\Delta$ (VRFT cost) | $1.04$ | $0.89$ | $935.42$
Table 2: Performance indices of the controllers evaluated in the optimization
test (double-lane-change with step-steer at $120\ km/h$).
For the optimization of $C_{\delta}$, we consider a double-lane change
maneuver, followed by a step steer, executed at $120\ km/h$.
To use VRFT, we execute the maneuver onto the digital twin (_i.e._ on the
upper branch in Fig. 7), while feeding the physical vehicle with a Pseudo
Random Binary Input signal (as in Radrizzani et al. (2020)), to excite the
dynamics to be controlled. These information is displayed in Fig. 9; the upper
plot depicts the command from DT $\tilde{s}_{cmd}$ and the one fed in open-
loop $s_{\delta}$. The lower plot depicts the system outputs $r$ and $\beta$,
as well as their combination $\epsilon$.
The learned controller is then used as a prior for global optimization; each
optimization iterates for $60$ steps. To minimize the effect of randomization
present in both CBO and SMGO-$\Delta$, we perform each optimization $10$
times. The results are displayed in Fig. 10: in the upper plot, the optimal
cost found at each iteration (averaged per the $10$ optimization runs) is
displayed.
The lower plot instead displays the number of infeasible experiments for each
controller, at each iteration, _i.e._ breaking the constraint on $\beta$. From
the upper plot, we note as VRFT is capable of delivering an almost optimal
solution, with just one experiment; BO and SMGO are able to approach VRFT cost
after $>10$ iterations on average. Figure 11 shows an highlight of the upper
plot: SMGO provides the better results in terms of cost function minimization,
while CBO is somewhat less performing. In general, both optimizers fail at
converging at the best point when VRFT prior is not provided. It is
interesting to note that these results confirm the findings in (Busetto et
al., 2023), for a different application.
If one looks at the bottom plot is clear as the increased performance achieved
with SMGO is traded with somewhat less cautious exploration, in that the
number of unfeasible attempts grows. This is consistent to what found in
(Busetto et al., 2023); when provided with VRFT prior, SMGO fastly switches
from exploitation mode to exploration mode (as the global minimum is in
practice already achieved by using VRFT), and this eventually translates into
breaking the constraints more easily.
Concerning the computation time, at each iteration CBO takes on average
333These results have been obtained with MATLAB 2022a, on a $16GB$ RAM Asus
Laptop, with an Intel Core i7-8750H $2.20GHz$ processor. $0.4442\ s$ –with
$0.0961\ s$ of standard deviation– to compute the next point to be evaluated;
SMGO-$\Delta$ is on the other hand significantly faster, taking $0.0238\ s$ on
average, with $0.0017\ s$ of standard deviation. In practice, SMGO-$\Delta$ is
almost two orders of magnitude faster than CBO for this optimization problem –
consistently to what found in (Sabug Jr et al., 2022), and this potentially
allows for the whole tuning procedure to be run on low cost computing units
and at real-time.
### 4.3 Time-domain results comparison
In the following, we compare the controllers optimized as above in the time-
domain. Being CBO and SMGO-$\Delta$ calibrations very similar, we show here
the second one. Also, given that the two global optimizers are able to
converge only by using VRFT control parameters as a prior, we consider the
corresponding SMGO-$\Delta$ calibration initialized with VRFT. We then show
the results obtained via VRFT calibration, and for the optimization maneuver,
we also show what would happen if explicitly using the VRFT cost function
within SMGO-$\Delta$.
#### 4.3.1 Double lane change with step steer - $120\ km/h$
Figure 12: Controller validation onto the optimization experiment: yaw-rate
and sideslip tracking. Figure 13: Controller validation onto the optimization
experiment: speed and steer profiles.
Controller | $rms(\tilde{r}-r)$ $\left[deg/s\right]$ | $rms(\tilde{\beta}-\beta)$ $\left[deg\right]$ | $rms(\dot{s}_{act})$ $\left[deg/s\right]$
---|---|---|---
MPC | $1.65$ | $1.46$ | $921.92$
TiL-SMGO-$\Delta$ | $0.75$ | $1.07$ | $462.35$
TiL-VRFT | $1.17$ | $1.25$ | $424.56$
Table 3: Performance of the controllers in the first validation test (double-
lane-change with step-steer at $140\ km/h$).
We show here the time-domain results when considering the optimization test,
_i.e._ the double lane change followed by a step steer, at $120\ km/h$. Figure
12 depicts the vehicle yaw-rate (upper plot) and sideslip angle (bottom plot).
The black dashed line $\tilde{r}$ is the reference behaviour obtained in
nominal conditions by using MPC onto the digital twin (see the scheme of Fig.
7). The nominal MPC exhibits some issues in the final part of the experiment,
and fails at maintaining the sideslip angle at an acceptable level, while also
badly tracking the yaw-rate. On the other hand, the TiL enhanced controller is
capable of smoothly recovering the system behaviour and guaranteeing good yaw-
rate tracking and minimization of the sideslip. Note that perfectly ”tracking”
the sideslip is not possible (nor it is required), since the cornering
stiffness is different and the physical vehicle has a different steady-state
response.
Looking at the performance obtained with VRFT in the time-domain confirms that
the method is adequate for learning a TiL controller, and significantly eases
the tuning phase, at the cost of a minimum performance reduction (note that
VRFT is achieving in a single iteration what BO or SMGO are able to do only
after 10 or 15 experiments). When considering SMGO with VRFT cost (_i.e._
filtering the output errors within the cost function), we note an oscillating
response. This is explainable, as the pre-filtering applied onto the measured
signals during the training process ”hides” some high- frequencies, and SMGO
is in fact over-fitting the controller onto a filtered version of the system,
poorly performing outside of that frequency range.
For the same experiment, Fig. 13 depicts speed (constant) and steer profiles
when using the same controllers. What observed from the oscillating yaw-rate
response in the SMGO with VRFT cost is further noted by the steer profile,
which is particularly aggressive. $s_{ref}$ is the driver steer request: one
can note as the controllers cut the request and apply a slightly different
steering angle, so as to guarantee the vehicle stability and avoid spinning
(_i.e._ too high sideslip angles). Finally, Table 2 quantifies the cost
indexes for the given experiment and the controllers.
#### 4.3.2 Double lane change with step steer - $140\ km/h$
Figure 14: Controller validation onto the first testing experiment: yaw-rate
and sideslip tracking. Figure 15: Controller validation onto the first testing
experiment: speed and steer profiles.
We show here the time-domain results when considering a double lane change
followed by a step steer, at $140\ km/h$. Figure 16 depicts the vehicle yaw-
rate (upper plot) and sideslip angle (bottom plot). As in the previous case,
TiL is able to make the real vehicle track the digital twin: both SMGO and
VRFT calibrations suffices for our purposes, and yield very similar results.
Figure 17 shows the speed and steer profiles in the same test, while Table 4
provides some quantitative metrics.
#### 4.3.3 Chikane test
Figure 16: Controller validation onto the second testing experiment: yaw-rate
and sideslip tracking. Figure 17: Controller validation onto the second
testing experiment: speed and steer profiles.
Controller | $rms(\tilde{r}-r)$ $\left[deg/s\right]$ | $rms(\tilde{\beta}-\beta)$ $\left[deg\right]$ | $rms(\dot{s}_{act})$ $\left[deg/s\right]$
---|---|---|---
MPC | $5.40$ | $3.73$ | $3.09\times 10^{3}$
TiL-SMGO-$\Delta$ | $2.30$ | $1.25$ | $1.39\times 10^{3}$
TiL-VRFT | $3.59$ | $1.92$ | $1.19\times 10^{3}$
Table 4: Performance of the controllers in the second validation test
(chikane)
We show here the time-domain results when considering the second validation
test, _i.e._ a chikane maneuver inspired from the Savelli curve (nr. 7 in
Mugello track). In this case, a realistic speed profile is considered
(displayed in the bottom plot of Fig. 15), with the driver pushing the gas
pedal at the middle of the curve, thus combining lateral with longitudinal
acceleration; this kind of profile if particularly challenging for the
considered MPC controller, which assumes the longitudinal speed is constant in
the prediction horizon. In fact, from Fig. 14, one can note the worsened yaw-
rate tracking performance, with respect to constant speed tests. Also in this
case, the nominal MPC exhibits a much worse behaviour than TiL-enhanced ones.
This is further noted from the steering command, in the upper plot of Fig. 15,
which shows significant oscillations. Table 3 gives the quantitative metrics
for the chikane test: it is interesting to note that although slightly less
performing, VRFT calibration is somewhat less aggressive in terms of
$rms(\dot{s}_{act})$.
## 5 Conclusions
In this paper, we took a challenging case study for the recently proposed
TiL-C architecture, namely, the yaw-rate tracking problem, for which a
baseline Model Predictive Controller is designed, and then enhanced with a
simple PID compensator. A thorough discussion on different tools for
optimizing the compensator gains is carried out: specifically, we compared the
widely popular Bayesian Optimization with the newly proposed Set Membership
Global Optimization. We also consider a philosophically different approach,
the Virtual Reference Feedback Tuning, which is a direct control design
method, requiring just one batch of open-loop data for obtaining the
compensator gains.
The comparisons show as VRFT is capable of getting very close to the global
minimum, with just one experiment, while BO and SMGO need at least $10-15$
tests to overcome it, and fail at converging at the best point if not provided
with the previously computed VRFT controller gains.
The time-domain results validate the calibrated controllers in different
scenarios, considering both constant and varying speed profiles, thus
confirming the effectiveness of TiL-C with respect to the baseline controller,
and showing the amount of improvement achieved by performing some iterations
of BO/SMGO, as compared to VRFT.
## Acknowledgments
We would like to thank companies VI-Grade GmbH and Ferrari S.p.A. for the
technical support provided during the work.
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|
Formalizing Relations in Type Theory]Formalizing Relations in Type Theory
F. Kachapova]Farida Kachapova
Department of Mathematical Sciences
Auckland University of Technology
New Zealand
Type theory plays an important role in foundations of mathematics as a framework for formalizing mathematics and a base for proof assistants providing semi-automatic proof checking and construction. Derivation of each theorem in type theory results in a formal term encapsulating the whole proof process. In this paper we use a variant of type theory, namely the Calculus of Constructions with Definitions, to formalize the standard theory of binary relations. This includes basic operations on relations, criteria for special properties of relations, invariance of these properties under the basic operations, equivalence relation, well-ordering, and transfinite induction. Definitions and proofs are presented as flag-style derivations.
[2020]Primary 03B30; Secondary 03B38
§ INTRODUCTION
First type theories were proposed by B. Russell [10] as a foundation of mathematics. Other important type theories are typed $\lambda$-calculus introduced by A. Church [2] and intuitionistic type theory introduced by P. Martin-Löf [7]. A higher-order typed $\lambda$-calculus known as Calculus of Constructions (CoC) was created by T. Coquand [3]. Variants of CoC make formal bases of proof assistants, which are computer tools for formalizing and developing mathematics. In particular, the well-known proof assistant Coq is based on the strong variant of CoC called the Calculus of Inductive Constructions (CIC).
Here we use the variant $\lambda D$ of CoC developed in [8]; $\lambda D$ is called the Calculus of Constructions with Definitions. We choose $\lambda D$ because of its following useful properties.
– In $\lambda D$, as in other variants of CoC, proofs are expressed as formal terms and thus are incorporated in the system.
– In $\lambda D$ type checking is decidable and therefore proof checking is decidable. So the correctness of a proof can be checked by an algorithm.
– $\lambda D$ is strongly normalizing, which implies the logical consistency of this theory, even with classical logic (when no extra axioms are added) - see [1].
The theory $\lambda D$ is weaker than CIC because $\lambda D$ does not have inductive types. This does not limit its capability for formalizing mathematics because in $\lambda D$ we can use axiomatic approach and higher-order logic to express the objects that CIC defines with inductive types.
In Section 2 we briefly describe the theory $\lambda D$, derived rules of intuitionistic logic in $\lambda D$, and the classical axiom of excluded third that can be added to $\lambda D$ if necessary; we also briefly explain the flag format derivation.
In Section 3 we describe the equality in $\lambda D$ and its derived properties.
In Section 4 we study binary relations in $\lambda D$, operations on relations, and their properties. In Section 5 we formally prove criteria of relexivity, symmetry, antisymmetry and transitivity, and study the invariance of these properties under some basic operations. In Section 6 we formally define partitions in $\lambda D$ and provide a proof of their correspondence with equivalence relations. In Section 6 we also provide an example of partial order with a formal proof, definition of well-ordering in $\lambda D$ and a formal proof of the principle of transfinite induction.
In our formalizations we aim to keep the language and theorems as close as possible to the ones of standard mathematics.
In definitions and proofs we use the flag-style derivation described in [8]. Long formal derivations are moved from the main text to Appendices for better readability.
§ TYPE THEORY $\LAMBDA D$
In [8] Nederpelt and Geuvers developed a formal theory $\lambda D$ and formalized some parts of logic and mathematics in it. Here we briefly describe main features of $\lambda D$.
§.§ Type Theory $\lambda D$
The language of $\lambda D$ described in [8] has an infinite set of variables, $V$, and an infinite set of constants, $C$; these two sets are disjoint. There are also special symbols $\square$ and $*$.
Expressions of the language are defined recursively as follows.
* Each variable is an expression.
* Each constant is an expression.
* Constant * is an expression.
* Constant $\square$ is an expression.
* (Application) If $A$ and $B$ are expressions, then $AB$ is an expression.
* (Abstraction) If $A$, $B$ are expressions and $x$ is a variable, then $\lambda x:A.B$ is an expression.
* (Dependent Product) If $A$, $B$ are expressions and $x$ is a variable, then $\Pi x:A.B$ is an expression.
* If $A_1,A_2,\ldots,A_n$ are expressions and $c$ is a constant, then $c\left(A_1,A_2,\ldots,A_n\right)$ is an expression.
An expression $A\rightarrow B$ is introduced as a particular type of Dependent Product from (7) when $x$ is not a free variable in $B$.
* A statement is of the form $M:N$, where $M$ and $N$ are expressions.
* A declaration is of the form $x:N$, where $x$ is a variable and $N$ is an expression.
* A descriptive definition is of the form:
\[\bar{x}:\bar{A}\rhd c(\bar{x}):=M:N,\]
where $\bar{x}$ is a list $x_1,x_2,\ldots,x_n$ of variables, $\bar{A}$ is a list $A_1,A_2,\ldots,A_n$ of expressions, $c$ is a constant, and $M$ and $N$ are expressions.
* A primitive definition is of the form:
\[\bar{x}:\bar{A}\rhd c(\bar{x}):=\Bot:N,\]
where $\bar{x}$, $\bar{A}$, and $c$ are described the same way as in (3), and $N$ is an expression. The symbol $\Bot$ denotes the non-existing definiens. Primitive definitions are used for introducing axioms where no proof terms are needed.
* A definition is a descriptive definition or a primitive definition.
* A judgement is of the form:
\[\Delta;\Gamma \vdash M:N,\]
where $M$ and $N$ are expressions of the language, $\Delta$ is an environment (a properly constructed sequence of definitions) and $\Gamma$ is a context (a properly constructed sequence of declarations).
For brevity we often use implicit variables in definitions, that is we omit the previously declared variables $\bar{x}$ in $c(\bar{x})$ in (3) and (4).
The following informally explains the meaning of expressions.
* If an expression $M$ appears in a derived statement of the form $M$ $:*$, then $M$ is interpreted as a type, which represents a set or a proposition.
Note: There is only one type $*$ in $\lambda D$. But informally we often use $*_p$ for propositions and $*_s$ for sets to make proofs more readable.
* If an expression $M$ appears in a derived statement of the form $M:N$, where $N$ is a type, then $M$ is interpreted as an object at the lowest level.
When $N$ is interpreted as a set, then $M$ is regarded as an element of this set.
When $N$ is interpreted as a proposition, then $M$ is regarded as a proof (or a proof term) of this proposition.
* The symbol $\square$ represents the highest level.
* Sort is $*$ or $\square$. Letters $s, s_1,s_2,\ldots$ are used as variables for sorts.
* If an expression $M$ appears in a statement of the form $M:\square$, then $M$ is called a kind.
$\lambda D$ contains the derivation rule:
\[\varnothing;\varnothing \vdash *:\square,\]
which is its (only) axiom because it has an empty environment and an empty context.
Further details of the language and derivation rules of the theory $\lambda D$ can be found in [8]. Judgments are formally derived in $\lambda D$ using the derivation rules.
§.§ Flag Format of Derivations
The flag-style deduction was introduced by Jaśkowski [5] and Fitch [4]. A derivation in the flag format is a linear deduction. Each "flag" (a rectangular box) contains a declaration that introduces a variable or an assumption; a collection of already introduced variables and assumptions makes the current context. The scope of the variable or assumption is established by the "flag pole". In the scope we construct definitions and proof terms for proving statements/ theorems in $\lambda D$. Each new flag extends the context and at the end of each flag pole the context is reduced by the corresponding declaration.
For brevity we can combine several declarations in one flag.
More details on the flag-style deduction can be found in [9] and [8].
§.§ Logic in $\lambda D$
The rules of intuitionistic logic are derived in the theory $\lambda D$ as shown in [8]. We briefly describe it here by showing the introduction and elimination rules for logical connectives and quantifiers.
§.§.§ Implication
The logical implication $A\Rightarrow B$ is identified with the arrow type $A\rightarrow B$. The rules for implication follow from the following general rules for the arrow type (we write them in the flag format):
*A:s_1 | B:s_2
*u:A→B | v:A
*λx:A.M : A→B
Here $x$ is not a free variable in $B$.
In $\lambda D$ arrows are right associative, that is $A\rightarrow B\rightarrow C$ is a shorthand for $A\rightarrow (B\rightarrow C)$.
§.§.§ Falsity and Negation
Falsity $\bot$ is introduced in $\lambda D$ by:
\[\bot:=\Pi A:*_p.A\;:\;*_p.\]
From this definition we get a rule for falsity:
The rule states that falsity implies any proposition.
As usual, negation is defined by: $\neg A:= A\rightarrow \bot$.
Other logical connectives and quantifiers are also defined using second order encoding. Here we only list their derived rules and names of the corresponding terms, without details of their construction.
The exact values of the terms can be found in [8].
Some of our flag derivations contain the proof terms that will be re-used in other proofs; such proof terms are written in bold font, e.g. $\boldsymbol{\wedge}\textbf{-in}$ in the first derived rule for conjunction as follows.
§.§.§ Conjunction
These are derived rules for conjunction $\wedge$:
*u:A | v:B
*∧-in(A,B,u,v) : A∧B
*∧-el_1(A,B,w) : A
*∧-el_2(A,B,w) : B
§.§.§ Disjunction
These are derived rules for disjunction $\vee$:
*∨-in_1(A,B,u) : A∨B
*∨-in_2(A,B,u) : A∨B
*u:A∨B | v:A⇒C | w:B⇒C
*∨-el(A,B,C,u,v,w) : C
§.§.§ Bi-implication
Bi-implication $\Leftrightarrow$ has the standard definition:
\[(A\Leftrightarrow B):=(A\Rightarrow B)\wedge (B\Rightarrow A).\]
We will often use this lemma to prove bi-implication $A\Leftrightarrow B$.
*u:A⇒B | v:B⇒A
*bi-impl(A,B,u,v):=∧-in(A⇒B,B⇒A,u,v) : A⇔B
§.§.§ Universal Quantifier
The universal quantifier $\forall$ is defined through the dependent product:
*S:*_s | P:S→*_p
*Definition ∀(S,P) :=Πx:S.Px : *_p
* Notation: (∀𝐱:𝐒.𝐏𝐱) for ∀(S,P)
§.§.§ Existential Quantifier
These are derived rules for the existential quantifier $\exists$.
*S:*_s | P:S→*_p
*y:S | u:Py
*∃-in(S,P,y,u) : (∃x:S.Px)
*u:(∃x:S.Px) | v:(∀x:S.(Px⇒C))
*∃-el(S,P,u,C,v) : C
Here $x$ is not a free variable in $C$.
§.§.§ Classical Logic
We use mostly intuitionistic logic. But sometimes classical logic is needed; in these cases we add the following Axiom of Excluded Third:
*exc-thrd(A):= : A∨A
This axiom implies the Double Negation theorem:
§.§ Sets in $\lambda D$
Here we briefly repeat some definitions from [8]
relating to sets, in particular, subsets of type $S$.
*ps(S):=S→*_pPower set of S
{x:S | xεV} for λx:S.Vx
*x: S
*Notation: xε_S V or xεV for element(S,x,V)
Thus, a subset $V$ of $S$ is regarded as a predicate on $S$ and $x\varepsilon V$ means $x$ satisfies the predicate $V$.
§ INTENSIONAL EQUALITY IN $\LAMBDA D$
Here we introduce intensional equality for elements of any type; we will call it just equality. In the next section we will introduce extensional equality and the axiom of extensionality relating the two types of equality.
*x,y: S
*eq(S,x,y):=ΠP:S→*_p. (Px⇒Py):*_p
* Notation: x=_Sy for eq(S,x,y)Intensional equality
§.§ Properties of Equality
§.§.§ Reflexivity
The following diagram proves the reflexivity property of equality in $\lambda D$.
*S:* | x: S
*P: S→*_p
Proof terms are constructed similarly for the following properties of Substitutivity, Congruence, Symmetry, and Transitivity (see [8]).
§.§.§ Substitutivity
Substitutivity means that equality is consistent with predicates of corresponding types.
*P: S→*_p
*x,y: S | u:x=_Sy | v:Px
§.§.§ Congruence
Congruence means that equality is consistent with functions of corresponding types.
*f: Q→S
*x,y: Q | u:x=_Qy
§.§.§ Symmetry
The following diagram expresses the symmetry property of equality in $\lambda D$.
*x,y: S | u:x=_Sy
§.§.§ Transitivity
The following diagram expresses the transitivity property of equality in $\lambda D$.
*x,y,z: S | u:x=_Sy | v:y=_Sz
§ RELATIONS IN TYPE THEORY
§.§ Sets in $\lambda D$
Here we briefly repeat some definitions from [8]
relating to sets, in particular, subsets of type $S$.
*ps(S):=S→*_pPower set of S
{x:S | xεV} for λx:S.Vx
*x: S
*Notation: xε_S V or xεV for element(S,x,V)
Thus, a subset $V$ of $S$ is regarded as a predicate on $S$ and $x\varepsilon V$ means $x$ satisfies the predicate $V$.
§.§ Defining Binary Relations in $\lambda D$
Binary relations are introduced in [8], together with the properties of reflexivity, symmetry, antisymmetry, and transitivity, and definitions of equivalence relation and partial order. We use them as a starting point for formalizing the theory of binary relations in $\lambda D$.
A relation on $S$ is a binary predicate on $S$, which is regarded in $\lambda D$ as a composition of unary predicates. For brevity we introduce the type $br(S)$ of all binary relations on $S$:
*S: *_s
*Definition br(S):= S→S→*_p : □
In the rest of the article we call binary relations just relations.
The equality of relations and operations on relations are defined similarly to the set equality and set operations.
Next we define the extensional equality of relations vs the intentional equality introduced in the previous section.
*S: *_s
*Definition ⊆(S,R,Q) :=(∀x,y:S.(Rxy⇒Qxy)) : *_p
* Notation: R⊆Q for ⊆(S,R,Q)
*Definition Exeq(S,R,Q) :=R⊆Q∧Q⊆R : *_p
* Notation: R=Q for Exeq(S,R,Q) Extensional equality
We add to the theory $\lambda D$ the following axiom of extensionality for relations.
*S: *_s
*extaxiom(S,R,Q,u) := : R=_br(S)QExtensionality Axiom
The axiom is introduced in the last line by a primitive definition with the symbol $\Bot$ replacing a non-existing proof term.
The Extensionality Axiom states that the two types of equality are the same for binary relations. So we will use the symbol = for both and we will not elaborate on details of applying the axiom of extensionality when converting one type of equality to the other.
§.§ Operations on Binary Relations
Using the flag format, we introduce the identity relation $id_S$ on type $S$ and converse $R^{-1}$ of a relation $R$.
*S: *_s
*Definition id_S :=λx,y:S.(x=_Sy) : br(S)Identity relation
*Definition conv(S,R) :=λx,y:S.(Ryx) : br(S)
* Notation: R^-1 for conv(S,R)Converse relation
Next we introduce the operations of union $\cup$, intersection $\cap$, and composition $\circ$ of relations.
*S: *_s
*Definition ∪(S,R,Q) :=λx,y:S.(Rxy∨Qxy) : br(S)
* Notation: R∪Q for ∪(S,R,Q)Union
*Definition ∩(S,R,Q) :=λx,y:S.(Rxy∧Qxy) : br(S)
* Notation: R∩Q for ∩(S,R,Q)Intersection
*Definition ∘(S,R,Q) :=λx,y:S.(∃z:S. (Rxz∧Qzy)) : br(S)
* Notation: R∘Q for ∘(S,R,Q)Composition
§.§ Properties of Operations
The following two technical lemmas will be used in some future proofs.
This lemma gives a shortcut for constructing an element of a composite relation.
*S: *_s | R,Q:br(S) | x,y,z:S
*u:Rxy | v:Qyz
*a:=∧-in (Rxy,Qyz,u,v) : Rxy∧Qyz
*prod-term (S,R,Q,x,y,z,u,v):=∃-in (S,λt.Rxt∧Qtz,y,a) : (R∘Q)xz
This lemma gives a shortcut for proving equality of two relations.
*S: *_s | R,Q:br(S)
*u:R⊆Q | v:Q⊆R
*relequal (S,R,Q,u,v):=∧-in (R⊆Q,Q⊆R,u,v) : R= Q
For relations $R,P$ and $Q$ on $S$ the following hold.
1) $(R^{-1})^{-1}=R.$
2) $(R\circ Q)^{-1}=Q^{-1}\circ R^{-1}.$
3) $(R\cap Q)^{-1}=R^{-1}\cap Q^{-1}.$
4) $(R\cup Q)^{-1}=R^{-1}\cup Q^{-1}.$
5) $R\circ(P\cup Q)=R\circ P\cup R\circ Q.$
6) $(P\cup Q)\circ R=P\circ R\cup Q\circ R.$
7) $R\circ(P\cap Q)\subseteq R\circ P\cap R\circ Q.$
8) $(P\cap Q)\circ R\subseteq P\circ R\cap Q\circ R.$
9) $(R\circ P)\circ Q=R\circ(P\circ Q).$
The formal proof is in part A of Appendix. The proof of part 2) has the form:
*S: *_s | R,Q:br(S)
*convprod(S,R,Q):=… : (R∘Q)^-1=Q^-1∘R^-1
Its proof term $conv\mhyphen prod(S,R,Q)$ will be re-used later in the paper.
§ PROPERTIES OF BINARY RELATIONS
The properties of reflexivity, symmetry, antisymmetry, transitivity, and the relations of equivalence and partial order are defined in [8] as follows.
*S:*_s | R:br(S)
*Definition refl(S,R) :=∀x:S.(Rxx) : *_p
*Definition sym(S,R) :=∀x,y:S.(Rxy⇒Ryx) : *_p
*Definition antisym(S,R) :=∀x,y:S.(Rxy⇒Ryx⇒x=_Sy) : *_p
*Definition trans(S,R) :=∀x,y,z:S.(Rxy⇒Ryz⇒Rxz) : *_p
*Definition equiv$-$relation(S,R) :=refl(S,R)∧sym(S,R)
∧ trans(S,R) : *_p
*Definition part$-$ord(S,R) :=refl(S,R)∧antisym(S,R)
∧ trans(S,R) : *_p
Suppose $R$ is a relation on type $S$. Then the following hold.
1) Criterion of reflexivity.
$R$ is reflexive $\Leftrightarrow id_S\subseteq R$.
2) First criterion of symmetry.
$R$ is symmetric $\Leftrightarrow R^{-1}\subseteq R$.
3) Second criterion of symmetry.
$R$ is symmetric $\Leftrightarrow R^{-1}= R$.
4) Criterion of antisymmetry.
$R$ is antisymmetric $\Leftrightarrow R^{-1}\cap R\subseteq id_S$.
5) Criterion of transitivity. $R$ is transitive $\Leftrightarrow R\circ R\subseteq R$.
The formal proof is in part B of Appendix. The proof of part 3) has the form:
*S: *_s | R:br(S)
*symcriterion(S,R):=… : sym(S,R)⇔R^-1=R
Its proof term $sym\mhyphen criterion(S,R)$ will be re-used later in the paper.
Relation $R$ on $S$ is reflexive, symmetric and antisymmetric $\Rightarrow R=id_S.$
The formal proof is in the following flag diagram.
*S:*_s | R:br(S)
*u_1:refl(S,R) | u_2:sym(S,R) | u_3:antisym(S,R)
*x,y:S | v:Rxy
*a_3:=λx,y:S.λv:Rxy.a_2 : (R⊆id_S)
*x,y:S | v:(id_S)xy
*Notation P:=λz:S.Rxz : S→*_p
*a_6:=λx,y:S.λv:(id_S)xy.a_5 : (id_S⊆R)
Invariance under converse operation. Suppose $R$ is a relation on type $S$. Then the following hold.
1) $R$ is reflexive $\Rightarrow R^{-1}$ is reflexive.
2) $R$ is symmetric $\Rightarrow R^{-1}$ is symmetric.
3) $R$ is antisymmetric $\Rightarrow R^{-1}$ is antisymmetric.
4) $R$ is transitive $\Rightarrow R^{-1}$ is transitive.
*S:*_s | R:br(S)
*a:=λx:S.ux : refl(S,R^-1)
*S:*_s | R:br(S)
*x,y:S | v:R^-1xy
*a_1:=uyxv : Rxy
*a_2:=λx,y:S.λv:R^-1xy.a_1 : sym(S,R^-1)
*S:*_s | R:br(S)
*x,y:S | v:R^-1xy | w:R^-1yx
*a_1:=uxywv : x=y
*a_2:=λx,y:S.λv:R^-1xy.λw:R^-1yx.a_1 : antisym(S,R^-1)
*S:*_s | R:br(S)
*x,y,z:S | v:R^-1xy | w:R^-1yz
*a_1:=uzyxwv : Rzx
*a_2:=λx,y,z:S.λv:R^-1xy.λw:R^-1yz.a_1 : trans(S,R^-1)
Invariance under intersection. Suppose $R$ and $Q$ are relations on type $S$. Then the following hold.
1) $R$ and $Q$ are reflexive $\Rightarrow R\cap Q$ is reflexive.
2) $R$ and $Q$ are symmetric $\Rightarrow R\cap Q$ is symmetric.
3) $R$ or $Q$ is antisymmetric $\Rightarrow R\cap Q$ is antisymmetric.
4) $R$ and $Q$ are transitive $\Rightarrow R\cap Q$ is transitive.
*S:*_s | R,Q:br(S)
*u:refl(S,R) | v:refl(S,Q)
*a_1:=ux : Rxx
*a_2:=vx : Qxx
*a_3:=∧-in (Rxx,Qxx,a_1,a_2) : (R∩Q)xx
*a_4:=λx:S.a_3 : refl(S,R∩Q)
*S:*_s | R,Q:br(S)
*u:sym(S,R) | v:sym(S,Q)
*x,y:S | w:(R∩Q)xy
*a_1:=∧-el_1(Rxy,Qxy,w) : Rxy
*a_2:=∧-el_2(Rxy,Qxy,w) : Qxy
*a_3:=uxya_1 : Ryx
*a_4:=vxya_2 : Qyx
*a_5:=∧-in (Ryx,Qyx,a_3,a_4) : (R∩Q)yx
*a_6:=λx,y:S.λw:(R∩Q)xy. a_5 : sym(S,R∩Q)
*S:*_s | R,Q:br(S)
*Notation A:=antisym(S,R):*_p
*Notation B:=antisym(S,Q):*_p
*Notation C:=antisym(S,R∩Q):*_p
*x,y:S | w_1:(R∩Q)xy | w_2:(R∩Q)yx
*a_1:=∧-el_1(Rxy,Qxy,w_1) : Rxy
*a_2:=∧-el_1(Ryx,Qyx,w_2) : Ryx
*a_3:=vxya_1a_2 : x=y
*[2]a_4:=λv:A.λx,y:S.λw_1:(R∩Q)xy.λw_2:(R∩Q)yx.a_3 : (A⇒C)
*x,y:S | w_1:(R∩Q)xy | w_2:(R∩Q)yx
*a_5:=∧-el_2(Rxy,Qxy,w_1) : Qxy
*a_6:=∧-el_2(Ryx,Qyx,w_2) : Qyx
*a_7:=vxya_5a_6 : x=y
*[2]a_8:=λv:B.λx,y:S.λw_1:(R∩Q)xy.λw_2:(R∩Q)yx.a_7 : (B⇒C)
*S:*_s | R,Q:br(S)
*u_1:trans(S,R) | u_2:trans(S,Q)
*x,y,z:S | v:(R∩Q)xy | w:(R∩Q)yz
*a_1:=∧-el_1(Rxy,Qxy,v) : Rxy
*a_2:=∧-el_2(Rxy,Qxy,v) : Qxy
*a_3:=∧-el_1(Ryz,Qyz,w) : Ryz
*a_4:=∧-el_2(Ryz,Qyz,w) : Qyz
*a_5:=u_1xyza_1a_3 : Rxz
*a_6:=u_2xyza_2a_4 : Qxz
*a_7:=∧-in (Rxz,Qxz,a_5,a_6) : (R∩Q)xz
*a_8:=λx,y,z:S.λv:(R∩Q)xy.λw:(R∩Q)yz.a_7 : trans(S,R∩Q)
Invariance under union. Suppose $R$ and $Q$ are relations on type $S$. Then the following hold.
1) $R$ or $Q$ is reflexive $\Rightarrow R\cup Q$ is reflexive.
2) $R$ and $Q$ are symmetric $\Rightarrow R\cup Q$ is symmetric.
*S:*_s | R,Q:br(S)
*u:refl(S,R) | x:S
*ux : Rxx
*a_1:=∨-in_1(Rxx,Qxx,ux) : (R∪Q)xx
*a_2:=∨-in_2(Rxx,Qxx,ux) : (Q∪R)xx
*a_3:=λu:refl(S,R).λx:S.a_1 : (refl(S,R)⇒refl(S,R∪Q))
*a_4(R,Q):=λu:refl(S,R).λx:S.a_2 : (refl(S,R)⇒refl(S,Q∪R))
*a_5:=a_4(Q,R) : (refl(S,Q)⇒refl(S,R∪Q))
*a_7:=∨el(refl(S,R),refl(S,Q),refl(S,R∪Q),u,a_3,a_5) : refl(S,R∪Q)
*S:*_s | R,Q:br(S)
*u_1:sym(S,R) | u_2:sym(S,Q)
*x,y:S | v:(R∪Q)xy
*a_1:=u_1xyw : Ryx
*a_2:=∨-in_1(Ryx,Qyx,a_1) : (R∪Q)yx
*a_3:=λw:Rxy. a_2 : (Rxy⇒(R∪Q)yx)
*a_4:=u_2xyw : Qyx
*a_5:=∨-in_2(Ryx,Qyx,a_4) : (R∪Q)yx
*a_6:=λw:Qxy. a_5 : (Qxy⇒(R∪Q)yx)
*a_7:=∨-el(Rxy,Qxy,(R∪Q)yx,v,a_3,a_6) : (R∪Q)yx
*a_8:=λx,y:S.λv:(R∪Q)xy. a_7 : sym(S,R∪Q)
Invariance under composition. Suppose $R$ and $Q$ are relations on type $S$. Then the following hold.
1) $R\circ R^{-1}$ is always symmetric.
2) $R$ and $Q$ are reflexive $\Rightarrow R\circ Q$ is reflexive.
3) Suppose $R$ and $Q$ are symmetric. Then
\[R\circ Q\text{ is symmetric }\Leftrightarrow R\circ Q=Q\circ R.\]
*S:*_s | R:br(S)
*x,y:S | u:(R∘R^-1)xy
* Notation P:=λz:S.Rxz∧R^-1zy : S→*_p
*z:S | v:Pz
*a_3:=prod-term (S,R,R^-1,y,z,x,a_2,a_1) : (R∘R^-1)yx
*a_4:=λz:S.λv:Pz. a_3 : (∀z:S.(Pz⇒(R∘R^-1)yx))
*a_5:=∃-el (S,P,u,(R∘R^-1)yx,a_4): (R∘R^-1)yx
*a_6:=λx,y:S.λu:(R∘R^-1)xy. a_5 : sym(S,R∘R^-1)
*S:*_s | R,Q:br(S)
*u:refl(S,R) | v:refl(S,Q)
*ux : Rxx
*vx : Qxx
*a_1:=prod-term (S,R,Q,x,x,x,ux,vx) : (R∘Q)xx
*a_2:=λx:S.a_1 : refl(S,R∘Q)
3) Here we use the proof term $sym\mhyphen criterion(S,R)$ from Theorem <ref>.3) for the second criterion of symmetry and the proof term $conv\mhyphen prod$ from
Theorem <ref>.2).
*a_1:=symcriterion(S,R) : sym(S,R)⇔(R^-1=R)
: (R^-1=R)⇒sym(S,R)
*R,Q:br(S) | u:sym(S,R) | v:sym(S,Q)
*a_6:=convprod(S,R,Q) : (R∘Q)^-1=Q^-1∘R^-1
* Notation P_1:=λK:br(S).((R∘Q)^-1=K∘R^-1) : br(S)→*_p
* Notation P_2:=λK:br(S).((R∘Q)^-1=Q∘K) : br(S)→*_p
*a_7:=eqsubs(br(S),P_1, Q^-1,Q,a_5,a_6) :
*a_8:=eqsubs(br(S),P_2, R^-1,R,a_4,a_7) :
* Notation A:=sym(S,R∘Q) : *_p
* Notation B:=(R∘Q=Q∘R) : *_p
*a_9:=a_2(R∘Q)w : (R∘Q)^-1=R∘Q
*a_10:=eqsym(br(S),(R∘Q)^-1,R∘Q,a_9) : R∘Q=(R∘Q)^-1
*a_11:=eqtrans(br(S),R∘Q,(R∘Q)^-1,Q∘R,a_10,a_8) : R∘Q=Q∘R
*a_12:=λw:A.a_11 : A⇒B
*a_13:=eqsym(br(S),R∘Q,Q∘R,w) : Q∘R=R∘Q
*a_14:=eqtrans(br(S),(R∘Q)^-1,Q∘R,R∘Q,a_8,a_13) : (R∘Q)^-1=R∘Q
*a_15:=a_3(R∘Q)a_14 : sym(S,R∘Q)
*a_16:=λw:B.a_15 : B⇒A
§ SPECIAL BINARY RELATIONS
§.§ Equivalence Relation and Partition
Invariance of equivalence relation under converse operation and intersection. Suppose $R$ and $Q$ are equivalence relations on type $S$. Then the following hold.
1) $R^{-1}$ is an equivalence relation on $S$.
2) $R\cap Q$ is an equivalence relation on $S$.
1) can easily be derived from Theorem
<ref>.1), 2), 4) using intuitionistic logic.
2) can easily be derived from Theorem <ref>.1), 2), 4) using intuitionistic logic.
We skip the formal proofs.
Next we formalize the fact that there is a correspondence between equivalence relations on $S$ and partitions of $S$.
Equivalence classes are introduced in [8] as follows.
*S:*_s | R:br(S) | u:equivrel(S,R)
*class(S,R,u,x):={y:S | Rxy}:ps(S)
*Notation [x]_R for class(S,R,u,x)
Next we define a partition of type $S$:
*S:*_s | R:S→ps(S)
As usual, we can regard a partition $R$ as a collection $Rx\,( x\in S)$ of subsets of $S$. From this point of view, the above diagram expresses the standard two facts for a partition:
* any element of $S$ belongs to one of subsets from the collection (namely $Rx$);
* if the intersection of two subsets $Rx$ and $Ry$ is non-empty, then they coincide.
(1) implies that each subset from the collection is non-empty and that the union of all subsets from the collection is $S$.
Any equivalence relation $R$ on type $S$ is a partition of $S$ and vice versa.
The type of partitions of $S$ is $S\rightarrow ps(S)$, which is $S\rightarrow S\rightarrow *_p$, and it is the same as the type $br(S)$ of relations on $S$. The proof consists of two steps.
Step 1. Any equivalence relation is a partition.
*S:*_s | R:S→S→*_p
*a_3:=λx:S.a_2 : (∀x:S.xεX)
This proves the first part of the definition of $partition(S,R)$ and the second part was proven in [8], pg. 291.
Step 2. Any partition is an equivalence relation.
*S:*_s | R:S→S→*_p
*Notation A:=∀x:S.(xεRx)
*Notation B:=∀x,y,z:S.(zεRx⇒zεRy⇒Rx=Ry)
*a_4:=λx:S.a_3 : refl(S,R)
*x,y:S | v:Rxy
*a_5:=a_1y : (yεRy)
*a_6:=a_2xyyva_5 : Rx=Ry
*a_7:=a_1x : (xεRx)
*a_8:=eqsubs(ps(S),λZ:ps(S).xεZ,Rx,Ry,a_6,a_7) : (xεRy)
*a_9:=λx,y:S.λv:Rxy.a_8 : sym(S,R)
*x,y,z:S | v:Rxy | w:Ryz
*a_10:=a_9yzw : Rzy
*a_11:=a_2zxya_10v : Rz=Rx
*a_12:=a_1z : (zεRz)
*a_13:=eqsubs(ps(S),λZ:ps(S).zεZ,Rz,Rx,a_11,a_12) : zεRx
*a_14:=λx,y,z:S.λv:Rxy.λw:Ryz.a_13 : trans(S,R)
: equivrel(S,R)
§.§ Partial Order
Invariance of partial order under converse operation and intersection. Suppose $R$ and $Q$ are partial orders on type $S$. Then the following hold.
1) $R^{-1}$ is a partial order on $S$.
2) $R\cap Q$ is a partial order on $S$.
1) can easily be derived from Theorem
<ref>.1), 3), 4) using intuitionistic logic.
2) can easily be derived from Theorem <ref>.1), 3), 4) using intuitionistic logic.
We skip the formal proofs.
$\subseteq$ is a partial order on the power set $ps(S)$ of type $S$.
This is the formal proof.
*Notation R:=λX,Y:ps(S).X⊆Y : br(ps(S))
*Notation A:=refl(ps(S),R)
*Notation B:=antisym(ps(S),R)
*Notation C:=trans(ps(S),R)
*a_1:=λx:S.λu:(xεX).u : X⊆X
*a_2:=λX:ps(S).a_1 : A
*X,Y:ps(S) | u:X⊆Y | v:Y⊆X
*a_3:=∧in(X⊆Y,Y⊆X,u,v) : X=Y
*a_4:=λX,Y:ps(S).λu:X⊆Y.λv:Y⊆X.a_3 : B
*X,Y,Z:ps(S) | u:X⊆Y | v:Y⊆Z
*x:S | w:xεX
*a_5:=uxw : (xεY)
*a_6:=vxa_5 : (xεZ)
*a_7:=λx:S.λw:(xεX).a_6 : X⊆Z
*a_8:=λX,Y,Z:ps(S).λu:X⊆Y.λv:Y⊆Z.a_7 : C
*a_9:=∧in(A∧B, C,∧in(A,B,a_2,a_4),a_8) : A∧B∧C
§.§ Well-Ordering and Transfinite Induction
We will use the notation $\leqslant$ for partial order. In the following diagram we define the strict order $<$, the least element of a partially ordered set, and well-ordering of type $S$.
*S:*_s | ⩽:br(S) | u:partord(S,⩽)
*Definition < :=λx,y:S.(x⩽y∧(x=y))
*X:ps(S) | x:S
*Definition least(S,⩽,X,x):=xεX∧∀y:S.(yεX⇒x⩽y)
*Definition wellord(S,⩽):=partord(S,⩽)
Transfinite Induction. Suppose $\leqslant$ is a well-ordering of type $S$. Then for any predicate $P$ on $S$:
\[\forall x:S.[(\forall y:S.(y<x\Rightarrow Py)\Rightarrow Px]\Rightarrow \forall x:S.Px.\]
Here is the formal proof.
*S:*_s | ⩽:br(S) | u_1:wellord(S,⩽) | P:S→*_p
*u_2 : ∀x:S.[∀y:S.(y<x⇒Py)⇒Px]
*Notation A:=partord(S,⩽)
*Notation B:=[∀X:ps(S).(∃x:S.xεX⇒∃x:S.least(S,⩽,X,x))]
*a_1:=∧el_1(A,B,u_1) :A
*a_2:=∧el_2(A,B,u_1) :B
: refl(S,⩽)∧antisym(S,⩽)
: antisym(S,⩽)
*Notation X:=λx:S.Px : ps(S)
*a_5:=a_2Xv_1 : [∃x:S.least(S,⩽,X,x)]
*x:S | v_2:least(S,⩽,X,x)
*a_6:=∧el_1(xεX,∀y:S.(yεX⇒x⩽y),v_2) : xεX
*a_7:=∧el_2(xεX,∀y:S.(yεX⇒x⩽y),v_2) : [∀y:S.(yεX⇒x⩽y)]
*y:S | w_1:y<x
*a_8:=∧el_1(y⩽x,(x=y),w_1) : y⩽x
*a_9:=∧el_2(y⩽x,(x=y),w_1) : (x=y)
*a_12:=a_9a_11 :
*a_13:=λw_2:Py.a_12 : Py
*a_14:=doubneg(Py)a_13 : Py
*a_15:=λy:S.λw_1:y<x.a_14 : [∀y:S.(y<x⇒Py)]
*a_16:=u_2xa_15 : Px
*a_17:=a_6a_16 :
*a_18:=λx:S.λv_2:least(S,⩽, X,x).a_17 : [∀x:S.(least(S,⩽, X,x)⇒)]
*a_19:=∃el(S,λx:S.least(S,⩽, X,x),a_5,,a_18) :
*a_20:=λv_1:(∃x:S.xεX).a_19 : (∃x:S.xεX)
*a_21:=∃in(S,λz:S.zεX,x,w) : (∃z:S.zεX)
*a_22:=a_20a_21 :
*a_23:=λw:Px.a_22 : Px
*a_24:=doubneg(Px)a_23 : Px
*a_25:=λx:S.a_24 : (∀x:S.Px)
Here we used (twice) the Double Negation theorem with the proof term $doub\mhyphen neg$. This is the only place in this paper where we use the classical (not intuitionistic) logic.
§ CONCLUSION
Starting with the definitions from [8] of binary relations and properties of reflexivity, symmetry, antisymmetry, and transitivity, we formalize in the theory $\lambda D$ (the Calculus of Constructions with Definitions) criteria for these properties and prove their invariance under operations of union, intersection, composition, and taking converse. We provide a formal definition of partition and formally prove correspondence between equivalence relations and partitions. We derive a formal proof that $\subseteq$ is a partial order on power set. Finally we formally prove the principle of transfinite inductions for a type with well-ordering.
The results can be transferred to the proof assistants that are based on
the Calculus of Constructions. Since binary relations are the abstract concepts used in many areas of mathematics, the results can be useful for further formalizations of mathematics in $\lambda D$. Our next direction of research is formalization of parts of probability theory in $\lambda D$ that we outlined in [6].
§ APPENDIX
§ PROOF OF THEOREM <REF>
*S: *_s | R:br(S)
*a_1:=λu:(R^-1)^-1xy.u : (R^-1)^-1xy⇒Rxy
*a_2:=λu:Rxy.u : Rxy⇒(R^-1)^-1xy
*a_3:=λx,y:S.a_1 : (R^-1)^-1⊆R
*a_4:=λx,y:S.a_2 : R⊆(R^-1)^-1
*a_5:=relequal (R^-1)^-1, R,a_3,a_4) : (R^-1)^-1=R
*S: *_s | R,Q:br(S)
* Notation A:=(R∘Q)^-1 :br(S)
* Notation B:=Q^-1∘R^-1 :br(S)
* Notation P_1:=λz:S.Ryz∧Qzx : S→*_p
* Notation P_2:=λz:S.Q^-1xz∧R^-1zy : S→*_p
*z:S | v:P_1z
*a_3:=prod-term (S,Q^-1,R^-1,x,z,y,a_2,a_1) : (Q^-1∘R^-1)xy
*a_4:=λz:S.λv:P_1z.a_3 : (∀z:S.(P_1z ⇒Bxy))
*a_5:=∃-el (S,P_1,u,Bxy,a_4):Bxy
λu:Axy.a_5 : A⊆B
*x,y:S | u:Bxy
*z:S | v:P_2z
*a_9:=prod-term (S,R,Q,y,z,x,a_8,a_7) : (R∘Q)yx
*a_10:=λz:S.λv:P_2z.a_9 : (∀z:S.(P_2z ⇒Axy))
*a_11:=∃-el (S,P_2,u,Axy,a_10):Axy
λu:Bxy.a_11 : B⊆A
*convprod(S,R,Q):=relequal (A,B,a_6,a_12) : (R∘Q)^-1=Q^-1∘R^-1
*S: *_s | R,Q:br(S)
* Notation A:=(R∩Q)^-1 :br(S)
* Notation B:=R^-1∩Q^-1 :br(S)
*x,y:S | u:Axy
* u:Ryx∧Qyx
*a_4:=λx,y:S.λu:Axy.a_3 : A⊆B
*x,y:S | u:Bxy
* u:R^-1xy∧Q^-1xy
*a_8:=λx,y:S.λu:Bxy.a_7 : B⊆A
*a_9:=relequal (A,B,a_4,a_8):(R∩Q)^-1= R^-1∩Q^-1
*S: *_s | R,Q:br(S)
* Notation A:=(R∪Q)^-1 :br(S)
* Notation B:=R^-1∪Q^-1 :br(S)
*x,y:S | u:Axy
* u:(R∪Q)yx
* u:Ryx∨Qyx
*a_2:=λv:Ryx.a_1 : Ryx⇒Bxy
*a_4:=λv:Qyx.a_3 : Qyx⇒Bxy
*a_5:=∨-el (Ryx,Qyx,Bxy,u,a_2,a_4) : Bxy
*a_6:=λx,y:S.λu:Axy.a_5 : A⊆B
*x,y:S | u:Bxy
* u:R^-1xy∨Q^-1xy
*a_8:=λv:R^-1xy.a_7 : R^-1xy⇒Axy
*a_10:=λv:Q^-1xy.a_9 : Q^-1xy⇒Axy
*a_11:=∨-el (R^-1xy,Q^-1xy,Axy,u,a_8,a_10) : Axy
*a_12:=λx,y:S.λu:Bxy.a_11 : B⊆A
*a_13:=relequal (A,B,a_6,a_12) : (R∪Q)^-1= R^-1∪Q^-1
*S: *_s | R,P,Q:br(S)
* Notation A:=R∘(P∪Q) :br(S)
* Notation B:=R∘P∪R∘Q :br(S)
* Notation P_0:=λz:S.Rxz∧(P∪Q)zy : S→*_p
*z:S | v:P_0z
*a_3:=prod-term (S,R,P,x,z,y,a_1,w) : (R∘P)xy
*a_5:=λw:Pzy.a_4 : Pzy⇒Bxy
*a_6:=prod-term (S,R,Q,x,z,y,a_1,w) : (R∘Q)xy
*a_7:=∨-in _2((R∘P)xy,(R∘Q)xy,a_6):Bxy
*a_8:=λw:Qzy.a_7 : Qzy⇒Bxy
*a_9:=∨-el (Pzy,Qzy,Bxy,a_2,a_5,a_8):Bxy
*a_10:=λz:S.λv:P_0z.a_9 : (∀z:S.(P_0z ⇒Bxy))
*a_11:=∃-el (S,P_0,u,Bxy,a_10):Bxy
λu:Axy.a_11 : A⊆B
* Notation P_1:=λz:S.Rxz∧Pzy : S→*_p
* Notation P_2:=λz:S.Rxz∧Qzy : S→*_p
*z:S | w:P_1z
*a_13:=∧el_1(Rxz,Pzy,w) : Rxz
*a_14:=∧el_2(Rxz,Pzy,w) : Pzy
*a_16:=prod-term (S,R,(P∪Q),x,z,y,a_13,a_15) : Axy
*a_17:=λz:S.λw:P_1z.a_16 : (∀z:S.(P_1z⇒Axy))
*a_18:=∃-el (S,P_1,v,Axy,a_17):Axy
*a_19:=λv:(R∘P)xy.a_18 : ((R∘P)xy⇒Axy)
*z:S | w:P_2z
*a_20:=∧-el_1(Rxz,Qzy,w) : Rxz
*a_21:=∧-el_2(Rxz,Qzy,w) : Qzy
*a_23:=prod-term (S,R,(P∪Q),x,z,y,a_20,a_22) : Axy
*a_24:=λz:S.λw:P_2z.a_23 : (∀z:S.(P_2z⇒Axy))
*a_25:=∃-el (S,P_2,v,Axy,a_24):Axy
*a_26:=λv:(R∘Q)xy.a_25 : ((R∘Q)xy⇒Axy)
*a_27:=∨-el((R∘P)xy,(R∘Q)xy, Axy,u,a_19, a_26) : Axy
*[2]a_28:=λx,y:S.λu:Bxy.a_27 : B⊆A
*a_29:=relequal (A,B,a_12,a_28) : R∘(P∪Q)=R∘P∪R∘Q
6) is proven similarly to 5).
*S: *_s | R,P,Q:br(S)
* Notation A:=R∘(P∩Q) :br(S)
* Notation B:=R∘P∩R∘Q :br(S)
* Notation P:=λz:S.Rxz∧(P∩Q)zy : *_p
* u:(∃z:S.Pz)
*z:S | v:Pz
*a_5:=prod-term (S,R,P,x,z,y,a_1,a_3) : (R∘P)xy
*a_6:=prod-term (S,R,Q,x,z,y,a_1,a_4) : (R∘Q)xy
*a_7:=∧-in ((R∘P)xy,(R∘Q)xy,a_5,a_6) : Bxy
*a_8:=λz:S.λv:Pz.a_7 : (∀z:S.(Pz⇒Bxy))
*a_9:=∃-el (S,P,u,Bxy,a_8):Bxy
*[2]a_10:=λx,y:S.λu:Axy.a_9 : R∘(P∩Q)⊆R∘P∩R∘Q
8) is proven similarly to 7).
*S: *_s | R,P,Q:br(S)
* Notation A:=(R∘P)∘Q :br(S)
* Notation B:=R∘(P∘Q) :br(S)
* Notation P_1(x,y):=λz:S.(R∘P)xz∧Qzy : S→*_p
* Notation P_2(x,y):=λz:S.Rxz∧(P∘Q)zy : S→*_p
* Notation P_3(x,y):=λz:S.Rxz∧Pzy : S→*_p
* Notation P_4(x,y):=λz:S.Pxz∧Qzy : S→*_p
*x,y:S | u:Axy
* u:(∃z:S.P_1(x,y)z)
*z:S | v:P_1(x,y)z
*z_1:S | w:P_3(x,z)z_1
*a_5:=prod-term (S,P,Q,z_1,z,y,a_4,a_2) : (P∘Q)z_1y
*a_6:=prod-term (S,R,(P∘Q),x,z_1,y,a_3,a_5) : Bxy
*a_7:=λz_1:S.λw:P_3(x,z)z_1.a_6 : (∀z_1:S.(P_3(x,z)z_1⇒Bxy))
*a_8:=∃-el (S,P_3(x,z),a_1,Bxy,a_7):Bxy
*a_9:=λz:S.λv:P_1(x,y)z.a_8 : (∀z:S.(P_1(x,y)z⇒Bxy))
*a_10:=∃-el (S,P_1(x,y),u,Bxy,a_9):Bxy
*a_11:=λx,y:S.λu:Axy.a_10 : A⊆B
*x,y:S | u:Bxy
* u:(∃z:S.P_2(x,y)z)
*z:S | v:P_2(x,y)z
*z_1:S | w:P_4(z,y)z_1
*a_16:=prod-term (S,R,P,x,z,z_1,a_12,a_14) : (R∘P)xz_1
*a_17:=prod-term (S,R∘P,Q,x,z_1,y,a_16,a_15) : Axy
*a_18:=λz_1:S.λw:P_4(z,y)z_1.a_17 : (∀z_1:S.(P_4(z,y)z_1⇒Axy))
*a_19:=∃-el (S,P_4(z,y),a_13,Axy,a_18):Axy
*a_20:=λz:S.λv:P_2(x,y)z.a_19 : (∀z:S.(P_2(x,y)z⇒Axy))
*a_21:=∃-el (S,P_2(x,y),u,Axy,a_20):Axy
*a_22:=λx,y:S.λu:Bxy.a_21 : B⊆A
*a_23:=relequal(A,B,a_11,a_22) : (R∘P)∘Q=R∘(P∘Q)
§ PROOF OF THEOREM <REF>
Each statement here is a bi-implication, so we use the proof term bi-impl from Lemma <ref>.
*S:*_s | R:br(S)
* Notation A:=refl(S,R):*_p
* Notation B:=id_s⊆R:*_p
*x,y:S | v:(id_S)xy
* Notation P:=λz:S.Rxz : S→*_p
*a_1:=eq$-$subs(S,P,x,y,v,ux) : Py
*a_2:=λx,y:S.λv:(id_S)xy.a_1 : (id_S⊆R)
*a_3:=λu:A.a_2 : (A⇒B)
*a_4:=eq$-$refl(S,x) : x=_Sx
*a_5:=uxxa_4 : Rxx
*a_6:=λx:S.a_5 : (∀x:S.Rxx)
*a_7:=λu:B.a_6 : (B⇒A)
*a_8:=biimpl (A,B,a_3,a_7) : refl(S,R)⇔id_s⊆R
2) and 3) are proven together as follows.
*S:*_s | R:br(S)
* Notation A:=sym(S,R):*_p
* Notation B:=R^-1⊆R:*_p
* Notation C:=R^-1= R:*_p
*x,y:S | v:R^-1xy
* uyx:(Ryx⇒Rxy)
*a_1:=uyxv : Rxy
*a_2:=λx,y:S.λu:R^-1xy.a_1 : (R^-1⊆R)
*x,y:S | v:Rxy
* uxy:(Rxy⇒Ryx)
*a_3:=uxyv : Ryx
*a_4:=λx,y:S.λu:Rxy.a_3 : (R⊆R^-1)
*a_5:=relequal(S,R^-1,R,a_2,a_4) : R^-1=R
*a_6:=λu:A.a_2 : A⇒B
*a_7:=λu:A.a_5 : A⇒C
*x,y:S | v:Rxy
*a_8:=uyxv : Ryx
*a_9:=λx,y:S.λv:Rxy.a_8 : sym(S,R)
*a_10:=λu:B.a_8 : (B⇒A)
*a_12:=a_10a_11 : A
*a_13:=λu:C.a_12 : (C⇒A)
*a_14:=biimpl (A,B,a_6,a_10) : sym(S,R)⇔R^-1⊆R
*symcriterion(S,R):=biimpl (A,C,a_7,a_13) : sym(S,R)⇔R^-1=R
*S:*_s | R:br(S)
* Notation A:=antisym(S,R):*_p
* Notation B:=R∩R^-1⊆id_S:*_p
*x,y:S | v:(R∩R^-1)xy
*a_3:=uxya_2a_1 : (x=y)
*a_4:=λx,y:S.λv:(R∩R^-1)xy.a_3 : (R∩R^-1⊆id_S)
*a_5:=λu:A.a_4 : (A⇒B)
*x,y:S | v:Rxy | w:Ryx
*a_7:=uxya_6 : (id_S)xy
*a_8:=λx,y:S.λv:Rxy.λw:Ryx. a_7 : antisym(S,R)
*a_9:=λu:B.a_8 : (B⇒A)
*S:*_s | R:br(S)
* Notation A:=trans(S,R):*_p
* Notation B:=R∘R⊆R:*_p
* Notation P:=λz:S.Rxz∧Rzy : S→*_p
*z:S | w:Pz
*a_3:=uxzya_1a_2 : Rxy
*a_4:=λz:S.λw:Pz.a_3 : (∀z:S.(Pz ⇒Rxy))
*a_5:=∃-el (S,P,v,Rxy,a_4) : Rxy
*[2]a_6:=λx,y:S.λv:(R∘R)xy.a_5 : (R∘R⊆R)
*a_7:=λu:A.a_6 : (A⇒B)
*x,y,z:S | v:Rxy | w:Ryz
*a_8:=prodterm(S,R,R,x,y,z,v,w) : (R∘R)xz
*a_9:=uxz : ((R∘R)xz⇒Rxz)
*a_10:=a_9a_8 : Rxz
*a_11:=λx,y,z:S.λv:Rxy.λw:Ryz.a_10 : trans(S,R)
*a_12:=λu:B.a_11 : (B⇒A)
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# Changing CO Bands in Near-IR Spectra of CP Cephei
Scott G. Call,1 Eric G. Hintz,1 Steve Ardern,2 Victoria Scowcroft,2 and
Timothy D. Morrell1
1Department of Physics and Astronomy, Brigham Young University, Provo, Utah,
84602, USA
2Department of Physics, University of Bath, Claverton Down, Bath, BA2 7AY, UK
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We present time-series near-infrared (NIR) spectra for the classical Cepheid,
CP Cephei, from the Astrophysical Research Consortium 3.5-m telescope and NIR
spectrograph, TripleSpec, at Apache Point Observatory, NM, USA. Spectral
observations were made at three points on the ascending portion of the visible
phase diagram for the star. Carbon monoxide (CO) was detected in absorption in
the 2.3-micron band head for each observation. We observed that the equivalent
width of the 3-1 transition of the CO band head decreased by half from our
first observation to the second, or slightly over one day out of the
17.867-day period. Our third observation occurred 54 days after the first
(slightly over three periods for the star) and showed similar CO levels to the
first observation, suggesting that the CO is in the stellar atmosphere and
varies with pulsation.
###### keywords:
stars: variables: Cepheids – infrared: stars – stars: abundances
††pubyear: 2023††pagerange: Changing CO Bands in Near-IR Spectra of CP
Cephei–Changing CO Bands in Near-IR Spectra of CP Cephei
## 1 Introduction
The period-luminosity relation of classical Cepheid variables, the Leavitt
law, has been shown to have less scatter at longer wavelengths (Madore &
Freedman, 1991). As we move beyond the optical region, temperature variation
of the stars and extinction due to the interstellar medium become smaller
obstacles to obtaining accurate measurements. The near and mid-infrared (NIR
and MIR) have become the priority regions for distance measurements using the
Leavitt law.
The conditions present in supergiant stars allow carbon monoxide (CO) to form
in the atmospheres (see Lancon & Rocca-Volmerange, 1992). Marengo et al.
(2010) found that variable CO was the cause of color variations in the
$4.5\,\mathrm{\mu m}$ passband for a selection of Cepheids, and modeled the CO
variation at different phases for one star. The dissociation and recombination
of CO in Cepheid atmospheres is the result of changes in temperature and thus,
correlated with pulsation phase (discussions of this are given in Scowcroft et
al., 2011; Monson et al., 2012; Scowcroft et al., 2016). As temperature
decreases and CO forms, the flux in the associated passband will be suppressed
due to greater absorption (see Figure 2 of Scowcroft et al., 2016).
In addition to the photometric detections in the MIR, Hamer et al. (2023)
observed CO in four classical Cepheids at sub-mm wavelengths. This represents
the first direct evidence of CO in Cepheid atmospheres. We present NIR
spectroscopic data for CP Cephei (hereafter CP Cep), one of the targets
observed by Hamer et al. (2023), confirming CO presence in the form of the
2.3-$\mathrm{\mu m}$ band head. We observed CP Cep three times all at similar
phases, and the CO strength in these observations confirm the variation is
correlated with pulsation of the star.
This work contains the initial spectroscopic observations of temperature
dependent CO in CP Cep. In Section 2 we discuss the instrumentation and data
reduction and provide context for the observations. Our brief analysis of
changes in CO strength and the results from the three observations are given
in Section 3. We conclude with a summary of our findings.
## 2 Observations
Near-infrared observations of CP Cephei (CP Cep) were obtained using the
Astrophysical Research Consortium 3.5-m telescope at Apache Point Observatory
on July 4th, July 5th, and August 27th, 2023 (UTC). The near-infrared
spectrograph, _TripleSpec_ , covers a range of
$0.95{\text{-}}2.46\,\mathrm{\mu m}$ with spectral resolution
$\mathrm{R}=3500$ (Wilson et al., 2004). Observations were “nodded”, or
dithered, between two points on the slit allowing for subtraction of
atmospheric emission lines and background. Standard stars were observed close
in time and airmass to CP Cep for flux calibration and correction of telluric
absorption.
Figure 1: Calibrated spectra for CP Cep for the three nights of observation.
Low signal-to-noise segments due to atmospheric absorption were removed. To
better show the differences in continuum levels, the July 4th and August 27th
spectra are both offset by $+10^{-13}\,\mathrm{erg\,s^{-1}\,cm^{-2}}$ Å-1. The
phase increases from top to bottom.
Table 1 details the observations and sky conditions. Spectral extraction and
telluric correction were performed using a modified version of _SpexTool_
(Cushing et al., 2004; Vacca et al., 2003). We had a small window to observe
CP Cep at an acceptable airmass ($<1.5$) on July 4th. The target was higher in
the sky during our observing run the following night, with almost 26 hours
between observations. The sky conditions on both July nights were mostly clear
with the possibility of thin clouds affecting the strength of signal. For the
August observation, clouds were intermittent at the beginning of the session
which led to a larger difference in airmass between our standard and CP Cep.
Flux values should not be considered absolute due to uncertainties stemming
from sky conditions.
Near-infrared spectra for CP Cep from the three nights are shown in Figure 1,
with the fluxes of two of the observations shifted for the sake of comparison.
The shape of each continuum is indicative of temperature, with the spectra
from July 4th as the coolest of the set with more of a curved continuum and
the July 5th as the hottest. At 2.3-$\mathrm{\mu m}$ three transitions of the
CO absorption feature are visible in the July 4th and August 27th
observations. In the July 5th spectra the CO feature is more difficult to
distinguish from the continuum.
Table 1: Table of observations. $T$ is the total exposure time, $\phi$ is the phase, $X$ is the average airmass, and $\Delta X$ is the difference in airmass between CP Cep and the standard. Obs Date | $T$ | $\phi$ | $X$ | $\Delta X$
---|---|---|---|---
UTC | s | | | std-obj
2023-07-04 | 120 | 0.832 | 1.371 | -0.0708
2023-07-05 | 450 | 0.893 | 1.137 | -0.0342
2023-08-27 | 450 | 0.853 | 1.243 | -0.1112
The phase diagram in Figure 2 was constructed using $K$-band magnitudes from
Monson & Pierce (2011) with the corresponding gloess fitted light curve
(Persson et al., 2004) and shifted magnitudes from TESS in 2022. We adopted
the period of 17.867373 days from _Gaia_ (Gaia Collaboration et al., 2016,
2023) and used maximum light from the TESS data as the epoch. Vertical lines
are overlaid showing the phase locality of our observations. Our three
observations of CP Cep happen after the bump on the ascending portion of the
visible light curve, with the first observation on the relatively flat section
and second on the steep rise. The phase of our third observation similar to
the first observation, but slightly farther along. These observations are near
minimum light in the $K$-band.
Figure 2: Phase diagram for CP Cep comprised of $K$-band photometry from
Monson & Pierce (2011) and 2022 TESS photometry shifted 1.5 magnitudes
brighter for the purpose of comparison. Vertical lines are our observations
relative to the phase, with solid orange, purple dotted-dash, and dotted green
representing July 4th, July 5th and August 27th respectively.
## 3 Results
Figure 3: The normalized spectra from each night for a segment of the $K$
region. Vertical lines represent the central wavelengths for CO transitions in
vacuum, and are labeled in the top plot. The bottom has July 5th plotted with
July 4th to show differences in depth and width.
The $2.3{\text{-}}\mathrm{\mu m}$ CO band head is strong enough on the first
and third nights of observations that it can be seen even in the full spectra
given in Figure 1. While the CO band head is there on July 5th, it is not as
deep. To visualize the decrease in strength, the normalized spectra for the
range covering four of the ro-vibrational transitions (2-0, 3-1, 4-2 and 5-3)
in the band head are shown in Figure 3, and the bottom plot has the normalized
spectra of the second night overlaid on the first night. While the CO features
decreased in strength between observations on July 4th and July 5th, other
features remained relatively constant such as the metal lines near
$2.28\,\mathrm{and}\,2.34\,\mathrm{\mu m}$.
Out of the four transitions, the 3-1 transition appears to be the best
indicator of the change in CO strength in our set of spectra. The 4-2
transition seems to have more contamination from metal lines in the stellar
atmosphere, and many telluric absorption lines exist in the area of the 5-3
transition. Also, the even-to-even transitions, 2-0 and 4-2, appear to be
greatly broadened on the second night while the 3-1 and 5-3 transitions are
not. There is some indication of this broadening on the August 27th
observation, with slight depressions to the right of the central wavelengths
for the 2-0 and 4-2 transitions.
Historically, the most common transition to measure in this CO feature is the
2-0. Indices for CO from Origlia et al. (1993) and Frogel et al. (2001) are
often used in extragalactic research, and the index from Kleinmann & Hall
(1986) is often used for late-type stars. We would intend to use the same
wavelength ranges for comparison, but the 2-0 broadening seen in the July 5th
data requires us to use the 3-1 transition.
To quantify the change in CO strength, equivalent widths of the 3-1 transition
were carefully measured using the _Specutils_ package of astropy (Astropy
Collaboration et al., 2022; Earl et al., 2023). We defined the 3-1 feature as
the range from 2.32061 to 2.33016 $\mathrm{\mu m}$. The results are shown in
Table 2 with the average signal-to-noise for the wavelength range and
calculated errors. For the 3-1 transition, the equivalent width decreased by a
factor of 2 from the first observation on the relatively flat portion of the
bump to the second observation about halfway up the ascending section (a phase
difference of $0.061$). From the first observation to the third (phase
difference of $0.012$), the equivalent width decreased by less than $0.1$ Å
and the errors overlap.
Table 2: Equivalent width (W) measurements for the 3-1 transition and mean signal-to-noise (SNR) for the wavelengths used in determining W. Associated errors were calculated using Cayrel (1988). Obs Date | W (Å) | SNR
---|---|---
2023-07-04 | $3.111\pm 0.0463$ | 391.3
2023-07-05 | $1.509\pm 0.1183$ | 153.3
2023-08-27 | $3.024\pm 0.0876$ | 202.8
## 4 Conclusions
Figure 4: Normalized spectra from July 4 with the $K$ and $K_{s}$ filter
profiles overlaid. The four CO band head transitions are labeled as is the
nearby hydrogen absorption line, Brackett 7-4.
We obtained NIR spectra for the classical Cepheid, CP Cep, at three different
phases. The transitions of the CO band head are present in each observation
but with differing depths and equivalent widths. The strengths of these
transitions appear to change with the pulsation of the star, which agrees with
previous work in the MIR (Scowcroft et al., 2016). The August observation was
at a similar phase to the July 4th observation and features similar depths and
equivalent widths, but these data were obtained three pulsation cycles later;
further suggesting that the CO absorption is in the star and not the
interstellar medium.
The effect the changing CO absorption levels has on the photometry is beyond
the scope of this paper, but we have included the Mauna Kea filter profiles in
the area of interest in Figure 4 (Tokunaga et al., 2002). The $K_{s}$ filter
was developed to minimize the effect of the atmosphere’s thermal background
for ground-based telescopes, and Persson et al. (1998) showed that the CO band
head was responsible for the scatter seen in $K-K_{s}$ for red stars. With
these filter profiles, we can see the $K_{s}$ filter captures the 2-0
transition at around $60\%$ transmission and the 3-1 transition close to
$0\%$. The $K$ filter however, captures the first three transitions in the
band head at almost full transmission. To determine whether this changing
opacity has a non-negligible effect on NIR photometric measurements, NIR
spectral observations should be taken at maximum and minimum CO strength.
We have confirmed that CO varies with pulsation with three observations of
classical Cepheid, CP Cep. One of those observations occurred several cycles
later and had a similar CO strength to the first observation confirming that
the CO is in the stellar atmosphere. To better understand this CO variation in
Cepheid atmospheres, more observations must be made at different phases for
this and other Cepheids.
## Acknowledgements
This paper includes data collected by the TESS mission, which are publicly
available from the Mikulski Archive for Space Telescopes (MAST). Based on
observations obtained with the Apache Point Observatory 3.5-meter telescope,
which is owned and operated by the Astrophysical Research Consortium.
## Data Availability
The data discussed in this paper will be made available upon reasonable
request.
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|
11institutetext: Yunnan Observatories, Chinese Academy of Sciences,Kunming
650216, China<EMAIL_ADDRESS>
22institutetext: University of Chinese Academy of Sciences, Beijing 100049,
China;
33institutetext: Key Laboratory for the Structure and Evolution of Celestial
Objects, Chinese Academy of Sciences, Kunming 650216, China
44institutetext: International Centre of Supernovae, Yunnan Key Laboratory,
Kunming 650216, China
55institutetext: South-Western Institute for Astronomy Research, Yunnan
University, Kunming 650500, China;
Received 20xx month day; accepted 20xx month day
# Prospects of the multi-channel photometric survey telescope in the
cosmological application of Type Ia supernovae
Zhenyu Wang 11223344 Jujia Zhang 113344 Xinzhong Er 55 Jinming Bai 113344
###### Abstract
The Multi-channel Photometric Survey Telescope (Mephisto) is a real-time,
three-color photometric system designed to capture the color evolution of
stars and transients accurately. This telescope system can be crucial in
cosmological distance measurements of low-redshift (low-$z$, $z$ $\lesssim
0.1$) Type Ia supernovae (SNe Ia). To optimize the capabilities of this
instrument, we perform a comprehensive simulation study before its official
operation is scheduled to start. By considering the impact of atmospheric
extinction, weather conditions, and the lunar phase at the observing site
involving the instrumental features, we simulate the light curves of SNe Ia
obtained by the Mephisto. The best strategy in the case of SN Ia cosmology is
to take the image at an exposure time of 130 s with a cadence of 3 days. In
this condition, Mephisto can obtain hundreds of high-quality SNe Ia to achieve
a distance measurement better than $4.5\%$. Given the on-time spectral
classification and monitoring of the Lijiang 2.4 m Telescope at the same
observatory, Mephisto, in the whole operation, can significantly enrich the
well-calibrated sample of supernovae at low-$z$ and improve the calibration
accuracy of high-$z$ SNe Ia.
###### keywords:
transients: supernovae; cosmology: cosmological parameters
## 1 Introduction
As the most reliable indicators of cosmological distance so far, Type Ia
supernovae (SNe Ia) play a crucial role in unveiling the accelerated expansion
of the universe (Riess et al. 1998; Perlmutter et al. 1999). Additionally, SNe
Ia have greatly contributed to the discovery of the Hubble tension (Dainotti
et al. 2021). Despite the proposal of dark energy as an explanation for the
accelerated expansion of the universe, our understanding of its fundamental
properties remains limited. As the most straightforward observational tool for
constraining the parameters associated with dark energy, SNe Ia achieve a high
level of accuracy when employed for distance measurements (Weinberg et al.
2013). However, the recent accuracy of SNe Ia (e.g., $\lesssim 7\%$, Abbott et
al. 2019) falls short in constraining cosmological parameters and determining
the nature of dark energy.
The crucial factor in constraining the range of cosmological parameter values
lies in accurately ascertaining the distances of SNe Ia at high-$z$ ( $z>$
0.5). The application of SNe Ia in cosmological distance measurement depends
on some empirical relations between the absolute peak brightness and shape of
light curves, for example, the width-luminosity relation (WLR, usually known
as the Phillips relation; Phillips 1993). The original form of the Phillips
relation is
$M(\lambda)_{max}=a+b\Delta m_{15}(\lambda),$ (1)
where $M(\lambda)_{max}$ represents the maximum absolute magnitude at a
particular band (usually using the $B$-band), and $\Delta m_{15}$ is the
decline rate from the peak luminosity to 15 days later. The coefficients $a$
and $b$ are measured by dozens of well-observed low-$z$ SNe Ia. Comparing with
the low-$z$ sample, one can derive the intrinsic brightness of SNe Ia at a
corresponding band for a given redshift and thus measure its luminosity
distance. For example, at $z$ = 0.01, the rest-frame 4310Å flux is observed in
the $B$-band, while at $z$ = 1.01 this same flux is roughly observed in the
$I$-band. To accurately determine the distances of SNe Ia at various
redshifts, it is imperative to possess knowledge regarding the wavelength-
dependent evolution of WLR. This calibration predicament in SN Ia cosmology is
the primary source of systematic error in constructing the SNe Ia Hubble
diagram. Consequently, enhancing calibration precision at low-$z$ becomes
indispensable for refining measurements of cosmological parameters using SNe
Ia. Moreover, low-$z$ supernova samples not only help to reduce the
uncertainties associated with high-$z$ SN Ia measurements but also provide
constraints to the Hubble constant (Riess et al. 2009, 2011, 2016, 2022),
which can alleviate the Hubble tension.
For low-$z$ samples, the current largest source of systematic error is the
calibration. The existing low-$z$ SNe Ia come from different telescopes, with
some using Bessel-like $UBVRI$ filters and some utilizing SDSS-like $ugriz$
filters. Differing filter systems present a significant challenge for
calibration at the level of a few percent. Even with the same filter system,
subtle differences can exist in the transmissions. The most direct approach to
reducing this error is to obtain additional samples using a consistent
telescope system. Extinction is another factor of systematic uncertainty. For
the purpose of calculating extinction, accurate color measurement is
essential. Additionally, variations in ignition positions (central versus off-
center) and explosion mechanisms (deflagration versus detonation) contribute
to the diversity of SN Ia observations.
Furthermore, WLR is not only a function of wavelength but also depends on the
spectral parameters of SNe Ia. Besides the outliers that cannot be
standardized by any kind of WLR, the normal SNe Ia can be further divided into
some subclasses depending on the spectral features around the peak brightness,
e.g., the equivalent width (EW, Branch et al. 2006), velocity (Wang et al.
2009), and temporal velocity gradient (Benetti et al. 2005) of Si II
$\lambda$6355\. The fine classification is an efficient way to improve the
accuracy of SNe Ia in distance measurement. For example, Wang et al. (2009)
divided the normal SNe Ia into two subclasses depending on the velocity of Si
II 6355 around the maximum brightness and improved the accuracy of the Hubble
diagram from 0.178 mag to 0.125 mag. Blondin et al. (2012) use different
coefficients to fit each subclasses of various SNe Ia depending on the EW of
Si II lines. Thus, to create a consistent sample, it is crucial to have three
spectra for every well-observed SN Ia that covers the spectral evolution from
a few days before to a few days after the $B$-band maximum.
In this paper, we investigate the possibility of the Multi-channel Photometric
Survey Telescope (Mephisto, Yuan et al. 2020) for low-$z$ calibration of SNe
Ia. Mephisto is a 1.6-meter wide-field high-precision telescope with a 2 deg2
field of view. It is led by Yunnan University and co-built by Yunnan
Observatories and Nanjing Institute of Astronomical Optics & Technology. This
telescope can simultaneously image the sky in three bands ($ugi$ or $vrz$) and
obtain real-time color information. It boasts high color calibration accuracy,
which can effectively reduce errors introduced by extinction and reddening.
Moreover, the available samples around $z=0.1$ so far are low (Abbott et al.
2019). As a 1.6 m-aperture wide-field survey telescope, it has the ability to
obtain a great number of high-quality samples. Based on multi-color
photometry, it can trigger the Lijiang 2.4 m Telescope (LJT, Fan et al. 2015)
to perform follow-up spectral observation. We can get the spectral evolution
to help us identify the subclass. Hence, we can remove SNe Ia with larger
deviations in the analysis, and thus reduce the uncertainty in the Hubble
diagram.
To improve the observation efficiency of Mephisto, we conduct observation
simulation and cosmological fitting to produce an appropriate observation
strategy. We use SNCosmo (Barbary et al. 2022) in this study to simulate the
light curves of SNe Ia for Mephisto. Section 2 provides a concise introduction
to the fundamental information of Mephitso and the Lijiang Observatory.
Section 3 delves into the simulation of SN Ia light curves and the photometric
accuracy under different exposure durations. The cosmological parameter
estimation method is detailed in Section 4. Section 5 is the presentation of
the results, and there we have provided appropriate observation strategies.
Section 6 presents the results of the estimation and provides suggestions for
future observations.
## 2 Mephisto and Lijiang Observatory
Mephisto innovatively uses a Ritchey-Chrétien (RC) system with correctors and
film-coated cubic prisms, so that the focal plane of the telescope is under
the main mirror and there is a large distance between the focal plane and the
main mirror. Therefore, the color separation system and multiple cameras can
be placed in the rear optical path while ensuring a high-quality image in the
entire field of view. The telescope is equipped with three high-quality CCD
cameras, which can capture real-time color images of the same patch of sky in
three different bands ($ugi$ or $vrz$), providing valuable color information.
The current total efficiency of Mephisto is depicted in Figure 1. The filters
used by Mephisto are not the most popular system among observers, e.g.,
$UBVRI$ of Johnson-Cousins or $ugriz$ of SDSS. The specific choice of these
filters depends largely on their scientific objectives. Mephisto’s survey mode
includes both W-survey and D/H/M-survey modes. The W mode focuses on 2*20
second exposures of observable areas in the northern hemisphere ($\delta>$15°,
26,000 square degrees). The D/H/M models, on the other hand, sample at daily,
hourly, and minute cadence, respectively, for time-domain astronomy
observations. The observation strategy adopted in this paper is based on D
mode.
Mephisto is situated at the Lijiang Observatory administered by Yunnan
Observatories, located in Gaomeigu, Lijiang City, in Yunnan Province, China,
at an altitude of 3200 meters. The site benefits from an effective observation
time of more than 2000 hours per year. The average seeing is about 1.17, with
25$\%$ of the time experiencing seeing below 0.91 and 75$\%$ of the time
experiencing seeing below 1.33 (Xin et al. 2020). Overall, the Lijiang
Observatory is an outstanding site for optical surveys.
Figure 1: The current total efficiency of Mephisto in $u,v,g,r,i$ and $z$
bands.
## 3 The simulation of SNe Ia
The simulation of SN Ia light curves is discussed in Section 3.1. We take
atmospheric extinction, seeing, telescope efficiency, weather, and lunar phase
into consideration. We explore the impact of exposure time on the accuracy of
light measurement and distance measurement in Section 3.2. In Section 3.3, we
list some well-observed spectra captured by LJT.
### 3.1 SN Ia light curve
The modified spectral adaptive light curve template (SALT2, Guy et al. 2007)
model produces a spectral energy distribution (SED) for the following
simulation. The flux is defined as follows
$F(t,\lambda)=x_{0}\times[M_{0}(t,\lambda)+x_{1}M_{1}(t,\lambda)]\times
exp[c\times CL(\lambda)],$ (2)
where $\lambda$ and t represent the rest-frame wavelength and time
respectively, $M_{0}$($t$, $\lambda$) and $M_{1}$(t, $\lambda$) describe the
temporal variation of SED, CL($\lambda$) is the average color correction law,
$x_{0}$, $x_{1}$ and $c$ are the fitting parameters of the light curves for
SNe Ia, $x_{0}$ is the flux normalization parameter, $x_{1}$ is the stretch
parameter, and $c$ is color. In addition to atmospheric extinction, seeing,
and telescope efficiency, we consider several other factors, including:
Figure 2: This figure simulates an SN Ia with $z$ = 0.1, taking atmospheric
extinction, seeing, weather, and lunar phase into account. The exposure time
is 130 s. The absolute magnitude in the $B$-band is -19.3 mag. The top panel
shows the observation data without the influence of the lunar phase, while the
bottom panel includes the lunar phase.
Weather: Lijiang experiences a notable rainy season from June to September.
The number of nights suitable for observation during this period is relatively
limited. This paper primarily focuses on analyzing the weather patterns during
the dry season. Based on the statistics of the weather conditions (Xin et al.
2020), usually there are about 200 nights in one year which are suitable for
observations. In this study, we adopt 200 observational nights per year in the
simulation. The occurrence of supernovae is uniformly and randomly distributed
among these 200 days. Furthermore, weather conditions are also taken into
account. Even during the dry season, there may be occasional instances where
observation is hindered due to factors such as precipitation or high humidity.
Therefore, a few data points are randomly omitted from the observed dataset to
account for these circumstances.
Lunar phase: The influence of the lunar phase on observation is predominantly
concentrated in the $u$ and $g$ bands, which correspond to the $u$, $v$, and
$g$ bands of Mephisto. We incorporated the effect of lunar phase into these
three bands, based on data from ref. Xin et al. (2020), as per the
corresponding lunar calendar during the time of observation. The observation
data with and without the influence of the lunar phase are presented in Figure
2. As we can see, the influence of the lunar phase decreases the number of
observable days in the $u$ and $v$ bands.
data: We generate a batch of SNe Ia with the following conditions. In our
simulation, the redshifts are less than 0.5. The volumetric rate of SNe Ia we
adopt in this simulation is $1.0\times 10^{-4}$ yr-1 Mpc-3 according to ref.
Rodney_2014. The redshift distribution of 4957 SNe Ia with 200 days of
observation time and a 180 $deg^{2}$ sky area is shown in the upper panel of
figure 5. The absolute magnitude follows a normal distribution with a mean of
-19.26 and a standard deviation of 0.2 (Richardson et al. 2014). The
distributions of the parameters $x_{1}$ and $c$ obey the following functions
(Scolnic & Kessler 2016):
$P(x)=\left\\{\begin{aligned}
Ae^{(-(x-\overline{x})^{2})/2\sigma_{1}^{2}},x<\overline{x};\\\
Ae^{(-(x-\overline{x})^{2})/2\sigma_{2}^{2}},x\geq\overline{x}.\\\
\end{aligned}\right.$ (3)
where $A=\sqrt{2/\pi}(\sigma_{1}+\sigma_{2})^{-1}$, and $\overline{x}$ =
1.142, $\sigma_{1}$ = 1.652, and $\sigma_{2}$ = 0.104 for $x_{1}$, and
$\overline{x}$ = -0.061, $\sigma_{1}$ = 0.023, $\sigma_{2}$=0.083 for c (Li et
al. 2023). With these distributions, we have compiled a comprehensive library
of SN Ia light curves, which serves as a valuable resource for further
analysis.
### 3.2 exposure time
Table 1: The fitting results under different exposure times, and $\mu$ represents the distance modulus. exposure time | $\mu$
---|---
20 s | $39.02\pm{0.21}$
60 s | $38.84\pm{0.11}$
80 s | $38.83\pm{0.10}$
130 s | $38.82\pm{0.09}$
We test the photometric accuracy at different exposure times. The results are
plotted in Figure 3. Due to the sensitivity of the SALT2 fitting to
observation points near the maximum brightness, we select a total of 10 data
points before and after the maximum and evaluate the quality of photometric
measurements based on their errors. Additionally, for bands with fewer than 10
data points, we include all available data in our analysis.
Figure 3: The horizontal axis is exposure time, and the vertical axis is the
average mag error over the 10 days before and after the $B$-band maximum. The
exposure time is from 10 to 200 s, with two decimal places retained for the
result. We select an SN Ia with a redshift of 0.1.
As expected, the photometric error generally decreases exponentially with
increasing exposure time. However, lower apparent magnitudes become visible in
the u-band with increasing exposure time, which leads to an overall increase
in the average photometric error in the $u$-band. Only data from the $g$ and
$r$ bands can be captured when the exposure time is limited to 10 s. Data of
$v,g,r$, and $i$ bands are available in all four bands when the exposure time
is longer than 20 s. Data from the $u$ band also becomes observable when the
exposure time is further increased to 60 s. Data from all six bands can be
detected when the exposure time exceeds 70 seconds. An increase of 10 s in
exposure time does not bring significant improvements in accuracy at 80 and
130 s. We take into account the SALT2 fitting accuracy under various exposure
times for further analysis. Table 1 displays the results of the fitting.
Fitting results show very small and close errors when the exposure time
exceeds 60 s. This indicates that the telescope can achieve good observation
accuracy for SNe Ia with redshifts near 0.1. The error in the distance modulus
decreases to $~{}0.1$ mag when the exposure time exceeds 80 s. This means the
distance uncertainty is $\lesssim 4.5\%$ according to the law of propagation
of uncertainties. We take 80 and 130 s into consideration for further
exploration.
### 3.3 Spectral diversity of well-observed SNe Ia
Spectral follow-up of LJT is crucial for achieving a homogeneous SN Ia sample
from Mephisto. For example, including three spectra at $t\sim-7,0,+7$ days
post-maximum is vital in obtaining essential information for diversity
studies, encompassing the examination of EW, velocity, and velocity
graduation. Consequently, we can effectively exclude subclass samples
exhibiting more significant dispersion, thereby reducing distance measurement
errors arising from scatter and enhancing the fitting accuracy of cosmological
parameters.
Table 2: The observation results of SNe Ia with a redshift of 0.1 under different cadences. SN Ia | $\Delta m_{15}(B)$ | $M_{B}$ | type(wang) | type(branch) | ref.
---|---|---|---|---|---
13dy | 0.90 | -19.65 | NV | CN | Zhai et al.2016
09ig | 0.90 | -19.46 | HV | $\cdots$ | Marion et al.2013
12fr | 0.85 | -19.49 | HV | SS | Zhang et al.2014
15bq | 0.82 | -19.68 | $\cdots$ | SS | Li et al.2022
19ein | 1.35 | -18.71 | HV | BL | Xi et al.2022
11fe | 1.18 | -19.40 | NV | CN | Zhang et al.2016
We present the photometric data of SNe Ia in Table 2, while Figure 4 shows
spectra of selected well-observed SNe Ia. Notably, supernovae with similar
$\Delta m_{15}$ values may exhibit varying absolute magnitudes, e.g., SN
2013dy and SN 2009ig. Through the measurement of the velocity of SiII
$\lambda$6355 SN 2013dy is classified into the normal-velocity (NV) type,
while SN 2009ig is classified into the high-velocity (HV) type in the Wang
diagram (Wang et al. 2009). The $\Delta m_{15}$ and $M_{B}$ of SNe 2012fr and
2015bq are similar. They are both divided into shallow silicon (SS) due to the
small EWs ($\leq 60\AA$) of Si II $\lambda$6355 (Branch et al. 2006). However,
SN 2012fr shows narrower Si II $\lambda$6355 and Ca II IRT profiles, with
generally deeper absorption lines than the 91T-like events. In the early
spectra, there are strong high-velocity features of Si II $\lambda$6355 that
are not seen in other 91T-like events (Zhang et al. 2014). SN 2019ein is a
91bg-like supernova. Its peak brightness is significantly lower than normal
SNe Ia. During maximum, the 91bg-like events show weak or absent Fe II lines
and strong absorption lines of intermediate elements. These supernovae do not
follow the WLR. SN 2011fe is a typical normal SN Ia. We also list some SNe Ia
with $z>0.05$, e.g., SN 2022zsy ($z\sim 0.06$), SN 2022aadh ($z\sim 0.076$),
and SN 2022adfs ($z\sim 0.088$).
The collaboration between LJT and Mephisto enables us to acquire a substantial
quantity of samples featuring spectra at $z\sim 0.1$ for subsequent
classification based on multidimensional spectral information. This
information facilitates the elimination of subclasses that significantly
deviate from the WLR, resulting in a more homogeneous sample set.
Additionally, distinct subclasses of supernovae exhibit varying Phillip
relationship coefficients, effectively mitigating the dispersion observed in
the SNe Ia Hubble diagram.
Figure 4: Temporal spectra of different subclasses of SNe Ia at a few days
before and after the $B$-band maximum. The spectrum of 13dy-7 is from ref
Zheng et al. (2013), and the others are from LJT.
## 4 Calibration and Constraints on Cosmological Parameters
To achieve higher accuracy, we conduct a preliminary screening on the data
before performing the light curve fitting. We remove some supernovae with a
limited number of observed data points. The filtering criteria are as follows:
1. 1)
signal-to-noise ratio (S/N) greater than 5;
2. 2)
The sum of all the bands is at least 15 data points;
3. 3)
Data are available in at least three different bands;
4. 4)
After applying the aforementioned selection criteria, we retained only the
supernovae with successful convergence in the SALT2 fitting.
We refer to SNe Ia that have passed the screening as good-samples. Table 3
presents the number of supernovae obtained under the observation conditions of
a 200 day time span and a 180 deg2 sky area. Mephisto has many scientific
goals, such as the search for extremely metal-poor stars and transient
sources. In our simulation, we focus on a 180 $deg^{2}$ sky area for our SNe
Ia survey, which accounts for about 20% of the observing time. Our analysis
primarily focuses on the cases with cadence of 2 days, 3 days, or 5 days,
considering exposure times of 80 s and 130 s respectively.
Table 3: The number of SNe Ia that can be observed after screening at different cadence and exposure times. The observation time is one year. Cadence | exposure time | Number
---|---|---
2 d | 130 s | 1094
2 d | 80 s | 762
3 d | 130 s | 979
3 d | 80 s | 616
5 d | 130 s | 566
5 d | 80 s | 312
Various factors can influence the details of an SN Ia explosion, including the
precise location of ignition, whether it is at the center or off-center, and
the dynamical burning mechanism (e.g., subsonic deflagration or supersonic
detonation). Additionally, the circumstellar environment in which the
explosion occurs plays a significant role. These factors collectively
contribute to the observational diversities and result in distinct light curve
parameters. The dispersion caused by these influential factors can
substantially impact accurately fitting cosmological parameters. Therefore,
prior data calibration becomes imperative before undertaking cosmological
parameter fitting procedures. Our study employed the minimizing $\chi^{2}$
methodology (Marriner et al. 2011) to produce the best fitting result.
$\chi^{2}(\alpha,\beta)=\sum\limits_{i=1}[m_{xi}-{\mu}(z_{i})+{\alpha}x_{1i}-{\beta}c_{i}-M_{bin}(z)]^{2}/({\sigma}_{i}^{2}+{\sigma}_{int}^{2}),$
(4)
where $m_{xi}$, $x_{1i}$, and $c_{i}$ are the best-fit parameters of the SALT2
model for the $i-th$ SN Ia. $M_{bin}$ is a constant in each redshift bin. The
number of bins exceeds 30, with each redshift bin having a width of 0.01. The
definition of $M_{bin}$ is as follows
$M_{bin}(z_{b})=m_{x}-{\mu}(z)+{\alpha}x_{1}-{\beta}c,$ (5)
where $m_{x}=-2.5logx_{0}$, $\mu(z)$ is the distance modulus, and the redshift
$z_{b}$ represents the central point of each bin. One of the key benefits of
this approach is its robustness against variations in wavelength bands. The
difference between $m_{x}$ and $m_{0}$ is about 10 mag. While a cosmological
model is employed to calculate $\mu(z)$ in this process, it has been shown in
ref. Marriner et al. (2011) that the values of $\alpha$ and $\beta$ are not
affected by the choice of cosmological parameters when a large number of bins
are used. The definition of $\sigma_{int}$ is as follows
${\sigma}_{int}^{2}=V_{m_{x}}+\alpha^{2}V_{x_{1}}+\beta^{2}V_{c}+2\alpha
V_{m_{x},x_{1}}-2\beta V_{m_{x},c}-2\alpha\beta V_{x_{1},c},$ (6)
where $V$ represents the intrinsic covariance matrix of parameters,
$V_{m_{x}}$, $V_{x_{1}}$, and $V_{c}$ are the diagonal elements of the
covariance matrix, $V_{m_{x},x_{1}}$, $V_{m_{x},c}$, and $V_{x_{1},c}$ are the
corresponding off-diagonal elements. $\sigma_{i}^{2}$ is defined as follows
${\sigma}_{i}^{2}=V_{m_{xi}}+\alpha^{2}V_{x_{1i}}+\beta^{2}V_{c_{i}}+2\alpha
V_{m_{xi},x_{1i}}-2\beta V_{m_{xi},c_{i}}-2\alpha\beta V_{x_{1i},c_{i}},$ (7)
Similar to the definition of $\sigma_{int}^{2}$, the covariance matrix
consists of $m_{x}$, $x_{1}$, and $x_{0}$ for the $i-th$ SN Ia. Restrictions
on cosmological parameters can be made based on the observed distance moduli
$\mu$ and the theoretically calculated values of $\mu_{th}$. We consider the
flat $\Lambda CDM$ model here. We used the MCMC (Foreman-Mackey et al. 2013)
method for parameter fitting, with particular focus on the values of $H_{0}$
and $\Omega_{M}$. The likelihood function is defined as follows
$lnP=-\frac{1}{2}\sum_{i=0}^{n}\frac{(\mu_{i}-\mu_{th}(z_{i},H_{0},\Omega_{M}))^{2}}{\sigma_{\mu_{i}}^{2}}$
(8)
where $\mu_{i}=m_{x}-M_{0}(z_{b})+\alpha x_{1_{i}}-\beta c{{}_{i}}$, and
$\mu_{th}$ is defined as follows
$\mu_{th}(z_{i},H_{0},\Omega_{M})=5log(d_{L}/Mpc)+25$ (9)
Considering a flat universe and discarding the radiation term, the luminosity
distance $d_{L}$ is defined by
$d_{L}=\frac{c(1+z)}{H_{0}}\int_{0}^{z}{\frac{dz^{\prime}}{\sqrt{\Omega_{M}(1+z^{\prime})^{3}+(1-\Omega_{M})}}}$
(10)
The absolute peak magnitude of SNe Ia in our simulation follows a normal
distribution with a mean of -19.26 and a deviation of 0.2 (Richardson et al.
2014). This can lead to the production of supernovae that are very bright or
very dim. We removed data that significantly deviated from the Hubble flow.
## 5 Result
Figure 5: The top panel shows the original number-redshift distribution of
simulated SNe Ia (initial data, blue histogram) and selected SNe Ia (selected
data, red step line). The bottom panel displays the moduli-redshift
distribution of selected SNe Ia (red points). The solid line in the bottom
panel represents the theoretical prediction. The figure illustrates a scenario
where the cadence is 3 days and the exposure time is 130 s for red points. The
observation time is one year for both initial data and selected data.
Figure 5 displays the redshift distribution of the screened data. The
observation strategy utilized a cadence of 2 days and an exposure time of 130
s. The top panel displays the distribution of the number of SNe Ia with
redshifts. The blue histogram represents the original redshift distribution of
simulated SNe Ia, while the red step line depicts the number distribution of
SNe Ia after the selection process. The bottom panel displays the distribution
of distance moduli with redshift. The solid line represents a flat $\Lambda
CDM$ model with $\Omega_{M_{0}}$=0.3, $\Omega_{\Lambda_{0}}$=0.7, and
$H_{0}$=70 km s-1Mpc-1, which are used to generate the SNe Ia data. The red
points in the chart represent the distribution of SNe Ia after the selection
process. Under this observational strategy, the telescope’s observational
limit extends slightly beyond a redshift of 0.3. For supernovae with a
redshift around 0.1, there is not only good observational accuracy but also no
significant selection bias. However, selection bias is apparent when the
redshift exceeds 0.26.
Table 4: The observation results of an SN Ia with a redshift of 0.1 under different cadences. Observation Strategy | $H_{0}$ | $\Omega_{M}$
---|---|---
cadence=2 days, t=130 s | $68.76^{+0.70}_{-0.72}$ | $0.44^{+0.07}_{-0.07}$
cadence=2 days, t=80 s | $68.83^{+0.91}_{-0.95}$ | $0.45^{+0.12}_{-0.10}$
cadence=3 days, t=130 s | $69.73^{+0.76}_{-0.77}$ | $0.36^{+0.08}_{-0.08}$
cadence=3 days, t=80 s | $69.70^{+1.01}_{-1.05}$ | $0.37^{+0.13}_{-0.12}$
cadence=5 days, t=130 s | $70.17^{+1.00}_{-1.03}$ | $0.31^{+0.11}_{-0.11}$
cadence=5 days, t=80 s | $70.58^{+1.36}_{-1.40}$ | $0.26^{+0.19}_{-0.17}$
Figure 6: The MCMC fitting results include high-redshift SNe Ia. The left
panel is cadence = 2 days, and the right panel is cadence = 3 days. The
exposure times are both 130 s with a year of observation time.
The fitting results are shown in Table 4. The obtained fit results do not
align with the parameters used to generate the SNe data, which will be
discussed in the next section. Here we focus on the errors rather than the fit
values. From the results, it can be seen that the fit values have the smallest
errors for a cadence of 2 days and an exposure time of 130 s. However,
compared to the strategy with a cadence of 3 days and the same exposure time,
it increases the observation time by 33$\%$, resulting in only a 12.5$\%$
decrease in the error of $\Omega_{M}$ and almost no change in the error of
$H_{0}$. We include 88 high-$z$ samples at redshifts higher than 0.5 (Riess et
al. 2007; Brout et al. 2019). The improvement in precision at this level will
be washed out when incorporating existing data that includes high-$z$ samples
(see Figure 6). Therefore, taking everything into consideration, we recommend
adopting the strategy of a cadence of 3 days and an exposure time of 130
seconds for the observations.
## 6 Discussion and conclusion
The results obtained by different strategies indicate a deviation from the
cosmological parameters in generating the supernovae. The discrepancy is
attributed to multiple factors. Firstly, a sample collected at low-$z$
provides limited constraints on cosmological parameters. Figure 7 illustrates
the Hubble diagram at lower redshifts for different values of $\Omega_{M}$ and
$\Omega_{\Lambda}$. It can be seen that at low redshifts, the curves predicted
with different $\Omega_{M}$ and $\Omega_{\Lambda}$ almost overlap, indicating
that the constraints on $\Omega_{M}$ and $\Omega_{\Lambda}$ are weak by the
low-$z$ samples. Even a slight deviation can result in significant
discrepancies in the fitting values, highlighting the sensitivity of the
process. Another factor to consider is the diversity exhibited by SNe Ia,
which stems from variations in ignition locations, explosion mechanisms, and
environments. This study does not incorporate spectral simulations, thus
making it difficult to ascertain the abnormal supernovae within the simulated
data. This lack of comprehensive classification of SNe Ia may introduce biases
into the fitting of cosmological parameters. Moreover, it is essential to
acknowledge that Mephisto filters are non-standard and may introduce certain
biases during fitting procedures. Lastly, while the light curves of SNe Ia
align with actual observations, any deviations between fitted and actual
values can be attributed to potential limitations or inaccuracies within the
utilized cosmological model for generating SNe Ia at varying distances;
however, these deviations do not significantly impact overall error
estimation.
Figure 7: Hubble diagram for different values of cosmological parameters. Red
points represent simulated SN Ia data, similar to the data in Figure 5.
This study serves as a preliminary investigation of the Mephisto survey. Once
Mephisto is fully operational, we can combine it with the LJT to obtain about
three spectra during the peak phase of well-observed SNe Ia. Currently, LJT
can identify 30–50 SNe Ia every year. With the addition of Mephisto, the
sample size of SNe Ia will be significantly increased. The fundamental
astrophysics of SNe Ia, including their progenitor scenarios and explosion
mechanisms, remains elusive (Liu et al. 2023). A larger sample size of SNe Ia
can significantly improve our ability to constrain the complex interplay
between the progenitor channels, explosion models, and observations.
Although Mephisto may not exhibit a numerical advantage in the quantity of SN
Ia discoveries compared to other large-scale survey telescopes, its
observational capabilities are formidable when detecting SNe Ia at lower
redshifts. Furthermore, the Mephisto project prides itself on its exceptional
color calibration precision and unique ability to conduct simultaneous
observations across three spectral bands. As a result, these features play a
significant role in partially mitigating systematic errors arising from
variations in site weather conditions.
We conducted a simulation for Mephisto to investigate the accuracy of
cosmological parameter estimation under various observation strategies, with a
fixed observation duration of 180 deg2 and 200 days. The optimal observation
strategy we have determined is an exposure duration of 130 seconds and a
frequency of three days. In our simulation, more than 900 SNe can be detected
in one year of observation. Implementing this strategy ensures commendable
photometric precision and enables us to detect numerous SNe Ia. Our simulation
has considered the influence of the lunar phase in the light curve and
assessed the photometric precision of SNe Ia at $z\sim$ 0.1 for various
exposure durations. The $\sigma_{\mu}$ of sources near redshift 0.1 are about
0.1 mag, with distance uncertainty 4.5%. Compared to the existing sample, the
distance measurement accuracy has improved from 7% to 4.5%. By incorporating
spectroscopic analysis to exclude abnormal SNe Ia, it is possible to achieve a
better fit of the Phillips relation coefficients and further enhance distance
measurement precision. We found that beyond an exposure time of 60 s, there is
a diminishing improvement in photometric accuracy with increasing duration.
Once Mephisto runs officially for survey purposes, the collaborative
observations with LJT will substantially increase the sample size of SNe Ia
and facilitate more comprehensive spectroscopic classification, thereby
mitigating the scatter induced by the diversities.
Mephisto and LJT can build an extensive SN Ia database, a valuable resource
for further analysis. This database will encompass light curves exhibiting
diverse shapes and brightness levels, along with spectra captured near the
peak of the light curve. This comprehensive dataset will offer a valuable tool
for investigating the heterogeneity of SN Ia events and comprehending their
utility as cosmological probes. Through meticulous examination of these data,
we can derive precise distance measurements, delve into the nature of dark
energy, and unveil fresh insights into the origin and evolution of our
universe.
###### Acknowledgements.
We thank the anonymous referee for constructive comments and suggestions. This
work was supported by the National Key R&D Program of China (2021YFA1600404),
the National Natural Science Foundation of China (12173082), and science
research grants from the China Manned Space Project (CMS-CSST-2021-A12), the
Yunnan Province Foundation (202201AT070069), the Top-notch Young Talents
Program of Yunnan Province, the Light of West China Program provided by the
Chinese Academy of Sciences, and the International Centre of Supernovae,
Yunnan Key Laboratory (202302AN360001). We acknowledge the support of the
staff of the LJT and Mephisto. Funding for the LJT has been provided by the
CAS and the People’s Government of Yunnan Province. The LJT is jointly
operated and administrated by YNAO and the Center for Astronomical Mega-
Science, CAS. Mephisto is mainly funded by the “Yunnan University Development
Plan for World-Class University” and “Yunnan University Development Plan for
World-Class Astronomy Discipline”, and obtained supports from the “Science &
Technology Champion Project” (202005AB160002) and from two “Team Projects” –
the “Innovation Team” (202105AE160021) and the “Top Team” (202305AT350002),
all funded by the “Yunnan Revitalization Talent Support Program”.
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|
# Hamilton cycles in a semi-random graph model
Alan Frieze† Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh PA 15213<EMAIL_ADDRESS>and Gregory B. Sorkin
Department of Mathematics, The London School of Economics and Political
Science, Houghton Street, London WC2A 2AE, England<EMAIL_ADDRESS>
(Date: 5 August 2022)
###### Abstract.
We show that whp we can build a Hamilton cycle after at most $1.85n$ rounds in
a particular semi-random model. In this model, in one round, we are given a
uniform random $v\in[n]$ and then we can add an _arbitrary_ edge
$\left\\{v,w\right\\}$. Our result improves on $2.016n$ in [5].
${\dagger}$ Research supported in part by NSF grant DMS1952285
## 1\. Introduction
We consider the following semi-random graph model. We start with $G_{0}$ equal
to the empty graph on vertex set $[n]$. We then obtain $G_{i+1}$ from
$G_{i},i\geq 0$ as follows: we are presented with a uniform random $v\in[n]$
and then we can choose to add an arbitrary edge $\left\\{v,w\right\\}$ to
$G_{i}$. This model was suggested by Peleg Michaeli and first explored in Ben-
Eliezer, Hefetz, Kronenberg, Parczyk, Shikhelman and Stojaković [1]. Further
research on the model can be found in Ben-Eliezer, Gishboliner, Hefetz and
Krivelevich [2]; Gao, Kamiński, MacRury and Prałat[3]; and Gao, Macrury and
Prałat[4, 5]. In particular [5] shows that whp one can construct a Hamilton
cycle in this model in at most $2.016n$ rounds.
In this short note we modify the algorithm of [5] and prove:
###### Theorem 1.1.
In the semi-random model, there is a strategy for constructing a Hamilton
cycle in at most $1.85n$ rounds.
## 2\. Outline and Algorithm
Our algorithm and analysis are largely similar to those of [5], so let us
recapitulate the broad strokes. They maintain a large and growing path, and a
set of isolated nodes. When an isolated node is presented they join it to the
tail of the path. When a path node $v$ is presented, they generate a
“stubedge” (our name, not theirs) to a random isolated node $w$; later, if a
path node $v^{\prime}$ adjacent to $v$ is presented, they generate edge
$\left\\{v^{\prime},w\right\\}$ and use it to insert the vertex $w$ into the
path between $v$ and $v^{\prime}$. These stubs are vital when the path is long
and there are few isolated vertices: at that point, isolated vertices are
rarely presented, while many stubs are generated. Note that by “birthday
paradox” reasoning, only $\Theta(\sqrt{n})$ stubs are needed before there is a
good chance of a neighboring vertex being presented.
As observed in [5], stubs can also be used to turn a Hamilton path into a
Hamilton cycle in $o(n)$ rounds. Assume w.l.o.g. that the path vertices are in
sequence $1,\ldots,n$. From each vertex $v$ presented, we generate a stubedge
randomly to vertex $1$ or $n$. If $v$ had a stubedge to $n$ and later $v+1$ is
presented, joining $v+1$ to $1$ creates a Hamilton cycle, using $1,\ldots,v$;
$v+1,\ldots,n$; the stubedge $\left\\{v,n\right\\}$; and the new edge
$\left\\{v+1,1\right\\}$. This takes expected time $O(1/\sqrt{n})$.
As in [5], we maintain a large and growing path and isolated nodes, but a key
difference is that we also maintain a set of _pairs_. When an isolated node
$v$ is presented, rather than joining it to the tail of the path, we join it
to another isolated node $v^{\prime}$ to make a pair. When a vertex $v$ in a
pair is presented, we join the pair to the tail of the path. Stubs are used to
incorporate into the path either an isolated node, just as in [5], or a pair:
If a stubedge goes from $v$ to a paired vertex $w$, and a path neighbor
$v^{\prime}$ of $v$ is presented, joining $v^{\prime}$ to the partner
$w^{\prime}$ of $w$ allows replacement of the edge
$\left\\{v,v^{\prime}\right\\}$ with the path $v,w,w^{\prime},v^{\prime}$.
The motivation for this is simple. If an isolated vertex $v$ is presented, the
number of components decreases by 1 whether $v$ is added to the path or paired
with another vertex $v^{\prime}$: in this sense, equal progress is made either
way. Ignoring the use of stubs, in the [5] algorithm, after $v$ is presented
and joined to the path, to join $v^{\prime}$ to the path we would have to wait
for $v^{\prime}$ to be presented. In the paired version, after $v$ is
presented and paired with $v^{\prime}$, to join the pair of them to the path
we wait until _either_ $v$ or $v^{\prime}$ is presented; this takes half as
long in expectation.
With regard to the stubs, the two versions are similar. As just noted, the
number of components (isolated vertices or pairs) needing to be incorporated
into the path (including by use of stubs) is the same either way. The only
drawback of the paired version is that the path’s growth is somewhat delayed,
so there are fewer early opportunities to create and use stubs.
## 3\. Algorithm and Differential Equations
Our algorithm is largely the same as the main “fully randomized algorithm” of
[5]. We do not employ any equivalent of their initial “degree-greedy” phase,
although doing so would probably improve our results slightly. Like them, we
run the main algorithm until the path is nearly but not quite complete, so
that it can be analysed by the differential equation method. We finish up by
appealing to the “clean-up” algorithm of [5, Lemma 2.5].
We now describe our algorithm in detail but briefly, then present the
corresponding differential equations. As said, we maintain a _path_ $P$, and
_non-path_ vertices $V$ consisting of _isolated_ vertices $V_{1}$ and _paired_
vertices $V_{2}$.
A _stubedge_ goes from a path vertex we call the _stubroot_ or simply _stub_
to a non-path vertex we call a _stubend_. We say a stubroot with $i$ stubedges
has _stub-degree_ $i$; we will also call it an _$i$ -stubroot_, and let
$S_{i}$ be the set of such vertices. We limit the stub-degree to at most 3, so
$S=S_{1}\cup S_{2}\cup S_{3}$ is the set of all stubroots. (There are never
more than about $0.001n$ 3-stubs, and restricting the degree to at most 2 as
in [5] only increases our completion time by about $0.002n$, from about
$1.8465n$ to $1.8482n$.)
Each path vertex is one of four types, and when focussing on type we will use
this font: a stub; a path-neighbor of a stub, called a stubneighbor; a clear
vertex, which if presented will become a stub; or a blocked vertex, which is
essentially useless.
In an abuse of notation, reusing the letters for the sets to denote their
cardinality, let $P=P(t)$ denote the number of vertices on the path at time
$t$, $V_{1}=V_{1}(t)$ the number of isolated vertices, $V_{2}=V_{2}(t)$ the
number of vertices in pairs, $S_{i}=S_{i}(t)$ the number of $i$-stubs (for
$i\in\left\\{1,2,3\right\\}$), $V(t)=V_{1}(t)+V_{2}(t)$, and
$S(t)=S_{1}(t)+S_{2}(t)+S_{3}(t)$. We explore the expected changes in these
quantities in one round.
Our algorithm will preserve the following property.
###### Property 3.1.
Within path-distance 2 of any stub there is no other stub nor any clear
vertex, and the total number of clear vertices is exactly $P-5S$.
We discuss this in case (C1) below.
### 3.1. Description of the Algorithm
We list the actions taken after a (random) vertex is presented. The
description below is valid as long as there remain at least 2 isolated
vertices, and we will stop the algorithm long before that is an issue.
1. (C1)
_The presented vertex $v$ is clear._ Choose a random non-path vertex $w$ and
create a stubedge from $v$ to $w$, making $v$ a 1-stubroot. Change the type of
$v$ from clear to stub, and if $v$ has path-distance 5 or more from other
stubs, change the types of its path neighbors and second-neighbors,
respectively, to stubneighbor and blocked, i.e., BNSNB. If $v$ is at distance
4 from the next stub to the right, make the types from $v$ to the next stub be
SNBNS, and if distance 3, then SNNS.
This changes 5 or fewer vertices from clear to another type. If fewer, then
artificially change the type of additional clear vertices to blocked to make
it exactly 5. This, and the fact that there is no clear nor stub vertex within
path-distance 2 of $v$, preserve Property 3.1. There is no constraint on where
the artificially blocked vertices should be, and they need not even stay fixed
from round to round.
2. (C2)
_The presented vertex $v$ is a stub with $i$ stubneighbors, $i=1,2$._ Choose a
random non-path vertex $w$ and create a stubedge from $v$ to $w$, making $v$
an $(i+1)$-stubroot.
3. (C3)
_The presented vertex $v$ is a stubneighbor of a stub vertex $u$._ By Property
3.1, $u$ is uniquely determined. Randomly choose one of $u$’s stubneighbors,
$w$. Lengthen the path by removing the edge $\left\\{v,u\right\\}$, then
adding the path $u,w,v$ (if $w\in V_{1}$, making the new edge
$\left\\{v,w\right\\}$) or $u,w,w^{\prime},v$ (if $w\in V_{2}$ and
$w,w^{\prime}$ is a pair, making the new edge
$\left\\{v,w^{\prime}\right\\}$). If $u$’s degree was 2 or 3, $u$ becomes an
$(i-1)$-stub.
If $u$’s degree was 1, the stub disappears: $u$ and its two associated
stubneighbor vertices become clear, as do its two associated blocked vertices
unless they must remain blocked by proximity to some other stub. If necessary,
change one or two other blocked vertices to clear to preserve Property 3.1.
The stubedge $\left\\{v,w\right\\}$ becomes a path edge, and all other
stubedges into $w$ (and $w^{\prime}$, if relevant) are deleted. This results
in reducing the stub-degrees of other stubroots, and possibly their deletion.
4. (C4)
_The presented vertex $v$ is on the path, but blocked or a stub of degree 3._
Do nothing.
5. (C5)
_The presented vertex $v$ is isolated._ Choose another isolated vertex
$v^{\prime}$ at random and make a pair $v,v^{\prime}$.
6. (C6)
_The presented vertex $v$ is one of a pair, with some $v^{\prime}$._ Add
$v,v^{\prime}$ to the tail of the path. As in (C3), delete all stubs to $v$
and $v^{\prime}$.
###### Lemma 3.2.
In (C6), each stubedge has probability $2/V(t)$ that its stubend is either $v$
or $v^{\prime}$, and these events are independent. In (C3) where $w$ was
isolated, each stubedge except $\left\\{v,w\right\\}$ has probability $1/V(t)$
that its stubend is $w$, and these events are independent. In (C3) where $w$
was paired with $w^{\prime}$, each stubedge except $\left\\{v,w\right\\}$ has
probability $2/V(t)$ that its stubend is either $w$ or $w^{\prime}$, and these
events are independent.
This is analogous to a claim within [5, Lemma 2.2]. For (C6) the Lemma is
immediate as the stubs to $v$ and $v^{\prime}$ are independent of their
getting paired or joining the path. In the (C3) cases, though, there is a
potential issue of size-biased sampling that is not explicitly addressed in
the proof in [5], and so we give a proof sketch. The issue is that $w$ being
the stubend of the chosen stubedge biases $w$ to have higher stub-degree
(e.g., $w$ could not have been selected if it had no stub edges), suggesting
that other stubedges are also more likely to have $w$ as stubend.
###### Proof.
Imagine that, when created, the stubedges are not revealed. They remain, then,
uniformly random between the stubroots (whose stub-degrees are “known”) and
non-path vertices. Only when the stubroot $v$ is determined and one of its
stubedges is chosen, reveal (or, indeed, generate) the stubedge: this
determines $w$. Only then, reveal (or generate) the other stubedges: each is
equally likely to lead to $w$ or any other non-path vertex (including
$w^{\prime}$, if relevant). So that we can apply the argument again in later
rounds, we can reveal just the stubedges incident to $w$ (and $w^{\prime}$ if
relevant): after deleting them, the other stubedges remain unrevealed and
uniformly random. ∎
Alternatively, one may argue from the perspective that if one sample is taken
from a population of i.i.d. Poisson $\lambda$ random variables, in proportion
to the variables’ values, the sampled value $X$ is not Poisson $\lambda$ (for
example, it cannot be 0), but $X-1$ is Poisson $\lambda$.
### 3.2. Derivation of the Equations
The following equations are valid as long as $P(t)\leq(1-\epsilon)n$ where
$\epsilon>0$ is arbitrarily small; anyway the differential equation method can
only be applied through such time. The error terms below are sometimes
naturally $O(1/n)$ and sometimes $O(1/V(t))$, but with this assumption we
always write them as $O(1/n)$.
#### 3.2.1. ${\bf P(t)}$
$\displaystyle\mathbb{E}(P(t+1)\mid
G_{t})=P(t)+\frac{2V_{2}(t)}{n}+\frac{2S(t)}{n}\cdot\frac{V_{1}(t)+2V_{2}(t)}{V(t)}+O(1/n).$
(1)
(C6):
$V_{2}(t)/n$ is the probability that a paired vertex is presented. The path
length increases by 2.
(C3):
$2S(t)/n$ is the probability that a stubneighbor is presented.111Actually, a
stubneighbor is presented with probability $2S(t)/n+O(1/n)$, because a stub at
either end of $P$ would have only one neighbor rather than the two we are
assuming. The $O(1/n)$ correction term in (1) covers this case. Similar
correction terms apply in subsequent cases and we will not explain the rest.
By Property 3.1 the stubroot $v$ is uniquely determined, and one of its
stubedges $\left\\{v,w\right\\}$ is chosen randomly. With probability
$V_{1}(t)/V(t)$, $w$ is isolated and the path length increases by 1; with
probability $V_{2}(t)/V(t)$, $w$ is paired and the path length increases by 2.
#### 3.2.2. ${\bf V_{1}(t)}$
$\displaystyle\mathbb{E}(V_{1}(t+1)\mid
G_{t})=V_{1}(t)-\frac{2V_{1}(t)}{n}-\frac{2S(t)}{n}\cdot\frac{V_{1}(t)}{V(t)}+O(1/n).$
(2)
(C5):
$V_{1}(t)/n$ is the probability that an isolated vertex is presented. The
vertex is paired with another isolated vertex and the number of isolated
vertices decreases by 2.
(C3):
$2S(t)/n$ is the probability that a stubneighbor is presented. As in section
3.2.1’s (C3), the chosen stubend of the stub is isolated with probability
$V_{1}(t)/V(t)$ and the number of isolated vertices decreases by 1.
#### 3.2.3. ${\bf V_{2}(t)}$
$\displaystyle\mathbb{E}(V_{2}(t+1)\mid
G_{t})=V_{2}(t)-\frac{2V_{2}(t)}{n}+\frac{2V_{1}(t)}{n}-\frac{2S(t)}{n}\cdot\frac{2V_{2}(t)}{V(t)}+O(1/n).$
(3)
(C6):
$V_{2}(t)/n$ is the probability that a paired vertex is presented. The path is
extended using this pair, and the number of paired vertices decreases by 2.
(C5):
$V_{1}(t)/n$ is the probability that an isolated vertex is presented. The
vertex is paired with another isolated vertex and the number of paired
vertices increases by 2.
(C3):
$2S(t)/n$ is the probability that a stubneighbor is presented. The chosen
stubend of the stub is paired with probability $V_{2}(t)/V(t)$, and the number
of paired vertices decreases by 2.
At this point the reader will observe that the expected change in
$n=P(t)+V_{1}(t)+V_{2}(t)$ is zero, as it should be.
#### 3.2.4. ${\bf S_{1}(t)}$
$\displaystyle\mathbb{E}(S_{1}(t+1)\mid G_{t})$
$\displaystyle=S_{1}(t)+\frac{P(t)-5S(t)}{n}-\frac{S_{1}(t)}{n}-\frac{2S_{1}(t)}{n}+\frac{2S_{2}(t)}{n}$
$\displaystyle\hskip
10.00002pt+\frac{2S(t)}{n}\cdot\frac{V_{1}(t)+2V_{2}(t)}{V(t)^{2}}\cdot(2S_{2}(t)-S_{1}(t))$
$\displaystyle\hskip
10.00002pt+\frac{V_{2}(t)}{n}\cdot\frac{2}{V(t)}\cdot(2S_{2}(t)-S_{1}(t))+O(1/n).$
(4)
(C1):
$(P(t)-5S(t))/n$ is the probability that a clear vertex is presented. It
becomes a 1-stub and $S_{1}(t)$ increases by 1.
(C2, $i=1$):
$S_{1}(t)/n$ is the probability that a 1-stub of the path is presented. It
becomes a 2-stub, and $S_{1}(t)$ decreases by 1.
(C3, $i=1$):
$2S_{1}(t)/n$ is the probability that a neighbor of a 1-stub is presented. The
stub is used and $S_{1}(t)$ decreases by 1.
(C3, $i=2$):
$2S_{2}(t)/n$ is the probability that a neighbor of a 2-stub is presented. The
stub is used and $S_{1}(t)$ increases by 1.
(C3):
$2S(t)/n$ is the probability that a stubneighbor is presented. As in previous
cases, the stub $v$ is determined and one of its stubedges
$\left\\{v,w\right\\}$ is chosen randomly. Edge $\left\\{v,w\right\\}$ becomes
a path edge, and is no longer a stubedge; this is captured by the previous two
cases.
All other stubedges into $w$, and its pair-partner $w^{\prime}$ if any, are
deleted. With probability $V_{1}(t)/V(t)$, $w$ was isolated, in which case by
Lemma 3.2 each stubedge has probability $1/V(t)$ of having $w$ as stubend.
With probability $V_{2}(t)/V(t)$, $w$ was paired with some $w^{\prime}$, in
which case by Lemma 3.2 each stubend has probability $2/V(t)$ of having $w$ or
$w^{\prime}$ as stubend. This gives the probability in the next term.
The effect in that term is that each $S_{2}$ vertex has 2 stubedges whose
potential deletion turns it into an $S_{1}$ vertex, increasing $S_{1}$ by 1,
while each $S_{1}$ vertex has 1 stubedge whose potential deletion turns it
into a clear vertex, decreasing $S_{1}$ by 1.
(C6):
$V_{2}(t)/n$ is the probability that a paired vertex is presented. As in the
preceding case, by Lemma 3.2 each stubedge has probability $2/V(t)$ of having
either element of the pair as stubend and thus being deleted. The effect is
that of the previous case.
#### 3.2.5. $\bf S_{2}(t)$
$\displaystyle\mathbb{E}(S_{2}(t+1)\mid G_{t})$
$\displaystyle=S_{2}(t)+\frac{S_{1}(t)}{n}-\frac{S_{2}(t)}{n}-\frac{2S_{2}(t)}{n}+\frac{2S_{3}(t)}{n}$
$\displaystyle\hskip
10.00002pt+\left[\frac{2S(t)}{n}\cdot\frac{V_{1}(t)+2V_{2}(t)}{V(t)^{2}}+\frac{V_{2}(t)}{n}\cdot\frac{2}{V(t)}\right]\cdot(3S_{3}(t)-2S_{2}(t))+O(1/n).$
(5)
(C2, $i=1$):
$S_{1}(t)/n$ is the probability that a 1-stubroot of the path is presented. It
becomes a 2-stubroot, and $S_{2}(t)$ increases by 1.
(C2, $i=2$):
$S_{2}(t)/n$ is the probability that a 2-stubroot of the path is presented. It
becomes a 3-stubroot, and $S_{2}(t)$ decreases by 1.
(C3, $i=2$):
$2S_{2}(t)/n$ is the probability that a neighbor of a 2-stubroot is presented.
The stub is used and $S_{2}(t)$ decreases by 1.
(C3, $i=3$):
$2S_{3}(t)/n$ is the probability that a neighbor of a 3-stubroot is presented.
The stub is used and $S_{2}(t)$ increases by 1.
(C3),(C6):
Analogous to (C3) and (C6) of section 3.2.4, here combined.
#### 3.2.6. $\bf S_{3}(t)$
$\displaystyle\mathbb{E}(S_{3}(t+1)\mid G_{t})$
$\displaystyle=S_{3}(t)+\frac{S_{2}(t)}{n}-\frac{2S_{3}(t)}{n}$
$\displaystyle\hskip
10.00002pt-\left[\frac{2S(t)}{n}\cdot\frac{V_{1}(t)+2V_{2}(t)}{V(t)^{2}}+\frac{V_{2}(t)}{n}\cdot\frac{2}{V(t)}\right]\cdot
3S_{3}(t)+O(1/n).$ (6)
(C2, $i=2$):
$S_{2}(t)/n$ is the probability that a 2-stub vertex of the path is presented.
It becomes a 2-stub, and $S_{3}(t)$ increases by 1.
(C3, $i=3$):
$2S_{3}(t)/n$ is the probability that a neighbor of a 3-stub is presented. The
stub is used and $S_{3}(t)$ decreases by 1.
(C3),(C6):
Analogous to the corresponding case of section 3.2.5.
The equations (1) – (3.2.6) lead to the following differential equations in
the usual way: we let $\tau=t/n$ and $p(\tau)=P(t)/n,v_{1}(\tau)=V_{1}(t)/n$
etc. The initial conditions are $v_{1}(0)=1,p(0)=v_{2}(0)=\cdots=s_{3}(0)=0$.
$\displaystyle\begin{split}p^{\prime}&=2v_{2}+\frac{2s(v_{1}+2v_{2})}{v}.\\\
v_{1}^{\prime}&=-2v_{1}-\frac{2sv_{1}}{v}.\\\
v_{2}^{\prime}&=-2v_{2}+2v_{1}-\frac{4sv_{2}}{v}.\\\
s_{1}^{\prime}&=p-5s-3s_{1}+2s_{2}+\left[\frac{2s(v_{1}+2v_{2})}{v^{2}}+\frac{2v_{2}(t)}{v}\right](2s_{2}-s_{1}).\\\
s_{2}^{\prime}&=s_{1}-3s_{2}+2s_{3}+\left[\frac{2s(v_{1}+2v_{2})}{v^{2}}+\frac{2v_{2}}{v}\right](3s_{3}-2s_{2}).\\\
s_{3}^{\prime}&=s_{2}-2s_{3}-\left[\frac{2s(v_{1}+2v_{2})}{v^{2}}+\frac{2v_{2}}{v}\right](3s_{3}).\\\
\end{split}$ (7)
A numerical simulation of the differential equations is shown in Figure 1. It
shows that $v_{1}(\tau^{*})+v_{2}(\tau^{*})\approx 0$ and $p(\tau^{*})\approx
1$ for $\tau^{*}\approx 1.85$. Justification of the use of the differential
equation method follows as in [5]. As in [5], we use the differential equation
method to analyse the algorithm until the path has length $(1-\epsilon)n$, for
some suitably small $\epsilon$. After this we apply the _clean-up_ algorithm
of [5, Lemma 2.5] to construct a Hamilton cycle in a further
$O(\sqrt{\epsilon}n+n^{3/4}\log^{2}n)$ rounds.
## 4\. Concluding Remarks
Our combining of isolated vertices into pairs leads to a substantial speedup
of the algorithm compared with [5], despite our skipping their first, “degree
greedy” phase. We allowed for stub degrees up to 3 where [5] goes up only to
2, but, observing that the number of degree-3 stubs is never more than about
$0.001n$, this seems to have been unimportant. Further improvements could
probably be made.
First, since pairs gave a big gain, it is natural to consider paths of 3
vertices (“triplets”) or more. We have not tried it, but it appears that this
cannot help. Specifically, if an isolated vertex $v$ is presented, there would
appear to be no advantage in using $v$ to extend a “pair” $Q$ to a 3-vertex
path, over concatenating $v$ to the main path. Either way, the number of
components is the same. Either way, $Q$ must eventually be brought into $P$,
either when one of its endpoints is presented (no difference in whether $p$ is
added to $Q$ or not, as either way $Q$ has two endpoints), or when a stubedge
to one of $Q$’s endpoints is used (again, with no advantage to $v$ over $Q$’s
earlier endpoint).
Other improvements, possibly challenging to analyse, would come from choices
intuitively more sensible than the uniformly random choices made by our
algorithm.
One such is to restore the “degree greedy” approach from [5] that we
discarded: when generating stubedges, let each go to a (random) non-path
vertex _of lowest stub-degree_.
Another, when generating stubedges, is to favour paired vertices over isolated
ones. We have some weak evidence that generating stubedges only to non-paired
vertices up to some time, then uniformly to all non-paired vertices, is better
than generating them uniformly throughout.
Another strategy is, in the case where a 2- or 3-stub is used, to select a
stubedge to a non-path vertex of low stub degree, and/or to favor an isolated
vertex over a paired one (or vice versa).
Returning to the idea of using “triples” as well as pairs, potentially, small
advantages could be found if, for example, we linked $v$ with $Q$ only if $v$
were the stubend of more stubedges than the $Q$-endpoint it extends.
It would be very satisfying in its own right to better understand the natural
stub process, where a presented vertex becomes a new stubroot unless it is
already a stubroot or stubneighbor, i.e., if it is at path distance at least 2
from every existing stub. This in contradistinction to the easier-to-analyse
process taken from [5] and described in (C1), where a presented vertex becomes
a stub only if it is at path distance at least 3 from every existing stub, and
not blocked. In the natural process, the number of stubneighbors will be
between 1 and 2 times the number of stubroots (not 2 times, as used in (C3))
but it is not clear how to find the typical number, nor give a good lower
bound. Presumably more stubroots will be produced, but it is not clear how to
control the likelihood that a presented vertex will become a stub; indeed,
nothing about the process is clear.
Figure 1. The differential equations simulated, as a function of time (rounds
divided by $n$). Path length $P$ (as a multiple of $n$) is shown in red,
isolated vertices $V_{1}$ in green, paired vertices $V_{2}$ in blue, and
degree 1-, 2-, and 3-stubs $S_{i}$ in yellow, turquoise, and magenta. (The
figure is clearer in some PDF viewers than others.)
## Acknowledgement
We thank Zachary Hunter for a spotting two oversights in the paper’s first
version.
## References
* [1] O. Ben-Eliezer, D. Hefetz, G. Kronenberg, O. Parczyk, C. Shikhelman, M. Stojaković (2020). Semi‐random graph process. Random Structures & Algorithms, 56, 648–675.
* [2] O. Ben-Eliezer, L. Gishboliner, D. Hefetz, M. Krivelevich. Very fast construction of bounded-degree spanning graphs via the semi-random graph process. Random Structures & Algorithms, 57(4), 892–919.
* [3] P. Gao, B. Kamiński, C. MacRury, and P. Prałat. 2022. Hamilton cycles in the semi-random graph process. Eur. J. Comb. 99, C (Jan 2022).
* [4] P. Gao, C. MacRury, and P. Prałat. Perfect Matchings in the Semirandom Graph Process. SIAM J. Discrete Mathematics 36(2) (2022).
* [5] P. Gao, C. MacRury, and P. Prałat. A Fully Adaptive Strategy for Hamiltonian Cycles in the Semi-Random Graph Process. 2022\. arXiv, https://arxiv.org/abs/2205.02350.
|
# Leidenfrost Flows : instabilities and symmetry breakings
E. Yim1,†, A. Bouillant2,3,†, D. Quéré2,3, F. Gallaire1 1LFMI, École
Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
2LadHyX, École polytechnique, 91128 Palaiseau, France
3PMMH, PSL-ESPCI, CNRS-UMR 7636, 75005 Paris, France
$\dagger$ Authors contributed equally to this work
(Received: XX 2021; Revised: XX XX 2021; Accepted: XX XX 2022)
###### Abstract
Leidenfrost drops were recently found to host strong dynamics. In the present
study, we investigate both experimentally and theoretically the flows
structures and stability inside a Leidenfrost water drop as it evaporates,
starting with a large puddle. As revealed by infrared mapping, the drop base
is warmer than its apex by typically 10∘C, which is likely to trigger bulk
thermobuoyant flows and Marangoni surface flows. Tracer particles unveil
complex and strong flows that undergo successive symmetry breakings as the
drop evaporates. We investigate the linear stability of the baseflows in a
non-deformable, quasi-static, levitating drop induced by thermobuoyancy and
effective thermocapillary surface stress, using only one adjustable parameter.
The stability analysis of nominally axisymmetric thermoconvective flows,
parametrized by the drop radius $R$, yields the most unstable, thus, dominant,
azimuthal modes (of wavenumber $m$). Our theory predicts well the radii $R$
for the mode transitions and cascade with decreasing wavenumbers from $m=3$,
$m=2$, down to $m=1$ (the eventual rolling mode that entails propulsion) as
the drop shrinks in size. The effect of the escaping vapor is not taken into
account here, which may further destabilize the inner flow and couple to the
liquid/vapor interface to give rise to motion (Bouillant et al., 2018b;
Brandão & Schnitzer, 2020).
Leidenfrost; Flow stability; Thermo-driven convection
## I Introduction
A water drop can levitate above a hot surface provided the solid temperature
exceeds the boiling temperature of the liquid by typically 100∘C. This effect
was first reported in 1756 by J.G. Leidenfrost (Leidenfrost, 1756) and it has
been ever since a source of scientific curiosity. Evaporation produces a vapor
film underneath the drop with typical thickness 50 $\mu$m that thermally
insulates the liquid from its substrate and suppresses boiling. The drop can
thus survive a few minutes to plate temperatures as high as 300∘C. Levitation
also prevents the liquid from wetting the surface. As a consequence, a drop
adopts a quasi-spherical shape when its radius $R$ is smaller than the
capillary length $\ell_{c}=\sqrt{{\sigma_{0}}/{\rho_{0}g}}$ $-$ where
$\sigma_{0}$ is the surface tension, $\rho_{0}$ is the liquid density and $g$
is the acceleration of gravity, that is $\ell_{c}\approx 2.5$ mm for water at
100∘C $-$ while it gets flattened by gravity to a height 2$\ell_{c}$ when
$R>\ell_{c}$. The absence of contact also produces a (quasi-)frictionless
situation by which the liquid drop becomes highly mobile. Thermal and
mechanical insulation have contrasted consequences. On the one hand, above the
Leidenfrost temperature, vapor compromises the liquid cooling properties and
pilots the transition from nucleate to film boiling, unwelcome in nuclear
reactor or in metallurgy (loss of control in quenching processes). On the
other hand, it produces a purely non-wetting situation, which can serve as a
canonical model for superhydrophobicity. Recent studies have illustrated the
richness of spontaneous dynamics related to Leidenfrost drops (Quéré, 2013),
including oscillations (Garmett, 1878; Holter & Glasscock, 1952; Brunet &
Snoeijer, 2011; Bouillant et al., 2021a), bouncing (Celestini et al., 2012;
Waitukaitis et al., 2017) and directed propulsion on asymmetrically textured
(Linke et al., 2006; Cousins et al., 2012) or on non-uniformly heated surfaces
(Sobac et al., 2017; Bouillant et al., 2021b), which could be exploited in the
design of micro-reactors (Raufaste et al., 2016), of heat pipes and
exchangers, as well as for lab-on-a-chip technologies.
Among those dynamics is the ability of Leidenfrost droplets to self-rotate and
self-propel, even in the absence of external fields (Bouillant et al., 2018b).
Despite the early discovery of the Leidenfrost effect, the existence of strong
inner flows has only been reported recently, with velocities as high as a few
cm/s and whose morphology switches from four counter-rotating swirling cells,
preserving the axial symmetry, to an unique asymmetric swirl as evaporation
proceeds (Bouillant et al., 2018a). The eventual solid-like asymmetric
rotation comes with a tilt at the droplet base by an angle $\alpha$ of
typically a few milliradians (Bouillant et al., 2018b), producing
accelerations that scale as $a\sim\alpha g$ (Dupeux et al., 2013) of a few
tens of mm/s2. The internal rolling actually couples with the vapor cushion in
a feedback loop to sustain the self-rolling and self-propelling motions
(Brandão & Schnitzer, 2020), which thereby explains the intrinsic mobility of
Leidenfrost droplets. The origin of inner flows as well as their structure
remain however unclear. They may arise from: i) a thermal scenario: the base
of the drop is close to the hot plate while its apex is exposed to cooler air.
Temperature differences in the liquid may give birth to convective flows,
either driven by thermocapillary effects, known as Marangoni effect
(variations of the surface tension along the temperature gradient as in
Scriven & Sternling (1960)) or driven in the liquid bulk by thermobuoyant
effects, known as Rayleigh-Bénard convection (resulting from the thermal
expansion of the liquid); ii) a hydrodynamic scenario: vapor produced at the
drop base escapes through the subjacent film exerting a viscous radial stress
that may draw liquid with it. Both scenarios a priori preserve axisymmetry,
which is contradicted by experimental observations. Indeed, Particle induced
velocimetry (PIV) has revealed that internal flow structures evolve in time
and therefore with the drop geometry (Bouillant et al., 2018b), suggesting a
symmetry selection mechanism as the drop shrinks in size. Similar selection
mechanism has been reported in a non-levitating drop deposited on a warm plate
(Tam et al., 2009; Dash et al., 2014). In this close configuration, the vapor
cushion is suppressed and the origin of the internal dynamics is purely
thermal. Temperature gradients being greater in Leidenfrost drop than for
sessile drop on warm plates, we anticipate enhanced thermoconvection,
prompting us to focus on the thermal scenario (i) rather than on the
hydrodynamics scenario relying on the escaping vapor (ii). We therefore
address the stability and the symmetry of thermo-induced flows inside a
Leidenfrost drop as it evaporates, neglecting the hydrodynamic effect from the
escaping vapor and without explicitly modelling the surrounding gas. We aim at
predicting the radii at which are expected the successive symmetry breakings
starting with $R<4\ell_{c}$ (to prevent chimney formation), that is in the
regime where drops are flatten by gravity ($R>\ell_{c}$), while approaching
the regime where drops get quasi-spherical ($R<\ell_{c}$). Note that we do not
specifically explore the transition to the rolling mode $m=1$ as in Bouillant
et al. (2018b); Brandão & Schnitzer (2020) but we explore higher modes $m>1$
observed in large puddles $R>\ell_{c}$. In this regime, vapors carve a blister
underneath the liquid, whose amplitude increases with $R$ and reaches the
entire height of the puddle, $2\ell_{c}$, when $R\lesssim 4\ell_{c}$. Viscous
entrainment being markedly weakened along this vapor pocket, vapors would only
draw liquid along the very narrow, peripheral neck (Pomeau et al., 2012;
Burton et al., 2012; Sobac et al., 2014). The peculiar geometry of the vapor
cushion underneath large puddles undermines the hydrodynamic scenario,
corroborating our assumptions. Hydrodynamic effects of the vapor cushion may
however shift our prediction for the bifurcation radii, especially when
$R\rightarrow\ell_{c}$. As shown by Brandão & Schnitzer (2020) in the case of
quasi-spherical drops, the internal rolling couples with the vapor cushion in
a feedback loop, which sustains the self-rolling and self-propelling motions.
Such a coupling could also reshape the vapor cushion when $R>\ell_{c}$, with
consequences on the drop mobility that are not captured by our model.
The selection mechanism for the flow symmetry is a common feature in both
buoyancy and Marangoni instabilities. These instabilities are known to be very
sensitive to the geometry and confinement; the preferred discrete azimuthal
mode wavenumber decreases with the liquid domain size. The stability analysis
on the Marangoni-Rayleigh-Bénard convection has been extensively studied for a
rectangular and cylindrical domain with various boundary conditions Pearson
(1958); Nield (1964); Vrentas et al. (1981); Rosenblat et al. (1982); Kuhlmann
& Rath (1993); Johnson & Narayanan (1999). Yet, the stability of a free liquid
drop subjected to a vertical temperature gradient has heretofore not received
the same attention. Therefore, Leidenfrost drops constitute a toy model where
the evaporation-driven confinement enables to quasi-steadily sweep the states
in a non-wetting drop heated from below. We restrict our parametric study in
$R$ to the limit of $R<5$ mm to prevent Leidenfrost chimneys and pulsating
stars to appear (Quéré, 2013). Leaning on the experimental observations, we
develop a numerical model, which implicitly decouples the evaporation
timescale from the meanflow evolution timescale. We thus look for the
stability of the nominally axisymmetric thermo-convective baseflow in
Leidenfrost drops in order to explain the symmetry breaking from 4 to 1
convective cells reported in Bouillant et al. (2018a), as well as the prior
transition from 6 to 4 cells.
We first characterize the successive symmetry breakings in the internal flows
(§II) and extract from experiments physical quantities relevant to the problem
such as the temperature difference at the drop interface. The governing
hydrodynamic and thermal equations, as well as the linear stability analysis
are then presented in §III and the results are shown in §IV. We obtain the
baseflow generated by the stratification within the liquid and search for the
effective Marangoni number, which best captures our experimental observations,
such as the surface temperature and the velocity field. Then, a stability
analysis is carried out for different drop radii $R$. We compare for a given
$R$ the stability properties of each symmetry breaking mode, the mode with the
highest growth rate – the most unstable mode – being expected to dominate the
flow structure. Our study predicts the successive inner flow symmetries as
drops shrink in size. It also provides the critical radii for the modes
transition in quantitative agreement with observations. We eventually discuss
and compare the numerical outcomes to experiments, and add a few concluding
remarks in §LABEL:sec:discuss.
## II Experimental observations
### II.1 Quasi-steady state assumption
Leidenfrost drops levitate above a thin layer of vapor, of good insulating
properties since $k_{a}<<k$, where $k_{a}$ and $k$ are the air and water
thermal conductivities, provided in Table 1 of the SI. Evaporation, which
mainly takes place at the drop base, is thus markedly reduced and we verify
here that the drop is at quasi-static equilibrium. A water drop with initial
radius $R_{0}$ initially close to $\approx 4$ mm is deposited on a plate
brought to 300∘C. We use a slightly curved substrate to immobilize the highly
mobile liquid. The drop is observed using a top-view high speed camera, from
which we extract $R$, the drop equatorial radius as evaporation proceeds.
Figure 1 shows that $R$ decreases linearly with time $t$ at a rate
$\mathrm{d}R/\mathrm{d}t\approx-22\rm\;\mu m/s$, so that the drop survives
about $\tau_{0}\approx 3$ minutes.
Figure 1: (a) Radius $R$ of a Leidenfrost drop levitating on a plate heated
at 350∘C as a function of time $t$. $R$ decreases as
$R(t)=R_{0}(1-t/\tau_{0})$ (eq.1), plotted as dotted line, denoting
$R_{0}=3.7\pm 0.1$ mm as the initial radius and $\tau_{0}=176.5$ s as the
lifetime. (b) Drop shape for some radii $R\in[0.9;4.5]$ mm readable in (a)
(see colored dots). Simulated shapes (full lines), obtained by numerically
integrating (2), are compared to experimental ones (dotted lines), obtained
for side-viewed drops pinned with a needle. $\ell_{c}$ is the capillary
length. Surface flows, viewed from the top, successively self-organize into
(c) 6 counter-rotating cells (mode $m=3$), (d) 4 counter-rotating cells (mode
$m=2$), and eventually a unique rolling cell (mode $m=1$). Drop keeps on
rolling but eventually stop. The consecutive snapshots are extracted from
movie SM1. (f-h) The horizontal median cut views of the most unstable mode
from the numerical stability analysis for some selected radii. The radii for
the inner flow symmetry transitions from experiments and from stability
analyses are plotted is (a) as black and orange lines, respectively.
Temporal variations in $R$ are best fitted by a linear law, plotted as the
blue dotted line and with equation:
$R(t)=R_{0}(1-t/\tau_{0}),$ (1)
where $R_{0}=3.7\pm 0.1$ mm and $\tau_{0}=176.5$ s. This time is much larger
than the characteristic time of the internal motion $R/V$, since tracers
inside the liquid and at the drop surface reveal flow velocities $V$ as high
as a few cm/s. The separation of time-scales $\tau_{0}\gg R/V\approx 0.02$ s
suggests that the evaporation-driven dynamics can be decoupled from the inner
dynamics. As a result, a Leidenfrost drop can be considered in quasi-static
equilibrium at any time, a key assumption to discuss the stability of
Leidenfrost inner flows. Moreover, as will be discussed in §IV.3.2, the
instability develops faster than $\tau_{0}$, which supports the quasi-static
stability analysis of the Leidenfrost drop.
### II.2 Leidenfrost drop shapes
A consequence of the timescale separation is that a given volume of liquid
adopts the static shape of a non-wetting drop (Roman et al., 2001). If we
denote $C(z)$ as the local curvature at a given height $z$ and $C_{0}$ as the
curvature at the apex, the balance of hydrostatic and Laplace pressures can be
written $C(z)=C_{0}+z/\ell_{c}^{2}$. We introduce the curvilinear abscissa
$s$, the horizontal radius at a given height $r(z)$ and the angle $\beta$
tangent to the interface, the previous equation can be recast into :
$\frac{\sin\beta}{r}+\frac{\mathrm{d}\beta}{\mathrm{d}s}=C_{0}+\frac{z}{\ell_{c}^{2}}.$
(2)
A numerical integration of eq.(2) for $\beta$ ranging from 0 to $\pi$ and
$\mathcal{C}_{0}\ell_{c}$ ranging from 0.5 to 10, provides the drop shape for
radii ranging from $R=0.9$ mm to 4.3 mm, plotted as full lines in Figure 1(b).
These theoretical shapes are found to match the shapes obtained for water
drops kept in place by a needle (see dotted lines), except at the drop north
pole, where the needle locally forms a meniscus. Increasing the drop size
tends to saturate the puddle height $H$ at its maximum value $H_{\max}\approx
2\ell_{c}$, that is roughly 5 mm for water at 100${}^{\circ}\mathrm{C}$. Drops
smaller than the capillary length $\ell_{c}$ are quasi-spherical while drops
larger than $\ell_{c}$ get flattened owing to gravity. As a consequence,
evaporation induces geometric changes, particularly on the drop aspect ratio
$2R/H$. The presence of strong internal flows could in principle deform the
liquid interface. The Reynolds number associated to the inner flows with
typical velocity $V\sim$ cm/s and kinematic viscosity $\nu$ writes as
$Re=RV/\nu$. For a millimetric drop, we have $Re\approx 10^{3}$, so that
inertia overcomes viscosity. The Weber number $We$, which compares inertia to
the resisting capillarity is expressed as $We=\rho_{0}V^{2}R/\sigma_{0}\sim
10^{-2}$. Capillary therefore outbalances inertia, which justifies that the
drop shape does not deviate from the static ones.
### II.3 Drop internal flow structure
The apparent quietness of Leidenfrost drops does not reflect what really
happens inside the liquid. Side-viewed PIV measurements performed in a median
plane of a water drop containing tracer particles have revealed strong
internal flows, with velocities of a few cm/s. As the drop shrinks owing to
evaporation, Leidenfrost flows undergo a series of successive symmetry
breakings. This is further evidenced by focusing on the drop top surface. A
Leidenfrost drop, kept on a concave substrate, is seeded with surface
particles that have a greater affinity for the air interface. These hollow
glass beads are i) pre-dispersed in water,ii) skimmed from the interface
(where particles accumulate owing to a wetting or shape defect), and iii)
introduced in a pre-dispensed Leidenfrost drop. Despite the apparent
axisymmetry of the experiment, interfacial flow structures emerge, as
illustrated by the top-views in Figures 1(c-e) and visible in the movie SM1).
When the drop has a radius $R>2.5$ mm, it hosts multiple vortices, which can
be described by the azimuthal wavenumber $m\geq 3$ in a cylindrical coordinate
system, by denoting ($r$, $z$, $\theta$) and using a periodic wavenumber
expansion in $\theta$ as $e^{im\theta}$, $m\in\mathbb{N}^{*}$. Within the
range $R=[1.8;2.5]$ mm, a mode $m=2$ clearly appears, while for smaller radius
$R<1.5$ mm, the droplet rolls in an asymmetric fashion, corresponding to a
mode $m=1$. Inner flows thus successively self-organize into 6 counter-
rotating cells ($m=3$, Figure 1(c)); 4 counter-rotating cells ($m=2$, Figure
1(d)), and eventually a unique rolling cell ($m=1$, Figure 1(e)). We can
notice that at some instance of movie SM1, the convective cells loose their
organization and coherence. The flow structures seems to be transiently
perturbed by the drop oscillation in the slightly curved well. However, the
dominant modes reappear within a second as a hint of the robustness of the
unstable modes. The transition from $m=2$ to $m=1$ can be also visualized by
side views, using PIV techniques, as in Bouillant et al. (2018b) (images
reproduced in Figure 6(a) and 5(a)) or even indirectly measured, as the drop
acceleration $a$ (extracted from top-views, for water drops initially at rest)
suddenly increases with the mode switching onto $m=1$. The experiment proposed
by (Bouillant et al., 2018b) is reproduced in the SI, for plate temperatures
ranging from $250$ to $450^{\circ}\mathrm{C}$. The accelerations $a$ of about
80 drops as a function of their radii $R$ exhibit similar jumps from $\sim$1
mm/s2 (in the regime where drops are flattened by gravity), up to 60 mm/s2 as
$R$ decreases to $\sim 1$ mm. This indeed corresponds to entering the self-
rotation and propelling regime (Bouillant et al., 2018a). The onset for the
transition from $m=2$ to $m=1$, referred as $R_{2\rightarrow 1}$, is found to
weakly depend on the plate temperature and to be within the interval $[1;1.5]$
mm (see §S6 of the SI).
### II.4 Temperature gradient at the drop interface
To test the aforementioned thermally-based instability scenario, we need to
specify the thermal boundary conditions at the drop surface. The drop base is
expected to be maintained at roughly the water boiling point
($T_{b}=100^{\circ}\mathrm{C}$), while its apex is cooled down by the ambient
air.
Figure 2: (a) Infrared side views of an evaporating Leidenfrost water drop
deposited on a slightly curved surface of brass heated at 350∘C. Images, taken
with a thermal camera give access to the surface temperature (calibration
range from $-40^{\circ}$C to $+150^{\circ}$C to focus on water surface).
Images are extracted from movie SM2. (b) Surface temperature $T$ of a given
water drop along its central vertical axis $z$ (white dotted line), showing
the change of temperature profile as the drop radius $R$ decreases.
The temperature field in a Leidenfrost drop is measured using an infrared
camera (FLIR A600 series), calibrated on the temperature range
$[-40;+150]^{\circ}\mathrm{C}$, only suitable to see water, and not the brass
substrate. Water being opaque to infra-red wavelengths, this measurement
provides the "skin" temperature $T$. Figure 2 shows at $t=0$, that is when
$R=3.5$ mm, $T$ linearly decreases with height $z$ (in millimeters) as
$T=-3.75z+97.0^{\circ}\mathrm{C}$, reaching a maximum
$T_{\max}=97.0^{\circ}\mathrm{C}$ at the drop base ($z=0$). A temperature
difference $\Delta T\approx 25^{\circ}\mathrm{C}$ thus develops along the
drop. At $t=16$ s, the gradient is slightly smaller with
$T=-3.53z+97.0^{\circ}\mathrm{C}$, and it keeps decreasing as $R$ decreases.
These observations are confirmed by introducing a thermocouple inside the
liquid (see Figure S1 in the (SI)). For $t=74$ s, when $R=1.8$ mm, $T$
suddenly becomes homogeneous with $T\approx 88^{\circ}$C. As best visible in
the supplementary movie SM1, this coincides with the moment where the flow
switches to a symmetry $m=1$. At this transition, the drop starts to vibrate,
which enhances mixing. For $t>100$ s, $T$ thus becomes roughly homogeneous,
with a minimum at the drop center (along the rolling axis), and a maximum at
its periphery since fluid are periodically brought close to the hot plate. In
this rolling state, the liquid temperature is $T\approx 80^{\circ}$C, a value
smaller than the boiling point of water, as consequences of i) the
intensifying evaporation-driven cooling; ii) a reduction of the flattened area
at the drop base from which water is heated, which scales as $R^{4}/l_{c}^{2}$
(Mahadevan & Pomeau, 1999); iii) the temperature homogenization due to the
rolling-enhanced mixing. We now try to link the existence of such temperature
distributions to the origin and structure of the internal flows.
## III Theory
### III.1 Problem formulation
Based on the experimental observations, we develop a minimal model assuming
that i) the liquid adopts at any time the static shape of a non-wetting drop;
ii) the baseflow inside a drop is steady; iii) the temperature at the bottom
is fixed to the liquid boiling temperature; iv) the interaction between the
liquid and the surrounding gas is not solved completely but modelled by heat
transfer correlation laws applied on the side boundary. We sketch in Figure
3(a) the problem, where we represent from the side a static drop provided by
eq.(2). We denote by $\Omega$ the liquid domain, and by $\partial\Omega$ the
boundaries, which are decomposed into $\partial\Omega_{F}$, the upper free
interface and $\partial\Omega_{S}$, the bottom interface of the drop. We also
introduce $\partial\Omega_{A}$ the vertical centerline of the drop, the axis
of symmetry. Hence, the numerical computation is done only on the half domain
$r=[0;R]$ as illustrated in Figure 3(b). The bottom interface
$\partial\Omega_{S}$ is assumed to be isothermal at the temperature
$T_{s}=100^{\circ}\mathrm{C}$ (the phase change of a pure body occurs at a
given constant temperature), while the temperature on the side of the drop
$\partial\Omega_{F}$ needs to be evaluated using the heat transfer balance.
Figure 3: Sketch of the problem. (a) The drops (domain $\Omega$) presents
boundaries $\partial\Omega$ including the upper free surface
$\partial\Omega_{F}$, the bottom interface $\partial\Omega_{S}$ and the axis
of symmetry $\partial\Omega_{A}$. (b) Thermal conditions.
### III.2 Governing equations
Under the Boussinesq approximation, the governing Navier-Stokes equation for
the velocity fields $\mathbf{u}=[u_{r},u_{\theta},u_{z}]^{\mathrm{T}}$ and the
temperature $T$ in the cylindrical coordinate $(r,\theta,z)$ defined in the
domain $\Omega$ of boundaries $\partial\Omega$ reads,
$\displaystyle\frac{\partial\mathbf{u}}{\partial
t}+\mathbf{u}\cdot\nabla\mathbf{u}$ $\displaystyle=-\frac{1}{\rho_{0}}\nabla
p-\frac{\rho(T)}{\rho_{0}}\mathbf{g}+\nu\nabla^{2}\mathbf{u}\quad$
$\displaystyle\mathrm{in}\ \Omega,$ (3a) $\displaystyle\nabla\cdot\mathbf{u}$
$\displaystyle=0\quad$ $\displaystyle\mathrm{in}\ \Omega,$ (3b)
$\displaystyle\frac{\partial T}{\partial t}+\mathbf{u}\cdot\nabla T$
$\displaystyle=\kappa\nabla^{2}T\quad$ $\displaystyle\mathrm{in}\ \Omega,$
(3c)
where $\mathbf{g}$ is the gravitational acceleration in $z$ direction, $\nu$
the liquid kinematic viscosity and $\kappa$ its thermal diffusivity. The
boundary conditions for $\partial\Omega_{F}$, $\partial\Omega_{S}$ and
$\partial\Omega_{A}$ are respectively:
$\displaystyle\mathbf{S}\cdot\mathbf{n}$
$\displaystyle=\mathbf{S}_{\mathrm{gas}}\cdot\mathbf{n}-\sigma(\nabla\cdot\mathbf{n})\mathbf{n}+(\mathbf{I}-\mathbf{nn})\cdot\nabla\sigma\quad$
$\displaystyle\mathrm{on}\ \partial\Omega_{F},$ (4a)
$\displaystyle\mathbf{u\cdot n}$ $\displaystyle=0\quad$
$\displaystyle\mathrm{on}\ \partial\Omega_{F},$ (4b) $\displaystyle
k\mathbf{n}\cdot\nabla T$
$\displaystyle=-h_{c}(T-T_{a})-n^{\prime}\mathcal{L}\quad$
$\displaystyle\mathrm{on}\ \partial\Omega_{F},$ (4c) $\displaystyle T$
$\displaystyle=T_{0}\ $ $\displaystyle\mathrm{on}\ \partial\Omega_{S},$ (4d)
with suitable boundary conditions on the flow axis of symmetry,
$\partial\Omega_{A}$, detailed in §III.3 for linear perturbations and reading
${u}_{r}={u}_{\theta}=0$ for the base axisymmetric case. $\mathbf{S}$ and
$\mathbf{S}_{\mathrm{gas}}$ are the stress tensors in the liquid and the gas,
respectively, $\sigma$ the surface tension, $\mathbf{n}$ the normal vector,
$k$ the conductivity, $h_{c}$ the convective heat transfer coefficient in air,
$n^{\prime}$ the evaporation rate and $\mathcal{L}$ the latent heat. Equation
4c expresses the heat flux balance at the interface stemming from the liquid
($j_{k}$) transferred into air ($j_{c}$) and into evaporative cooling
($j_{e}$), as schematized in Figure 3(b).
Both the liquid density $\rho$ and the surface tension $\sigma$ are assumed to
vary linearly with the temperature: $\rho(T)=\rho_{0}(1-\beta_{w}(T-T_{0})),\
\sigma=\sigma_{0}-\sigma_{1}(T-T_{0}),$ where $\beta_{w}$ is the water thermal
expansion coefficient, $\beta_{w}={\rho_{0}}^{-1}{\partial\rho}/{\partial T}$
and $\sigma_{1}$ the surface tension variation with the temperature,
$\sigma_{1}=-{\partial\sigma}/{\partial T}$. We decompose now the temperature
$T$ as:
$T=T_{0}-\Theta(r,z),$ (5)
where $T_{0}$ is the water boiling temperature taken as reference
($T_{0}=100^{\circ}\mathrm{C}$) at $z=0$ and $\Theta$ the deviation from
$T_{0}$. The reference density $\rho_{0}$ and the surface tension $\sigma_{0}$
are defined at $T_{0}$. With these definitions, the density and the surface
tension write:
$\displaystyle\rho(T)=\rho_{0}\left(1-\beta_{w}\Theta\right),\qquad\sigma=\sigma_{0}-\sigma_{1}\Theta.$
(6)
Denoting $p_{0}$ the hydrostatic pressure, the pressure $p$ inside the liquid
decomposes as $p=p_{0}(z)+p_{1}(r,z)$. The time, velocity and length are
scaled with $H^{2}/\kappa,\kappa/H$ and $H$, respectively where $H$ is the
drop height. The temperature is scaled with the temperature difference $\Delta
T(=|\Theta_{z=0}-\Theta_{z=H}|)$ and the pressure is scaled with
$\rho_{0}\nu\kappa/H^{2}$, which naturally appears when plugging eq. 7 in eq.
3a, and comparing the viscous term to the pressure term,
$r=H\hat{r},\quad z=H\hat{z},\quad
t=\frac{H^{2}}{\kappa}\hat{t},\quad\mathbf{u}=\frac{\kappa}{H}\hat{\mathbf{u}},\quad{\Theta}=\Delta
T\hat{\Theta},\quad
p=\frac{\rho_{0}\nu\kappa}{H^{2}}\hat{p},\quad\nabla=\frac{1}{H}\hat{\nabla}.$
(7)
We provide in the SI a Table S1 with the physical parameters relative to
water, vapor, air and their interface relevant to describe our problem. We
introduce the dimensionless numbers, Prandtl, Rayleigh, Marangoni, Biot and
Sherwood numbers as:
${Pr}=\frac{\nu}{\kappa},\quad{Ra}=\frac{\beta_{w}g\Delta
TH^{3}}{\nu\kappa},\quad Ma=\frac{\sigma_{1}\Delta
TH}{\rho_{0}\nu\kappa},\quad Bi=\frac{h_{c}H}{k},\quad
Sh=\frac{h_{m}H}{D_{va}}.$ (8)
The governing equation (3) can be now recast into
$\displaystyle
Pr^{-1}\left(\frac{\partial\hat{\mathbf{u}}}{\partial\hat{t}}+\hat{\mathbf{u}}\cdot\hat{\nabla}\hat{\mathbf{u}}\right)$
$\displaystyle=-\hat{\nabla}\hat{p}_{1}+Ra\hat{\Theta}\mathbf{e}_{z}+\hat{\nabla}^{2}\hat{\mathbf{u}}\quad$
$\displaystyle\mathrm{in}\ \Omega,$ (9a)
$\displaystyle\hat{\nabla}\cdot\hat{\mathbf{u}}$ $\displaystyle=0\quad$
$\displaystyle\mathrm{in}\ \Omega,$ (9b)
$\displaystyle\frac{\partial\hat{\Theta}}{\partial
t}+\hat{\mathbf{u}}\cdot\hat{\nabla}\hat{\Theta}$
$\displaystyle=\hat{\nabla}^{2}\hat{\Theta}\quad$ $\displaystyle\mathrm{in}\
\Omega,$ (9c)
with the boundary conditions:
$\displaystyle-\hat{p}_{1}\mathbf{n}+(\hat{\nabla}\hat{\mathbf{u}}+(\hat{\nabla}\hat{\mathbf{u}})^{T})\cdot\mathbf{n}$
$\displaystyle=-Ma(\mathbf{I}-\mathbf{nn})\cdot\hat{\nabla}\hat{\Theta}\ $
$\displaystyle\mathrm{on}\ \partial\Omega_{F},$ (10a)
$\displaystyle\hat{\mathbf{u}}\cdot\mathbf{n}$ $\displaystyle=0\ $
$\displaystyle\mathrm{on}\ \partial\Omega_{F},\partial\Omega_{S},$ (10b)
$\displaystyle\mathbf{n}\cdot\hat{\nabla}\hat{\Theta}$
$\displaystyle=-Bi\left(\hat{\Theta}+\frac{T_{0}-{T}_{a}}{\Delta
T}\right)-\frac{n^{\prime}\mathcal{L}H}{k}\ $ $\displaystyle\mathrm{on}\
\partial\Omega_{F},$ (10c) $\displaystyle\hat{\Theta}$ $\displaystyle=0\ $
$\displaystyle\mathrm{on}\ \partial\Omega_{S}.$ (10d)
and suitable symmetry conditions on $\partial\Omega_{A}$ (see 3.3 for
details). The evaporation rate $n^{\prime}$ is linked to the $Sh$ number,
which represents the evaporative heat transfer coefficient. The heat exchanges
on $\partial\Omega_{F}$, which determine $Bi$ and $Sh$ are modelled using the
Ranz-Marshall correlation (Ranz & Marshall, 1952; Bergman et al., 2011) as
detailed in the SI (§S2), where the input ambient air temperature $T_{a}$ is
also measured and provided in SI (SI) (§S4).
### III.3 Stability analysis
The steady toroidal baseflow ($\mathbf{u}_{b},p_{b},\Theta_{b}$) is obtained
by solving the nonlinear steady state solution of (3) satisfying the boundary
condition (4) using the Newton method. The baseflow is solved using the
dimensional equations since some dimensionless numbers depend on $\Delta T$,
which is also an unknown of the thermal problem.111Note that the non-
dimensional equation can be resolved by defining $Ra$ and $Ma$ with the known
parameters, i.e. $T_{s}$. However, we kept the classical definition of $Ra$
and $Ma$ which are with $\Delta T$. We first compute the baseflows, estimate
$\Delta T$ and deduce the dimensionless numbers. This approach differs from
thermoconvective studies on a flat plate or in a cylinder. In these
situations, there is no radial pressure gradient and the vertical gradient is
solely balanced with the density, without inducing any velocity field. The
temperature difference $\Delta T$ and thermal dimensionless parameters are
control parameters. In contrast, when the radial pressure gradient is nonzero
(as in the Leidenfrost configuration, owing to the boundary conditions), it
induces a steady flow with nonzero velocity. Both $\Delta T$ and the
dimensionless parameters are solutions of the problem. Assuming infinitesimal
perturbations on the baseflow with the complex frequency $\omega$ and the
azimuthal wavenumber $m$, the dimensionless flow field, pressure and
temperature are decomposed using the normal mode expansion:
$[\hat{u}_{r},\hat{u}_{\theta},\hat{u}_{z},\hat{p}_{1},\hat{\Theta}](r,\theta,z,t)=[{u}_{r},{u}_{\theta},{u}_{z},{p},{\Theta}](r,z)\exp(\mathrm{i}m\theta-\mathrm{i}\omega
t)+c.c.,$ (11)
where $c.c.$ indicates complex conjugate. The linearized equation (9) becomes
$\displaystyle
Pr^{-1}\left(\mathrm{i}\omega\mathbf{u}+\mathbf{u}_{b}\cdot\nabla_{m}+\mathbf{u}\cdot\nabla_{0}\mathbf{u}_{b}\right)$
$\displaystyle=-\nabla_{m}{p}+Ra\Theta\mathbf{e}_{z}+\nabla^{2}_{m}\mathbf{u}\quad$
$\displaystyle\mathrm{in}\ \Omega,$ (12a)
$\displaystyle{\nabla}_{m}\cdot\mathbf{u}$ $\displaystyle=0\quad$
$\displaystyle\mathrm{in}\ \Omega,$ (12b)
$\displaystyle\mathrm{i}\omega\Theta+\mathbf{u}_{b}\cdot{\nabla_{m}}\Theta+\mathbf{u}\cdot{\nabla_{0}}\Theta_{b}$
$\displaystyle={\nabla}_{m}^{2}\Theta\quad$ $\displaystyle\mathrm{in}\
\Omega,$ (12c)
where $\nabla_{m}$ represents the derivative in $\theta$ is replaced by
$\mathrm{i}m$. The boundary conditions on $\partial\Omega_{F}$ are
$\displaystyle-{p}\mathbf{n}+({\nabla}_{m}\mathbf{u}+({\nabla}_{m}\mathbf{u})^{T})\cdot\mathbf{n}$
$\displaystyle=-Ma(\mathbf{I}-\mathbf{nn})\cdot{\nabla}_{m}{\Theta}\quad$
$\displaystyle\mathrm{on}\ \partial\Omega_{F},$ (13a)
$\displaystyle\mathbf{u}\cdot\mathbf{n}$ $\displaystyle=0\quad$
$\displaystyle\mathrm{on}\ \partial\Omega_{F},$ (13b)
$\displaystyle\mathbf{n}\cdot{\nabla}_{m}{\Theta}$
$\displaystyle=-Bi{\Theta}\quad$ $\displaystyle\mathrm{on}\
\partial\Omega_{F}.$ (13c)
Note that the perturbation temperature is now only affected by the $Bi$ as the
other heat transfer coefficients are constant and contribute only in the zero
order baseflow equation. One could also linearize the last term in (4c) and
include in the stability analysis, but the effect on the stability results is
negligible (Yim et al., 2021). The condition prescribed on the drop axis of
symmetry $\partial\Omega_{A}$ (illustrated in figure 3) depends on $m$, in
particular on the mode symmetry. The following conditions are thus used as in
Batchelor & Gill (1962):
$\displaystyle m=0,$ $\displaystyle\quad u_{r}=u_{\theta}=\frac{\partial
u_{z}}{\partial r}=\frac{\partial\Theta}{\partial r}=0\quad$
$\displaystyle\mathrm{on}\ \partial\Omega_{A},$ (14a) $\displaystyle m=1,$
$\displaystyle\quad u_{z}=\Theta=p=\frac{\partial u_{r}}{\partial
r}=0=\frac{\partial u_{\theta}}{\partial r}=0\quad$ $\displaystyle\mathrm{on}\
\partial\Omega_{A},$ (14b) $\displaystyle m\geq 2,$ $\displaystyle\quad
u_{r}=u_{\theta}=u_{z}=\Theta=p=0\quad$ $\displaystyle\mathrm{on}\
\partial\Omega_{A}.$ (14c)
Finally, we prescribe on $\partial\Omega_{S}$, a Dirichlet condition for the
temperature and a free slip condition for the velocity since the bottom
surface is not in contact with the plate:
$\displaystyle\mathbf{u}\cdot\mathbf{n}$ $\displaystyle=0\quad$
$\displaystyle\mathrm{on}\ \partial\Omega_{S},$ (15a) $\displaystyle\Theta$
$\displaystyle=0$ $\displaystyle\mathrm{on}\ \partial\Omega_{S}.$ (15b)
### III.4 Numerical methods
All the numerical analyses are performed using FreeFEM++ software (Hecht,
2012) for axisymmetric cylindrical coordinates $(r,z)$. The velocity, pressure
and temperature are discretized with Taylor-Hood P2, P1 and P2 elements,
respectively. The typical number of triangles is $\sim 10^{4}$. The linear
equations and the eigenvalue problem are solved using UMFPACK library and
ARPACK shift-invert method, respectively. Starting with the initial radius
$R=3.5$ mm, the nonlinear solution of (3) is solved for the given water
properties. Once the solution for one radius is found, it is used as the
initial guess for the smaller radius. Within 5 iterations, the L2 norm
residual becomes smaller than $1\cdot 10^{-8}$. The zero normal velocity
condition on the free surface is applied using the Lagrange multiplier method
(Babuška, 1973; Yim et al., 2021).
## IV Numerical results
### IV.1 Pure buoyancy induced flow ($Ma=0$)
#### IV.1.1 Baseflow ($Ma=0$)
Figure 4: Baseflows for $Ma=0$: (a) $R=2$ mm, (b) $R=1.3$ mm and (c) $R=0.8$
mm. The temperature and velocity fields are shown in color and with arrows.
(d) The spectrum of growth rate $\omega_{i}$ with $R$.
Let us first consider the case of pure buoyant flows, neglecting
thermocapillary effects. The baseflow is thus computed following (3) while
setting the superficial stress on $\partial\Omega_{F}$ to zero. We restrict
this parametric study in $R$ to the limit $R\lesssim 2$ mm, since according to
IR measurements (Figure 2), the drop surface temperature tends to homogenize,
suppressing Marangoni surfaces flows. Figure 4 shows the baseflow obtained in
this limit of $Ma=0$, for drop radii $R=2$ mm, $R=1.3$ mm and $R=0.8$ mm,
respectively. In the absence of surface tension gradient, the baseflow
exhibits pure thermal convection: the flow rises along the center axis, warmer
since it is insulated from the drop interface, and descends along the side of
the drop, where the drop is cooled. The inner velocities for purely buoyant
flows ($\sim 1$ cm/s) underestimate the experimental observation ($\sim 5$
cm/s) of the similar size of drop (see Figure 5a,c for the experimental
measurements).
#### IV.1.2 Stability analysis ($Ma=0$)
The stability of purely thermobuoyant flows is herein considered. Figure 4(d)
shows the growth rate $\omega_{i}$ (imaginary part of the complex frequency
$\omega$) as a function of $R$ for the modes $m=0,1,2,3$. A positive growth
rate indicates the grow of the perturbations leading to the instability. As
shown in Figure 4(d), the $m=1$ mode is only unstable mode for $R>0.6$ mm. The
frequency $\omega_{r}$ of this mode (not shown) is zero, corresponding to a
steady unstable mode. The corresponding Rayleigh number $Ra$ decreases from
$20000$ to 10 as $R$ decreases from 2 to 0.5 mm (see Fig. S8b in SI), reaching
the value $Ra\sim 630$ when $R\sim 0.6$ mm, for which flows get stable.
Interestingly, this limit also corresponds to the radius where the droplet
ability to self-rotate and propel disappears, as noticeable in the last stage
of movie SM1. Both the measured propelling acceleration and the droplet base
asymmetry vanish below $R\lesssim 0.6$ mm until it stops (the measurement in
Figure 1(a) then ceases). This suggests that thermobuoyant effects become
stable to non-axisymmetric disturbances and flows stabilize as the drop size
reduces below a critical value. Figure 5 compares the $m=1$ unstable mode to
the flow fields obtained in a Leidenfrost drop with $R\sim 0.9$ mm. The mode
$m=1$ with a structure describes well the solid-like rolling motion in the
experimental observation.
Figure 5: (a,c) Velocity fields within a droplet with $R\sim 0.9$ mm deduced
from PIV measurements. (b, d) Corresponding flow fields deduced from the
numerical stability analysis in the absence of Marangoni effects ($m=1$,
$Ma=0$, $Ra=6.1\cdot 10^{3}$). Velocity arrows are plotted within the lateral
(a,b) and horizontal (c,d) planes. The red dashed line in (c) indicates the
area of the flattened base of the drop on which the bottom camera focuses.
Color in (b,d) indicates the perturbed temperature field (normalized with its
maximum value).
Although the $m=1$ unstable mode with $Ma=0$ represents well the rolling
motion of small drops observed in the experiments, it fails to describe the
presence of higher azimuthal modes for large drops. This prompts us to look at
the $Ma\neq 0$ case for larger radii.
### IV.2 Reduced Marangoni approximation
A major challenge for computing the baseflow in a Leidenfrost drop is to be
able to predict the surface tension distribution. When we take the exact
surface tension temperature dependence $\sigma_{1}$ provided in the Table S1
(SI), the Marangoni flows are largely overestimated compared to experimental
observations (see Figure S3 of the SI). It is known however that Marangoni
effects are very often markedly reduced (Hu & Larson, 2005a, 2006;
Dhavaleswarapu et al., 2010) or even almost absent (Marin et al., 2011;
Gelderblom et al., 2012; Dash et al., 2014).Based on measured quantities, we
first try to correct the surface tension temperature dependence $\sigma_{1}$.
To that end, we evaluate the Rayleigh $Ra$ and Marangoni $Ma$ numbers, which
compare the buoyancy and surface tension stresses in comparison to inertia,
respectively, as defined in (8). Using the parameters documented in Table S1
of the SI, they culminate to $Ra\sim 1.8\cdot 10^{5}$ and $Ma\sim 3.6\cdot
10^{5}$ for $\Delta T=25^{\circ}\mathrm{C}$, $H=4$ mm, $R=3.5$ mm. Both values
greatly exceed expected critical values for the onset for Rayleigh-Bénard and
Marangoni instabilities (see Table S2 in the SI) – typically $Ra_{c}\sim
O(10^{3})$ and $Ma\sim O(10^{2})$, for similar geometries (Chandrasekhar,
1961). As detailed in §S7 of the SI, the effective surface tension variation
$\sigma_{1,\rm eff}$ (and thus $Ma_{\rm\rm eff}$) is determined using
numerical analysis. We select $\sigma_{1,\rm eff}$ that best describes the
temperature difference $\Delta T$ within the liquid (Fig 2), as well as the
flow velocities, typically 5 cm/s. The temperature profiles reported in Figure
2 (left panels, $R=3.5$ mm and $R=3.0$ mm) are best represented by the curve
with $\sigma_{1,\rm eff}=4\cdot 10^{-5}\sigma_{1}$ as shown in Figure S7 of
the SI for the surface temperature and Figure 6 for velocity field. The
surface stress seems to be reduced by a few order of magnitude. This has also
been reported in Savino et al. (2002); Hu & Larson (2002, 2005b, 2006), where
it is ascribed to surface contamination, or to the fact that at large
Marangoni surface stress, it gets moderated by dissipation or by transport-
limited properties of the fluid, reducing the achievable velocities (see Fig.6
of the SI). In the following stability analysis, we thus use this reduced
surface tension variation by adjusting $\sigma_{1}$ to $\sigma_{1,\rm
eff}=4\cdot 10^{-5}\sigma_{1}$, which is the only tunable parameter of our
study (all other physical parameters are kept exact values of given thermal
properties, provided in the Table S1).
### IV.3 Thermocapillary flow
#### IV.3.1 Baseflow with effective Ma
The baseflow in Leidenfrost drops is now computed as in IV.1.1, adding to
thermobuoyant effects reduced thermocapillary effects ($\sigma_{1,\rm
eff}=4\cdot 10^{-5}\sigma_{1}$ and $Ma_{\rm eff}$) and shown in Figure 6
compared to the experimental measurement (Figure 6a). As the surface tension
gradient induces Marangoni flow from low surface tension to the higher one
along the surface boundary, the flow direction is opposed than purely buoyant
flow: it sinks on the center-line and rises along the surface. The typical
velocity magnitude is about $\sim 5$ m/s which is similar to the experimental
observation,which corroborates the choice of effective thermocapillary
gradient done by tuning the temperature distribution in the previous section
IV.2.
Figure 6: (a) PIV measuments in a drop with $R=2.5$ mm (Bouillant et al.,
2018b). (b) Baseflow for $Ma_{\rm eff}=11.3$ ($\sigma_{1,\rm eff}=4\cdot
10^{-5}\sigma_{1}$). The colormap and arrows give the inner temperature and
velocity.
#### IV.3.2 Stability analysis with effective Ma
Figure 7: Growth rates and the corresponding frequencies for (a,b) $m=1$,
(c,d) $m=2$ and (e,f) $m=3$ for the two least unstable modes as a function of
decreasing radius. The dominant modes for each $m$: (g) growth rate and (h)
corresponding frequency.
Figure 7 shows the dominant eigenvalues as a function of decreasing radius
(with $\sigma_{1,\rm eff}=4\cdot 10^{-5}\sigma_{1}$). Compared to the pure
buoyant flow, there exist several unstable azimuthal modes ranging from $m=1$
to 3. Figures 7(a,c,e) show the two least stable growth rates $\omega_{i}$ and
their corresponding frequencies $\omega_{r}$ are shown in Figures 7(b,d,f) for
azimuthal wavenumbers $m=1,2$ and 3. For all modes $m=1,2,3$, there exist two
branches of unstable modes: one with $\omega_{r}=0$ (steady) and other with
$\omega_{r}\neq 0$ (unsteady). The $m=1$ mode (Figure 7a,b) is unstable both
at large radius $R>2.8$ mm and small radius $R<1.3$ mm, but it shows a window
of stability for intermediate values of the radius. The prevailing unstable
mode for $R>2.8$ mm is unsteady, with frequency $\omega_{r}\sim 10\
\mathrm{s^{-1}}$, while for $R<1.3$ mm, it becomes steady.
Mode $m=2$ (Figure 7c,d) is only unstable in the intermediate radius range
$R=[2.3;1.3]$ mm. At radius $R\sim 2$ mm, the unsteady branch dominates, yet,
with a small frequency, but the steady branch takes over at smaller $R$. Mode
$m=3$ (Figure 7e,f) is unstable when $R<2.8$ mm. Its steady branch dominates
at large $R$ but it becomes unsteady as the radius gets closer to $R\approx
2.8$ mm. The unsteady branch of mode $m=3$ reaches a maximum growth rate
around $R=2.4$ mm.
Finally, Figures 7g,h collect and retain only the most unstable mode $m$ and
their corresponding frequencies. We note in Figure 7g that the axisymmetric
mode $m=0$ is always stable ($\omega_{i}<0$). The dependence of the dominant
azimuthal mode on radius thus becomes explicit: for $R>2.1$ mm, the mode $m=3$
is the most unstable (first steady and then oscillatory). As the radius
decreases further, the $m=3$ mode becomes stable around $R=2$ mm. At $R=2.2$
mm, the mode $m=2$ starts to grow and becomes the only unstable mode in the
range of radii $R=[2;1.3]$ mm, with the maximum growth rate at $R=1.6$ mm. For
$R<1.3$ mm, the steady mode $m=1$ takes over as the dominant unstable mode.
This trend is very similar to the apparent mode transition in experiments,
which is $m\geq 3$ for $R\gtrsim 2.8$ mm, $m=2$ for $R\approx[2.5;1.8]$ mm and
$m=1$ for $R\approx 1.5$ mm, as shown in Figure 1.
Figure 8: Eigenmodes of the most unstable modes at radii (a,d) $R=3$ mm
($Ma_{\rm eff}=12,Ra=1.5\cdot 10^{5}$), (b,e) $R=1.5$ mm ($Ma_{\rm
eff}=9.6,Ra=5\cdot 10^{4}$) and (c,f) $R=1$ mm ($Ma_{\rm eff}=8,Ra=2.4\cdot
10^{4}$). Colors indicate temperature perturbations and the arrows show the
in-plane velocity perturbations normalized with their maximum real values. The
top view is a plane cut at the maximum radius $R=R_{\max}$.
A growth rate $\omega_{i}\sim 1~{}\mathrm{s^{-1}}$ implies that within 2.3 s,
the perturbation amplitude becomes 10 times larger than its initial value. For
all modes, we verify that the growth rate is much faster than the drop
evaporation rate $\omega_{i}\gg(1/R)\mathrm{d}R/\mathrm{d}t$, underlying the
quasi-steady assumption in §II.1.
Figure 8 illustrates the eigenvectors of the most unstable mode at radii
$R=3,\ 1.5$ and $1$ mm. The top panel displays side-views while the right one
shows top-views: $xy$ plane cut at the maximum radius $R=R_{\max}$. The colors
indicate temperature perturbations and the arrows are the in-plane velocity
perturbations. For $R=3$ mm (Figure 8ad), the mode $m=3$ dominates.
Temperature perturbations are localized along the interface and the central
axis of the drop while velocity perturbations are localized where the
temperature perturbations are the weakest. The top view shows three counter-
rotating vortex pairs, very similar to our observations (Figure 1c). For
$R=1.5$ mm (Figure 8be), the mode $m=2$ becomes the most unstable thus
dominant mode. Temperature perturbations are maximum near the central axis,
while velocity perturbations remain localized where temperature perturbations
are weak. The top view shows 4 vortex cells (or 2 vortex pairs), in close
agreement to the experiments (Figure 1d). For $R=1$ mm (Figure 8cf), the
displacement mode $m=1$ is the most unstable. Temperature perturbations are
null on the central axis, where velocity perturbations are maximum. However,
its structure does not match the solid-like rolling motion reported in
Bouillant et al. (2018b) and illustrated in the top of Figure 1e. We interpret
this at a consequence of the temperature homogenization evidenced at small $R$
in Figure 2. The temperature difference at the liquid surface indeed tends to
vanish as $R$ becomes smaller than 1.8 mm. We thus expect thermocapillary
effects to weaken and eventually vanish. We provide in Figure S8 (SI), the
(reduced) Marangoni and Rayleigh numbers with $R$, showing that both effects
weaken as the drop shrinks in size, with a more abrupt decrease of
thermocapillary effects than thermobuoyant effects, prompting us to use our
predictions for the case $Ma=0$ when $R$ becomes sufficiently small. The
observed rolling motion is thus better captured by the eigenmode for $Ma=0$ as
shown in Figure 5 than the one with $Ma\sim O(10)$ in Figure 8e. Moreover, the
drop shape is, as yet, steady, but the internal flows could be coupled to the
drop envelop deformation. This extension of our model would enable the
exploration of the limit of even larger drops ($R\gtrsim 5$ mm), where star-
pulsations appear. Finally, it would be interesting to see how our model
applies to the inverse-Leidenfrost situation (Gauthier et al., 2019), for
which a rolling mode $m=1$ seems dominant in the millimetric, quasi-spherical
droplet. Their configuration is however essentially different since
temperature gradients are reversed, the drop shape remains quasi-spherical on
the deformable bath, and, as soon as the drop freezes, both the thermo-buoyant
and thermo-capillary flows should extinguish.
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|
# The Dynamics of Quantum Fluids
Henri Godfrin and Eckhard Krotscheck
###### Abstract
We review experimental and theoretical progress in the physical description of
the dynamics of the quantum fluids 4He and 3He. Historically, the elementary
excitations in these systems have been identified as phonons and rotons and,
in 3He, collective zero sound and spin-fluctuations. Both recent high-
precision measurements and theoretical methods have shown that the dynamics of
these systems is actually very rich as will be discussed in detail in the body
of this contribution.
Keywords— Quantum Fluids, Liquid Helium, Fermions, Bosons, Neutron Scattering,
X-Ray scattering, Elementary Excitations, Phonons, Rotons, Ripplons
## 1 The Helium Liquids
Helium (4He and the less common isotope 3He) and hydrogen are simple condensed
matter systems, amenable to fundamental quantum mechanical descriptions.
Helium remains liquid even at the absolute zero of temperature, an effect
explained by Heisenberg’s uncertainty principle, which has the consequence
that light atoms confined in a small volume have a large kinetic energy, as is
the case in a liquid or a solid, where each atom is confined by several
surrounding atoms. For this reason, in helium, kinetic energies are larger
than the weak interatomic potential energy. Even at the lowest temperatures,
substantial atomic motion is present, and the system remains liquid, unless
pressures on the order of 25 MPa are applied, and solidification is achieved.
Another quantum effect plays an important role in the behavior of liquid
helium: as the temperature is reduced below a few degrees Kelvin, the thermal
de Broglie wave-length of the atoms becomes larger than the interatomic
distance. The waves describing the helium atoms overlap significantly, and a
description in terms of individual wave-packets of distinguishable particles
becomes inappropriate, it must be replaced by a quantum many-body description
in terms of indistinguishable particles, governed by quantum statistics.
Quantum Mechanics and Statistical Physics [1, 2] explain well the properties
of Quantum Gases, i.e. ensembles of non-interacting particles, which
necessarily belong to one of the two possible classes: Bosons having an
integer spin quantum number s), or fermions with half-integer spin. 4He atoms
($s$=0) are bosons, and 3He atoms ($s$=$\frac{1}{2}$) are fermions.
The wave-functions of particles confined in a volume $V$ are quantized plane
waves characterized by their wave-vector ${\bf k}$. As shown by A. Einstein,
the ground state of a Bose gas is the BEC (Bose-Einstein condensate), where
all particles occupy the same microscopic state with ${\bf k}$=0. The ground
state of a Fermi gas is very different, because Pauli’s principle forbids
double occupancy for Fermions: the particles form a Fermi sphere in wave-
vector space, by completing successive energy levels of increasing wave-
vector, up to the Fermi wave-vector $k_{\rm F}$. The energy $E_{\rm F}$ of the
last one, called the Fermi level, can be considerable.
The excitation spectrum $\epsilon(k)$ of these quantum gases is simply given
by the kinetic energies of the individual particles,
$\epsilon(k)=\hbar^{2}k^{2}/2m$, where $m$ is the particle mass and $k$ the
the wave number. For bosons, at low temperatures, particles are excited from
the common ground state. The energy spectrum is the usual parabolic spectrum
of free particles described above. This behavior, however, is profoundly
modified by interparticle interactions, i.e., as is the case in Quantum Fluids
[1, 2, 3, 4].
The Helium quantum fluids are unique in the sense that they are very dense and
very quantum, making them a very challenging problem for theory. The issue is
made quantitative by looking at the simple Lennard-Jones model of the helium
fluids. In this model, the helium atoms interact via the interaction
$V(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]\equiv\epsilon
v\left(\frac{r}{\sigma}\right)$
where $\epsilon=10.22\,$K is the well depth of the potential and
$\sigma=2.556\,$Å is the core size [5].
The interaction is not the very best (quantitative calculations are in fact
made using the Aziz potential [6]), but it reproduces the ground state
properties of the liquids within a few percent. Measuring all energies in
units of the well depth, and all lengths in units of the core size, $x\equiv
r/\sigma$, the Hamiltonian of the system is
$\displaystyle\frac{1}{\epsilon}H({\bf x}_{1}\ldots,{\bf x}_{N})=$ (1)
$\displaystyle-\frac{\Lambda^{2}}{8\pi^{2}}\sum_{i}\nabla_{{\bf
x}_{i}}^{2}+\sum_{i<j}v(|{\bf x}_{i}-{\bf x}_{j}|)$
where
$\Lambda=\left(\frac{h}{m\epsilon\sigma^{2}}\right)^{\frac{1}{2}}$
is de Boer’s “quantum parameter” [7] $\Lambda$, giving the ratio between the
de Broglie wavelength and a typical diameter of the molecule. $\Lambda\approx
2.67$ for 4He, $\Lambda\approx 3.09$ for 3He, but $\Lambda\approx 1.3-1.7$ for
H2, HD, D2, and $\Lambda<0.1$ for heavy rare gases.
Moreover, the density of 4He at zero pressure is
$\rho_{0}=0.02185\,$Å${}^{-3}=0.365/\sigma^{-3}$ and for 3He
$\rho_{0}=0.0163\,$Å${}^{-3}=0.273/\sigma^{-3}$, in other words the average
particle distance is of the order of the core size.
Figure 1: Normalized Lennard-Jones “6-12” interatomic potential, as a function
of the normalized interatomic distance. The normalization factors for
distances ($\sigma$) and potential energies ($\epsilon$) are given in the
text. We also show, for comparison, the Aziz potential [6] which is the
currently accepted most accurate static 2-body interaction.
## 2 4He in 3D
Liquid helium (4He) is the archetype of a Bose Liquid. Immediately after the
discovery of the superfluidity of helium [8, 9] by Kapitza (Moscow) and Allen
and Misener (Toronto), F. London [10] understood that the system was
undergoing Bose-Einstein condensation, and L. Tisza proposed a “two-fluids
model” featuring a coexistence of a normal fluid and a superfluid [4]. A Bose-
Einstein condensed gas is not superfluid, however, and therefore the
superfluidity observed in helium is an additional effect, associated to the
interactions. A major step in the understanding of the physics of helium was
made by L.D. Landau, who found a deeper interpretation of Tisza’s two-fluids
model. Landau postulated that the “normal fraction”, which carries the
entropy, is in fact the ensemble of thermal excitations of the superfluid
ground state. This interpretation removed the unphysical consideration of two
categories within a set of indistinguishable atoms. Landau considered
elementary excitations of the fluid, in the form of quantized density
fluctuations (“phonons”), similar to the Debye phonon modes of a crystalline
solid. In order to calculate the thermodynamic properties by a Debye-like
approach, where the energy is obtained by a sum over all modes, weighted by
the Bose factor, Landau postulated that the dispersion relation $\epsilon(k)$
had a linear part $\epsilon=\hbar ck$, where $c$ is the speed of sound, and
another part displaying a minimum, the “roton gap” of energy $\Delta$.
Landau’s intuition was guided by specific heat measurements, showing a cubic
temperature dependence at low temperatures (below 0.5 K), as expected from the
linear part of the dispersion relation, as well a an exponential contribution
at higher temperatures (on the order of 1K), characteristic of an energy gap.
In a first article [11], Landau proposed a spectrum with two branches, with a
parabolic branch centered at k=0 in addition to the linear branch. This
spectrum was not found satisfactory by Landau himself, the presence of two
branches being reminiscent of Tisza’s “two kinds of atoms”.
In a second publication [12], he presented his celebrated “Landau dispersion
relation”, where a single branch of excitations characterizes the full
spectrum (Fig. 2). With two parameters, the sound velocity $c$ and the roton
gap $\Delta$, both obtained from the temperature dependence of the specific
heat, Landau’s theory provides an excellent account of the experimental
properties of superfluid helium at low temperatures. The existence of a
“second sound” collective mode, where the superfluid and normal components
oscillate with opposite phases while keeping the total density constant, as
proposed by Tisza, corresponds in Landau’s model to propagating entropy (and
hence temperature) waves.
Figure 2: The dispersion relation proposed by L. D. Landau in 1947 to describe
the elementary excitations in superfluid 4He at zero pressure.
Another fundamental observation due to Landau, is that the particular shape of
the dispersion relation leads to the superfluid behavior. A simple argument of
energy and momentum conservation shows that the creation of excitations by the
displacement of an object immersed in the fluid is not possible, unless the
object’s velocity exceeds a critical value. The linearity of the dispersion
relation at low wave vectors is essential for this to happen: in a Bose gas,
where the dispersion relations is parabolic, the critical velocity is zero,
the system remains normal even at the absolute zero of temperature. The
evolution of the dispersion relation from the parabolic shape characteristic
of a gas, to the linear behavior observed as the interaction is switched on,
has been explained by Bogoliubov (1947) [13] in the weak interaction limit.
The dispersion relation of the elementary excitations plays therefore a major
role in the determination of the thermal properties, and also in the
superfluid nature of the liquid’s ground state.
Landau’s phenomenological theory introduced the fruitful concept of
“quasiparticles”, describing in a simple way the elementary excitations of
complex many-body systems. The concept was of course already present in the
Debye theory of solids, but the generalization to helium, a non-periodic
system of strongly interacting particles, opened a new field of physics.
There were however two serious interrogations after Landau’s theory was
released. The first was obviously related to the necessity of an experimental
observation of the proposed dispersion curve. And the second, to the need of a
microscopic theory providing some insight in the superfluid helium behavior,
and able to test the validity of Landau’s model.
Experimentally, the evidence for a Landau-like dispersion relation was
missing: only indirect evidence obtained form thermodynamic measurement
(specific heat, thermal conductivity, viscosity, fountain pressure, etc.) was
available. Feynman and Cohen suggested the use of inelastic neutron
scattering, a technique already used to measure phonon dispersion curves in
crystals, to try to observe the excitation dispersion curve of superfluid
helium. The first proof of the existence of a roton minimum was reported by H.
Palevsky, K. Otnes, K. E. Larsson, R. Pauli and R. Stedman (1957) [16, 17],
clearly showing that the elementary excitations in superfluid helium are
extremely sharp, compared to those of normal helium. These exciting results
motivated extensive investigations of the dynamics of helium, covering the low
energy region, i.e. the Landau-like dispersion relation, the multi-excitations
region at higher energies, as well as the very high energy region of the
spectrum, where BEC can be investigated [18].
Figure 3: The time-of-flight neutron scattering spectrometer IN5 at the
Institut Laue-Langevin.
The principle of the inelastic neutron scattering is simple, but measurements
are demanding, and high flux neutron sources (reactors, spallation sources)
are needed. The helium sample is placed in an incident beam of “monochromatic”
(of a given energy) neutrons, which are scattered by the helium sample, and
then detected at some distance from the sample, as a function of angle and
time of arrival, in the so-called time-of-flight technique. Other methods
(triple-axis, back-scattering, spin-echo, etc.) have also been used, offering
a different coverage of the momentum-energy plane, and also different
resolutions in energy and momentum.
Energy and momentum conservation allow the determination of the energy
transfer and the momentum transfer from the neutron to the helium sample,
which correspond to the energy $\epsilon=\hbar\,\omega$ and momentum
$\hbar\vec{k}=\hbar\vec{Q}$ of the density fluctuations created in the helium.
The incident neutron has an energy $E_{i}$ and a wave-vector $\vec{k_{i}}$,
and, after scattering, a final energy $E_{f}$ and a wave-vector $\vec{k_{f}}$;
the wave-vector transfer is $\vec{Q}=\vec{k_{i}}-\vec{k_{f}}$, and the energy
transfer $\hbar\,\omega=E_{i}-E_{f}$. The first microscopic theory of
superfluid helium, due to R. P. Feynman (1954) [14], is a wonderful example of
physical intuition. He found a general form for a variational wave-function
which successfully explained the general shape of the dispersion relation,
including the roton minimum. It was soon completed by Feynman and Cohen [15],
who introduced, improving the variational wave-functions, the concept of
“backflow”: the flow of helium atoms around a moving helium atom.
Dealing with many-body correlations, including dynamic effects, has been
central to the development of microscopic theories, which have been
considerably refined in the last decades, reaching now a quantitative power of
prediction, as shown below.
Figure 4: Neutron scattering: an incident neutron of wavevector $\vec{k_{i}}$
creates an excitation of momentum $\hbar\vec{k}=\hbar\vec{Q}$ in the
superfluid. Q is the wavevector transfer (see text)
The theory of neutron scattering [19, 20, 18] relates the double differential
scattering cross section per target atom, which is the parameter measured by a
scattering experiment, to the dynamic structure factor $S(Q,\omega)$ and the
dynamic susceptibility $\chi(Q,\omega)$, the latter being the functions
calculated by microscopic theories. They are given by the following
expression, where $b_{c}$ is the coherent scattering length of helium:
$\frac{\partial^{2}\sigma}{\partial\Omega~{}\partial{E_{f}}}=\frac{b^{2}_{c}}{\hbar}\frac{k_{f}}{k_{i}}S(Q,\omega)$
(2)
Results are generally presented in terms of $S(Q,\omega)$, where sharp peaks
of high intensity are observed, as expected from the creation of Landau
“single-excitations”, as well as a more complex, broad landscape of multi-
excitations. A vast literature on the subject exists, as described in the
review articles by Woods and Cowley, Stirling, Glyde (see [18]), the book by
Glyde [20], and recent works [21, 22], which benefited from the substantial
progress of neutron scattering facilities and instruments.
The dispersion relation $\epsilon(k)$ of the sharp single-excitations, as
measured in recent inelastic neutron scattering experiments, is shown in Fig.
5. It closely resembles the curve predicted by Landau, with a linear part at
low wave-vectors (“phonons”), and a deep gap (“roton” minimum) at a wave-
vector $k$$\sim$2Å-1. The name roton is historical, no special rotation takes
place; the minimum in energy reflects an incipient localization (short-range
order), which eventually will lead to a liquid-solid transition under a
pressure $P$$\sim$2.5 MPa. The dispersion curve becomes flat for $k$$\geq$2.8
Å-1 as the energy reaches twice the roton gap, forming “Pitaevskii’s plateau”.
---
---
Figure 5: The dispersion relation $\epsilon(k)$ of excitations in superfluid
4He at zero pressure. Top: Experimental data, displaying the phonon, roton,
maxon, and plateau regions of the spectrum. The red dots are recent high-
accuracy, highly pixelized data (seen at this scale as a continuous line, the
dispersion is only visible at the highest wave-vectors) from Ref. [22], the
blue dots with error bars are from the original work by Cowley and Woods [23]
and the green dots from the high-momentum measurements by Glyde et al. [24].
Bottom: theoretical results, from Feynman’s model [14] to modern calculations
such as Brillouin-Wigner perturbation theory with correlated wave function
(BW, Ref. [25, 26]), diffusion (DMC [27, 28]) and path-integral (PIMC, [29])
Monte Carlo calculations and dynamic many-body theory (DMBT) [30, 21],
compared to the experimental curve [22] .
The calculation of the low temperature ($T<1.3\,K$) thermodynamic properties
of helium at low temperatures only involves the statistical physics of phonons
and rotons of low energy. That is, the properties of the strongly interacting
bosonic system are simply given by those of a collection of non-interacting
bosonic “quasi-particles” described by the phonon-roton curve [22]. At high
temperatures, where the number of rotons becomes very large, roton-roton
interactions start playing a role. At the “lambda point” (T=2.17 K), 4He
becomes a normal fluid.
Theoretical studies of the dynamic structure function in 4He began with the
work of Feynman [14] and Feynman and Cohen [15], the Feynman theory of
elementary excitations was developed in a systematic Brillouin-Wigner
perturbation theory by Jackson and Feenberg [31, 25, 32, 33] in terms of a
basis of Feynman excitation states
$\left\\{\left|{\bf k}\right\rangle\right\\}=\left\\{S^{-1}(k)\hat{\rho}_{{\bf
k}}\left|\Psi_{0}\right\rangle\right\\}$ (3)
where $\hat{\rho}_{\bf k}$ is the density operator and
$\left|\Psi_{0}\right\rangle$ is the ground state. An important contribution
was the identification of classes of theories for the dynamic structure
function [34] that satisfy the $\omega^{0}$ and $\omega^{1}$ sum rules
exactly. The most complete evaluation of the phonon–roton dispersion relation
in terms of Brillouin–Wigner perturbation theory was carried out by Lee and
Lee [26] who obtained an impressive agreement with the experimental
phonon–roton spectrum up the wave number of $Q=2.5$Å-1. The major drawback
with these old calculations was that the required input – pair- and three-body
distribution functions – were poorly known. Manousakis and Pandharipande [35]
tried to generalize the input states of the Brillouin-Wigner perturbation
theory to include “backflow” correlations in the spirit of Feynman and Cohen
[15]. Through the gradient operator acting on the wave function, in principle
dynamic correlations are introduced to all orders. The “backflow–function” is,
however, chosen per physical intuition rather than by fundamental principles,
and the evaluation of the perturbative series becomes very complicated.
Topologically, diagrams similar to those of Lee and Lee [26] were included.
While the accuracy of the theoretical roton energy is comparable to that of
Lee and Lee, one can clearly see an inconsistency since the energy of the
Pitaevskii-plateau [36] lies below twice the energy of the roton gap. More
recent progress [37, 30] used a hybrid approach of Brillouin–Wigner
perturbation theory and equations of motion for time–dependent multiparticle
correlation functions to derive a self- consistent theory of the dynamic
density–density response of 4He.
In a nutshell, the Feynman theory of excitations is generalized by writing the
dynamic wave function as
$\displaystyle\Psi({\bf r}_{1},\ldots,{\bf r}_{N};t)$ $\displaystyle=$
$\displaystyle\exp(U(t))\Psi_{0}({\bf r}_{1},\ldots,{\bf r}_{N})$ (4a)
$\displaystyle U(t)$ $\displaystyle=$ $\displaystyle\sum_{i}\delta u_{1}({\bf
r}_{i};t)+\sum_{i<j}\delta u_{2}({\bf r}_{i},{\bf r}_{j};t)\ +\ldots\,.$ (4b)
The Feynman form of the wave function is obtained by omitting all $n$-body
fluctuations $\delta u_{n}({\bf r}_{1},\ldots{\bf r}_{n};t)$ for $n\geq 2$.
The different theoretical descrptions basically differ by the way the $n$-body
fluctuations are determined.
Recent novel numerical methods [28, 27, 38, 39, 40] give access to dynamic
properties of quantum fluids. These are important algorithmic developments
that may ultimately aid in the demanding elimination of background and
multiple-scattering events from the raw data. Of course, it is generally
agreed upon that the model of static pair potentials like the Aziz interaction
[6] describes the helium liquids accurately. Hence, given sufficiently
elaborate algorithms and sufficient computing power, such calculations must
reproduce the experimental data. The aim of the works cited above is
different: The identification of physical effects like phonon-phonon, phonon-
roton, roton-roton, maxon-roton …couplings that lead to observable features in
the dynamic structure function is, from simulation data, only possible
a-posteriori whereas the semi-analytic methods permit a direct identification
of these effects, their physical mechanisms, their relationship to the ground
state structure, and their consequences on the analytic properties of the
dynamic structure function, directly from the theory.
The dynamic structure of 4He is actually quite rich as seen in Fig. 6 [30,
21]: “multi-excitations” are observed at energies above the single-excitation
dispersion curve discussed above. They can be related to combinations of
single-excitations, some of them are indicated in Fig. 6 by colored ellipses:
in blue, an extension of the linear phonon region (“ghost phonon”) deep into
the continuum; in red, an extension of the Pitaevskii-Plateau to small wave-
vectors; in green, above the roton minimum, a broadening of the dispersion
relation due to a Cherenkov effect; a red dashed ellipse at high energies
indicates a region of maxon-roton couplings. Multi-excitations are well
reproduced by recent theories. The mechanisms behind these are mostly the
same: It is kinematically possible that a perturbation of energy/momentum
$(E,{\bf k})$ can decay into two sharp quasiparticle excitations. For example,
the “ghost phonon” is caused by the effect that a perturbation can decay into
two almost collinear phonons, hence the effect disappears at about twice the
energy and wave number at which the dispersion relation $\epsilon(k)$ ceases
to be linear. Similarly, the “Pitaevskii-Plateau” is caused by the effect that
a perturbation of energy $2\Delta$ can decay into two rotons of energy
$\Delta$ and wave number $k_{\Delta}$. Hence, energy and momentum conservation
dictate that the plateau ends at a wave number of $2k_{\Delta}$, in this case
the two roton wave vectors are parallel. Perturbations with the same energy
and smaller wave numbers can decay into two rotons that are not collinear.
That means that the plateau can in principle extend to zero wave number; in
that case the perturbation decays into two anti-parallel rotons.
The Cherenkov effect is particularly interesting; it is caused by the fact
that the group velocity of a quasiparticle excitation beyond the roton
($R_{+}$ roton) is larger than the sound velocity. The effect is observed only
at low pressures.
Figure 6: Experimental (left) and theoretical (right) determinations of the
dynamic structure factor of superfluid 4He at zero pressure (upper graphs) and
near-solidification pressure (lower graphs) [30, 21]. In addition to the the
intense single-excitations dispersion seen in Fig. 5 (red lower curves), much
lower intensity multi-excitations (see color scale) can be observed here.
Colored ellipses indicate excitation couplings described in the text.
## 3 3He in 3D
The investigation of the dynamics of the 3He Fermi Liquid has naturally
followed that of the 4He Bose fluid [41, 20]. From the theoretical point of
view, the problem is notoriously difficult, due to the antisymmetry
requirement for the wave-functions of many-body fermionic systems [1, 2, 3].
Obtaining accurate results for the ground state is a challenge, even using
very sophisticated variational wave functions.
The dynamics is treated analogously to the boson case (4b); the local
excitation functions $\delta u_{n}({\bf r}_{1},\ldots,{\bf r}_{n};t)$ are
replaced by particle-hole operators
$\displaystyle\sum_{\genfrac{}{}{0.0pt}{1}{p_{1},\ldots,p_{n}}{h_{1}\ldots
h_{n}}}u_{p_{1},\ldots,p_{n};h_{1}\ldots h_{n}}(t)\times$
$\displaystyle{\vspace{-1cm}\quad\times a^{\dagger}_{p_{1}}\dots
a^{\dagger}_{p_{n}}a_{h_{n}}\dots a^{\dagger}_{h_{1}}}$
where the $p_{i}$ are the quantum numbers of unoccupied (“particle”) states
and the $h_{i}$ those of occupied (“hole”) states. Restricting the excitation
amplitudes to one-particle-one-hole components leads to a correlated version
of the random phase approximation (RPA) [42], the quantitative understanding
of the experiments requires at least 2-particle-2-hole amplitudes, see Refs.
[43, 44].
From the experimental point of view, even though the same inelastic neutron
techniques successfully applied to superfluid 4He can be used to investigate
liquid 3He, in practice the experiments are extremely difficult due to the
huge neutron absorption cross-section of the 3He nucleus [20]. In typical 4He
experiments, sample thicknesses are larger than 1 cm, while absorption limits
the maximum thickness in a liquid 3He measurement, to about 0.1 mm. In
addition, due to the lower excitation energies, experiments in 3He must be
performed at much lower temperatures ($\approx 0.1$ K) than in 4He ($\approx
1$ K), to remain in the low temperature limit, where few excitations are
present. For these reasons, in spite of the considerable interest in strongly
interacting fermions, accurate theoretical and experimental results have been
available only recently [45].
In order to understand the dynamics of a Fermi liquid, we begin by considering
the excitations of a Fermi gas. In the latter, excitations are created by
removing a particle from an occupied state inside the Fermi sphere, and
placing it outside the sphere, in a higher energy free state. The resulting
states are confined to a region of the energy vs wave-vector space called the
“particle-hole band”. The interacting fermionic system, i.e. the Fermi Liquid,
behaves in a similar way. A particle-hole band can be observed in bulk 3He
(see Fig 7), but in addition, collective modes are present. A density mode,
named “zero-sound” or “collision-less sound”, is observed. Its physical
origin, as noted by D. Pines [46], resides in the strong interactions, and
hence this mode is analogous to the phonon-roton mode described above for
bosons; statistics play here a minor role. Its substantial energy width
results from the possibility to decay into particle-hole excitations as it
enters the particle-hole band (Landau-damping mechanism).
The thermodynamic properties of bulk liquid 3He are often described, at very
low temperatures, by the phenomenological Landau Fermi Liquid model, where the
properties of the interacting system are related to those of the Fermi gas by
the introduction of an interaction function (“quasiparticle interaction”),
leading to a renormalization of parameters (effective mass, susceptibility
enhancement, etc.). Landau theory predicts, among other physical effects, the
existence of collective modes, the zero-sound and paramagnon modes.
Landau’s Fermi Liquid Theory is only applicable at very low temperatures. In
liquid 3He, where the Fermi energy, expressed in temperature units, is of the
order of several kelvins, the theory is valid below 0.1 K.
Figure 7: (Color online) Dynamic structure factor of bulk liquid 3He. Measured
density excitations [47] (triangles, upper branch) and spin-density
excitations [45] (squares, lower branch) are clearly visible (bars on the data
points indicate the observed width of the branch). Color intensity map, from
light blue to dark red: DMBT microscopic theory including exchange [44]. The
solid lines indicate the boundaries of the particle-hole band.
The reason becomes clear if one observes the actual excitation spectrum shown
in Fig. 7. The low energy mode, observed below the zero-sound excitation, is a
spin-density mode, completely immersed in the particle-hole band, and hence
severely Landau-damped. The origin of this mode, analogous to a dampened spin-
wave and often called “paramagnon”, can be traced back directly to the
antisymmetry of the many-body wave function.
There are also some interesting X-fay studies of the dynamic structure
function at high momentum transfers [48] that have led to a discussion about
the location of the particle-hole band [49, 50]. The authors of Ref. [48]
state that The obtained results show no evidence of such a decay: the zero-
sound mode remains well defined in the whole explored wave number range.
According to that, the particle-hole continuum should be much lower than the
continuum of the non-interacting Fermi system. An analysis of both the data
and the theoretical predictions within DMBT [51] shows that there is, when the
theoretical data are convoluted with the experimental resolution, actually no
contradiction to the conventional interpretation of the scenario.
Figure 8: The figure shows the normalized dynamic structure function
$S(k,\omega)/S(k)$ of 3He at a density of $\rho=0.0166$Å-3, for a sequence of
wave numbers. The theoretical results obtained by DMBT [51] are compared with
the X-ray scattering data [48] (squares) at saturated vapor pressure. Also
shown are the theoretical results convoluted with the experimental resolution
(dashed lines). The gray-shaded area shows the particle-hole continuum of a
non-interacting Fermi fluid. $t_{\rm F}=\hbar^{2}k_{\rm F}^{2}/2m$ is the
Fermi energy of the non-interacting system.
Details are shown in Fig. 8. To facilitate the comparison with experiments,
the theoretical spectra were convoluted with the experimental resolution.
Also, the results were scaled by $1/S(k)$ such that the integrated strength is
1 for all momentum transfers. The same scaling was applied to the experimental
data.
## 4 4He in reduced dimensions
The dynamics of quantum fluids has also been investigated in reduced
dimensions. Two-dimensional (2D) systems are obtained experimentally by
adsorption of gases onto solid substrates, usually graphite. One-dimensional
(1D) systems can also be created, they are obtained by confining the gases
inside silica or graphite nano-tubes [52, 53, 54]. There is also theoretical
interest [55, 56, 57, 58, 59] in the properties of 4He in one-dimension.
Extensive theoretical studies have clarified the nature of the excitations in
such systems. Superfluid 4He films of atomic thicknesses, for instance,
display in-plane density modes, as well as capillary waves.
Let us first discuss the dynamics of rigorously 2D 4He. Basically we see the
same effects as in 3D: a linear “phonon” branch as well as, at high momentum
transfers, a “roton minimum”.
Figure 9: (Color online) The figure shows contour plots of the dynamic
structure function of two-dimensional liquid 4He for a sequence of areal
densities as shown in the legends. The colors have been chosen to highlight
the prominent features, darker colors correspond to higher values of
$S(k,\hbar\omega)$. We also show the Feynman spectrum, and the simulation data
of Ref. [60]. From Ref. [61].
One can also see the somewhat finer details discussed above: At low densities,
the “ghost-phonon”, and at high densities the plateau coming down and
extending to long wave lengths. A second striking feature is the appearance of
a sharp mode below the plateau. We stress the difference: normally, the
plateau is a threshold above which a wave of energy/momentum $(\hbar\omega,k)$
can decay into two rotons. This has the consequence that the imaginary part of
the self-energy $\Sigma(k,\hbar\omega)$ is a step function and the real part
has a logarithmic singularity [36]. A collective mode is, on the other hand,
characterized by a singularity of the $S(k,\hbar\omega)$. Figs. 9 show, for
the two highest densities, the appearance of a sharp discrete mode below the
plateau. At a wave number of $k\approx 2.6\,$Å-1, the collective mode is still
merged into the continuum. With increasing wave number, we see, however, a
clearly distinguishable but rather weak ($Z(k)\approx 0.02$) mode about 0.3 K
below the plateau. We have mentioned already above that the roton should be
seen as an emergence of short–ranged order, the “ghost of a Bragg Spot” [62,
63]. If this is the case, then one might see a second Bragg spot. Such a thing
is not seen, but location of the secondary minimum in 2D corresponds indeed to
the position of a second Bragg spot of a triangular lattice.
Density modes are analogous to the phonons and rotons already mentioned in the
case of bulk helium, but due to the adsorption potential of the substrate, the
fluid is inhomogeneous, and solidification of the first two or three atomic
layers is observed in thick films. Layered-excitations are observed in this
regime.
Capillary waves, also called “ripplons”, have a different nature: these are
surface waves, which do not (in first approximation) correspond to
compression/expansion of the fluid, but to a change of height in the external
potential (gravitational in the case of a bulk helium surface, or substrate
potential in adsorbed films).
A typical example is shown in Fig. 10. It corresponds to a multilayer film of
about 6 atomic layers, where both density excitations and ripplons are visible
with comparable intensity [64]. The experimental curve is broader than the
theoretical one, an effect largely due to the mosaic spread (angular
distribution) of the graphite substrate, a powder made of microscopic flat
crystals.
Figure 10: Experimental [64] (top) and theoretical (bottom) [65]
determinations of the dynamic structure factor of a multilayer film of 4He,
displaying density and ripplon excitations.
Finally, in the sub-monolayer coverage regime, one can observe gaseous, fluid
and solid phases, as well as “commensurate phases” where the atoms adopt a
periodic arrangement commensurate with the substrate atomic lattice.
## 5 3He in reduced dimensions
Figure 11: Experimental (top) and theoretical (left) determinations of the
dynamic structure factor of two-dimensional liquid 3He [66]. The theoretical
calculation shows the density and spin-density modes; the latter is not
visible experimentally, it is masked by a strong elastic signal due to the
substrate. A comparison with Fig. 7 shows that the roton-like excitation in
the 2D system is found outside the particle-hole band (the region limited by
the blue solid lines), thus avoiding Landau damping.
The excitations in 3He films are shown in Fig. 11. They are very similar to
those discussed above for bulk liquid, with the remarkable exception [66] that
the roton-like excitations are at the edge of the particle-hole band and hence
not strongly affected by Landau damping. These excitations can therefore
propagate, suggesting interesting physical effects specific to 2D Fermi
liquids.
## 6 Summary
The investigation of the dynamics of quantum fluids has been performed
experimentally at very low temperatures using various techniques, mainly
specific heat, compressibility, neutron and X-ray scattering and, for 3He,
nuclear magnetic susceptibility. Theoretical calculations are based on semi-
phenomenological approaches [67, 68, 69], dynamic many body theory (DMBT) [43,
30, 44] as well as different versions of Quantum Monte Carlo numerical
calculations (PIMC, DMC) [27, 29, 60, 57] to cite only a few. These have
brought a very detailed understanding of these systems. It is sometimes very
difficult to perform experimental measurements on some particular systems (for
instance in reduced dimensionality), or even impossible to create quite
arbitrary systems (”mathematical models” of interaction or external
potentials, etc.). In this case the combination of numerical and microscopic
techniques is essential.
The results find a natural application to many different physical systems,
like nuclear matter, neutron stars, particle physics and cosmology, where
strongly interacting particles are investigated. Particles themselves, in
fact, can be interpreted as quantized excitations of an underlying field. The
helium liquids have been, in some sense, the Ariadne thread in these
investigations: Their interactions are well established [70, 71] and, compared
to nucleon interactions, quite simple. The density of these systems is very
high and they are very “quantum”, hence they do not permit the simple
treatments that have become popular by the experimental success to generate
and investigate cold quantum gases. The experimental challenge is equally high
and only the very best of the experimental equipment and the experimental
techniques can stand up to the challenge. Nevertheless, these challenges have
been met on both the theoretical and experimental side.
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|
# Beyond Helly graphs: the diameter problem on absolute retracts
Guillaume Ducoffe National Institute for Research and Development in
Informatics, Romania University of Bucharest, Romania
###### Abstract
Characterizing the graph classes such that, on $n$-vertex $m$-edge graphs in
the class, we can compute the diameter faster than in ${\cal O}(nm)$ time is
an important research problem both in theory and in practice. We here make a
new step in this direction, for some metrically defined graph classes.
Specifically, a subgraph $H$ of a graph $G$ is called a retract of $G$ if it
is the image of some idempotent endomorphism of $G$. Two necessary conditions
for $H$ being a retract of $G$ is to have $H$ is an isometric and isochromatic
subgraph of $G$. We say that $H$ is an absolute retract of some graph class
${\cal C}$ if it is a retract of any $G\in{\cal C}$ of which it is an
isochromatic and isometric subgraph. In this paper, we study the complexity of
computing the diameter within the absolute retracts of various hereditary
graph classes. First, we show how to compute the diameter within absolute
retracts of bipartite graphs in randomized $\tilde{\cal O}(m\sqrt{n})$ time.
For the special case of chordal bipartite graphs, it can be improved to linear
time, and the algorithm even computes all the eccentricities. Then, we
generalize these results to the absolute retracts of $k$-chromatic graphs, for
every fixed $k\geq 3$. Finally, we study the diameter problem within the
absolute retracts of planar graphs and split graphs, respectively.
## 1 Introduction
One of the most basic graph properties is the diameter of a graph (maximum
number of edges on a shortest path). It is a rough estimate of the maximum
delay in order to send a message in a communication network [34], but it also
got used in the literature for various other purposes [2, 82]. The complexity
of computing the diameter has received tremendous attention in the Graph
Theory community [1, 16, 20, 22, 29, 28, 26, 31, 32, 33, 38, 48, 49, 50, 45,
53, 56, 73]. Indeed, while this can be done in ${\cal O}(nm)$ time for any
$n$-vertex $m$-edge graph, via a simple reduction to breadth-first search,
breaking this quadratic barrier (in the size $n+m$ of the input) happens to be
a challenging task. In fact, under plausible complexity assumptions such as
the Strong Exponential-Time Hypothesis (SETH), the optimal running time for
computing the diameter is essentially in ${\cal O}(nm)$ — up to sub-polynomial
factors [79]. This negative result holds even if we restrict ourselves to
bipartite graphs or split graphs [1, 15]. However, on the positive side,
several recent works have characterized important graph classes for which we
can achieve for the diameter problem ${\cal O}(m^{2-\epsilon})$ time, or even
better ${\cal O}(mn^{1-\epsilon})$ time, for some $\epsilon>0$. Next, we focus
on a few such classes that are most relevant to our work. Specifically, we
call $G=(V,E)$ a Helly graph if every family of pairwise intersecting balls of
$G$ (of arbitrary radius and center) have a nonempty common intersection. The
Helly graphs are a broad generalization of many better-known graph classes,
such as: trees, interval graphs, strongly chordal graphs and dually chordal
graphs [5]. Furthermore, a celebrated theorem in Metric Graph Theory is that
every graph is an isometric (distance-preserving) subgraph of some Helly graph
[44, 64]. Other properties of Helly graphs were also thoroughly investigated
in prior works [8, 9, 11, 25, 36, 37, 39, 41, 69, 77, 78]. In particular, as
far as we are concerned here, there is a randomized $\tilde{\cal
O}(m\sqrt{n})$-time algorithm in order to compute the diameter within
$n$-vertex $m$-edge Helly graphs with high probability [48].
Larger classes, related to the Helly graphs, have been considered recently.
For instance, $G=(V,E)$ is a $k$-Helly graph if every family of $k$-wise
intersecting balls of $G$ have a nonempty common intersection (Helly graphs
are exactly the $2$-Helly graphs). For every fixed $k$, there is a randomized
$\tilde{\cal O}(m\sqrt{n})$-time algorithm in order to compute the radius
(minimum eccentricity of a vertex) within $k$-Helly graphs [46]. The Helly-gap
of $G=(V,E)$ is the least $\alpha$ such that, for every family of pairwise
intersecting balls of $G$, if we increase all the radii by $\alpha$ then this
family has a nonempty common intersection [23, 42]. It also follows from [48]
that the radius and the diameter of a graph with bounded Helly-gap can be
approximated up to some additive constant, that only depends on its Helly-gap.
The latter result generalizes prior work on diameter and center approximations
within hyperbolic graph classes [28]. Finally, the graphs of bounded “distance
VC-dimension” were introduced in [30], where it was observed that, by a result
from [72], they satisfy certain “fractional” Helly property. Many interesting
graph classes have bounded distance VC-dimension, such as: proper minor-closed
graph classes [30], interval graphs [50] and bounded clique-width graphs [17].
For all the aforementioned sub-classes, there exist algorithms in ${\cal
O}(mn^{1-\epsilon})$ time, for some $\epsilon>0$, in order to compute all the
eccentricities, and so, the diameter [47, 50, 73]. Partial results also have
been obtained for the diameter problem on all graph classes of bounded
distance VC-dimension [50, 48].
Recall that an endomorphism of a graph $G$ is an edge-preserving mapping of
$G$ to itself. A retraction is an idempotent endomorphism. If $H$ is the image
of $G$ by some retraction (in particular, $H$ is a subgraph of $G$) then, we
call $H$ a retract of $G$. The notion of retract has applications in some
discrete facility location problems [62], and it is useful in characterizing
some important graph classes. For instance, the median graphs are exactly the
retracts of hypercubes [4]. We here focus on the relation between retracts and
Helly graphs, that is as follows. For some class ${\cal C}$ of reflexive
graphs (i.e., with a loop at every vertex), let us define the absolute
retracts of ${\cal C}$ as those $H$ such that, whenever $H$ is an isometric
subgraph of some $G\in{\cal C}$, $H$ is a retract of $G$. Absolute retracts
find their root in Geometry, where they got studied for various metric spaces
[67]. In the special case of the class of all reflexive graphs, the absolute
retracts are exactly the Helly (reflexive) graphs [63]. Motivated by this
characterization of Helly graphs, and the results obtained in [48] for the
diameter problem on this graph class, we here consider the following notion of
absolute retracts, for irreflexive graphs. – Unless stated otherwise, all
graphs considered in this paper are irreflexive. – Namely, let us first recall
that a subgraph $H$ of a graph $G$ is isochromatic if it has the same
chromatic number as $G$. Then, given a class of (irreflexive) graphs ${\cal
C}$, the absolute retracts of ${\cal C}$ are those $H$ such that, whenever $H$
is an isometric and isochromatic subgraph of some $G\in{\cal C}$, $H$ is a
retract of $G$. We refer the reader to [6, 7, 10, 62, 61, 66, 68, 70, 76, 74,
75], where this notion got studied for various graph classes.
#### Our results.
In this paper, we prove new structural and algorithmic properties of the
absolute retracts of various hereditary graph classes, such as: bipartite
graphs, $k$-chromatic graphs (for any $k\geq 3$), split graphs and planar
graphs. Our focus is about the diameter problem on these graph classes but, on
our way, we uncover several nice properties of the shortest-path distribution
of their absolute retracts, that may be of independent interest.
* •
First, in Sec. 2, we consider the absolute retracts of bipartite graphs and
some important subclasses of the latter. We observe that in the square of such
graph $G$, its two partite sets induce Helly graphs. This result complements
the known relations between Helly graphs and absolute retracts of bipartite
graphs [7]. Then, we show how to compute the diameter of $G$ from the diameter
of both Helly graphs (actually, from the knowledge of the peripheral vertices
in these graphs, i.e., those vertices with maximal eccentricity). Recently
[40], we announced an ${\cal O}(m\sqrt{n})$-time algorithm in order to compute
all the eccentricities in a Helly graph. However, extending this result to the
absolute retracts of bipartite graphs appears to be a more challenging task.
We manage to do so for the subclass of chordal bipartite graphs, for which we
achieve a linear-time algorithm in order to compute all the eccentricities.
For that, we prove the stronger result that in the square of such graph, its
two partite sets induce strongly chordal graphs. Here also, our result
complements the known relations between both graph classes [18, 52].
* •
In Sec. 3, we generalize our above framework to the absolute retracts of
$k$-chromatic graphs, for any $k\geq 3$. Our proofs in this part are more
technical and intricate than in Sec. 2. For instance, we cannot extract a
Helly graph from each colour class anymore. Instead, we define a partial
eccentricity function for each colour (i.e., by restricting ourselves to the
distances between vertices of the same colour), and we prove that the latter
functions almost have the same properties as the eccentricity function of a
Helly graph.
* •
Our positive results in Sec. 2 and 3 rely on some Helly-type properties of the
graph classes considered. However, our hardness result in Sec. 4 hints that
the weaker property of being an absolute retract of some well-structured graph
class is not sufficient on its own for faster diameter computation.
Specifically, we prove that under SETH, there is no ${\cal
O}(mn^{1-\epsilon})$-time algorithm for the diameter problem, for any
$\epsilon>0$, on the class of absolute retracts of split graphs. This negative
result follows from an elegant characterization of this subclass of split
graphs in [66].
* •
Finally, in Sec. 5, we briefly consider the absolute retracts of planar
graphs. While there now exist several truly subquadratic-time algorithms for
the diameter problem on all planar graphs [22, 50, 56] – with the best-known
running time being in $\tilde{\cal O}(n^{5/3})$ – the existence of a quasi
linear-time algorithm for this problem has remained so far elusive, and it is
sometimes conjectured that no such algorithm exists [22]. We give evidence
that finding such algorithm for the absolute retracts of planar graphs is
already a hard problem on its own. Specifically, we prove that every planar
graph is an isometric subgraph of some absolute retract of planar graphs. This
result mirrors the aforementioned property that every graph isometrically
embeds in a Helly graph [44, 64]. It implies the existence of some absolute
retracts of planar graphs with treewidth arbitrarily large and inner vertices
of degree three. Doing so, we rule out two general frameworks in order to
compute the diameter in quasi linear time on some subclasses of planar graphs
[29, 51].
Let us mention that all graph classes considered here are polynomial-time
recognizable. For the absolute retracts of $k$-chromatic graphs, the best-
known recognition algorithms have superquadratic running-time (even for $k=2$)
[6, 10]. Fortunately, we do not need to execute these recognition algorithms
before we can compute the diameter of these graphs. Indeed, our algorithms in
Sec. 2 and 3 are heuristics which can be applied to any graph. Sometimes,
these algorithms may fail in outputting a value, that certifies the input
graph is not an absolute retract of $k$-chromatic graphs, for some $k$.
Conversely, if the input graph is an absolute retract of $k$-chromatic graphs,
then the algorithm always succeeds in outputting a value and this value is
exactly the diameter. We consider our framework to be especially useful for
all subclasses of the absolute retracts of $k$-chromatic graphs that admit
quasi linear-time recognition algorithms. In this respect, we stress that the
absolute retract of bipartite graphs are a superclass of cube-free modular
graphs, and so, of chordal bipartite graphs, cube-free median graphs and
covering graphs of modular lattices of breadth at most two [6].
#### Notations.
We mostly follow the graph terminology from [14, 35]. All graphs considered
are finite, simple (i.e., without loops nor multiple edges), unweighted and
connected. For a graph $G=(V,E)$, let the (open) neighbourhood of a vertex $v$
be defined as $N_{G}(v)=\\{u\in V\mid uv\in E\\}$ and its closed neighbourhood
as $N_{G}[v]=N_{G}(v)\cup\\{v\\}$. Similarly, for a vertex-subset $S\subseteq
V$, let $N_{G}(S)=\bigcup_{v\in S}N_{G}(v)\setminus S$, and let
$N_{G}[S]=N_{G}(S)\cup S$. The distance between two vertices $u,v\in V$ equals
the minimum number of edges on a $uv$-path, and it is denoted $d_{G}(u,v)$. We
also let $I_{G}(u,v)$ denote the vertices “metrically” between $u$ and $v$,
i.e., $I_{G}(u,v)=\\{w\in V\mid d_{G}(u,v)=d_{G}(u,w)+d_{G}(w,v)\\}$. The ball
of center $v$ and radius $r$ is defined as $N_{G}^{r}[v]=\\{u\in V\mid
d_{G}(u,v)\leq r\\}$. Furthermore, let the eccentricity of a vertex $v$ be
defined as $e_{G}(v)=\max_{u\in V}d_{G}(u,v)$. The diameter and the radius of
a graph $G$ are defined as $diam(G)=\max_{v\in V}e_{G}(v)$ and
$rad(G)=\min_{v\in V}e_{G}(v)$, respectively. A vertex $v\in V$ is called
central if $e_{G}(v)=rad(G)$, and peripheral if $e_{G}(v)=diam(G)$. Note that
we sometimes omit the subscript if the graph $G$ is clear from the context. We
introduce additional terminology where it is needed throughout the paper.
## 2 Bipartite graphs
The study of the absolute retracts of bipartite graphs dates back from Hell
[60], and since then many characterizations of this graph class were proposed
[6]. This section is devoted to the diameter problem on this graph class. In
Sec. 2.1, we propose a randomized $\tilde{\cal O}(m\sqrt{n})$-time algorithm
for this problem. Then, we consider the chordal bipartite graphs in Sec. 2.2,
that have been proved in [6] to be a subclass of the absolute retracts of
bipartite graphs. For the chordal bipartite graphs, we present a deterministic
linear-time algorithm in order to compute all the eccentricities.
Let us introduce a few additional terminology. For a connected bipartite graph
$G$, we denote its two partite sets by $V_{0}$ and $V_{1}$. A half-ball is the
intersection of a ball with one of the two partite sets of $G$. Finally, for
$i\in\\{0,1\\}$, let $H_{i}$ be the graph with vertex-set $V_{i}$ and an edge
between every two vertices with a common neighbour in $G$.
### 2.1 Faster diameter computation
We start with the following characterization of the absolute retracts of
bipartite graphs:
###### Theorem 1 ([6]).
$G=(V,E)$ is an absolute retract of bipartite graphs if and only if the
collection of half-balls of $G$ satisfies the Helly property.
This above Theorem 1 leads us to the following simple, but important for what
follows, observation about the internal structure of the absolute retracts of
bipartite graphs:
###### Lemma 1.
If $G=(V_{0}\cup V_{1},E)$ is an absolute retract of bipartite graphs then
both $H_{0}$ and $H_{1}$ are Helly graphs.
###### Proof.
For each $i\in\\{0,1\\}$ the balls of $H_{i}$ are exactly the half-balls of
$G$ that intersect $V_{i}$ and have as their center a vertex of $V_{i}$.
Therefore, by Theorem 1, the collection of balls of $H_{i}$ satisfies the
Helly property, i.e., $H_{i}$ is a Helly graph. ∎
Next, we prove that in order to compute $diam(G)$, with $G$ an absolute
retract of bipartite graphs, it is sufficient to compute the peripheral
vertices of the Helly graphs $H_{0}$ and $H_{1}$.
###### Lemma 2.
If $G=(V_{0}\cup V_{1},E)$ is an absolute retract of bipartite graphs such
that $diam(H_{0})\leq diam(H_{1})$ then,
$diam(G)\in\\{2diam(H_{1}),2diam(H_{1})+1\\}$. Moreover, if $diam(G)\geq 3$
then we have $diam(G)=2diam(H_{1})+1$ if and only if:
* •
$diam(H_{1})=1$;
* •
or $diam(H_{0})=diam(H_{1})$ and, for some $i\in\\{0,1\\}$, there exists a
peripheral vertex of $H_{i}$ whose all neighbours in $G$ are peripheral
vertices of $H_{1-i}$.
###### Proof.
Let $i\in\\{0,1\\}$. Clearly, $diam(G)\geq\max_{u,v\in
V_{i}}d_{G}(u,v)=2diam(H_{i})$. Furthermore, since $G$ is connected, $V_{i}$
is a dominating set of $G$, and therefore, every vertex of $V_{i}$ has
eccentricity at most $2diam(H_{i})+1$. Overall, since we assume
$diam(H_{1})\geq diam(H_{0})$, we get as desired
$diam(G)\in\\{2diam(H_{1}),2diam(H_{1})+1\\}$. Note that this result holds for
any bipartite graph.
In what follows, we further assume $diam(G)\geq 3$. If $diam(H_{1})\leq 1$
then, $diam(G)\leq 2diam(H_{1})+1\leq 3$. Therefore, $diam(H_{1})=1$ and
$diam(G)=2diam(H_{1})+1$. From now on, $diam(H_{1})\geq 2$.
Let us first assume $diam(G)=2diam(H_{1})+1$. Observe that for each
$i\in\\{0,1\\}$, every vertex of $V_{i}$ is at a distance at most
$1+2diam(H_{1-i})$ from every vertex of $V_{1-i}$ (in order to see that, just
take any neighbour of this vertex in $V_{1-i}$). In particular, if
$diam(H_{0})<diam(H_{1})$, then every vertex of $V_{0}$ has eccentricity at
most $2diam(H_{0})+1<2diam(H_{1})$, while every vertex of $V_{1}$ has
eccentricity at most $\max\\{2diam(H_{1}),2diam(H_{0})+1\\}=2diam(H_{1})$.
Hence, in order to have $diam(G)=2diam(H_{1})+1$, we must have
$diam(H_{0})=diam(H_{1})$. Furthermore, for some fixed $i\in\\{0,1\\}$, let
$v\in V_{i}$ be a peripheral vertex of $G$. Since $2diam(H_{1})+1=e_{G}(v)\leq
2e_{H_{i}}(v)+1$, we must have $e_{H_{i}}(v)=diam(H_{i})=diam(H_{1})$. In the
same way, for any neighbour $u\in N_{G}(v)$, we have
$e_{G}(v)\leq\max\\{2diam(H_{i}),2e_{H_{1-i}}(u)+1\\}$, and therefore, we must
also have $e_{H_{1-i}}(u)=diam(H_{1-i})=diam(H_{1})$. It implies the existence
of a peripheral vertex of $H_{i}$ whose all neighbours are peripheral vertices
of $H_{1-i}$.
Conversely, let us assume that $diam(H_{0})=diam(H_{1})$ and that, for some
$i\in\\{0,1\\}$, there exists a $v\in V_{i}$ such that:
$e_{H_{i}}(v)=diam(H_{i})=diam(H_{1})$; for every $u\in N_{G}(v)$,
$e_{H_{1-i}}(u)=diam(H_{1-i})=diam(H_{1})$. Suppose by contradiction
$e_{G}(v)<2diam(H_{1})+1$. Then, since $G$ is bipartite, all the vertices of
$V_{1-i}$ are at a distance at most $2diam(H_{1})-1$ from vertex $v$.
Furthermore, for every $x,y\in V_{1-i}$, since we have
$d_{G}(x,y)=2d_{H_{1-i}}(x,y)$, we obtain that for every even $\ell\geq
d_{H_{1-i}}(x,y)$ the half-balls $N^{\ell}_{G}[x]\cap V_{1-i}$ and
$N^{\ell}_{G}[y]\cap V_{1-i}$ intersect. In particular, we may choose
$\ell=2(diam(H_{1})-1)$ because $diam(H_{1})\geq 2$ implies
$2(diam(H_{1})-1)\geq diam(H_{1})\geq d_{H_{1-i}}(x,y)$. But then, the half-
balls $N_{G}(v)$ ($=N_{G}^{1}[v]\cap V_{1-i}$) and
$N_{G}^{2(diam(H_{1})-1)}[w]\cap V_{1-i}$, for every $w\in V_{1-i}$, pairwise
intersect. By Theorem 1, there exists a $u\in N_{G}(v)$ s.t.
$e_{H_{1-i}}(u)\leq diam(H_{1})-1$. This is a contradiction because such
neighbour $u$ cannot be peripheral in $H_{1-i}$. ∎
The remaining of Sec. 2.1 is devoted to the computation of all the peripheral
vertices in both Helly graphs $H_{0}$ and $H_{1}$. While there exists a truly
subquadratic-time algorithm for computing the diameter of a Helly graph [48],
we observe that in general, we cannot compute $H_{0}$ and $H_{1}$ in truly
subquadratic time from $G$. Next, we adapt [48, Theorem 2], for the Helly
graphs, to our needs.
###### Lemma 3.
If $G=(V_{0}\cup V_{1},E)$ is an absolute retract of bipartite graphs then,
for any $k$, we can compute in ${\cal O}(km)$ time the set of vertices of
eccentricity at most $k$ in $H_{0}$ (resp., in $H_{1}$).
###### Proof.
By symmetry, we only need to prove the result for $H_{0}$. Let $U=\\{v\in
V_{0}\mid e_{H_{0}}(v)\leq k\\}$ be the set to be computed. We consider the
more general problem of computing, for any $t$, a partition ${\cal
P}_{t}=(A^{t}_{1},A^{t}_{2},\ldots,A^{t}_{p_{t}})$ of $V_{0}$, in an arbitrary
number $p_{t}$ of subsets, subject to the following constraints:
* •
For every $1\leq i\leq p_{t}$, let $C^{t}_{i}:=\bigcap_{v\in
A^{t}_{i}}N^{t}_{G}[v]$. Let $B_{i}^{t}:=C^{t}_{i}\cap V_{0}$ if $t$ is even
and let $B_{i}^{t}:=C^{t}_{i}\cap V_{1}$ if $t$ is odd (for short,
$B_{i}^{t}=C_{i}^{t}\cap V_{t\pmod{2}}$). _We impose the sets $B_{i}^{t}$ to
be nonempty and pairwise disjoint_.
Indeed, under these two conditions above, we have $U\neq\emptyset$ if and only
if, for any partition ${\cal P}_{2k}$ as described above, $p_{2k}=1$.
Furthermore if it is the case then $U=B^{2k}_{1}$.
The algorithm. We construct the desired partition by induction over $t$. If
$t=0$ then, let $V_{0}=\\{v_{1},v_{2},\ldots,v_{p_{0}}\\}$. We just set ${\cal
P}_{0}=(\\{v_{0}\\},\\{v_{1}\\},\ldots,\\{v_{p_{0}}\\})$ (each set is a
singleton), and for every $1\leq i\leq p_{0}$ let
$B_{i}^{0}=A_{i}^{0}=\\{v_{i}\\}$. Else, we construct ${\cal P}_{t}$ from
${\cal P}_{t-1}$. Specifically, for every $1\leq i\leq p_{t-1}$, we let
$W_{i}^{t}:=N_{G}(B_{i}^{t-1})$. Then, starting from $j:=0$ and ${\cal
F}:={\cal P}_{t-1}$, we proceed as follows until we have ${\cal F}=\emptyset$.
We pick a vertex $u$ s.t. $\\#\\{i\mid A_{i}^{t-1}\in{\cal F},\ u\in
W_{i}^{t}\\}$ is maximized. Then, we set $A_{j}^{t}:=\bigcup\\{A_{i}^{t-1}\mid
A_{i}^{t-1}\in{\cal F},\ u\in W_{i}^{t}\\}$ and
$B_{j}^{t}:=\bigcap\\{W_{i}^{t}\mid A_{i}^{t-1}\in{\cal F},\ u\in
W_{i}^{t}\\}$. We add the new subset $A_{j}^{t}$ to ${\cal P}_{t}$, we remove
all the subsets $A_{i}^{t-1},u\in W_{i}^{t}$ from ${\cal F}$, then we set
$j:=j+1$.
Correctness. The base case of our above induction is trivially correct. In
order to prove correctness of our inductive step, we need the following two
intermediate claims.
###### Claim 1.
$W_{i}^{t}=V_{t\pmod{2}}\cap\left(\bigcap_{v\in
A^{t-1}_{i}}N^{t}_{G}[v]\right)$.
Proof. Recall that $B_{i}^{t-1}=V_{t-1\pmod{2}}\cap\left(\bigcap_{v\in
A^{t-1}_{i}}N^{t-1}_{G}[v]\right)$, and that $W_{i}^{t}=N_{G}(B_{i}^{t-1})$.
Therefore by construction, $W_{i}^{t}\subseteq
V_{t\pmod{2}}\cap\left(\bigcap_{v\in A^{t-1}_{i}}N^{t}_{G}[v]\right)$.
Conversely, let $w\in V_{t\pmod{2}}\cap\left(\bigcap_{v\in
A^{t-1}_{i}}N^{t}_{G}[v]\right)$ be arbitrary. Let $a\in A_{i}^{t-1}$ be
arbitrary. There are two cases. If $a\neq w$ then, by considering any $x\in
N_{G}(w)\cap I_{G}(a,w)$, we get that the half-balls $N_{G}(w)$
($=N_{G}^{1}[w]\cap V_{t-1\pmod{2}}$) and $N_{G}^{t-1}[a]\cap V_{t-1\pmod{2}}$
intersect. Otherwise, $a=w$ and then, $t$ is even. Here also the half-balls
$N_{G}(w)$ and $N_{G}^{t-1}[a]\cap V_{t-1\pmod{2}}$ ($=N_{G}^{t-1}[a]\cap
V_{1}$) intersect because we have $N_{G}(w)=N_{G}(a)\subseteq
N_{G}^{t-1}[a]\cap V_{t-1\pmod{2}}$. Overall, in all the cases, the half-balls
$N_{G}(w)$ and $N_{G}^{t-1}[a]\cap V_{t-1\pmod{2}}$, for every $a\in
A_{i}^{t-1}$, pairwise intersect. Then, by Theorem 1, vertex $w$ has a
neighbour in $B_{i}^{t-1}$. $\diamond$
It follows from this above claim that, for each subset $A_{j}^{t}$ created at
step $t$, we have $B_{j}^{t}=V_{t\pmod{2}}\cap\left(\bigcap_{v\in
A^{t}_{j}}N^{t}_{G}[v]\right)$, as desired. Observe that all the subsets
$B_{j}^{t}$ are nonempty since they at least contain the vertex $u\in
V_{t\pmod{2}}$ that is selected in order to create $A_{j}^{t}$. What now
remains to prove is that all the subsets $B_{j}^{t}$ are pairwise disjoint.
The latter easily follows from our next intermediate claim, namely:
###### Claim 2.
Let $u\in V_{t\pmod{2}}$ be a vertex maximizing $\\#\\{i\mid u\in
W_{i}^{t}\\}$. For every index $i^{\prime}$ s.t. $u\notin W_{i^{\prime}}^{t}$,
we have $W_{i^{\prime}}^{t}\cap\left(\bigcap_{u\in
W_{i}^{t}}W_{i}^{t}\right)=\emptyset$.
Proof. It directly follows from the maximality of $\\#\\{i\mid u\in
W_{i}^{t}\\}$. $\diamond$
We are done applying this above claim at each creation of a new subset
$A_{j}^{t}$.
Complexity. The base case of our induction requires ${\cal O}(n)$ time. Let us
prove that each inductive step requires ${\cal O}(m)$ time. First, since by
the hypothesis the sets $B_{i}^{t-1}$ are pairwise disjoint, we can compute
the sets $W_{i}^{t}$ in total linear time. Then, we create an array of
$p_{t-1}$ lists, numbered from $1$ to $p_{t-1}$. For each vertex $u\in
V_{t\pmod{2}}$ s.t. $\\{i\mid u\in W_{i}^{t}\\}\neq\emptyset$, we put it in
the list numbered $\\#\\{i\mid u\in W_{i}^{t}\\}$. Since it only requires to
scan the $W_{i}^{t}$’s once, it takes ${\cal O}(m+n)$ time. We scan the lists
in decreasing order (i.e., from $p_{t-1}$ downto $1$), going to the next list
each time the current one is empty. When the current list is nonempty, we pick
any vertex $u$ of this list in order to create the next subset $A_{j}^{t}$.
Note that both subsets $A_{j}^{t}$ and $B_{j}^{t}$ can be computed in ${\cal
O}(\sum\\{|A_{i}^{t-1}|+|W_{i}^{t}|\mid A_{i}^{t-1}\in{\cal F},\ u\in
W_{i}^{t}\\})$ time. Finally, we need to discard the vertices of $B_{j}^{t}$
from the list in which they are currently contained, while for every other
$w\in\left(\bigcup\\{W_{i}^{t}\mid A_{i}^{t-1}\in{\cal F},u\in
W_{i}^{t}\\}\right)\setminus B_{j}^{t}$ we need to update
$\\#\\{i^{\prime}\mid A_{i^{\prime}}^{t-1}\in{\cal F},w\in
W_{i^{\prime}}^{t}\\}$ and the corresponding list. If, for each vertex, we
store a pointer to its position in the corresponding list, we can also do the
latter in ${\cal O}(\sum\\{|W_{i}^{t}|\mid A_{i}^{t-1}\in{\cal F},\ u\in
W_{i}^{t}\\})$ time. Overall, since each set $A_{i}^{t-1}$ gets removed once
from ${\cal F}$, the total running time for the inductive step is in ${\cal
O}(m+n)$. ∎
We use this above Lemma 3 when the respective diameters of $H_{0}$ and $H_{1}$
are in ${\cal O}(\sqrt{n})$. For larger values of diameters, we use a
randomized procedure (Algorithm 1).
Algorithm 1 Diameter computation in Helly graphs.
0: A Helly graph $H$ s.t. $diam(H)>3k=\omega(\log{|V(H)|})$.
1: Set $p=c\frac{\log{|V(H)|}}{k}$, for some sufficiently large constant $c$.
2: Let $U(p)$ contain every $v\in V(H)$ independently with probability $p$.
3: for all $v\in V(H)$ do
4: if $\forall u\in U(p),\ d_{H}(u,v)>k$ then
5: Set $\bar{e}(v):=0$.
6: else
7: Set $\bar{e}(v):=\min\\{d_{H}(u,v)+e_{H}(u)\mid u\in U(p),\ d_{H}(u,v)\leq
k\\}$.
###### Lemma 4 (Theorem 3 in [48]).
With high probability, Algorithm 1 runs in $\tilde{\cal
O}(|E(H)|\cdot|V(H)|/k)$ time, we have $diam(H)=\max_{v\in V(H)}\bar{e}(v)$,
and the peripheral vertices of $H$ are exactly the vertices $v\in V(H)$ which
maximize $\bar{e}(v)$.
We are now ready to prove the main result of this section, namely:
###### Theorem 2.
If $G=(V_{0}\cup V_{1},E)$ is an absolute retract of bipartite graphs then,
with high probability, we can compute $diam(G)$ in $\tilde{\cal O}(m\sqrt{n})$
time.
###### Proof.
We may assume $diam(G)\geq 3$. Indeed, a bipartite graph has diameter at most
two if and only if it is complete bipartite. First, we compute the peripheral
vertices of $H_{0}$ and $H_{1}$. For that, we start computing a
$2$-approximation of $diam(G)$ in linear time (e.g., by computing the
eccentricity of an arbitrary vertex). Let $D$ be the resulting value. There
are two cases.
* •
If $D<\sqrt{n}$ then, we compute the least $k$ s.t. all vertices of $H_{0}$
(resp., of $H_{1}$) have eccentricity at most $k$. For that, it is sufficient
to perform a one-sided binary search where, at each step, we apply Lemma 3.
Note that this value $k$ computed is in fact $diam(H_{0})$ (resp.,
$diam(H_{1})$). In particular, the total running time is in $\tilde{\cal
O}(mk)=\tilde{\cal O}(mD)=\tilde{\cal O}(m\sqrt{n})$. Then, in order to
compute the peripheral vertices of $H_{0}$ (resp., of $H_{1}$), it is
sufficient to apply Lemma 3 one more time in order to compute the vertices of
eccentricity at most $k-1$.
* •
Otherwise, $D\geq\sqrt{n}$, and we apply Algorithm 1 to both $H_{0}$ and
$H_{1}$. Since by Lemma 1, both $H_{0}$ and $H_{1}$ are Helly graphs, we have
by Lemma 4 that the output of Algorithm 1 is correct with high probability.
Furthermore, since any breadth-first search in either $H_{0}$ or $H_{1}$ can
be simulated with a breadth-first search in $G$, the total running time for
executing Algorithm 1 is with high probability (by Lemma 4) in $\tilde{\cal
O}(mn/D)=\tilde{\cal O}(m\sqrt{n})$.
Finally, in order to compute $diam(G)$ from the peripheral vertices of $H_{0}$
and $H_{1}$, we apply the criterion of Lemma 2. For that, it is sufficient to
scan the neighbourhood of each peripheral vertex of $H_{0}$ and $H_{1}$, and
therefore it can be done in linear time. ∎
### 2.2 Chordal bipartite graphs
We improve Theorem 2 for the special case of chordal bipartite graphs. Recall
(amongst many characterizations) that a bipartite graph is chordal bipartite
if and only if every induced cycle has length four [58]. It was proved in [6]
that every chordal bipartite graph is an absolute retract of bipartite graphs.
###### Theorem 3.
If $G=(V,E)$ is chordal bipartite then we can compute all the eccentricities
(and so, the diameter) in linear time.
The remaining of this section is devoted to a proof of Theorem 3. For that, we
subdivide our proof into four main steps (undefined terminology below is
introduced step by step in this section):
1. 1.
We base ourselves on the results from [19, 43] in order to prove that $H_{0}$
and $H_{1}$ – the two Helly graphs induced by the partite sets of $G$ in its
square – are strongly chordal graphs.
2. 2.
The same as in Sec. 2.1, in general we cannot compute $H_{0}$ and $H_{1}$ from
$G$ in subquadratic time. In order to overcome this issue, we explain how to
compute a clique-tree for these two graphs.
3. 3.
Then, we present an algorithm in order to compute all the eccentricities for
strongly chordal graphs, being given as input the clique-tree of such graph
(i.e., instead of its adjacency list).
4. 4.
Finally, we present an “all eccentricities” version of Lemma 2, and we explain
how to solve the corresponding algorithmic problem for chordal bipartite
graphs.
#### The chordal structure of the partite sets.
A graph is chordal if it has no induced cycle of length more than three. It is
strongly chordal if it is chordal and it does not contain any $n$-sun ($n\geq
3$) as an induced subgraph [52]. A dually chordal graph is a Helly graph in
which the intersection graph of balls is chordal [5, 19]; for other
characterizations of this graph class, see [19]. The relation between dually
chordal graphs and strongly chordal graphs is as follows:
###### Lemma 5 ([19]).
A graph is strongly chordal if and only if each induced subgraph is dually
chordal.
The $k^{th}$-iterated neighbourhood of a vertex $V$, denoted $N^{k}_{G}(v)$,
is defined recursively as: $N_{G}^{1}(v)=N_{G}(v)$ (open neighbourhood) and
$N_{G}^{k+1}(v)=N_{G}(N_{G}^{k}(v))$ [43]. The following observation was used
implicitly in Lemma 3:
###### Lemma 6.
If $G=(V_{0}\cup V_{1},E)$ is bipartite then, for every $k\geq 1$ and $v\in
V_{0}$, $N_{G}^{2k}(v)=N_{H_{0}}^{k}[v]$.
###### Proof.
By induction, for every $k\geq 1$, $N_{G}^{2k-1}(v)=N_{G}^{2k-1}[v]\cap V_{1}$
and $N_{G}^{2k}(v)=N_{G}^{2k}[v]\cap V_{0}$. The lemma follows since we have
$N_{H_{0}}^{k}[v]=N_{G}^{2k}[v]\cap V_{0}$. ∎
For a vertex $v$ in $G$, a maximum neighbour111This terminology is sometimes
used, with a different meaning, for dually chordal graphs [19]. is a vertex
$u\in N_{G}(v)$ such that, for any other $w\in N_{G}(v)$, we have
$N_{G}(w)\subseteq N_{G}(u)$. A vertex $v^{\prime}\neq v$ such that
$N_{G}(v)\subseteq N_{G}(v^{\prime})$ is said to cover $v$. Finally a maximum
neighbourhood ordering of $G$ is a total ordering
$V=(v_{1},v_{2},\ldots,v_{n})$ of its vertex-set such that, for every $1\leq
i\leq n-2$, the vertex $v_{i}$ has a maximum neighbour and is covered in the
induced subgraph $G_{i}:=G\setminus\\{v_{1},v_{2},\ldots,v_{i-1}\\}$. We do
not use the existence of a maximum neighbourhood ordering directly in our
proofs, but rather the following two related results:
###### Lemma 7 ([43]).
$G=(V,E)$ is chordal bipartite if and only if every induced subgraph of $G$
has a maximum neighbourhood ordering.
###### Lemma 8 (items (iii) and (v) of the main theorem in [43]).
$G$ has a maximum neighbourhood ordering if and only if the system of all
iterated neighbourhoods has the Helly property and its intersection graph is
chordal.
The next result now follows by combining Lemmas 5–8.
###### Lemma 9.
If $G=(V_{0}\cup V_{1},E)$ is chordal bipartite, then $H_{0}$ and $H_{1}$ are
strongly chordal.
###### Proof.
By symmetry, it suffices to prove the result for $H_{0}$. For that, by Lemma
9, it is sufficient to prove that every induced subgraph of $H_{0}$ is dually
chordal. Let $U\subseteq V_{0}$ be arbitrary. We define $G_{U}$ as the
subgraph induced by $N_{G}[U]$ (the union of $U$ and of all the vertices of
$V_{1}$ with a neighbour in $U$). By construction, $H_{0}[U]$ is exactly the
graph with vertex-set $U$ and an edge between every two vertices with a common
neighbour in $G_{U}$. Furthermore, since the class of chordal bipartite graphs
is hereditary, $G_{U}$ is chordal bipartite, and so, by [6], an absolute
retract of bipartite graphs. It thus follows from Lemma 1 that $H_{0}[U]$ is a
Helly graph. In order to prove that $H_{0}[U]$ is dually chordal, it now
suffices to prove that the intersection graph of its balls is chordal. By the
combination of Lemmas 7 and 8, the intersection graph $I_{U}$ of all iterated
neighbourhoods of $G_{U}$ is chordal. By Lemma 6, the intersection graph of
all balls of $H_{0}[U]$ is an induced subgraph of $I_{U}$ and therefore, it is
also chordal. ∎
#### Computation of a clique-tree.
For a graph $H=(V,E)$, a clique-tree is a tree $T$ whose nodes are the maximal
cliques of $H$ and such that, for every $v\in V$, the maximal cliques of $H$
containing $v$ induce a connected subtree $T_{v}$ of $T$. It is well-known
that $H$ is chordal if and only if it has a clique-tree [21, 55, 81]. Our
intermediate goal is, given a chordal bipartite graph $G=(V_{0}\cup V_{1},E)$,
to compute a clique-tree for $H_{0}$ and $H_{1}$ (that are chordal graphs by
the above Lemma 9).
A hypergraph ${\cal H}=(X,{\cal R})$ is called a dual hypertree if there
exists a tree $T$ whose nodes are the hyperedges in ${\cal R}$ and such that,
for every $x\in X$, the hyperedges containing $x$ induce a connected subtree
$T_{x}$ of $T$ (such tree $T$ is called a join tree of ${\cal H}$). Note that
dual hypertrees can be recognized in linear time [80]. Furthermore, we have:
###### Lemma 10 ([43]).
If $G=(V_{0}\cup V_{1},E)$ is chordal bipartite then, both hypergraphs
$(V_{0},\\{N_{G}(v)\mid v\in V_{1}\\})$ and $(V_{1},\\{N_{G}(v)\mid v\in
V_{0}\\})$ are dual hypertrees.
###### Corollary 1.
If $G=(V_{0}\cup V_{1},E)$ is chordal bipartite then, we can compute a clique-
tree for $H_{0}$ and $H_{1}$ in linear time.
###### Proof.
By symmetry, we only prove the result for $H_{0}$. Consider the hypergraph
${\cal H}=(V_{0},\\{N_{G}(v)\mid v\in V_{1}\\})$. Note that the underlying
graph of ${\cal H}$ (obtained by adding an edge between every two vertices
that are contained in a common hyperedge of ${\cal H}$) is exactly $H_{0}$.
Since by Lemma 10, ${\cal H}$ is a dual hypertree, every maximal clique of the
underlying graph $H_{0}$ must be a hyperedge of ${\cal H}$ [12]. Then, let us
reduce ${\cal H}$, i.e., we remove all hyperedges that are strictly contained
into another hyperedge. Again since ${\cal H}$ is a dual hypertree, the
resulting reduced hypergraph ${\cal H}^{\prime}$ can be computed in linear
time [80]. Furthermore, by construction, the hyperedges of ${\cal H}^{\prime}$
are exactly the maximal cliques of $H_{0}$. Let us construct a join tree of
${\cal H}^{\prime}$. It can be done in linear time [80]. We are done as such
join tree is a clique-tree of $H_{0}$. ∎
#### Computation of all the eccentricities in the partite sets.
Next, we propose a new algorithm in order to compute all the eccentricities of
a strongly chordal graph $H$, being given a clique-tree. We often use in our
proof the clique-vertex incidence graph of $H$, i.e., the bipartite graph
whose partite sets are the vertices and the maximal cliques of $H$, and such
that there is an edge between every vertex of $H$ and every maximal clique of
$H$ containing it.
Let us first recall the following result about the eccentricity function of
Helly graphs:
###### Lemma 11 ([36]).
If $H=(V,E)$ is Helly then, for every vertex $v$ we have
$e_{H}(v)=d_{H}(v,C(H))+rad(H)$, where $C(H)$ denotes the set of central
vertices of $H$.
Hence, by Lemma 11, we are left computing $C(H)$. It starts with computing one
central vertex. Define, for every vertex $v$ and vertex-subset $C$,
$d_{H}(v,C)=\min_{c\in C}d_{H}(v,c)$. Following [27], we call a set $C$ gated
if, for every $v\notin C$, there exists a vertex $v^{*}\in
N_{H}^{d_{H}(v,C)-1}[v]\cap\left(\bigcap\\{N_{H}(c)\mid c\in C,\
d_{H}(v,c)=d_{H}(v,C)\\}\right)$ (such vertex $v^{*}$ is called a gate of
$v$).
###### Lemma 12 ([24]).
Every clique in a chordal graph is a gated set.
###### Lemma 13 ([48]).
If $T$ is a clique-tree of a chordal graph $H$ then, for every clique $C$ of
$H$, for every $v\notin C$ we can compute $d_{H}(v,C)$ and a corresponding
gate $v^{*}$ in total ${\cal O}(w(T))$ time, where $w(T)$ denotes the sum of
cardinalities of all the maximal cliques of $H$.
For every $u,v\in V$ and $k\leq d_{H}(u,v)$, the set $L_{H}(u,k,v)=\\{x\in
I_{H}(u,v)\mid d_{H}(u,x)=k\\}$ is called a slice. We also need the following
result about slices in chordal graphs:
###### Lemma 14 ([24]).
Every slice in a chordal graph is a clique.
Now, consider the procedure described in Algorithm 2 in order to compute a
central vertex.
Algorithm 2 Computation of a central vertex.
0: A strongly chordal graph $H$.
1: $v\leftarrow$ an arbitrary vertex of $H$
2: $u\leftarrow$ a furthest vertex from $v$, i.e., $d_{H}(u,v)=e_{H}(v)$
3: $w\leftarrow$ a furthest vertex from $u$, i.e., $d_{H}(u,w)=e_{H}(u)$
4: for all $r\in\\{\left\lceil
e_{H}(u)/2\right\rceil,\left\lceil(e_{H}(u)+1)/2\right\rceil,1+\left\lceil
e_{H}(u)/2\right\rceil\\}$ do
5: Set $C:=L(w,r,u)$ //$C$ is a clique by Lemma 14
6: for all $v\notin C$ do
7: Compute $d_{H}(v,C)$ and a corresponding gate $v^{*}$ //whose existence
follows from Lemma 12
8: Set $S:=\\{v^{*}\mid d_{H}(v,C)=r\\}$ //gates of vertices at max. distance
from $C$
9: for all $c\in C$ do
10: if $S\subseteq N_{H}(c)$ then
11: return $c$
###### Lemma 15 (special case of Theorem 5 in [48]).
Algorithm 2 outputs a central vertex of $H$.
###### Lemma 16.
If $T$ is a clique-tree of a strongly chordal graph $H$ then, we can implement
Algorithm 2 in order to run in ${\cal O}(w(T))$ time, where $w(T)$ denotes the
sum of cardinalities of all the maximal cliques of $H$.
###### Proof.
Lines 1–3 require executing breadth-first searches in $H$. It can be done in
${\cal O}(w(T))$ time by executing breadth-first searches in the clique-vertex
incidence graph $I_{H}$ of $H$. Note that we can compute $I_{H}$ in ${\cal
O}(w(T))$ time from the clique-tree $T$. Now, consider any of the at most
three executions of the for loop starting at Line 4. The subset $C$ at Line 5
can also be computed in ${\cal O}(w(T))$ time using breadth-first searches in
$I_{H}$. Then, for implementing the computation of all the gates, at Line 6–7,
we call Lemma 13. We are left explaining how to implement the internal for
loop, starting at Line 9, so that it runs in total ${\cal O}(w(T))$ time. For
that, we assign a counter $g(c)$ for every $c\in C$ (initially equal to $0$),
whose final value must be $|N_{H}(c)\cap S|$. Doing so, the test at Line 10
boilds down to verifying whether $g(c)=|S|$. In order to correctly set the
values of the counters $g(c),\ c\in C$, we root the clique-tree $T$
arbitrarily and then we perform a breadth-first search of this tree $T$
starting from the root. For each maximal clique $K$ of $H$, let $K^{\prime}$
be the common intersection with its father node in $T$ (in particular,
$K^{\prime}=\emptyset$ if $K$ is the root). For every $c\in C\cap(K\setminus
K^{\prime})$, we increase the counter $g(c)$ by exactly $|K\cap S|$. However,
for every $c\in C\cap K^{\prime}$, we only increase the counter $g(c)$ by
$|S\cap(K\setminus K^{\prime})|$; indeed, for these vertices, the contribution
of the gates in $S\cap K^{\prime}$ was already counted earlier during the
breadth-first search. Overall, we just need to scan each maximal clique of $H$
once, and so, the running time is in ${\cal O}(w(T))$, as desired. ∎
Then, given a central vertex $c$ of $H$, we explain how to compute $C(H)$ by
local search in the neighbourhood at distance two around $c$. For that, we
need one more structural result about the center of strongly chordal graphs,
namely:
###### Lemma 17 ([36, 37]).
If $H$ is strongly chordal then, its center $C(H)$ induces a strongly chordal
graph of radius $\leq 1$.
We also need the following nice characterization of strongly chordal graphs,
namely:
###### Lemma 18 ([18, 52]).
$H$ is strongly chordal if and only if its clique-vertex incidence graph
$I_{H}$ is chordal bipartite.
###### Proposition 1.
If $T$ is a clique-tree of a strongly chordal graph $H=(V,E)$ then, we can
compute its center $C(H)$ in ${\cal O}(w(T))$ time.
###### Proof.
Let $c\in C(H)$ be a fixed central vertex of $H$. By the combination of Lemmas
15 and 16, it can be computed in ${\cal O}(w(T))$ time. We also compute the
clique-vertex incidence graph $I_{H}$ from $T$, in ${\cal O}(w(T))$ time. Let
$r:=rad(H)$ (computable in ${\cal O}(w(T))$ time by using a breadth-first
search rooted at $c$ in $I_{H}$). Note that if $r\leq 2$, then in order to
compute $C(H)$, we are left computing all the vertices of eccentricity at most
$r$ in $H$. For that, since by Lemma 18 $I_{H}$ is chordal bipartite, we can
use Lemma 3. It takes time linear in the size of $I_{H}$, and so it takes
${\cal O}(w(T))$ time. From now on, we assume $r\geq 3$. By Lemma 17,
$C(H)\subseteq N_{H}^{2}[c]$. In particular, If $v\in V$ is such that
$d_{H}(v,c)\leq r-2$ then it is trivially at a distance $\leq r$ from every
vertex of $N_{H}^{2}[c]$. Thus, in order to compute $C(H)$ from
$N_{H}^{2}[c]$, we only need to consider the subsets $A_{r-1}:=\\{v\in V\mid
d_{H}(v,c)=r-1\\}$ and $A_{r}:=\\{v\in V\mid d_{H}(v,c)=r\\}$. We prove as an
intermediate claim that $N_{H}[c]$ is gated. Indeed, for every $v\notin
N_{H}[c]$, the closest vertices to $v$ in this closed neighbourhood are those
in the slice $L(v,d_{H}(v,c)-1,c)$, and thus they induce a clique by Lemma 14.
Then, the claim follows from Lemma 12. Furthermore, for every $v\notin
N_{H}[c]$, we can compute a corresponding gate $v^{*}$ as follows: we contract
in $T$ the subtree $T_{c}$ (induced by all the maximal cliques containing $c$)
to a single node representing $N_{H}[c]$, then we apply Lemma 13. It takes
${\cal O}(w(T))$ time.
Now, let $v\in A_{r-1}\cup A_{r}$. If the closed neighbourhood of any vertex
$u$ intersects $N_{H}[c]\cap N_{H}^{r-1}[v]$ then, $d_{H}(u,v)\leq r$.
Conversely, we claim that the closed neighbourhood of any central vertex must
intersect $N_{H}[c]\cap N_{H}^{r-1}[v]$. Indeed, observe that for every
$c^{\prime}\in C(H)$, $N_{H}[c^{\prime}]\cap N_{H}^{r-1}[v]\neq\emptyset$. By
Lemma 17, the balls $N_{H}[c^{\prime}],\ c^{\prime}\in C(H)$ also pairwise
intersect. Therefore, by the Helly property, there exists a $x\in
N^{r-1}_{H}[v]\cap\left(\bigcap_{c^{\prime}\in C(H)}N_{H}[c^{\prime}]\right)$.
We are done as in this situation $x\in N_{H}[c]\cap N_{H}^{r-1}[v]$. We now
analyse the following two cases:
Case $v\in A_{r}$. Let $v^{*}$ be the corresponding gate. Note that
$d_{H}(v^{*},c)=d_{H}(v,N_{H}[c])-1=d_{H}(v,c)-2=r-2$. In particular, every
vertex of $N_{H}^{2}[v^{*}]$ is at a distance $\leq r$ from vertex $v$.
Conversely, the closed neighbourhood of every central vertex must intersect
$N_{H}[c]\cap N^{r-1}_{H}[v]=L(v,r-1,c)\subseteq N_{H}[v^{*}]$. As a result,
$C(H)\subseteq N_{H}^{2}[v^{*}]$.
Case $v\in A_{r-1}$. Let $Z=\\{z\in N_{H}[c]\mid d_{H}(v,c)=r-1\\}$. Note that
$c\in Z$. Since the balls $N^{r-2}_{H}[v]$ and $N_{H}[z],\ z\in Z$ pairwise
intersect, by the Helly property there exists a vertex $v^{\prime}\in
N_{H}^{r-2}[v]\cap\left(\bigcap\\{N_{H}(z)\mid z\in Z\\}\right)$. Observe that
$v^{\prime}\in N_{H}^{r-2}[v]\cap N_{H}[c]=L(v,r-2,c)$. In particular, every
vertex of $N_{H}^{2}[v^{\prime}]$ is at a distance $\leq r$ from $v$.
Conversely, since by Lemma 14 the set $L(v,r-2,c)$ is a clique,
$N_{H}^{r-1}[v]\cap N_{H}[c]=L(v,r-2,c)\cup Z\subseteq N_{H}[v^{\prime}]$.
Thus, $C(H)\subseteq N_{H}^{2}[v^{\prime}]$.
Note that we may choose as our $v^{\prime}$ any $y\in N_{H}(v^{*})$ that
maximizes $|N_{H}[y]\cap N_{H}[c]|$, where $v^{*}$ is the gate computed for
$v$. In order to compute such vertex $v^{\prime}$, for every $v\in A_{r-1}$,
we use a similar trick as for Lemma 16. Namely, for every $y\in V$, we assign
a counter $h(y)$ (initially equal to $0$), whose final value must be
$|N_{H}[y]\cap N_{H}[c]|$. In order to correctly set the values of these
counters, we root the clique-tree $T$ arbitrarily and then we perform a
breadth-first search of this tree $T$ starting from the root. For each maximal
clique $K$ of $H$, let $K^{\prime}$ be the common intersection with its father
node in $T$ (in particular, $K^{\prime}=\emptyset$ if $K$ is the root). For
every $y\in K\setminus K^{\prime}$, we increase the counter $h(y)$ by exactly
$|K\cap N_{H}[c]|$. However, for every $y\in K^{\prime}$, we only increase the
counter $h(y)$ by $|N_{H}[c]\cap(K\setminus K^{\prime})|$. Overall, we just
need to scan each maximal clique of $H$ once, and so, the running time is in
${\cal O}(w(T))$. Finally, we scan each maximal clique $K$ once more and, to
every vertex of $K$, we assign a vertex $y\in K$ maximizing $h(y)$. For every
$v\in A_{r-1}$, we choose for $v^{\prime}$ any vertex $y$ assigned to $v^{*}$
and maximizing $h(y)$.
Overall, let $B:=\\{c\\}\cup\\{v^{*}\mid v\in A_{r}\\}\cup\\{v^{\prime}\mid
v\in A_{r-1}\\}$. By the above case analysis, $C(H)=\\{x\in V\mid\forall b\in
B,\ d_{H}(b,x)\leq 2\\}$. Since $I_{H}$ is chordal bipartite (Lemma 18) we can
adapt the technique of Lemma 3 in order to compute $C(H)$ (i.e., we set $k=2$,
and then compute partitions of the subset $B$ instead of computing partitions
for the full half-set $V$). It takes time linear in the size of $I_{H}$, and
so, it can be done in ${\cal O}(w(T))$ time. ∎
#### Computation of all the eccentricities in $G$.
Before proving Theorem 3, we need a final ingredient. Let us first generalize
Lemma 2 as follows.
###### Lemma 19.
If $G=(V_{0}\cup V_{1},E)$ is an absolute retract of bipartite graphs then,
the following holds for every $i\in\\{0,1\\}$ and $v\in V_{i}$:
* •
If $e_{H_{i}}(v)\leq rad(H_{1-i})-1$ then,
$e_{G}(v)=2e_{H_{i}}(v)+1=2rad(H_{1-i})-1$.
* •
If $e_{H_{i}}(v)=rad(H_{1-i})$ then, $e_{G}(v)=2rad(H_{1-i})$ if and only if
$N_{G}(v)\subseteq C(H_{1-i})$ and, for every $u\in V_{1-i}$, we have
$d_{H_{1-i}}(u,N_{G}(v))\leq rad(H_{1-i})-1$ (otherwise,
$e_{G}(v)=2rad(H_{1-i})+1$).
* •
If $e_{H_{i}}(v)\geq rad(H_{1-i})+1$ then, $e_{G}(v)=2e_{H_{i}}(v)$ if and
only if we have $e_{H_{1-i}}(u)<e_{H_{i}}(v)$ for some neighbour $u\in
N_{G}(v)$ (otherwise, $e_{G}(v)=2e_{H_{i}}(v)+1$).
###### Proof.
Let us first consider the case $e_{H_{i}}(v)\leq rad(H_{1-i})-1$. In
particular, every neighbour $u\in N_{G}(v)$ is at a distance $\leq
1+2e_{H_{i}}(v)+1\leq 2rad(H_{1-i})$ from any vertex of $V_{1-i}$. Therefore,
$e_{H_{i}}(v)=rad(H_{1-i})-1$, and every vertex of $N_{G}(v)$ is central in
$H_{1-i}$. Observe that $v$ must be at a distance $\geq 2rad(H_{1-i})-1$ from
at least one vertex of $V_{1-i}$ since otherwise, the eccentricity of all the
vertices of $N_{G}(v)$, in $H_{1-i}$, would be $<rad(H_{1-i})$. As a result,
$e_{G}(v)=2rad(H_{1-i})-1$.
Now, consider the case $e_{H_{i}}(v)=rad(H_{1-i})$. Then, since $G$ is
bipartite, we have $e_{G}(v)=2rad(H_{1-i})$ if and only if every vertex of
$V_{1-i}$ is at a distance $\leq 2rad(H_{1-i})-1$ from vertex $v$ (otherwise,
$e_{G}(v)=2rad(H_{1-i})+1$). Equivalently, $e_{G}(v)=2rad(H_{1-i})$ if and
only if, for every $u\in V_{1-i}$, we have $d_{G}(u,N_{G}(v))\leq
2rad(H_{1-i})-2$. Again, since $d_{H_{1-i}}(u,N_{G}(v))=d_{G}(u,N_{G}(v))/2$,
the latter is equivalent to have $d_{H_{1-i}}(u,N_{G}(v))\leq rad(H_{1-i})-1$.
Note that since $N_{G}(v)$ is a clique of $H_{1-i}$, this last equivalence
also implies that $N_{G}(v)\subseteq C(H_{1-i})$.
Finally, consider the case $e_{H_{i}}(v)\geq rad(H_{1-i})+1$. If furthermore
there is a neighbour $u\in N_{G}(v)$ s.t. $e_{H_{1-i}}(u)<e_{H_{i}}(v)$ then,
vertex $v$ is at a distance at most $2e_{H_{1-i}}(u)+1<2e_{H_{i}}(v)$ from
every vertex of $V_{1-i}$, and so, $e_{G}(v)=2e_{H_{i}}(v)$. Now, let us
assume to have $e_{H_{1-i}}(u)\geq e_{H_{i}}(v)$ for every neighbour $u\in
N_{G}(v)$. Suppose for the sake of contradiction $e_{G}(v)<2e_{H_{i}}(v)+1$.
In particular, every vertex of $V_{1-i}$ must be at a distance at most
$2e_{H_{i}}(v)-1$ from vertex $v$. Equivalently for any $x\in V_{1-i}$, the
half-balls $N_{G}(v)$ and $N_{G}^{2e_{H_{i}}(v)-2}[x]\cap V_{1-i}$ intersect.
Note that $2e_{H_{i}}(v)-2\geq 2rad(H_{1-i})\geq diam(H_{1-i})$. Hence, the
half-balls $N_{G}^{2e_{H_{i}}(v)-2}[x]\cap V_{1-i},\ x\in V_{1-i}$ also
pairwise intersect. But then, by Theorem 1, there is a neighbour $u\in
N_{G}(v)$ such that $e_{H_{1-i}}(u)\leq(2e_{H_{i}}(v)-2)/2=e_{H_{i}}(v)-1$. A
contradiction. ∎
Of the three cases in the above Lemma 19, the real algorithmic challenge is
the case $e_{H_{i}}(v)=rad(H_{1-i})$, for some $i\in\\{0,1\\}$. Finally, we
prove that such case can be solved in linear time for chordal bipartite
graphs.
###### Proof of Theorem 3.
By Lemma 9, $H_{0}$ and $H_{1}$ are strongly chordal graphs. We compute
clique-trees $T_{0}$ and $T_{1}$ for both $H_{0}$ and $H_{1}$, that takes
linear time by Corollary 1. Note that in particular, for each $i\in\\{0,1\\}$
we get $w(T_{i})={\cal O}(m+n)$, with $w(T_{i})$ the sum of the cardinalities
of the maximal cliques of $H_{i}$. Then, we compute all the eccentricities of
$H_{0}$ (resp, of $H_{1}$), that is ${\cal O}(w(T_{0}))$-time equivalent to
computing $C(H_{0})$ according to Lemma 11 (resp., ${\cal O}(w(T_{1}))$-time
equivalent to computing $C(H_{1})$). By Proposition 1, the center can be
computed in ${\cal O}(w(T_{0}))$ time (resp., in ${\cal O}(w(T_{1}))$ time).
Overall, for each $i\in\\{0,1\\}$ and every $v\in V_{i}$, we so computed
$e_{H_{i}}(v)$. It takes total linear time. Doing so, we can also compute
$rad(H_{0})$ and $rad(H_{1})$ within the same amount of time. Finally, we are
left explaining how to deduce from the latter the eccentricities $e_{G}(v),\
v\in V_{0}$ (the case $v\in V_{1}$ is symmetric to this one). By Lemma 19, if
$e_{H_{0}}(v)\neq rad(H_{1})$ then, we can compute $e_{G}(v)$ by scanning the
neighbour set $N_{G}(v)$. Therefore, in total ${\cal O}(m)$ time, we can
compute $e_{G}(v)$ for every $v\in V_{0}$ s.t. $e_{H_{0}}(v)\neq rad(H_{1})$.
In the same way, for every $v\in V_{0}$ s.t. $e_{H_{0}}(v)=rad(H_{1})$ and
$N_{G}(v)\not\subseteq C(H_{1})$, we set directly $e_{G}(v)=2rad(H_{1})+1$,
that is correct by Lemma 19, and it also takes ${\cal O}(m)$ time in total.
Let $S:=\\{v\in V_{0}\mid e_{H_{0}}(v)=rad(H_{1}),\ N_{G}(v)\subseteq
C(H_{1})\\}$. By Lemma 19, every vertex of $S$ has eccentricity either
$2rad(H_{1})$ or $2rad(H_{1})+1$ in $G$. We may further assume $rad(H_{1})\geq
3$ because otherwise, we can compute the eccentricity of all the vertices of
$S$ in total ${\cal O}(m)$ time by applying the technique of Lemma 3. In this
situation, we define a set $W\subseteq V_{1}$ with the following property: for
every $v\in S$, we have $e_{G}(v)=2rad(H_{1})$ if and only if for every $w\in
W$ we have $d_{G}(v,w)\leq 3$. Note that doing so, we may compute the
eccentricity of all the vertices of $S$ in ${\cal O}(m)$ time by applying the
same technique as for Lemma 3 (i.e., we set $k=3$, then we compute partitions
for $W$ rather than for the full half-set $V_{1}$). Furthermore, in order to
compute this set $W$, we proceed in a quite similar fashion as for Proposition
1. We detail this procedure next.
The algorithm. Let $r=rad(H_{1})$. First, we compute a vertex $c\in C(H_{1})$
that is adjacent to all the central vertices of $H_{1}$ (such vertex is
guaranteed to exist by Lemma 17). For every $u\in V_{1}$ s.t.
$d_{H_{1}}(u,c)\geq r-1$, we compute a vertex $u^{*}\in
N_{H_{1}}^{r-2}[u]\cap\left(\bigcap\\{N_{H_{1}}[x]\mid x\in N_{H_{1}}[c]\cap
N^{r-1}_{H_{1}}[u]\\}\right)$ and we add this gate $u^{*}$ to the set $W$.
Correctness. By Lemma 19, for every $v\in S$, we have $e_{G}(v)=2rad(H_{1})$
if and only if, for every $u\in V_{1},\ d_{H_{1}}(u,N_{G}(v))\leq r-1$. Note
that if $d_{H_{1}}(u,c)\leq r-2$ then, the distance in $H_{1}$ between $u$ and
any vertex of $C(H_{1})$ is at most $r-1$. So, we are only interested in those
$u$ s.t. $d_{H_{1}}(u,c)\geq r-1$. Assume the existence for such vertex $u$ of
a $u^{*}\in N_{H_{1}}^{r-2}[u]\cap\left(\bigcap\\{N_{H_{1}}[x]\mid x\in
N_{H_{1}}[c]\cap N^{r-1}_{H_{1}}[u]\\}\right)$. If $d_{G}(v,u^{*})\leq 3$
then, $d_{G}(u,v)\leq 3+2(r-2)=2r-1$, as desired. Conversely, if
$e_{G}(v)=2rad(H_{1})$ then there must be a $c_{u}\in N_{G}(v)\subseteq
C(H_{1})$ s.t. $d_{H_{1}}(c_{u},u)\leq r-1$. In particular, $c_{u}\in
N_{H_{1}}[c]\cap N^{r-1}_{H_{1}}[u]\subseteq N_{H_{1}}[u^{*}]$ and therefore,
$d_{G}(v,u^{*})\leq
d_{G}(v,c_{u})+d_{G}(c_{u},u^{*})=1+2d_{H_{1}}(c_{u},u^{*})\leq 3$. The
existence of a $u^{*}$ as above was already proved in Proposition 1, but we
repeat here the arguments for completeness. There are two cases. If
$d_{H_{1}}(u,c)=r$ then, $N_{H_{1}}[c]\cap
N^{r-1}_{H_{1}}[u]=L_{H_{1}}(u,r-1,c)$, and the result follows from the
combination of Lemmas 14 and 12. Otherwise, $d_{H_{1}}(u,c)=r-1$, and let
$Z=\\{z\in N_{H_{1}}[c]\mid d_{H_{1}}(u,z)=r-1\\}$. Since the balls
$N_{H_{1}}^{r-2}[u]$ and $N_{H_{1}}[z],\ z\in Z$ pairwise intersect, by the
Helly property, there exists a $u^{*}\in
N_{H_{1}}^{r-2}[u]\cap\left(\bigcap\\{N[z]\mid z\in Z\\}\right)$. Observe that
by construction, $u^{*}\in L_{H_{1}}(u,r-2,c)$, that is a clique according to
Lemma 14. We are done as $N^{r-1}_{H_{1}}[u]\cap
N_{H_{1}}[c]=L_{H_{1}}(u,r-2,c)\cup Z\subseteq N_{H_{1}}[u^{*}]$.
Complexity. The constructive proof of Proposition 1 yields an ${\cal
O}(w(T_{1}))$-time algorithm in order to compute $W$. Therefore, this set $W$
can be constructed in linear time. ∎
## 3 $k$-chromatic graphs
Recall that a proper $k$-coloring of $G=(V,E)$ is any mapping
$c:V\to\\{1,2,\ldots,k\\}$ such that $c(u)\neq c(v)$ for every edge $uv\in E$.
The chromatic number of $G$ is the least $k$ such that it has a proper
$k$-coloring, and a $k$-chromatic graph is a graph whose chromatic number is
equal to $k$. Note in particular with this definition that a $(k-1)$-chromatic
graph is not $k$-chromatic. We study the diameter within the absolute retracts
of $k$-chromatic graphs, for every fixed $k\geq 3$.
Our approach requires such graphs to be equipped with a proper $k$-coloring.
While this is a classic NP-hard problem for every $k\geq 3$ [65], it is known
that it can be done in polynomial time for absolute retracts of $k$-chromatic
graphs [10]. We remind this colouring algorithm in Sec. 3.1 where we observe
it can be implemented in order to run in linear time. Our general framework
for diameter computation (somehow mimicking what we did in Sec. 2.1) is
presented in Sec. 3.2. We complete our approach in Sec. 3.3, before concluding
this section with its main result in Sec 3.4.
### 3.1 Colouring algorithm
We start with a reminder of the Colouring algorithm presented in [10]
(Algorithm 3). The description below is exactly the same as in [10], where an
${\cal O}(n^{2})$ running-time was claimed.
Algorithm 3 Colouring algorithm.
0: A graph $G=(V,E)$.
1: Pick a vertex $u$ and let $K$ be a maximal clique containing $u$.
2: Let $k:=|K|$ and let $c:K\to\\{1,2,\ldots,k\\}$ be a proper $k$-coloring.
For each $v\in K\setminus\\{u\\}$, colour the common neighbours of
$K\setminus\\{v\\}$ with $c(v)$. Then, let $L$ be the set of vertices coloured
so far. For each neighbour $v$ of $u$ not in $L$, there is a unique colour $i$
such that $u$ does not have a neighbour in $L$ with colour $i$ (otherwise, $G$
is not an absolute retract). Assign $c(u):=i$. If $N_{G}[u]$ includes all of
$G$, then goto Step 5.
3: For each vertex $v\in V$ s.t. $d_{G}(u,v)=2$, there is a unique colour $i$
not occurring in $N_{G}(u)\cap N_{G}(v)$ or in $\\{u\\}\cup(N_{G}(u)\cap
N_{G}(v))$ (otherwise, $G$ is not an absolute retract). Assign $c(v):=i$. If
$N_{G}^{2}[u]$ includes all of $G$, then goto Step 5, otherwise assign
$\ell:=3$.
4: For each vertex $v\in V$ s.t. $d_{G}(u,v)=\ell$, there is a unique colour
$i$ not occurring in $N_{G}^{\ell-1}[u]\cap N_{G}(v)$ (otherwise, $G$ is not
an absolute retract). Assign $c(v):=i$. If $N_{G}^{\ell}[u]$ does not yet
contain all of $G$, then assign $\ell:=\ell+1$ and start Step 4 again.
5: If $c(u)=c(v)$ for some edge $uv$, then $G$ is not an absolute retract.
Otherwise, $c$ is a proper $k$-coloring of $G$, and $G$ is $k$-chromatic.
We refer to [10] for a correctness proof of Algorithm 3. Our modest, but
important contribution for our claimed running-times is as follows:
###### Proposition 2.
There is a linear-time algorithm such that, for every $k\geq 3$, if the input
$G$ is an absolute retract of $k$-chromatic graphs, then it computes a proper
$k$-coloring of $G$.
###### Proof.
Consider the following modified version of Algorithm 3:
1. 1.
We start from an arbitrary vertex $u$ and we greedily compute a maximal clique
$K$ containing vertex $u$. It takes linear time. Furthermore, let $k:=|K|$ and
let $c:K\to\\{1,2,\ldots,k\\}$ be a proper $k$-coloring (i.e., obtained by
numbering the vertices of $K$ from $1$ to $k$).
2. 2.
For every vertex $x\in V$, we compute $|N_{G}(x)\cap K|$. It can be done in
linear time by scanning once the neighbourhood of each vertex $v\in K$. We
consider the vertices $v\in N_{G}(u)\setminus K$ by non-increasing value of
$|N_{G}(x)\cap K|$. Note that such ordering of $N_{G}(u)\setminus K$ can be
computed using a linear-time sorting algorithm. As in the standard greedy
coloring algorithm, we assign to $v$ the least color $i$ not present in its
neighbourhood. If $i\geq k+1$ then, $G$ is not an absolute retract and we
stop. Doing so, it takes ${\cal O}(|N_{G}(v)|)$ time in order to color $v$,
and so, this whole step takes ${\cal O}(m)$ time.
3. 3.
We perform a breadth-first search rooted at $u$. Then, we consider the
vertices $v\in V$ s.t. $d_{G}(u,v)=2$. First, we search for the least color
$i$ such that $i\neq c(u)$ and there is no neighbour of $v$ coloured $i$. If
$i\leq k$ then, we set $c(v):=i$. Otherwise, we assign to $v$ the least color
$i$ not present in $N_{G}(v)$ (possibly, $i=c(u)$). If $i\geq k+1$ then, $G$
is not an absolute retract and we stop. This whole step also takes ${\cal
O}(m)$ time.
4. 4.
Finally, we consider all the remaining vertices $v,\ d_{G}(u,v)\geq 3$, by
non-decreasing value of $d_{G}(u,v)$ (these distances were computed during the
breadth-first search). Here also, we can compute such ordering of the
remaining vertices by using a linear-time sorting algorithm. We apply the
classic greedy coloring procedure, assigning to the current vertex $v$ the
least colour $i$ not present in its neighbourhood. If $i\geq k+1$ then $G$ is
not an absolute retract, and we stop. Overall, the whole algorithm indeed runs
in linear time.
In order to prove correctness of this above algorithm, it suffices to prove
that if $G$ is an absolute retract, then it computes the same coloring as
Algorithm 3. This is clear at Step 1, where we only colour the vertices of the
maximal clique $K$. Then, at Step 2, we start coloring the neighbours of $u$
with exactly $k-1$ neighbours in $K$. Note that the only possible color
amongst $\\{1,\ldots,k\\}$ to assign to such vertex $x$ is $c(v)$, where $v$
is the unique non-neighbour of $x$ in $K$. This is also what Algorithm 3 does.
Let $L$ be the set of vertices coloured so far. We continue coloring the
remaining neighbours $v\in N_{G}(u)\setminus L$. Algorithm 3 exploits the
property that such vertices have exactly one color $i\in\\{1,\ldots,k\\}$ that
is not present amongst their neighbours in $L$. But then, the classic greedy
coloring procedure also assigns this color $i$ to $v$. Next, consider a vertex
$v$ s.t. $d_{G}(u,v)=2$. If there is a unique available color $i$ amongst
$\\{1,\ldots,k\\}$ that is not present in $N_{G}(u)\cap N_{G}(v)$ then,
Algorithm 3 assigns this color to $v$; so does the classic greedy coloring
procedure. Otherwise, Algorithm 3 assigns to $v$ the unique color amongst
$\\{1,\ldots,k\\}$ that is not present in $\\{u\\}\cup(N_{G}(u)\cap
N_{G}(v))$; so does our modified greedy coloring procedure, where we first
exclude color $c(u)$ from the range of possibilities. Finally, similar
arguments apply to the vertices $v$ s.t. $d_{G}(u,v)\geq 3$. ∎
In the remainder of the section, we always assume the input graph $G$ to be
given with a proper $k$-coloring. We sometimes use implicitly the fact that,
for an absolute retract, such proper $k$-coloring is unique up to permuting
the colour classes [76].
### 3.2 General framework
The present section aims at introducing the necessary results and terminology
for Sec 3.3. Recall that in a graph $G=(V,E)$, a vertex $v$ is covered by
another vertex $w$ if $N_{G}(v)\subseteq N_{G}(w)$ (a covered vertex is called
embeddable in [76]). We now introduce a first characterization of absolute
retracts:
###### Theorem 4 ([76]).
Let $k\geq 3$. The graph $G=(V,E)$ is an absolute retract of $k$-chromatic
graphs if and only if for any proper $k$-coloring $c$, every peripheral vertex
$v$ is adjacent to all vertices $u$ with $c(u)\neq c(v)$, or it is covered and
$G\setminus v$ is an absolute retract of $k$-chromatic graphs.
We highlight the following special case of this characterization, that allows
us to deal with the diameter-two case separately from the general case,
namely:
###### Proposition 3 ([76]).
Let $G=(V,E)$ be an absolute retract of $k$-chromatic graphs for some $k\geq
3$, and let $c$ be a corresponding proper $k$-coloring. Then, $diam(G)\leq 2$
if and only if for every $1\leq i\leq k$, there exists a $z_{i}\in V$ with
$z_{i}v\in E$ for all $v\in V$ s.t. $c(v)\neq i$.
###### Corollary 2.
If $G=(V,E)$ is an absolute retract of $k$-chromatic graphs for some $k\geq
3$, then we can decide whether $diam(G)\leq 2$ in linear time.
###### Proof.
By Proposition 2, a proper $k$-coloring $c:V\to\\{1,\ldots,k\\}$ can be
computed in linear time. Then, we apply the criterion of Proposition 3. For
that, for each colour $i$ we choose as our candidate vertex for $z_{i}$ a
maximum-degree vertex of colour $i$. ∎
Thus, in what follows, we focus on the case $diam(G)\geq 3$. For that, we use
in combination to Theorem 4 yet another characterization of absolute retracts
(we refer the reader to [10] for other characterizations, leading to
polynomial-time recognition algorithms):
###### Theorem 5 ([10]).
Let $k\geq 3$. The graph $G=(V,E)$ is an absolute retract of $k$-chromatic
graphs if and only if the following conditions hold for any proper
$k$-coloring $c:V\to\\{1,2,\ldots,k\\}$:
1. 1.
For each colour $i$, any family of balls intersects in colour $i$ whenever
each pair of them does.
2. 2.
Every maximal clique of $G$ has cardinality exactly $k$.
3. 3.
For each colour $i$ and any two non-adjacent vertices $u$ and $v$ such that:
$c(v)\neq i$, and either $c(u)\neq i$ or $d_{G}(u,v)\geq 3$, there is a
neighbour $x$ of $v$ on a shortest $uv$-path s.t. $c(x)=i$.
Let $G=(V,E)$ be equipped with a proper $k$-coloring $c$. For every colour
$i$, let $V_{i}:=\\{v\in V\mid c(v)=i\\}$ be called a colour class. We define,
for every $v\in V_{i}$, $e_{i}(v):=\max\\{d_{G}(u,v)\mid u\in V_{i}\\}$. A
vertex $v\in V_{i}$ is $i$-peripheral if it maximizes $e_{i}(v)$. Finally, let
$d_{i}:=\max\\{e_{i}(v)\mid v\in V_{i}\\}$. We now generalize Lemma 2, as
follows:
###### Lemma 20.
Let $G=(V,E)$ be an absolute retract of $k$-chromatic graphs for some $k\geq
3$, and let $c$ be a corresponding proper $k$-coloring. Then, $\max_{1\leq
i\leq k}d_{i}\leq diam(G)\leq 1+\max_{1\leq i\leq k}d_{i}$. Moreover, if
$diam(G)\geq 3$, then we have $diam(G)=1+\max_{1\leq i\leq k}d_{i}$ if and
only if:
* •
either $\max_{1\leq i\leq k}d_{i}=2$;
* •
or, for some $i\neq j$ s.t. $d_{i}=d_{j}$ is maximized, there is some
$i$-peripheral vertex whose all neighbours coloured $j$ are $j$-peripheral.
###### Proof.
Clearly, we have $\max_{1\leq i\leq k}d_{i}\leq diam(G)$. In order to prove
that we also have $diam(G)\leq 1+\max_{1\leq i\leq k}d_{i}$, it suffices to
prove that each colour class $V_{i}$ is a dominating set of $G$. For that,
consider any vertex $v\in V\setminus V_{i}$. Let $K$ be a maximal clique
containing $v$. By Theorem 5, we have $|K|=k$, and therefore, $K\cap
V_{i}\neq\emptyset$. In particular, $N_{G}(v)\cap V_{i}\neq\emptyset$, as
desired.
In what follows, $diam(G)\geq 3$. Then, if $\max_{1\leq i\leq k}d_{i}=2$, we
must have $diam(G)=1+\max_{1\leq i\leq k}d_{i}=3$. From now on, we assume
$diam(G)\geq\max_{1\leq i\leq k}d_{i}\geq 3$.
Let us first assume $diam(G)=1+\max_{1\leq i\leq k}d_{i}$. Consider some
peripheral vertex $v$ of $G$, and let $c(v)=i$. Since $V_{i}$ is a dominating
set of $G$, $e_{G}(v)\leq 1+e_{i}(v)\leq 1+d_{i}$. In particular, we must have
$d_{i}$ is maximized and $v$ is $i$-peripheral (otherwise,
$e_{G}(v)<diam(G)$). We pick some $u\in V$ such that $d_{G}(u,v)=diam(G)$, and
let $c(u)=j$. Note that $j\neq i$ (otherwise, $d_{G}(u,v)\leq d_{i}<diam(G)$).
Let $x\in N_{G}(v)\cap V_{j}$ be arbitrary. Observe that we always have
$d_{G}(u,v)\leq 1+d_{G}(u,x)\leq 1+e_{j}(x)\leq 1+d_{j}$. As a result,
$d_{j}=d_{i}$ is maximized, and any neighbour $x$ of vertex $v$ with $c(x)=j$
is $j$-peripheral.
Conversely, suppose by contradiction this above necessary condition for having
$diam(G)=1+\max_{1\leq i\leq k}d_{i}$ is not sufficient on its own for the
absolute retracts of $k$-chromatic graphs. Without loss of generality, amongst
all the absolute retracts of $k$-chromatic graphs, $G$ is a minimum counter-
example for which the condition holds and $diam(G)=\max_{1\leq i\leq k}d_{i}$.
For some $i\neq j$ such that $d_{i}=d_{j}$ is maximized, let $v$ be a
$i$-peripheral vertex whose all neighbours coloured $j$ are $j$-peripheral,
and let $u\in N_{G}(v)$ be such that $c(u)=j$. Then,
$e_{G}(u)=d_{j}=d_{i}=diam(G)$. Observe that $u$ cannot be adjacent to all the
vertices coloured $i$ (otherwise, $d_{i}=2<3$). Therefore, by Theorem 4, $u$
is covered and $G^{\prime}=G\setminus u$ is an absolute retract of
$k$-chromatic graphs. Let
$V_{1}^{\prime},V_{2}^{\prime},\ldots,V_{k}^{\prime}$ be the colour classes of
$G^{\prime}$ (induced by the coloring $c$ restricted on $V\setminus\\{u\\}$).
Observe that for $p\neq j$ we have $V_{p}^{\prime}=V_{p}$, while
$V_{j}^{\prime}=V_{j}\setminus\\{u\\}$. Furthermore, let
$d_{1}^{\prime},d_{2}^{\prime},\ldots,d_{k}^{\prime}$ be the largest distances
between vertices in a same colour class of $G^{\prime}$. Since $G^{\prime}$ is
isometric, $d_{p}^{\prime}\leq d_{p}$ for all $p$. But, since
$V_{i}^{\prime}=V_{i}$, we get $d_{i}^{\prime}=d_{i}$, and therefore,
$\max_{p}d_{p}^{\prime}=\max_{p}d_{p}=d_{i}=diam(G)$. It implies
$diam(G^{\prime})=diam(G)=\max_{p}d_{p}^{\prime}$. By minimality of $G$, the
subgraph $G^{\prime}$ is not a counter-example. Hence, vertex $v$ must have
some neighbour $u^{\prime}\neq u$ coloured $j$ s.t.
$\max\\{d_{G}(u^{\prime},x)\mid x\in V_{j}\setminus\\{u\\}\\}\leq d_{i}-1$.
However, it implies
$e_{j}(u^{\prime})\leq\max\\{d_{G}(u^{\prime},u),d_{i}-1\\}=\max\\{2,d_{i}-1\\}=d_{i}-1$.
A contradiction. ∎
### 3.3 Computation of the $d_{i}$’s
Our strategy is as follows. First, we prove in Lemma 21 that we can reduce our
study to the case $k=3$. In Lemmas 22 and 25, respectively, we deal with the
cases $d_{i}={\cal O}(\sqrt{n})$ and $d_{i}=\Omega(\sqrt{n})$, respectively.
#### Reduction.
The next Lemma 21 allows us to reduce the general case to $k=3$.
###### Lemma 21.
Let $G=(V,E)$ be an absolute retract of $k$-chromatic graphs for some $k\geq
3$, and let $c$ be a corresponding proper $k$-coloring. For every distinct
colours $i_{1},i_{2},i_{3}$, the subgraph $H:=G[V_{i_{1}}\cup V_{i_{2}}\cup
V_{i_{3}}]$ is isometric. Moreover, $H$ is an absolute retract of
$3$-chromatic graphs.
###### Proof.
Suppose by contradiction $H$ is not isometric in $G$. Let $u,v\in V(H)$ be
such that no shortest $uv$-path of $G$ is contained in $H$, and $d_{G}(u,v)$
is minimum for this property. Since $H$ is induced, $d_{G}(u,v)>1$. But then,
let $j\in\\{i_{1},i_{2},i_{3}\\}\setminus\\{c(u),c(v)\\}$. By Theorem 5, there
is a neighbour $x$ of $v$ that is on a shortest $uv$-path of $G$ and such that
$c(x)=j$. In particular, $x\in V(H)$. Since $H$ is induced, we have $vx\in
E(H)$. However, by minimality of $d_{G}(u,v)$ we also have
$d_{H}(x,u)=d_{G}(x,u)$, and therefore $d_{H}(u,v)\leq
1+d_{H}(x,u)=1+d_{G}(x,u)=d_{G}(v,u)$. A contradiction.
Now, let us prove that $H$ is an absolute retract of $3$-chromatic graphs. For
that, it suffices to prove that $H$ satisfies the three properties stated in
Theorem 5. – Note that, since $c$ is the unique proper $k$-coloring of $G$,
the restriction of $c$ to $H$ is the unique proper $3$-coloring of this
subgraph. –
1. 1.
Let $j\in\\{i_{1},i_{2},i_{3}\\}$ be fixed. Consider some family of balls in
$H$, denoted $N_{H}^{r_{1}}[v_{1}],N_{H}^{r_{2}}[v_{2}],\ldots,\\\
N_{H}^{r_{q}}[v_{q}]$. Let us assume these balls pairwise intersect in colour
$j$, i.e., for every $1\leq a,b\leq q$ we have $N_{H}^{r_{a}}[v_{a}]\cap
N_{H}^{r_{b}}[v_{b}]\cap V_{j}\neq\emptyset$. Since $H$ is a subgraph of $G$,
the balls
$N_{G}^{r_{1}}[v_{1}],N_{G}^{r_{2}}[v_{2}],\ldots,N_{G}^{r_{q}}[v_{q}]$ also
pairwise intersect in colour $j$ (in $G$). By Theorem 5, there exists a vertex
$z_{j}\in V_{j}\cap\left(\bigcap\\{N_{G}^{r_{a}}[v_{a}]\mid 1\leq a\leq
q\\}\right)$. Since $V_{j}\subseteq V(H)$, and $H$ is isometric in $G$, we get
$z_{j}\in V_{j}\cap\left(\bigcap\\{N_{H}^{r_{a}}[v_{a}]\mid 1\leq a\leq
q\\}\right)$.
2. 2.
Let $K$ be a maximal clique of $H$. We have $K\subseteq K^{\prime}$ where
$K^{\prime}$ is a maximal clique of $G$. By Theorem 5, $|K^{\prime}|=k$. It
implies that every colour class intersects $K^{\prime}$. Hence,
$|K|=|K^{\prime}\cap(V_{i_{1}}\cup V_{i_{2}}\cup V_{i_{3}})|=3$.
3. 3.
Finally, let $u,v\in V(H)$ and let $j\in\\{i_{1},i_{2},i_{3}\\}$ be such that
$c(v)\neq j$ and either $c(u)\neq j$ or $d_{H}(u,v)\geq 3$. Since $H$ is
isometric, either $c(u)\neq j$ or $d_{G}(u,v)\geq 3$. By Theorem 5, there
exists a neighbour $x$ of $v$ coloured $j$ that is on a shortest $uv$-path in
$G$. Since $x\in V(H)$ and $H$ is isometric, $x$ is also on a shortest
$uv$-path in $H$.
Overall, all three properties of Theorem 5 are satisfied by $H$. It implies
that $H$ is an absolute retract of $3$-chromatic graphs. ∎
#### Small-diameter case.
Next, we deal with the case when $d_{i}$ is sufficiently small.
###### Lemma 22.
Let $G=(V,E)$ be an absolute retract of $3$-chromatic graphs, and let $c$ be a
corresponding proper $3$-coloring. For each colour $i$ and $D\geq 2$, we can
compute in ${\cal O}(Dm)$ time the set $U_{i}:=\\{v\in V_{i}\mid e_{i}(u)\leq
D\\}$.
###### Proof.
It suffices to prove the result for $i=1$. For that, we generalize the
techniques used in Lemma 3. Specifically, for every $1\leq t\leq D$, the
objective is to compute a partition of $V_{1}$, denoted ${\cal P}_{t}$,
subject to the following property: for each colour $i$, the $|{\cal P}_{t}|$
subsets $B_{t}[A,i]:=V_{i}\cap\left(\bigcap\\{N_{G}^{t}[a]\mid a\in
A\\}\right)$, for all $A\in{\cal P}_{t}$, must be pairwise disjoint and, if
$i\neq 1$ or $t\geq 2$, these sets must also be nonempty. Being given any such
partition ${\cal P}_{D}$, we observe that $U_{1}\neq\emptyset$ if and only if
${\cal P}_{D}=(V_{1})$ and, if it is the case, then $U_{1}=B_{D}[V_{1},1]$.
The algorithm. For the base case $t=1$, let $Y:=V_{1}$. While
$Y\neq\emptyset$, we select some vertex $x\in V_{2}$ such that $|N_{G}(x)\cap
Y|$ is maximized, we add $N_{G}(x)\cap Y$ as a new group in ${\cal P}_{1}$,
then we set $Y:=Y\setminus N_{G}(x)$. Finally, we enumerate all groups
$A\in{\cal P}_{1}$, and we scan the neighbourhoods of all vertices $a\in A$ in
order to compute the sets $B_{1}[A,i]$ for each colour $i$ (note that
$B_{1}[A,1]=A$ if $|A|=1$ and $B_{1}[A,1]=\emptyset$ otherwise).
At step $t+1$, for each group $A\in{\cal P}_{t}$, let us define:
$W(A,1):=N_{G}(B_{t}[A,2])\cap V_{1},$ $W(A,2):=N_{G}(B_{t}[A,3])\cap V_{2},$
$W(A,3):=N_{G}(B_{t}[A,2])\cap V_{3}.$
Let ${\cal F}:={\cal P}_{t}$. While ${\cal F}\neq\emptyset$, we proceed as
follows. We pick some $v\in V_{1}$ such that $\\#\\{A\in{\cal F}\mid v\in
W(A,1)\\}$ is maximized. Let $A^{\prime}:=\bigcup\\{A\in{\cal F}\mid v\in
W(A,1)\\}$. We add $A^{\prime}$ to ${\cal P}_{t+1}$, for each colour $i$ we
set $B_{t+1}[A^{\prime},i]:=\bigcap\\{W(A,i)\mid A\in{\cal F},\ v\in
W(A,1)\\}$, and we set ${\cal F}:={\cal F}\setminus\\{A\in{\cal F}\mid v\in
W(A,1)\\}$.
Correctness. We first consider the base case $t=1$. Clearly, the sets
$B_{1}[A,1]$ (equal to either $A$, if it is a singleton, and to the empty set
otherwise) are pairwise disjoint. Since at every step, in order to create a
new group, we pick a vertex of $V_{2}$ with a maximum number of unselected
vertices in $V_{1}$, it follows by construction that the sets $B_{1}[A,2]$ are
nonempty. By maximality of the vertex $V_{2}$ selected at each step, it also
follows that all these sets $B_{1}[A,2]$ are pairwise disjoint (this is
exactly the same argument as the one used for Claim 2, in Lemma 3). The next
claim shows that both properties (i.e., being nonempty and pairwise disjoint)
also hold for the sets $B_{1}[A,3]$.
###### Claim 3.
For $A\subseteq V_{1}$, the neighbour-sets $N_{G}(a),\ a\in A$ intersect in
colour $2$ if and only if they intersect in colour $3$.
Proof. Let us assume the balls $N_{G}[a],\ a\in A$ intersect in colour $2$
(resp., in colour $3$). In particular, the vertices of $A$ are pairwise at
distance two. By the third property of Theorem 5, the balls $N_{G}[a],\ a\in
A$ pairwise intersect in colour $3$ (resp., in colour $2$), and therefore the
claim now follows from the first (Helly-type) property of this Theorem 5.
$\diamond$
From now on, we consider the inductive step (i.e., from $t$ to $t+1$). We need
the following claim for our analysis:
###### Claim 4.
For each colour $i$, $t\geq 1$ and $A\in{\cal P}_{t}$, we have
$W(A,i)=V_{i}\cap\left(\bigcap\\{N_{G}^{t+1}[a]\mid a\in A\mid\\}\right)$.
Proof. By construction, $W(A,i)\subseteq
V_{i}\cap\left(\bigcap\\{N_{G}^{t+1}[a]\mid a\in A\\}\right)$. Conversely, let
$u\in\bigcap\\{N_{G}^{t+1}[a]\mid a\in A\\}$ be such that $c(u)=i$. Let $j$ be
the least colour available amongst $\\{2,3\\}\setminus\\{i\\}$ (i.e., $j=2$ if
$i\neq 2$ and $j=3$ otherwise). We prove as a subclaim that, for every $a\in
A$, the balls $N_{G}^{t}[a]$ and $N_{G}[u]$ intersect in colour $j$. For that,
we need to consider three cases:
* •
Case $u=a$. This can only happen if $i=1$. Then, $N_{G}[u]\subseteq
N_{G}^{t}[u]=N_{G}^{t}[a]$. Since $u$ has at least one neighbour coloured $j$,
the balls $N_{G}^{t}[a]$ and $N_{G}[u]$ trivially intersect in colour $j$.
* •
Case $ua\in E$. By Theorem 5, every maximal clique of $G$ has cardinality $3$.
It implies that $u,a$ have a common neighbour coloured $j$.
* •
Case $d_{G}(u,a)\geq 2$. By Theorem 5, there exists a neighbour $x$ of $u$ on
a shortest $au$-path such that $c(x)=j$.
Furthermore, since $B_{t}[A,j]\neq\emptyset$, the balls $N_{G}^{t}[a],\ a\in
A$, also pairwise intersect in colour $j$. By Theorem 5, there exists a $x\in
N_{G}(u)\cap B_{t}[A,j]$. In order to conclude the proof, we just need to
observe that $u\in N_{G}(B_{t}[A,j])\cap V_{i}=W(A,i)$. $\diamond$
When a new set $A^{\prime}$ is created, as the union of sets in some nonempty
family ${\cal Q}\subseteq{\cal P}_{t}$, we define for each colour $i$,
$B_{t+1}[A^{\prime},i]=\bigcap\\{W(A,i)\mid A\in{\cal Q}\\}$. By the above
Claim 4,
$B_{t+1}[A^{\prime},i]=V_{i}\cap\left(\bigcap\\{N_{G}^{t+1}[a^{\prime}]\mid
a^{\prime}\in A^{\prime}\mid\\}\right)$, as desired. It now remains to prove
that all sets $B_{t+1}[A^{\prime},i],\ A^{\prime}\in{\cal P}_{t+1}$ are
nonempty and pairwise disjoint. Observe that in order to create a new group,
we repeatedly pick a vertex $v\in V_{1}$ with a maximum number of unselected
groups $A\in{\cal P}_{t}$ such that $v\in W(A,1)$. To ensure that all groups
$A\in{\cal P}_{t}$ are eventually selected, we prove that:
###### Claim 5.
For each $t\geq 1$ and $A\in{\cal P}_{t}$, we have $W(A,1)\neq\emptyset$.
Proof. Let $x\in B_{t}[A,2]$. Since $G$ is an absolute retract, $N_{G}(x)\cap
V_{1}\neq\emptyset$. We are now done as we have by construction $N_{G}(x)\cap
V_{1}\subseteq W(A,1)$. $\diamond$
By maximality of the vertex of $V_{1}$ selected at each step, the sets
$B_{t+1}[A^{\prime},1],\ A^{\prime}\in{\cal P}_{t+1}$ are nonempty and
pairwise disjoint. Finally, the next claim shows that both properties also
hold for the sets $B_{t+1}[A^{\prime},i]$, for any colour $i$:
###### Claim 6.
For each colour $i\neq 1$, $A\subseteq V_{1}$ and $t\geq 2$, the balls
$N_{G}^{t}[a],\ a\in A$ intersect in colour $1$ if and only if they also
intersect in colour $i$.
Proof. First assume that the balls $N_{G}^{t}[a],\ a\in A$ intersect in colour
$1$. Let $a,a^{\prime}\in A$ and let $v\in N_{G}^{t}[a]\cap
N_{G}^{t}[a^{\prime}]$ such that $c(v)=1$. W.l.o.g., $a\neq v$. For
$j\in\\{2,3\\}\setminus\\{i\\}$, by Theorem 5 there exists a neighbour $x$ of
$v$ on a shortest $av$-path such that $c(x)=j$. We divide our analysis into
three cases:
* •
Case $xa^{\prime}\notin E$. By Theorem 5, there exists a neighbour $y$ of $x$
on a shortest $xa^{\prime}$-path such that $c(y)=i$. Note that $y\in
N_{G}^{t}[a]\cap N_{G}^{t}[a^{\prime}]$.
* •
Case $xa,xa^{\prime}\in E$. By Claim 3, the neighbour-sets $N_{G}(a),\
N_{G}(a^{\prime})$ intersect in colour $i$. Since $t\geq 2$, so do the balls
$N_{G}^{t}[a],\ N_{G}^{t}[a^{\prime}]$.
* •
Case $xa\notin E,\ xa^{\prime}\in E$. By Theorem 5, there exists a neighbour
$y$ of $x$ on a shortest $ax$-path such that $c(y)=i$. Note that $y\in
N_{G}^{t-2}[a]\cap N_{G}^{2}[a^{\prime}]\subseteq N_{G}^{t}[a]\cap
N_{G}^{t}[a^{\prime}]$.
Overall, the balls $N_{G}^{t}[a],\ a\in A$ pairwise intersect in colour $i$,
and therefore by Theorem 5 there is a vertex coloured $i$ in their common
intersection.
Conversely, assume that the balls $N_{G}^{t}[a],\ a\in A$ intersect in colour
$i$. Let $a,a^{\prime}\in A$ and let $u\in N_{G}^{t}[a]\cap
N_{G}^{t}[a^{\prime}]$ such that $c(u)=i$. Here also, we divide our analysis
into several cases:
* •
Case $ua,ua^{\prime}\notin E$. Let $j\in\\{2,3\\}\setminus\\{i\\}$. By Theorem
5 (applied twice) there exist $x,y\in N_{G}(u)$ that are on a shortest
$au$-path and a shortest $a^{\prime}u$-path respectively, such that
$c(x)=c(y)=j$. If $x=y$ then, any neighbour $z\in N_{G}(x)$ coloured $1$ is in
$N_{G}^{t}[a]\cap N_{G}^{t}[a^{\prime}]$. Otherwise, $d_{G}(x,y)=2$. By
Theorem 5, there is a shortest $xy$-path whose internal node $z$ has colour
$1$. Here also, $z\in N_{G}^{t}[a]\cap N_{G}^{t}[a^{\prime}]$.
* •
Case $ua,ua^{\prime}\in E$. In this situation, $d_{G}(a,a^{\prime})=2$, and
the balls $N_{G}^{2}[a]\subseteq N_{G}^{t}[a]$ and
$N_{G}^{2}[a^{\prime}]\subseteq N_{G}^{t}[a^{\prime}]$ intersect in colour
$1$.
* •
Case $ua\in E,\ ua^{\prime}\notin E$ (The case $ua\notin E,\ ua^{\prime}\in E$
is symmetrical to this one). If $d_{G}(a,a^{\prime})\leq t$ then, the balls
$N_{G}^{t}[a]$ and $N_{G}^{t}[a^{\prime}]$ intersect in colour $1$. Otherwise,
$d_{G}(a^{\prime},u)=t$ and $d_{G}(a,a^{\prime})=t+1$. If furthermore
$d_{G}(u,a^{\prime})=t\geq 3$ then, by Theorem 5, there exists a neighbour $x$
of $u$ on a shortest $a^{\prime}u$-path such that $c(x)=1$. In this situation,
$x\in N_{G}^{2}[a]\cap N_{G}^{t-1}[a^{\prime}]\subseteq N_{G}^{t}[a]\cap
N_{G}^{t}[a^{\prime}]$. Thus, we are left considering the special subcase
$t=2$. Let $j\in\\{2,3\\}\setminus\\{i\\}$, and let $v\in
N_{G}(a^{\prime})\cap N_{G}(u)$. Observe that we have $c(v)=j$. Furthermore,
by Theorem 5, vertices $a$ and $u$ must have a common neighbour $y$ coloured
$j$ (i.e., the third vertex in a maximal clique containing $a$ and $u$). By
construction, $d_{G}(y,v)=2$. Then, again by Theorem 5, there exists a common
neighbour $z\in N_{G}(y)\cap N_{G}(v)$ such that $c(y)=1$. In this situation,
$y\in N_{G}^{2}[a]\cap N_{G}^{2}[a^{\prime}]\subseteq N_{G}^{t}[a]\cap
N_{G}^{t}[a^{\prime}]$.
Overall, the balls $N_{G}^{t}[a],\ a\in A$ pairwise intersect in colour $1$,
and therefore by Theorem 5 there is a vertex coloured $1$ in their common
intersection. $\diamond$
Complexity. We revisit the approach taken in Lemma 3. For the base case $t=1$
we compute in ${\cal O}(m)$ time, for each vertex of $V_{2}$, its number of
neighbours in $V_{1}$. We create an array of $|V_{1}|$ lists, numbered from
$1$ to $|V_{1}|$. Each vertex $u\in V_{2}$ is put in the list numbered
$|N_{G}(u)\cap V_{1}|$. It takes ${\cal O}(n)$ time. Then, we scan the lists
in decreasing order (i.e., from $|V_{1}|$ downto $1$), going to the next list
each time the current one is empty. When the current list is nonempty, we pick
any vertex $u$ of this list in order to create the next subset $A\in{\cal
P}_{1}$. Note that $A$ can be constructed in ${\cal O}(|N_{G}(u)|)$ time.
Since each vertex $u$ gets used for the creation of at most one group
(otherwise, there would exist $A,A^{\prime}\in{\cal P}_{1}$ such that
$B_{1}[A,2]\cap B_{1}[A^{\prime},2]\neq\emptyset$), the total running time in
order to create all the groups of ${\cal P}_{1}$ is in ${\cal O}(n+m)$.
Furthermore, after a group $A$ is created, we need to actualize the number of
unselected vertices in $V_{1}$ for each $x\in V_{2}$ (discarding all such
vertices whose all neighbours in $V_{1}$ are already selected). For that, we
first scan $N_{G}(a),a\in A$ in order to update, for each $x\in V_{2}$, its
number of unselected vertices in $V_{1}$. Then, if each vertex of $V_{2}$
keeps a pointer to its position in the unique list in which it is contained,
we can update the list contents in ${\cal O}(\sum_{a\in A}|N_{G}(a)|)$ time
(maximum number of vertices of $V_{2}$ that need to be moved). Overall, this
phase also takes total ${\cal O}(n+m)$ time. Finally, since ${\cal P}_{1}$ is
a partition of $V_{1}$, for each colour $i$ we can create the sets
$B_{1}[A,i],i\in A$ in ${\cal O}(m)$ time, simply by scanning the neighbour-
sets of each vertex of $V_{1}$. In order to complete our complexity analysis,
it remains to prove that each inductive step (from $t$ to $t+1$) also requires
${\cal O}(m)$ time. For that, since by the hypothesis the sets $B_{t}[A,i],\
A\in{\cal P}_{t}$ are pairwise disjoint, we can compute the sets $W(A,i)$ in
total linear time. Then, we proceed as for the first case (see also Lemma 3),
but we now consider the vertices of $V_{1}$ rather than $V_{2}$, and for these
vertices, we keep track of the number of sets $W(A,i)$ they belong to rather
than keeping track of their degrees. We get a running time proportional to
$\sum_{A\in{\cal P}_{t}}W(A,i)={\cal O}(n+m)$. ∎
#### Large-diameter case.
Finally, we address the case when $d_{i}$ is large. For that, we start with a
simple intermediate lemma. Recall that, for every two vertices $u$ and $v$ and
any $\ell\leq d(u,v)$, we can define the slice $L(u,\ell,v):=\\{x\in
I(u,v)\mid d(u,x)=\ell\\}$.
###### Lemma 23.
Let $G=(V,E)$ be an absolute retract of $k$-chromatic graphs for some $k\geq
3$, and let $c$ be a corresponding proper $k$-coloring. For each colour $i$
and $u,v\in V_{i}$ such that $u\neq v$ and $d_{G}(u,v)\neq 3$, there exists a
$x\in L(u,2,v)$ coloured $i$.
###### Proof.
If $d_{G}(u,v)=2$ then, $x=v$. Otherwise, $d_{G}(u,v)\geq 4$. By Theorem 5,
for any $j\neq i$ there exists a neighbour $y$ of $u$ coloured $j$ on a
shortest $uv$-path. Since $d_{G}(y,v)\geq 3$, again by Theorem 5, there exists
a neighbour $x$ of $y$ coloured $i$ on a shortest $yv$-path, and so, on a
shortest $uv$-path. ∎
A function is called unimodal if every local minimum is also a global minimum.
It is known that the eccentricity function of a Helly graph is unimodal [36],
and this property got used in [40] in order to compute all the eccentricities
in this graph class in subquadratic time. Next, we prove that a similar, but
weaker property holds for each colour class of absolute retracts, namely:
###### Lemma 24.
Let $G=(V,E)$ be an absolute retract of $k$-chromatic graphs for some $k\geq
3$, and let $c$ be a corresponding proper $k$-coloring. For each colour $i$
and any $u\in V_{i}$ s.t. $e_{i}(u)\geq(d_{i}+5)/2\geq 7$, there exists a
$u^{\prime}\in V_{i}$ s.t. $d_{G}(u,u^{\prime})=2$ and
$e_{i}(u^{\prime})=e_{i}(u)-2$.
###### Proof.
Let $X_{i}=\\{x\in V_{i}\mid d_{G}(u,x)\geq 4\\}$. By Lemma 23, for every
$x\in X_{i}$, the balls $N_{G}^{2}[u]$ and $N_{G}^{e_{i}(u)-2}[x]$ intersect
in colour $i$. Furthermore, let $x,x^{\prime}\in X_{i}$ be arbitrary. If
$d_{G}(x,x^{\prime})\leq e_{i}(u)-2$ then the balls $N_{G}^{e_{i}(u)-2}[x]$
and $N_{G}^{e_{i}(u)-2}[x^{\prime}]$ intersect in colour $i$. Otherwise, let
$\ell:=d_{G}(x,x^{\prime})-(e_{i}(u)-2)$, and let $2t\in\\{\ell,\ell+1\\}$ be
even. We prove by finite induction the existence of vertices
$x_{0},x_{1},\ldots,x_{t}\in I_{G}(x,x^{\prime})$ s.t., for every $0\leq p\leq
t$, we have $d_{G}(x,x_{p})=2p$ and $c(x_{p})=i$. For the base case $p=0$, we
have $x_{0}=x$. For $0<p\leq t$, we observe that:
$\displaystyle d_{G}(x_{p-1},x^{\prime})$
$\displaystyle=d_{G}(x,x^{\prime})-2(p-1)=d_{G}(x,x^{\prime})-2p+2$
$\displaystyle\geq d_{G}(x,x^{\prime})-2t+2\geq
d_{G}(x,x^{\prime})-(\ell+1)+2$
$\displaystyle=d_{G}(x,x^{\prime})-\ell+1=e_{i}(u)-1>3.$
Then, the existence of $x_{p}$ follows from Lemma 23 (applied to the pair
$x_{p-1},x^{\prime}$). Furthermore,
$\displaystyle 2t$ $\displaystyle\leq\ell+1=d_{G}(x,x^{\prime})-e_{i}(u)+3\leq
d_{i}-e_{i}(u)+3$ $\displaystyle\leq
d_{i}-(d_{i}+5)/2+3=(d_{i}+1)/2=(d_{i}+5)/2-2$ $\displaystyle\leq e_{i}(u)-2.$
Thus, in this situation, $x_{2t}\in N_{G}^{e_{i}(u)-2}[x]\cap
N_{G}^{e_{i}(u)-2}[x^{\prime}]$. Overall, we obtain that the balls
$N_{G}^{2}[u]$ and $N_{G}^{e_{i}(u)-2}[x],\ x\in X_{i}$ pairwise intersect in
colour $i$. By Theorem 5, there exists a $u^{\prime}\in
N_{G}^{2}[u]\cap\left(\bigcap\\{N_{G}^{e_{i}(u)-2}[x]\mid x\in
X_{i}\\}\right)$ such that $c(u^{\prime})=i$. Note that $u^{\prime}\neq u$
because there must be a $x\in X_{i}$ s.t. $d_{G}(u,x)=e_{i}(u)\geq 4$. We
obtain that $d_{G}(u,u^{\prime})=2$ and that, for every $x\in V_{i}\setminus
X_{i}$ we have $d_{G}(u^{\prime},x)\leq 2+d_{G}(u,x)\leq 5$. As a result,
$e_{i}(u^{\prime})\leq\max\\{5,e_{i}(u)-2\\}=e_{i}(u)-2$. Since
$d_{G}(u,u^{\prime})=2$, this must be an equality. ∎
We end up applying this almost-unimodality property to the computation of the
$d_{i}$’s (assuming these values to be at least in $\Omega(\sqrt{n})$):
###### Lemma 25.
Let $G=(V,E)$ be an absolute retract of $k$-chromatic graphs for some $k\geq
3$, let $c$ be a corresponding proper $k$-coloring, and let $i$ be such that
$d_{i}\geq 8D+5=\omega(\log{n})$. Then, with high probability, we can compute
in total $\tilde{\cal O}(mn/D)$ time the value $d_{i}$ and the $i$-peripheral
vertices.
###### Proof.
We modify Algorithm 1 as follows.
The algorithm. We set $p=\alpha\frac{\log{n}}{D}$, for some sufficiently large
constant $\alpha$. Then, let $U_{i}(p)$ contain every $v\in V_{i}$
independently with probability $p$. For all $v\in V_{i}$, let
$\bar{e}_{i}(v):=\min\\{d_{G}(u,v)+e_{i}(u)\mid u\in U_{i}(p),d_{G}(u,v)\leq
D\\}$ (with the understanding that, if no vertex of $U_{i}(p)$ is at a
distance $\leq D$ from $v$, then $\bar{e}_{i}(v)=0$). We output
$d_{i}=\max\\{\bar{e}_{i}(v)\mid v\in V_{i}\\}$, and we identify as
$i$-peripheral all vertices $v\in V_{i}$ such that $\bar{e}_{i}(v)$ is
maximized.
Complexity. Since we have to perform a breadth-first search from every vertex
of $U_{i}(p)$, the running time of the algorithm is in ${\cal
O}(m|U_{i}(p)|)$. Let us prove that with high probability,
$|U_{i}(p)|=\tilde{\cal O}(n/D)$. First note that we have $\mathbb{E}\left[\
|U_{i}(p)|\ \right]=\tilde{\Theta}(|V_{i}|/D)$. We claim to have
$|V_{i}|=\Omega(D)$. In order to see that, consider $x,y\in V_{i}$ s.t.
$d_{G}(x,y)=d_{i}$. Recall that $d_{i}=\Omega(D)$. Thus, by applying Lemma 23
$\Omega(D)$ times, we get the existence of a shortest $xy$-path with
$\Omega(D)$ vertices coloured $i$. The latter proves, as claimed,
$|V_{i}|=\Omega(D)$. Then, $\mathbb{E}\left[\ |U_{i}(p)|\
\right]=\Omega(\log{n})$. By Chernoff bounds, we have $|U_{i}(p)|=\tilde{\cal
O}(|V_{i}|/D)=\tilde{\cal O}(n/D)$ with high probability.
Correctness. Let $v\in V_{i}$ be arbitrary. We divide our analysis into two
cases.
* •
Case $e_{i}(v)<(d_{i}+5)/2+2D$. For any $u\in U_{i}(p)$ s.t. $d_{G}(u,v)\leq
D$, $e_{i}(u)<(d_{i}+5)/2+3D$. Therefore,
$\bar{e}_{i}(v)<(d_{i}+5)/2+4D=d_{i}/2+(8D+5)/2\leq d_{i}$.
* •
Case $e_{i}(v)\geq(d_{i}+5)/2+2D$. In particular, we always fall in this case
if $v$ is $i$-peripheral. Then, by applying Lemma 24 $D$ times, we get the
existence of vertices $x_{0}=v,x_{1},x_{2},\ldots,x_{D}$ such that, for every
$j>0$, $d_{G}(x_{j-1},x_{j})=2$ and $e_{i}(x_{j})=e_{i}(x_{j-1})-2$. Observe
that we have $d_{G}(v,x_{j})\leq 2j$ and $e_{i}(x_{j})=e_{i}(v)-2j$, and
therefore $d_{G}(v,x_{j})=2j$. It implies that, if $x_{j}\in U_{i}(p)$ for
some $j$, $\bar{e}_{i}(v)=e_{i}(v)$. Let us prove this happens with high
probability. Indeed:
$\displaystyle\mathbb{P}r[U_{i}(p)\cap\\{x_{0},x_{1},\ldots,x_{D}\\}=\emptyset]<(1-p)^{D}=(1-p)^{\frac{\alpha\log{n}}{p}}\leq
n^{-\alpha}.$
Summarizing both cases above, for all $i$-peripheral vertices $v$ we have
$\bar{e}_{i}(v)=e_{i}(v)=d_{i}$, whereas for every $v\in V_{i}$ that is not
$i$-peripheral, $\bar{e}_{i}(v)<d_{i}$. ∎
### 3.4 Main result
We end up gathering all previous results in this section, in order to prove
the following:
###### Theorem 6.
If $G=(V,E)$ is an absolute retract of $k$-chromatic graphs, for some $k\geq
3$, then we can compute its diameter with high probability in $\tilde{\cal
O}(m\sqrt{n})$ time.
###### Proof.
We may assume $diam(G)>1$ (trivial case). By Corollary 2, we can decide in
linear time whether $diam(G)\leq 2$. Thus, from now on, let us assume
$diam(G)\geq 3$. We compute a proper $k$-coloring of $G$, that takes linear-
time according to Proposition 2. Then, by Lemma 20, we can compute $diam(G)$
in linear time if, for each colour $i$, we are given $d_{i}$ and the
corresponding $i$-peripheral vertices. In order to compute this information,
let $V_{1},V_{2},\ldots,V_{k}$ be the colour classes. Up to repeating $V_{1}$
($=V_{k+1}$) and $V_{2}$ ($=V_{k+2}$) at most once we may assume the number of
colour to be a multiple of three, and then we partition the colour classes in
disjoint triples $V_{i}\cup V_{i+1}\cup V_{i+2}$. By Lemma 21, each subgraph
$H_{i}:=G[V_{i}\cup V_{i+1}\cup V_{i+2}]$ is isometric in $G$ and is an
absolute retract of $3$-chromatic graphs. Therefore, we may restrict ourselves
to $H_{i}$ in order to compute the values $d_{i},d_{i+1},d_{i+2}$. Let
$n_{i}:=|V(H_{i})|$ and $m_{i}:=|E(H_{i})|$. By construction,
$\sum_{i}n_{i}=\Theta(n)$ and $\sum_{i}m_{i}=\Theta(m)$. Hence, if for each
$i$ our computations require $\tilde{\cal O}(m_{i}\sqrt{n_{i}})$ time, the
total running time is in $\tilde{\cal O}(m\sqrt{n})$. Let us prove it is the
case for $H_{1}$ (and so, by symmetry, for every $i$). Since there are only
three colour classes in $H_{1}$, it is sufficient to prove that we can compute
$d_{1}$ and all the $1$-peripheral vertices in $\tilde{\cal
O}(m_{1}\sqrt{n_{1}})$ time. For that, we first compute $D:=e_{1}(v)$ for some
arbitrary $v\in V_{1}$. It takes ${\cal O}(m_{1})$ time and, by the triangular
inequality, it is a $2$-approximation of $d_{1}$. There are now two cases. If
$D\leq 16\sqrt{n_{1}}+10$ then, we compute by one-sided binary search the
smallest $t$ such that $\forall v\in V_{1},\ e_{1}(v)\leq t$, that is exactly
$d_{1}$. Note that at each step of the binary search we apply Lemma 22, that
results in a running time in $\tilde{\cal O}(m_{1}d_{1})=\tilde{\cal
O}(m_{1}D)=\tilde{\cal O}(m_{1}\sqrt{n_{1}})$. Furthermore, we can compute all
the $1$-peripheral vertices within the same amount of time, simply with one
more call to Lemma 22 (for $d_{1}-1$). Otherwise, $d_{1}\geq D/2\geq
8\sqrt{n_{1}}+5$, and we apply Lemma 4. With high probability, the running
time is in $\tilde{\cal O}(m_{1}\sqrt{n_{1}})$. ∎
## 4 Split graphs
The graphs studied in the two previous Sec. 2 and 3 are exactly the absolute
retracts of the class of all (irreflexive) graphs. In this section, we show
that by considering the absolute retracts of more restricted graph classes, we
may derive negative results about faster diameter computation. Specifically,
$G=(V,E)$ is a split graph if its vertex-set $V$ can be partitioned into a
clique $K$ and a stable set $S$. In what follows, we always denote such a
bipartition by $K+S$. We also use the standard notations $\omega(G)$ and
$\alpha(G)$ for, respectively, the clique number (maximum cardinality of a
clique) and the independence number (maximum cardinality of a stable set) of
$G$.
Let us recall the following hardness result about the diameter problem on
split graphs:
###### Lemma 26 ([15]).
For any $\epsilon>0$, there exists a $c(\epsilon)$ s.t., under SETH, we cannot
compute the diameter in ${\cal O}(n^{2-\epsilon})$ time on the split graphs of
order $n$ and clique-number at most $c(\epsilon)\log{n}$.
Our main result in this section is that this above hardness result also holds
for the absolute retracts of split graphs (Theorem 8). For that, we need a few
preparatory lemmas, namely:
###### Lemma 27 ([57]).
Let $G=(K+S,E)$ be a split graph. Exactly one of the following conditions
holds:
1. 1.
$|K|=\omega(G)$ and $|S|=\alpha(G)$ (in this case the partition $K+S$ is
unique);
2. 2.
$|K|=\omega(G)-1$ and $|S|=\alpha(G)$ (in this case there exists an $y\in S$
s.t. $K+\\{y\\}$ is complete);
3. 3.
$|K|=\omega(G)$ and $|S|=\alpha(G)-1$ (in this case there exists an $x\in K$
s.t. $S+\\{x\\}$ is stable).
Recall that $G=(V,E)$ is a complete split graph if there exists a partition of
its vertex-set into a clique $K$ and a stable set $S$ s.t. every vertex of $S$
is adjacent to every vertex of $K$. In particular, a complete split graph has
diameter at most two.
###### Theorem 7 ([66]).
A split graph is an absolute retract of split graphs if and only if it is a
complete split graph or it has a unique partition of its vertex-set into a
clique and a stable set.
We are now ready to prove the main result of this section, namely:
###### Theorem 8.
There is a linear-time reduction from the diameter problem on split graphs to
the same problem on the absolute retracts of split graphs.
In particular, for any $\epsilon>0$, there exists a $c(\epsilon)$ s.t., under
SETH, we cannot compute the diameter in ${\cal O}(n^{2-\epsilon})$ time on the
absolute retracts of split graphs of order $n$ and clique-number at most
$c(\epsilon)\log{n}$.
###### Proof.
Let $G=(V,E)$ be a split graph. First, we check in linear time whether
$diam(G)=0$ (i.e., $G$ is a singleton) or $diam(G)=1$ (i.e., $G$ is a complete
graph). From now on, let us assume $diam(G)\geq 2$. Since $G$ is a split
graph, computing the diameter is equivalent to deciding whether $diam(G)=3$.
For that, we compute a partition of $V$ into a clique $K$ and a stable set
$S$. It can be done in linear time [57]. For every vertex $v$, let
$deg(v):=|N(v)|$. We apply the following pruning rules until no more vertex
can be removed:
* •
If there exists a $x\in K$ s.t. $deg(x)=|K|-1$, then we remove $x$ from $G$.
* •
If there exists a $y\in S$ s.t. $deg(y)=|K|$, then we remove $y$ from $G$.
Complexity. Let us explain how both rules above can be applied exhaustively in
total linear time. For that, we maintain an array of $n-1$ lists, numbered
from $1$ to $n-1$, so that each vertex of $S$ of degree $i$ must be contained
into the $i^{th}$ such list. We proceed similarly for the vertices of $K$ (but
in a separate array of lists). Note that such a data structure can be
initialized in linear time, simply by computing the degree sequence of the
graph. If we further store, for each vertex, a pointer to its position in the
unique list in which it is contained, then every time we remove a vertex $v$,
we can update the structure in a time proportional to $deg(v)$. The latter
results in a total update time in ${\cal O}(m)$. Finally, if at each step we
maintain the cardinality $|K|$ of the clique $K$, then in order to check
whether one of the two pruning rules applies, we are left testing whether at
most two lists of our data structure are nonempty.
Correctness. In order to prove correctness of these above pruning rules, let
us first define a super-simplicial vertex as any vertex $v$ s.t.
$N(v)=K\setminus\\{v\\}$. Note that if $x\in K$ is s.t. $deg(x)=|K|-1$ then,
$N(x)=K\setminus\\{x\\}$, and so, $x$ is super-simplicial. In the same way if
$y\in S$ is s.t. $deg(y)=|K|$ then, $N(y)=K$, and so, $y$ is super-simplicial.
Summarizing the above, we can only prune super-simplicial vertices. Now, we
claim that, for a split graph $G$ and a super-simplicial vertex $v$ of $G$, we
have $diam(G)=3$ if and only if $diam(G\setminus\\{v\\})=3$. Indeed, since
$G\setminus\\{v\\}$ is an isometric subgraph of $G$,
$diam(G\setminus\\{v\\})=3$ implies $diam(G)\geq 3$, hence (since $G$ is a
split graph) $diam(G)=3$. Conversely, if $diam(G)=3$ then, there exist
$y,y^{\prime}\in S$ s.t. $d(y,y^{\prime})=3$. In order to prove that
$diam(G\setminus\\{v\\})=3$, it suffices to prove that none of $y$ and
$y^{\prime}$ can be a super-simplicial vertex. That is indeed the case
because, since $K$ is a dominating set of $G$, any super-simplicial vertex has
eccentricity at most two.
Finally, let $G^{\prime}$ be the resulting subgraph after no more vertex can
be removed. Observe that we have reduced the computation of $diam(G)$ to the
computation of $diam(G^{\prime})$. We shall prove that there exists a unique
partition of $V(G^{\prime})$ into a clique $K^{\prime}$ and a stable set
$S^{\prime}$. By Theorem 7, this will imply that $G^{\prime}$ is an absolute
retract of split graphs. Let $K^{\prime}$ and $S^{\prime}$ be a partition of
$V(G^{\prime})$ into a clique and a stable set, and suppose for the sake of
contradiction this partition is not unique. By Lemma 27, either there exists
$y\in S^{\prime}$ s.t. $K^{\prime}+\\{y\\}$ is complete, or there exists $x\in
K^{\prime}$ s.t. $S^{\prime}+\\{x\\}$ is a stable set. In the former case,
$deg(y)=|K^{\prime}|$, while in the latter case, $deg(x)=|K^{\prime}|-1$. But
then, we could still have applied one of our two pruning rules above, a
contradiction. ∎
## 5 Planar graphs
Our last (non-algorithmic) section is about the absolute retracts of planar
graphs The latter have been characterized in [61], assuming the Four-color
conjecture. Since this is now a theorem [3], the characterization of absolute
retracts of planar graphs is complete:
###### Theorem 9 ([61]).
A planar graph $G$ is an absolute retract of planar graphs if and only if it
is maximal planar and, in an embedding of $G$ in the plane, any triangle
bounding a face of $G$ belongs to a subgraph of $G$ isomorphic to $K_{4}$.
To our best knowledge, there has been no relation uncovered between the
absolute retracts of planar graphs and other important planar graph
subclasses. We make a first step in this direction. Specifically, the
Apollonian networks are a subclass of planar graphs that can be defined
recursively, as follows. The triangle $K_{3}$ is an Apollonian network. If $G$
is an Apollonian network, and $f$ a triangular face in a plane embedding of
$G$, then the graph $G^{\prime}$, obtained by adding a new vertex adjacent to
the three ends of $f$, is also an Apollonian network. The Apollonian networks
of order at least four (all Apollonian networks but the triangle) have some
important alternative characterizations, namely they are exactly the maximal
planar chordal graphs of order at least four [71], the planar $3$-trees [13],
and the uniquely $4$-colorable planar graphs [54].
###### Proposition 4.
Every Apollonian network with at least four vertices is an absolute retract of
planar graphs.
###### Proof.
Let $G$ be an Apollonian network of order at least four. In particular, $G$ is
maximal planar. Consider any triangular face $f$ in a plane embedding of $G$.
We also have that $G$ is a $3$-tree, and so, all its maximal cliques have four
vertices. In particular, there exists a vertex adjacent to all three vertices
of $f$. By Theorem 9, $G$ is an absolute retract of planar graphs. ∎
However, this above inclusion is strict (e.g., see Fig. 1 for a counter-
example). Indeed, our main result in this section is as follows:
Figure 1: An absolute planar retract of treewidth $\geq 5$.
###### Theorem 10.
Every connected planar graph is an isometric subgraph of some absolute planar
retract. In particular, there are absolute retracts of planar graphs with
arbitrarily large treewidth.
###### Proof.
Let $G=(V,E)$ be a connected planar graph. The result is trivial if $|V|\leq
2$. From now on, we assume $|V|\geq 3$. We first prove the existence of a
biconnected planar graph $H_{1}$ in which $G$ isometrically embeds. If $G$ is
already biconnected, then we set $H_{1}:=G$. Otherwise, let $u$ be a cut-
vertex of $G$. In any fixed planar embedding of $G$, consider a clock-wise
ordering of $N_{G}(u)$. There must be two consecutive neighbours $x,y$ in
different connected components of $G\setminus u$. We add a new vertex $z\notin
V$ such that $N(z)=\\{u,x,y\\}$. Since $x,y$ are consecutive in the clock-wise
ordering of $N_{G}(u)$, the resulting graph $G^{\prime}:=G+u$ is still planar.
It is also easy to check that $G$ isometrically embeds in $G^{\prime}$ because
$u,x,y$ are pairwise at distance at most two in $G$. Overall, by iterating
this above operation, we obtain the desired biconnected planar graph $H_{1}$
in which $G$ isometrically embeds.
Note that because $H_{1}$ is biconnected, in any plane embedding of $H_{1}$,
the boundary of each face is a simple cycle [35, Proposition 4.2.6]. Then, we
claim that $H_{1}$ is an isometric subgraph of some planar $H_{2}$ such that,
in some plane embedding of $H_{2}$ all the faces have length at most five. In
order to prove the claim, fix a plane embedding of $H_{1}$ and let the simple
cycle $C$ be the boundary of some face of length $\geq 6$ (if no such face
exists then, we are done by setting $H_{2}:=H_{1}$). Pick some path $[u,v,w]$
of length two on the cycle $C$. Let $C^{\prime}$ be obtained from $C$ by
contracting $[u,v,w]$ to a single vertex $x$. We add a copy of $C^{\prime}$ in
the face of which $C$ is a boundary, we add an edge between $x$ and all of
$u,v,w$, and finally we add an edge between every $y\in C\setminus\\{u,v,w\\}$
and its copy in $C^{\prime}$. Note that doing so, we replace the face bounded
by $C$ by a new face bounded by $C^{\prime}$ and by some new triangular and
quadrangular faces. Let $H_{1}^{\prime}$ be the resulting planar graph. We
prove as an intermediate subclaim that $H_{1}$ isometrically embeds in
$H_{1}^{\prime}$. For that, fix a shortest-path in $H_{1}^{\prime}$ between
some two vertices $a,b\in V(H_{1})$. Assume there is a subpath
$P^{\prime}=[c_{1}^{\prime},\ldots,c_{p}^{\prime}]$ fully in $C^{\prime}$
(otherwise, we are done). There is a natural mapping between the latter and
the path $P$ of $C$ that is obtained by uncontracting the path $[u,v,w]$. Note
that the length of $P$ is at most the length of $P^{\prime}$ plus two.
Furthermore, let $x,y\in C$ be the respective neighbours of
$c_{1}^{\prime},c_{p}^{\prime}$ on the shortest $ab$-path. By construction,
$x,y\in V(P)$. Hence, we may replace $[x,P^{\prime},y]$ by $P$ on the shortest
$ab$-path. By repeating this operation until there is no more vertex of
$C^{\prime}$ on the shortest $ab$-path, we prove as claimed that $H_{1}$
isometrically embeds in $H_{1}^{\prime}$. Overall, we repeat the above
operations until there is no more face of length $\geq 6$ in the plane
embedding. Doing so, we obtain $H_{2}$.
We further claim that $H_{2}$ is an isometric subgraph of some plane
triangulation $H_{3}$. This is easily achieved by adding, in the middle of
each face, a new vertex adjacent to all the vertices of the boundary. Note
that $H_{2}$ isometrically embeds in $H_{3}$ because all its faces have length
at most five, and so, weak diameter at most two.
Finally, let us prove that every plane triangulation $H_{3}$ isometrically
embeds in some absolute planar retract $H_{4}$. The construction is the same
as in the previous step, namely, we add in the middle of each (triangular)
face a new vertex adjacent to all the vertices of the boundary. Doing so,
since each face of $H_{4}$ shares exactly one edge with a triangular face of
$H_{3}$, we satisfy the following property: any triangle bounding a face of
$H_{4}$ belongs to a subgraph isomorphic to $K_{4}$. Furthermore, $H_{4}$ is a
plane triangulation of order $\geq|V|\geq 3$, and so, it is maximal planar
[35, Propositions 4.2.8 and 4.4.1]. By Theorem 9, $H_{4}$ is an absolute
planar retract. ∎
We stress that the proof of Theorem 10 is constructive, and that it leads to a
polynomial-time algorithm in order to construct an absolute planar retract in
which the input planar graph $G$ isometrically embeds. In contrast to our
result, the smallest Helly graph in which a graph $G$ isometrically embeds may
be exponential in its size [59].
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|
# Knowledge-injected Prompt Learning for Chinese Biomedical Entity
Normalization
Songhua Yang School of Computer and Artificial Intelligence
Zhengzhou University
Zhengzhou, China
<EMAIL_ADDRESS>Chenghao Zhang School of Computer and Artificial
Intelligence
Zhengzhou University
Zhengzhou, China
<EMAIL_ADDRESS>Hongfei Xu School of Computer and Artificial
Intelligence
Zhengzhou University
Zhengzhou, China
<EMAIL_ADDRESS>Yuxiang Jia* *Corresponding author. School of Computer
and Artificial Intelligence
Zhengzhou University
Zhengzhou, China
<EMAIL_ADDRESS>
###### Abstract
The Biomedical Entity Normalization (BEN) task aims to align raw, unstructured
medical entities to standard entities, thus promoting data coherence and
facilitating better downstream medical applications. Recently, prompt learning
methods have shown promising results in this task. However, existing research
falls short in tackling the more complex Chinese BEN task, especially in the
few-shot scenario with limited medical data, and the vast potential of the
external medical knowledge base has yet to be fully harnessed. To address
these challenges, we propose a novel Knowledge-injected Prompt Learning (PL-
Knowledge) method. Specifically, our approach consists of five stages:
candidate entity matching, knowledge extraction, knowledge encoding, knowledge
injection, and prediction output. By effectively encoding the knowledge items
contained in medical entities and incorporating them into our tailor-made
knowledge-injected templates, the additional knowledge enhances the model’s
ability to capture latent relationships between medical entities, thus
achieving a better match with the standard entities. We extensively evaluate
our model on a benchmark dataset in both few-shot and full-scale scenarios.
Our method outperforms existing baselines, with an average accuracy boost of
12.96% in few-shot and 0.94% in full-data cases, showcasing its excellence in
the BEN task.
###### Index Terms:
Biomedical Entity Normalization, Prompt Learning, Knowledge Enhancement, Few-
shot Learning
## I Introduction
Biomedical Entity Normalization (BEN), also referred to as biomedical entity
linking, is integral to aligning medical and biological terms with
standardized entities. In the context of medicine, this process resolves
ambiguity, vagueness, and misspellings, promoting uniformity in domain-
specific textual terminology. With the exponential growth of medical data –
encompassing electronic health records and scholarly medical literature –
there lies a wealth of valuable knowledge for clinical decision-making,
disease diagnosis, and patient management. However, the diversity and non-
standardization of entities in medical texts, exacerbated by the intricacies
of Chinese medical terminology and language, pose challenges for downstream
tasks and applications. In response, BEN technology emerges to enhance medical
entity standardization and facilitate efficient utilization of textual medical
data.
Figure 1: An example of handling Chinese BEN task: the top left shows the non-
standard original entity, which is matched through the standard library on the
right. The correct standardized entity is output in the lower left. All
medical knowledge contained in the entity text listed alongside.
While the field of BEN has garnered extensive attention over the years, its
evolution has been shaped by advances in machine learning and natural language
processing (NLP). Initially, rule-based approaches gauged textual similarity
or curated dictionaries for entity matching, but struggled to scale. Machine
learning methods split BEN into candidate word matching and classification
stages, using classifiers and feature engineering. However, these methods
faced limitations in semantic understanding and the need for extensive
annotated data. Deep neural networks and word embedding algorithms
subsequently improved BEN through enhanced generalization and semantic
information capture. Pre-trained language models (PLMs), especially those
trained on medical corpora, brought efficiency to BEN by refining task
performance with minimal fine-tuning.
Despite notable strides, the complexity of biomedical entities, particularly
in the Chinese context, persists as a challenge for BEN. The specialized
nature of medical annotations incurs high costs, and the constant evolution of
medical categories limits available annotated data. Thus, the importance of
few-shot learning in BEN becomes evident, offering a pragmatic solution to
real-world problems.
Prompt learning, a burgeoning paradigm in natural language processing,
empowers PLMs to solve specific tasks using designed prompt templates. This
approach enhances task understanding, reduces overfitting risks, and optimizes
performance with scarce annotated data. While prompt learning has gained
traction in biomedical entity normalization, Chinese BEN remains less
explored, with untapped potential in leveraging external medical knowledge.
In this paper, we propose a novel Knowledge-injected Prompt Learning method
(PL-Knowledge) to address challenges in Chinese BEN. Our approach involves
candidate entity matching followed by prompt learning-based classification.
The model comprises four stages: knowledge extraction, encoding, injection,
and prediction. External medical knowledge refines entity embeddings,
synergizing with PLMs through an intricate prompt template. Through extensive
experiments on a benchmark Chinese BEN dataset, our PL-Knowledge method
demonstrates substantial improvements, excelling in both few-shot and full-
scale scenarios. Furthermore, our strategy underscores the crucial role of
medical domain knowledge and our infusion strategy in enhancing model
performance.
The main contributions of this work are as follows:
1) We are the first to apply prompt learning to Chinese biomedical entity
normalization task and demonstrate its effectiveness in few-shot scenarios.
2) We propose a knowledge-injected prompt learning method that leverages rich
external medical domain knowledge base to enhance the performance of our model
for BEN task.
3) Through experimental evaluation on a benchmark Chinese BEN dataset, our
method achieves significant improvement over baseline methods in both few-shot
and full-scale settings.
## II Related Work
The evolution of BEN techniques spans several decades, adapting to
advancements in NLP and the burgeoning volume of biomedical text data. The
approaches employed fall into distinct categories, including rule-based,
machine learning-based, deep learning-based, and PLM-based methods. In earlier
stages, rule-based methods were prevalent. Examples include MetaMap [1], which
maintained dictionaries, and systems such as [2], [3], and [4], which employed
algorithms like edit distance and semantic similarity. While efficient, rule-
based methods faced challenges in covering diverse cases and capturing nuanced
information. Machine learning methods addressed these limitations by
introducing candidate word matching and classification stages [5]. Approaches
like [6] and [7] harnessed SVMs and TF-IDF for entity resolution, leveraging
Bag of Words and ranking learning. However, the insufficiency of labeled data
and the incapability to grasp word semantics remained concerns. Deep neural
network algorithms emerged next, leveraging word embedding techniques like
Word2Vec and GloVe. Notable instances include [8], [9], and [10], deploying
models such as CNNs and LSTMs to enhance entity resolution and coherence.
More recently, the advent of PLMs revolutionized BEN by enabling minimal fine-
tuning. This encompassed approaches like [11], retraining PLMs like BioBERT
and ClinicalBERT [12, 13], and optimizing models for Chinese BEN [14, 15].
These methods exhibited substantial performance gains but had limitations in
domain specificity and effective knowledge utilization. Prompt learning, a
fourth paradigm in NLP, gained traction for few-shot and zero-shot tasks [16].
While some approaches integrated prompt learning with BEN [17, 18, 19], they
overlooked the complexities of Chinese BEN and the potential of external
medical knowledge.
In summary, as BEN methods have evolved, limitations remain. Our proposed
knowledge-injected prompt learning introduces innovative strategies to address
biomedical entity normalization, bridging gaps in current methodologies.
## III Approach
In this section, we provide a detailed explanation of our proposed model. We
begin by presenting a formal definition of the BEN task and then describe how
we address this task in two stages, along with the specific processes and
methods used in each stage.
### III-A Problem Definition
In the biomedical domain, an entity phrase $e$ may have multiple different
expressions that are easily confused. To address this, there exists a standard
entity name repository $E=\\{e_{1},e_{2},...,e_{k}\\}$ containing the unique
correct names for all entities. Given a list of $n$ biomedical entity phrases
$O=\\{o_{1},o_{2},...,o_{n}\\}$, the objective of the BEN task is to
accurately match the correct standard term $s_{j}$ from the standard base $S$
for each original term $o_{i}$. Notably, numerous one-to-many, many-to-many,
and many-to-one relationships exist between the original term $o$ and the
standard term $s$, as shown in Table I.
Original Entity | Standard Entity | Relationship
---|---|---
| 肝动脉化疗栓塞术
---
transcatheter arterial chemoembolization
| 经导管肝动脉栓塞术##动脉化疗栓塞
---
transcatheter arterial embolization ## chemoembolization
one to many
| 左侧腮腺肿块切除##浅叶切除##面神经松解减压术
---
left parotid mass excision ## superficial lobe excision
## facial nerve decompression surgery
| 面神经减压术##腮腺病损切除术
---
facial nerve decompression surgery ## parotid lesion excision
many to many
| 宫腔镜检查##诊刮术
---
hysteroscopy ## diagnostic curettage
| 宫腔镜诊断性刮宫术
---
hysteroscopic diagnostic curettage
many to one
TABLE I: Some representative examples of the relationship between the original
entity and the standard entity.
### III-B Preprocessing and Candidate Matching
Figure 2: Processing of representative examples during candidate term matching
in the first stage. The three English text segments in the middle-left of the
figure correspond to Case2.
Medical professionals may use irregularities when writing original terms in
real medical scenarios. Therefore, preprocessing of text data is necessary to
improve the performance of entity normalization task. We manually remove
irrelevant characters in the text, such as incorrect separators and
punctuation marks. Moreover, in many-to-many situations between original and
standard terms, delimiters between multiple entity words may be uncertain. We
standardize them to easily split and merge entity phrases. We have devised a
set of rules to flexibly handle many-to-many issues, as shown in Figure 2. For
the output of the classification model, if multiple standard terms are
predicted, we merge the predicted result words and compare them with the
standard terms.
Following the approach of previous studies [8, 20, 18], we divide the BEN task
into two stages. In the first stage, we employ a text similarity algorithm to
quickly match candidate entities. In particular, we segment all Chinese
medical entities and calculate the Jaccard similarity coefficient between each
original entity and each entity in the standard library. Based on the
relevance scores, we select the top few entity phrases as candidate entities.
The Jaccard similarity is a widely used text similarity calculation method
that is simple, efficient, scalable, and insensitive to text length.
For each original entity $o$ and its corresponding candidate entities
$C=\\{c_{1},c_{2},...,c_{m}\\}$ in the dataset, we form a triplet
$\\{o,c_{i},l\\}$ as the input for the next classification model, where
$l\in\\{0,1\\}$ is the true label between $o$ and $c_{i}$. In one-to-many and
many-to-many situations, the labels between the original entity containing
multiple sub-words and the corresponding standard entity sub-words are set to
1. In this way, our model can ingeniously solve complex many-to-many
situations through a binary classification approach.
### III-C Prompt Learning
We will now present a detailed explanation of our second-stage classification
model based on prompt learning, as shown in Figure 3.
Figure 3: The overall architecture of our model, featuring a carefully
designed knowledge-injected prompt template in the middle. The knowledge
extraction, knowledge encoding, and knowledge injection modules are shown
moving from left to right at the bottom, while the final prediction output
process is at the top.
#### III-C1 Overview
The core idea of prompt learning is to construct a piece of natural language
text (i.e., a template) and input it into the encoder of a Masked Language
Model (MLM), converting the classification problem into a cloze-style task.
Prompt learning models consist of a template and a verbalizer. Formally, given
an original entity $o$ and its corresponding candidate entities
$C=\\{c_{1},c_{2},...,c_{m}\\}$ as input, a tailor-made template
$T(\cdot,\cdot)$ is used to create a text sequence $T(o,c)$, as shown in Table
II. The templates must include at least one [MASK] token. Let $\mathcal{M}$ be
an MLM model, and $\mathcal{M}$ outputs the probability
$P_{\mathcal{M}}\left([\mathrm{MASK}]=v\mid T(o,c)\right)\mid v\in\mathcal{V}$
of each word in the vocabulary $\mathcal{V}$ of $\mathcal{M}$ being filled in
the [MASK] token.
The mapping from the predicted masked words to the labels indicating whether
$o$ and $c$ are synonymous is called the verbalizer, which transforms the
masked prediction task into a classification task. The performance of prompt
learning is significantly influenced by the verbalizer’s configuration. A
verbalizer can be defined as $f:\mathcal{V}_{y}\mapsto\mathcal{Y}$, where
$\mathcal{V}_{y}$ is the well-designed label words, a subset of the overall
vocabulary $\mathcal{V}$, and $\mathcal{Y}$ represents the corresponding
output labels (i.e., $y_{0}:0$ and $y_{1}:1$ for BEN). The final probability
output of the classification model can be formulated as:
$P(y\in\mathcal{Y}\mid T(o,c))=g(P_{\mathcal{M}}([\mathrm{MASK}]=v\mid
T(o,c))\mid v\in\mathcal{V}_{y})$ (1)
where $g$ is the function corresponding to the verbalizer $f$, which
calculates the final label probability score by the probability of label words
in $\mathcal{V}_{y}$, thus obtaining the final classification result.
The prompt learning model’s overall architecture and a representative example
are illustrated in Figure 3. The original term ”胃静脉套扎术”(gastric variceal
ligation) and its corresponding candidate term ,”胃静脉曲张结扎术”(gastric variceal
banding) in this case, are inserted into the knowledge-injected template
provided in Table II. After converting the text into a vector representation
and feeding it into the MLM $\mathcal{M}$, the probability distribution of
label words for all candidate terms is obtained. The verbalizer decodes the
output label probability distribution, where words like ”不”, ”没” (no)
correspond to $y_{0}$, while ”是”, ”对” (yes) correspond to $y_{1}$. The
candidate term with a label probability of $P(y_{1})>P(y_{0})$ is the final
predicted standard term.
#### III-C2 Template Construction
Template | Content
---|---
Manual Template | [Original Entity] 和 [Candidate Entity] 意思相同吗？
| Is the meaning of [Original Entity] the same as [Candidate Entity]?
Mixed Template | [Original Entity] [soft] [soft:和] [soft] [Candidate Entity] [soft: 的] 意思相同吗？
| Is the meaning of [Original Entity] [soft] [soft:and] [soft] [Candidate
Entity] [soft: the] same?
Knowledgable Template | [Original Entity] [know] [soft:和] [Candidate Entity] [know] [soft: 的] 意思相同吗？
| Is the meaning of [Original Entity] [know] [soft:and] [Candidate Entity]
[know] [soft: the] same?
TABLE II: Demonstration of our three different types of prompt templates. ”[]”
denotes placeholders, where the original and candidate entities are filled in
their respective positions. ”[soft]” represents learnable soft tokens, with
the subsequent characters indicating initialization with the corresponding
vector; if not present, it is randomly initialized. ”[know]” indicates that
the vector at this position comes from knowledge injection.
The construction of templates is a critical factor in determining the
performance of prompt learning. In order to maximize the ability of PLMs to
perform classification task, we design a tailor-made knowledge-injected
template, which can incorporates knowledge effectively. In addition, we also
attempt the intuitive manual template and the automatically learnable mixed
template for comparison. These templates are detailed in Table II.
The manual template is the most straightforward type of template, containing
only human-readable natural text. It adopts a hard-encoding approach, where
the token embeddings in the template sequence are fixed to their corresponding
vector representations. On the other hand, the mixed template combines both
hard-encoding and soft-encoding approaches. Soft-encoding refers to encoding
tokens as dynamic, learnable word vectors. For example, placeholders like
[soft: 和] and [soft] in Table II represent soft tokens. The embedding of the
former token is initialized to the encoding vector corresponding to ”和” (and),
while the latter token is randomly initialized. These soft token embeddings
can be continuously optimized during the training process through gradient
calculations. Soft tokens are more flexible and adaptive compared to hard-
encoding, guiding the model to learn task-related key information and
supporting more accurate classification. Several studies [21, 16, 22] have
demonstrated that a prompt template with special learnable tokens can make
prompt learning more effective. Therefore, we incorporate some soft-tokens in
both Mixed Template and Knowledgeable Template to enhance model performance.
#### III-C3 External Knowledge Injection
In this section, we will elaborate on the detailed process of prompt learning
with knowledge injection, as shown in Figure 3. We find that although medical
entities may have some degree of irregular writing, they contain a wealth of
medical knowledge, such as diseases, surgeries, and anatomical locations. This
medical knowledge can act as crucial information to assist in matching
original entities and standard entities. For example, in the case shown in the
figure, we can see that both items contain ”胃, 部位”(stomach, site), ”静脉”(vein),
and ”静脉曲张”(varicose veins) are related, and ”套扎术”(ligation procedure) and
”结扎术”(ligation procedure) are often synonymous. Therefore, we exploit a
knowledge injection module to integrate medical knowledge from external
knowledge base into the prompt template, which includes knowledge extraction,
knowledge encoding, and knowledge injection.
We leverage the CMeKG [23] Chinese medical knowledge graph, which contains
knowledge about diseases, locations, and surgeries, to build an external
knowledge base comprising 7,249 items denoted as
$K=\\{(k_{1},r_{1}),(k_{2},r_{2}),...,(k_{n},r_{n})\\}$. To extract the
corresponding knowledge item sequences $o\rightarrow
o_{k}=\\{(k_{1},r_{1}),(k_{2},r_{2}),...,(k_{p},r_{p})\\}$ and $c\rightarrow
c_{k}=\\{(k_{1},r_{1}),(k_{2},r_{2}),...,(k_{q},r_{q})\\}$ from the knowledge
base $K$, we use the longest overlap matching method for the input original
entity $o$ and the corresponding candidate entity $c$. To encode all knowledge
items, we construct an additional vocabulary, treating each knowledge item and
relationship type as an entity that can be encoded.
After encoding the vocabulary, we concatenate the two knowledge item sequences
$o_{k}$ and $c_{k}$ to form a text sequence, which is then input into the
subsequent embedding layer. For this layer, we use the encoder of a pre-
trained medical domain PLM, MC-BERT [24], as shown below:
$\displaystyle
h_{o}=Embedding((k_{1},r_{1})\parallel(k_{2},r_{2})\parallel...\parallel(k_{p},r_{p}))$
(2) $\displaystyle
h_{c}=Embedding((k_{1},r_{1})\parallel(k_{2},r_{2})\parallel...\parallel(k_{q},r_{q}))$
(3)
To better capture the internal relationships and contextual information
between knowledge item sequences, we incorporate a BiGRU module to refine the
encoding of hidden vectors output by the Embedding layer. The BiGRU captures
both forward and backward contextual information. We add the output forward
and backward hidden state vectors $h^{\prime}_{of}$, $h^{\prime}_{ob}$,
$h^{\prime}_{cf}$, and $h^{\prime}_{cb}$, and align the dimension of the
resulting vector representation with the template through a Multi-Layer
Perceptron (MLP). The final output vectors $h^{\prime}_{ok}$ and
$h^{\prime}_{ck}$ are the knowledge encoding results to be injected into the
template. The entire process can be formalized as follows:
$\displaystyle
h^{\prime}_{of},h^{\prime}_{ob}=BiGRU(h_{o}),h^{\prime}_{cf},h^{\prime}_{cb}=BiGRU(h_{c})$
(4) $\displaystyle
h^{\prime}_{ok}=MLP(h^{\prime}_{of}+h^{\prime}_{ob}),h^{\prime}_{ck}=MLP(h^{\prime}_{cf}+h^{\prime}_{cb})$
(5)
Finally, we substitute the knowledge embedding vectors $h^{\prime}_{ok}$ and
$h^{\prime}_{ck}$ into the corresponding positions of the [know] placeholders
in the template $T(o,c)$ near the original entity and candidate entity
representations. In this way, the additional knowledge serves as a supplement
to the entity information, assisting the model in making classification
judgments.
## IV Experiments
### IV-A Dataset
To evaluate the effectiveness of our model, we conduct extensive experiments
on the dataset provided by the Clinical Terminology Standardization Evaluation
Task (CHIP 2019) [25]. . This publicly available BEN dataset is derived from
actual Chinese electronic medical records and manually annotated using the
”ICD9-2017 Clinical Edition” surgical standard terminology. The statistics are
shown in Table III:
| Train | Dev | Test
---|---|---|---
one to one | 3596 | 897 | 1857
one to many | 37 | 7 | 23
many to many | 162 | 43 | 76
many to one | 205 | 53 | 44
total | 4000 | 1000 | 2000
TABLE III: The statistics of CHIP2019 dataset.
To construct our external medical knowledge base, we utilize the Chinese
Medical Knowledge Graph CMeKG [23], which is a large-scale knowledge graph
containing 1.56 million conceptual relations and attribute triplets in the
medical domain. We focus on extracting disease, location, and surgery-related
knowledge items that may be relevant to the task dataset, and after filtering,
screening, and deduplication, we obtain a total of 7,249 unique items. Given
the complex relationships between the triples in the knowledge graph, we only
retain the subject and predicate of each triple in our knowledge base.
Model | 16-shot | 64-shot | 256-shot | 1024-shot | All(3500)
---|---|---|---|---|---
Acc | Acc | Acc | Acc | Acc
SVM(TF-IDF) | 23.87 | 29.56 | 37.35 | 42.21 | 47.45
Edit-Distance | 24.87 | 30.47 | 38.05 | 44.18 | 48.36
TextCNN | 25.34 | 37.57 | 52.21 | 76.46 | 86.12
BERT-based | 29.86 | 41.98 | 61.38 | 83.20 | 85.53
BERT-Knowledge | 25.87 | 45.52 | 62.16 | 84.10 | 87.04
[14] | — | — | — | — | 89.30
[15] | 32.47 | 48.29 | 68.79 | 88.44 | 91.14
PL-Manual | 59.83(+27.36) | 64.24 | 74.79 | 88.35 | 90.67
PL-Mixed | 54.39 | 61.94 | 73.59 | 88.20 | 90.83
PL-Knowledge | 46.40 | 71.19(+22.90) | 80.21(+11.42) | 89.61(+2.17) | 92.08(+0.94)
TABLE IV: Final experimental results of our model and baseline models on the
CHIP 2019 BEN dataset. Bold in each column indicates the best result for that
sample size. 16, 64, 256, and 1024 are the sample sizes; ”All” refers to the
full-scale evaluation, with accuracy as the evaluation metric for the models.
### IV-B Baselines
To provide a comprehensive evaluation of our model, we compare it to some
baseline methods that cover a range of representative classification methods
from machine learning, deep learning, to PLM-based models. Here is a brief
description of each method:
SVM (TF-IDF): SVM [26] is a classic machine learning classification algorithm
that is combined with the TF-IDF [27] feature extraction method to vectorize
text representation. This combination enables SVM to quickly capture key
information in medical entities and provide more distinctive feature
representations for classification task.
Edit-distance: A classic text similarity calculation method that can quickly
calculate the similarity between two medical entities.
TextCNN: A Convolutional Neural Network (CNN) [28] suitable for text
processing is used as a deep learning classification method baseline. TextCNN
captures local contextual features between entity texts and ensures strong
translation invariance.
BERT-based: A sequence classification method based on the pre-trained BERT
model [29]. Concatenating the original entity and standard entity into a
sequence input model in a specified format for ranking.
BERT-Knowledge: In classification stage, concatenating the knowledge embedding
with the two entities vectors with the last layer of BERT, and feeding this
representation into a linear classifier. This is a common fine-tuning
knowledge injection method [30] and is compared with our knowledge-injected
prompt learning model.
[14]: Adopting an end-to-end sequence generative framework with category-based
constraint decoding, model-refining mechanism, and beam search techniques,
achieving promising results in BEN task.
[15]: Using a multitask learning framework to simultaneously perform many-to-
many prediction and entity classification subtasks, it is the previous best
model to solve Chinese BEN task.
### IV-C Experimental Settings
We employe the retrained MC-BERT111https://huggingface.co/freedomking/mc-bert
as the PLM for fine-tuning, prompt learning, and knowledge injection.
Utilizing the Adam optimizer and a 1e-5 learning rate scheduler, we set 10
epochs, a batch size of 32, a maximum sequence length of 128, and a 0.3
dropout probability. We assess the model on the validation set after each
epoch, selecting the best one for final testing on the test set. To evaluate
performance in data-scarce scenarios, we conduct few-shot experiments with 16,
64, 256, and 1024 training samples, simulating real-world few-shot issues. We
incorporate early stopping, terminating training if no significant improvement
occurred within two epochs. For reliability, each test result was
independently run five times using different random seeds, with the average
value as the final result.
Model | 16-shot | 64-shot | 256-shot | 1024-shot | All(3500)
---|---|---|---|---|---
PL-K | 46.40 | 71.19 | 80.21 | 89.61 | 92.08
PL-K(w) | 44.35(-2.05) | 65.08(-6.11) | 76.47(-3.74) | 86.54(-3.07) | 90.79(-1.29)
PL-K(r) | 42.17(-4.23) | 63.85(-7.34) | 73.60(-6.61) | 83.32(-6.29) | 88.23(-3.85)
TABLE V: Comparison of experimental results for PL-Knowledge using different
knowledge strategies.
### IV-D Main Results
| Template
---|---
#1 | [Original Entity][know][soft:和][Candidate Entity][know][soft:的]意思相同吗？
| Is the meaning [soft:of] [Original Entity][know] [soft:and] [Candidate
Entity] [know] the same?
#2 | [Original Entity][know]和[Candidate Entity] [know]相同吗？
| Is [Original Entity] [know] the same as [Candidate Entity] [know]?
#3 | [Original Entity][know][soft:和][Candidate Entity][know][soft:是][soft:不][soft:是][soft:相][soft:同]？
| [soft:Is] [Original Entity] [know] [soft:and] [Candidate Entity] [know]
[soft:the] [soft:same] [soft:or] [soft:not]?
#4 | 在医学术语中，[Original Entity][know][soft:和][Candidate Entity][know][soft:的] 意思相同吗？
| In medical terminology, is the meaning [soft:of] [Original Entity] [know]
[soft:and] [Candidate Entity] [know] the same?
#5 | 从专业医学的角度看，[Original Entity][know]与[Candidate Entity] [know]是否指代同一概念？
| From a professional medical , Do [Original Entity] [know] [soft:and]
[Candidate Entity] [know] refer to the same concept?
TABLE VI: Five different template settings. Template | 16-shot | 64-shot | 256-shot | 1024-shot | All(3500)
---|---|---|---|---|---
Acc | Acc | Acc | Acc | Acc
#1 | 46.40 | 71.19 | 80.21 | 89.61 | 92.08
#2 | 45.57 | 69.34 | 78.83 | 84.68 | 91.23
#3 | 52.25 | 67.53 | 78.01 | 87.48 | 91.19
#4 | 47.51 | 71.37 | 79.18 | 88.25 | 91.39
#5 | 53.73 | 72.28 | 79.35 | 89.03 | 91.61
TABLE VII: Experimental results for different templates, with numbers
corresponding to the template settings in Table VI. Bold values in each column
indicate the best experimental results.
In this section, we present the experimental results of our model and various
baseline methods on the CHIP 2019 BEN dataset, with Accuracy (Acc) used as the
evaluation metric. Our prompt learning models based on manual templates, mixed
templates containing soft tokens, and knowledge-injected templates are denoted
as PL-Mannual, PL-Mixed, and PL-Knowledge, respectively. Based on the results
presented in Table IV, the following observations can be made:
Prompt learning models significantly outperform baseline methods in few-shot
scenarios. As shown in the table, our prompt learning models consistently
outperform the baseline methods in all-scale scenarios, especially in few-shot
settings. For instance, our best prompt learning method achieves a notable
improvement of 27.36%, 22.90%, and 11.42% in the 16, 64, and 256-shot cases,
respectively, which greatly surpasses the baseline methods. This result
showcases the remarkable generalization capabilities of our prompt fine-tuning
model in the few-shot learning task.
Insufficient training data can impair the effectiveness of knowledge infusion.
We also observe that the effectiveness of knowledge infusion could be
undermined by inadequate training data. In the 16-shot case, both the PL-
Manual and BERT-based models outperformed the knowledge-injected PL-Knowledge
and BERT-Knowledge. This implies that while the knowledge infusion strategy
may enhance the model’s generalization abilities, it may not be sufficient to
counterbalance the negative effects of data scarcity when the training data is
extremely limited.
Knowledge enhancement greatly improves model performance in few-shot
scenarios. Our results demonstrate that, with the exception of the extremely
scarce 16-shot case, knowledge-injected methods achieve notable performance in
both few-shot and full-sample evaluation, indicating that the integration of
external knowledge significantly enhances the model’s understanding of the BEN
task. The most significant improvement is observed in the 64-shot and 256-shot
cases, suggesting that, with a moderately small sample size, the model can
effectively incorporate the infused knowledge items and substantially enhance
its performance. For the larger 1024-shot case, the performance improvement of
knowledge-enhanced models is less pronounced.
Our model maintains a slight advantage in all-data evaluation. The results of
the all-scale evaluation indicate that PL-Knowledge performs on par with the
best baseline method in terms of performance, with a slight advantage.
Template strategies have their pros and cons. We observed that in the
1024-shot and full-scale cases, PL-Mixed outperformed PL-Manual, while in
other cases, PL-Manual surpassed PL-Mixed. This implies that mixed templates
incorporating soft tokens can provide a more flexible learning approach for
the model, allowing it to better capture the correlation information between
entities. However, for too few samples, soft tokens may not fully exploit
their advantages, and PL-Manual may be better suited for few-shot scenarios.
Therefore, in practical applications, the choice of template strategy should
be based on factors such as sample size and data complexity.
Baseline methods still hold reference value. Among the baseline methods, [15]
achieved the highest accuracy, demonstrating their excellent performance on
the BEN task. Although the accuracy of early TF-IDF, Edit-Distance, and
TextCNN methods was relatively low, they provided an important benchmark for
evaluating the performance differences between our methods and traditional
machine learning and neural network methods.
In summary, the experimental results demonstrate that our model achieves good
performance in various scenarios, particularly in the few-shot learning task,
proving the strong generalization capabilities and practicality of our method.
### IV-E Impact of Knowledge Injection Strategies
We further investigated the impact of different knowledge injection strategies
on model performance by modifying the embedding method of knowledge items in
the PL-Knowledge model to use Word2vec and random initialization. Med-word2vec
[31] is an early word vector table pre-trained on a moderately sized Chinese
medical corpus, with weaker representation capabilities in the medical domain
compared to MC-BERT. On the other hand, random initialization does not
introduce any actual knowledge, requiring the model to learn on its own.
Therefore, we can observe the impact on model performance when introducing
limited knowledge and not introducing it at all.
As shown in Table V, when the knowledge embedding method adopts Word2vec and
random initialization, the model performance declines in both cases,
especially with random initialization. This phenomenon confirms the
effectiveness of our knowledge injection strategy in the entity normalization
task, as external knowledge can help the model capture associations between
entities, thereby improving the model’s generalization ability in this task.
Furthermore, this result also reveals the potential negative impact of
randomly initialized knowledge embeddings on model performance. Randomly
initialized embeddings cannot provide effective knowledge information to the
model, causing a certain degree of disturbance during the learning process.
Compared to MC-BERT, word2vec has a weaker representation capability.
### IV-F Analysis of Different Templates
In this section, we analyze and compare the impact of different template
settings in the PL-Knowledge method on the performance of the BEN task. A
well-designed template can help PLMs better understand task requirements and
improve model performance. In addition to the three token types of hard, soft,
and knowledge-injected, the length and specific statements in the template are
also essential factors in prompt learning. Appropriate template length can
significantly improve model performance, and some specific statements can
better induce PLMs to output correct results. These factors need to be
experimented with and adjusted based on specific tasks and datasets [21, 16,
32, 33].
Considering the above factors, we designed five different templates in Table
VI based on token type, template length, and specific statements. Table VII
shows the experimental results of different templates under various sample
sizes. Although each template performs well in specific scenarios, Template #1
consistently achieves the highest accuracy in the 256-shot, 1024-shot, and
full-scale scenarios. On the other hand, Template #5 performs best in the
small sample scenarios of 16-shot and 64-shot, indicating that this template
is better suited for such situations. Notably, although both #4 and #5
templates contain similar specific prefixes, their performance differences are
significant, further confirming the significant impact of specific statements
on model performance. In all scenarios, the completely manual #2 template and
the entirely soft token-based #3 template perform relatively poorly,
emphasizing the need for the appropriate incorporation of soft tokens to
improve prompt fine-tuning performance. These findings provide essential
guidance for selecting appropriate template strategies and directions in
practical applications.
## V Conclusion
In this study, we addressed the challenges of the Chinese BEN task, focusing
on data-limited few-shot scenarios. Our proposed PL-Knowledge method combines
an external medical knowledge base with prompt learning to enhance BEN
performance. Empirical results on a benchmark Chinese BEN dataset showcase the
efficacy of our model, outperforming existing baselines in both few-shot and
full-data scenarios. Through ablation and comparison experiments, we highlight
the pivotal role of knowledge injection and template design in boosting model
performance.
Future work will delve into more effective strategies for integrating external
knowledge to further enhance task performance, and will explore maintaining
model effectiveness in zero-shot scenarios. Additionally, our focus on the
Chinese context limits our exploration. In forthcoming research, we aim to
validate our model on English datasets and bridge this gap in our analysis.
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|
# Revealing interactions between HVDC cross-area flows and frequency stability
with explainable AI
Sebastian Pütz<EMAIL_ADDRESS>Forschungszentrum Jülich, Institute for
Energy and Climate Research - Systems Analysis and Technology Evaluation (IEK-
STE), 52428 Jülich, Germany Institute for Theoretical Physics, University of
Cologne, 50937 Köln, Germany Benjamin Schäfer Karlsruhe Institute of
Technology, Institute for Automation and Applied Informatics (IAI),76344
Eggenstein-Leopoldshafen, Germany Dirk Witthaut Forschungszentrum Jülich,
Institute for Energy and Climate Research - Systems Analysis and Technology
Evaluation (IEK-STE), 52428 Jülich, Germany Institute for Theoretical
Physics, University of Cologne, 50937 Köln, Germany Johannes Kruse
Forschungszentrum Jülich, Institute for Energy and Climate Research - Systems
Analysis and Technology Evaluation (IEK-STE), 52428 Jülich, Germany Institute
for Theoretical Physics, University of Cologne, 50937 Köln, Germany
###### Abstract
The energy transition introduces more volatile energy sources into the power
grids. In this context, power transfer between different synchronous areas
through High Voltage Direct Current (HVDC) links becomes increasingly
important. Such links can balance volatile generation by enabling long-
distance transport or by leveraging their fast control behavior. Here, we
investigate the interaction of power imbalances - represented through the
power grid frequency - and power flows on HVDC links between synchronous areas
in Europe. We use explainable machine learning to identify key dependencies
and disentangle the interaction of critical features. Our results show that
market-based HVDC flows introduce deterministic frequency deviations, which
however can be mitigated through strict ramping limits. Moreover, varying HVDC
operation modes strongly affect the interaction with the grid. In particular,
we show that load-frequency control via HVDC links can both have control-like
or disturbance-like impacts on frequency stability.
## Introduction
The transition to a renewable energy supply challenges the operation and
stability of electric power systems witthaut2021collective . Wind and solar
power generation are determined by the weather and are thus intrinsically
fluctuating and uncertain anvariShortTermFluctuations2016 ;
collinsImpactsInterannualWind2018 . Hence it becomes increasingly difficult to
balance the generation and consumption of electric power
elsner2015flexibilitatskonzepte . Furthermore, the location of renewable power
generation is partly determined by natural conditions and not by consumer
needs. For instance, wind turbines are preferably installed near the coast,
while the centers of the load may be far away inland, leading to higher
transmission needs and grid loads peschImpactsTransformationGerman2014 . To
master these challenges, we need new infrastructures, but also new concepts
for power system control and operation poollaOptimalPlacementVirtual2017 ;
milanoFoundationsChallengesLowInertia2018 , as well as a better understanding
of the complex inter-dependencies of different parts of the energy system.
This article contributes to this understanding by a data-based analysis of the
interplay of power system control and cross-area power flows.
Load-frequency control is the central method to balance generation and load on
short time scales. Although extreme imbalances, e.g. after contingencies, are
rare, system balancing becomes a regular challenge due to large fluctuations,
unscheduled events, or forecasting errors schaferNonGaussianPowerGrid2018 ;
krusePredictabilityPowerGrid2020 ; kruse2021revealing . Any imbalance
manifests in the grid frequency, which decreases from its reference value in
case of scarcity of generation and vice versa anderson2003 ;
machowskiPowerSystemDynamics2008 . In the Central European power grid, the
number of large frequency deviations has been increasing in the last decade
such that frequency stability has become a matter of increasing importance
entso-eaisblReportDeterministicFrequency2019 . The power balance is restored
by activating different layers of reserves, which are generally activated
based on deviations of the grid frequency Primary control or frequency
containment reserve (FCR) is activated within seconds and can roughly be
described by a proportional control law. Secondary control or frequency
restoration reserve (FRR) is activated within minutes and can roughly be
described by an integral control law.
Power flows between different synchronous areas play an increasingly important
role due to the energy transition. The increasing demand for long-distance
power transmission is covered by a variety of grid extension projects
Netzentwicklungsplan . This includes in particular new high-voltage direct
current (HVDC) links within and between different synchronous areas. Within
the last decade, several new links have been established in Europe, and
further ones are planned or under construction. Links from and to the
Scandinavian power grid are particularly promising due to the large resources
of (controllable) hydropower and pumped hydro storage in the Nordic area
tellefsen2020norwegian . The operation of such cross-area HVDC links is mainly
determined in advance on the market. However, HVDC links and converter
stations can also be used in load-frequency control, either optimizing the
frequency in both areas or only on one site entso-HVDCLinksOperationReport .
Then, power flows are adjusted on short terms and thus deviate from the
previously scheduled values, which is why unscheduled cross-area flows are
inherently connected to frequency control through HVDC links. Until now, pilot
projects and simulation studies have demonstrated the additional use of
frequency control via HVDC links between synchronous areas
jingyahuangHVDCbasedFastFrequency2017 ; langwasserEnhancedGridFrequency2020 ;
dijokasFrequencyDynamicsNorthern2021 ; dehaanStabilisingSystemFrequency2016 .
The European network of transmission system operators (ENTSO-E) currently
revises the standards for the application of HVDC links within the two
implementation projects for coordination of aFRR and mFRR throughout Europe,
PICASSO and MARI picasso ; mari .
In spite of these developments, cross-area flows on HVDC links can also have
unexpected and even unfavorable effects on frequency quality, especially on
the Rate of Change of Frequency (RoCoF). Firstly, scheduled power flows on
HVDC links can be ramped up or down very fast, which can lead to deterministic
frequency deviations entso-eaisblReportDeterministicFrequency2019 . As fast
HVDC ramps cause RoCoFs beyond the systems’ control capabilities, technical
studies even suggest ramp limits entso-HVDCLinksOperationReport . While the
effect of fast generation ramps on frequency quality is well-known
weissbachHighFrequencyDeviations2009 ; kruse2021revealing , the effects of
scheduled HVDC ramps are still under investigation entso-
eaisblReportDeterministicFrequency2019 . Secondly, ramp limits and other
technical parameters limit the effect of HVDC links on the grid frequency
jingyahuangHVDCbasedFastFrequency2017 . Here the question arises, how such
constraints affect frequency deviations in reality. Thirdly, frequency control
on one side of an HVDC link can act as a power disturbance on the other side
langwasserEnhancedGridFrequency2020 , which might cause unfavorable effects on
frequency stability. Understanding expected as well as unfavorable effects in
historic operational data of HVDC links and grid stability is therefore of
great importance. However, multiple factors affect grid frequency deviations
in large synchronous areas thus making it hard to isolate the interaction with
HVDC power flows by simple data analysis, such as correlation studies
kruse2021revealing .
In this article, we use eXplainable Artificial intelligence (XAI) for a data-
based analysis of HVDC operation, frequency stability, and their
interrelations. Machine Learning (ML) allows us to model and disentangle
different effects in the data hastieElementsStatisticalLearning2016 , while
methods from XAI enable us to gain insights into the dependencies identified
by the model barredoarrietaExplainableArtificialIntelligence2020 . In
particular, we use SHapley Additive exPlanations (SHAPs), which have highly
desirable mathematical properties lundbergUnifiedApproachInterpreting2017 ;
lundbergLocalExplanationsGlobal2020a . As a case study, we apply these methods
to HVDC flows between the major synchronous areas within Europe. Using five
years of publicly available data transnetbwfreqdata ; nationalgridfreqdata ;
fingridfreqdata ; ENTSOETransparencyPlatform , we extract the interactions
between HVDC flows and the grid frequency, both for the entire grid as well as
for individual HVDC links.
The article is organized as follows. First, we describe our methodology, which
we then apply to two different models. The stability model aims to predict
indicators for frequency stability from various techno-economic features. We
focus on the impact of cross-area flows which are extracted via SHAP values.
Then, we introduce a second model, the flow model, that targets unscheduled
cross-areas flows on individual links, including frequency stability
indicators as features. This approach reflects the fact that interactions
between flows and frequency can be both-way; in particular, frequency
deviations may cause unscheduled flows due to frequency control. Finally, we
close our study with a discussion.
## Explainable machine learning for the analysis of complex energy systems
We aim to extract the relation between frequency stability and the flows on
HVDC lines, that connect different synchronous areas within the European power
system. Frequency stability depends on a variety of drivers that display
complex interactions kruse2021revealing such that it is extremely difficult
to single out the role of one or even a few key influence factors based on
ordinary correlation analysis. Instead, we need a method that can disentangle
the contributions of different factors in a mathematically consistent way.
Machine learning provides powerful tools to model such complex dependencies.
We refer to our code github and data zenodo for details on our model
implementation and data preparation that go beyond the main text.
Figure 1: (a) Indicators of frequency stability in hourly resolution. The
RoCoF is the slope of the frequency at the beginning of the hour while the
Nadir is the largest frequency deviation within one hour. We also considered
the Mean Square Deviation (MSD) and the integral for each hourly frequency
trajectory. (b) Map of northern Europe including the HVDC links we considered
in this study. The differently colored regions reflect the five synchronous
areas adjoining these HVDC connections.
### Quantifying frequency stability
Frequency stability is quantified by four indicators (Fig. 1 (panel a)), which
are evaluated on an hourly basis reflecting the basic time interval for
electricity trading and scheduling. All indicators are extracted from
frequency time series $f(t)$ with a 1-second resolution obtained from
transnetbwfreqdata ; nationalgridfreqdata ; fingridfreqdata . First, the Rate
of Change of Frequency (RoCoF) is defined as the gradient of $f(t)$ at the
beginning of the hour after the dispatch has been adapted. Second, we consider
the integral of $f(t)$ which reflects a sustained frequency deviation and thus
a sustained power imbalance. Finally, we consider the largest deviation from
the reference frequency during the hour, the nadir, and the mean square
deviation. Details are described in ref. kruse2021revealing .
### HVDC cross-area flows
This study investigates the relations of non-embedded HVDC link operation to
frequency stability. Therefore, we included the time series of scheduled
commercial exchanges and physical flows in the analysis, which was neglected
in a previous model kruse2021revealing . The ENTSO-E transparency platform
provides the in and outflows between two bidding zones, control areas, or
countries ENTSOETransparencyPlatform .
To include HVDC flows in our models, we engineer two different datasets with
different purposes.
First, we use an aggregated dataset to investigate the impact of flows on the
frequency stability indicators. For each synchronous area, we construct a time
series summarizing the net inflow from each neighboring synchronous area.
Second, we use a link-resolved dataset to investigate how the flows depend on
frequency deviations. We construct time series from the transmission data
between two bidding zones or control areas, which we can attribute to a
specific HVDC link or link group. Time periods where links are malfunctioning
or unavailable are excluded from the analysis as we want our models to explain
the day-to-day behavior. To this end, we utilized the transmission grid
unavailability data on the ENTSO-E transparency platform
ENTSOETransparencyPlatform . We found that omitting all data from listed time
periods shorter than two months yielded a good trade-off between eliminating
exceptional behavior while maintaining sufficient data for training and
testing.
In addition, we constructed time series of unscheduled flows by subtracting
physical flows from scheduled commercial exchanges for both datasets.
Overall, we included transmission data for 13 HVDC links connecting five
different synchronous areas, as illustrated in Fig. 1 (panel b). The
transmission data between Belgium and the United Kingdom was generally omitted
as the Nemo Link was just commissioned towards the end of the period
considered in this paper.
### Stability model and flow model
To investigate the relationship between frequency deviations and HVDC flows,
we used two models that differ in their inputs and targets. The stability
model, which we will look at first, targets the four frequency indicators
introduced above for the synchronous power grids of Great Britain (GB),
Northern Europe (Nordic), and Continental Europe (CE) by using multiple
techno-economic features. The set of features contains the aggregated inflows
from other synchronous areas as described above and several features
characterizing generation and load. As in ref. kruse2021revealing , we
included time series of day-ahead load forecast, day-ahead wind and solar
forecast, day-ahead electricity prices, day-ahead scheduled generation per
production type as well as the actual load and generation per type. All time
series were engineered from publicly available day-ahead and ex-post data
obtained from the ENTSO-E transparency platform ENTSOETransparencyPlatform .
As we want to model hourly frequency indicators, the time series with higher
time resolution were downsampled to hourly resolution by taking hourly
averages. We aggregated the time series for each synchronous area following
the procedure in ref. kruse2021revealing . Beyond the aggregated features we
engineered ramp features (gradients from time $t-1$ to $t$), forecast errors
(day ahead - actual), and an inertia proxy inspired by ref.
ulbigImpactLowRotational2014 .
The second model, the flow model, then targets the unscheduled flows of
individual HVDC links. We use one model class for each of the four border
regions, which differ in their input feature set. For links connecting GB and
CE as well as CE and Nordic, data from both terminal sides are available. The
feature sets contain all time series on generation, load, and prices from both
sides as described for the stability model. In addition, the feature set
includes the planned commercial exchange and the four stability indicators
from both terminal sides. For links connecting the Nordic and the Baltic grid
as well as the GB and the Irish grid, only data from one terminal side is
included due to the unavailability of data from the Irish and the Baltic grid.
### Model training, testing, and explanation
We used the LightGBM framework ke2017lightgbm to train gradient boosted trees
to our datasets consisting of hourly time series for the years 2015-2020.
Gradient Boosted Trees are considered among the highest performing methods for
tabular datasets chenXGBoostScalableTree2016 while they also allow for fast
and efficient computation of SHAP values which enable us to explain the model
lundbergLocalExplanationsGlobal2020a . Before training, we shuffled our data
and split it into a training set (64%), a validation set (16%), and a test set
(20%). To optimize the hyperparameters of each model we conducted a grid
search combined with a 5-fold cross-validation on the training set while
applying early stopping of the boosting rounds utilizing the validation set.
Subsequently, we tested the model performance on an unseen test set.
Finally, the model is interpreted and analyzed in terms of SHAP values which
allow us to disentangle each model prediction into the contribution of the
individual features lundbergLocalExplanationsGlobal2020a . Assume that we have
a set of $n$ features. Given the feature values $x_{1},\ldots,x_{n}$, the
model yields the output $f(x_{1},\ldots,x_{n})$. Using the SHAP framework,
this prediction can be attributed additively to the individual features
$j\in\\{1,\ldots,n\\}$. That is, the output can be written as a sum
$f(x_{1},\ldots,x_{n})=\phi_{0}(f)+\sum_{j=1}^{n}\phi_{j}(f,x_{1},\ldots,x_{n}),$
where the $\phi_{j}$ is the SHAP value of the $j$th feature and the base value
$\phi_{0}$ is the expected value of $f$.
However, SHAP values are not limited to the explanation of individual inputs,
but can also be used to interpret global feature importances and model
behavior by combining many local explanations. The absolute value of a
feature’s SHAP value reflects how much the model relies on this feature for
the corresponding input. Therefore, we can quantify global feature importance
within one model by averaging over a feature’s SHAP values for all inputs. We
then obtain a normalized feature importance by dividing the feature
contribution of one feature $k$ by the total sum of all feature contributions:
$\frac{\left<|\phi_{k}(f,x_{1},\ldots,x_{n})|\right>_{\text{inputs}}}{\sum_{j=1}^{n}\left<|\phi_{j}(f,x_{1},\ldots,x_{n})|\right>_{\text{inputs}}}.$
Consequently, a feature importance of one would imply that the model solely
relies on this single feature while a zero feature importance implicates that
the model does not consider the feature at all. Moreover, feature importances
for a single model sum to one. We also utilize SHAP dependency plots, which
are able to provide much deeper insights than classical dependence plots,
especially when SHAP interactions are also taken into account. For instance,
SHAP dependence plots are capable of providing much deeper insights than
classical dependence plots, especially when SHAP interactions are also taken
into account. Nevertheless, it is important to keep in mind that SHAP in its
own right is not able to unveil any causal relationship. Therefore any results
have to be interpreted using domain knowledge to yield meaningful knowledge
discovery.
## Modeling and understanding frequency stability indicators
We first consider the stability models, which predict the four different
indicators for power systems frequency deviations. We focus on the role of the
features describing cross-area flows via HVDC links, which had been discarded
in previous models in ref. kruse2021revealing .
Figure 2: Including HVDC transmission data improves the modeling of frequency
stability indicators. For three European synchronous areas, we present Machine
Learning (ML) models that predict frequency stability indicators from techno-
economic features such as generation per type, load, and day-ahead electricity
prices. We measure their performance by the $R^{2}$ score, which quantifies
the share of variance explained by the model (colored bars). As a benchmark,
we also provide the $R^{2}$ score for the daily profile predictor, which
predicts the targets purely based on their daily average evolution (grey
bars). The ML models outperform the benchmark by a large margin, showing the
overall importance of techno-economic features for frequency stability.
Compared to previous models without HVDC features (cf. kruse2021revealing ),
the inclusion of cross-area flows improves the performance by a factor of up
to 1.8 (indicated by the numbers above the red bars). The benefits of
including HVDC flows are particularly large in GB and the Nordic area.
### Importance of HVDC flows for frequency stability
We start our investigation by looking at the overall importance of HVDC flows
for frequency stability. The importance is quantified in terms of ML models,
comparing the performance with and without the HVDC flows in the feature set.
The performance is measured by the R2-score, which reflects the share of
variance explained by the model, and evaluated for all four frequency
indicators and three different synchronous areas (Fig. 2).
We find that the importance of cross-area flows is very different for the
three major European grids. In Great Britain, the performance of the ML models
increases strongly, up to a factor of 1.8, if cross-area flows are included.
In absolute terms, however, the performance remains mediocre with values
$R^{2}\approx 0.4$. In contrast, the performance is substantially higher for
the Central European grid and remains largely unaffected when cross-area flows
are included or not. This may be attributed to the fact that the Central
European grid is much larger (factor of approximately eight
rydingorjaoOpenDatabaseAnalysis2020 ), such that HVDC converter stations take
a much smaller relative share in generation or load.
Figure 3: Feature importances of HVDC flows vary among the different
synchronous areas. We measure the feature importances in our ML models by the
mean absolute SHAP values. The importances for each model are normalized such
that they sum to one as described in the main text. For each synchronous area
and each indicator, the eight most important features are depicted. In the
Nordic and GB areas, the HVDC features are among the most important features,
but in CE the model did not identify cross-area flows as highly important.
The general finding is confirmed by a detailed analysis of the feature
importances reported in Fig. 3. In the GB grid, unscheduled, physical, and
scheduled flows on HVDC links are among the three most important features for
different targets, while HVDC features occur in the Nordic model less
prominently. In the CE model, HVDC features are not among the eight most
important features. This underlines the importance of HVDC power flows in GB
and to a lesser extent the Nordic grid, while HVDC flows have a relatively
small impact in CE.
### The effect of scheduled HVDC flows
Some feature importances reveal relevant effects of scheduled HVDC flows,
which are related to deterministic frequency deviations kruse2021exploring .
Fig. 3 shows a strong impact of scheduled flow ramps on the RoCoF in GB, while
the other stability indicators are only weakly affected. This relates to the
market-based schedule of flows on HVDC links, which changes mostly in
intervals of one hour, thus affecting the hourly RoCoF predominately.
Schedule-based ramps on HVDC links introduce temporary power imbalances and
thus lead to deterministic deviations in the grid frequency entso-
eaisblReportDeterministicFrequency2019 . This is similar to the deterministic
effect of fast, scheduled generation ramps, which are a well-known driver of
deterministic frequency deviations weissbachHighFrequencyDeviations2009 ;
kruse2021revealing . In the Nordic grid, scheduled flows ramps are ranked as
the sixth most important feature for the RoCoF, indicating weak deterministic
effects through scheduled HVDC ramps.
Interestingly, the direction of these ramping effects varies among the grids
(Fig. 4). While increasing scheduled ramps leads to larger RoCoFs in GB, the
relation is the opposite in the Nordic grid. Following the discussion in ref.
kruse2021revealing , we thus identify the ramping effect in GB as RoCoF-
driving, while in the Nordic grid HVDC ramps are RoCoF-offsetting, i.e. larger
scheduled ramps are rather related to a RoCoF reduction.
We explain this observation with ramp speeds of HVDC converters relative to
the speed of other generation types in the respective synchronous area. We
start from typical values of the rate of change of power (RoCoP) for every
type. Then we compute the ratio with respect to the fastest RoCoP in the grid,
yielding the relative ramp speed $s\in[0,1]$ similar to the procedure in ref.
kruse2021revealing .
In GB, scheduled HVDC ramps are equally fast ($s=0.14$) as other RoCoF-driving
generation types such as pumped hydro generation ($s=0.15$), thus causing a
step-like behavior, which drives the deterministic frequency deviations. In
the Nordic grid, fast reservoir hydro ramps are the dominating RoCoF-driving
technology ($s=1$) and inflow ramps from CE are relatively slow ($s=0.04$).
This is most probably due to the strict ramping limits on HVDC links imposed
by Nordic TSOs entso-eaisblReportDeterministicFrequency2019 . In this way,
HVDC ramps rather smooth the generation curve and dampen the temporary power
imbalance, which explains the different dependencies in Fig. 4. In CE,
scheduled HVDC ramping speeds are also very slow ($s=0.03-0.09$) as large
generation ramps dominate the overall RoCoP due to the size of the area. Thus,
scheduled HVDC ramps are not strongly important for the RoCoF in CE and do not
appear in Fig. 3.
Figure 4: Different effects of scheduled HVDC ramps on the Rate of Change of
Frequency (RoCoF). We show the modeled interdependencies between the ramps of
scheduled HVDC flow to Continental Europe and the hourly RoCoF in the GB
(left) and Nordic (right) synchronous areas. The dependencies have opposite
directions, which is confirmed by the Kendall correlation coefficient $\tau$
of the scattered data. We explain these effects with different relative ramp
speeds $s$, which we calculate based on the ramp rates $r$ of the HVDC links
entso-eaisblReportDeterministicFrequency2019 . On the CE-GB border, a cable
with 1000 MW capacity is allowed to ramp 100 MW/min, i.e., the ramp rate is
$r=0.1/\rm{min}$ (share of full load per minute). On the CE-Nordic border, a
cable such as Kontek has 600 MW capacity and Nordic ramp restrictions allow
600 MW per hour, i.e., a ramp rate of $r=0.017/\rm{min}$. Using the mean
$\Delta P$ of the absolute hourly flow changes on the links, we estimate the
typical Rate of Change of Power as RoCoP$=\Delta P\cdot r$, which results in
different relative ramp speeds $s$ (see main text). These different ramping
speeds most probably explain the different effects on the hourly RoCoF.
Figure 5: Unscheduled flows and the frequency integral show different
dependencies among the European grids. The model-independent scatter plots
(top row) of unscheduled outflows and frequency integrals do not exhibit clear
dependencies. Therefore we display SHAP dependency plots (bottom row), which
depict the relationship extracted by the stability model thus isolating the
effect of a single feature. The integral in the Nordic (a) and the GB area (b)
show a positive relation to unscheduled flows, which corresponds to the effect
of frequency control, while the CE unscheduled flows have a negative effect on
the Nordic integral (c) thus showing a disturbance-like behavior. The
scheduled flows (color code) demonstrate that Nordic-CE and GB-CE flows are
mostly uni-directional, while CE-Nordic flows exhibit both directions (see
main text for discussion).
### The effect of unscheduled HVDC flows
In addition to scheduled flows, unscheduled HVDC flows, i.e., deviations from
the market-based schedule, play an important role in frequency deviations.
Figure 3 reports CE unscheduled flows to be the most important feature for
three stability indicators in GB. In the Nordic grid, unscheduled flows and
the related physical flows are also of high importance. The importance of
unscheduled flows in frequency dynamics might relate to the effect of
unforeseen outages on the HVDC links, but these are rare events entso-
NordicBalticHVDCStatistics , which cannot strongly influence the dependencies
within our model. Another, more realistic explanation for the importance of
unscheduled flows is load-frequency control, which can be applied through HVDC
links entso-HVDCLinksOperationReport and always introduces an unscheduled
change in the power flow on the link.
To further investigate this hypothesis, we exemplarily examine the effect of
unscheduled flows on the frequency integral for three different HVDC
connections (Fig. 5). We focus on the frequency integral, as it most closely
reflects the need for frequency control via HVDC links in our dataset. The
data only includes hourly averaged power flows, so that only net hourly
balancing actions through HVDC links are recorded. These are most closely
related to the frequency integral as it reflects the net power imbalance
within the hour. Remarkably, the scatter plots of flows and integral values
(top row Fig. 5) do not identify clear dependencies. In contrast, the SHAP
dependency plots (bottom row) isolate the effect of the individual feature and
show distinct relations, thus confirming the advantages of XAI methods over
simple correlations analysis (cf. kruse2021revealing ).
The observed effects of unscheduled HVDC flows vary among the grids, showing
either a control- or a disturbance-like behavior. A control-like effect
resembles the behavior of a frequency controller, i.e., positive frequency
deviations lead to positive outflows to balance the oversupply within the
system. Contrary, a disturbance-like effect entails a negative dependency,
where a positive frequency deviation is triggered by a sudden negative
outflow, i.e., the sudden inflow of power into the system. In Fig. 5 (bottom
row), unscheduled outflows to the Baltic area have a positive relation to the
Nordic frequency integral (panel a), which is similar to CE unscheduled flows
in GB (panel b). Thus, these effects on the Nordic and British frequency show
a control-like behavior. In contrast, unscheduled outflows from the Nordic
grid to CE (panel c) show a negative dependency, which resembles a
disturbance-like effect. Interestingly, also the physical flows (color code)
differ strongly among the three examples in Fig. 5, which we will discuss
within the next section.
Finally, we note that unscheduled HVDC flows are the result of explicit
operation strategies by the adjoining TSOs, which are in control of the DC
power flow on the links entso-HVDCLinksOperationReport . To further understand
the varying effects of unscheduled HVDC flows, we thus have to examine the
individual operation modes on different links between the synchronous areas.
## Modeling and understanding unscheduled flows between synchronous areas
Our data-based analysis shows a strong relation between unscheduled flows and
frequency stability. To further understand this interaction, we now switch the
perspective and investigate unscheduled flows on individual HVDC links as
targets. For this purpose, we use our second model, the flow model, which
predicts unscheduled flows on single HVDC links based on techno-economic
features and the frequency indicators.
Figure 6: With our flow model, we predict the unscheduled flows on single
HVDC links between different synchronous areas using techno-economic features
such as prices, loads, or generation per type as well as frequency stability
indicators. The models successfully predict unscheduled flows with
performances similar to those of the stability model (cf. Fig. 2).
### Importance of power imbalances for unscheduled flows
The flow model can explain a large share of unscheduled flows on various HVDC
links (Fig. 6). As a recap, the flow model uses features from the stability
model, such as the load or scheduled HVDC flows, as well as the frequency
indicators from areas that are available in our data set. Based on these
features, we achieve similar performances as in the stability models
($R^{2}\sim$ 0.25…0.85), which are acceptable values given the stochastic
nature and our limited access to all influencing variables. Among the feature
set, the role of the frequency integral is particularly interesting, as it
closely relates to the control need within the hour.
The frequency integral plays a major role for some HVDC links, which can be
related to load-frequency control delivered by these links. We first focus on
the feature importance, which is indicated in Fig. 7 on the y-axis, and the
feature rank, represented by black numbers.
The Britned link is most prominent, as the British frequency integral is the
most important feature in the respective model (blue color). The Nordic
integral (orange color) is ranked among the ten most important features for
Estlink, Kontiskan, and Storebaelt. In contrast, the Nordned link and
particularly the Baltic cable show very low feature importances of the
integral.
Our findings are fully consistent with operation modes of HVDC links reported
by the transmission system operators and the ENTSO-E. Kontiskan, Storebaelt,
and Estlink NordicSOA2019-LFCannex ; NordicSOA2006 as well as Britned entso-
HVDCLinksOperationReport are used in load-frequency control. As the time
series for physical and unscheduled flows correspond to the hourly average,
the integral is the relevant feature of the frequency time series. The
situation is very different for the Nordned link, which is not used in load-
frequency control as reported in ref. entso-HVDCLinksOperationReport .
Consistently, the frequency integral exhibits very low feature importance in
the model.
Our studies complement and augment the sparsely available public information
on HVDC operation. First, public information is entirely lacking for some of
the HVDC links. For instance, we are not aware of any official report
describing the operation of the Baltic cable. Our results show that this
connection is likely not used for load-frequency control. Second, most
available documents report only the general participation in load-frequency
control without further details. Our results allow us to quantify this
dependency as we will discuss in the following section.
### Control-like and disturbance-like effects
Turning from feature importances to dependencies, we reveal control-like and
disturbance-like interactions between unscheduled flows and frequency
stability (Fig. 7). The correlation $\tau$ between the SHAP values of the
frequency integral and the unscheduled flows quantifies the direction of this
dependency (similar to the correlation in Fig. 4). We note that we define the
direction of the unscheduled flows such that positive values correspond to an
outflow of electric energy. This holds for all connections such that a flow is
counted differently for the two terminal sides. For instance, an unscheduled
flow from CE to GB would be counted as positive for CE and as negative for GB.
In that way, control-like dependencies appear in the right half of the plot
($\tau>0$), where positive frequency deviations lead to more outflows, while
disturbance-like effects lie in the left half ($\tau<0$). We now discuss these
effects for the four links exhibiting the highest feature importances of the
integral, i.e., Britned, Estlink, Konstiskan, and Storebaelt.
Control-like effects are most pronounced for Britned and Estlink, which show
high feature importance of the GB and the Nordic integral and a strong
positive correlation. In contrast, the CE integral in the Britned model is
negatively correlated thus showing a disturbance-like effect. (Note that the
Baltic integral is not included in the Estlink model due to missing data).
These observations are consistent with reports and TSO agreements. Britned is
used only for uni-directional control, i.e. only for frequency support in GB
entso-HVDCLinksOperationReport , and Estlink is also dominantly used by the
Nordic side for load-frequency control, according to the 2019 System Operator
Agreement (SOA) NordicSOA2019-LFCannex . Thus, on the GB and Nordic sides, we
only have a control effect, which introduces a positive dependency between
frequency deviations and unscheduled outflows. On the other side of the link,
the control actions behave like a disturbance, which explains the negative
correlation of the CE integral in the Britned model.
Disturbance-like effects are most pronounced for Kontiskan and Storebaelt,
which exhibit a negative correlation for the Nordic integral, but with
relatively low feature importance. In contrast to Britned, the integral on
both sides of the Kontiskan and Storebaelt links show a negative dependency
(red and orange symbols), although the CE integral exhibits even lower feature
importance. Reports and TSO agreements suggest, that both links are used bi-
directionally for control, i.e., both sides can receive and supply frequency
support NordicSOA2019-LFCannex ; NordicSOA2006 . This might explain both the
low feature importance as well as the observed dependencies. Both areas
experience the control and the disturbance-like effect, which might cancel
each other out. This might partly explain the lower importance of the
frequency integral for both links. However, the disturbance-like effects
dominate thus yielding the observed negative correlation for both the Nordic
and the CE integral. One explanation can be the control delay, particularly if
links are used for slow tertiary control (mFRR), which is slower than primary
and secondary control. On the receiving side, a frequency deviation might
trigger a delayed control action that partly falls into the next hour. On the
supporting side, the disturbance-like effect of this action appears
immediately and thus might introduce a stronger dependency between the hourly
features and targets. However, also other effects of HVDC operation might play
a role here.
Figure 7: Relation between frequency integrals and unscheduled flows indicates
different HVDC operation modes. We show the feature importance of the
frequency integrals, which is the mean absolute SHAP value normalized by the
sum of all such values, i.e. it becomes one if the model uses only this single
feature. The numbers indicate the feature rank within the respective model.
For a specific area, the values $\tau$ depict the correlation between the SHAP
values of a frequency integral and the unscheduled outflows from this area,
i.e., disturbance-like relations appear on the left side and control-like
interactions on the right side. The strong control-like effects in the Britned
and Estlink model probably relate to their uni-directional control scheme,
while the weaker disturbance-like effects for Kontiskan and Storebaelt can be
a result of the bi-directional application of frequency control. Figure 8:
Multiple factors affect the regulation of unscheduled flows on single HVDC
links. We show the dependency plots for the first six most important features
(measured by mean absolute SHAP values) in the flow models of Britned and
Kontiskan as examples. The reference area for unscheduled outflows is GB (for
Britned) and the Nordic area (for Kontiskan). While the frequency integral
dominates the Britned model, other features such as the scheduled HVDC flows
and actual load and generation also affect the unscheduled power flows on the
two links.
### Other effects of HVDC operation modes
The frequency integrals, i.e., systematic power imbalances, do not alone
describe the unscheduled HVDC flows as demonstrated by their low feature
importance for multiple links (Fig. 7). However, most flow models still
exhibit a high performance (Fig. 6) thus pointing to other highly predictive
features. These effects can relate to other properties of HVDC operation,
which we discuss using the Britned and Kontiskan links as examples (Fig. 8).
The scheduled flows play a major role for both Kontiskan and Britned (green
color). For Kontiskan, this feature is even the most important one, with the
integral (red color) only following at rank six. Scheduled inflow on Kontiskan
to the Nordic area is associated with an increase in the amount of unscheduled
outflow. For Britned the impact of scheduled flows is nearly zero if the flows
direct to GB, which is the dominating case for this border (cf. the physical
flows in Fig. 5). A strong negative impact on the unscheduled outflows is only
observed for a few data points with scheduled outflows from GB, i.e.,
scheduled outflows from GB systematically overestimate the physical outflow.
The different effects of scheduled flows on Britned and Kontiskan might relate
to different operation modes. On Britned the link overload capacity is used
for frequency support entso-HVDCLinksOperationReport , such that frequency
control can operate nearly independently of scheduled flows, which is
consistent with a nearly vanishing impact of scheduled flows for the majority
of time steps (Fig. 8). Only in the rare situation of strong outflows to CE,
this situation changes. In contrast, no capacity is reserved for frequency
control on Kontiskan NordicSOA2006 . Control can only be provided if the
market-based schedule leaves free capacity in the right direction, such that
scheduled inflows to the Nordic area allow for increased unscheduled outflows
as depicted by the dependency plot (Fig. 8).
If we look back at the stability models and Fig. 5, we can even see
indications for these effects using the aggregated flow data. For the
connections Nordic-Baltic and CE-GB, the scheduled flow is oriented in the
same direction for most hours of the year as indicated in the color code in
Fig. 5. GB mostly imports power via HVDC links, such that positive control
power requires a further increase of the flows beyond the commercially
scheduled values. Obviously, this is possible only if capacity is available,
which is again consistent with ref. entso-HVDCLinksOperationReport and our
interpretation of Fig. 8. In contrast, links between the Nordic and CE areas
are used for power flows in both directions.
Other important features are the actual generation of different types, actual
load, and forecast errors (blue color). Most of these dependencies exhibit a
strong vertical dispersion, which indicates strong interactions with other
features lundbergLocalExplanationsGlobal2020a . Actual load, generation, and
forecast errors directly relate to actual power imbalances, which can explain
their importance for unscheduled HVDC flows. These features can indirectly
also reflect market situation, e.g., intra-day prices or reserve energy
prices, which are not included in the model. These market-based features can
also influence unscheduled HVDC flows, as control via these links might only
be applied if it is cheaper than other domestic operational reserves.
All in all, Britned is used for uni-directional control in GB, which in most
cases does not depend much on the market-based schedule of the link, thus
introducing a strong control-like interaction with the British grid frequency.
This strong control effect on Britned most probably explains the control-like
dependency, which we observed for the aggregated stability model in Fig. 5
(panel b). In contrast, Kontiskan exhibits disturbance-like interactions with
the grid frequency on both sides, which probably relates to a bi-directional
control setting. Control actions are further constrained by scheduled flows,
thus diminishing the importance of the grid frequency for unscheduled flows.
Similar observations can be made for other HVDC links between the Nordic and
the CE area (Fig. 7), thus explaining the disturbance-like effect observed in
our aggregated stability model in Fig. 5 (panel c).
## Conclusion and Outlook
In summary, we revealed important dependencies between non-embedded HVDC
operation and grid frequency stability using explainable machine learning.
Using our publicly available data set and model github ; zenodo , we extracted
such dependencies both for the entire synchronous areas as well as for
individual HVDC links between synchronous areas.
First, we highlighted the important role of HVDC flows to model frequency
stability indicators, particularly for the GB and Nordic synchronous areas.
Between CE and GB, flow ramps drive the British RoCoF as they are relatively
fast in the power grid compared to other generation types. In the Nordic grid,
scheduled flow ramps rather offset the hourly RoCoF, which is most probably a
result of strict ramp limits. This suggests that the ramp limits imposed by
the Nordic TSOs are successful in preventing stability problems through fast
HVDC ramps.
Second, we successfully modeled unscheduled flows on individual HVDC links
based on techno-economic features, including frequency stability indicators.
For two links in the GB and Nordic areas, we highlighted how frequency
stability needs drive unscheduled HVDC flows, which is most probably related
to their important role in load-frequency control.Meanwhile, unscheduled flows
on most other lines seem to be driven by factors not directly related to
frequency stability. Importantly, if balancing power is provided from one area
to the other through HVDC links, power is lost on the supporting side and the
frequency drops. This disturbance-like effect was particularly strong in the
Nordic grid, which supplies control power to Continental Europe. In the
British grid, the control-like effect dominated, i.e. the British side was
stabilized, which probably relates to the fact that balancing is done only on
the British side of the link.
Currently, these varying effects are a direct consequence of the differences
in operation concepts of the different HVDC lines, which follow bilateral
contracts between the respective TSOs NordicSOA2019-LFCannex ; NordicSOA2006 ;
entso-HVDCLinksOperationReport . ENTSO-E is currently developing PICASSO and
MARI to provide a unified concept for aFRR and mFRR throughout Europe,
including the application of HVDC in load-frequency control picasso ; mari .
In this context, effects such as the decrease of frequency quality due to
frequency support via HVDC are important for the evaluation of HVDC-based
load-frequency control concepts. In the future, data-driven methods, such as
our approach, can provide valuable tools for analyzing and monitoring the
changes due to new control concepts and regulations.
## Acknowledgements
The authors thank Thomas Dalgas Fechtenburg for fruitful discussions.
## Funding
J.K. and D.W. gratefully acknowledge support from the Helmholtz Association
via the Helmholtz School for Data Science in Life, Earth and Energy (HDS-LEE).
B.S. gratefully acknowledges funding from the Helmholtz Association under
grant no. VH-NG-1727.
## Availability of data and materials
Our data set, including the techno-economic features, stability indicators,
and HVDC flows as well as the results of our hyper-parameter optimization, is
available on Zenodo zenodo . The code is accessible on GitHub github .
## Author’s contributions
S.P.: Investigation, Formal analysis, Visualization. B.S.: Supervision,
Writing, Conceptualization. D.W.: Project administration, Funding acquisition,
Supervision, Writing, Conceptualization. J.K.: Project administration,
Supervision, Writing, Conceptualization, Methodology.
## Competing interests
The authors declare that they have no competing interests.
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|
# Ecological Data Reveal Imbalances in Collision Avoidance Due to Groups’
Social Interaction
Adrien Gregorj Okayama University, Okayama, Japan Zeynep Yücel Okayama
University, Okayama, Japan Advanced Telecommunication Research Institute
International, Kyoto, Japan Francesco Zanlugo Okayama University, Okayama,
Japan Advanced Telecommunication Research Institute International, Kyoto,
Japan Osaka International Professional University, Osaka, Japan Takayuki
Kanda Advanced Telecommunication Research Institute International, Kyoto,
Japan Kyoto University, Kyoto, Japan
###### Abstract
The relative dynamics in collision avoidance between individual pedestrians
and dyads has been recently studied, and it was shown that individuals may
intrude and disrupt those dyads that are not socially interacting, but not the
ones that are engaged in interaction. Building on this, our current study
examines how much each party contributes to collision avoidance in the
absolute sense, i.e. by measuring deviations from their intended paths.
Our findings suggest that individuals prioritise trajectory efficiency in
undisturbed situations, but prioritise safety when encountering dyads, by
deviating more from their intended path. Not socially interacting dyads
present a similar behavior, although their trajectories appear to be even more
efficient than those of individuals in undisturbed situations, and their
deviations during encounters less pronounced. On the other hand, socially
interacting dyads are not very efficient in undisturbed situations, and their
behavior is mostly unaffected by encounters.
These results strongly suggest that group dynamics affects in two ways the
behavior of pedestrians, namely it has a dynamical and a social effect. The
dynamical effect, i.e. the necessity to keep a spatial vicinity to one’s
partner, stabilises their trajectory, while the social one, by redirecting
one’s attention to the partner, decreases the ability to focus on the external
environment, and thus leads to reduced efficiency and safety.
Another finding concerns the tendency of individuals to avoid in a more
prominent way the interacting dyads as compared to non-interacting ones. This
suggests that individuals not only observe their surroundings to anticipate
the future paths of others, but may also assess others’ contribution to
collision avoidance.
An impact parameter analysis reveals that collision risk influences path
deviations in pedestrian encounters. For individuals, larger behavioral
differences between low and high interaction levels of the dyad occur both
when the collision risk is high and during less critical encounters. For
dyads, the deviation differences between low and high interaction levels are
most pronounced when the individual is on course to pass close to the dyad.
## 1 Introduction
Human walking motion has been a research subject in various disciplines, each
with distinct scopes and objectives. Crowd dynamics primarily studies
collective human movement [1], often with the goal of ensuring safety in
large-scale event management (e.g. Olympic games, Hajj) [2], prevention of
crowd accidents (e.g. during festivals) [3], and evacuation planning (e.g.
during fire, terror attacks etc.) [4], among others. On the other hand,
architecture and urban design extend this focus to everyday life settings [5],
prioritizing safety and comfort. This includes architectural designs for
enhancing business in malls [6], landscape designs [7, 8] for recreational
spaces, and urban infrastructure planning for efficient travel and minimal
detours [9]. In addition, robotics directs its attention towards individuals
[10] with the objective of achieving natural and comfortable motion at the
actuator and/or navigation levels. Applications range from assisting human
limbs [11] to seamless integration into human crowds [12] for services such as
museum guiding or companionship [13]. Furthermore, kinesiology examines motion
from a biomechanical standpoint, aiming to explain the fundamentals of motor
control [14]. In contrast, cognitive science predominantly explores the
neurophysiological perspective, with a specific focus on such aspects as
perception, visuo-motor coordination, and embodied cognition [15]. Despite its
brevity, we believe that the above overview already provides a glimpse into
the multifaceted nature of research on human walking motion and the vast
variety of scopes, objectives and methodologies.
In this study, we choose to view human walking motion in its simplest form,
namely as the most fundamental means of transportation [16] and as an
essential activity for an independent life and daily tasks (e.g. trips to the
market, the bank etc.). In such settings, individuals maneuver among fellow
pedestrians and obstacles with a focus on safety and fluidity, a concept known
as collision avoidance. These localised interactions are thought to give rise
to self-organization within a local-to-global framework [17]. For capturing
the properties of this sort of navigation behaviour, a large variety of
computational models have been proposed [18], where the early models were
inspired by physics and used repulsive forces to reproduce collision avoidance
[19]. While these models have been successful in generating coherent patterns
at the collective level and have greatly contributed to our understanding of
pedestrian motion [19], they often fall short in accurately capturing the
realistic attributes of human trajectories [17].
In that respect, this study will focus on human walking behavior in an
ordinary public space aiming to unveil nuanced patterns in decision-making and
path selection. Specifically, we will focus on collision avoidance between
individual pedestrians and social groups of two people. By dissecting such
encounters according to the group’s level of engagement in social interaction,
we aim to illustrate the emergence of an intriguing pattern, namely a degree
of involvement in collision avoidance and possibly an understanding, by
participants not absorbed in social interaction, of the others’ involvement.
In a previous study, we studied dyad-individual collision avoidance in a
relative sense —specifically, in terms of how closely they approach each other
during collision avoidance [20]— and demonstrated that individuals tend to
intrude and disrupt a dyad when its members are not socially interacting, but
they refrain from doing so when the dyad is socially interacting or appears to
have a strong social bond (e.g. couples). Additionally, we have shown that the
collision avoidance dynamics between the dyad and individual can be modeled
using an “interaction potential”, obtained through an analogy with the two-
body scattering problem, increasing rapidly as the distance decreases, and
depending on the level of interaction or the nature of the social relation of
the dyad.
In the current study, we aim to build upon our previous research by examining
individual-dyad collision avoidance in an absolute sense, i.e. quantifying
each peer’s contribution to collision avoidance based on their deviations from
intended paths. Firstly, we will illustrate, using ecological data, that
social interaction within groups leads to less efficient path adjustments.
Specifically, when groups move without other pedestrians nearby, their
deviations from a straight path increase with higher levels of interaction,
resulting in longer and less economical trajectories. Conversely, when groups
encounter individuals, their deviations decrease with higher interaction
levels, indicating reduced responsiveness in collision avoidance.
On the other hand, individual pedestrians clearly modify their behavior in
presence of dyads. Additionally, individuals show sensitivity to the level of
interaction, albeit not significantly so. This result, when combined with the
statistically significant result reported in [20], namely that relative
collision avoidance is stronger between individuals and socially interacting
dyads with respect to the one between individuals and not socially interacting
dyads, suggests that pedestrians anticipate the diminished involvement of
groups in collision avoidance and adjust their deviations from intended paths.
Additionally, we will investigate the role of collision risk in shaping path
deviations during pedestrian encounters. The concept of impact parameter,
borrowed from physics and used in [20], will be employed to quantify the
collision risk between individuals and dyads. We will show that collision risk
influences path deviations in pedestrian encounters, with larger behavioral
differences between low and high interaction levels of the dyad occurring both
when the collision risk is high and during less critical encounters. The
particularly straight trajectories of individuals encountering non-interacting
dyads echoes the findings of [20], suggesting that are more susceptible to
pass through such dyads. We will also demonstrate that the deviation
differences between low and high interaction levels are most pronounced when
the individual is on course to pass close to the dyad.
Furthermore, a comparison between the behavior of individuals, dyads with and
without social interaction shows that group dynamics presents both a dynamical
and a social effect, suggesting that this aspect should be introduced also in
crowd models to properly asses the presence of groups and their effect on
overall dynamics.
The adaptive behavior observed suggests a nuanced understanding of social cues
and an ability to adjust behavior accordingly. Moreover, it sheds light on the
complexity of human interaction within crowded environments, where individuals
must constantly explore path choices/adjustments and negotiate space alongside
others. Understanding these dynamics not only enhances our comprehension of
pedestrian behavior but also has implications for urban planning, crowd
management, and the design of intelligent systems aimed at facilitating smooth
and safe movement in public spaces.
In the following section, we will frame our work and the scenario in focus and
explain related literature concerning the factors relevant for shaping human
motion in those settings.
## 2 Background
The concept of “community ambulation” as described in the literature refers to
individuals’ ability to move independently in public spaces [21]111A more
precise definition given in [22] specifies that community ambulation entails
walking a defined distance (e.g., 800 m) and navigating stairs without
assistance. . Community ambulation, by definition, necessitates the capacity
to integrate walking with a diverse range of demands arising from the dynamic
nature of the surrounding public environment. Patla et al. precisely identify
factors contributing to this integration as walking distance and speed,
ambient conditions, physical load, terrain variations, postural transitions,
traffic density, and attentional demands [23].
In order to meet these demands, individuals must constantly navigate decision-
making processes while engaged in ongoing activities [24]. These embodied
decisions necessitate rapid and continuous processing of multi-modal sensory
information [25] as well as evaluating all possibilities in parallel,
culminating in the execution of the anticipated optimal choice. Although there
is ongoing debate regarding the exact mechanisms underlying these decision-
making processes [26, 27], the complexity of this cognitive process is widely
accepted.
We consider a typical urban setting to be characterised by low to medium
density, plain geometry, and even terrain. Such an environment is often
populated by individuals from diverse backgrounds, spanning various ages,
occupations, and other demographic factors. In this study, as an adequate
example of urban environment, we use the DIAMOR pedestrian trajectory data set
[28] (see Sec. 3.1 for details). The use of ecological data is essential in
our study, as it captures real-world complexities and nuances inherent in
human behaviour within urban environments. Unlike simulated or laboratory-
based data, ecological data reflect the genuine interactions and responses of
individuals navigating through authentic urban settings. By leveraging such
data, we hope to better understand the intricacies of pedestrian dynamics,
including the influence of social interactions, and individual behaviours.
Even though there is an inherent lack of control over the factors influencing
navigation choices of pedestrians in this naturalistic environment [29], we
can assess extrinsic factors in accordance with Patla and Shumway-Cook’s
guidelines [23]. Regarding walking speed, the environment imposes no
significant constraints, as evidenced by empirical velocity distributions that
align with ranges observed in other ecological studies on community ambulation
[28, 30] (see Sec. 3.2). Analyzing a sufficiently large sample size allows for
the proper representation of interpersonal variations. Regarding postural
transitions, the absence of curves, slopes, or stairs in the environment
indicates minimal need for significant adjustments, a conclusion further
supported by video recordings. In terms of physical load, analysis of the same
data reveals that pedestrians predominantly carry either no load or a
seemingly light handbag or backpack, typical for urban travel222Human
annotations of video data show that pedestrians mainly consist of students,
workers, or shoppers commuting to schools, workplaces, or commercial centers..
Furthermore, the terrain and ambient conditions within the data set remain
constant, whereas density undergoes minute changes (see Sec. 3.2 and Sec.
3.3). Regarding distance, as detailed in Sec. 3.3, specific preprocessing
steps are employed to ensure the comparability of path lengths across
pedestrians333However, we cannot comment on how long the pedestrians might
have walked before entering our observation space.. This leaves us with the
last extrinsic factor of community ambulation, namely, attentional demands,
which may actually be quite challenging to assess.
While navigating in public environments, pedestrians need to allocate
attentional resources for the observation of their surroundings to identify
potential hazards (e.g. stairs, static obstacles) and remain vigilant to
changes [31]. Nevertheless, the underlying principles of anticipatory
locomotor adaptations used to circumvent such challenges is not completely
understood [32]. Shumway-Cook et al. identify three key factors influencing
attentional demands in community ambulation: familiarity with the trip
location, environmental distractions, and the presence or absence of travel
companions [22]. In our case, information regarding pedestrians’ familiarity
with the environment is unavailable. Nevertheless, given the straightforward
nature of the corridor shown in Fig. 1, it is unlikely that familiarity with
such an environment would significantly affect route finding. Additionally,
with no visual or auditory distractions such as shops or music present,
environmental distractors are expected to be minimal and consistent across all
observed pedestrians. This allows us to isolate the effect of the presence or
absence of travel companions and analyze its specific impact.
The distinction between physical crowds, comprised of individuals in close
spatial proximity but lacking a shared psychological identity, and
psychological crowds, characterised by a collective sense of identity or
purpose among its members, has been introduced in [33, 34]. Pedestrians
observed in the DIAMOR data set are likely to belong to the former category,
as they do not appear to have a collective identity. Nevertheless, our data
set contains instances of small social groups and individuals. Here, by groups
we refer to 2 or more pedestrians travelling together towards a shared
destination and engaged in a social relationship [35, 36]. Conversely,
pedestrians who are not part of a group are termed individuals444It is
important to acknowledge that while members of a group are individuals, in the
context of this research, their group affiliation is considered to
significantly influence their navigation and, thus, they are identified by
this aspect of their social dynamics throughout the study.. Additionally, we
use the term interaction, in a specific context, referring exclusively to
social interaction characterised by verbal communication, gestures, gaze,
physical contact, etc., among members of a group. In this respect, we
emphasise that we strictly separate social interaction from collision
avoidance.
Note that while individuals need to allocate a certain portion of their
attentional resources for the inspection of their surroundings and collision
avoidance, they do not need to allocate any resources for coordinating their
motion with a partner or for social interaction. Consequently, we posit that
cognitive load is likely to be lower for individuals, as well as relatively
consistent across different individuals.
On the other hand, groups are likely to experience higher mental workload
compared to individuals, since they need to invest effort in managing group’s
internal dynamics as well as social interaction. Concerning the former, group
members need to maintain a cohesive pace and orientation with their partners
while ensuring group consistency [37, 38]. In addition, if they also carry out
social interaction, they need to ensure its smoothness through practices like
mutual gaze (on a common target or as eye contact), facial expressions, hand
gestures, head/body pose, back-channeling etc. Furthermore, the level of
effort may vary depending on factors such as group size, hierarchy intricacy,
and the level of group members’ engagement in social interaction [39].
Therefore, we cannot simply assume comparable attentional demands for all
groups.
At this point, in order to dissociate the attentional demands of groups from
their size and hierarchy, we choose to focus on two-people groups, known as
dyads. We argue that this approach is not oversimplified, as the majority of
groups in crowds consist of two people [40] and larger groups are shown to
break down into sub-groups of two or three people, making dyads a fundamental
building block of crowds [30, 41]. In addition, by breaking down these dyads
into sub-categories according to their level of involvement in social
interaction [39], we can also address the non-uniformity of attentional
demands and approximate the gradation of mental workload.
Regardless of whether they travel as an individual or as part of a group, to
ensure safe community ambulation, humans need to attentively monitor the
continuously changing environment. To that end, among the five sensory
information channels, the visual one emerges as being the most important.
Generally speaking, humans have high quality perceptual access to the world
[25], although not flawless [42]. In community ambulation, such visual
information is processed online [14] under the influence of visual conspicuity
[43, 25] and task demands [44]. The principal role of visual channel in
locomotion is providing an understanding of location of the self, the goal,
and the environment (e.g. dimensions, terrain features etc.), which are
essential in the control of adaptive locomotion [45] as demonstrated by
studies contrasting open loop obstacle avoidance to full visual sampling [14].
Studies [46, 47] have demonstrated pedestrians’ reliance on visual cues for
efficient path planning around static obstacles. This process entails
utilizing visual information to locate the target destination and assess the
surrounding path area. Additionally, pedestrians evaluate the necessary
magnitude of the deviation to navigate obstacles effectively, ensuring a
seamless progression towards their intended destination.
In addition to static obstacles, pedestrians must also be aware of moving
targets (e.g. other pedestrians, bicycles, cars etc.) and anticipate potential
collisions, which present greater challenges compared to stationary obstacles
due to their momentum and unpredictable motion [48]. In this study, we focus
on human-human collision avoidance and thus consider as moving obstacles the
other pedestrians in the environment.
Previous research on gaze analysis of pedestrians encountering other moving
people has revealed several noteworthy findings. It was observed that gaze
fixations occur at consistent frequencies regardless of increases in
pedestrian density, albeit with a narrower scanning range of the street [49].
This suggests that pedestrians tend to focus more on people directly in front
of them, particularly those in closer proximity. Moreover, pedestrians in the
central and right positions were fixated at greater distances compared to
those on the left, indicating a modulation of gaze behaviour based on the
location and direction of pedestrians in the community environment [50].
Additionally, studies have found that during encounters, pedestrians tend to
focus around the chest of the oncoming pedestrian [51], and initially scanning
various parts of their body, providing evidence that body motion cues serve as
a significant source of visual information during such situations [52, 53].
Hessels et al. also concluded that walkers tend to look at different body
parts based on the behaviour of the other pedestrian, with more attention
directed towards individuals who address or direct themselves towards them,
with the exception of pairs engaged in conversation, who are likely to be
looked at despite not being directed towards the observer [54].
Pedestrians use visual information not only for anticipating others’ path, but
also for estimating certain personal features [55, 56]. Some person-specific
information which can be observed by visual inspection (even without detailed
scrutiny) involve height, gender, and group relation. Uninstructed pedestrians
circumventing around standing people are found to maintain a greater distance
from males than females, from groups than individuals, and from an attractive
person than an unattractive one [55]. In addition, role dependent strategies
in collision avoidance (i.e. who passes first and who gives way) are shown to
depend on gender and height [56]555Knorr et al. failed to observe any
significance on gender and height, possibly due to the lack of cognitive load
and oblique (i.e. not frontal) crossing situation [57].. Other person-specific
information which cannot be directly observed but which can be estimated
visually involve age, personality and mood. Interestingly, it has been shown
that people can estimate such traits as domination, boldness, easygoingness,
happiness, youthfulness with quite high agreement, even by watching only
walkers’ point light displays [58, 59]. To the best of our knowledge, no study
has explored whether pedestrians perceive social interactions within the
groups they encounter. However, given our findings, we believe it is probable
that they do notice such interactions.
In addition, eye gaze is shown to serve not only for collecting information
from one’s surroundings, but also for delivering information about one’s
intended path to others [54, 60]. In particular, Nummenmaa et al. [60]
interestingly shed light on the communicative efficacy of “gaze aversion” and
demonstrated that diverting gaze to one side can effectively signal a planned
path to others, suggesting steering in that direction to avoid collision,
thereby prompting the other pedestrian to maneuver in the opposite direction.
Generally speaking, studies on the Theory of Mind Model (ToMM) suggest that
eye gaze plays a crucial role in understanding others’ intentions [61], and
the absence of gaze alternation can result in a failure to anticipate others’
attentional focus and intentions in static scenarios, referred to as “mind
blindness” [62]. In addition to gaze direction, head orientation, known to
indicate focus of attention [63], serves as a cue for path selection, while
body orientation is associated with a “potential to move” [12]. Yet, such
findings imply that the brain, responsible for orchestrating swift and
successful behavior in such settings, relies not only on spontaneous sensory
inputs but also on internal representations of surrounding pedestrians to
fulfill this role [15].
On the other hand, the auditory channel can be considered as the second most
important sensory channel during community ambulation. Although dual-task
studies suggest that environmental sounds (e.g. construction noise, music
etc.) are likely to be distractive rather than supportive [21], holistically
speaking, the audio channel plays a significant role in shaping the overall
perception of the surroundings. Specifically, task load on the auditory
channel has been shown to interfere with locomotion, leading to lower walking
speed or collisions, and a preference for maintaining a larger personal space
[22, 32]666The concept of personal space can be interpreted as a zone
pedestrians maintain for themselves and others, ensuring a comfortable
distance [64]. More recently, researchers have proposed an alternative
interpretation of personal space as a buffer zone that enables people to
perceive risks and plan trajectory adaptations [32, 65]. Some recent works
advocate for a combined approach, incorporating both a social influence field
and a collision avoidance field [12].
Furthermore, in dual-task scenarios, individuals tend to move at lower speeds,
and the increased workload on the auditory channel may lead to changes in
visual scanning patterns. Specifically, eye gaze is seen to be directed more
frequently towards pedestrians posing a higher risk of collision, and
fixations are observed more commonly on upper body segments, possibly since
the upper body is more informative regarding future walking directions [21,
12].
In addition to securing a certain degree of awareness about the environment,
group members may also need to observe their partners for ensuring group
cohesion and possibly for orchestrating social interactions. This monitoring
process often involves utilizing primarily visual and auditory channels, akin
to a dual-task mobility scenario. Specifically, social interaction in a group
is likely to require visual resources for tracking focus of attention (e.g.
gaze on partner or a mutual gaze on a target) and elicitation of emotions
(e.g. facial expression, gestures) etc. [66]. In addition, audio channel is
essential for comprehension of speech, detection of adjustments on pitch,
loudness, intonation etc. of the interlocutor, as well as turn management
(signaling turn taking, holding, giving or skipping), communicating feedback
(e.g. agreement, surprise etc.) or acknowledgement (i.e. back-channeling).
Note that since humans have limited attentional resources and group members
may already need to devote part of those to the afore-mentioned commitments
towards their partners, they are likely to be left with less resources for
navigation planning than individuals777One may also expect them to require a
larger personal space to make the trajectory adjustments necessary for
collision avoidance [32].. Intuitively, assessing the likelihood of collision
with moving obstacles, such as other pedestrians, is more challenging compared
to navigating around stationary obstacles [31] and requires an anticipation of
others’ trajectory. At this point, it is highly likely that humans consider
other pedestrians around them not simply as obstacles that can move, but as
humans like themselves [61]. Therefore, they possibly also monitor and assess
whether others are engaged in the same visual exploration and this meta-
awareness can play a crucial role in their own planning and, in turn, visual
sampling.
The current study aims to contribute beyond existing research by showcasing
how social interaction affects the ability of pedestrians to focus on the
surrounding environment, with possible effects on efficiency and safety.
Furthermore, it attempts to demonstrate that humans possess awareness not only
of others’ planned paths inferred from subtle actions, but also of the absence
of such indications. Specifically, we believe individuals are capable of
recognizing the lack of these subtle communicative actions or understanding
that they are not directed towards an implication of a planned path (but, for
instance, addressed at a travel companion accompanying their social
interaction). In such cases, they proactively anticipate a limited
contribution to collision avoidance from those pedestrians and take
compensatory measures on their part.
## 3 Methods
### 3.1 Data set
The data set utilised in this study is the DIAMOR data set [28], which was
previously employed in [67, 39] for the purpose of group recognition and
pedestrian dynamics modeling. The recordings were made in an underground
pedestrian street network located in a commercial district of Osaka, Japan.
Specifically, the data set comprises recordings from two straight corridors
within this street network, and our focus centers on one of these. With
several train stations, business centers, and shopping malls accessible from
the recording location, there is a diversity in pedestrian profiles. The
recording area is roughly 200 m2 and allows continuous tracking along
approximately 50 m and the recording spans eight hours in a weekday.
We would like to highlight that this data set is collected from uninstructed
pedestrians in their ecological environment, which we consider a valuable
asset for observing naturalistic behaviour888Experimentation has been reviewed
and approved by ATR ethics board with document number 10-502-1. Posters
explaining that an experiment concerning pedestrian tracking was being held
were present in the environment. The data are publicly available [68] and
contain anonymous trajectories derived from range data [69].. Notably, studies
involving non-human animals have revealed intriguing distinctions and
qualitative differences in behaviour between constrained tasks and
uncontrolled settings [70, 71, 72]. Similarly, in human studies, the
phenomenon of modifying of one’s behaviour in response to the awareness of
being observed has even been given a name, the “Hawthorne effect”999Although
it is now largely agreed that such effect was less significant than originally
thought in the scenario from which it takes its name. [73, 74]. This effect
has been observed in various settings, including when assessing the quality of
care provided by trained practitioners [75], or the energy awareness of
consumers [76]. In the specific context of human locomotion, Farhan et al.
have shown that observed participants exhibit lower variability in gait
parameters [77], and Friesen et al. have demonstrated that locomotion
parameters (e.g. speed, step length) were impacted by the number of
researchers present in the room [78, 77]. In this respect, the ecological data
studied in the upcoming sections are considered to be largely devoid of
experimental or behavioural bias, or subconscious alterations in behaviour, or
at least minimally affected by such factors 101010 Note that with the above
arguments, we by no means intend to assert that the outcomes derived from
traditional, meticulously controlled experimental paradigms are inaccurate or
invalid. These approaches, proficient at dissecting intricate behaviours into
their individual components, have significantly contributed to our
understanding of the fundamental processes that govern behaviour. Nonetheless,
their reductionist approach is likely to constrain their capacity to elucidate
naturalistic behaviour comprehensively, since pristine experiences are more of
an exception than the norm in real-world settings and it is ideal to study
humans in naturalistic settings for ultimately explaining their real-world
behaviour [29]. .
The data include both depth and video information. The depth information is
utilised to derive the trajectories of pedestrians [69], which can be freely
downloaded from [68]. As a result of this tracking process, we obtain the
normalised cumulative density map shown in Fig. 1 (refer also to [20]). The
map is obtained by dividing the recording area into a grid of 10 cm $\times$
10 cm cells and counting the number of pedestrians that have been in each cell
at any point in time. The counts are then normalised by dividing by the
maximum count in the grid. Darker areas indicate higher pedestrian density.
The video data served as the basis for establishing the ground truth regarding
dyads and the intensity of interaction. To assess errors arising from coding
fatigue and coder bias, each relation (belonging to a group and intensity) was
labeled by two different coders. In the first stage of coding, coders observed
walking patterns, age, gender, and clothing etc. to determine which
pedestrians formed a group. In the second stage, focusing solely on
pedestrians labeled as dyads in the first stage, coders assessed the intensity
of interaction. This reduced the amount of data each coder had to view, thus
enhancing coding efficiency.
Coders were asked to label the intensity of interaction for pedestrian groups
using a 0-3 subjective scale. To avoid biasing the assessment, only the
resolution (i.e. the number of interaction levels) was predefined (four
levels), with no guidelines provided on what constituted weak, mild, or strong
interaction intensity. Instead, coders carried out free-viewing, watching
three hours of video footage of groups to intuitively grasp variations in
interaction intensity, before the actual coding task began.
The agreement between coders for group relation labeling was evaluated using
Cohen’s $\kappa$ coefficient, resulting in a high value of $\kappa$ = 0.96,
indicating strong agreement [79]. For interaction intensity labeling, the
reliability was assessed using Krippendorff’s $\alpha$ coefficient, yielding a
value of $\alpha$ = 0.67, which is usually considered sufficiently high [80].
Figure 1: The normalised cumulative density map for the DIAMOR data set. It is
obtained by dividing the recording area into a grid of 10 cm $\times$ 10 cm
cells and counting the number of pedestrians that have been in each cell at
any point in time. The counts are then normalised by dividing by the maximum
count in the grid. Darker areas indicate higher pedestrian density.
### 3.2 Metadata
To further clarify the context of the study, we provide some metadata about
the studied corridor. The density of pedestrians in the recording area is
shown in Fig. 2. We compute it by counting the number of pedestrians in the
recording area during 1 min time windows and dividing it by the area of the
recording area ($40$ m $\times$ $7$ m $=280$ m2). The density of pedestrians
in the recording area varies between 0 and 0.06 pedestrians/m2, with an
average value of 0.04 pedestrians/m2, which can be considered as a low
density.
The probability density function of the velocity of individuals and dyads in
the recording area is shown in Fig. 3. The dyads are categorised based on
their level of interaction, as defined in Sec. 3.1. We observe the effect of
interaction previously reported in [39], namely that strongly interacting
dyads have a lower average velocity compared to weakly interacting dyads. In
addition, we observe that the velocity of non-interacting dyads is comparable
to that of individuals.
(a)
(b)
Figure 2: Density of pedestrians in the DIAMOR data set. (a) Density of
pedestrians in the recording area over time. (b) Histogram of the density of
pedestrians. Density is calculated as the number of pedestrians in the
recording area during a 1 min time window, divided by the area of the
recording area. Figure 3: The probability density function of the velocity of
individuals and dyads.
### 3.3 Data preparation
The raw trajectories in the DIAMOR data set are sampled at a non-uniform rate,
with a frequency varying between 20 and 50 Hz [68] due to holes in the data
caused by occlusions or tracking errors. To ensure that the analysis is not
affected by this, we resample the trajectories at a constant rate $f_{s}$ of
33 Hz with cubic spline interpolation [57].
As mentioned in Sec. 1, the focus of this study lies on understanding the
dynamics of typical individual-dyad collision avoidance in public settings. To
achieve this, we choose to keep only the trajectories that align with typical
walking speeds in public spaces and disregard those that do not. For
establishing the common velocity range of urban pedestrians, we refer to
literature on human locomotion and, basing our reasoning on the findings of
[28], we consider the trajectories with an average velocity falling within the
range of $[0.5,3]$ m/sec to belong to typical urban walking motion and the
rest to be associated with other states (e.g. standing/running or tracking
artifacts).
Following this, to address the impact of sensing noise and natural swaying
resulting from human gait, we applied filtering to the trajectories. Commonly,
low-pass filtering is employed for this purpose [81, 82, 17], and in this
study we opted for a Savitzky-Golay filter [83], which is known to be well-
suited for smoothing noisy data. Specifically, we adjusted the polynomial
order used to fit the samples to 2 and the length of filter window to 3 s.
This decision was guided by the observation that typical gait cycles last
between 1 and 2 seconds [84, 85].
In what follows, we will use the notation $\mathbf{p}(t)$ to denote the
position of a pedestrian at time $t$, and $\mathbf{v}(t)$ to denote the
velocity of the pedestrian at time $t$. We will also use the notation
$\mathbf{p}_{i}(t)$ and $\mathbf{v}_{i}(t)$ to denote the position and
velocity of the individual $i$ at time $t$, and $\mathbf{p}_{d}(t)$ and
$\mathbf{v}_{d}(t)$ to denote the position and velocity of the dyad $d$ at
time $t$111111Note that when this notation is used, we consider the dyad as a
single entity and use the average position and velocity of the dyad members.
In other situations, we will consider the dyad members separately to avoid
biasing the results with an artificial smoothing caused by averaging..
A trajectory $T$ is defined as the sequence of positions $\mathbf{p}(t_{k})$
and velocities $\mathbf{v}(t_{k})$ of a pedestrian, where $t_{k}$ is the time
at which the positions are recorded, with $k\in[0,N-1]$ and $N$ being the
number of samples.
The velocity $\mathbf{v}(t_{k})$ is derived from the positions using a simple
forward Euler difference, i.e.
$\mathbf{v}(t_{k})=\begin{cases}\frac{\mathbf{p}(t_{k+1})-\mathbf{p}(t_{k})}{t_{k+1}-t_{k}}&\text{if
}k<N-1\\\ \mathbf{v}(t_{k-1})&\text{if }k=N-1\end{cases}$ (1)
$T=\left[(\mathbf{p}(t_{0}),\mathbf{v}(t_{0})),(\mathbf{p}(t_{1}),\mathbf{v}(t_{1})),\ldots,(\mathbf{p}(t_{N-1}),\mathbf{v}(t_{N-1}))\right]$
(2)
### 3.4 Intended direction of motion
Our primary assumption in assessing the trajectory deviation is that
pedestrians aim to minimise the distance traveled and will therefore select
the straightest path to reach their destination whenever possible. This
assumption, notably introduced in [86] and adopted by numerous other studies
[21, 87], posits that at the tactical level, where pedestrians make decisions
about their desired area and route, they do so by minimizing a cost function.
This function takes into account factors such as distance traveled, trajectory
comfort, or anticipated encounters with other pedestrians [86]. In the context
of a straight corridor, such as the one examined in our study, a straight line
is reasonably expected to be the optimal route to cross the
corridor121212Assuming that the pedestrian is not aiming to exit the corridor
through a side passage.,131313It is worth noting that if the corridor is
sufficiently wide, the optimal path may still be straight, but not perfectly
aligned with the corridor axis, as pedestrians may cross it diagonally..
To compute the intended straight line trajectory, we first need to identify
the intended direction of motion of the pedestrian. We define it as the line
going through $\mathbf{p}(t_{0})$ and guided by $\mathbf{v_{0}}$, which is the
average velocity vector over a 0.5 s window starting at $t_{0}$. At the
sampling frequency of 33 Hz, this corresponds to $N_{e}=\lfloor 33\times
0.5\rfloor=16$ samples and141414Since the velocity vectors are computed using
a forward Euler difference, the velocity time at $t_{k}$ is a vector pointing
from $\mathbf{p}(t_{k})$ to $\mathbf{p}(t_{k+1})$. Therefore, the average
velocity vector $\mathbf{v_{0}}$ is a vector pointing from $\mathbf{p}(t_{0})$
to $\mathbf{p}(t_{N_{e}})$.
$\mathbf{v_{0}}=\frac{1}{N_{e}}\sum_{k=0}^{N_{e}-1}\mathbf{v}(t_{k}).$ (3)
We believe that $0.5$ s is a reasonable window size for this purpose, as,
since it is approximately the time it takes for a pedestrian to take one step,
it is large enough to capture the general direction of motion, while being
small enough to avoid capturing the effects of the collision avoidance
behavior.
The intended direction of motion $L_{0}$ is therefore formally defined as
$L_{0}=\\{\mathbf{p}(t_{0})+\lambda\mathbf{v_{0}}\mid\lambda\in\mathbb{R}\\}.$
(4)
We argue that this line better represents the intended motion of the
pedestrian than taking the line going through $\mathbf{p}(t_{0})$ and
$\mathbf{p}(t_{N-1})$, as the latter would be influenced by the trajectory
deviations that we aim to quantify, particularly if the pedestrian does not
return to its original intended path after the deviation.
In addition, the straight line trajectory $T_{0}$ is defined as the trajectory
that the pedestrian would follow, should she maintain her desired motion
direction walking at a constant speed. The points of the straight line
trajectory are all on the line $L_{0}$ and verify
$\left\\{\begin{array}[]{@{}l@{}}\mathbf{\tilde{p}}(t_{k})=\mathbf{p}(t_{0})+\mathbf{v_{0}}t_{k}\\\
\mathbf{\tilde{v}}(t_{k})=\mathbf{v_{0}}\end{array}\right.\quad\forall
k\in[0,N-1]$ (5)
We emphasise that the straight line trajectory, composed of a sequence of $N$
discrete positions and velocities, differs from the desired direction of
motion, which represents a line with an infinite number of points.
### 3.5 Situations of interest
Once again, drawing from the terminology introduced in [86], at the
operational level -where pedestrians execute their selected route- they
deviate from the straight line trajectory, possibly influenced by such factors
as gait characteristics [88] and the presence of other pedestrians or
obstacles. In this study, since we aim to quantify and compare trajectory
deviations of individuals and dyads, magnifying on the impact of groups’
social interaction (and the amount of attentional resources available for
navigation planning), we examine two types of situations: (1) encounters,
where a dyad and an individual are on a frontal collision or close-to-
collision course, and (2) undisturbed segments, where neither the individual
nor the dyad encounters any other pedestrian in a reasonably large area around
them, allowing them to move freely without needing to perform avoidance
behaviour. In the following discussion, we begin by providing detailed
insights into the undisturbed case, since it serves as a baseline representing
the deviation in the absence of specific collision avoidance maneuvers.
#### 3.5.1 Undisturbed situations
As explained in Sec. 3.4, we argue that although pedestrians aim to minimise
the traveled distance and will therefore select the straightest path to reach
their destination when possible, it is unrealistic to anticipate them to walk
on a perfectly straight line, even when there are no other pedestrians
present. This meandering [88] can be explained by the natural swaying
resulting from the human gait and the impact of factors such as the cognitive
load [89] and was modeled using a Langevin-like model in [90].
In this respect, we wish to obtain a baseline of straightness for both
individuals and dyads, when they are not forced to perform collision avoidance
behavior. To that end, we define undisturbed segments as the portions of a
trajectory where no other pedestrian is located at less than 4 m away [91,
52], the same window size adopted in the computation of trajectory deviation
during encounters.
However, although the encounters defined in Sec. 3.5.2 are by definition
spatially bounded151515Usually these portions measure 3 to 4 m, depending on
the lateral distance and speed of the two parties., undisturbed segments can
be arbitrarily long161616In less densely populated environments, longer
undisturbed segments can be observed.. To enable comparability of segments
from the two scenarios, we must select undisturbed segments of similar lengths
to those in the spatially constrained encounter cases. To that end, we extract
undisturbed segments of 4 m, ensuring there are no overlaps between
them171717In addition, we require that the direction of motion at the
beginning and end of the segment are aligned with the horizontal axis. This is
ensured by verifying that the absolute value of the angles of the velocity
vector in 0.5 s windows at the beginning and end of the segment, wrapped in
the range $[-\pi,\pi]$ are less than $\frac{\pi}{8}$ or greater than
$\frac{7\pi}{8}$ in more than 90% of all $N_{e}$ time steps in these windows.
This is to ensure that the pedestrian is not turning at the beginning or end
of the segment..
According to the above, one individual or one dyad may have multiple
undisturbed segments (in particular if they are observed for a long time and
if there are few other pedestrians around). Since these segments might not be
independent, given that the pedestrians’ behavior in one segment may be
influenced by the behavior in the previous one, we will consider the average
deviation over all undisturbed segments of an individual or dyad as a single
data point in the analysis. In Tab. 1-(a), we show the number of individuals
and dyads in undisturbed situations and in Tab. 1-(b), we show a breakdown of
the number of undisturbed dyads according to the intensity of interaction of
the dyad.
Table 1: (a) Number of individuals and dyads in undisturbed situations and number of encounters. (b) Breakdown of the number of undisturbed dyads and encounters according to the intensity of interaction of the dyad. (a) | (b)
---|---
| Undisturbed | Encounter
---|---
Individuals | Dyads |
1966 | 457 | 609
| Intensity of interaction | Undisturbed | Encounter
---|---|---
0 | 18 | 45
1 | 60 | 88
2 | 299 | 380
3 | 80 | 96
#### 3.5.2 Encounters
We define encounters as situations where a dyad $d$ and an individual $i$ are
moving towards each other and are on a collision or close-to-collision course.
In such scenarios, it is likely that one or both parties will engage in
collision avoidance behaviour to ensure a comfortable passage.
Although the superposition assumption, commonly employed in many models
(e.g.[19]) suggests that the collective effects on a pedestrian from multiple
neighbors can be linearly combined [17], there is ongoing debate regarding
whether neighborhood is determined by metric or topological distances (e.g.
degree of neighborhood). In this work, we choose to use metric distance, since
the density associated with our data set is not high enough to result in
collective behaviour [92]. In that respect, we consider only those dyads and
individuals, who approach each other sufficiently close. Specifically,
sufficiency is assessed by the condition $\exists t\mid d_{di}(t)\leq 4$m,
where $t$ represents time and $d_{di}$ denotes the instantaneous distance
between the dyad and the individual.
The choice of a 4 m threshold is grounded in prior research on collision
avoidance. Cinelli and Patla found that the “safety zone,” which is the
distance individuals allow a moving object to approach before initiating an
avoidance behaviour, averages around 3.73 m [91]. In addition, Gérin-Lajoie et
al. showed that anticipatory locomotor phase starts with an initial path
deviation which occurs about 4.5 m from the obstacle [32]. Moreover, Kitazawa
et al. demonstrated that pedestrians focus their gaze most intensely on
approaching individuals when they are, on average, approximately 3.97 m away,
seldom directing their attention to pedestrians at greater distances [52].
Among dyad-individual pairs that approach sufficiently close, we solely
consider those engaging in frontal encounters, namely moving in opposite
directions. There are two main reasons for this selective approach. Firstly,
given our focus on collision avoidance inside an environment with
predominantly bi-directional flow, we contend that encounters involving
pedestrians from opposite flows are more pertinent than those within the same
flow. Secondly, since we aim to discern the implications of social interaction
level (and in turn speculate on the allocation of attentional demands), only
those instances where involved parties can visually examine each other
(particularly in which the individual can judge groups relation and assess
dyad’s level of interaction) are considered [57, 56]. In contrast, non-frontal
encounters are omitted, since collision avoidance is less prominent within
low-density bi-directional flow settings, and individuals are not likely to
react to dyads’ characteristics due to limited observation capabilities.
To address this, we compute the predominant relative motion direction of $d$
and $i$ at the beginning of the encounter by calculating the cosine of the
angle between their velocity vectors,
$c_{di}(t)=\frac{\mathbf{v}_{d}(t)\cdot\mathbf{v_{i}}(t)}{||\mathbf{v}_{d}(t)||||\mathbf{v}_{i}(t)||}.$
(6)
We classify an encounter as frontal if, during a 0.5 second window starting at
$t_{0}$, the cosine of the angle between the velocity vectors is smaller than
$-\cos(\frac{\pi}{8})$ for at least $90\%$ of all $N_{e}$ time steps
(indicating an angle range of $[\frac{7\pi}{8},\frac{9\pi}{8}]$). This
condition is formally expressed as
$\frac{1}{N_{e}}\sum_{k=0}^{N_{e}-1}\mathbbm{1}_{\\{c_{di}(t_{k})<-\cos(\frac{\pi}{8})\\}}\geq
0.9,$ (7)
where $\mathbbm{1}$ is the indicator function that is equal to 1 if the
condition inside the brackets is true and 0 otherwise.
For further ensuring anticipatory locomotor adjustments during frontal
encounters, we calculate extrapolated straight line trajectories (see Eq. 5)
at the initial instant of the encounter and require the closest approach
distance on such paths to be less than 2 m181818Note that despite starting the
encounter at a distance of 4 m, it is possible that the dyad and the
individual have sufficient lateral distance to comfortably clear each other,
rendering such encounters irrelevant for the scope of this work..
Specifically, we derive the approach distance using the average velocity of
the dyad and the individual over a 0.5 s window (the same time window used for
determining the relative motion direction, see also Eq. 7), employing a method
akin to computing the impact parameter for examining the interaction of
charged particles.
Finally, we ensure that both the dyad and the individual’s behaviour is
captured even after they have laterally passed each other. This conditioning
is motivated by the findings reported in [93], where the authors noted that,
in cases of very close encounters, lateral distance kept increasing even after
ensuring avoidance. Hence, we necessitate that both the individual and the
dyad clear each other entirely, with adequately long trajectories preceding
and following the clearance. To be precise, we specify that they must maintain
a distance of at least 3 m apart at the beginning and end of the encounter,
demonstrating an initial approach succeeded by subsequent distancing.
After applying the conditions described above, the number of encounters which
will be subject to an analysis in the upcoming sections, turn out to be as
illustrated in Tab. 1-(b).
### 3.6 Measures of deviation
In this section, we provide the definitions of a set of measures for
quantifying the deviation of an actual trajectory from an intended straight
line trajectory (or, equivalently, its dissimilarity to such path) together
with a discussion on their specifications. While most of the measures are
compiled from existing literature on collision avoidance and trajectory
clustering [94, 95], we also introduced several original measures to capture
various aspects of deviation. Additionally, while some measures from the
literature are directly applied to our data, others are adapted to suit our
specific case.
When computing the deviation of a dyad, we consider the deviation of both
members separately. This ensures that the deviation of the dyad is not
artificially reduced by averaging the positions of the two members.
#### 3.6.1 Position based measures
In collision avoidance literature, it is common to assess avoidance behaviour
as deviation from a straight line. In studies with handcrafted experimental
settings, such as frontal encounters in a corridor [96, 97, 98], such straight
line is considered directly as the environment axis and deviation is measured
along the direction orthogonal to that (typically denoted as $y$-axis). In our
case, rather than using the environment (i.e. corridor) axis, we derive an
intended direction of motion for each individual and group (see Eq. 4 and Eq.
5 and $L_{0}$, for instance, in Fig. 7). We compare an observed trajectory to
a straight line along that intended direction of motion in various ways as
explained below.
##### 3.6.1.1 Euclidean deviation $\delta_{E}$
The Euclidean distance $\delta_{E}$ [99] is probably one of the most intuitive
and straightforward measures of deviation. It is computed as the average
distance between the points of the observed trajectory $T$ and the points on
the straight line trajectory $T_{0}$, i.e.
$\delta_{E}=\frac{1}{N}\sum_{k=0}^{N-1}||\mathbf{p}(t_{k})-\mathbf{\tilde{p}}(t_{k})||.$
(8)
In Fig. 4, we illustrate the Euclidean deviation $\delta_{E}$ for some
hypothetical $T$ and $T_{0}$.
Figure 4: Illustration of the Euclidean deviation $\delta_{E}$, which is the
average of the distances between the observed trajectory of the pedestrian $T$
and the straight line trajectory $T_{0}$.
##### 3.6.1.2 Lockstep maximum deviation $\delta_{max}$
The lockstep maximum deviation $\delta_{max}$ is defined as the maximum
distance between simultaneous pairs of points of the trajectory and the
straight line trajectory. While the Euclidean deviation $\delta_{E}$ provides
an average measure of deviation across the entire trajectory, the lockstep
maximum deviation $\delta_{max}$ captures the maximum deviation, at a certain
time, between the observed trajectory $T$ of the pedestrian and her straight
line trajectory $T_{0}$.
Formally,
$\delta_{max}=\max_{k\in[0,N-1]}||\mathbf{p}(t_{k})-\mathbf{\tilde{p}}(t_{k})||.$
(9)
Note that this measure is sometimes called the lockstep Euclidean distance in
the literature [99].
In Fig. 5, we show an example for the computation of lockstep maximum
deviation $\delta_{max}$.
Figure 5: Illustration of the lockstep maximum deviation $\delta_{max}$, which
is the maximum distance between simultaneous pairs of points of the trajectory
$T$ and the straight line trajectory $T_{0}$.
##### 3.6.1.3 Discrete Fréchet deviation $\delta_{F}$
We derive the discrete Fréchet deviation from the standard discrete Fréchet
distance between two trajectories. Originally, Fréchet distance is a measure
of similarity between two continuous curves and is often colloquially referred
to as the “dog leash” distance, as it can be interpreted as the length of the
shortest leash that would allow a dog to walk along one trajectory, while its
owner walks along the other trajectory (without backtracking, but allowing to
stop). The discrete Fréchet distance is its discrete version and assesses the
dissimilarity of two discrete curves [100]191919In the figures, we illustrate
$T$ and $T_{0}$ as piecewise continuous for making them easier to understand.
However, we use only the end points of the line segments in evaluating the
measures.. To that end, an optimal mapping between the points of the two
discrete curves is identified as the one minimizing the maximum distance
between the matched points. Given such mapping, discrete Fréchet deviation
$\delta_{F}$ is the maximum of the distances between each pair of matched
points along $T$ and $T_{0}$.
In Fig. 6, we show the discrete Fréchet distance $\delta_{F}$ for some
hypothetical $T$ and $T_{0}$ and illustrate the two points in space that
correspond to the $\delta_{F}$ of such $T$ and $T_{0}$.
Figure 6: Illustration of the discrete Fréchet deviation $\delta_{F}$, which
is the maximum distance in the optimal mapping (shown in black dashed lines)
between the observed trajectory of the pedestrian $T$ and the straight line
trajectory $T_{0}$.
##### 3.6.1.4 Maximum lateral deviation $d_{max}$
The maximum lateral deviation $d_{max}$ is defined as the maximum distance
between a point on the trajectory of the pedestrian and the intended direction
of motion $L_{0}$ (see Fig. 7).
Formally,
$d_{max}=\max_{k\in[0,N-1]}|d_{k}|$ (10)
where $d_{k}$ is the signed distance between the point $\mathbf{p}(t_{k})$ and
its projection $\mathbf{h}(t_{k})$ on $L_{0}$ computed as
$d_{k}=\left(\left(\mathbf{p}(t_{k})-\mathbf{p}(t_{0})\right)\times\frac{\mathbf{v_{0}}}{||\mathbf{v_{0}||}}\right)_{z}$
(11)
where $\times$ denotes the cross product202020The cross product is defined in
3D space, while we are working in 2D space. However, we can consider the cross
product of two 2D vectors $\mathbf{a}=(a_{x},a_{y})$ and
$\mathbf{b}=(b_{x},b_{y})$ as
$(\mathbf{a}\times\mathbf{b})_{z}=a_{x}b_{y}-a_{y}b_{x}$, i.e. the
$z$-component of the 3D cross product of the two 3D vectors $(a_{x},a_{y},0)$
and $(b_{x},b_{y},0)$. It can also be interpreted as the determinant of the
matrix $\begin{pmatrix}a_{x}&b_{x}\\\ a_{y}&b_{y}\end{pmatrix}$..
The maximum lateral deviation $d_{max}$ and variables involved in computing it
are illustrated in Fig. 7.
We draw the reader’s attention to the fact that the maximum lateral deviation
is not the same as the lockstep maximum deviation $\delta_{max}$, which is the
maximum distance between simultaneous pairs of points of the observed
trajectory and the straight line trajectory. Although it is slightly more fine
grained, as it does not require the discretization of the intended direction
of motion, it also does not take into consideration velocity (magnitude)
information.
Figure 7: Illustration of the maximum lateral deviation $d_{max}$. The
intended direction of motion $L_{0}$ is shown with a solid line. The maximum
lateral deviation $d_{max}$ is the maximum distance between a point on $T$ and
$L_{0}$.
##### 3.6.1.5 Integral of lateral deviation $\Delta$
The integral of lateral deviation is defined as the integral of the distance
between a pedestrian’s observed trajectory $T$ and the intended direction of
motion $L_{0}$. Using the definitions of $d_{k}$ and $\mathbf{h}(t_{k})$ given
above, it can be approximated with the trapezoidal rule as
$\Delta=\sum_{k=0}^{N-2}||\mathbf{h}(t_{k+1})-\mathbf{h}(t_{k})||\frac{d_{k}+d_{k+1}}{2}.$
(12)
One notable difference of the integral of lateral deviation with the maximum
lateral deviation is that deviation on both sides of the intended direction of
motion will tend to cancel each other out (since $d_{k}$ is signed, see also
Eq. 11). This measure is therefore more sensitive to the overall deviation of
the trajectory of the pedestrian from the intended direction of motion (over
the entire course), rather than to the maximum deviation (at a single point in
space).
In Fig. 8, we show an example for the computation of integral of lateral
deviation $\Delta$, which corresponds to area of the shaded region.
Figure 8: Illustration of the integral of lateral deviation $\Delta$. The
intended direction of motion $L_{0}$ is shown with a solid line. The integral
of lateral deviation $\Delta$ is the sum of the areas of the shaded region.
##### 3.6.1.6 Dynamic time warping deviation $\delta_{DTW}$
The dynamic time warping (DTW) deviation is computed using the time warped
distance between a pedestrian’s observed trajectory $T$ and her straight line
trajectory $T_{0}$. Notably, it allows for non-linear alignment of points
[101], where each point on $T$ is matched to a point on $T_{0}$, ensuring that
the mapping is monotonically increasing in time (i.e. not going back in time).
Additionally, the first (resp. last) point of one trajectory has to be matched
to the first (resp. last) point of the other one. The cost of such mapping is
computed as the sum of the distances between the matched points, and the DTW
deviation $\delta_{DTW}$ is simply the cost of the optimal mapping and can
efficiently be computed using dynamic programming.
In Fig. 9, we show a hypothetical optimal mapping between a pedestrian’s
trajectory $T$ and her straight line trajectory $T_{0}$. For this example,
$\delta_{DTW}$ is the sum of the lengths of the dashed lines.
Figure 9: Illustration of the dynamic time warping deviation $\delta_{DTW}$.
The optimal mapping between the trajectory of the pedestrian and the straight
line trajectory is shown with black dashed lines. The dynamic time warping
deviation $\delta_{DTW}$ is the sum of the distances between the matched
points.
##### 3.6.1.7 Longest common subsequence deviation $\delta_{LCSS}$
Originally, longest common subsequence is defined for assessing the similarity
of two sequences of symbols (e.g. text strings). Specifically, it is computed
as the length of the longest subsequence that is common to these sequences,
where a subsequence can be derived from a sequence by deleting some elements
without changing the order of the remaining ones.
In the context of trajectories, the longest common subsequence [102]
concerning two trajectories $T,T^{\prime}$ denoted as $l_{LCSS}(T,T^{\prime})$
is defined as the length of the longest subsequence of points that are
considered close enough, i.e. the distance between the points is less than a
given threshold $\epsilon$.
The choice of threshold is crucial. Namely, a threshold that is too small
would make the measure overly sensitive to any residual noise that is not
filtered out during preprocessing, while a threshold that is too large would
result in considering all points as close enough, rendering
$l_{LCSS}(T,T^{\prime}))$ equal to the length of the trajectories, which would
not be informative. Given that deviations of pedestrians are usually of the
order of a few tens of centimeters, we consider that two points are close
enough, if the distance between them is less than 5 cm. This value was
empirically selected to be small enough to capture the deviations of
pedestrians, while being large enough to remain robust to noise (a position
accuracy of 34 mm was reported for the tracking system used in this work in
[67]).
Finally, we turn this measure of similarity into a distance with the
normalization proposed in [99], and apply it on a pedestrian’s observed
trajectory $T$ and her corresponding straight line trajectory $T_{0}$,
$\delta_{LCSS}(T,T_{0})=1-\frac{l_{LCSS}(T,T_{0})}{N}.$ (13)
In Fig. 10, we illustrate the computation of the longest common subsequence
deviation. The circles of radius $\epsilon$ represent the threshold under
which two points are considered close enough. Points matched when computing
the longest common subsequence are shown with dashed lines.
Figure 10: Illustration of the longest common subsequence deviation
$\delta_{LCSS}$. The circles of radius $\epsilon$ represent the threshold on
vicinity. Points connected with black dashed lines are considered close
enough.
##### 3.6.1.8 Levenshtein deviation $\delta_{Lev}$
Levenshtein distance [103] $d_{Lev}$ (also referred to as edit distance on
real sequence, or simply as edit distance, even though the latter actually
refers to a family of distance metrics) is a measure of similarity between two
trajectories. It is defined as the minimum number of edit operations that
transforms one trajectory into the other, with possible edit operations being
insertions or deletions of points. Similar to the longest common subsequence
distance, we consider that two points can be mapped, provided that the
distance between them is less than $\epsilon=5$ cm.
Levenshtein distance and LCSS are closely related and in some situations
complementary (when no points is mapped to more than one point,
$N=l_{LCSS}+d_{Lev}$, as the number of deletion/insertion is the number of
points that are not matched by the longest common subsequence).
The Levenshtein deviation $\delta_{Lev}$ is defined as the Levenshtein
distance between a pedestrian’s observed trajectory $T_{0}$ and her straight
line trajectory, i.e. $\delta_{Lev}=d_{Lev}(T,T_{0})$.
##### 3.6.1.9 Deviation index $\tilde{\tau}$
The straightness index [104] of a discrete trajectory is defined as the ratio
of the net displacement to the gross displacement (thus, also referred to as
the net-to-gross displacement ratio) [105], where the net displacement $D$, is
simply
$D=\left\|\mathbf{p}(t_{N-1})-\mathbf{p}(t_{0})\right\|,$ (14)
and the gross displacement $L$ is the sum of the distances between consecutive
positions:
$L=\sum_{k=1}^{N-1}\left\|\mathbf{p}(t_{k})-\mathbf{p}(t_{k-1})\right\|.$ (15)
The straightness index is then defined as
$\tau=\frac{D}{L}.$ (16)
From Eq. 14 and Eq. 15, one can see that $\tau$ takes values between 0 and 1.
In particular, for an infinitely tortuous trajectory ($D$ fixed, $L\to\infty$)
or a closed curve ($D=0$, $L$ fixed), $\tau$ assumes the value of 0, whereas
for a perfectly straight trajectory (i.e. $D=L$), it leads to the value of 1.
In order to be consistent with other measures of deviation, i.e. to have a
lower value for straighter trajectories, we define the deviation index
$\tilde{\tau}$ as $1-\tau$.
Fig. 11 illustrates the computation of the deviation index. The net
displacement $D$ is shown in orange and all the distances between consecutive
positions are shown in green. The gross displacement $L$ is the sum of these
distances.
Figure 11: Illustration of deviation index $\tilde{\tau}$. The net
displacement $D$ is shown in orange and the gross displacement $L$ is the sum
of the distances between consecutive positions show in green. The deviation
index is $1-\frac{D}{L}$.
#### 3.6.2 Orientation based measures
In addition to the position based measures, we introduce a set of measures
that quantify the deviation of the trajectory of a pedestrian from a straight
line trajectory based on the orientation of the velocity vectors, i.e. the
direction of motion.
##### 3.6.2.1 Maximum cumulative turning angle $\theta_{max}$
To deviate from an intended trajectory, pedestrians naturally have to turn. We
can therefore quantify the deviation of the trajectory by looking at the
amount of turning performed. The cumulative turning angle until time $t_{k}$
is defined as the sum of the turning angles between consecutive velocity
vectors until time $t_{k}$212121Note that is not the same as computing the
angle between $\mathbf{v}(t_{0})$ and $\mathbf{v}(t_{k})$, since it is
theoretically possible to obtain an angle larger than $2\pi$ if the pedestrian
makes a full turn. Nonetheless, this situation should not occur in practice..
Formally, it is defined as
$\theta_{k}=\sum_{j=0}^{k-1}d\theta_{j},$ (17)
where $d\theta_{j}$ is the signed angle between the velocity vectors
$\mathbf{v}(t_{j})$ and $\mathbf{v}(t_{j+1})$,
$d\theta_{j}=\angle(\mathbf{v}(t_{j}),\mathbf{v}(t_{j+1}))$.
The turning angles being signed, the cumulative turning angle can be positive
or negative, depending on the direction of the turning (see Fig. 12). We are
interested in the maximum cumulative turning angle $\theta_{max}$, which is
the maximum of the absolute value of the cumulative turning angles over the
trajectory.
$\theta_{max}=\max_{k\in[0,N-1]}|\theta_{k}|$ (18)
In Fig. 12, we illustrate the computation of the maximum cumulative turning
angle. The left part of the figure shows the turning angles $d\theta_{j}$ and
the segment of the trajectory where the cumulative turning angle is maximum.
In the right part of the figure, we show a graph of the turning angles
$d\theta_{j}$, the cumulative turning angles $\theta_{k}$, $|\theta_{k}|$ and
the maximum cumulative turning angle $\theta_{max}$.
We note that, in contrast with the position based measures, the maximum
cumulative turning angle is an early indicator of the deviation of the
trajectory. Typically, this maximum value is not reached at the same position,
for instance, as the maximum lateral deviation. Specifically, by the time the
pedestrian has begun turning back towards their intended direction of motion,
the cumulative turning angle starts to decrease.
(a)
(b)
Figure 12: Illustration of the variables used in computing maximum cumulative
turning angle. (a) The turning angles $d\theta_{j}$ are shown in red. The
segment of the trajectory for which the cumulative turning angle is maximum is
drawn in purple. (b) Corresponding values of turning angles $d\theta_{j}$,
cumulative turning angles $\theta_{k}$, their absolute values $|\theta_{k}|$
and the maximum cumulative turning angle $\theta_{max}$.
##### 3.6.2.2 Average cumulative turning angle $\Theta$
In the same way that we defined the Euclidean deviation as the average of the
deviations, we can define the average cumulative turning angle $\Theta$ as the
average of the cumulative turning angles over the trajectory. Formally,
$\Theta=\frac{1}{N}\sum_{k=0}^{N-1}|\theta_{k}|.$ (19)
##### 3.6.2.3 Sinuosity $S$
Sinuosity was introduced in [106] as a way to quantify the randomness of an
animal’s path (especially in the case of foraging patterns, for instance when
studying how ants locate and collect food). It is a measure proportional to
the standard deviation of the distribution of the turning angles for a given
trajectory and is formally defined as
$S=1.18\frac{\sigma_{q}}{\sqrt{q}},$ (20)
where $q$ is the step size of the trajectory. According to the definition of
[106], the step size has to be constant across the trajectory. Thus, $S$
requires the trajectory $T$ to be re-discretised into another trajectory
$\tilde{T}$ with equal spacing between all the points. $\sigma_{q}$ is the
(circular) standard deviation of the turning angles $\theta$. The value $1.18$
has been numerically computed by the authors of [106] using simulations of
correlated random walks.
This measure is based on the assumption that the turning angles are drawn from
a normal distribution wrapped on a circle (with some correlation between
consecutive draws, to account for the forward tendency of the locomotion).
In Fig. 13, we illustrate the re-discretization of the trajectory and the
computation of the turning angles.
Figure 13: Illustration of sinuosity $S$. The trajectory is re-discretised to
obtain a trajectory $\tilde{T}$ with equal spacing $q$ between all the points.
The turning angles are then computed.
##### 3.6.2.4 Energy curvature $E_{\kappa}$
In two dimensions, curvature is a measure of how much a curve deviates from a
straight line. It describes the rate of change of the tangent angle of the
curve and can be computed as
$\kappa(t_{k})=\frac{||\mathbf{v}(t_{k})\times\mathbf{a}(t_{k})||}{||\mathbf{v}(t_{k})||^{3}},$
(21)
where $\mathbf{v}(t_{k})$ and $\mathbf{a}(t_{k})$ are the velocity and
acceleration vectors of the trajectory at times $t_{k}$ and
$\mathbf{a}(t_{k})$ is derived in a similar way to $\mathbf{v}(t_{k})$ as in
Eq. 1.
Another way to understand curvature is to consider the circle that best
approximates the curve at each point, i.e. the osculating circle. The
curvature at a point is then defined as the reciprocal of the radius of the
osculating circle.
In Fig. 14, we illustrate the computation of curvature. The color of the point
on the trajectory corresponds to the signed curvature at that point. We also
show the osculating circle at a point of the trajectory.
Figure 14: Illustration of curvature $\kappa$. The color of a point on the
trajectory correspond to the signed curvature at that point. The osculating
circle at a point of the trajectory is shown (the curvature at that point is
the reciprocal of the radius of the circle).
The energy curvature is defined as the integral of the square of the curvature
over the trajectory. The integration is performed using the trapezoidal rule,
and the energy curvature is given by
$E_{\kappa}=\sum_{k=0}^{N-2}(t_{k+1}-t_{k})\frac{\kappa(t_{k})^{2}+\kappa(t_{k+1})^{2}}{2}.$
(22)
In path planning, minimising the energy curvature is often considered as a way
to ensure smooth trajectories [107].
#### 3.6.3 Mixed measures
Some works have proposed measures that combine position and orientation
information to quantify the deviation of a trajectory. In this section, we
present two such measures: turn intensity and suddenness of turn.
##### 3.6.3.1 Turn intensity $i$
The turn intensity was introduced in [108]. In the authors’ experiment,
participants were moving and crossing each other along a straight elongated
path ($x$-axis) and they consider instants at which the motion along the
orthogonal axis ($y$-axis) changes direction, which they refer to as “turning”
instants. A “step” is defined as the motion between two consecutive turning
instants. The “step length” is defined as the $y$ component of a step and the
“step angle” is defined as the absolute value of the angle deviation of a step
from the horizontal axis. The turn intensity is then defined as the product of
the step length and step angle.
In our case, since pedestrians might not be moving along the environment axis
(i.e. $x$ axis, as detailed in Sec. 3.4), we adapt the definitions of the
turning instants and steps to consider the deviation from the intended
direction of motion. In particular, we consider the signed angle $\psi_{k}$
between the initial velocity vector $\mathbf{v_{0}}$ and the velocity vector
$\mathbf{v}(t_{k})$ at each time step $t_{k}$,
$\psi_{k}=\angle(\mathbf{v_{0}},\mathbf{v}(t_{k}))$. The turning instants
$t_{s}$ are then defined as the instants at which $\psi_{k}$ changes sign.
We then consider the $s$-th step angle $\omega_{s}$ to be the absolute value
of the angle deviation of a step from the intended direction of motion and the
step length $\lambda_{s}$ to be the orthogonal distance between the position
of the pedestrian at the end of the step and the intended direction of motion.
$\omega_{s}=\Big{|}\angle(\mathbf{v_{0}},(\mathbf{p_{s+1}}-\mathbf{p_{s}}))\Big{|},$
(23)
$\lambda_{s}=\frac{||(\mathbf{p_{s+1}}-\mathbf{p_{s}})\times\mathbf{v_{0}}||}{||\mathbf{v_{0}}||}.$
(24)
The turn intensity is then defined as average value of the product of the step
lengths and step angles.
$i=\frac{\sum_{s=0}^{N_{S}-1}\omega_{s}\lambda_{s}}{N_{S}},$ (25)
where $N_{S}$ is the number of steps.
In Fig. 15, we illustrate the variables used in computing turn intensity $i$.
Figure 15: Illustration of the variables used in computing turn intensity $i$.
The steps are shown in orange, step angles $\omega$ are shown in green and
step lengths $\lambda$ are shown in purple.
##### 3.6.3.2 Suddenness of turn $\sigma$
The suddenness of turn was introduced in [108]. It is defined as the absolute
value of the product of the angle between the velocity vector at a given time
$t$ and the horizontal axis (since the participants were moving along the
$x$-axis in the experiments of [108]) and the change in speed, i.e. the
difference between the speed at time $t$ and the speed at time $t-dt$, $dt$
being the sampling period.
Similar to the previous measures, we adapt this definition to consider the
intended direction of motion rather than the environment (i.e. $x$) axis. In
particular, we use the angles $\psi_{k}$ defined in the previous section and
the suddenness of turn $\sigma$ is defined as
$\sigma=\frac{1}{N-2}\sum_{k=1}^{N-2}\left|\psi_{k}(||\mathbf{v}(t_{k})||-||\mathbf{v}(t_{k-1})||)\right|.$
(26)
### 3.7 Toy trajectories
To demonstrate how various deviation measures evaluate different trajectory
shapes, we examine a set of 7 artificially generated toy trajectories (see
Fig. 16). These trajectories, designed to be comparable with the actual
undisturbed and encounter paths studied in this work, were generated by
defining handcrafted waypoints and interpolating between them using a cubic
spline, such that a constant velocity of 1.2 m/s is maintained at a sampling
frequency of 33 Hz.
a. Straight line:
perfectly straight line.
b. Small deviation without recovery:
the trajectory starts straight, then deviates to the left (up to 30 cm) and
does not recover (i.e. the trajectory does not return to the straight line).
c. Small deviation with recovery:
the trajectory starts straight, then deviates to the left (up to 30 cm) and
recovers (i.e. the trajectory returns to the straight line).
d. Big deviation without recovery:
the trajectory starts straight, then deviates to the left (up to 60 cm) and
does not recover.
e. Big deviation with recovery:
the trajectory starts straight, then deviates to the left (up to 60 cm) and
recovers.
f. Fast deviation with recovery:
the trajectory starts straight, then deviates to the left (up to 30 cm) and
recovers, but the deviation is performed over a shorter distance than the
previous cases.
g. Deviation on both sides:
the trajectory starts straight, then deviates to the left (up to 30 cm), then
to the right (up to 60 cm) and recovers.
Figure 16: Toy trajectories, representing different types and levels of
deviation.
We calculate the value of deviation measures outlined in Sec. 3.6 for each of
these toy trajectories and present the results in Tab. 2. As anticipated, we
observe that all deviation measures yield a 0 or very small value for a,
identifying it as the one with least deviation222222While this information may
seem trivial, it serves as a sort of sanity check. To facilitate the
comparison of deviation values, in each column of Tab. 2 we underline the
second smallest value and highlight the highest value in bold. Additionally,
the rank of each value in a column is provided in parentheses on its right.
Just by observing the underlined and bold values, significant discrepancies
between different deviation measures become apparent. For example, “fast
deviation with recovery” f is identified to have very small deviation by
measures such as $\delta_{F}$ and $d_{max}$, but it is identified as having
the largest deviation by measures like $S$, $E_{\kappa}$, and $\sigma$. Upon
closer examination of the ranks and comparison across different columns, we
notice that trajectories other than a can be evaluated quite differently by
various deviation measures.
In particular, three trajectories (d, f, and g) are identified as having the
largest deviation by our deviation measures. Since distance-based measures are
primarily influenced by the distance from the straight-line trajectory and the
number of points significantly deviating from it, they consequently rank “big
deviation without recovery” d and “deviation on both sides” g as the most
deviated. Furthermore, suddenness of turn $\sigma$ is highest for “fast
deviation with recovery” f. In addition, since angular measures are mainly
affected by the number of turning points and the magnitude of the turning
angles, they rank “deviation on both sides” g as having the largest deviation.
Moreover, deviation index $\tilde{\tau}$ also ranks it as the most deviated
due to its highest gross displacement among all toy trajectories.
By examining the second smallest (underlined) value in each column, we notice
that position-based measures (with the exception of $\delta_{E}$, albeit with
a small difference between second and third places) rank “fast deviation with
recovery” f as the second least deviated. Since these measures are primarily
influenced by the distance to the straight line trajectory and the number of
points deviating, they evaluate f to be relatively straight, as it deviates
over the shortest distance. In contrast, position based measure
$\tilde{\tau}$, orientation based measures such as $\theta_{max}$,
$E_{\kappa}$, $S$, and mixed measures $i$ and $\sigma$ rank “small deviation
without recovery” b as the second least deviated. This is consistent with the
fact that b has the smallest gross displacement and undergoes a slower
deviation.
We highlight the fact that “fast deviation with recovery” f is ranked as both
the most deviated and the least deviated (ignoring the straight line) by
different measures. This shows that the measures capture different aspects of
the deviation, and no single measure is expected to fully capture the
complexity of deviations of real (human) trajectories. In the specific
scenarios investigated in this study, we nevertheless anticipate the
deviations not to be excessively abrupt (i.e. with low acceleration and jerk)
due to the density and geometry of the environment.
Table 2: Measures of deviation for the toy trajectories. The maximum value for
each measure is highlighted in bold and the second smallest value is
underlined (since the straight line trajectory is the smallest value for all
the measures). The ranking of the trajectories (from the straightest to the
most deviated) is also indicated in parenthesis.
Measures $\delta_{E}$ $\delta_{max}$ $\delta_{F}$ $d_{max}$ $\Delta$
$\delta_{DTW}$ $\delta_{LCSS}$ $\delta_{Lev}$ $\tilde{\tau}$ $\theta_{max}$
$\Theta$ $S$ $E_{\kappa}$ $i$ $\sigma$ a $7.61\times 10^{-17}$ (1) $4.55\times
10^{-16}$ (1) $4.55\times 10^{-16}$ (1) $0$ (1) $0$ (1) $7.61\times 10^{-17}$
(1) $8.93\times 10^{-3}$ (1) $0$ (1) $0$ (1) $0$ (1) $0$ (1) $0$ (1) $0$ (1)
$0$ (1) $0$ (1) b $1.65\times 10^{-1}$ (4) $3.37\times 10^{-1}$ (4)
$3.36\times 10^{-1}$ (4) $3.36\times 10^{-1}$ (4) $6.50\times 10^{-1}$ (5)
$1.65\times 10^{-1}$ (4) $7.14\times 10^{-1}$ (4) $7.90\times 10$ (4)
$\underline{1.76\times 10^{-3}}$ (2) $\underline{1.97\times 10^{-1}}$ (2)
$\underline{1.32\times 10^{-1}}$ (2) $\underline{1.32\times 10^{-1}}$ (2)
$\underline{4.21\times 10^{-5}}$ (2) $\underline{6.58\times 10}$ (2)
$\underline{7.31\times 10^{-2}}$ (2) c $\underline{1.35\times 10^{-1}}$ (2)
$\underline{3.18\times 10^{-1}}$ (2) $3.16\times 10^{-1}$ (3) $3.16\times
10^{-1}$ (3) $4.85\times 10^{-1}$ (4) $1.24\times 10^{-1}$ (3) $5.75\times
10^{-1}$ (3) $6.60\times 10$ (3) $1.89\times 10^{-2}$ (4) $3.77\times 10^{-1}$
(4) $1.74\times 10^{-1}$ (5) $5.18\times 10^{-1}$ (4) $6.48\times 10^{-4}$ (4)
$1.20\times 10^{2}$ (3) $2.22$ (4) d $2.98\times 10^{-1}$ (5) $6.38\times
10^{-1}$ (5) $\mathbf{6.34\times 10^{-1}}$ (7) $\mathbf{6.34\times 10^{-1}}$
(7) $\mathbf{1.17}$ (7) $\mathbf{2.98\times 10^{-1}}$ (7) $7.35\times 10^{-1}$
(5) $8.20\times 10$ (5) $7.10\times 10^{-3}$ (3) $2.62\times 10^{-1}$ (3)
$1.44\times 10^{-1}$ (3) $2.56\times 10^{-1}$ (3) $1.62\times 10^{-4}$ (3)
$1.81\times 10^{2}$ (5) $5.83\times 10^{-1}$ (3) e $3.10\times 10^{-1}$ (6)
$\mathbf{6.45\times 10^{-1}}$ (7) $6.28\times 10^{-1}$ (6) $6.28\times
10^{-1}$ (6) $9.41\times 10^{-1}$ (6) $2.55\times 10^{-1}$ (6) $7.48\times
10^{-1}$ (6) $9.50\times 10$ (6) $6.81\times 10^{-2}$ (6) $6.93\times 10^{-1}$
(5) $3.28\times 10^{-1}$ (6) $9.41\times 10^{-1}$ (5) $2.36\times 10^{-3}$ (5)
$4.60\times 10^{2}$ (6) $1.40\times 10$ (5) f $1.37\times 10^{-1}$ (3)
$3.23\times 10^{-1}$ (3) $\underline{3.00\times 10^{-1}}$ (2)
$\underline{3.00\times 10^{-1}}$ (2) $\underline{1.00\times 10^{-1}}$ (2)
$\underline{4.76\times 10^{-2}}$ (2) $\underline{1.78\times 10^{-1}}$ (2)
$\underline{2.60\times 10}$ (2) $5.91\times 10^{-2}$ (5) $9.11\times 10^{-1}$
(6) $1.64\times 10^{-1}$ (4) $\mathbf{2.84}$ (7) $\mathbf{2.71\times 10^{-2}}$
(7) $1.70\times 10^{2}$ (4) $\mathbf{6.09\times 10}$ (7) g $\mathbf{3.39\times
10^{-1}}$ (7) $6.39\times 10^{-1}$ (6) $5.43\times 10^{-1}$ (5) $5.43\times
10^{-1}$ (5) $2.04\times 10^{-1}$ (3) $2.39\times 10^{-1}$ (5)
$\mathbf{7.70\times 10^{-1}}$ (7) $\mathbf{1.04\times 10^{2}}$ (7)
$\mathbf{1.27\times 10^{-1}}$ (7) $\mathbf{9.97\times 10^{-1}}$ (7)
$\mathbf{4.40\times 10^{-1}}$ (7) $1.57$ (6) $7.32\times 10^{-3}$ (6)
$\mathbf{7.98\times 10^{2}}$ (7) $4.20\times 10$ (6)
### 3.8 Impact parameter
In this section, we briefly introduce the concept of the impact parameter,
which we previously used in [20].
In the scattering of particles in physics, the impact parameter is the
distance between the path of an incoming particle and the target particle. To
apply this concept to the study of pedestrian trajectories, we treat the
deviation of an individual from a dyad as a scattering event. We start by
transforming the trajectories of the dyad $d$ and the individual $i$ into a
reference frame that moves with the dyad. Specifically, at each time instant
the positions of the dyad and the individual are translated so that the dyad’s
center of mass is positioned at the origin, and their velocities are rotated
such that the dyad’s velocity is aligned with the positive x-axis232323In
physical scattering theory there is no need to build a special frame, since
only relative positions and velocities are used. Nevertheless in our
computation, since we use a box (see Fig. 17) which is not rotation invariant,
the definition of frame became important. The reason we use a box is that our
environment has walls and preferred directions.. In this reference frame, the
dyad remains fixed at the origin, while the individual moves towards it,
analogous to a particle approaching a target in a scattering event.
We denote the position of the individual in the reference frame as
$\mathbf{\hat{p}}_{i}$ and its velocity as $\mathbf{\hat{v}}_{i}$.
In this reference frame, the impact parameter $r_{b}$ is computed as the
distance from the dyad (positioned at the origin) to the line guided by the
individual’s velocity vectors at the beginning of the encounter (as in Sec.
3.4, we average the velocity vectors over $N_{e}$ time instants to alleviate
the impact of orientation noise).It is a measure of how close the individual
and the dyad would have passed each other if there were no collision
avoidance. We illustrate the computation of the impact parameter in Fig. 17.
Let $\mathbf{\hat{v}}_{i_{0}}$ be the average velocity vector of the
individual over the first $N_{e}$ time instants of the encounter,
$\mathbf{\hat{v}}_{i_{0}}=\frac{1}{N_{e}}\sum_{k=0}^{N_{e}-1}\mathbf{\hat{v}}_{i}(t_{k}),$
(27)
where $N_{e}$ is the same as in the computation of the desired direction of
motion (see Sec. 3.4).
We can compute $r_{b}$ as
$r_{b}=\frac{||\mathbf{\hat{v}}_{i_{0}}\times\mathbf{\hat{p}}_{i}(t_{0})||}{||\mathbf{\hat{v}}_{i_{0}}||}.$
(28)
As detailed in [20], we scale the impact parameter by the width of the dyad
(i.e. the average distance between the members of dyads with that level of
interaction) to obtain a dimensionless measure $\bar{r}_{b}$ that better
captures the relative distance between the individual and the member of the
dyad. In particular, a value of $\bar{r}_{b}$ smaller than $0.5$ indicates
that the individual would have passed through the dyad.
Figure 17: Illustration of the impact parameter. The individual’s trajectory
is transformed into a reference frame that moves with the dyad. The impact
parameter $r_{b}$ is the distance from the dyad to the line guided by the
individual’s velocity vectors at the beginning of the encounter.
## 4 Results
### 4.1 Undisturbed situations
We begin by comparing the deviation of individuals and dyads in undisturbed
situations, which we argue to constitute a baseline for the amount of
deviation that can be expected in the absence of any encounter. As noted in
Sec. 3.5.1, undisturbed situations occur when individuals or dyads are walking
without any other pedestrian within a distance of at least 4 m along or
perpendicular to their direction of motion.
In Tab. 3 $\sim$ Tab. 17, we present the average deviation for dyads
categorised by levels of interaction ranging from 0 to 3, all dyads, and
individuals, with respect to each measure introduced in Sec. 3.6.
For convenient comparison with individuals, we also display the ratio of the
average deviation of the dyads to the average deviation of the individuals.
Additionally, we present the Kruskal-Wallis test $p$-value for the difference
of means between dyads with various level of interaction, and the Welch T-test
$p$-value for the difference of means between individuals and all dyads [109,
110].
The initial noteworthy finding is that, across the vast majority of measures,
deviation is significantly higher for dyads (averaged across all levels of
interaction) compared to individuals. However, to accurately interpret this
observation, it is essential to delve into the allocation of attentional
resources during social interaction, achieved by contrasting the deviations of
dyads with varying levels of interaction. Examining the breakdown of
normalised deviation values according to the level of interaction of the dyad,
we observe a trend where the level of interaction correlates with an increase
in deviation. Interestingly, the deviations of non-interacting dyads appear to
be smaller to those of individuals, while interacting dyads which exhibit
comparable or higher deviations.
### 4.2 Encounters
We now turn to the encounter situations, which correspond to the situations in
which dyads and individuals pass each other frontally at a distance less than
4 m [32, 91, 52]. We first contrast deviations during encounters with
undisturbed situations. Subsequently, we conduct a closer examination of
encounters, comparing deviations of individuals and dyads during these cases.
Finally, we provide insights into the effect of dyads’ level of interaction on
both their own deviation and the deviation of the individuals involved.
#### 4.2.1 Deviations during encounters
In Tab. 18 $\sim$ Tab. 32 we show the amount of deviation of individuals and
dyads for each measure introduced in Sec. 3.6. It is crucial to emphasise that
when reporting deviations related to encounter situations, we categorise not
only the values of dyads but also those of individuals, with respect to the
level of social interaction of the dyad involved in the encounter.
We notice that individuals have a tendency to deviate more when encountering
dyads with high or medium interaction levels (i.e. levels 3, 2) compared to
when they encounter non- or weakly interacting ones (i.e. levels 0, 1). In
particular, the deviation when encountering a strongly interacting dyad (level
3) is almost always242424Except for $\Delta$, $\delta_{LCSS}$ an $\sigma$.
larger than for other levels of interaction. However, upon subjecting the data
to a Kruskal-Wallis test, we see that the differences in deviation for the
individual when encountering dyads with varying levels of interaction are not
statistically significant for most metrics (all except energy curvature
$E_{\kappa}$ and sinuosity $S$). Conversely, the variation in dyad deviation
is statistically significant but does not exhibit a clear pattern, with dyads
of interaction levels 0 and 3 deviating more than those with interaction
levels of 1 and 2. Finally, averaging across all interaction levels,
individuals deviate more than dyads in a statistically significant manner for
all measures.
#### 4.2.2 Comparison of deviations during encounters and undisturbed
situations
The results from Tab. 18 $\sim$ Tab. 32 can be better understood when
contrasted with undisturbed situations. To facilitate this comparison, Tab. 33
$\sim$ Tab. 47 provide the ratio of average deviations during encounters to
the corresponding values observed in undisturbed scenarios.
One notable observation regarding individuals is the consistent and steady
increase in deviation during encounters across the majority of measures (see
ratios of individuals in Tab. 33 $\sim$ Tab. 47). Specifically, the ratios are
consistently greater than 1 for all measures except suddenness of turn
$\sigma$ and energy curvature $E_{\kappa}$, which we believe to be caused by
the noise introduced by further differentiation when deriving the
accelerations.
A similar increase is noted for dyads with 0 interaction levels, exhibiting
generally large ration ($>1.5$), whereas higher interaction levels show a
ratio close to 1, suggesting minimal change. Furthermore, we observe a
decrease in the ratios with increasing interaction level, with statistically
significant differences between various interaction levels. Additionally, a
statistically significant difference is evident between the ratios of
individuals and dyads after averaging across all interaction levels (see
T-test $p$-values in the bottom line of Tab. 33 $\sim$ Tab. 47)
These findings are further clarified by Tab. 48 $\sim$ Tab. 62, which provide
$p$-values for the difference between undisturbed situations and encounters
for both individuals and dyads across all interaction levels. Upon averaging
over interaction levels, both individuals and dyads exhibit a statistically
significant difference between undisturbed and encounter scenarios for all
measures. However, while this disparity holds true for most measures for
individuals even when considering the interaction level of the encountered
dyad, it appears that only dyads with 0 interaction levels demonstrate a
distinct behavior in encounters compared to undisturbed situations.
Finally, we show in Tab. 63 $\sim$ Fig. 77 the $p$-values for the difference
of means between dyads with various level of interaction and individuals
encountering these dyads. We see that the differences in the deviation between
the dyads and the individuals are often significant for interaction (levels 1,
2 and 3), but not for the non-interacting dyads.
The results presented thus far do not take into account the initial risk of
collision, as we only set a threshold of 4 m to qualify an encounter. Among
these encounters, some might have been more critical than others, with
individual and dyad even facing a risk of collision. In the following section,
we investigate the impact of the initial risk of collision on the deviation of
individuals and dyads, by considering the impact parameter.
### 4.3 Effect of the impact parameter
As detailed in Sec. 3.8, the impact parameter is a measure of how close the
individual would have passed the dyad if there were no collision avoidance
behavior involved. In particular, $\bar{r}_{b}$ is a dimensionless measure
that indicates the initial risk of collision, since it measures the distance
between the individual and the dyad relative to the dyad’s width252525The
width of the dyad is the average distance between the members of dyads with
that level of interaction..
We chose to bin the values of $\bar{r}_{b}$ into 4 bins of equal size, which
we interpret as follows: a value of $\bar{r}_{b}$ smaller than 1 indicates
that the individual is on track to pass through or collide with the dyad.
Between 1 and 2, the individual is close to the dyad, but not on track to pass
through it. Between 2 and 3, the individual is further away from the dyad and
may not need to deviate significantly to pass comfortably. Finally, a value of
$\bar{r}_{b}$ larger than 3 indicates that the individual is far from the dyad
and does not need to deviate.
In Fig. 20 $\sim$ Fig. 34, we illustrate the ratio of the average deviation
during encounters to the average deviation in undisturbed situations for
individuals and dyads with respect to the normalised impact parameter
$\bar{r}_{b}$ for each measure. We also provide the T-test $p$-values for the
difference in the ratio between dyads with low interaction levels (0 and 1)
and dyads with high interaction levels (2 and 3) for each bin of
$\bar{r}_{b}$.
Because the binning necessarily reduces the amount of data, in particular for
interaction with already limited data (e.g. levels 0 and 3), we chose to
separate encounters into two categories based on the level of interaction of
the dyad: one with interaction levels 0 and 1, and another with levels 2 and
3. This helps balancing the number of data points in each and to get
comparable sizes of samples for both classes. We argue that this
categorisation is not unreasonable, as it allows to contrast low and high
interaction levels.
For individuals, we observe that the ratio is generally262626Except for the
suddenness of turn $\sigma$. higher when encountering dyads with higher
interaction levels (2 and 3) compared to those with lower interaction levels
(0 and 1). This is consistent with the results presented in Sec. 4.2.2, where
we observed that the deviation of individuals was higher when encountering
dyads with higher interaction levels. We observe that the associated T-test
$p$-values generally follow a similar pattern for the position-based
measures272727Except for $\delta_{LCSS}$ and $\delta_{Lev}$., where the
difference in the ratio between the two classes of dyads is statistically
significant (or close to being significant) in the first and third bins of
$\bar{r}_{b}$ (i.e. when the individual is on track to pass through the dyad
or when the individual is relatively far from the dyad).
For the dyads, in accordance with the results of Sec. 4.2.2, we observe that
the ratio of the lower interaction levels (0 and 1) is systematically higher
than that of the higher interaction levels (2 and 3) for all measures and all
values of $\bar{r}_{b}$. For all measures except $\sigma$, the difference in
the ratio between the two classes of dyads is statistically significant in the
second bin of $\bar{r}_{b}$ (i.e. when the individual is close to the dyad but
not on track to pass through it).
### 4.4 Correlation between measures
In Sec. 3.7, we demonstrated that different measures of deviation may capture
distinct aspects of deviation. Some measures may be more sensitive to the
magnitude of deviation, while others may be more sensitive to its abruptness.
In this section, our objective is to assess the consistency of deviation
measures to some extent.
To that end, we illustrate Spearman correlation between the measures of
deviation for all trajectories (dyads and individuals) in both undisturbed and
encounter situations in Fig. 18. It is evident from this figure that the
deviation measures generally exhibit correlation, with coefficients exceeding
0.5 for most measures.
Notably, sinuosity, curvature, and suddenness of turn display lower
correlations with other measures. We attribute this observation to the fact
that their computation involves some form of differentiation, such as deriving
turning angles from velocity vectors for $S$, and calculating $\sigma$ and
$E_{\kappa}$ using velocity and acceleration values.
We also note that the correlation between the measures is generally larger for
the undisturbed situations than for the encounters. We believe this result to
be partly due to the fact that the number of encounters is smaller than the
number of undisturbed situations, which might make the correlation less
reliable.
To further investigate the correlation between the measures, we visualise the
dependence of the different measures against the lockstep maximum deviation in
Fig. 19. The deviation $\delta_{max}$ is binned in 16 bins of equal size, and
the average value of the different measures is computed for each bin. We can
see that the measures are all increasing with the lockstep maximum deviation,
which is consistent with the idea that the measures are correlated. We see
that although most measures seem to be linearly correlated with the lockstep
maximum deviation, the rate of increase for $\delta_{LCSS}$ and $\delta_{Lev}$
slows down for large values of $\delta_{max}$. The main reason is that these
two measures are dimensionless values (see Sec. 3.6) which are impacted by the
number of points in the trajectory that are far away from the undisturbed
trajectory rather than the actual amplitude of the deviation.
Figure 18: Spearman correlation between the measures of deviation. Figure 19:
Average value of the different measures of deviation as a function of the
lockstep maximum deviation. Values are binned in 16 bins of equal size. The
scale of the y-axis is logarithmic for $E_{\kappa}$.
## 5 Discussion
We interpret the presented results considering two kinds of effects: the
dynamic stability of the considered entity (individual or dyad) and the amount
of awareness regarding the environment. We argue that the amount of deviation
observed is a result of the interplay between these two factors.
The deviation of the non- and weakly interacting dyads is seen to be
significantly smaller than the deviation of the individuals, which implies
that dyads keep a more straight trajectory than individuals. This suggests
that dyads maintain a more consistent trajectory compared to individuals,
potentially due to their inclination to remain physically close, thereby
constraining deviations from the intended path. Using a physical analogy, the
inertia of the system composed of the two members of the dyad can be expected
to be larger than the inertia of the individual, which would make the dyad
more stable.
In addition, awareness about the changes in the environment is argued to be
directly related to the level of interaction of the dyad. Namely, it is
reasonable to assume that non interacting dyads in undisturbed situations are
more attentive to their surroundings, since they do not need to allocate
attention towards their social interaction. Consequently, in undisturbed
scenarios, these dyads are more aware of the availability of economical
straight paths and are more likely to adopt them compared to interacting
dyads, which may deviate due to their internal -social interaction- dynamics.
During encounters, the fact that all dyads deviate significantly less than the
individuals suggests that they contribute less to the avoidance process. In
addition, the deviation of non-interacting dyads is higher than that of
interacting dyads, which suggests that the level of interaction affects the
dyad’s ability to focus on the environment. This is further supported by the
fact that the deviation of individuals increases significantly during
encounters, regardless of the level of interaction of the dyad. This indicates
that individuals anticipate (or react to) the diminished involvement of dyads
in collision avoidance and adjust their deviations accordingly282828Despite
their simplicity, force based models such as the traditional social force
model [19] might predict such results, since the repulsive force perceived by
the individual would be the sum of the repulsive forces perceived by the two
members of the dyad.
All the used measures show an increase in individuals’ deviation when
encountering dyads with higher interaction levels, although no measure
suggests statistical significance. Nevertheless, our prior investigation
focusing on relative dynamics [20] revealed an interaction-level dependent
statistical difference in encounters, particularly when such encounters could
lead to a collision without deviation. By combining these findings with those
of the current study, which show that in encounters between an interacting
dyad and an individual, the latter bears the burden of collision avoidance, we
can deduce that the individual’s behavior is influenced by the dyad’s level of
interaction.
The impact parameter analysis provides further insights into the dynamics of
the encounters. The observed patterns in the ratios of deviation during
encounters to undisturbed situations for individuals and dyads with respect to
the impact parameter suggest that the initial risk of collision influences the
deviation of both individuals and dyads.
Concerning the individual, the difference between high and low interaction
levels is most pronounced when the individual is on track to pass close to the
dyad, with the deviation of the high interaction levels being significantly
higher than that of the low interaction levels. We believe that is an effect
of the intrusion phenomena that we observed in [20], where the individual is
more likely to pass through the dyad when the dyad is less reactive,
essentially non-deviating and maintaining a straight trajectory. In situations
where the encounter is expected to be close but not on a collision course, the
deviation of the individual is less dependent on the dyad’s level of
interaction, which might be due to the fact that the individual has to deviate
regardless of the dyad’s involvement in social interaction. For less critical
situations, the deviation of the individuals is again more dependent on the
dyads’ level of interaction. We believe that this is due to the fact that in
such situations the individuals mostly do not need to avoid low interacting
dyads, since they walk on straight paths, while they more often need to avoid
high interacting ones, due to their wandering behaviour. Finally, for high
values of the impact parameter, the deviation of the individual does not
depend on the dyad’s level of interaction, since there is not much risk of
collision and the individual essentially doesn’t need to deviate.
For the dyad, the difference between high and low interaction levels is most
pronounced when the individual and dyad are on track to pass close to each
other, with the deviation of the low interaction levels being significantly
higher than that of the high interaction levels. We hypothesise that this is
where the effect of the dyad’s awareness of the environment is most
pronounced, as the dyad with low interaction levels is more likely to be
focused on the environment and react to the individual’s presence. For very
low values of the impact parameter, there is no difference between the
deviation of the dyads with different interaction levels, which might arise
from the fact that the situation is so critical that the dyad has to react
regardless of its involvement in social interaction by either opening up or
deviating. At higher values of the impact parameter, the deviation of the
dyads with different interaction levels is also similar, which might be due to
the fact that the situation is less critical and the dyad can afford to be
less reactive and the stability takes over.
While previous research has suggested that utilizing angular variables [111,
108] or velocity adjustments [97, 112] to assess deviations from the intended
path could offer a promising approach, our study faced limitations in this
regard. As discussed in Sec. 4.4, the trajectory data we utilised may not
adequately support differentiation, resulting in certain measures (such as
$\sigma$, $E_{\kappa}$, and $S$) being less effective in capturing deviations
compared to others. While controlled experiments can employ various wearable
sensors to accurately register position or acceleration [113, 114],
implementing such methods in real-world settings poses challenges.
Environmental sensors may be sparse or susceptible to measurement noise and
clutter, making it difficult to derive accurate values in naturalistic
environments. Although higher-quality tracking data can be obtained in
laboratory settings using wearable sensors, it remains uncertain whether the
collected data accurately reflects the behaviour of human subjects in
naturalistic settings, as discussed in Sec. 1 [29]. Moreover, with more
detailed data, such as limb orientations, it could become feasible to
investigate how pedestrians manage deviations. This could involve analyzing
adjustments in step lengths or cephalocaudal reorientations [32, 115].
Evidence from ethology suggests that interactions among collectively moving
non-human animals, such as bird flocks, are primarily influenced by
topological distance rather than metric distance [92]. While there is ongoing
debate regarding whether neighborhood in human motion is determined in a
topological or metric manner [17], it is important to consider that the
conclusions drawn from ethological studies may not directly apply to our
scenario. This is because collective behaviour, which is characterised by
emergent complex patterns, typically arises in situations involving a large
number of individuals, whereas our scenario involves low density.
Research on collision avoidance often examines metrics such as the time or
distance to the predicted point of collision [116, 117]. For instance, Olivier
et al. define the Minimal Predicted Distance (MDP) as the anticipated closest
distance between participants at a given time, assuming no adjustments are
made to their velocity vectors [87]. Similarly, Bhojwani et al. introduce the
Theoretical Point of Collision (TPC), representing the point where a collision
with an approaching virtual pedestrian would occur if no locomotor adjustments
are made [21]. These metrics, akin to $d_{max}$, primarily focus on the point
of collision rather than the deviation from the intended path. They also
consider the involved parties in the encounter in a more integrated manner
compared to our isolated approach. In future research, it would be beneficial
to explore how deviation metrics evolve over time and their relationship with
these collision distance measures.
Another aspect of pedestrian dynamics that should be investigated is the
behaviour of overtaking or following others, whether they are individuals or
groups. Given the typically faster and more flexible nature of individuals
compared to groups, it is reasonable to expect that individuals would engage
in overtaking more frequently than groups. Moreover, individuals may tend to
overtake groups more frequently than other individuals, again due to their
greater speed and maneuverability. Interestingly, the decision of individuals
to overtake groups may not necessarily depend on the social relation or
interaction within the group. Since individuals approaching from behind may
have limited visibility of the group’s social features or intentions, their
overtaking behaviour may primarily be driven by considerations such as speed
and convenience rather than social dynamics.
## 6 Conclusion
Over the last few decades, numerous pedestrian collision avoidance models have
emerged, offering insights into various aspects of pedestrian behavior [118,
117]. However, despite this extensive body of work, most microscopic models
predominantly focus on one-on-one collision avoidance scenarios, neglecting
the dynamics involving groups [119]. This oversight is notable considering
that groups represent a substantial portion of crowds in real-world settings
[120, 40]. Thus, understanding the mechanisms underlying group-individual
collision avoidance has remained a gap in pedestrian dynamics research.
Addressing this gap, our study aims to contribute to the understanding of how
groups and individuals navigate and avoid collisions in crowded environments,
shedding light on this understudied aspect of pedestrian behavior. In
particular, by analyzing ecological data and considering various deviation
measures, this study presents insights into the stability and deviation
dynamics of dyads and individuals.
In particular, concerning individuals, we demonstrate a heightened inclination
to avoid interacting dyads compared to non-interacting ones. This suggests
that individuals not only anticipate the future paths of others but also
assess their contribution to collision avoidance. It is probable that upon
detecting a lack of involvement, individuals adjust their behavior based on
their internal model of human navigation.
Regarding dyads, our findings suggest that, in undisturbed situations, non-
interacting ones, exhibit small deviations from their intended trajectories,
indicating a high level of efficiency and stability. Conversely, interacting
dyads show larger deviations, possibly attributable to being more consumed
with their internal dynamics. Additionally, interacting dyads are less
impacted by encounters compared to individuals, while non-interacting dyads
display a similar pattern to individuals.
Based on these observations, we believe that being part of a group affects the
behavior of a pedestrian in two, possibly contrasting, ways. A dynamical
effect, i.e. the tendency to move together, appears to increase the stability
and economy of motion. On the other hand, a social effect diverts attention
from the environment, resulting in a less economical movement, in a way
similar to how the usage of smart phones affects pedestrian behavior [108].
We believe that this study not only provides valuable insights into the
complex dynamics of group-individual collision avoidance, but it also has
tangible implications and has the potential to open up new avenues of
technical improvement. In particular, it holds practical significance in
various real-world scenarios, such as urban planning and crowd management.
Namely, with such insights, we can inform the development of more effective
pedestrian flow management strategies, improve the design of urban spaces to
leverage safety and mobility efficiency, and enhance the planning of events or
gatherings to reduce congestion and potential hazards. Additionally, our
results can help the development of intelligent systems for autonomous
vehicles, robots, or ubiquitous technology, providing safer and more seamless
interactions with human pedestrians in diverse smart environments.
Table 3: Average and standard deviation of the Euclidean distance $\delta_{E}$ (in m), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{E}$ | Normalized | $p$-value
---|---|---|---
0 | $6.74\times 10^{-2}\pm 4.54\times 10^{-2}$ | $0.80$ | $\mathbf{3.21\times 10^{-4}}$
1 | $8.71\times 10^{-2}\pm 5.71\times 10^{-2}$ | $1.03$
2 | $9.93\times 10^{-2}\pm 8.97\times 10^{-2}$ | $1.18$
3 | $1.22\times 10^{-1}\pm 7.54\times 10^{-2}$ | $1.44$
All dyads | $1.00\times 10^{-1}\pm 8.31\times 10^{-2}$ | $1.19$ | $\mathbf{2.18\times 10^{-4}}$
Individuals | $8.44\times 10^{-2}\pm 7.86\times 10^{-2}$ | $1.00$
Table 4: Average and standard deviation of the lockstep maximum deviation $\delta_{max}$ (in m), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{max}$ | Normalized | $p$-value
---|---|---|---
0 | $1.89\times 10^{-1}\pm 1.32\times 10^{-1}$ | $0.77$ | $\mathbf{1.01\times 10^{-3}}$
1 | $2.49\times 10^{-1}\pm 1.66\times 10^{-1}$ | $1.01$
2 | $2.69\times 10^{-1}\pm 2.07\times 10^{-1}$ | $1.09$
3 | $3.21\times 10^{-1}\pm 1.83\times 10^{-1}$ | $1.31$
All dyads | $2.72\times 10^{-1}\pm 1.97\times 10^{-1}$ | $1.11$ | $\mathbf{9.37\times 10^{-3}}$
Individuals | $2.46\times 10^{-1}\pm 1.92\times 10^{-1}$ | $1.00$
Table 5: Average and standard deviation of the Fréchet deviation $\delta_{F}$ (in m), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{F}$ | Normalized | $p$-value
---|---|---|---
0 | $1.89\times 10^{-1}\pm 1.32\times 10^{-1}$ | $0.77$ | $\mathbf{1.21\times 10^{-3}}$
1 | $2.48\times 10^{-1}\pm 1.66\times 10^{-1}$ | $1.01$
2 | $2.65\times 10^{-1}\pm 1.98\times 10^{-1}$ | $1.08$
3 | $3.17\times 10^{-1}\pm 1.80\times 10^{-1}$ | $1.29$
All dyads | $2.69\times 10^{-1}\pm 1.91\times 10^{-1}$ | $1.10$ | $\mathbf{1.60\times 10^{-2}}$
Individuals | $2.45\times 10^{-1}\pm 1.92\times 10^{-1}$ | $1.00$
Table 6: Average and standard deviation of the maximum lateral deviation $d_{max}$ (in m), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $d_{max}$ | Normalized | $p$-value
---|---|---|---
0 | $1.52\times 10^{-1}\pm 1.30\times 10^{-1}$ | $0.75$ | $\mathbf{2.27\times 10^{-2}}$
1 | $1.99\times 10^{-1}\pm 1.50\times 10^{-1}$ | $0.98$
2 | $2.02\times 10^{-1}\pm 1.36\times 10^{-1}$ | $1.00$
3 | $2.37\times 10^{-1}\pm 1.50\times 10^{-1}$ | $1.17$
All dyads | $2.06\times 10^{-1}\pm 1.41\times 10^{-1}$ | $1.02$ | $6.76\times 10^{-1}$
Individuals | $2.03\times 10^{-1}\pm 1.49\times 10^{-1}$ | $1.00$
Table 7: Average and standard deviation of the integral of lateral deviation $\Delta$ (in m2), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\Delta$ | Normalized | $p$-value
---|---|---|---
0 | $2.07\times 10^{-1}\pm 1.76\times 10^{-1}$ | $0.79$ | $\mathbf{1.39\times 10^{-2}}$
1 | $2.62\times 10^{-1}\pm 1.95\times 10^{-1}$ | $0.99$
2 | $2.81\times 10^{-1}\pm 2.02\times 10^{-1}$ | $1.07$
3 | $3.37\times 10^{-1}\pm 2.26\times 10^{-1}$ | $1.28$
All dyads | $2.85\times 10^{-1}\pm 2.07\times 10^{-1}$ | $1.08$ | $\mathbf{4.22\times 10^{-2}}$
Individuals | $2.63\times 10^{-1}\pm 2.05\times 10^{-1}$ | $1.00$
Table 8: Average and standard deviation of the dynamic time warping deviation $\delta_{DTW}$ (in m), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{DTW}$ | Normalized | $p$-value
---|---|---|---
0 | $5.88\times 10^{-2}\pm 4.70\times 10^{-2}$ | $0.78$ | $\mathbf{6.02\times 10^{-3}}$
1 | $7.58\times 10^{-2}\pm 5.55\times 10^{-2}$ | $1.00$
2 | $8.23\times 10^{-2}\pm 6.35\times 10^{-2}$ | $1.09$
3 | $9.71\times 10^{-2}\pm 6.17\times 10^{-2}$ | $1.28$
All dyads | $8.31\times 10^{-2}\pm 6.21\times 10^{-2}$ | $1.10$ | $\mathbf{2.64\times 10^{-2}}$
Individuals | $7.57\times 10^{-2}\pm 7.36\times 10^{-2}$ | $1.00$
Table 9: Average and standard deviation of the longest common subsequence deviation $\delta_{LCSS}$ (dimensionless), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{LCSS}$ | Normalized | $p$-value
---|---|---|---
0 | $3.08\times 10^{-1}\pm 2.10\times 10^{-1}$ | $0.77$ | $\mathbf{3.63\times 10^{-2}}$
1 | $4.01\times 10^{-1}\pm 1.63\times 10^{-1}$ | $1.01$
2 | $4.16\times 10^{-1}\pm 1.70\times 10^{-1}$ | $1.05$
3 | $4.53\times 10^{-1}\pm 1.65\times 10^{-1}$ | $1.14$
All dyads | $4.17\times 10^{-1}\pm 1.72\times 10^{-1}$ | $1.05$ | $\mathbf{4.17\times 10^{-2}}$
Individuals | $3.98\times 10^{-1}\pm 1.77\times 10^{-1}$ | $1.00$
Table 10: Average and standard deviation of the Levenshtein deviation $\delta_{Lev}$ (dimensionless), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{Lev}$ | Normalized | $p$-value
---|---|---|---
0 | $3.31\times 10\pm 2.38\times 10$ | $0.80$ | $\mathbf{4.81\times 10^{-4}}$
1 | $4.33\times 10\pm 1.99\times 10$ | $1.04$
2 | $5.03\times 10\pm 2.65\times 10$ | $1.21$
3 | $5.84\times 10\pm 2.76\times 10$ | $1.41$
All dyads | $5.01\times 10\pm 2.64\times 10$ | $1.21$ | $\mathbf{<10^{-4}}$
Individuals | $4.14\times 10\pm 2.19\times 10$ | $1.00$
Table 11: Average and standard deviation of the deviation index $\tilde{\tau}$ (dimensionless), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\tilde{\tau}$ | Normalized | $p$-value
---|---|---|---
0 | $8.40\times 10^{-4}\pm 8.84\times 10^{-4}$ | $0.58$ | $\mathbf{5.46\times 10^{-3}}$
1 | $1.30\times 10^{-3}\pm 1.59\times 10^{-3}$ | $0.89$
2 | $1.73\times 10^{-3}\pm 5.65\times 10^{-3}$ | $1.19$
3 | $1.77\times 10^{-3}\pm 1.63\times 10^{-3}$ | $1.21$
All dyads | $1.65\times 10^{-3}\pm 4.66\times 10^{-3}$ | $1.13$ | $4.19\times 10^{-1}$
Individuals | $1.46\times 10^{-3}\pm 3.85\times 10^{-3}$ | $1.00$
Table 12: Average and standard deviation of the maximum cumulative turning angle $\theta_{max}$ (in rad), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\theta_{max}$ | Normalized | $p$-value
---|---|---|---
0 | $1.18\times 10^{-1}\pm 6.98\times 10^{-2}$ | $0.81$ | $\mathbf{3.92\times 10^{-3}}$
1 | $1.37\times 10^{-1}\pm 6.87\times 10^{-2}$ | $0.95$
2 | $1.56\times 10^{-1}\pm 1.92\times 10^{-1}$ | $1.08$
3 | $1.83\times 10^{-1}\pm 1.38\times 10^{-1}$ | $1.27$
All dyads | $1.57\times 10^{-1}\pm 1.69\times 10^{-1}$ | $1.08$ | $1.44\times 10^{-1}$
Individuals | $1.45\times 10^{-1}\pm 1.24\times 10^{-1}$ | $1.00$
Table 13: Average and standard deviation of the integral cumulative turning angle $\Theta$ (in rad), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\Theta$ | Normalized | $p$-value
---|---|---|---
0 | $5.12\times 10^{-2}\pm 3.93\times 10^{-2}$ | $0.77$ | $\mathbf{9.48\times 10^{-3}}$
1 | $6.52\times 10^{-2}\pm 4.58\times 10^{-2}$ | $0.97$
2 | $6.91\times 10^{-2}\pm 6.78\times 10^{-2}$ | $1.03$
3 | $8.53\times 10^{-2}\pm 8.10\times 10^{-2}$ | $1.27$
All dyads | $7.07\times 10^{-2}\pm 6.74\times 10^{-2}$ | $1.06$ | $2.58\times 10^{-1}$
Individuals | $6.69\times 10^{-2}\pm 5.37\times 10^{-2}$ | $1.00$
Table 14: Average and standard deviation of the sinuosity $S$ (in rad/m${}^{\frac{1}{2}}$), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $S$ | Normalized | $p$-value
---|---|---|---
0 | $1.41\times 10^{-1}\pm 4.63\times 10^{-2}$ | $1.15$ | $\mathbf{1.03\times 10^{-2}}$
1 | $1.40\times 10^{-1}\pm 3.75\times 10^{-2}$ | $1.15$
2 | $1.73\times 10^{-1}\pm 2.42\times 10^{-1}$ | $1.42$
3 | $1.83\times 10^{-1}\pm 1.02\times 10^{-1}$ | $1.50$
All dyads | $1.69\times 10^{-1}\pm 2.01\times 10^{-1}$ | $1.39$ | $\mathbf{<10^{-4}}$
Individuals | $1.22\times 10^{-1}\pm 1.04\times 10^{-1}$ | $1.00$
Table 15: Average and standard deviation of the energy curvature $E_{\kappa}$ (in s/m2), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $E_{\kappa}$ | Normalized | $p$-value
---|---|---|---
0 | $5.79\times 10^{-8}\pm 4.68\times 10^{-8}$ | $5.85\times 10^{-5}$ | $\mathbf{2.53\times 10^{-3}}$
1 | $5.64\times 10^{-8}\pm 4.55\times 10^{-8}$ | $5.71\times 10^{-5}$
2 | $1.99\times 10^{-3}\pm 3.43\times 10^{-2}$ | $2.01$
3 | $4.48\times 10^{-4}\pm 2.95\times 10^{-3}$ | $0.45$
All dyads | $1.38\times 10^{-3}\pm 2.78\times 10^{-2}$ | $1.40$ | $7.80\times 10^{-1}$
Individuals | $9.88\times 10^{-4}\pm 2.34\times 10^{-2}$ | $1.00$
Table 16: Average and standard deviation of the turn intensity $i$ (in deg$\times$cm), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $i$ | Normalized | $p$-value
---|---|---|---
0 | $3.38\times 10\pm 5.04\times 10$ | $0.59$ | $\mathbf{6.51\times 10^{-3}}$
1 | $5.59\times 10\pm 1.06\times 10^{2}$ | $0.98$
2 | $5.50\times 10\pm 8.71\times 10$ | $0.96$
3 | $7.42\times 10\pm 9.49\times 10$ | $1.30$
All dyads | $5.77\times 10\pm 9.04\times 10$ | $1.01$ | $9.29\times 10^{-1}$
Individuals | $5.72\times 10\pm 1.05\times 10^{2}$ | $1.00$
Table 17: Average and standard deviation of the suddenness of turn $\sigma$ (in deg$\times$cm/s), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\sigma$ | Normalized | $p$-value
---|---|---|---
0 | $5.04\times 10^{3}\pm 6.02\times 10^{3}$ | $0.75$ | $1.38\times 10^{-1}$
1 | $5.10\times 10^{3}\pm 4.30\times 10^{3}$ | $0.76$
2 | $5.15\times 10^{3}\pm 4.30\times 10^{3}$ | $0.76$
3 | $5.29\times 10^{3}\pm 3.30\times 10^{3}$ | $0.79$
All dyads | $5.17\times 10^{3}\pm 4.22\times 10^{3}$ | $0.77$ | $\mathbf{<10^{-4}}$
Individuals | $6.74\times 10^{3}\pm 6.39\times 10^{3}$ | $1.00$
Table 18: Average and standard deviation of the Euclidean distance $\delta_{E}$ (in m), for dyads and individuals during encounters. Kruskal-Wallis $p$-values for the difference of means between dyads (and individuals encountering dyads) with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. | Individual | Dyad
---|---|---
Intensity of | $\delta_{E}$ | Kruskal-Wallis | $\delta_{E}$ | Kruskal-Wallis
interaction | | $p$-value | | $p$-value
0 | $1.25\times 10^{-1}\pm 8.73\times 10^{-2}$ | $9.58\times 10^{-1}$ | $1.02\times 10^{-1}\pm 5.90\times 10^{-2}$ | $\mathbf{2.61\times 10^{-2}}$
1 | $1.16\times 10^{-1}\pm 7.02\times 10^{-2}$ | $8.91\times 10^{-2}\pm 6.56\times 10^{-2}$
2 | $1.30\times 10^{-1}\pm 1.14\times 10^{-1}$ | $9.65\times 10^{-2}\pm 7.05\times 10^{-2}$
3 | $1.31\times 10^{-1}\pm 1.00\times 10^{-1}$ | $1.09\times 10^{-1}\pm 8.43\times 10^{-2}$
All | $1.28\times 10^{-1}\pm 1.05\times 10^{-1}$ | | $9.78\times 10^{-2}\pm 7.17\times 10^{-2}$ |
T-test $p$-value | $\mathbf{<10^{-4}}$ |
FedPot: A Quality-Aware Collaborative and Incentivized Honeypot-Based Detector for Smart Grid Networks
Abdullatif Albaseer, Member, IEEE,
Nima Abdi,
Mohamed Abdallah, Senior Member, IEEE, Marwa Qaraqe Senior Member, IEEE, Saif Al-Kuwari, Senior Member, IEEE
Abdullatif Albaseer,Nima Abdi, Mohamed Abdallah, Marwa Qaraqe, and Saif Alkuwari are with Division of Information and Computing Technology, College of Science and Engineering, Hamad Bin Khalifa University, Doha, Qatar (e-mail:{aalbaseer, niab52126, moabdallah, mqaraqe, smalkuwari}@hbku.edu.qa).
* Preliminary results in this work are presented at the IEEE CCNC Conference, 2023 [1]
Honeypot technologies provide an effective defense strategy for the Industrial Internet of Things (IIoT), particularly in enhancing the Advanced Metering Infrastructure's (AMI) security by bolstering the network intrusion detection system. For this security paradigm to be fully realized, it necessitates the active participation of small-scale power suppliers (SPSs) in implementing honeypots and engaging in collaborative data sharing with traditional power retailers (TPRs). To motivate this interaction, TPRs incentivize data sharing with tangible rewards. However, without access to an SPS's confidential data, it is daunting for TPRs to validate shared data, thereby risking SPSs' privacy and increasing sharing costs due to voluminous honeypot logs. These challenges can be resolved by utilizing Federated Learning (FL), a distributed machine learning (ML) technique that allows for model training without data relocation. However, the conventional FL algorithm lacks the requisite functionality for both the security defense model and the rewards system of the AMI network. This work presents two solutions: first, an enhanced and cost-efficient FedAvg algorithm incorporating a novel data quality measure, and second, FedPot, the development of an effective security model with a fair incentives mechanism under an FL architecture. Accordingly, SPSs are limited to sharing the ML model they learn after efficiently measuring their local data quality, whereas TPRs can verify the participants' uploaded models and fairly compensate each participant for their contributions through rewards. Moreover, the proposed scheme addresses the problem of harmful participants who share subpar models while claiming high-quality data through a two-step verification approach. Simulation results, drawn from realistic mircorgrid network log datasets, demonstrate that the proposed solutions outperform state-of-the-art techniques by enhancing the security model and guaranteeing fair reward distributions.
AMI, Honeypot-Based Detector, Security Model, Machine Learning, Incentive Mechanism, Collaborative Learning
SGSmart Grid
IIoTIndustrial Internet of Things
SPSsSmall-Scale Power Suppliers
TPRsTraditional power retailers
AMIAdvanced Metering Infrastructure
DDoSDistributed Denial-of-Service
NIDSNetwork Intrusion Detection System
MLMachine Learning
DLDeep Learning
GDPRGeneral Data Protection Regulation
FLFederated Learning
VDDValid Defense Data
CPUCentral Processing Unit
IRIndividual Rationality
ICIncentive Compatibility
LDICLocal Downward Incentive Compatibility
LUICLocal Upward Incentive Compatibility
PLCsProgrammable Logic Controllers
RTUsRemote Terminal Units
TPRateTrue Positive Rate
TNRTrue Negative Rate
IIDIndependent and Identically Distributed
$M$number of connected SPSs
$\theta_m$SPS type: $\theta_1\le \ldots \le \theta_m \le \ldots \le \theta_M$
$\boldsymbol{V}_m$local model of SPS $m$
${\Pi}$Incentive set for all types
$\pi_m$the model and its rewards of type $m$
$\boldsymbol{V}(z)$global model at $z$-th training round
$R_m$the reward given to SPS $m$
$\boldsymbol{V}_m(j)$the local model update at $j$ local iteration
$F_m(\boldsymbol{V}_{m}(j))$the local loss function at $j$ local iteration
$F_m(\boldsymbol{V}_{m})$the updated local loss function
${F_{i}(\mathbf{\boldsymbol{V}_m}, x_{i}, y_{i}})$loss function on sample $i$
$E$number of local training epochs
$\eta$the learning rate
$\mathcal{S}_z$the selection set at round $z$
$S^m_z$the selection binary variable
$\xi$the defense model size
$C_d$the honeypot deploying cost
$C_u$the uploading cost of the local defense model
$C_t$the local training cost to update the defense model
$C_m$the local deployment, training, and uploading costs
$T^{max}$the maximum latency in each round
$U_m$the utility of SPS $m$
$U_{TPR}$the utility of TPR (i.e., utility company)
$G(V_{m})$the revenue given by each SPS $m$
$\Lambda(x_{i}, \delta)$an open ball space with a radius of $\delta$ centered at $x_i$
$\varphi(\mathcal{D}_m)$the estimation of the local VDD quality
§ INTRODUCTION
A Smart Grid (SG) is an advanced electricity system that leverages digital tech, IIoT, and networking to boost efficiency, reliability, and sustainability. It achieves this through a cyber-physical system for the bidirectional flow of power and information, automating supply-demand balance with real-time data [2]. SG allows individuals to become small-scale power suppliers (SPSs) using renewable sources, reducing the burden on traditional power retailers (TPRs) and fostering advanced metering infrastructure (AMI) integration [3, 4].
However, integrating different network technologies to interconnect SPSs and TPRs, which lack a proper defense system, is raising concerns about the security and privacy of SG systems. Adversaries have access to numerous vulnerabilities, including ways to launch destructive attacks and access sensitive information such as a homeowner's residential address, social security number, daily habits, and any information related to unauthorized electricity consumption or disruption in the network. As an example of such attacks, the 2015 Ukrainian power outage illustrates the critical vulnerabilities to cyber-attacks in both the control center and the smart devices employed for managing and observing the electrical system [5].
In addition, SG is subject to denial-of-service (DoS) attacks, which can flood the network with traffic, causing delays in data transmission and processing. This can lead to disruptions in the system's normal operation and potentially cause critical elements of the energy system to fail [6].
To protect the communication infrastructure, implementing a Network Intrusion Detection System (NIDS) is crucial. NIDS serves as a robust shield, detecting numerous threats that extend beyond the capacity of traditional firewalls. The protective capability of NIDS has been notably enhanced by the advent of Machine Learning (ML) and Deep Learning (DL)-based approaches, pushing their performance to a significantly higher level. This interconnected relationship between NIDS, ML, and DL demonstrates a synergistic blend of network security and advanced computational methodologies [7].
Building upon this security paradigm, incorporating honeypot technology further enriches the protective capabilities of ML/DL-based NIDSs. Honeypots empower these systems to meticulously map the attack surface, discern patterns, and thwart malicious actions by delivering detailed insights into potential intruder behavior.
Within the realm of SG, a honeypot mimics the regular operations of a meter with the intent to deceive, misdirect, and analyze the activities of potential intruders. Employing such tactics allows SPSs to develop independent, streamlined security measures. Furthermore, it facilitates the exchange of defensive information with TPRs, eliminating the need for TPRs to purchase costly security models from security retailers.
This approach amplifies the protective layers of AMI, creating a more robust defense mechanism. Simultaneously, it alleviates the financial strain associated with defense strategies for TPRs, presenting a dual advantage by strengthening network security while managing costs effectively [8].
Significant effort in the literature has been devoted to developing incentive mechanisms and designing contracts to encourage SPSs to implement honeypots and share protective information with TPRs while maintaining a balance such that SPSs do not reap excessive benefits beyond their due. For example, in [9, 10], the authors proposed information asymmetry-based contract theory approaches considering different communication systems. Concentrating on the SG network, the work in [11] introduced a recent motivation-driven approach where TPR motivates SPSs to deploy honeypots and exchange defensive information to enhance the security framework. This approach relies on a range of essential prerequisites, including the dimensions of the shared data and the related expenses.
However, all the prior work may violate the SPSs’ privacy
while increasing the cost of sharing the obtained raw data.
The recent enforcement of stringent data privacy regulations, such as the General Data Protection Regulation (GDPR) [12], further underscores the importance of maintaining data privacy.
Additionally, the large volume of honeypot logs can lead to excessive transmission costs and network congestion. It is also worth noting that the shared honeypot logs may not always yield benefits, as they could contain redundant information that does not enhance the existing protection strategy.
Given the aforementioned concerns, federated learning (FL) [13] has become an effective distributed ML/DL method to maintain privacy and minimize communication costs by exclusively sharing the ML model while retaining the raw data in its original repository. This seamlessly connects the need for privacy preservation and cost-efficiency with the advantages of shared data for network security.
Current research on FL typically assumes that entities are willing to participate in the FL training process and use the collected data to improve the shared model [14, 15]. In reality, entities may be reluctant to join without properly designed incentives (i.e., contracts) because FL consumes significant resources (i.e., computation and communication costs). Furthermore, entities in FL are autonomous actors who decide when and how to interact. When dealing with different reward strategies from different alliances, participants may use different training techniques, influencing the performance of the resulting models. To that end, it is critical to develop an efficient reward mechanism to encourage entities to participate in FL while maintaining the required level of data quality.
Researchers have recently presented several incentive schemes aimed at compensating the involved parties (i.e., SPSs) using financial incentives according to the magnitude of their data contributions [16, 17, 18]. Utilizing current FL methodologies, all participants acquire the same model upon the end of the training phase, regardless of the produced data quality (i.e., honeypot logs) or the impact of the submitted local models. Consequently, certain participants possessing large datasets may provide inadequate contributions yet attain a higher proficiency level than others who possess high-quality data. This scenario gives rise to a challenging issue called the FL free-rider problem.
§.§ Contribution
Considering the previously mentioned remarks, we introduce FedPot, an FL framework initially designed for SG networks but validated across diverse scenarios, including IEC 104 and N-BaIoT datasets. This architecture incorporates refined, efficient, and resilient aggregation and averaging techniques complemented by a fair rewards system based on data quality. While our primary case study is centered on SGs, our approach of combining honeypot logs and FL has shown adaptability and effectiveness across multiple domains.
We introduce novel schemes for local data quality, participant selection, and global model averaging, where the TPR resolves a convex optimization problem that prioritizes data quality over data size. Each SPS fine-tunes the global model received from the TPR using its honeypot logs and transmits back the model parameters. After that, the TPR combines and enhances the defensive model employing the proposed approaches, as detailed later. To overcome the FL free-rider problem within AMI networks, we propose a novel metric to measure local data quality and contributions instead of data size, which may contain redundant information that does not improve the security defense model. We devised a two-step verification process to tackle malicious or poorly performing participants. The TPR verifies submitted models, then updates the global model and allocates incentives based on contribution.
In summary, the primary contributions of this work are:
* Develop an effective architecture for privacy-preserving honeypot-based detectors, FedPot, that protects user privacy while considering data quality, an efficient global model, and fair incentives. The proposed solution handles and ensures a reliable FL training process based on valid defense data acquired by implementing honeypots on the SPS side.
* Formulate the problem as an optimization problem, then provide solutions incorporating (i) problem reformulation and transformation, (ii) the prior quality determination of the local data through novel metrics, and (iii) two schemes for reliable global averaging and contribution-based reward distributions.
* In response to the challenge of adversarial perturbations in model uploads, we introduce a universally applicable two-step verification system. This robust approach is designed to ensure the integrity of model contributions.
* Carry out comprehensive simulations using realistic log data from various datasets (i.e., N-BaIoT, IEC 104, and IEC MMS datasets). The results affirm that our proposed framework outperforms existing state-of-the-art techniques in multiple application domains.
§.§ Organization
The remainder of the paper is structured as follows.
Section <ref> briefly reviews the relevant literature on NIDSs and honeypot deployment.
The system models, including the learning, cost, and reward, are presented in <Ref>. The problem is formulated in <Ref>, and the proposed solution, including the problem reformulation and transformation, is presented in <Ref>. In <Ref>, we discuss the experimental setup and present the numerical results. Finally, <Ref> concludes this paper and suggests possible directions for further research.
§ RELATED WORK
With the proliferation of microgrids, major concerns about cyberattacks on such systems via smart meters have arisen. The United States Department of Homeland Security reported 224 destructive cyber intrusions against local electric utilities between 2013 and 2014 [19].
AMI Security
Security concerns have been intensively investigated in the past few years as a key component of the IIoT. Authors in [20] studied the security approach to mitigate cyber-attacks in the context of the IIoT. The main assumption was that the attackers have sufficient tools to identify advanced vulnerabilities that enable them to attack IIoT systems. In [21], Li et al. used consortium blockchain technology to overcome transaction constraints in the IIoT. However, fewer studies have been conducted on the security of various components of AMI systems. The work in [22] introduced a security protocol that preserves AMI private information while securely delivering control packets at the exact time. In [23], Fasial et al. investigated the feasibility of employing data stream mining to improve AMI cybersecurity via NIDS. Yan et al. [24] describe an SG AMI security framework where different security concerns associated with AMI deployment are considered.
ML/DL-based and FL-based NIDSs
Many ML algorithms have been utilized to boost the NIDS in the past years to understand complicated network traffic better [25, 26, 27]. ML/DL-based NIDSs are used to identify unknown intrusions by analyzing the statistical characteristics of the network traffic. However, DL-based solutions have shown better performance, especially in extracting knowledge from complicated features rather than the shallow features in ML-based detectors [25, 26, 27]. Recently, FL has been increasingly employed in the realm of NIDSs for the collaborative design of ML and DL-based detectors [28]. Specifically, the work presented in [29] developed a cooperative detector capable of identifying zero-day botnet attacks in oT networks using FL. Extending FL's utility to energy systems, [30] introduced a privacy-preserving and communication-efficient FL-based energy predictor for net-metering systems. This approach combines a hybrid DL model for high-accuracy energy forecasting with an Inner-Product Functional Encryption scheme to encrypt local model parameters, maintaining data privacy.
Honeypot Deployment Based Incentive Mechanisms
Honeypots are practical security tools that deceive cyber attackers by acting as vulnerability traps [31]. It has been widely used to enhance defensive performance on different systems [8, 32]. Tian et al. [32] developed a honeypot system to defend against APT attacks in the SG, mainly in the bus nodes. In their system, low-interaction and high-interaction honeypots were applied. Similarly, Wang et al. [33] proposed a honeypot scheme with various mixed distributions to detect the AMI network traffic. However, Wang et al. failed to consider how using a honeypot system could reduce TPR defensive pressure while increasing AMI defensive effectiveness. Using a honeypot, Du and Wang [8] captured distributed denial of service attacks (DDoS) attacks in AMI. The implemented methodology uses incomplete information static game and honeypot deployment to investigate non-cooperative problems. Householders, for example, install small wind turbines for small enterprises to sell excess electricity and get a profit from utility companies [3].
The work in [11] proposed a game theory-based approach to designing an incentive mechanism that allows SPSs to share their defense data with TPRs; thus, the TPR can pay the rewards accordingly.
To summarize, significant effort has been made in the literature to overcome the challenge of deploying the honeypot and exchanging the obtained defensive data in AMI.
However, most of these studies focused on log size and overlooked data redundancy that may be received from multiple SPSs.
Furthermore, the privacy of the SPSs was completely ignored, posing an undesirable privacy risk to target consumers.
Finally, there are additional costs associated with uploading large logs.
Thus, in this paper, we propose FedPot. This efficient framework not only focuses on all these challenges but also solves advanced issues associated with malicious SPSs and ensures a more reliable defense model.
§ SYSTEM MODEL
Micro-grid system where $M$ SPSs deploy honeypot and transfer security model with a single TPR.
This paper considers singular TPR within an AMI network, which gathers defensive data from $M$ interconnected SPSs, as depicted in <Ref>.
In our microgrid system, SPSs can acquire defense data by implementing and deploying honeypots. This includes mimicking real and different services (i.e., IEC 104 and GOOSE), applications, and typical vulnerabilities. The honeypot remains an attractive target while collecting diverse attack patterns to evolve with the changing landscape of threats. In this work, we consider two types of traffic captured by honeypots: fake traffic, which enables us to understand attacker tactics without data centralization, and legitimate traffic, which provides valuable insights into user behavior patterns processed in a decentralized manner to maintain privacy. To foster a diverse range of defense data, TPR may provide incentives to stimulate SPSs to establish honeypots, gather cyber-attack intelligence, encapsulate the defensive data, and transmit it to the TPR.
Though SPSs can exchange defense data in the AMI network, there is an information imbalance between the TPR and SPS. This discrepancy arises because the SPS knows its valid defense data (VDD) while the TPR is not. The VDD includes the undisclosed attack event logs gathered by honeypots and used by the currently deployed TPR defensive system to improve the existing architecture.
Practically, the honeypot logs amassed are extensive and could include users' sensitive information, leading to concerns surrounding computation and communication costs and potential privacy violations.
FL can mitigate these issues by utilizing ML algorithms to extract knowledge from the local VDD of each SPS and share only the model attributes with the TPR.
Nevertheless, traditional FL Integration algorithms and incentive techniques are not ideally suited for the security defense framework within the AMI network. As a result, we introduce a privacy-preserving framework for effective honeypot-based detection. Our objective is to develop a method that promotes effective assurance of local model quality and provides incentives to bolster the defense system of the AMI network. We presume the SPSs hold $M$ distinct data types that are separately distributed.
Let $\theta_1\le \ldots \le \theta_m \le \ldots \le \theta_M$ denote the VDD type for each SPS, with a higher type signifying superior quality. To conduct FL model training, the TPR must ensure the desired quality at every global training round by devising an incentive set (i.e., contract) ${\Pi}=\{\pi_m=(\boldsymbol{V}_m, R_m)|m\in \{1,\ldots, M\}\}$, which establishes the relationship between rewards and local model quality based on the type, where $\boldsymbol{V}_m$ denotes the local model attributes of $m$ VDD type and $R_m$ represents the incentives provided by the TPR.
For participant selection, the TPR needs initial information such as how many SPSs are available to participate in defense model training, effort duration, declared data quality, and associated costs.
To guarantee an efficient training procedure, the TPR imposes a time constraint for every global round $z$, beyond which the model modifying and transferring must be finalized. The chosen SPSs employ their VDD to refine the global model and subsequently transfer the upgraded model to the TPR. According to [14], the local model is formulated as follows:
\begin{equation}
\boldsymbol{V}_{m}(j)= \boldsymbol{V}_{m}(j-1)-\eta \nabla F_m(\boldsymbol{V}_{m}(j)),
\end{equation}
where $j = 1, 2, ...,$ $E\frac{D_m}{b}$ represents the index of local update for batch $b$ and the epoch count $E$, and
\begin{equation}
F_m(\mathbf{\boldsymbol{V}_m}) \triangleq \frac{1}{{D}_{m}}\sum_{i\in\mathcal{D}_{m}}{F_{i}(\mathbf{\boldsymbol{V}_m}, x_{i}, y_{i}}),
\end{equation}
denotes the local loss function, quantifying the model's error concerning local data samples, $D_m$ signifies the quantity of data examples in the gathered records (i.e., traffic logs), and $\eta$ stands for the learning rate. Upon completion of local training by all chosen SPSs, their updates are transmitted to the TPR. Then, the TPR consolidates these updates and modifies the global model given by:
\begin{equation}
{\mathbf{\boldsymbol{V}(z)}={\sum_{m=1}^{M}\frac{D_m}{D} \mathbf{\boldsymbol{V}}_{m}}\label{eq:globalAverage},}
\end{equation}
D = \sum_{m=1}^M D_m,
represents the aggregate samples across all SPSs.
Moreover, to guarantee that the attained accuracy is generalized, the TPR evaluates the uploaded models and provides incentives only to SPSs exceeding the threshold, $\psi_m$, as determined by the quality assessment algorithm discussed subsequently. SPSs that pass the quality evaluation receive rewards.
The TPR iterates this procedure for $Z$ global iterations until the shared security model reaches the target accuracy level.
In this paper, we highlight that the incentive set is constructed every global round, where the TPR aims to sustain the desired accuracy while allowing for a slight increment in succeeding rounds.
§.§ Participation Willingness
In addition to the cost of deploying the honeypot, the suppliers bear the cost of training and uploading the local models. This cost includes the time and energy consumption for the honeypot deployment as well as training and uploading their local models. As such, the TPR cannot serve all existing devices due to insufficient bandwidth sub-channels. Thus, in every FL round, just a subset of the participants $\mathcal{S}_z$ can send their updates. The chosen set is described as follows:
\begin{equation}
\mathcal{S}_z=\{m \mid S^m_z=1, m=1,2,...,M\},
\end{equation}
where $S^m_z=1$ denotes that device $m$ is in the selected set $\mathcal{S}_z$, otherwise $S^m_z=0$.
The latency among the chosen set consists of two parts: uploading latency and computing latency.
For the uploading latency, all devices have the same defense model structure determined by TPR, with a size ${{\boldsymbol{v}}_z}$ denoted by $\xi $. Generally, we consider that the orthogonal frequency division multiple access (OFDMA) scheme is used for the communication between the BS and the devices, where each $m$-th device is given bandwidth of size $\lambda^m_z B$. Additionally, based on the size of the model, the system bandwidth is split into N sub-channels given by:
N = \frac{B}{\xi},
where each participant is assigned an $1$-th sub-channel of size $\lambda^m_z B$ . As a result, the m-th device is capable of achieving a data rate of
$ \alpha_m^z=\lambda^m_z B\ln{\left( 1+\frac{P^{m}_z {|h^{m}_z|^2}}{N_0} \right)},
$ where $h^{m}_z$ represents the channel gain between the BS and client $m$, while $P^{m}_z$ denotes the transmit power of the $m$-th client, and $N_0$ denotes background noise (i.e, Additive White Gaussian Noise (AWGN)). Hence, the uploading latency is given by:
T_m^{trans}= \frac{\xi }{ \alpha_m^z}, \cdot
and the related transmission cost is defined as:
C_u = T_m^{trans} p_m^{trans},
where $p_m^{trans}$ is the transmit power.
Regarding the calculated latency, every device requires a time duration defined as:
T^m_{cmp}= E\frac{\phi_m D_m}{f_m},
to train its local model, where $\phi_m$ (cycles/sample) denotes the number of processing cycles to execute one sample, and $f_m$ (cycles/second) is the central processing unit (CPU) frequency. Accordingly, the training cost is defined as:
C_t = \hat{\kappa}_mf_m^3 T^m_{cmp},
where ${\kappa}_m$ is the capacitor's coefficient related to the chip.
As a result, the combined latency for computation and uploading for each $m$-th participant during the $z$-th round is expressed as:
T^m_{total}= T^m_{trans} + T^m_{cmp}.
In a real FL scenario, the coordinating server (i.e., TPR) establishes a time constraint, i.e., round deadline constraint, ensuring that every SPS participant completes their tasks within this specified timeframe.
Specifically, this time can be identified based on the latency of the slowest chosen SPS $m \in \mathcal{S}_z$, which is defined as:
\begin{equation}
\label{eq:deadline}
T^{max} = \max\{S^m_z(T^m_{trans} + T^m_{cmp})\}.
\end{equation}
It is worth emphasizing that while SPSs do benefit from the globally trained model, this alone does not ensure a model tailored to their unique threat landscape. Incentives encourage SPSs to actively contribute quality data, keeping the model updated and effective against evolving cybersecurity threats, leading to solving the free-rider problem in FL. Incentives also offset the operational costs of maintaining honeypots and promote diverse participation. This results in a more robust and tailored global model, rewarding contributors with a system better suited to their security needs.
§.§ SPS and TPR Utilities
To calculate the SPS utility, which involves implementing the honeypot and exchanging information associated with the VDD data (i.e., the updated local model), we initially have to compute the total incurred cost in the following manner:
\begin{equation}
C_{m} = C_d + C_t + C_u,
\end{equation}
where $C_d$ represents the cost related to running the honeypot, $C_t$ is the cost related to updating the local model utilizing the gathered logs, and $C_u$ is the cost related to uploading the local update. Accordingly, the utility of each SPS is expressed as:
\begin{equation}\label{SPS_utlity}
U_{m}=\ln(\theta_{m} R_{m}) - C_{m}.
\end{equation}
Equ. (<ref>) implies that while the reward increases, the utility does not increase at the same rate due to the diminishing returns property of the logarithmic function.
Once all SPSs taking part in the updating and uploading of the global defense model finish their tasks, the utility of the TPR is computed as follows:
\begin{equation}\label{Utility_server}
U_{TPR} = \sum_{m=1}^{M}( \theta_{m}G(V_{m})-R_{m}),
\end{equation}
where $G(V_{m})$ represents the revenue obtained from each SPS's update. Specifically, each SPS shares its model, $V_{m}$, which will add a profit, $G(V_{m})$ to TPR with associated costs.
§.§ Adversarial Model and Assumption
In this research paper, we analyze two stages of adversarial behavior using the N-BaIoT dataset. The first stage involves deploying botnets to carry out DDoS attacks by introducing networked zombies into the SPSs. The SPS implements a honeypot system to counter these threats, collect valuable information about the attack surface, and document all activities. These logs are then integrated into a collaborative model training approach, enabling information exchange with the TPR and enhancing understanding of the attacker's behavior. The second stage of adversarial conduct occurs during the FL process when a malicious SPS falsely claims superior local data quality while contributing subpar updates to the model. To develop a secure and efficient FL process that can mitigate such issues, we put forward two distinct aggregation and averaging techniques. These methods will be discussed in greater detail later in the paper, providing insight into their practical applications and advantages.
§ PROBLEM FORMULATION
The TPR aims to optimize data quality to boost the performance of the target model and assures fair incentives to all participating SPSs. This is equivalent to the optimization problem given by:
\begin{align}\label{Problem_1}
\textbf{P1.} \quad & \max _{\boldsymbol{\Pi}}: \sum_{z=1}^{Z}\ \sum_{m=1}^{M} S^m_z( \theta_{m}G(V_{m})-R_{m})
\end{align}
such that
\begin{align}
%s.t. & \nonumber\\
C1: &\ln(\theta_{m}R_{m}) - C_{m} \ge 0, \forall m \in \{1, 2, ..., M\}, \\
C2: &\ln(\theta_{m}R_{m}) - C_{m} \ge \theta_{m}R_{m'} - (C_{m}^{m'}), \nonumber\\
& \forall m'\neq m, \ m,m' \in \{1, \dots, M\}, \\
C3: & \sum_{z=1}^{Z}\sum_{m=1}^{M} R_{m} \leq B, \\
C4: & T_m^{total} < T^{max}, \quad \forall m \in \{1, 2, ..., M\},\\
C5: & R_m > 0,
\end{align}
where $C_{m}^{m'}$ is the cost of the supplier in type $m$, selecting a contract of type $m'$, and $B$ is the allocated total budget for upgrading the security defense model. In (<ref>), C1 is the individual rationality (IR) constraint in which each supplier has to gain non-negative utility. Constraint C2 is incentive compatibility (IC), in which each supplier should choose the exact incentive aligned with its type. In C3, the retailer guarantees that the rewards delivered to the participants do not surpass the allocated budget. The delay constraint given by C4 ensures that participants complete the assigned tasks in a defined period. In C5, each participating SPS should receive non-negative rewards. It is worth noting that the TPR can pay for the model upgrade from the security market if the required rewards are high and exceed its allocated budget.
Finding the direct solution for (<ref>) is extremely intricate due to the need for previous knowledge of all participants over all rounds, which is nearly unattainable. Additionally, the contracts must be adjusted based on the SPSs' participation over the training period since contributions decrease over time, and the model may converge to various stationary points, resulting in slower convergence. To address these challenges, we propose to transform the problem into an online problem in which the retailer can redesign the contracts every round. The problem is reformulated as:
\begin{align}\label{Problem_2}
\textbf{P2.} \quad & \max _{\boldsymbol{\Pi}}: \sum_{m=1}^{M} S^m_z( \theta_{m}G(V_{m})-R_{m})
\end{align}
such that
\begin{align}
% s.t. & \nonumber\\
C1: &\ln(\theta_{m}R_{m}) - C_{m} \ge 0, \forall m \in \{1, \dots, M\}, \\
C2: &\ln(\theta_{m}R_{m}) - C_{m} \ge \ln(\theta_{m}R_{m'}) - C_{m}^{m'}, \nonumber\\
&\forall m'\neq m, \ m,m' \in \{1, \dots, M\}, \\
C3: & \sum_{m=1}^{M} R_{m} \leq B_z, \\
C4: & T_m^{total} < T^{max} \quad \forall m \in \{1, 2, ..., M\},\\
C5: & R_m > 0,
\end{align}
where $B_z$ is the budget allocated for round $z$. For example, $B_z = \frac{B}{Z}$, if we aim to allocate a fixed budget in each round, it can be adjusted depending on the TPR's gain.
Clearly, (<ref>) is intractable, and the constraints are coupled. In particular, constraint C1 in (<ref>) related to IR and $M(M-1)$ constraints related to IC make finding a direct solution for (<ref>) very challenging. Thus, we start by reducing the IR and IC constraints. Then, we propose FedPot, a framework that includes tractable solutions to these challenges. FedPot is divided into two blocks, one on the SPS side and one on the TPR side. On the TPR side, we initially use the local data quality to ensure maximum utility for the TPR. Then we relax the problem to enable the TPR to select the optimal participating SPS. Next, we propose two averaging schemes considering the security aspects. To ensure fair incentives, we also proposed two reward schemes based on the claimed data quality and the contribution to the global model.
§ PROPOSED SOLUTION
This section introduces the proposed solutions, including how we reduced the constraints' complexity, the prior quality determination of the local VDD, the trusted and untrusted model averaging schemes, and the rewards distribution.
The proposed solution for the SPS side starts by deploying a honeypot on the SPS side to record all network traffic into log files. The logs are then transformed into a readable format (i.e., CSV) and cleaned with an extensive preprocessing step. It is worth mentioning that the practice of anonymizing data (i.e., CSV files) is a commonly used method to protect sensitive information. However, several factors motivate us to use FL rather than simply anonymizing the data. First, FL mitigates the risk of data re-identification, a vulnerability in anonymization. Second, unlike anonymized CSV files, FL only shares aggregated model parameters, preserving data patterns while reducing exposure risks. Third, FL minimizes the chance of large-scale breaches by avoiding centralized data storage. Lastly, FL enables collaborative learning across distributed honeypots without centralization, reducing communication and storage costs. We propose a local evaluation scheme in which each SPS evaluates its local VDD to determine whether to participate in the FL learning process. It is important to mention that the SPS is willing to participate only if the required quality is achieved. <Ref> illustrates the whole procedure performed at both the SPS and TPR.
FedPot Architecture.
§.§ Constraints complexity reduction
As stated previously, it is almost intractable to directly solve (<ref>). To simplify, we present the subsequent lemmas.
Given the budget for each round $B_z$, for any feasible contract $(\boldsymbol{V}_m, R_m)$, $R_m \geq R_{m'}$ $\iff$ $m \geq m'$ $\forall \quad m, m' \in \{1, 2, ..., M\}$
we first prove that if $\theta_m \geq \theta_{m'}$ where $m \geq m',$ then $R_m \geq R_{m'}$. Adding the IC constraints for both types $\theta_m$ and $\theta_{m'}$, yields:
\begin{equation}
\label{pro1}
\ln(\theta_m R_m ) - C_m \geq \ln(\theta_m R_{m'}) - C_{m'},
\end{equation}
\begin{equation}
\label{pro2}
\ln(\theta_{m'} R_{m'}) - C_{m'} \geq \ln(\theta_{m'} R_{m}) - C_{m},
\end{equation}
We add both (<ref>) and (<ref>), we have:
\begin{equation}
\label{pro3}
\ln(\theta_m - \theta_{m'})(R_m-R_{m'}) \geq 0.
\end{equation}
Thus, $R_m \geq R_{m'}$.
From <Ref>, we note that more rewards will be given to more participants. This means that <Ref> is monotonic. Hence, this analysis of IC constraints reduces the IR constraints as indicated in the following lemma.
Given the $B_z$, and <Ref> related to IC constraints and the sorted participants based on their data quality, the IR condition can be reduced as:
\begin{equation}
\ln(\theta_{1}R_{1}) - C_{1} \ge 0
\end{equation}
Given the sorted SPSs based on their types as defined in <Ref>, the IC constraints are used as follows:
\begin{equation}
\ln(\theta_m R_m ) - C_m \geq \ln(\theta_m R_1) - C_1 \geq \ln(\theta_1 R_1) - C_1 \geq 0.
\end{equation}
From (<ref>), we note that if the first SPS's type meets the IR constraint, all other SPSs' types will automatically meet other IR constraints. Therefore, the IR of type $1$ is sufficient to achieve all other IR constraints.
According to <Ref>, we can transform the IC constraint into local downward incentive compatibility (LDIC) as follows [34]:
\begin{equation}
\label{eq:LDIC}
\ln(\theta_{m}R_{m}) - C_{m} \ge \ln(\theta_{m}R_{m-1}) - C_{m-1}, \forall n \in \{2, 3, ..., M\},
\end{equation}
and local upward incentive compatibility (LUIC) as follows:
\begin{equation}
\label{eq:LUIC}
\ln(\theta_{m}R_{m}) - C_{m} \le \ln(\theta_{m}R_{m+1}) - C_{m+1}, \forall n \in \{1, 2, ..., M-1\}.
\end{equation}
From <Ref>, we note that participants should be given rewards based solely on their contributions. Moreover, we observe that the objective function in (<ref>) is decreasing w.r.t $R_m$ and increasing w.r.t $G(V_{m})$.
§.§ Quality Measure of the SPSs' VDD
The SPSs asymmetrically exchange information, sharing only the local model updated using the collected VDD with the TPR, rather than the VDD itself. Hence, the TPR is necessary to validate the uploaded models.
The quality of the local model depends on several factors, including the variety of the data, the number of contained classes, and the number of updates performed.
In this regard, we first model the SPS's data by considering the data quality, and , then by assessing the impact of the local models on the generalization of the global model. This strategy allows the TPR to guarantee a robust model even in the presence of malicious participants.
Each local dataset $\mathcal{D}m$ generally comprises traffic data with input-output pairs of $\{\mathbf{x}_{i,d}^{(m)},y_i^{(m)}\}^{D_m}_{i=1}$, where $\mathbf{x}_{i,d}^{(m)} \in \mathbb{R}^d$ denotes the input holding $d$ attributes, and $y_i^{(m)} \in \mathbb{R}$ denotes the matching class label.
Given that every SPS has its own network traffic patterns and a varying response according to its activity, the honeypot logs produce data of various sizes and follow a non-identical and independently distributed (non-i.i.d.) manner.
Furthermore, the logs in each SPS may have distinct attack types, and some of the SPSs might not have any malicious samples.
Let $\mathcal{D} = \{\mathcal{D}_1 \cup \mathcal{D}_2 \cup \ldots, \cup~\mathcal{D}_M\}$ represent the aggregated dataset from all SPSs. We define the probability of a sample, $i$, being included in the local logs as:
\begin{equation}
\rho(\mathcal{D}_m|i) = \left\{
\begin{array}{l}
1: \mathrm{i \in \mathcal{D}_m}\\
0: \mathrm{otherwise}
\end{array}.
\right.
\label{eq1}
\end{equation}
With a predetermined radius, the coverage of data collection through sampling is obtained by:
\begin{equation}
%C_{D}^{\mu}(\epsilon) =
\rho(\mathcal{D}_m,\delta)= \mathcal{D} \cap \cup_{x_{i} \in \mathcal{D}_m} \Lambda(x_{i}, \delta),
\end{equation}
where $\Lambda(x_{i}, \delta)$ is an open ball space with a radius of $\delta$ centered at $x_i$.
Considering the data space among all SPSs is an Euclidean space, the range of $\delta$ is confined to the interval $[0,\sqrt{d}]$.
, $\varphi(\mathcal{D}_m)$ gives the estimation of the local VDD quality measurements defined as:
\begin{equation}
\varphi(\mathcal{D}_m)=\frac{1}{\sqrt{d}}\int_{0}^{\sqrt{d}} \rho(\mathcal{D}_m,\delta) \ {\rm d}\delta
\end{equation}
The value of $\varphi(\mathcal{D}_m)$ can indicate the quality of the local VDD. A more increased value of $\varphi(\mathcal{D}_m)$ implies improved data diversity, resulting in a better quality of the locally updated model. This is because greater spaces between traffic samples grasp more patterns during model training, enhancing the overall generalization of the global model. Let $\phi={\varphi_1,...\varphi_{M}}$ be the quality set of all SPS with $M$ types, and all SPSs with $\varphi(\mathcal{D}_m)\in[\frac{m-1}{M},\frac{m}{M}]$ hold a data of type $m$.
To solve <Ref>, the TPR can leverage each participant's previous performance, which might be time-consuming since assessing all upgraded models is a post-processing action. As a result, the quantity of $\varphi(\mathcal{D}_m)$ can be utilized to measure local VDD quality before a given SPS is selected.
This approach allows the TPR to select high-quality data and avoid using models from participants with low-quality data, which can negatively impact the global model's performance. By employing this technique, the TPR can ensure that the final model is robust, even in the presence of malicious participants.
We can substitute $\pi_m=(\boldsymbol{V}_m, R_m)$ in <Ref> by $\pi_m=(\boldsymbol{\Omega}, R_m)$. Accordingly, we can further reformulate the optimization problem in (<ref>) as follows:
\begin{align}\label{Problem_3}
\textbf{P3.} \quad & \max _{\boldsymbol{\Pi}}: \sum_{m=1}^{M} S^m_z\varphi(\mathcal{D}_m)
\end{align}
such that
\begin{align}
% s.t. & \nonumber\\
C1: &\ln(\theta_{1}R_{1}) - C_{1} \ge 0, \\
C2: & \ln(\theta_{m}R_{m}) - C_{m} \ge \ln(\theta_{m}R_{m-1}) - C_{m-1}, \forall m, \\
C3: & \ln(\theta_{m}R_{m}) - C_{m} \le \ln(\theta_{m}R_{m}) - C_{m+1}, \forall m, \\
C4: & \sum_{m=1}^{M} R_{m} \leq B_z, \\
C5: & T_m^{total} \leq T^{max} \quad \forall m \in \{1, 2, ..., M\},\\
C6: & R_m > 0.
\end{align}
It is worth mentioning that the optimization problem (P3) considers the lure score to adaptively place honeypots in the network to maximize their luring potential. As in our proposed approach, the lure score can be calculated based on various factors such as network traffic patterns, historical attack vectors, and even machine learning models trained for this specific purpose. <Ref> illustrates how the luring score can be determined. The system ensures that honeypots are effectively deployed by considering the lure score, which will most likely capture meaningful data and contribute to better FL. Specifically, high-quality data, signifying that the user’s honeypot has successfully deceived the attackers, is defined by a luring score. This leads to the development of more accurate and comprehensive models, which are essential for detecting and mitigating sophisticated cyber threats.
Schematic illustrating how the luring score can be determined
However, we note that the problem in (<ref>) is still challenging to be solved directly due to the monotonic constraints in C1, C2, and C3. The authors in [34] proposed relaxing such constraints, enabling the problem to be directly solvable as an optimization problem. However, in the context of FL, it is not practical to apply such solutions due to the following:
* The server cannot adequately analyze local updates even if all selected SPSs are trusted. The VDD quality metric can only be utilized as described in <Ref> only if all participants are completely trusted.
* Similar data may have varying contributions during the global training rounds, wherein it could initially make a higher contribution but gets reduced over time. For instance, the training process slows down if the model is closer to its stationary point, regardless of the associated cost or local data quality.
Thus, the rewards should be given based on the value added to the global model every round, not as a fixed contract.
* The server should ensure fairness between the participants where the higher cost does not imply higher quality.
In the following section, we solve this problem iteratively by implementing a smooth FL process, including novel selection, aggregation, and averaging schemes. Then, we design a fair and efficient postprocessing rewards mechanism that fully ensures fairness so that the constraints (36-39) can be implicitly satisfied. The introduced solutions incorporate three phases; the first phase (i.e., the Preprocessing Phase) aims to solve an optimized relaxation problem to select the proper participants. The selected participants receive the global model, update it locally, and upload it to the TPR. In the second phase, the locally upgraded models by all participating SPSs are aggregated. The TPR evaluates the received models using two schemes (discussed below) and assigns a contribution rank for each model. In the third phase, the enhanced FedAvg based on adopted weights is applied under two scenarios when the participants are fully trusted and when some participants are malicious. The rewards are then given based on the contribution rank given to each participant (i.e., based on the uploaded model).
§.§ Quality-Assurance Model Averaging
The phases of the proposed solution are as follows. By resolving (<ref>) during the initial phase (i.e., the preprocessing phase), it attempts to identify the appropriate SPSs. The identified participants will then receive the global model for local updating. The second phase combines the local updates from all SPSs to mitigate the challenges arising using the traditional FedAvg approach being assessed in two different settings.
Trust-based Model Averaging
The TPR trusts all participants, and the weight of each participant is determined according to their reported data quality:
\begin{equation}
\label{eq:trusted}
v_m = \frac{\varphi(\mathcal{D}_m)}{\sum_{m=1}^{M}\varphi(\mathcal{D}_m)}
\end{equation}
Following that, the global model is generated and updated given by
\begin{equation}
{\mathbf{\boldsymbol{V}(z)}={\sum_{m=1}^{M}v_m \mathbf{\boldsymbol{V}}_{m}}\label{eq:globalAveragemode1}.}
\end{equation}
In contrast to the conventional FedAvg, which utilizes the amount of data samples, this enhanced FedAvg does model averaging by weighting the local models according to the local data quality. Practically, weighting the model parameters according to the amount of data is futile since the data may contain redundancies and slightly affect the global model. Further details can be found in <Ref>, where the conventional FedAvg approach demonstrates inferior performance.
§.§ Two-steps verification mechanism
In this approach, the retailer initially selects the participating suppliers based on their local data quality. As explained in <Ref>, the higher the data quality, the better the model update. Thus, the server adopts the value of $\varphi(\mathcal{D}_m)$ to prioritize the suppliers and select the best set. However, the malicious supplier may claim higher data quality. Therefore, the server needs to apply a further verification step by testing the model update before confirming the supplier's reward. This can be done by estimating the Euclidean distance between two consecutive updates of the global model and the updated local model, respectively, or by having global data samples and feeding them into the uploaded models one by one before conducting the global averaging.
Untrust-based Model Averaging In this scenario, some participating SPSs might declare to have valuable data, yet the shared models may be perturbed or generated randomly. Consequently, a two-step verification is proposed, where the TPR initially assesses the obtained model employing the Euclidean distance or utilizing generalized test data evaluating every model independently. Assuming the latter is utilized, the adjusted averaging weights can be determined using the formula as follows:
\begin{equation}
\label{eq:untrusted}
w_m = \frac{G(v_{m})}{\sum_{m=1}^{M}G(v_{m})}
\end{equation}
Following that, the model is generated and updated based on
§.§ Incentive Reward design
We propose applying the Soft-max function to determine the incentives provided to the selected SPSs after computing the authentic average weights employing one of the methods outlined in the preceding section. This will guarantee equality according to the contribution provided by each SPS. It can be expressed as follows:
\begin{equation}
\label{eq:rewardcal}
% \upsilon_m = \frac{e^{v_m}}{\sum_{m=1}^{M}
w_m = \frac{e^{v_m}}{\sum_{m=1}^{M}
\end{equation}
It is important to note that for each round $z$, the total rewards will be less than the budget assigned. As a result, the participants receive the earned rewards from:
\begin{equation}
\label{eq:rewads_given}
R_m = w_m B_z.
\end{equation}
We may infer from (<ref>) and (<ref>) that the obtained rewards ensure equality for all SPSs participants and completely meet the assigned budget. It is worth noting that our approach considers the heterogeneity of threats and their distribution implying that most SPSs, over a reasonable time frame, are exposed to some kind of malicious activity. It might be accurate that at certain times, specific SPS face a higher volume or intensity of attacks, but when viewed over an extended period, the distribution becomes relatively even. We have designed the reward mechanism to factor in not just the volume but also the quality of the data. This way, even if an SPS does not face a high number of attacks, the uniqueness or novelty of a single quality data point they contribute could be of immense value, ensuring they are adequately rewarded. Alg. <ref> outlines the overall process of the suggested approach.
In Alg. <ref>, the TPR starts by initializing the model and the hyper-parameters (step 1), then gathering prior information from all SPSs (step 2). From steps 3–17, the TPR runs the FL algorithm by determining the deadline for each round (step 4) and solving the relaxed P3 (step 5) to select the best participants while accounting for the stated data quality. The global model is then sent (step 6), and all selected participants undertake local training and submit all local models once finished (steps 7-10). The TPR aggregates all models (step 11) before deciding on one of the proposed schemes. If the TPR can confirm the trustworthiness of the participating SPSs, the solution in <Ref> is applied (steps 12 and 13), and the solution in <Ref> is applied (steps 15 and 16) if TPR can not. Finally, as explained in <Ref>, the rewards are fairly given to the participants efficiently.
FedPot Framework
All available SPSs $M$
Upgraded Security Model $V$
starting global model $V_0$, local epochs $\varepsilon$, step size $\eta$, rounds $Z$, and assigned budget for every global round
TPR collects previous information (e.g, $\varphi(\mathcal{D}_m)$, from existing SPSs $M$.
$z=1$ to $Z$
The TPR establishes the time limit (i.e., deadline)
TPR addresses the optimization problem, P3, to choose the best entities for participating in the model update
TPR shares the model $V_{z-1}$ to chosen SPSs
Each SPS $m \in M$ synchronously
SPS $m$ gets $V_{z-1}$
SPS $m$ updates $m$ employing its VDD obtained data for $E$ epochs
SPS $m$ uploads $V_m$ to TPR
The TPR gathers all submitted updates from participating SPSs
Every participant is trusted
TPR employs Equation (<ref>) to recompute the weight associated with each update.
TPR assesses each model using the generalization test data
TPR utilizes Equation (<ref>) to recompute the weight for every update
TPR utilizes equation (39) to generate a modified global model.
§.§ Architectural Mapping of the Proposed Solution
It is worth noting that our control framework is designed to emulate the hierarchical architecture commonly observed in modern SG systems. It comprises three main layers: the Data Acquisition Layer, the Control Layer, and the Decision Layer.
Data Acquisition Layer: This is the base layer where all sensors and IoT devices (as represented by our use of the N-BaIoT dataset in Section VI) are located. These devices are responsible for collecting real-time information such as voltage, current, and frequency from different points in the grid.
Control Layer: where the real-time data is analyzed and control signals are generated. This layer integrates the security protocols as studied through the IEC 104 dataset in section VI-C. The layer includes controllers like Remote Terminal Units (RTUs) and programmable logic controllers (PLCs), which interact directly with the devices in the Data Acquisition Layer. The control layer continuously monitors and analyzes real-time data, identifying potential cyber threats or operational anomalies. This layer generates signals, executes responses to mitigate threats, and integrates various security protocols. A key feature is the strategic deployment of honeypots within the network. These honeypots serve as decoy systems designed to attract cyber attackers. This allows for an in-depth analysis of attack strategies, significantly enhancing the grid's cyber resilience and providing valuable insights into potential vulnerabilities.
Decision Layer: This is the topmost layer, consisting of control centers or cloud-based systems where higher-level decision-making processes occur. Here, our proposed anomaly detection algorithms and security measures are implemented (i.e., ML-based detector).
To generate realistic grid scenarios, we employed both the N-BaIoT and IEC 104 datasets, creating a diverse set of operating conditions and cyber-attack vectors. Our framework is also designed to be scalable and robust, which allows for the integration of additional sensors and control units as required. The proposed incentive-based model is implemented at the Decision Layer, ensuring that it benefits from the real-time data collected and analyzed at the lower layers and addressing the free-rider problem. This facilitates more effective and timely decision-making. It is worth noting that in this layer of the context of cybersecurity, test datasets cannot remain static. When a new type of attack is detected by a honeypot, it may initially be evaluated through heuristics, expert rules, and anomaly detection methods for interim validation. Contextual information provided by the SPS also helps the TPR to assess the contribution's potential impact. To corroborate new threats, we propose sharing them (anonymously) with a subset of trusted SPS for additional verification. As these new types of attacks are validated, they will be incorporated into future iterations of the test data, ensuring a relevant benchmark for subsequent evaluations.
§.§ Theoretical Analysis of Robustness Against Malicious SPSs
To validate the resilience of our Quality-Assurance Model Averaging and Two-step Verification Mechanism against malicious activity, we introduce some theoretical metrics and analysis.
Let \( \Delta V(z) \) denote the deviation of the global model \(\boldsymbol{V}(z)\) under the Trust-based Model Averaging scheme given by (<ref>):
\begin{equation}
\Delta V(z) = \left\| \boldsymbol{V}(z) - \boldsymbol{V}^* \right\|
\end{equation}
Here, \(\boldsymbol{V}^*\) represents the optimal global model that would have been achieved without any malicious activity. We aim to bound \(\Delta V(z)\):
\begin{equation}
\Delta V(z) \leq f\left(\varphi(\mathcal{D}_m), M, v_m, \ldots \right)
\end{equation}
Here, \(f\) is a function that encapsulates the contributions of local data quality \(\varphi(\mathcal{D}_m)\), the total number of participants \(M\), and their respective weights \(v_m\), among other parameters. Similarly, we define \(\Delta W(z)\) as the deviation under the Untrust-based Model Averaging scheme given by (<ref>).
\begin{equation}
\Delta W(z) = \left\| \boldsymbol{V}(z) - \boldsymbol{V}^* \right\|
\end{equation}
We aim to bound \(\Delta W(z)\) as follows:
\begin{equation}
\Delta W(z) \leq g\left( G(v_m), M, w_m, \ldots \right)
\end{equation}
Here, \(g\) is a function capturing the effects of the generalized test \(G(v_m)\), total number of participants \(M\), and the adjusted weights \(w_m\), among others. In essence, the bounded nature of \(\Delta V(z)\) and \(\Delta W(z)\) implies that our proposed schemes are robust against a variable number of malicious SPSs, maintaining the global model's efficacy.
§ PERFORMANCE EVALUATION
In this section, we establish an experimental setting to evaluate the efficacy of our proposed methods for improving the security defense model and allocating fair incentives.
§.§ Experimental Setup
Label distribution among the logs of the first three devices.
We employ the N-BaIoT [35] as well as IEC 104 and IEC MMS datasets[36]. N-BaIoT was initially designed to cover the security aspects of IoT devices. This dataset is crucial for modern SG systems increasingly relying on IoT sensors and actuators for efficient and intelligent grid management. It includes data from benign operations as well as from a range of attacks like Mirai and Gafgyt (Bashlite). Mirai attacks focus on different flooding techniques using ACK, SYN, UDP, and other protocols while performing device vulnerability scans. Gafgyt attacks, similarly, engage in device scanning and deploy a mix of attack strategies, including TCP and UDP flooding. As in <Ref> for the first three devices, it consists of eleven classes, one benign, while the remaining ten represent adversarial classes (i.e., indicating malicious data), with each sample including 115 attributes. These datasets, in particular, include DDoS, which causes networks to become overloaded with traffic; bot attacks that take advantage of device flaws; man-in-the-middle attacks that intercept communications; SQL injection that manipulates databases; and XSS which injects malicious scripts. Firmware flashing, physical tampering, and side-channel attacks also represent advanced threats. In SG, DDoS attacks can disrupt grid communications, Bot attacks may compromise grid control systems, and MitM attacks pose risks to data integrity. SQL injection and XSS are relevant where SGs use web interfaces or databases. Firmware flashing and physical Tampering highlight the need to secure grid hardware, while side-channel attacks demonstrate the importance of protecting against indirect data leaks.
We compare our proposed method with the traditional FedAvg technique, where the server allocates weights to the local model updates based on the amount of traffic data samples they include. Furthermore, we evaluate the incentives-based algorithm against a customized-based algorithm.
We employ a supervised deep neural network (DNN) model consisting of 115 input layers followed by 115, 62, and 32 hidden layers in which $\frac{1}{2}$, $\frac{1}{4}$, and $\frac{1}{8}$ inputs follow each hidden layer, respectively. The cross-entropy is used as a loss function.
We initially consider the N-BaIoT dataset's initial device count, consisting of nine devices exhibiting non-i.i.d. characteristics.
Similar parameters are adopted for all experiments. There are eight SPSs as participants in the FL configurations, with each SPS possessing one out of the nine devices present in the N-BaIoT datasets, including both benign and adversarial samples. One device is solely utilized to evaluate the overall model developed by each of the other eight participants. We conduct each experiment five times to ensure the accuracy of the findings. Then, 50 devices are added to the data distribution. Every device uses a $10$ local epoch and a $32$ local batch size. We employ an adjustable learning rate with a default value of $0.01$.
We assess the efficiency of our proposed scheme using the following metrics accuracy, loss, true positive rate (TPRate), false positive rate (FPR), and F1-measure given by:
TPRate =\frac{TP}{TP+FN},
$ $
TNR = \frac{TN}{TN+FP}
$, and $
F1{-}score = \frac{2 * Precision * Recall}{Precision + Recall}
$. To assess the model's efficacy, we utilize the testing accuracy, which measures the proportion of accurately identified used incursions as a performance indicator.
Finally, we use rewards distribution fairness to showcase the efficiency of the proposed rewards allocation framework.
§.§ Numerical Results on BaIoT dataset
In this section, we perform our numerical analysis on IDD and non-i.i.d assumptions.
§.§.§ The IID Setting
<Ref> provides an overview of the trust-based model averaging framework performance under several distinct settings: complete data offloading to TPR (centralized training), the conventional FedAvg, and the proposed enhanced-FedAvg (FedAvg based on data quality).
The results of the proposed algorithm demonstrate that it outperforms the conventional FL approach, which can be attributed to the effect of assigning model weights according to the local data quality instead of the data size as in conventional FedAvg. Furthermore, the enhanced FedAvg scheme can attain almost traditional centralized performance. In contrast, the traditional FedAvg algorithm needs more rounds to obtain the required accuracy, which increases the cost, leading to surpassing the budget assigned.
Comparative Analysis of Test Accuracy: Enhanced FedAvg, Conventional FedAvg, and Centralized-Based Algorithms.
9 SPSs
50 SPSs
100 SPSs
Test Accuracy Comparison of Proposed Schemes in the Presence of Malicious SPSs Claiming High-Quality Data While Uploading Defective Models (i.i.d. Data Distribution, N-BaIoT dataset).
<Ref> shows the impact of the untrust-based model averaging framework under two scenarios: when all participants are completely trusted and when some are malicious. Our observations reveal that depending solely on the data quality asserted by SPSs, the presence of corrupted models leads to subpar models. This outcome arises due to malicious participants who assert high-quality data yet disperse it arbitrarily or manipulate local models.
The suggested untrusted model averaging approach may efficiently remove detrimental models by assigning weights to the shared models according to the generalization test data.
This causes the effects of adversarial or disturbed models to vanish during the averaging phase or at least reduces their negative impact on the designed global model utilized for NIDS.
Contrarily, malicious participants may significantly reduce the defensive model's effectiveness, despite the data quality they claimed throughout the selection process. We repeat similar experiments while splitting the data between 50 and 100 SPSs to showcase the effectiveness of the proposed approaches.
As shown in <Ref>, we use similar settings but distribute the data across 50 and 100 SPSs and keep the labels in an i.i.d fashion. We follow the original data distribution, where the labels across the SPSs follow the same distribution. We observe that even if the data is i.i.d, the presence of some malicious participants affects the performance of the models, especially as the number of SPSs increases. Nevertheless, the untrust-based averaging scheme still performs better than the trust-based scheme. Yet, the accuracy is slightly decreased as the number of SPSs increases.
Furthermore, we assess the efficacy of the proposed approaches regarding the fairness of the reward using the true positive rate (TPRate) and the true negative rate (TNR), critical performance indicators for the rewards allocation and detection rate, and in the security defense model. <Ref> shows that, whether malicious participants are present or not, both proposed approaches exceed the traditional FedAvg. Nonetheless, in the presence of malicious participants, the proposed trust-based model averaging (PTrusted) performance exhibits a considerable decline in TPRate, TNR, and fairness of rewards. This degradation occurs as we solely depend on the claimed data quality.
On the other hand, the proposed untrust-based model averaging ensures adequate outcomes despite malicious participants. This is achieved through a two-step verification process that excludes altered models and accordingly assigns rewards, thereby achieving a high level of fairness. Meanwhile, FedAvg distributes the incentives randomly depending on the data size.
Comparative Results for TPRate and TNR of Conventional FedAvg and Proposed Solutions Under Attack and Non-Attack Scenarios (i.i.d. Data Distribution, N-BaIoT dataset).
Alg. 2p0.1cm Con. FedAvg 2|p0.1cm P-Trusted 2|p0.1cm P-Untrusted
Malicious SPSs No Yes No Yes No Yes
TPRate 89.7 52.9 92.95 76.3 95 93
TNR 62.1 30.02 85 61.9 90.85 88
Fairness 50.01 31 93.9 55 95.1 94.7
Test Accuracy Comparison of Proposed Schemes in the Presence of Malicious SPSs Claiming High-Quality Data While Uploading Defective Models (non-I.I.D. Data Distribution, N-BaIoT dataset).
§.§.§ The Non-IID setting
For the non-i.i.d. setting, we assume that each SPS holds only a maximum of 2 types of attacks (i.e., Mirai UDP and combo). We consider three scenarios regarding the number of devices: 9, 50, and 100. We assign only a maximum of two classes for each SPS to ensure that the datasets amongst SPSs are non-i.i.d. The default data distribution is considered non-i.i.d. However, as illustrated in <Ref>, the labels' distribution across all devices is almost identical, even for those holding only 6 classes. Hence, in this work, we consider a realistic non-i.i.d. distribution to evaluate the performance in the worst situations.
As shown in <Ref>, we initially run the first scenario when the number of devices is only 9. One can see that the performance in general drops by approximately 15% for both proposed averaging schemes. However, when some participants maliciously claimed high-quality local logs during the selection phase and uploaded perturbed models, the performance of the first proposed scheme (i.e., trust-based model averaging) dramatically dropped by almost 50% to reach 46% accuracy. This is due to adopting inaccurate weights during the evaluation phase, thereby impacting the performance of the global model. In contrast, the second proposed scheme (i.e., untrust-based model averaging) shows immunity against the perturbed models even when the data is non-i.i.d.
In <Ref>, we repeat the same experiments, distributing the data across 50 and 100 SPSs, respectively. From a security perspective, the performance in both figures almost matches the performance in <Ref>, where the second scheme outperforms the first averaging scheme when attacks are present. However, the convergence becomes slightly slower as the number of SPSs grows.
In the case of massively distributed data due to the increased number of SPSs, more rounds are required to capture patterns from all available SPSs.
Comparative Results for TPRate and TNR of Conventional FedAvg and Proposed Solutions Under Attack and Non-Attack Scenarios (Non-i.i.d. Data Distribution, N-BaIoT Dataset).
Alg. 2p0.1cm Con. FedAvg 2|p0.1cm P-Trusted 2|p0.1cm P-Untrusted
Malicious SPSs No Yes No Yes No Yes
TPRate 78.3 31.6 84.5 34.2 85.2 81.7
TNR 56.4 31.4 82.6 46.2 82.3 79.1
Fairness 51.2 19.7 97.1 41.2 98 96.9
We also investigate the effect of all proposed schemes, including the reward mechanisms, on the overall performance in terms of detection rate (i.e., TPRate and TNR) and the fairness of reward distribution. From <Ref>, it can be shown that proposed averaging methods significantly surpass the conventional FedAvg, whether or not adversarial participants are present. However, when particular participants are adversarial, the proposed trust-based model averaging (P-Trusted) suffers drastically in terms of TPRate, TNR, and reward fairness, which is unsurprising as these are heavily based on the data quality. In contrast, the proposed untrust-based model averaging successfully filters the contaminated uploaded models using a two-stage verification process. It distributes the rewards appropriately, achieving a high degree of fairness despite the presence of adversarial participants. FedAvg, on the other hand, distributes its incentives randomly according to the amount of data gathered. Nevertheless, all schemes are affected by the data distribution amongst SPSs, as we note when comparing the i.i.d. to non-i.i.d. results.
§.§ Numerical Results on IEC 104 Dataset
The IEC 104 Dataset was released by Brno University in March 2022 and represents a significant resource in the field of SG security [36]. It comprises IEC 104 and IEC MMS headers and is primarily designed for anomaly detection and security monitoring. The dataset captures a broad spectrum of attack scenarios across different folders. For instance, the but-iec104-i folder contains various attacks, including Denial of Service (DoS), Injection, and Man-in-the-Middle (MITM) attacks. These offer a comprehensive view of the threat landscape in the Industrial.
Control Systems (ICS) communications. Another folder, vrt-iec104, presents a different set of intriguing attacks, such as value change and masquerading attacks. It is worth mentioning that The IEC 60870-104 (IEC 104) and IEC 61850 (MMS) datasets, produced by the "Security monitoring of communication in the smart grid (Bonnet)" project at the Brno University of Technology, Czech Republic (2019-2022), include CSV traces from PCAP files [36]. These datasets, derived from both real device observations and virtual application monitoring, provide a comprehensive view of normal and attack traffic patterns within smart grid environments. This dual approach, blending real-world operational traffic with simulated attack scenarios, ensures the datasets support robust model training and validation, preparing the models to detect and mitigate both existing and emerging threats effectively.
Despite its richness, the dataset was originally unlabeled and required substantial preprocessing. This step was critical to ensure that our subsequent analyses were based on accurate, well-defined data. The dataset is organized into multiple folders, each containing data related to either IEC 104 or IEC MMS headers. Each folder has a readme.txt file that provides valuable information, including data types and timestamps related to the attacks. This level of detail enabled us to better understand the structure and implications of the data.
We performed preprocessing steps through a multi-stage process to adequately prepare the data for our experimental framework. This involved labeling the samples into categories such as benign, switching, scanning, or communication interruption; cleaning the data to remove any irrelevant or inaccurate information, and finally transforming it into a format suitable for our machine learning algorithms. The encoding was significant for nominal or categorical data. The following section focuses on the numerical analysis conducted using the IEC 104 dataset. We examine performance under two conditions: the i.i.d. and non-i.i.d settings.
Test Accuracy Comparison of Proposed Schemes in the Presence of Malicious SPSs Claiming High-Quality Data While Uploading Defective Models (I.I.D. Data Distribution, IEC 104 dataset).
Comparative Results for TPRate and TNR of Conventional FedAvg and Proposed Solutions Under Attack and Non-Attack Scenarios (Non-i.i.d Data Distribution and IEC 104 Dataset).
Alg. 2p0.1cm Con. FedAvg 2|p0.1cm P-Trusted 2|p0.1cm P-Untrusted
Malicious SPSs No Yes No Yes No Yes
TPRate 81.2 28.2 86 23.8 92.7 87.3
TNR 63.1 25.1 86.2 37.4 85.3 83.6
Fairness 69.8 14.3 98.6 32.7 97.8 94.6
Test Accuracy Comparison of Proposed Schemes in the Presence of Malicious SPSs Claiming High-Quality Data While Uploading Defective Models (non-I.I.D. Data Distribution, IEC 104 dataset).
§.§.§ The IID Setting
We consider three scenarios featuring varying numbers of substations: 20, 50, and 100. <Ref> display the performance of our framework, which includes both trust-based and untrust-based schemes when the data is IID (i.e., all clients have the same data distribution). Our results clearly demonstrate the superiority of the enhanced FedAvg, which achieves performance levels nearly identical to centralized models. This can be attributed to our focus on local data quality for model weight assignment as opposed to merely considering data volume, as in traditional FedAvg. This advantage holds even when malicious participants are involved. Performance significantly drops and fluctuates when only the trust-based scheme is employed. This drop is due to the exclusive reliance on local data quality. However, our untrust-based averaging scheme effectively mitigates this by leveraging generalization test data to allocate weights, neutralizing the impact of corrupted local models.
§.§.§ The Non-IID Setting
In this setting, we assume that each SPS is limited to specific types of IEC 104 traffic patterns. <Ref> show that our proposed untrust-based averaging scheme performs exceptionally well, even in the presence of malicious activity. Trust-based model averaging struggles in this context, experiencing a significant decline in accuracy due to its reliance on claimed data quality.
Furthermore, <Ref> reveals that our approaches outperform traditional FedAvg in critical security metrics such as TPRate and TNR, especially when adversarial participants are present. The trust-based model sees a decline in these metrics due to its dependence on claimed data quality. In contrast, our untrust-based model averaging maintains robust performance, demonstrating its value in real-world scenarios where malicious activity is a concern. Notably, our proposed approaches exhibit superior performance in non-IID settings. This is due to our emphasis on evaluating the quality of returned contributions rather than just data quantity, leading to a more generalizable global model.
In summary, our numerical evaluation of the IEC 104 dataset validates the effectiveness of our proposed methodologies, particularly in non-i.i.d settings and when malicious participants are involved. The insights gained from this analysis will be crucial for refining and optimizing our models for intrusion detection in ICS.
Test Accuracy Comparison of Proposed Schemes in the Presence of Malicious SPSs Claiming High-Quality Data While Uploading Defective Models (non-I.I.D. Data Distribution, IEC 61850 (MMS) dataset).
§.§ Verfication with IEC 61850 Manufacturing Message Specification (MMS) dataset
To further verify our results, we carry out experiments using the IEC 61850 (MMS) dataset. IEC 61850 (MMS) is extensively employed in electric utility companies, predominantly for substation automation and ensuring interoperability between different manufacturers' systems. This global standard is important for streamlining the communication infrastructure of electrical substations and facilitating the integration, operation, and maintenance of diverse devices and systems in SG. Its usage spans various applications, including real-time monitoring, control of substation components, and ensuring robust, reliable data exchange. The dataset used includes both benign and malicious data samples. Remarkably, the performance trends observed with this dataset closely mirror those identified using the IEC 104 protocol data. Specifically, under non-IID settings as a most challenging scenario, our enhanced FedAvg framework, which includes both trust-based and untrust-based schemes, continued to demonstrate superior performance, closely approximating that of centralized models. This consistency highlights the robustness of our approach, particularly our novel method of weighting model updates based on the quality of local data rather than its volume. Including the IEC 61850 dataset not only verifies our initial findings but also broadens the applicability of our framework. It shows that our untrust-based averaging scheme, which assesses the quality of contributions through generalization on test data, can effectively neutralize the influence of corrupted local models, ensuring stable and reliable model performance even in the face of malicious activity. This result demonstrates the generalizability and robustness of our approach across different SG communication protocols, further validating the effectiveness of our method in enhancing security and reliability in real-world, heterogeneous SG network environments.
§ CONCLUSION
In this paper, we introduced FedPot, a novel quality assurance honeypot-based FL framework designed for network security in SG. FedPot incorporates a novel, efficient, and resilient aggregation and averaging schemes coupled with a fair rewards mechanism. We presented novel schemes for local data quality, participant selection, and global model upgrading using the N-BaIoT, IEC 104, and IEC MMS datasets. In FedPot, the TPR addresses a convex optimization problem, prioritizing data quality over data size. Each SPS optimizes the global model with its honeypot logs and transmits the model updates back to the TPR. Subsequently, the TPR enhances the defensive model using the approaches proposed in this study.
To mitigate the free-rider issue prevalent in AMI networks within the FL framework, we proposed a new metric to gauge local data quality and contributions, eliminating the need to rely on data size. We also devised a two-step verification process to address the challenge of adversaries or underperforming SPSs. Additionally, we introduced an improved FedAvg scheme for local model aggregations.
The results obtained from extensive simulations with realistic log data attest to the effectiveness of our proposed scheme, which outperforms current state-of-the-art techniques. As a direction for future research, investigating the real-time implementation and assessment of FedPot in streaming, diverse, and larger-scale environments would be insightful. Lastly, adapting the FedPot framework to address other cybersecurity threats within various IIoT applications could also be advantageous.
§ ACKNOWLEDGEMENT
This publication was made possible by NPRP Cluster project (NPRP-C) Twelve (12th) Cycle grant # NPRP12C-33905-SP-67 from the Qatar National Research Fund (a member of Qatar Foundation). The findings herein reflect the work, and are solely the responsibility, of the authors.
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[Online]. Available: <https://www.fit.vut.cz/research/project/1303/.en>
[] Abdullatif Albaseer (Member, IEEE) received an M.Sc. degree in computer networks from King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, in 2017 and a Ph.D. degree in computer science and engineering from Hamad Bin Khalifa University, Doha, Qatar, in 2022. He is a Postdoctoral Research Fellow with the Smart Cities and IoT Lab at Hamad Bin Khalifa University. He has authored and co-authored over thirty conference and journal papers in IEEE ICC, IEEE Globecom, IEEE CCNC, IEEE WCNC, and IEEE Transactions. He also has six US patents in the area of the wireless network edge. His current research interests include AI for Networking, AI for Cybersecurity, Distributed AI, and Edge LLMs.
[] Nima Abdi
received her B.Sc. in Electrical Engineering from Qatar University in 2020 and is currently pursuing an M.Sc. in Data Science and Engineering at Hamad Bin Khalifa University (HBKU). Her research focus is on the application of Artificial Intelligence on smart grid security, specifically the physical layer.
Mohamed Abdallah (Senior Member, IEEE) received his B.Sc. degree from Cairo University, Giza, Egypt, in 1996, and his M.Sc. and Ph.D. degrees from the University of Maryland at College Park, College Park, MD, USA, in 2001 and 2006, respectively.,From 2006 to 2016, he held academic and research positions with Cairo University and Texas A & M University in Qatar, Doha, Qatar. He is currently a Founding Faculty Member with the rank of Associate Professor with the College of Science and Engineering, Hamad Bin Khalifa University, Doha. He has published more than 150 journals and conferences and four book chapters and co-invented four patents. His current research interests include wireless networks, wireless security, smart grids, optical wireless communication, and blockchain applications for emerging networks. He is a recipient of the Research Fellow Excellence Award at Texas A& M University in Qatar in 2016, the Best Paper Award in multiple IEEE conferences, including IEEE BlackSeaCom 2019 and the IEEE First Workshop on Smart Grid and Renewable Energy in 2015, and the Nortel Networks Industrial Fellowship for five consecutive years, 1999–2003. His professional activities include an Associate Editor of the IEEE Transactions on Communications and the IEEE Open Access Journal of Communications, the Track Co-Chair of the IEEE VTC Fall 2019 Conference, the Technical Program Chair of the 10th International Conference on Cognitive Radio-Oriented Wireless Networks, and a technical program committee member of several major IEEE conferences.
Marwa Qaraqe (Senior Member, IEEE) is an Associate Professor within the Division of Information and Communication Technology at Hamad Bin Khalifa University's College of Science and Engineering. She completed her bachelor’s degree in Electrical Engineering at Texas A&M University in Qatar in 2010, followed by her MSc and PhD in Electrical Engineering at Texas A&M University in College Station, TX, USA, in August 2012 and May 2016, respectively. Dr. Qaraqe's research focuses on various aspects of wireless communication, signal processing, and machine learning, with applications spanning multidisciplinary areas such as security, IoT, and health. Her specific interests lie in physical layer security, federated learning across wireless networks, and employing machine learning techniques for enhancing wireless communication, security, and healthcare systems. She has been actively involved in developing physical layer security protocols for IoT networks and has secured a NATO SPS grant for her work in this domain. Additionally, Dr. Qaraqe is engaged in research exploring emerging technologies like RIS (Reconfigurable Intelligent Surfaces) and reinforcement learning to advance the capabilities of wireless communication, particularly in the context of enabling efficient and highly secure communication infrastructures for smart cities.
Saif Al-Kuwari (Senior Member, IEEE) received a Bachelor of Engineering in Computers and Networks from the University of Essex (UK) in 2006 and two PhD’s from the University of Bath and Royal Holloway, University of London (UK) in Computer Science, both in 2011. He is currently a faculty at the College of Science and Engineering at Hamad Bin Khalifa University and the director of the Qatar Center for Quantum Computing (QC2). His current research interests include, mainly, quantum cryptography and quantum machine learning. He is IET and BCS fellow, and IEEE and ACM senior member.
|
Then, $K^{-1}A$ is a compact subset of $X^{+}\times X^{+}$ which does not
contain the diagonal, so there is $\delta>0$ such that all $(x,y)\in K^{-1}A$
satisfy $d(x,y)\geq\delta$. It follows for any $(x,y)\in X_{0}\times X_{0}$
with $d(x,y)<\delta$, $d(kx,ky)<\epsilon$ for all $k\in K$. We see that
$\mathrm{prop}(h(T_{t}))\to 0$ uniformly in $h\in K$ as desired. The
continuous case is proved in the same way. ∎
###### Remark 8.15.
The inclusion $\rho\colon RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G\to
C_{L,u}^{*}(\tilde{H}_{X}\otimes B)_{\mathrm{Gcpt}}^{G}$ is never surjective
unless $G$ is finite. For example, if $G$ is compact and if $X$ is a point,
then the inclusion is identified as the inclusion
$C_{b,u}([1,\infty),\mathfrak{K}(H_{X})\otimes B)\rtimes_{r}G\to
C_{b,u}([1,\infty),(\mathfrak{K}(H_{X})\otimes B)\rtimes_{r}G)$
which is not surjective unless $G$ is finite even if $H_{X}=\mathbb{C}$ and
$B=\mathbb{C}$.
Now we go back to the classical setting when $G$ is discrete. Here is the
promised comparison between the crossed product algebra
$RL^{*}_{u}(H_{X})\rtimes_{r}G$ and the localized equivariant Roe algebra
$C^{*}_{L,u}(H_{X})^{G}$. A generalization with coefficient $B$ is possible
but we only consider $B=\mathbb{C}$ here. The continuous case can be shown too
but with an extra effort.
###### Theorem 8.16.
Let $G$ be a countable discrete group, $X$ be a proper $G$-space which is
$G$-equivariantly homotopic to a $G$-$CW$ complex and $H_{X}$ be any ample
$X$-$G$-module. Then, the inclusion $\rho\colon
RL^{*}_{u}(H_{X})\rtimes_{r}G\to
C_{L,u}^{*}(\tilde{H}_{X})_{\mathrm{Gcpt}}^{G}$ induces an isomorphism
$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X})\rtimes_{r}G)\cong
K_{\ast}(C_{L,u}^{*}(\tilde{H}_{X})_{\mathrm{Gcpt}}^{G}).$
In particular, when $X$ is $G$-compact, the inclusion $\rho\colon
RL^{*}_{u}(H_{X})\rtimes_{r}G\to C_{L,u}^{*}(\tilde{H}_{X})^{G}$ induces an
isomorphism
$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X})\rtimes_{r}G)\cong
K_{\ast}(C_{L,u}^{*}(\tilde{H}_{X})^{G}).$
###### Proof.
It suffices to show the case when $X$ is $G$-compact. For any $X$-$G$-module
$H_{X}$, let $L^{*}(H_{X})^{G}$ be the equivariant localization algebra as
defined in [48, Definition 6.5.1]. The $C^{*}$-algebra $L^{*}(H_{X})^{G}$ is
the norm completion of the $\ast$-algebra $\mathbb{L}[H_{X}]^{G}$ of all
bounded functions $T_{t}$ on $[1,\infty)$ to $\mathfrak{L}(H_{X})$ such that
1. (1)
$T_{t}$ is $G$-equivariant,
2. (2)
for any compact subset $K$ of $X$, there exists $t_{K}\geq 1$ such that for
all $t\geq t_{k}$, $\chi_{K}T_{t}$, $T_{t}\chi_{K}$ are compact, and the
functions $t\mapsto\chi_{K}T_{t}$, $t\mapsto T_{t}\chi_{K}$ are uniformly
norm-continuous when restricted to $[t_{k},\infty)$,
3. (3)
for any open neighborhood of the diagonal in $X^{+}\times X^{+}$, there exists
$t_{U}\geq 1$ such that for all $t>t_{U}$, $\mathrm{supp}(T_{t})\subset U$.
It is proved in [48, Proposition 6.6.2] that the natural inclusion induces an
isomorphism
$K_{\ast}(C_{L,u}^{*}(H_{X})^{G})\cong K_{\ast}(L^{*}(H_{X})^{G})$
whenever $H_{X}$ is ample. Thus, to prove our claim, it suffices to show that
when $X$ is $G$-compact, for any ample $X$-$G$-module $H_{X}$, the natural
inclusion
$\rho\colon RL^{*}_{u}(H_{X})\rtimes_{r}G\to C_{L,u}^{*}(\tilde{H}_{X})^{G}\to
L^{*}(\tilde{H}_{X})^{G}$
induces an isomorphism on K-theory. For this, we consider more generally, for
any not-necessary $G$-compact $X$, the equivariant localization algebra
$L^{*}(H_{X})^{G}_{\mathrm{Gcpt}}$ with $G$-compact support to be the
completion of $\mathbb{L}[H_{X}]_{\mathrm{Gcpt}}^{G}$ which is the subalgebra
of $\mathbb{L}[H_{X}]^{G}$ consisting of $T_{t}$ that has eventually uniform
$G$-compact support, that is there is $t_{0}\geq 1$ and a $G$-compact subset
$X_{0}$ of $X$ such that for all $t\geq t_{0}$, $\mathrm{supp}(T_{t})\subset
X_{0}\times X_{0}$. Then, our claim is that the natural inclusion
$\rho\colon RL^{*}_{u}(H_{X})\rtimes_{r}G\to
C_{L,u}^{*}(\tilde{H}_{X})_{\mathrm{Gcpt}}^{G}\to
L^{*}(\tilde{H}_{X})^{G}_{\mathrm{Gcpt}}$
induces an isomorphism on K-theory. Now, the point is that just as in the case
of our functor $\mathbb{D}_{\ast}^{G}$, the assignment
$X\mapsto
RK_{\ast}^{G}(X)=K_{\ast}(L^{*}(H_{X})^{G}_{\mathrm{Gcpt}}),\,\,\,f\mapsto
RK_{\ast}^{G}(f)=\mathrm{Ad}_{V^{f}_{t}\ast}$
becomes a functor from $\mathcal{PR}^{G}$ to $\mathcal{GA}$, where $H_{X}$ is
a chosen ample $X$-$G$-module for a proper $G$-space $X$ and $V^{f}_{t}$ is a
chosen $G$-equivariant continuous cover of a $G$-equivariant continuous map
$f\colon X\to Y$. As in the case of $\mathbb{D}^{G}_{\ast}$ (see the proof of
Proposition 6.2), one shows the composition law for not-necessarily proper
maps by reducing it to the $G$-compact setting after showing the
representability of the functor $RK^{G}_{\ast}$. Moreover, the functor
$RK_{\ast}^{G}$ satisfies the five listed properties in Theorem 6.4. Since
this does not involve any new idea, we explain this very briefly. The first
part of the second property (2) (induction from a finite subgroup) follows
from Proposition [48, Proposition 6.5.13] and from (3) the representability of
$RK^{G}_{\ast}$. The fourth property (4) (the Mayer–Vietoris sequence) can be
shown just as in the case of $\mathbb{D}_{\ast}^{G}$ by using the quotient
$L_{Q}^{*}(H_{X})^{G}_{\mathrm{Gcpt}}$ of $L^{*}(H_{X})^{G}_{\mathrm{Gcpt}}$
by the ideal $L_{0}^{*}(H_{X})^{G}_{\mathrm{Gcpt}}$ which is the completion of
the subalgebra of $\mathbb{L}[H_{X}]_{\mathrm{Gcpt}}^{G}$ consisting of
$T_{t}$ such that for any compact subset $K$ of $X$,
$\chi_{K}T_{t},T_{t}\chi_{K}=0$ eventually. The quotient
$L_{Q}^{*}(H_{X})^{G}_{\mathrm{Gcpt}}$ is a $C_{0}(X/G)$-algebra (see [48,
Lemma 6.4.18]) and the quotient map induces an isomorphism
$K_{\ast}(L^{*}(H_{X})^{G}_{\mathrm{Gcpt}})\cong
K_{\ast}(L_{Q}^{*}(H_{X})^{G}_{\mathrm{Gcpt}})$
because $K_{\ast}(L_{0}^{*}(H_{X})^{G}_{\mathrm{Gcpt}})=0$ (see [48, Lemma
6.5.12]). For the fifth property (5) (the homotopy invariance), the proof for
$\mathbb{D}_{\ast}^{G}$ works verbatim. It remains to prove the second part of
(2), that is
$RK^{H}_{\ast}(\mathrm{point})\cong K_{\ast}(C^{*}_{r}(H))$
for any finite group $H$. Using $K_{\ast}(C_{L,u}^{*}(H_{X})^{G})\cong
K_{\ast}(L^{*}(H_{X})^{G})$, we just need to show
$K_{\ast}(C_{b,u}([1,\infty),\mathfrak{K}(l^{2}(\mathbb{N})\otimes
l^{2}(H)))^{H})\cong K_{\ast}(C^{*}_{r}(H)),$
but since $\mathfrak{K}(l^{2}(\mathbb{N})\otimes
l^{2}(H))^{H}\cong\mathfrak{K}(l^{2}(\mathbb{N}))\otimes C^{*}_{r}(H)$, this
follows because the evaluation
$\mathrm{ev}_{1}\colon
C_{b,u}([1,\infty),\mathfrak{K}(l^{2}(\mathbb{N}))\otimes
C^{*}_{r}(H))\to\mathfrak{K}(l^{2}(\mathbb{N}))\otimes C^{*}_{r}(H)$
at $t=1$ is an isomorphism on K-theory.
Now, the inclusion
$\rho\colon RL^{*}_{u}(H_{X})\rtimes_{r}G\to
L^{*}(\tilde{H}_{X})_{\mathrm{Gcpt}}^{G}$
induces a natural transformation of the functors $\mathbb{D}^{G}_{\ast}$ to
$RK_{\ast}^{G}$. We can see that this is an isomorphism when $X=G/H$ for any
finite subgroup $H$ of $G$, by considering
$\rho\colon RL^{*}_{u}(H_{X})\rtimes_{r}G\to C_{L,u}^{*}(\tilde{H}_{X})^{G}$
instead. Indeed, at the level of K-theory, this inclusion is isomorphic to the
inclusion
$C_{b,u}([1,\infty),\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y}))\rtimes_{r}G\to
C_{b,u}([1,\infty),\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y})\rtimes_{r}G)$
where $\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y})$ is the
$G$-$C_{0}(G/H)$-algebra with fiber $\mathfrak{K}(H_{Y})$ at the coset $H$, or
more precisely
$\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y})=\\{f\in
C_{0}(G,\mathfrak{K}(H_{Y}))\mid f(gh)=h^{-1}(f(g))\,\,\text{for $h\in H$}\\}$
equipped with the left translation $G$-action. It is now easy to see that the
above inclusion is an isomorphism on K-theory because both are, at the level
of K-theory, isomorphic to
$\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y})\rtimes_{r}G$ via the evaluation at
$t=1$. Of course, we have
$K_{\ast}(\mathrm{Ind}_{H}^{G}\mathfrak{K}(H_{Y})\rtimes_{r}G)\cong
K_{\ast}(C^{*}_{r}(H))$. It follows since both functors
$\mathbb{D}^{G}_{\ast}$, $RK_{\ast}^{G}$ satisfy properties (1) - (5) in
Theorem 6.4, the natural transformation
$\rho\colon\mathbb{D}^{G}_{\ast}(X)\to RK^{G}_{\ast}(X)$
is an isomorphism for all $X$ which is $G$-equivariantly homotopic to a $G$-CW
complex. Going back to the original question, we just showed that the
inclusion $\rho$ induces an isomorphism
$K_{\ast}(RL^{*}_{u}(H_{X})\rtimes_{r}G)\cong
K_{\ast}(C_{L,u}^{*}(\tilde{H}_{X})_{\mathrm{Gcpt}})$
for all such $X$. ∎
## 9\. The forget-control map and the Baum–Connes assembly map
Let $G$ be a (second countable) locally compact group and $B$ be a separable
$G$-$C^{*}$-algebra. Let $X$ be a proper $G$-space and $H_{X}$ be an
$X$-$G$-module.
The evaluation map
$\mathrm{ev}_{1}\colon RL^{\ast}_{c}(H_{X}\otimes
B)\to\mathfrak{K}(H_{X})\otimes B$
at $1$, which we call the forget-control map, induces a $\ast$-homomorphism
$\mathrm{ev}_{1}\colon RL^{\ast}_{c}(H_{X}\otimes
B)\rtimes_{r}G\to(\mathfrak{K}(H_{X})\otimes
B)\rtimes_{r}G\cong\mathfrak{K}(H_{X})\otimes(B\rtimes_{r}G).$
Here, the isomorphism on the right is obtained by trivializing the inner
$G$-action on $\mathfrak{K}(H_{X})$.
It induces a group homomorphism (the forget-control map)
(9.1) $\mathcal{F}=\mathrm{ev}_{1\ast}\colon
K_{\ast}(RL^{\ast}_{c}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}(B\rtimes_{r}G).$
###### Proposition 9.1.
The forget-control map $\mathcal{F}$ (9.1) is functorial in $X$ and in $B$ in
a sense that the following diagrams commute. For any $G$-equivariant
continuous map $f\colon X\to Y$ and for any (if exists) $G$-equivariant
continuous cover $(V_{t}\colon H_{X}\to H_{Y})_{t\in[1,\infty)}$ of $f$,
$\textstyle{K_{\ast}(RL^{\ast}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\mathrm{Ad}_{V_{t}\ast}}$$\textstyle{K_{\ast}(B\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{K_{\ast}(RL^{\ast}_{c}(H_{Y}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B\rtimes_{r}G),}$
and for any $G$-equivariant $\ast$-homomorphism $\pi\colon B_{1}\to B_{2}$,
$\textstyle{K_{\ast}(RL^{\ast}_{c}(H_{X}\otimes
B_{1})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}}$$\scriptstyle{\pi_{\ast}}$$\textstyle{K_{\ast}(B_{1}\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\rtimes_{r}1_{\ast}}$$\textstyle{K_{\ast}(RL^{\ast}_{c}(H_{X}\otimes
B_{2})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B_{2}\rtimes_{r}G).}$
In particular, $\mathcal{F}$ $\eqref{eq_forgetK}$ induces a group homomorphism
(9.2) $\mathcal{F}\colon\mathbb{D}^{B,G}_{\ast}(X)\to K_{\ast}(B\rtimes_{r}G)$
which is natural in $X$ and in $B$.
###### Proof.
The first diagram commutes since $\mathrm{Ad}_{V_{t}}$ on
$(\mathfrak{K}(H_{X})\otimes B)\rtimes_{r}G$ is the identity on K-theory. The
second diagram commutes at the level of $\ast$-homomorphisms. ∎
###### Definition 9.2.
We call the group homomorphism $\mathcal{F}$ in (9.2), the forget-control map
for the functor $\mathbb{D}_{\ast}^{B,G}$.
In the rest of this section, our goal is to show that the forget-control map
$\mathcal{F}$ naturally factors through the Baum–Connes assembly map (see [6],
[47], [19])
(9.3) $\mu_{X}^{B,G}\colon\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)\to K_{\ast}(B\rtimes_{r}G)$
via a group homomorphism
$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)$
which we will define first. We will also show that $\rho_{X}$ is an
isomorphism for any $B$ when $G$ is a discrete group and $X$ is
$G$-equivariantly homotopy equivalent to a $G$-CW complex.
To obtain these, for technical reasons, we will use equivariant $E$-theory
$E^{G}$ of Guentner, Higson and Trout ([21], see also [24, Chapter 2]) in
place of equivariant $KK$-theory $KK^{G}$ of Kasparov [30]. For our purpose,
this is no problem because the canonical natural transformation ([32,
Appendix], see also [26, Definition 7.2])
(9.4) $KK_{\ast}^{G}(A,B)\to E_{\ast}^{G}(A,B)$
is an isomorphism when $A$ is a proper, nuclear $G$-$C^{*}$-algebra (more
generally when $A\mapsto KK_{\ast}^{G}(A,B)$ is half-exact), in particular
when $A=C_{0}(X)$ for a proper $G$-space $X$ ([32, Corollary A.3, A.4]) and
because the Baum–Connes assembly map in equivariant $E$-theory (see [21])
(9.5) $\mu^{B,G}_{X}\colon\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)\to K_{\ast}(B\rtimes_{r}G)$
is known to be equivalent to the one (9.3) in $KK^{G}$ via (9.4) (see [32,
Remark A.5]).
We first recall some materials from [21]. For any (not necessarily separable)
$G$-$C^{*}$-algebras $A$ and $B$, we let
$\mathfrak{T}(B)=C_{b}([1,\infty),B)_{\mathrm{Gcont}},\,\,\,\mathfrak{T}_{0}(B)=C_{0}([1,\infty),B).$
The asymptotic algebra of $B$ is defined as
$\mathfrak{A}(B)=\mathfrak{T}(B)/\mathfrak{T}_{0}(B).$
An (equivariant) asymptotic morphism from $A$ to $B$ is an equivariant
$\ast$-homomorphism from $A$ to $\mathfrak{A}(B)$. A homotopy of asymptotic
morphisms is given by an asymptotic morphism from $A$ to $BI=B\otimes C[0,1]$.
The set of homotopy equivalence classes of asymptotic morphisms from $A$ to
$B$ is denoted by $[[A,B]]_{1}$. More generally, $[[A,B]]_{n}$ is the set of
$n$-homotopy classes of equivariant $\ast$-homomorphisms from $A$ to
$\mathfrak{A}^{n}(B)$ where $\mathfrak{A}^{n}$ is the $n$-fold composition of
the functor $\mathfrak{A}$ with itself and an $n$-homotopy is given by an
equivariant $\ast$-homomorphism from $A$ to $\mathfrak{A}^{n}(BI)$ (see [21,
Definition 2.6]). The set $[[A,B]]$ is defined as the natural inductive limit
of $[[A,B]]_{n}$ (see [21, Definition 2.7]). If $A$ is separable, the set
$[[A,B]]$ can be naturally identified as $[[A,B]]_{1}$ ([21, Theorem 2.16]).
For any $G$-$C^{*}$-algebras $A,B,C$, we have a well-defined associative
composition law
$[[A,B]]\times[[B,C]]\to[[A,C]]$
[21, Proposition 2.12]. If $A$ is separable, the composition of asymptotic
morphisms $(\phi_{t})_{t\in[1,\infty)}\colon A\to\mathfrak{A}(B)$ and
$(\psi_{t})_{t\in[1,\infty)}\colon B\to\mathfrak{A}(C)$ can be represented by
an asymptotic morphism $\psi_{s(t)}\circ\phi_{t}\colon A\to\mathfrak{A}(C)$
where $(t\mapsto s(t))$ is an increasing function on $[1,\infty)$ such that
$s(t)\to\infty$ as $t\to\infty$ sufficiently fast [15] or alternatively by an
asymptotic morphism $\psi_{t}\circ\phi_{r(t)}\colon A\to\mathfrak{A}(C)$ where
$(t\mapsto r(t))$ is a continuous function on $[1,\infty)$ such that
$r(t)\to\infty$ sufficiently slowly [21, Lemma 2.17, Claim 2.18] (both ways of
representing the compositions are homotopic).
For any $G$-$C^{*}$-algebras $A,B,C$, we have a well-defined functor given by
the maximal tensor product with the identity
$\otimes_{\mathrm{max}}\mathrm{id}_{C}\colon[[A,B]]\to[[A\otimes_{\mathrm{max}}C,B\otimes_{\mathrm{max}}C]]$
[21, Proposition 4.4] and the maximal descent
$[[A,B]]\to[[A\rtimes_{\mathrm{max}}G,B\rtimes_{\mathrm{max}}G]]$
[21, Theorem 4.12].
Let $\mathcal{H}_{G}=l^{2}(\mathbb{N})\otimes L^{2}(G)$ and
$\Sigma=C_{0}(\mathbb{R})\otimes$ be the suspension functor. For any (not
necessarily separable) $G$-$C^{*}$-algebras $A,B$, the equivariant $E$-theory
group $E^{G}(A,B)$ [21, Definition 6.8] is defined as
$E^{G}(A,B)=[[\Sigma A\otimes\mathfrak{K}(\mathcal{H}_{G}),\Sigma
B\otimes\mathfrak{K}(\mathcal{H}_{G})]].$
If $A,B$ are separable, we define for $i=0,1$,
$E_{i}^{G}(A,B)=E^{G}(\Sigma^{i}A,B).$
For any $G$-$C^{*}$-algebras $A,B$, $E^{G}(A,B)$ is an abelian group and the
composition law
$E^{G}(A,B)\times E^{G}(B,C)\to E^{G}(A,C)$
is bilinear. In this way, we have the additive category $E^{G}$ whose objects
are $G$-$C^{*}$-algebras and the morphisms groups are $E^{G}(A,B)$ [21,
Theorem 6.9]. There are natural isomorphisms $K_{0}(B)\cong E(\mathbb{C},B)$,
$K_{1}(B)\cong E(\Sigma,B)$ for any $B$ (with trivial $G$-action). [21,
Theorem 6.24].
The Bott periodicity theorem and the half-exactness of the bi-functor
$(A,B)\mapsto E^{G}(A,B)$ are only proved for separable $A,B$. On the other
hand, as we recalled above, the composition law, the (maximal) tensor product
and the (maximal) crossed product are defined for general $A,B$. We have a
bilinear map
$E^{G}(A,D_{1}\otimes_{\mathrm{max}}B)\times
E^{G}(B\otimes_{\mathrm{max}}D_{2},C)\to
E^{G}(A\otimes_{\mathrm{max}}D_{2},D_{1}\otimes_{\mathrm{max}}C)$
given by the maximal tensor product with the identity on $\mathrm{id}_{D_{2}}$
on the first slot and with the identity on $\mathrm{id}_{D_{1}}$ on the second
slot, followed by the composition law. The maximal decent defines a group
homomorphism
$j^{G}_{\mathrm{max}}\colon E^{G}(A,B)\to
E(A\rtimes_{\mathrm{max}}G,B\rtimes_{\mathrm{max}}G)$
which is functorial in both variables. When $A$ is a proper algebra (more
generally, if $A\rtimes_{r}G=A\rtimes_{\mathrm{max}}G$), we define
$j^{G}_{r}\colon E^{G}(A,B)\to E(A\rtimes_{r}G,B\rtimes_{r}G)$
by the composition of $j^{G}_{\mathrm{max}}$ and the map
$E(A\rtimes_{r}G,B\rtimes_{\mathrm{max}}G)\to E(A\rtimes_{r}G,B\rtimes_{r}G)$
induced by the quotient map $B\rtimes_{\mathrm{max}}G\to B\rtimes_{r}G$.
For any separable $G$-Hilbert space $H$, any asymptotic morphism $\phi\colon
A\to\mathfrak{A}(B\otimes\mathfrak{K}(H))$ defines an element in $E^{G}(A,B)$.
This is given by the suspension of the tensor product
$\phi\otimes\mathrm{id}_{\mathfrak{K}(\mathcal{H}_{G})}\colon
A\otimes\mathfrak{K}(\mathcal{H}_{G})\to\mathfrak{A}(B\otimes\mathfrak{K}(H)\otimes\mathfrak{K}(\mathcal{H}_{G}))\cong\mathfrak{A}(B\otimes\mathfrak{K}(\mathcal{H}_{G}))$
using any isomorphism $\mathcal{H}\otimes\mathcal{H}_{G}\cong\mathcal{H}_{G}$.
With these in mind, we do some preparation for constructing a group
homomorphism
$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B).$
###### Lemma 9.3.
For any $X$-$G$-module $H_{X}$ and for any $T\in C_{L,c}^{*}(H_{X}\otimes
B)_{\mathrm{Gcpt}}^{G}$
$\phi T\in C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B)),\,\,[\phi,T]\in
C_{0}([1,\infty),\mathfrak{K}(H_{X}\otimes B))$
holds for any $\phi\in C_{0}(X)$.
###### Proof.
The first condition follows since $T_{t}$ is locally compact for each $t$. The
second one holds because for any $T$ in the dense subalgebra
$\mathbb{C}_{L,c}(H_{X}\otimes B)_{\mathrm{Gcpt}}^{G}$ which satisfies the
second condition (2) in Definition 8.10, we have $\lVert[\phi,T_{t}]\rVert\to
0$ as $t\to\infty$. This follows from Lemma 2.6. ∎
An important example of asymptotic morphisms is obtained by the (maximal)
tensor product of two asymptotically commuting $\ast$-homomorphisms:
###### Lemma 9.4.
Let $A_{1},A_{2}$ and $B$ be $G$-$C^{*}$-algebras and let
$\phi_{i}\colon A_{i}\to C_{b}([1,\infty),M(B))$
(i=1, 2) be equivariant $\ast$-homomorphisms such that
$\phi_{1}(a_{1})\phi_{2}(a_{2})\in
C_{b}([1,\infty),B),\,\,[\phi(a_{1}),\phi(a_{2})]\in C_{0}([1,\infty),B)$
for any $a_{1}\in A$ and $a_{2}\in A_{2}$. Then, there is a (unique)
equivariant asymptotic morphism
$\phi_{1}\otimes\phi_{2}\colon
A_{1}\otimes_{\mathrm{max}}A_{2}\to\mathfrak{A}(B)$
such that the image of $a_{1}\otimes a_{2}$ is represented by
$\phi_{1}(a_{1})\phi_{2}(a_{2})\in C_{b}([1,\infty),B).$
###### Proof.
This is trivial. We just note here that the image of any equivariant
$\ast$-homomorphism from a $G$-$C^{*}$-algebra consists of $G$-continuous
elements. ∎
###### Lemma 9.5.
(c.f. [16, Section 5], [48, Construction 6.7.5]) For any $X$-$G$-module
$H_{X}$, the natural map
$\pi_{X}\colon C_{0}(X)\to C_{b}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$
and the inclusion
$\iota\colon C_{L,c}^{*}(H_{X}\otimes B)_{\mathrm{Gcpt}}^{G}\subset
C_{b}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$
induce an asymptotic morphism
$\pi_{X}\otimes\iota\colon C_{0}(X)\otimes C_{L,c}^{*}(H_{X}\otimes
B)_{\mathrm{Gcpt}}^{G}\to\mathfrak{A}(\mathfrak{K}(H_{X}\otimes B))$
such that the image of $\phi\otimes T$ is represented by
$\phi T\in C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B)).$
###### Proof.
This follows from the previous two lemmas. ∎
We recall from the previous section that the right-regular representation
$\rho\colon C_{b}([1,\infty),\mathfrak{L}(H_{X}\otimes
B))_{\mathrm{Gcont}}\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes B))$
restricts to
$\rho\colon RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{L,c}^{*}(\tilde{H}_{X}\otimes B)_{\mathrm{Gcpt}}^{G}.$
Thus, we obtain the following asymptotic morphism.
###### Proposition 9.6.
For any $X$-$G$-module $H_{X}$, the natural map
$\pi_{X}\colon C_{0}(X)\to C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes
B))$
and the right-regular representation
$\rho\colon RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes B))$
induce an asymptotic morphism
$\pi_{X}\otimes\rho\colon C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\to\mathfrak{A}(\mathfrak{K}(\tilde{H}_{X}\otimes B))$
such that the image of $\phi\otimes T$ is represented by
$\phi\rho(T)\in C_{b}([1,\infty),\mathfrak{K}(\tilde{H}_{X}\otimes B)).$
###### Definition 9.7.
For any $X$-$G$-module $H_{X}$, the element
$[\pi_{X}\otimes\rho]\in E^{G}(C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G),B)$
is defined by the asymptotic morphism $\pi_{X}\otimes\rho$ in Proposition 9.6.
###### Definition 9.8.
We define a group homomorphism
$\rho_{X}\colon K_{i}(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\to
E_{i}^{G}(C_{0}(X),B)$
for $i=0,1$, by sending
$K_{i}(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\cong
E(\Sigma^{i},RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)$
to $E_{i}^{G}(C_{0}(X),B)$ by the composition with the class
$[\pi_{X}\otimes\rho]$ under the composition law
$E^{G}(\Sigma^{i},RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\times
E^{G}(C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G),B)$ $\to
E^{G}(\Sigma^{i}C_{0}(X),B).$
###### Lemma 9.9.
The group homomorphism $\rho_{X}$ is natural with respect to any
$G$-equivariant $\ast$-homomorphism $\pi\colon B_{1}\to B_{2}$ in a sense that
the following diagram commutes
$\textstyle{K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B_{1})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\scriptstyle{\rho_{X}}$$\textstyle{E_{\ast}^{G}(C_{0}(X),B_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\textstyle{K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B_{2})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\textstyle{E_{\ast}^{G}(C_{0}(X),B_{2}).}$
###### Proof.
This follows because the following diagram commutes
$\textstyle{C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes
B_{1})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{X}\otimes\rho}$$\scriptstyle{\mathrm{id}_{C_{0}(X)}\otimes\pi}$$\textstyle{\mathfrak{A}(\mathfrak{K}(\tilde{H}_{X}\otimes
B_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes
B_{2})\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{X}\otimes\rho}$$\textstyle{\mathfrak{A}(\mathfrak{K}(\tilde{H}_{X}\otimes
B_{2})).}$
∎
Now suppose that a $G$-equivariant continuous map $f\colon X\to Y$ is proper,
so that we have $f^{\ast}\colon C_{0}(Y)\to C_{0}(X)$. It defines
$f_{\ast}=(f^{\ast})^{\ast}\colon E_{\ast}^{G}(C_{0}(X),B)\to
E_{\ast}^{G}(C_{0}(Y),B).$
Let $H_{X}$ and $H_{Y}$ be an $X$-$G$-module and a $Y$-$G$-module respectively
and suppose that there is an equivariant continuous cover $(V_{t}\colon
H_{X}\to H_{Y})_{t\in[1\infty)}$ of $f$.
###### Lemma 9.10.
The following diagram commutes for any proper $f\colon X\to Y$ and for any
equivariant continuous cover $(V_{t}\colon H_{X}\to H_{Y})_{t\in[1\infty)}$ of
$f$:
$\textstyle{K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ad}_{V_{t},\ast}}$$\scriptstyle{\rho_{X}}$$\textstyle{E_{\ast}^{G}(C_{0}(X),B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\ast}}$$\textstyle{K_{\ast}(RL^{*}_{c}(H_{Y}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{Y}}$$\textstyle{E_{\ast}^{G}(C_{0}(Y),B).}$
###### Proof.
Recall that $V_{t}\colon H_{X}\to H_{Y}$ is a cover of $f$ if and only if it
is a cover of the identity when we view $H_{X}$ as a $Y$-module $(H_{X})_{Y}$
via $f^{\ast}$. It is clear from the definition of $\rho$ that the diagram
commutes when $V_{t}$ is the identity map $H_{X}\to(H_{X})_{Y}$. Thus, it is
enough to show the claim when $X=Y$ and $(V_{t}\colon H^{0}_{X}\to
H^{1}_{X})_{t\in[1\infty)}$ is a cover of the identity map on $X$. For
$i=0,1$, we have
$\pi_{X}^{i}\otimes\rho^{i}\colon C_{0}(X)\otimes RL^{*}_{c}(H^{i}_{X}\otimes
B)\rtimes_{r}G\to\mathfrak{A}(\mathfrak{K}(\tilde{H}_{X}^{i}\otimes B)),$
defined by the two asymptotically commuting $\ast$-homomorphisms
$\pi^{i}_{X}\colon C_{0}(X)\to
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}^{i}_{X}\otimes B)),$ $\rho^{i}\colon
RL^{*}_{c}(H^{i}_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}^{i}_{X}\otimes B)).$
We consider $H_{X}^{0}\oplus H_{X}^{1}$ and
$\tilde{H}_{X}^{0}\oplus\tilde{H}_{X}^{1}$. We show that two asymptotic
morphisms
$\pi_{X}^{0}\otimes\rho^{0},\,\pi_{X}^{1}\otimes(\rho^{1}\circ\mathrm{Ad}_{V_{t}})\colon
C_{0}(X)\otimes RL^{*}_{c}(H^{0}_{X}\otimes
B)\rtimes_{r}G\to\mathfrak{A}(\mathfrak{K}((\tilde{H}_{X}^{0}\otimes
B\oplus\tilde{H}_{X}^{1}\otimes B))$
are homotopic. Note that $V_{t}\colon H^{0}_{X}\to H^{1}_{X}$ defines an
$G$-equivariant isometry $\tilde{V}_{t}=V_{t}\otimes
1\colon\tilde{H}^{0}_{X}\to\tilde{H}^{1}_{X}$ in
$C_{b}([1,\infty),\mathfrak{L}(\tilde{H}^{0}_{X}\otimes
B\oplus\tilde{H}^{1}_{X}\otimes B))$ which conjugates $\rho^{0}$ to
$\rho^{1}\circ\mathrm{Ad}_{V_{t}}$. Furthermore, since $V_{t}$ is a cover of
the identity on $X$, it satisfies
$\pi_{X}^{1}(\phi)\tilde{V}_{t}-\tilde{V}_{t}\pi_{X}^{0}(\phi)\in
C_{0}([1,\infty),\mathfrak{L}(\tilde{H}^{0}_{X}\otimes
B\oplus\tilde{H}^{1}_{X}\otimes B)),$
and so
$\pi_{X}^{1}(\phi)\tilde{V}_{t}\tilde{V}_{t}^{\ast}-\tilde{V}_{t}\pi_{X}^{0}(\phi)\tilde{V}_{t}^{\ast}\in
C_{0}([1,\infty),\mathfrak{L}(\tilde{H}^{0}_{X}\otimes
B\oplus\tilde{H}^{1}_{X}\otimes B)).$
From this, we see that the isometry $\tilde{V}_{t}$ asymptotically conjugates
$\pi_{X}^{0}\otimes\rho^{0}$ to
$\pi_{X}^{1}\otimes(\rho^{1}\circ\mathrm{Ad}_{V_{t}})=\pi_{X}^{1}\otimes(\tilde{V}_{t}\rho^{0}\tilde{V}_{t}^{\ast})$
in a sense that
$\mathrm{Ad}_{\tilde{V}_{t}}(\pi_{X}^{0}\otimes\rho^{0})=\pi_{X}^{1}\otimes(\rho^{1}\circ\mathrm{Ad}_{V_{t}})$
in $\mathfrak{A}(\mathfrak{K}((\tilde{H}_{X}^{0}\otimes
B\oplus\tilde{H}_{X}^{1}\otimes B))$. It follows that the two asymptotic
morphisms are homotopic. The claim follows from this.
∎
Thanks to the previous lemma, and since $X\to E^{G}(C_{0}(X),B)$ is functorial
for proper $f\colon X\to Y$, the following definition is well-defined.
###### Definition 9.11.
For any $G$-compact proper $G$-space $X$, we define a group homomorphism
$\rho_{X}\colon\mathbb{D}^{B,G}_{\ast}(X)\to E_{\ast}^{G}(C_{0}(X),B)$
by
$\rho_{X}\colon K_{\ast}(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\to
E_{\ast}^{G}(C_{0}(X),B)$
where $H_{X}$ is the chosen universal $X$-$G$-module. More generally for any
proper $G$-space $X$, we define a group homomorphism
$\rho_{X}\colon\mathbb{D}^{B,G}_{\ast}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)$
so that the following diagram commutes:
$\textstyle{\mathbb{D}^{B,G}_{\ast}(X)\cong\varinjlim_{Y\subset
X,\mathrm{Gcpt}}\mathbb{D}^{B,G}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\textstyle{\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)}$$\textstyle{\mathbb{D}^{B,G}_{\ast}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{Y}}$$\textstyle{E_{\ast}^{G}(C_{0}(Y),B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$
###### Theorem 9.12.
The group homomorphisms
$\rho_{X}\colon\mathbb{D}^{B,G}_{\ast}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)$
define a natural transformation of functors from the category
$\mathcal{PR}^{G}$ of (second countable, locally compact) proper $G$-spaces to
the category $\mathcal{GA}$ of graded abelian groups. Furthermore, the
transformation is natural with respect to a $G$-equivariant
$\ast$-homomorphism $\pi\colon B_{1}\to B_{2}$ in a sense that the following
diagram commutes
$\textstyle{\mathbb{D}^{B_{1},G}_{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\scriptstyle{\rho_{X}}$$\textstyle{\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{\ast}}$$\textstyle{\mathbb{D}^{B_{2},G}_{\ast}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\textstyle{\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B_{2}).}$
###### Proof.
The first assertion follows from Lemma 9.10. The second assertion follows from
Lemma 9.9. ∎
The next goal is to show the following:
###### Theorem 9.13.
The forget-control map $\mathcal{F}\colon\mathbb{D}_{\ast}^{B,G}(X)\to
K_{\ast}(B\rtimes_{r}G)$ factors through the Baum–Connes assembly map
$\mu_{X}^{B,G}$ via $\rho_{X}$ in a sense that the following diagram commutes:
$\textstyle{\mathbb{D}_{\ast}^{B,G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B\rtimes_{r}G)}$$\textstyle{\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{B,G}_{X}}$
For this, we need some preparation. Recall that for any $G$-Hilbert $B$-module
$\mathcal{E}$, a Hilbert $B\rtimes_{r}G$-module $\mathcal{E}\rtimes_{r}G$ is
defined [30, Defintion 3.8]. For simplicity (for our purpose, this is enough)
we will only consider $\mathcal{E}$ of the form $H\otimes B$ where $H$ is a
$G$-Hilbert space and in this case $\mathcal{E}\rtimes_{r}G=H\otimes
B\rtimes_{r}G$ is canonically identified as the Hilbert $B\rtimes_{r}G$-module
$H\otimes(B\rtimes_{r}G)$. Kasparov defined for any $G$-equivariant
$\ast$-homomorphism $\pi\colon A\to\mathfrak{L}(\mathcal{E})$, its crossed
product $\pi\rtimes_{r}1\colon
A\rtimes_{r}G\to\mathfrak{L}(\mathcal{E}\rtimes_{r}G)$. In the case when
$\mathcal{E}=H\otimes B$, the $\ast$-homomorphism $\pi\rtimes_{r}1$ is defined
by sending $a\in A$ to
$\pi(a)\in\mathfrak{L}(\mathcal{E})\subset\mathfrak{L}(\mathcal{E}\rtimes_{r}G)$
and $g\in G$ to
$(g\otimes u_{g}\colon v\otimes f\mapsto gv\otimes
u_{g}f)\in\mathfrak{L}(H\otimes B\rtimes_{r}G)$
where $u_{g}\in M(B\rtimes_{r}G)$ is the unitary corresponding to $g$.
###### Lemma 9.14.
(c.f. [38, Proposition 5.2]) Let $X$ be a $G$-compact proper $G$-space. Let
$H_{X}$ be an $X$-$G$-module and let $\tilde{H}_{X}$ be the $X$-$G$-module as
in Definition 8.4. Consider
$\pi_{X}\colon C_{0}(X)\to\mathfrak{L}(\tilde{H}_{X}\otimes B),$
the structure map for the $X$-$G$-module $\tilde{H}_{X}\otimes B$ and consider
its Kasparov’s crossed product (see the explanation right before the lemma)
$\pi_{X}\rtimes_{r}1\colon
C_{0}(X)\rtimes_{r}G\to\mathfrak{L}(\tilde{H}_{X}\otimes B\rtimes_{r}G).$
Let $p_{c}(g)=\Delta(g)^{-1/2}cg(c)$ be the cut-off projection in
$C_{0}(X)\rtimes_{r}G$ for a cut-off function $c\in C_{c}(X)$ and let
$\rho\colon\mathfrak{L}(H_{X}\otimes
B)_{\mathrm{Gcont}}\rtimes_{r}G\to\mathfrak{L}(\tilde{H}_{X}\otimes
B)\subset\mathfrak{L}(\tilde{H}_{X}\otimes B\rtimes_{r}G)$
be the right-regular representation as in Definition 8.5. There is an
adjointable isometry
$U_{c}\colon H_{X}\otimes B\rtimes_{r}G\to\tilde{H}_{X}\otimes B\rtimes_{r}G$
of Hilbert $B\rtimes_{r}G$-modules such that
$U_{c}U_{c}^{\ast}=(\pi_{X}\rtimes_{r}1)(p_{c})$
and such that the c.c.p. map
$U_{c}^{\ast}\rho U_{c}\colon\mathfrak{L}(H_{X}\otimes
B)_{\mathrm{Gcont}}\rtimes_{r}G\to\mathfrak{L}(H_{X}\otimes B\rtimes_{r}G)$
is identified as the c.c.p. map on $\mathfrak{L}(H_{X}\otimes
B)_{\mathrm{Gcont}}\rtimes_{r}G$ (naturally viewed as a subalgebra of
$\mathfrak{L}(H_{X}\otimes B\rtimes_{r}G)$ via Kasparov’s crossed product)
defined by the $G$-equivariant c.c.p. map
$\mathfrak{L}(H_{X}\otimes B)\ni T\mapsto T^{\prime}=\int_{g\in
G}g(c)Tg(c)d\mu_{G}(g)\in\mathfrak{L}(H_{X}\otimes B).$
###### Proof.
An isometry $U_{c}\colon H_{X}\otimes B\rtimes_{r}G\to\tilde{H}_{X}\otimes
B\rtimes_{r}G$ is defined by sending $v\in H_{X}\otimes B\rtimes_{r}G$ to
$(G\ni h\mapsto\Delta(h)^{-1/2}(\pi_{X}\rtimes_{r}1)(c)(h\otimes u_{h})v\in
H_{X}\otimes B\rtimes_{r}G)$
in $L^{2}(G)\otimes H_{X}\otimes B\rtimes_{r}G$. Its adjoint $U_{c}^{\ast}$ is
defined by sending
$(G\ni h\mapsto v_{h}\in H_{X}\otimes B\rtimes_{r}G)\in L^{2}(G)\otimes
H_{X}\otimes B\rtimes_{r}G$
to
$\int_{h\in G}\Delta(h)^{-1/2}(h^{-1}\otimes
u_{h^{-1}})((\pi_{X}\rtimes_{r}1)(c)v_{h})d\mu_{G}(h)$
which converges weakly in $H_{X}\otimes B\rtimes_{r}G$. We can first see that
$U_{c}$ is well-defined and $\lVert U_{c}\rVert=1$ (as an a-priori not
necessarily adjointable map) and from which the weak convergence of the
formula of $U_{c}^{\ast}$ can be deduced. The equalities $U_{c}^{\ast}U_{c}=1$
and $U_{c}U_{c}^{\ast}=(\pi_{X}\rtimes_{r}1)(p_{c})$ can be checked directly.
Now, the following equalities can be checked directly,
$U_{c}^{\ast}\rho(T)U_{c}=\int_{h\in
G}h(c)Th(c)d\mu_{G}(h)\in\mathfrak{L}(H_{X}\otimes
B)\subset\mathfrak{L}(H_{X}\otimes B\rtimes_{r}G)$
for any $T\in\mathfrak{L}(H_{X}\otimes B)$ and
$U_{c}^{\ast}\rho_{g}U_{c}=g\otimes u_{g}\in\mathfrak{L}(H_{X}\otimes
B\rtimes_{r}G)$
for any $g\in G$. ∎
###### Lemma 9.15.
Let $X$ be a $G$-compact proper $G$-space. Let $H_{X}$ be any $X$-$G$-module.
Consider the element
$[\pi_{X}\otimes\rho]\in E^{G}(C_{0}(X)\otimes(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G),B).$
Let
$j^{G}_{r}([\pi_{X}\otimes\rho])\in
E(C_{0}(X)\rtimes_{r}G\otimes(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G),B\rtimes_{r}G)$
be its reduced crossed product (which is defined since the first variable is a
proper $G$-$C^{*}$-algebra). Let
$[p_{c}]\in E(\mathbb{C},C_{0}(X)\rtimes_{r}G)$
be the element corresponding to the cut-off projection $p_{c}$ in
$C_{0}(X)\rtimes_{r}G$ for a cut-off function $c$ on $X$. Then, the class
$j^{G}_{r}([\pi_{X}\otimes\rho])\circ[p_{c}]\in E(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G,B\rtimes_{r}G)$
is equal to the class associated to the natural inclusion map
$RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B)\rtimes_{r}G).$
###### Proof.
The crossed product $j^{G}_{r}([\pi_{X}\otimes\rho])$ is represented by the
product of the two asymptotically commuting representations
$\pi_{X}\rtimes_{r}1\colon
C_{0}(X)\rtimes_{r}G\to\mathfrak{L}(\tilde{H}_{X}\otimes B\rtimes_{r}G)\subset
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes B\rtimes_{r}G))$
and
$\rho\colon RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes B))\subset
C_{b}([1,\infty),\mathfrak{L}(\tilde{H}_{X}\otimes B\rtimes_{r}G)).$
The class $j^{G}_{r}([\pi_{X}\otimes\rho])\circ[p_{c}]$ is represented by the
asymptotic morphism from $RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G$ to
$\mathfrak{A}(\mathfrak{K}(\tilde{H}_{X}\otimes B\rtimes_{r}G))$ which is
represented by the c.c.p. map
$\pi_{X}\rtimes_{r}1(p_{c})\rho\pi_{X}\rtimes_{r}1(p_{c})\colon
RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{K}(\tilde{H}_{X}\otimes B\rtimes_{r}G)).$
In view of Lemma 9.14, this map can be identified as the c.c.p. map
$RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B\rtimes_{r}G))$
defined as the natural inclusion
$RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B\rtimes_{r}G))$
preceded by the c.c.p. map on $RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G$ induced
by the $G$-equivariant c.c.p. map
$T_{t}\mapsto\int_{h\in G}h(c)T_{t}h(c)d\mu_{G}(h)$
on $RL_{c}^{\ast}(H_{X}\otimes B)$. Thus, to prove our claim, we just need to
show that for any $T\in RL^{*}_{c}(H_{X}\otimes B)$,
$\lVert T_{t}-\int_{h\in G}h(c)T_{t}h(c)d\mu_{G}(h)\rVert\to 0$
as $t\to\infty$. We may assume $T$ has uniform compact support in $X$. In this
case, $h(c)T_{t}=0$ for $h\in G\backslash K$ for some compact subset $K$ of
$G$ and $\lVert[h(c),T_{t}]\rVert\to 0$ uniformly in $h\in K$ as $t\to\infty$.
We see the last assertion holds so we are done. ∎
###### Lemma 9.16.
Let $X$ be a $G$-compact proper $G$-space and $H_{X}$ be an $X$-$G$-module.
The forget-control map $\mathcal{F}\colon K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}(B\rtimes_{r}G)$ factors through
$E_{\ast}^{G}(C_{0}(X),B)$ via $\rho_{X}$. That is, the following diagram
commutes,
$\textstyle{K_{\ast}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B\rtimes_{r}G)}$$\textstyle{E_{\ast}^{G}(C_{0}(X),B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\mu^{G}_{X}}$
###### Proof.
Let $[\phi]\in K_{i}(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G)=E(\Sigma^{i},RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)$. Using
the functoriality of the crossed product functor and the composition law, we
see that the assembly map $\mu^{G}_{X}$ sends $\rho_{X}([\phi])$ to the
composition of $[\phi]$ with the element
$j^{G}_{r}([\pi_{X}\otimes\rho])\circ[p_{c}]\in E(RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G,B\rtimes_{r}G)$
under the composition law
$E(\Sigma^{i},RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G)\times
E(RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G,B\rtimes_{r}G)\to
E(\Sigma^{i},B\rtimes_{r}G).$
By Lemma 9.15, the element $j^{G}_{r}([\pi_{X}\otimes\rho])\circ[p_{c}]$ is
represented by the natural inclusion
$RL^{*}_{c}(H_{X}\otimes B)\rtimes_{r}G\to
C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B)\rtimes_{r}G).$
That is $j^{G}_{r}([\pi_{X}\otimes\rho])\circ[p_{c}]$ is represented by a
continuous family $(\mathrm{ev}_{t})_{t\in[1,\infty)}$ of $\ast$-homomorphisms
(evaluation at $t$). On the other hand, such a continuous family of
$\ast$-homomorphisms is homotopic (as an asymptotic morphism) to the constant
one $\mathrm{ev}_{1}$. It is now clear that the element
$\mu^{G}_{X}\circ\rho_{X}([\phi])$ coincides in
$E_{\ast}(\mathbb{C},B\rtimes_{r}G)$ with the one represented by the
composition of $\phi$ with the evaluation map $RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G\to\mathfrak{K}(H_{X})\otimes B\rtimes_{r}G$ at $t=1$ so we are
done. ∎
###### Proof of Theorem 9.13.
The theorem follows from Lemma 9.16. ∎
As the last thing in this section, we prove that the natural transformation
$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)$
is an isomorphism when $G$ is discrete and if $X$ is $G$-equivariantly
homotopic to a $G$-CW complex. Recall that for any open subgroup $H$ of $G$
and for any proper $H$-space $Y$, if $X$ is the balanced product
$G\times_{H}Y$, we have a natural isomorphism (see Theorem 7.4 (2))
$\mathbb{D}_{\ast}^{B,G}(X)\cong\mathbb{D}_{\ast}^{B,H}(Y).$
We also have a natural isomorphism ([21, Lemma 12.11], [13, Proposition 5.14])
$E_{\ast}^{G}(C_{0}(X),B)\cong E_{\ast}^{H}(C_{0}(Y),B).$
The rightward map is obtained by the restriction to the $H$-$C^{*}$-subalgebra
$C_{0}(Y)\subset C_{0}(X)$.
###### Lemma 9.17.
The natural transformation $\rho_{X}$ commutes with the induction. That is,
for any open subgroup $H$ of $G$, for any proper $H$-space $Y$ and for
$X=G\times_{H}Y$, the following diagram commutes.
$\textstyle{\mathbb{D}_{\ast}^{B,G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\rho_{X}}$$\textstyle{E^{G}(C_{0}(X),B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{D}_{\ast}^{B,H}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{Y}}$$\textstyle{E^{H}(C_{0}(Y),B).}$
###### Proof.
Recall that the isomorphism
$\mathbb{D}_{\ast}^{B,H}(Y)\cong\mathbb{D}_{\ast}^{B,G}(X)$ is obtained by the
canonical inclusion
$RL^{*}_{c}(H_{Y}\otimes B)\rtimes_{r}H\to RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G.$
With this in mind, the claim is direct to check. ∎
###### Theorem 9.18.
Let $G$ be a countable discrete group and $X$ be a proper $G$-space which is
$G$-equivariantly homotopy equivalent to a $G$-CW complex. Then, the group
homomorphism
$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)\cong\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)$
is an isomorphism for any separable $G$-$C^{*}$-algebra $B$.
###### Proof.
Since both functors satisfy the axioms (1)-(5) in Theorem 7.4 and since
$\rho_{X}$ is a natural transformation which commutes with the induction from
a finite subgroup (more generally from an open subgroup), it is enough to
prove that $\rho_{X}$ is an isomorphism when $G$ is a finite group $H$ and $X$
is a point. On the other hand, we have the following commutative diagram
$\textstyle{\mathbb{D}_{\ast}^{B,H}(\mathrm{point})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B\rtimes_{r}H)}$$\textstyle{E_{\ast}^{H}(\mathbb{C},B).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{r}^{B,H}}$
We know that the assembly map $\mu_{r}^{B,H}$ is an isomorphism for any finite
group $H$. We also know that the forget-control map is an isomorphism for any
finite group $H$ when $X$ is a point (see Proposition 6.5). Thus, $\rho_{X}$
is an isomorphism. Of course, we may also directly show that $\rho_{X}$ is an
isomorphism. ∎
###### Theorem 9.19.
Let $G$ be a countable discrete group $G$ and $X$ be a proper $G$-space which
is $G$-equivariantly homotopy equivalent to a $G$-CW complex. The forget-
control map $\mathcal{F}\colon\mathbb{D}_{\ast}^{B,G}(X)\to
K_{\ast}(B\rtimes_{r}G)$ is naturally equivalent to the Baum–Connes assembly
map $\mu_{X}^{B,G}$ for any separable $G$-$C^{*}$-algebra $B$.
###### Proof.
This follows from Theorem 9.13 and Theorem 9.18. ∎
Let $RL^{0}_{c}(H_{X}\otimes B)$ be the kernel of the evaluation map
$\mathrm{ev}_{1}$ on $RL^{*}_{c}(H_{X}\otimes B)$. The short exact sequence
$0\to RL^{0}_{c}(H_{X}\otimes B)\to RL^{*}_{c}(H_{X}\otimes
B)\to\mathfrak{K}(H_{X})\otimes B\to 0$
admits a $G$-equivariant c.c.p. splitting (by extending constantly and by
multiplying a bump function) and thus it descends to the short exact sequence
$0\to RL^{0}_{c}(H_{X}\otimes B)\rtimes_{r}G\to RL^{*}_{c}(H_{X}\otimes
B)\rtimes_{r}G\to(\mathfrak{K}(H_{X})\otimes B)\rtimes_{r}G\to 0.$
###### Corollary 9.20.
For any countable discrete group $G$, the Baum–Connes assembly map
$\mu^{B,G}_{r}$ is an isomorphism if and only if
$K_{\ast}(RL^{0}_{c}(H_{X}\otimes B)\rtimes_{r}G)=0$
for a universal (or ample) $X$-$G$-module $H_{X}$ for $X=\underline{E}G$.
We also have the following short exact sequence
$0\to(RL^{0}_{c}(H_{X})\otimes B)\rtimes_{r}G\to(RL^{*}_{c}(H_{X})\otimes
B)\rtimes_{r}G\to(\mathfrak{K}(H_{X})\otimes B)\rtimes_{r}G\to 0.$
Recall that the natural transformation $\mathbb{D}^{\otimes
B,G}_{\ast}(X)\to\mathbb{D}^{B,G}_{\ast}(X)$ is an isomorphism if $G$ is
discrete and $X$ is $G$-equivariantly homotopy equivalent to a proper $G$-CW
complex (Theorem 7.9). Hence, we also have the following:
###### Theorem 9.21.
Let $G$ be a countable discrete group $G$ and $X$ be a proper $G$-space which
is $G$-equivariantly homotopy equivalent to a $G$-CW complex. The forget-
control map $\mathcal{F}\colon\mathbb{D}_{\ast}^{\otimes B,G}(X)\to
K_{\ast}(B\rtimes_{r}G)$ is naturally equivalent to the Baum–Connes assembly
map $\mu_{X}^{B,G}$ for any separable $G$-$C^{*}$-algebra $B$.
###### Corollary 9.22.
For any countable discrete group $G$, the Baum–Connes assembly map
$\mu^{B,G}_{r}$ is an isomorphism if and only if
$K_{\ast}((RL^{0}_{c}(H_{X})\otimes B)\rtimes_{r}G)=0$
for a universal (or ample) $X$-$G$-module $H_{X}$ for $X=\underline{E}G$.
## 10\. $\rho_{X}$ is an isomorphism, part I
We begin our proof of the following:
###### Theorem.
Let $G$ be a locally compact group. The natural transformation
$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)\cong\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)$
is an isomorphism for any proper $G$-space $X$ and for any separable
$G$-$C^{*}$-algebra $B$.
The proof will be given over the next four sections. We will entirely focus on
the case when $X$ is $G$-compact. Note that the general case follows from
this. Here is a rough idea of the proof. Let $X$ be a $G$-compact proper
$G$-space. We first show (in this section) that the right-regular
representation induces an isomorphism
$\rho\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})$
whenever $H_{X}$ is a universal $X$-$G$-module. Then, we prove an isomorphism
$K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})\cong
E_{\ast}^{G}(C_{0}(X),B)\cong KK_{\ast}^{G}(C_{0}(X),B)$
following the idea of [16].
Let $X$ be a $G$-compact proper $G$-space and $H_{X}$ be an $X$-$G$-module. We
recall that the equivariant Roe algebra $C^{*}(H_{X}\otimes
B)^{G}=C^{*}(H_{X}\otimes B)^{G}_{\mathrm{Gcpt}}$ is defined as the completion
of the $\ast$-algebra $\mathbb{C}(H_{X}\otimes B)^{G}$ consisting of
$G$-equivariant, locally compact operators in $\mathfrak{L}(H_{X}\otimes B)$
that are properly supported. Note if $X_{0}$ is any compact subset of $X$ such
that $GX_{0}=X$ then a $G$-equivariant operator $T$ on $H_{X}\otimes B$ is
properly supported if and only if there is a compact subset $X_{1}$ of $X$ so
that $\chi_{X_{0}}T=\chi_{X_{0}}T\chi_{X_{1}}$ and
$T\chi_{X_{0}}=\chi_{X_{1}}T\chi_{X_{0}}$. We also recall that the localized
equivariant Roe algebra $C^{*}_{L,u}(H_{X}\otimes
B)^{G}=C^{*}_{L,u}(H_{X}\otimes B)^{G}_{\mathrm{Gcpt}}$ is the norm completion
of the $\ast$-subalgebra $\mathbb{C}_{L,u}(H_{X}\otimes B)^{G}$ of
$C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$ consisting of
$\mathbb{C}(H_{X}\otimes B)^{G}$-valued functions $T_{t}$ with
$\mathrm{prop}(T_{t})\to 0$ with respect to a (any) fixed metric $d$ on
$X^{+}$.
Let $H_{X}$ be any $X$-$G$-module. We introduce several $C^{*}$-subalgebras of
$C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$ containing the localized
equivariant Roe algebra. Let
$\pi\colon C_{0}(X)\to\mathfrak{L}(H_{X}\otimes B)$
be the structure map for the $X$-$G$-module $H_{X}\otimes B$. We let
$\mathcal{C}(\pi)^{G}$ to be the $C^{*}$-subalgebra of
$\mathfrak{L}(H_{X}\otimes B)$ consisting of $G$-equivariant, locally compact
operators. We have an inclusion
$C^{*}(H_{X}\otimes B)^{G}\subset\mathcal{C}(\pi)^{G}.$
* •
$\mathcal{C}_{L}(H_{X}\otimes B)^{G}$ is the $C^{*}$-subalgebra of
$C_{b,u}([1,\infty),C^{*}(H_{X}\otimes B)^{G})$ consisting of $T_{t}$ such
that $\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$ for any $\phi\in C_{0}(X)$.
* •
$\mathcal{C}_{L}(\pi)^{G}$ is the $C^{*}$-subalgebra of
$C_{b,u}([1,\infty),\mathcal{C}(\pi)^{G})$ consisting of functions $T_{t}$
such that $\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$ for any $\phi\in
C_{0}(X)$.
We have inclusions
$C^{*}_{L,u}(H_{X}\otimes B)^{G}\subset\mathcal{C}_{L}(H_{X}\otimes
B)^{G}\subset\mathcal{C}_{L}(\pi)^{G}.$
Let $c$ be a cut-off function on $X$. A $G$-equivariant u.c.p. map $\psi_{c}$
on $\mathfrak{L}(H_{X}\otimes B)$ defined as
$\psi_{c}\colon T\mapsto\int_{g\in G}g(c)Tg(c)d\mu_{G}(g)$
extends to a $G$-equivariant u.c.p. map on
$C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$. Note that $\psi_{c}$ sends
any $G$-equivariant, locally compact operator to a $G$-equivariant, locally
compact operator which is properly supported. Thus, $\psi_{c}$ maps
$\mathcal{C}(\pi)^{G}$ to $\mathbb{C}(H_{X}\otimes B)^{G}\subset
C^{*}(H_{X}\otimes B)^{G}$. Moreover, $\psi_{c}$ sends
$\mathcal{C}_{L}(\pi)^{G}$ to $\mathcal{C}_{L}(H_{X}\otimes B)^{G}$.
###### Lemma 10.1.
Let $T\in\mathcal{C}_{L}(\pi)^{G}$. The following are equivalent:
1. (1)
$\lim_{t\to\infty}\lVert T_{t}-\psi_{c}(T_{t})\rVert=0$ for any cut-off
function $c$ on $X$.
2. (2)
$\lim_{t\to\infty}\lVert T_{t}-\psi_{c}(T_{t})\rVert=0$ for some cut-off
function $c$ on $X$.
3. (3)
There is $S$ in $\mathcal{C}_{L}(H_{X}\otimes B)^{G}$ such that
$\lim_{t\to\infty}\lVert T_{t}-S_{t}\rVert=0$ and $S$ is properly supported in
a sense that for any compact subset $A$ of $X$, there is a compact subset $B$
of $X$ such that $\chi_{A}S=\chi_{A}S\chi_{B}$ and
$S\chi_{A}=\chi_{B}S\chi_{A}$.
4. (4)
There is $S$ in $\mathcal{C}_{L}(\pi)^{G}$ such that $\lim_{t\to\infty}\lVert
T_{t}-S_{t}\rVert=0$ and $S$ is properly supported.
###### Proof.
(1) $\implies$ (2): Obvious. (2) $\implies$ (3): This is because $\psi_{c}(T)$
is properly supported for any $T$. (3) $\implies$ (4): Obvious. (4) $\implies$
(1): It is enough to show that if $S$ in $\mathcal{C}_{L}(\pi)^{G}$ is
properly supported, then $\lim_{t\to\infty}\lVert
S_{t}-\psi_{c}(S_{t})\rVert=0$ for any cut-off function $c$ on $X$. Since $S$
is properly supported, there is $\chi\in C_{c}(X)$ such that $c=c\chi$ and
$cS=cS\chi$. We have
$S_{t}-\psi_{c}(S_{t})=\int_{g\in G}g(c)^{2}S_{t}-g(c)S_{t}g(c)d\mu_{G}(g)$
$=\int_{g\in G}g(c)^{2}S_{t}g(\chi)-g(c)S_{t}g(c)g(\chi)d\mu_{G}(g)=\int_{g\in
G}g(c)[g(c),S_{t}]g(\chi)d\mu_{G}(g).$
We have
$\lim_{t\to\infty}\lVert[g(c),S_{t}]\rVert=\lim_{t\to\infty}\lVert[c,S_{t}]\rVert=0$
(uniformly in $g$ in $G$). It follows $\lim_{t\to\infty}\lVert
S_{t}-\psi_{c}(S_{t})\rVert=0$. ∎
It is easy to see that the conditions (3) and (4) are preserved by taking
sums, products and adjoint and that the conditions (1) and (2) pass to the
norm limit. We define some $C^{*}$-subalgebras of
$\mathcal{C}_{L}(H_{X}\otimes B)^{G}$ and $\mathcal{C}_{L}(\pi)^{G}$ as
follows:
* •
$\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ is the
$C^{*}$-subalgebra of $\mathcal{C}_{L}(H_{X}\otimes B)^{G}$ consisting of $T$
satisfying the four equivalent conditions in Lemma 10.1.
* •
$\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}}$ is the $C^{*}$-subalgebra of
$\mathcal{C}_{L}(\pi)^{G}$ consisting of $T$ satisfying the four equivalent
conditions in Lemma 10.1.
We have inclusions
$\textstyle{\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(H_{X}\otimes
B)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\pi)^{G}.}$
We have the following commutative diagram of short exact sequences
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{0}([1,\infty),C^{*}(H_{X}\otimes
B)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L,Q}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{0}([1,\infty),\mathcal{C}(\pi)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L,Q}(\pi)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
In particular, the inclusion induces an isomorphism
$K_{\ast}(\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}})\cong
K_{\ast}(\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}}).$
###### Proposition 10.2.
The two $C^{*}$-subalgebras $C^{*}_{L,u}(H_{X}\otimes B)^{G}$ and
$\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ in
$C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$ are identical.
###### Proof.
$C^{*}_{L,u}(H_{X}\otimes B)^{G}\subset\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$: Take any $T\in
C_{b,u}([1,\infty),\mathbb{C}(H_{X}\otimes B)^{G})$ such that
$\mathrm{prop}(T_{t})\to 0$ with respect to a fixed metric $d$ on $X^{+}$. Let
$X_{0}$ be a compact subset of $X$ such that $GX_{0}=X$ and $X_{1}$ be any
compact neighborhood of $X_{0}$ in $X$. Since $\mathrm{prop}(T_{t})\to 0$,
there is $t_{0}\geq 1$ such that for any $t>t_{0}$,
$\chi_{X_{0}}T_{t}=\chi_{X_{0}}T_{t}\chi_{X_{1}}$ and
$T_{t}\chi_{X_{0}}=\chi_{X_{1}}T_{t}\chi_{X_{0}}$. This implies that $T$
satisfies the condition (3) of Lemma 10.1. We already know
$C^{*}_{L,u}(H_{X}\otimes B)^{G}\subset\mathcal{C}_{L}(H_{X}\otimes B)^{G}$ so
we have $C^{*}_{L,u}(H_{X}\otimes B)^{G}\subset\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$.
$\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}\subset
C^{*}_{L,u}(H_{X}\otimes B)^{G}$: Let $T\in\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$. It is enough to show that $\psi_{c}(T)\in
C^{*}_{L,u}(H_{X}\otimes B)^{G}$ for a cut-off function $c$ on $X$ since both
algebras contain the common ideal $C_{0}([1,\infty),C^{*}(H_{X}\otimes
B)^{G})$. Since $\lim_{t\to\infty}\lVert[T_{t},\phi]\rVert=0$ for any $\phi\in
C_{0}(X)$, there is $S\in C_{b,u}([1,\infty),\mathfrak{L}(H_{X}))$ such that
$\lim_{t\to\infty}\lVert T_{t}-S_{t}\rVert=0$ and such that
$\mathrm{prop}(S_{t})\to 0$ as $t\to\infty$ for some (any) fixed metric $d$ on
$X^{+}$. We can arrange it so that $S_{t}$ a $G$-continuous, locally compact
operator in $\mathfrak{L}(H_{X})$ for each $t\geq 1$. In this case, since
$\lim_{t\to\infty}\lVert T_{t}-S_{t}\rVert=0$ with $T$ $G$-equivariant, we see
that $S$ is $G$-continuous in $C_{b,u}([1,\infty),\mathfrak{L}(H_{X}))$. Now
consider $\tilde{S}\in C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$
defined (weakly) by
$\tilde{S}_{t}=\int_{g\in G}g(c)g(S_{t})g(c)d\mu_{G}(g).$
Note this is uniformly continuous in $t$. Indeed $S\mapsto\tilde{S}$ is a
u.c.p. map on $C_{b,u}([1,\infty),\mathfrak{L}(H_{X}\otimes B))$ ($S$ does not
have to be $G$-continuous for this to be well-defined). We have
$\psi_{c}(T_{t})-\tilde{S}_{t}=\int_{g\in
G}g(c)(T_{t}-g(S_{t}))g(c)d\mu_{G}(g).$
Together with $\lim_{t\to\infty}\lVert
T_{t}-g(S_{t})\rVert=\lim_{t\to\infty}\lVert T_{t}-S_{t}\rVert=0$ (uniformly
in $g$ in $G$), we see that
$\lim_{t\to\infty}\lVert\psi_{c}(T_{t})-\tilde{S}_{t}\rVert=0$. Since
$\tilde{S}_{t}\in\mathbb{C}(H_{X}\otimes B)^{G}\subset C^{*}(H_{X}\otimes
B)^{G}$ for any $t\geq 1$, to show $\psi_{c}(T)\in C^{*}_{L,u}(H_{X}\otimes
B)^{G}$ it is enough to show $\tilde{S}\in C^{*}_{L,u}(H_{X}\otimes B)^{G}$.
For this, it suffices to show $\mathrm{prop}(\tilde{S}_{t})\to 0$ as
$t\to\infty$ with respect to the fixed metric $d$ on $X^{+}$. It suffices to
show $\mathrm{prop}(\chi_{X_{0}}\tilde{S}_{t})\to 0$ and
$\mathrm{prop}(\tilde{S}_{t}\chi_{X_{0}})\to 0$ as $t\to\infty$ for any
compact subset $X_{0}$ of $X$. On the other hand,
$\chi_{X_{0}}\tilde{S}_{t}=\int_{g\in
G}\chi_{X_{0}}g(c)g(S_{t})g(c)d\mu_{G}(g)$
and $\chi_{X_{0}}g(c)=0$ for any $g\in G\backslash K$ for some compact subset
$K$ of $G$. The claim $\mathrm{prop}(\chi_{X_{0}}\tilde{S}_{t})\to 0$ now
follows since $\mathrm{prop}(g(S_{t}))\to 0$ uniformly in $g\in K$. We also
have $\mathrm{prop}(\tilde{S}_{t}\chi_{X_{0}})\to 0$ for the same reason, so
we are done. ∎
Note we have a natural inclusion ($\ast$-homomorphism)
$\iota\colon\mathcal{C}_{L}(\pi)^{G}\to M(RL^{*}_{u}(H_{X}\otimes B))\subset
M(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)$
just because $T\in\mathcal{C}_{L}(\pi)^{G}$ satisfies
$\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$. Moreover, for any
$T\in\mathcal{C}_{L}(\pi)^{G}$, $\phi$ in $C_{0}(X)$, we have
$\phi T\in RL^{*}_{u}(H_{X}\otimes B)$
because $T_{t}$ is locally compact and
$\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$. We let
$\pi\rtimes_{r}1\colon C_{0}(X)\rtimes_{r}G\to M(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)$
be the crossed product of the structure map $\pi\colon C_{0}(X)\to
M(RL^{*}_{u}(H_{X}\otimes B))$. Let $c\in C_{c}(X)$ be a cut-off function on
$X$ and $p_{c}\in C_{0}(X)\rtimes_{r}G$ be the associated cut-off projection.
We have a c.c.p. map
$(\pi\rtimes_{r}1)(p_{c})\iota(\pi\rtimes_{r}1)(p_{c})\colon\mathcal{C}_{L}(\pi)^{G}\to
RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G$
which sends $T\in\mathcal{C}_{L}(\pi)^{G}$ to the compression
$(\pi\rtimes_{r}1)(p_{c})\iota(T)(\pi\rtimes_{r}1)(p_{c})$ in
$RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G$. For any
$T\in\mathcal{C}_{L}(\pi)^{G}$, we have
$[\iota(T),(\pi\rtimes_{r}1)(p_{c})]\in RL^{*}_{0}(H_{X}\otimes
B)\rtimes_{r}G.$
Thus, after passing to the quotient $RL^{*}_{u,Q}(H_{X}\otimes
B)\rtimes_{r}G$, the c.c.p. map
$(\pi\rtimes_{r}1)(p_{c})\iota(\pi\rtimes_{r}1)(p_{c})$ becomes a
$\ast$-homomorphism
$(\pi\rtimes_{r}1)(p_{c})\iota(\pi\rtimes_{r}1)(p_{c})\colon\mathcal{C}_{L}(\pi)^{G}\to
RL^{*}_{u,Q}(H_{X}\otimes B)\rtimes_{r}G.$
Recall that the quotient map induces an isomorphism
$K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(RL^{*}_{u,Q}(H_{X}\otimes B)\rtimes_{r}G).$
###### Definition 10.3.
We define a group homomorphism
$\iota_{p_{c}}\colon K_{\ast}(\mathcal{C}_{L}(\pi)^{G})\to
K_{\ast}(RL^{*}_{u}(H_{X})\rtimes_{r}G)$
as the composition of the group homomorphism
$(\pi\rtimes_{r}1)(p_{c})\iota(\pi\rtimes_{r}1)(p_{c})_{\ast}\colon
K_{\ast}(\mathcal{C}_{L}(\pi)^{G})\to
K_{\ast}(RL^{*}_{u,Q}(H_{X})\rtimes_{r}G)$
and the inverse of the isomorphism
$K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(RL^{*}_{u,Q}(H_{X}\otimes B)\rtimes_{r}G)$
induced by the quotient map. If $A$ is any of the subalgebras
$\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}},\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}},\mathcal{C}_{L}(H_{X}\otimes
B)^{G}$
of $\mathcal{C}_{L}(\pi)^{G}$, we set
$\iota_{p_{c}}\colon K_{\ast}(A)\to K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)$
be the restriction of $\iota_{p_{c}}$ to the subalgebra $A$.
Now, for an $X$-$G$-module $H_{X}$, let $\tilde{H}_{X}=H_{X}\otimes L^{2}(G)$
be the $X$-$G$-module as before (see Definition 8.4). Let
$\tilde{\pi}\colon C_{0}(X)\to\mathfrak{L}(\tilde{H}_{X}\otimes B)$
be the structure map. We have the right-regular representation (see
Proposition 8.14)
$\rho\colon RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G\to
C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}$
and the inclusions
$\textstyle{C^{*}_{L,u}(\tilde{H}_{X}\otimes
B)^{G}=\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\tilde{\pi})^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{L}(\tilde{\pi})^{G}.}$
###### Definition 10.4.
If $A$ is any of the algebras
$\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}},\mathcal{C}_{L}(\tilde{\pi})^{G}_{\mathrm{proper}},\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G},\mathcal{C}_{L}(\tilde{\pi})^{G},$
we set
$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}(A)$
to be the group homomorphism induced by the right-regular representation
$\rho\colon RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G\to
C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}$
followed by the inclusion from $C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}$ to
$A$.
We are now interested in computing the compositions
$\iota_{p_{c}}\circ\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})\to
K_{\ast}(RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G)$
and
$\rho_{\ast}\circ\iota_{p_{c}}\colon K_{\ast}(C^{*}_{L,u}(H_{X}\otimes
B)^{G})\to K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}).$
Let us first compute the first one. We remind that a proper $G$-space $X$ has
been assumed to be $G$-compact from the beginning of this section.
###### Lemma 10.5.
For any (not-necessarily separable) $G$-$C^{*}$-algebra $A$ equipped with a
non-degenerate representation of the $G$-$C^{*}$-algebra $C_{0}(X)$ to $M(A)$,
the right regular representation
$\rho\colon A\rtimes_{r}G\to M(A\otimes\mathfrak{K}(L^{2}(G))),\,\,A\ni
a\mapsto(g(a))_{g\in G},\,\,G\ni g\mapsto\rho_{g},$
is an isomorphism onto the $C^{*}$-algebra
$M(A\otimes\mathfrak{K}(L^{2}(G)))^{G,c}$ defined as the completion of the
$\ast$-algebra consisting of $G$-equivariant, locally compact, properly
supported operators in
$M(A\otimes\mathfrak{K}(L^{2}(G)))\cong\mathfrak{L}(A\otimes L^{2}(G))$.
###### Proof.
The proof of Proposition 8.7 works verbatim. ∎
###### Remark 10.6.
The algebra $M(A\otimes\mathfrak{K}(L^{2}(G)))^{G,c}$ is known as a
generalized fixed point algebra and this lemma is proved in a more general
setting ([11, Corollary 3.25, Remark 3.26] see also [10] [12] for a more
general theory).
###### Lemma 10.7.
The composition $\iota_{p_{c}}\circ\rho_{\ast}\colon
K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}(RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G)$ coincides with the
map induced by the strict cover $V_{c}\colon H_{X}\to\tilde{H}_{X}$ of the
identity map on $X$ defined as
$V_{c}\colon H_{X}\ni v\mapsto(g\mapsto g(c)v)\in\tilde{H}_{X}=H_{X}\otimes
L^{2}(G).$
###### Proof.
A c.c.p. map
$\iota^{1}_{p_{c}}\colon RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G\to(RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G$
is defined as the compression of the right-regular representation
$RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G\ni T\to\rho(T)\in
M(RL^{*}_{u}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G)))$
by the projection $p^{(1)}_{c}$ which is the image of the cut-off projection
$p_{c}$ in $C_{0}(X)\rtimes_{r}G$ by the map
$\pi^{1}\rtimes_{r}1\colon C_{0}(X)\rtimes_{r}G\to M((RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G)$
induced by the natural $G$-equivariant representation
$\pi^{1}\colon C_{0}(X)\to M(RL^{*}_{u}(H_{X}))\to M(RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G))).$
Passing to the quotient, the c.c.p. map $\iota^{1}_{p_{c}}$ becomes a
$\ast$-homomorphism
$\iota^{1}_{p_{c}}\colon RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G\to(RL^{*}_{u,Q}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G.$
Composing its induced map on K-theory with the inverse of the isomorphism (the
quotient map)
$K_{\ast}((RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G)\cong
K_{\ast}((RL^{*}_{u,Q}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G),$
we obtain a group homomorphism
$\iota^{1}_{p_{c}}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}((RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G).$
Here, we used that the reduced crossed product preserves the exact sequence
$0\to RL^{*}_{0}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G))\to
RL^{*}_{u}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G))$ $\to
RL^{*}_{u,Q}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G))\to 0$
since the quotient is a proper $G$-$C^{*}$-algebra.
The composition $\iota_{p_{c}}\circ\rho_{\ast}$ factors through the natural
inclusion
$(RL^{*}_{u}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G\to
RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G$
via the group homomorphism $\iota^{1}_{p_{c}}$.
We now consider
$\mathrm{Ad}_{V_{c}\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}(RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G).$
Let
$p_{c}^{(2)}=V_{c}V_{c}^{\ast}\in\mathfrak{L}(\tilde{H}_{X})\subset\mathfrak{L}(\tilde{H}_{X}\otimes
B)$. The $G$-equivariant projection $p^{(2)}_{c}$ is the image of $p_{c}\in
C_{0}(X)\rtimes_{r}G$ by the right-regular representation
$C_{0}(X)\rtimes_{r}G\to\mathfrak{L}(\tilde{H}_{X})=\mathfrak{L}(H_{X}\otimes
L^{2}(G)),C_{0}(X)\ni\phi\mapsto(g(\phi))_{g\in G},\,\,G\ni g\mapsto\rho_{g}.$
We claim that the $\ast$-homomorphism
$\mathrm{Ad}_{V_{c}}\colon RL^{*}_{u}(H_{X}\otimes B)\ni T\mapsto
V_{c}TV_{c}^{\ast}\in RL^{*}_{u}(\tilde{H}_{X}\otimes B)$
and the $G$-equivariant c.c.p. map
$\iota^{2}_{p_{c}}\colon RL^{*}_{u}(H_{X}\otimes B)\ni T\mapsto
p^{(2)}_{c}(T)_{g\in G}p^{(2)}_{c}\in RL^{*}_{u}(\tilde{H}_{X}\otimes B)$
coincide after passing to the quotient $RL^{*}_{u,Q}(\tilde{H}_{X}\otimes B)$.
Indeed, for $S$ in $\mathfrak{L}(H_{X}\otimes B)$, we have
$V_{c}^{\ast}(S)_{g\in G}V_{c}=\int_{g\in G}g(c)Sg(c)d\mu_{g}(g).$
Thus, for any $T\in RL^{*}_{u}(H_{X}\otimes B)$, we have
$\lim_{t\to\infty}\lVert T_{t}-V_{c}^{\ast}(T_{t})_{g\in G}V_{c}\rVert=0.$
We can see this for example by considering $T$ which has uniform compact
support.
The $G$-equivariant c.c.p. map $\iota^{2}_{p_{c}}$ naturally factors through
the inclusion
$RL^{*}_{u}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G))\to
RL^{*}_{u}(\tilde{H}_{X}\otimes B).$
Overall, in order to show
$\iota_{p_{c}}\circ\rho_{\ast}=\mathrm{Ad}_{V_{c}\ast}$ on
$K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)$, it suffices to show that
the two group homomorphisms
$\iota^{1}_{p_{c}},\iota^{2}_{p_{c}}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}((RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G)$
induced by the c.c.p. maps $\iota^{1}_{p_{c}},\iota^{2}_{p_{c}}$ (which become
$\ast$-homomorphisms after passing to the quotient) coincide. On the other
hand, by viewing
$(RL^{*}_{u}(H_{X}\otimes B)\otimes\mathfrak{K}(L^{2}(G)))\rtimes_{r}G\cong
M(RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G))\otimes\mathfrak{K}(L^{2}(G)))^{G,c}$
as in Lemma 10.5, we can see that the two map $\iota^{1}_{p_{c}}$ and
$\iota^{2}_{p_{c}}$ are conjugate by the unitary $U$ in the multiplier algebra
of
$M(RL^{*}_{u}(H_{X}\otimes
B)\otimes\mathfrak{K}(L^{2}(G))\otimes\mathfrak{K}(L^{2}(G)))^{G,c}$
defined as the flip on $G\times G$, i.e. $U\in\mathfrak{L}(L^{2}(G\times G))$
defined by $Uf(g_{1},g_{2})=f(g_{2},g_{1})$ for $f\in L^{2}(G\times G)$ so we
are done. ∎
###### Corollary 10.8.
The composition
$\iota_{p_{c}}\circ\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}(RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G)$
is an isomorphism whenever $H_{X}$ is a universal $X$-$G$-module.
We now study the other composition.
###### Lemma 10.9.
The composition
$\rho_{\ast}\circ\iota_{p_{c}}\colon K_{\ast}(C^{*}_{L,u}(H_{X}\otimes
B)^{G})\to K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})$
coincides with the map induced by the strict cover $V_{c}\colon
H_{X}\to\tilde{H}_{X}$ of the identity map on $X$ defined as
$V_{c}\colon H_{X}\ni v\mapsto(g\mapsto g(c)v)\in\tilde{H}_{X}=H_{X}\otimes
L^{2}(G).$
###### Proof.
We can directly see that the composition is induced by the c.c.p. map (which
becomes a $\ast$-homomorphism after passing to the quotient),
$p_{c}\cdot p_{c}\colon C^{*}_{L,u}(H_{X}\otimes B)^{G}\ni T\mapsto
p_{c}(T)_{g\in G}p_{c}\in C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}$
where the cut-off projection $p_{c}$ is represented by the right-regular
representation of $C_{0}(X)\rtimes_{r}G$:
$C_{0}(X)\rtimes_{r}G\to\mathfrak{L}(\tilde{H}_{X})=\mathfrak{L}(H_{X}\otimes
L^{2}(G)),C_{0}(X)\ni\phi\mapsto(g(\phi))_{g\in G},\,\,G\ni g\mapsto\rho_{g}.$
As before, more precisely, this c.c.p. map induces the map on K-theory as the
composition of the $\ast$-homomorphism
$p_{c}\cdot p_{c}\colon C^{*}_{L,u}(H_{X}\otimes B)^{G}\to
C^{*}_{L,Q}(\tilde{H}_{X}\otimes B)^{G}=C^{*}_{L,u}(\tilde{H}_{X}\otimes
B)^{G}/C_{0}([1,\infty),C^{*}(\tilde{H}_{X}\otimes B)^{G})$
and the inverse of the isomorphism (the quotient map)
$K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})\to
K_{\ast}(C^{*}_{L,Q}(\tilde{H}_{X}\otimes B)^{G}).$
We have $p_{c}=V_{c}V_{c}^{\ast}$ and as before, for
$S\in\mathfrak{L}(H_{X}\otimes B)$,
$V_{c}^{\ast}(S)_{g\in G}V_{c}=\int_{g\in G}g(c)Sg(c)d\mu_{g}(g).$
To show that $p_{c}\cdot p_{c}$ and $\mathrm{Ad}_{V_{c}\ast}$ coincide on
$K_{\ast}(C^{*}_{L,u}(H_{X}\otimes B)^{G})$, it is enough to show that for any
$T\in C^{*}_{L,u}(H_{X}\otimes B)^{G}$, we have $\lim_{t\to\infty}\lVert
T_{t}-V^{\ast}_{c}(T_{t})_{g\in G}V_{c}\rVert=0$ but this is precisely the
condition (2) of Lemma 10.1, which is satisfied for any $T$ in
$\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}}$ in particular for any $T\in
C^{*}_{L,u}(H_{X}\otimes B)^{G}=\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$. ∎
The same proof shows the following:
###### Lemma 10.10.
The composition
$\rho_{\ast}\circ\iota_{p_{c}}\colon
K_{\ast}(\mathcal{C}_{L}(\pi)^{G}_{\mathrm{proper}})\to
K_{\ast}(\mathcal{C}_{L}(\tilde{\pi})^{G}_{\mathrm{proper}})$
coincides with the map induced by the strict cover $V_{c}\colon
H_{X}\to\tilde{H}_{X}$ of the identity map on $X$.
Before jumping to the conclusion, we make an important remark here. The
equivariant Roe algebra $C^{*}(H_{X}\otimes B)^{G}$, the localized equivariant
Roe algebra $C^{*}_{L,u}(H_{X}\otimes B)^{G}$ as well as all the other
algebras such as $\mathcal{C}(\pi)^{G}$ and $\mathcal{C}_{L}(\pi)^{G}$ are not
well-behaved with respect to an arbitrary equivariant continuous cover
$(V_{t}\colon H_{X}\to H_{Y})_{t\in[1,\infty)}$ of a $G$-equivariant
continuous map $f\colon X\to Y$ even if $f$ is the identity map on a
$G$-compact space. This is because a continuous cover $V_{t}$ may not be
properly supported in general and hence the conjugation by $V_{t}$ may not
preserve locally compact operators. On the other hand, they are functorial
with respect to a strict cover $V$, if exists, of the identity map.
More relevantly, if we focus on $X$-$G$-modules of the form
$\tilde{H}_{X}\otimes B$ for an $X$-$G$-module $H_{X}$, the equivariant Roe
algebra $C^{*}(\tilde{H}_{X}\otimes B)^{G}$, the localized equivariant Roe
algebra $C^{*}_{L,u}(\tilde{H}_{X}\otimes
B)^{G}=\mathcal{C}_{L}(\tilde{H}_{X}\otimes B)^{G}_{\mathrm{proper}}$ as well
as $\mathcal{C}_{L}(\tilde{H}_{X}\otimes B)^{G}$ are functorial with respect
to a $G$-equivariant continuous cover of the identity map of the form
$\tilde{V}=V\otimes 1\colon\tilde{H}_{X}\to\tilde{H}^{\prime}_{X}$ for a
$G$-equivariant continuous cover $V\colon H_{X}\to H_{X}^{\prime}$. To see
this, first, we have the following commutative diagram
$\textstyle{C^{*}(\tilde{H}_{X}\otimes
B)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ad}_{\tilde{V}}}$$\textstyle{C^{*}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}}$$\textstyle{\mathfrak{K}(H_{X}\otimes
B)\rtimes_{r}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ad}_{V}}$$\scriptstyle{\cong}$$\scriptstyle{\rho}$$\textstyle{\mathfrak{K}(H^{\prime}_{X}\otimes
B)\rtimes_{r}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\rho}$
for any $G$-equivariant isometry $V\colon H_{X}\to H_{X}^{\prime}$, which in
particular says that $\mathrm{Ad}_{\tilde{V}}$ maps
$C^{*}(\tilde{H}_{X}\otimes B)^{G}$ to $C^{*}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}$. Now, if $(V_{t}\colon H_{X}\to H^{\prime}_{X})_{t\in[1,\infty)}$ is a
$G$-equivariant continuous cover of the the identity map on $X$, we can see
that $\mathrm{Ad}_{\tilde{V}_{t}}$ maps $\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}$ to $\mathcal{C}_{L}(\tilde{H}^{\prime}_{X}\otimes B)^{G}$ and maps
$C_{L,u}^{*}(\tilde{H}_{X}\otimes B)^{G}=\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}}$ to $C_{L,u}^{*}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}=\mathcal{C}_{L}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}_{\mathrm{proper}}$. To see the latter, let $X_{0}$ be a compact subset
of $X$ with $X=GX_{0}$. If $S\in\mathcal{C}_{L}(\tilde{H}_{X}\otimes B)^{G}$
is properly supported, then
$S^{\prime}_{t}=\tilde{V}_{t}S_{t}\tilde{V}_{t}^{\ast}$ satisfies
$\chi_{X_{0}}S^{\prime}_{t}=\chi_{X_{0}}S^{\prime}_{t}\chi_{X_{1}},\,\,\,S^{\prime}_{t}\chi_{X_{0}}=\chi_{X_{1}}S^{\prime}_{t}\chi_{X_{0}}$
for some compact subset $X_{1}$ of $X$ for large enough $t$. This already
implies that $S^{\prime}$ satisfies the condition (3) of Lemma 10.1.
Similarly, for any two covers $(V_{i,t}\colon H_{X}\to
H^{\prime}_{X})_{t\in[1,\infty)}$ for $i=1,2$,
$\tilde{V}_{1,t}\tilde{V}_{2,t}^{\ast}$ multiplies
$C_{L,u}^{*}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}=\mathcal{C}_{L}(\tilde{H}^{\prime}_{X}\otimes
B)^{G}_{\mathrm{proper}}$. Using this, we see that the induced map
$\mathrm{Ad}_{\tilde{V}_{t}\ast}\colon
K_{\ast}(C_{L,u}^{*}(\tilde{H}_{X}\otimes B)^{G})\to
K_{\ast}(C_{L,u}^{*}(\tilde{H}^{\prime}_{X}\otimes B)^{G})$
on $K$-theory is independent of the covers and it is an isomorphism whenever
$H_{X}$ and $H_{X}^{\prime}$ are universal $X$-$G$-modules.
We say that an $X$-$G$-module $H_{X}$ is of infinite multiplicity if
$H_{X}\cong H^{0}_{X}\otimes l^{2}(\mathbb{N})$ as an $X$-$G$-module for some
$X$-$G$-module $H^{0}_{X}$.
###### Lemma 10.11.
For any universal $X$-$G$-module $H_{X}$ of infinite multiplicity, the right-
regular representation $\rho$ induces an isomorphism
$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G}).$
###### Proof.
By Corollary 10.8, we already know that the composition
$\iota_{p_{c}}\circ\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\to K_{\ast}(RL^{*}_{u}(\tilde{H}_{X}\otimes B)\rtimes_{r}G)$
is an isomorphism. It thus suffices to show that the composition
$\rho_{\ast}\circ\iota_{p_{c}}\colon K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes
B)^{G})\to K_{\ast}(C^{*}_{L,u}(\tilde{H}^{(2)}_{X}\otimes B)^{G})$
is injective (isomorphism) where $\tilde{H}^{(2)}_{X}=\tilde{H}^{\prime}_{X}$
for $H^{\prime}_{X}=\tilde{H}_{X}$. By Lemma 10.9, this composition is induced
by the isometry $V_{c}\colon H_{X}^{\prime}\to\tilde{H}_{X}^{\prime}$ which is
a strict cover of the identity map. On the other hand, $H_{X}^{\prime}\cong
H^{0}_{X}\otimes l^{2}(\mathbb{N})\otimes L^{2}(G)$ and
$\tilde{H}_{X}^{\prime}=H_{X}^{\prime}\otimes L^{2}(G)$ are isomorphic as
$X$-$G$-modules. From this, we see that
$\rho_{\ast}\circ\iota_{p_{c}}=\mathrm{Ad}_{V_{c}\ast}$ is an isomorphism. ∎
###### Theorem 10.12.
For any universal $X$-$G$-module $H_{X}$, the right-regular representation
$\rho$ and the natural inclusion induce isomorphisms
$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})\cong
K_{\ast}(\mathcal{C}_{L}(\tilde{\pi})^{G}_{\mathrm{proper}}).$
###### Proof.
The first isomorphism follows from the previous lemma using the functoriality
of both $K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)$ and
$K_{\ast}(C^{*}_{L,u}(\tilde{H}_{X}\otimes B)^{G})$ with respect to a
$G$-equivariant continuous cover $(V_{t}\colon H_{X}\to
H^{\prime}_{X})_{t\in[1,\infty)}$ of the identity map on $X$, i.e. we can
assume that the universal $X$-$G$-module $H_{X}$ is of infinite-multiplicity.
We always have the second isomorphism. ∎
## 11\. $\rho_{X}$ is an isomorphism, part II
In this section, we study a $G$-equivariant analogue of [16]. We let $X$ be a
$G$-compact proper $G$-space and $H_{X}$ be an $X$-$G$-module which will be
assumed to be of infinite-multiplicity at some places. We recall that
$\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}=C^{*}_{L,u}(H_{X}\otimes B)^{G}$ is the
$C^{*}$-subalgebra of $C_{b,u}([1,\infty),C^{*}(H_{X}\otimes B)^{G})$
consisting of functions $T$ such that
$\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$ and such that $T-\psi_{c}(T)\in
C_{0}([1,\infty),C^{*}(H_{X}\otimes B)^{G})$ for some (any) cut-off function
$c$ on $X$, where
$\psi_{c}(T)=\int_{g\in G}g(c)Tg(c)d\mu_{G}(g).$
We introduce the following $C^{*}$-algebras (c.f. [16, Section 3]).
* •
$D^{*}(H_{X}\otimes B)^{G}$ is the $C^{*}$-subalgebra of
$\mathfrak{L}(H_{X}\otimes B)$ generated by $G$-equivariant, properly
supported, pseudo-local operators. Here, an operator $T$ in
$\mathfrak{L}(H_{X}\otimes B)$ is pseudo-local if
$[\phi,T]\in\mathfrak{K}(H_{X}\otimes B)$ for any $\phi$ in $C_{0}(X)$.
* •
$\mathcal{D}_{L}(H_{X}\otimes B)^{G}$ is the $C^{*}$-subalgebra of
$C_{b,u}([1,\infty),D^{*}(H_{X}\otimes B)^{G})$ consisting of functions $T$
such that $\lim_{t\to\infty}\lVert[\phi,T_{t}]\rVert=0$.
* •
$\mathcal{D}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ is the
$C^{*}$-subalgebra of $\mathcal{D}_{L}(H_{X}\otimes B)^{G}$ consisting of
functions $T$ such that $T-\psi_{c}(T)\in C_{0}([1,\infty),D^{*}(H_{X}\otimes
B)^{G})$ for some (any) cut-off function $c$ on $X$. Similarly to Lemma 10.1,
the second condition is equivalent to that there is a properly supported
function $S$ in $\mathcal{D}_{L}(H_{X}\otimes B)^{G}$ such that $T-S\in
C_{0}([1,\infty),D^{*}(H_{X}\otimes B)^{G})$.
* •
$\mathcal{C}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ is the
$C^{*}$-subalgebra of $C_{b,u}([1,\infty),C^{*}(H_{X}\otimes B)^{G})$
generated by functions $T$ which are properly supported.
* •
$\mathcal{D}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ is the
$C^{*}$-subalgebra of $C_{b,u}([1,\infty),D^{*}(H_{X}\otimes B)^{G})$
generated by functions $T$ which are properly supported.
We have inclusions (each vertical map is an inclusion as an ideal)
$\textstyle{\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}},}$
as well as inclusions (horizontal maps are inclusions as constant functions)
$\textstyle{C^{*}(H_{X}\otimes
B)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D^{*}(H_{X}\otimes
B)^{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}.}$
###### Lemma 11.1.
We have
$\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\cap\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}=\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}.$
###### Proof.
Let $T\in\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\cap\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$. Then, we have $T\in
C_{b,u}([1,\infty),C^{*}(H_{X}\otimes B)^{G})$ and
$\lim_{t\to\infty}\lVert[T_{t},\phi]\rVert=0$ and $\lim_{t\to\infty}\lVert
T_{t}-\psi_{c}(T_{t})\rVert=0$ for any cut-off function $c$ on $X$. Thus,
$T\in\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$. The converse is
trivial. ∎
###### Lemma 11.2.
(c.f. [16, Lemma 4.1]) Let $D\subset C_{b,u}([1,\infty),D^{*}(H_{X}\otimes
B)^{G})$ be a separable subset. Assume that the set $D$ is properly supported
in a sense that for any compact subset $A$ of $X$, there is a compact subset
$B$ so that $\chi_{A}T=\chi_{A}T\chi_{B}$ (and $T\chi_{A}=\chi_{B}T\chi_{A}$)
for any $T\in D$. There is $x\in C_{b,u}([1,\infty),C^{*}(H_{X}\otimes
B)^{G})$ which is properly supported such that
* •
$[x,\phi]\in C_{0}([1,\infty),\mathfrak{K}(H_{X}\otimes B))$ for any $\phi\in
C_{0}(X)$,
* •
$(1-x)[T,\phi]\in C_{0}([1,\infty),\mathfrak{K}(H_{X}\otimes B))$ for any
$\phi\in C_{0}(X)$ and $T\in D$.
###### Proof.
Let $c$ be a cut-off function on $X$ and $X_{0}$ be the support of $c$ which
is a compact subset of $X$ with $GX_{0}=X$. Since $D$ is properly supported,
there is $\chi\in C_{c}(X)$ such that $c=c\chi$ and $cT=cT\chi$ for any $T\in
D$. Let $X_{1}$ be the support of $\chi$. Let $K$ be a compact subset of $G$
so that $X_{0}\cap gX_{1}=\emptyset$ for $g\in G\backslash K$. Let
$(\phi_{n})_{n}$ be a countable subset of $C_{0}(X)$ such that
$\mathrm{supp}(\phi_{n})\subset X_{0}$ for all $n$ and such that
$G(\phi_{n})_{n}$ is total in $C_{0}(X)$. Let $(T^{(m)})_{m\geq 1}$ be a
countable dense subset of $D$.
Let $y_{n}$ be an approximate unit in $\mathfrak{K}(H_{X}\otimes B)$ quasi-
central with respect to $C_{0}(X)$ so that we have
$\lim_{n\to\infty}\lVert[\phi,y_{n}]\rVert=0$ for any $\phi\in C_{0}(X)$ and
such that
$\lVert(1-y_{n})c[T^{(m)}_{t},g^{-1}(\phi_{k})]\rVert<1/n$
for any $g\in K$, $t\in[1,n+1]$, $1\leq m\leq n$, and $1\leq k\leq n$. Let
$x_{n}=\int_{g\in G}g(c)g(y_{n})g(c)d\mu_{G}(g)\in C^{*}(H_{X}\otimes B)^{G}.$
Then, we have $(n\mapsto[\phi,x_{n}])\in
C_{0}(\mathbb{N},\mathfrak{K}(H_{X}\otimes B))$ for any $\phi$ in $C_{0}(X)$.
Take any $\phi$ in $C_{0}(X)$ with support contained in $X_{0}$. For any $T\in
D$, we have
$(1-x_{n})[T_{t},\phi]=\int_{g\in
G}g(c)(1-g(y_{n}))g(c)[T_{t},\phi]g(\chi)d\mu_{G}(g)$ $=\int_{g\in
K}g(c)(1-g(y_{n}))g(c)[T_{t},\phi]g(\chi)d\mu_{G}(g),$
and
$\lVert(1-g(y_{n}))g(c)[T_{t},\phi]\rVert=\lVert(1-y_{n})c[T_{t},g^{-1}(\phi)]\rVert.$
In particular, we have
$\lVert(1-g(y_{n}))g(c)[T^{(m)}_{t},\phi_{k}]\rVert\leq 1/n$
for any $g\in K$, $t\in[1,n+1]$, $1\leq m\leq n$ and $1\leq k\leq n$ and so
$\lVert(1-x_{n})[T^{(m)}_{t},\phi_{k}]\rVert\leq C/n$
for any $t\in[1,n+1]$, $1\leq m\leq n$ and $1\leq k\leq n$, where the constant
$C>0$ only depends on fixed functions $c$ and $\chi$. A desired function $x$
can be obtained by
$x_{t}=(n+1-t)x_{n}+(t-n)x_{n+1}$
for $t\in[n,n+1]$.
∎
###### Lemma 11.3.
(c.f. [39, Proposition 2.3], [16, Proposition 4.2]) The natural map
(inclusion)
$\eta\colon\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\ \to\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$
is bijective (isomorphism).
###### Proof.
It is enough to show that for any $T\in C_{b,u}([1,\infty),D^{*}(H_{X}\otimes
B)^{G})$ which is properly supported, there is
$S\in\mathcal{D}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ such that
$T-S\in\mathcal{C}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$.
We let $x$ as given by Lemma 11.2 for $D=\\{T\\}$. Set $S=(1-x)T$. Then, we
have
$T-S=xT\in\mathcal{C}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$
since $x\in\mathcal{C}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$. We claim
that $S\in\mathcal{D}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$. Since $x$
and $T$ are properly supported, $S=(1-x)T$ is properly supported. Thus, it is
enough to show that $\lim_{t\to\infty}\lVert[\phi,S_{t}]\rVert=0$ but this
follows from
$[\phi,S]=-[\phi,x]T+(1-x)[\phi,T]$
and from the property of $x$. ∎
###### Lemma 11.4.
(c.f. [39, Proposition 3.5], [16, Proposition 4.3]) Assume that $H_{X}$ is of
infinite multiplicity. We have $K_{\ast}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})=0$. Hence the boundary map
$\partial\colon K_{\ast}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\to K_{\ast+1}(\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})$
is an isomorphism.
###### Proof.
We have $H_{X}\cong H^{(0)}(X)\otimes l^{2}(\mathbb{N})$ and let $U_{n}$ be
isometries on $l^{2}(\mathbb{N})$ such that
$\sum_{n\geq 0}U_{n}U_{n}^{\ast}=1.$
The following map is well-defined on $\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$,
$\alpha\colon T\mapsto\sum_{n\geq 0}U_{n}T_{t+n}U_{n}^{\ast}.$
Indeed, it maps the ideal $C_{0}([1,\infty),D^{*}(H_{X}\otimes B)^{G})$ to
itself and maps properly supported functions in $\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ to themselves. Once we have $\alpha$ well-defined,
it is routine to show that $K_{\ast}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})=0$. ∎
###### Lemma 11.5.
(c.f. [39, Proposition 3.6], [16, Proposition 4.3]) Assume that $H_{X}$ is of
infinite multiplicity. The evaluation map at $t=1$ induces an isomorphism
$\mathrm{ev}_{1\ast}\colon K_{\ast}(\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\cong K_{\ast}(D^{*}(H_{X}\otimes
B)^{G}/C^{*}(H_{X}\otimes B)^{G}).$
###### Proof.
Let $\mathcal{D}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ be the kernel
of the evaluation map $\mathrm{ev}_{1}$ on $\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ and let $\mathcal{C}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}=\mathcal{D}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\cap\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$. We have a split short exact sequence
$0\to\mathcal{D}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ $\to\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\to D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes
B)^{G}\to 0.$
Thus, it suffices to show that $K_{\ast}(\mathcal{D}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})=0$ and $K_{\ast}(\mathcal{C}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})=0$.
Let $U_{n}$ be isometries on $H_{X}\otimes B$ as in the proof of Lemma 11.4.
The following map is well-defined on $\mathcal{C}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ and on $\mathcal{D}^{0}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$,
$\alpha\colon T\mapsto\sum_{n\geq 0}U_{n}T_{t-n}U_{n}^{\ast}$
where functions $T$ on $[1,\infty)$ vanishing at $t=1$ are constantly extended
to the left by zero. Indeed, it maps properly supported functions in
$\mathcal{C}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$, resp. in
$\mathcal{D}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$, vanishing at
$t=1$ to themselves and it is not to hard to see that they form a dense
subalgebra of $\mathcal{C}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$,
resp. of $\mathcal{D}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$.
Once we have $\alpha$ well-defined, it is routine to show that
$K_{\ast}(\mathcal{C}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}})=0$ and
$K_{\ast}(\mathcal{D}^{0}_{T}(H_{X}\otimes B)^{G}_{\mathrm{proper}})=0$. ∎
Let $\pi\colon C_{0}(X)\to\mathfrak{L}(H_{X}\otimes B)$ be the structure map
for the $X$-$G$-module $H_{X}\otimes B$. We define:
* •
$C^{*}(\pi)^{G}$ is the $C^{*}$-subalgebra of $\mathfrak{L}(H_{X}\otimes B)$
consisting of $G$-equivariant, locally compact operators.
* •
$D^{*}(\pi)^{G}$ is the $C^{*}$-subalgebra of $\mathfrak{L}(H_{X}\otimes B)$
consisting of $G$-equivariant, pseudo-local operators.
The following is a standard fact (c.f. [27, Lemma 6.2], [40, Lemma 5.8], [28,
Lemma 12.3.2]).
###### Lemma 11.6.
The natural map
$\eta_{0}\colon D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes B)^{G}\to
D^{*}(\pi)^{G}/C^{*}(\pi)^{G}$
is bijective (isomorphism).
###### Proof.
Surjectivity: Let $T\in D^{*}(\pi)^{G}$. For a cut-off function $c$ on $X$,
consider
$T^{\prime}=\int_{g\in G}g(c)Tg(c)d\mu_{G}(g).$
Then, we have $T^{\prime}\in D^{*}(H_{X}\otimes B)^{G}$ and $T-T^{\prime}\in
C^{*}(\pi)^{G}$. Injectivity: we need to show that $D^{*}(H_{X}\otimes
B)^{G}\cap C^{*}(\pi)^{G}=C^{*}(H_{X}\otimes B)^{G}$. Note that this is not a
trivial consequence of the definitions. We first claim that $T-T^{\prime}\in
C^{*}(H_{X}\otimes B)^{G}$ for any $T\in D^{*}(H_{X}\otimes B)^{G}$. In fact,
if $T$ is properly supported, this is easy to see since then, $T-T^{\prime}$
is $G$-equivariant, locally compact and properly supported. The general case
follows from the continuity of the map $T\mapsto T-T^{\prime}$. Let $T\in
D^{*}(H_{X}\otimes B)^{G}\cap C^{*}(\pi)^{G}$. Then, $T\in C^{*}(H_{X}\otimes
B)^{G}$ follows from $T^{\prime}\in C^{*}(H_{X}\otimes B)^{G}$ which is easy
to see since $T^{\prime}$ is $G$-equivariant, locally compact and properly
supported. ∎
###### Proposition 11.7.
Assume that $H_{X}$ is of infinite multiplicity. We have the following
sequence of isomorphisms.
$\textstyle{K_{\ast}(D^{*}(\pi)^{G}/C^{*}(\pi)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta_{0}^{-1}}$$\textstyle{K_{\ast}(D^{*}(H_{X}\otimes
B)^{G}/C^{*}(H_{X}\otimes
B)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\iota}$$\textstyle{K_{\ast}(\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta^{-1}}$$\textstyle{K_{\ast+1}(\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
K_{\ast}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})}$$\scriptstyle{\cong}$$\scriptstyle{\partial}$
where $\iota$ is the natural inclusion as constant functions.
###### Proof.
Combine Lemma 11.3, Lemma 11.4, Lemma 11.5 and Lemma 11.6. ∎
## 12\. $\rho_{X}$ is an isomorphism, part III
Let us consider, in general, an $X$-$G$-Hilbert $B$-module $\mathcal{E}$, that
is a (countably generated) $G$-Hilbert $B$-module equipped with a non-
degenerate representation of the $G$-$C^{*}$-algebra $C_{0}(X)$. The support
and the propagation of operators between two $X$-$G$-Hilbert $B$-modules are
defined in an obvious way. We let $\tilde{\mathcal{E}}$ be an $X$-$G$-Hilbert
$B$-module $\mathcal{E}\otimes L^{2}(G)$ equipped with a unitary
representation
$G\ni g\mapsto u_{g}\otimes\lambda_{g}$
where $u_{g}$ is the unitary on $\mathcal{E}$ corresponding to $g\in G$ and
$\lambda_{g}$ is the left-translation by $g$ and with the structure map
$C_{0}(X)\ni\phi\mapsto\phi\otimes 1.$
For any cut-off function $c$ on $X$, we have a $G$-equivariant isometry
$V_{c}\in\mathfrak{L}(\mathcal{E},\tilde{\mathcal{E}})$ defined by sending
$v\in\mathcal{E}$ to
$(V_{c}(v)\colon h\mapsto h(c)v)\in\tilde{\mathcal{E}}=L^{2}(G,\mathcal{E}).$
The isometry $V_{c}$ is a strict cover of the identity, i.e. it commutes with
$C_{0}(X)$.
###### Lemma 12.1.
Let $G$ be a locally compact group and $X$ be a $G$-compact, proper $G$-space.
Let $H_{X}$ be an $X$-$G$-module which is ample as an $X$-module and let
$\mathcal{E}$ be the $X$-$G$-Hilbert $B$-module $\tilde{H}_{X}\otimes B$. For
any $X$-$G$-Hilbert $B$-module $\mathcal{E}_{0}$, there is a sequence $V_{n}$
of $G$-equivariant isometries in $\mathfrak{L}(\mathcal{E}_{0},\mathcal{E})$
such that for any $\phi\in C_{0}(X)$,
* •
$[V_{n},\phi]\in\mathfrak{K}(\mathcal{E}_{0},\mathcal{E})$,
* •
$\lim_{n\to\infty}\lVert[V_{n},\phi]\rVert=0$.
###### Proof.
Let $c\in C_{c}(X)$ be a cut-off function and $X_{0}$ be the support of $c$.
Let $W_{n}\in\mathfrak{L}(\mathcal{E}_{0},H_{X}\otimes B)$ be a sequence of
isometries satisfying the following three conditions: for any $\phi$ in
$C_{0}(X)$,
* •
$[W_{n},\phi]\in\mathfrak{K}(\mathcal{E}_{0},H_{X}\otimes B)$,
* •
$\lim_{n\to\infty}\lVert[W_{n},\phi]\rVert=0$,
* •
$W_{n}c=\chi W_{n}c$ for some $\chi\in C_{c}(X)$.
We explain how we can find such $W_{n}$. Fix a metric $d$ on $X^{+}$ and let
$\delta>0$ be small enough so that the $\delta$-neighborhood of $X_{0}$ is
relatively compact in $X$. Let $(\phi_{i})_{i\in S}$ be a finite partition of
unity in $C(X^{+})$ so that the support of each $\phi_{i}$ is contained in a
ball of radius $\delta/2$. Let $X_{i}$ be the support of $\phi_{i}$ and let
$\mathcal{E}_{0}^{(i)}=C(X^{+})\phi_{i}\mathcal{E}_{0}$ which is naturally an
$X_{i}$-module. Then, we have an isometry
$\Phi\colon\mathcal{E}_{0}\to\bigoplus_{i\in
S}\mathcal{E}_{0}^{(i)},\,\,\,v\mapsto(\phi^{1/2}_{i}v).$
Let $H_{X_{i}}=C(X^{+})\phi_{i}H_{X}$ which is naturally an ample
$X_{i}$-module. We take a sequence
$W_{n}^{(i)}\in\mathfrak{L}(\mathcal{E}^{(i)}_{0},H_{X_{i}}\otimes B)$ of
isometries satisfying,
* •
For each $n$, $(W_{n}^{(i)})_{i\in S}$ have mutually orthogonal ranges in
$H_{X}\otimes B$,
* •
$[W^{(i)}_{n},\phi]\in\mathfrak{K}(\mathcal{E}_{0},H_{X_{i}}\otimes B)$ for
any $\phi$ in $C(X_{i})$,
* •
$\lim_{n\to\infty}\lVert[W^{(i)}_{n},\phi]\rVert=0$ for any $\phi$ in
$C(X_{i})$.
Such isometries exist because $H_{X_{i}}$ is an ample $C(X_{i})$-module (see
[16, Theorem 2.3]). Let $W_{n}$ be the composition of $\Phi$ and the
(orthogonal) sum of $W_{n}^{(i)}$. We can see that $W_{n}$ are isometries
satisfying the three conditions above.
We now define a sequence $V_{n}$ of $G$-equivariant isometries in
$\mathfrak{L}(\mathcal{E}_{0},\mathcal{E})$ as the composition
$V_{n}=\tilde{W}_{n}V_{c}\colon\mathcal{E}_{0}\to\tilde{\mathcal{E}}_{0}\to\tilde{H}_{X}\otimes
B$
where a $G$-equivariant isometry
$\tilde{W}_{n}\colon\tilde{\mathcal{E}}_{0}\to\tilde{H}_{X}\otimes B$ is of
the form
$(\tilde{W}_{n}\colon g\mapsto g(W_{n}))\in
C_{b,\mathrm{SOT}^{*}}(G,\mathfrak{L}(\mathcal{E}_{0},H_{X}\otimes B)).$
Here, $g(W_{n})=u^{\prime}_{g}W_{n}u^{-1}_{g}$ where $u^{\prime}_{g}$ (resp.
$u_{g}$) is the unitary on $H_{X}\otimes B$, (resp. on $\mathcal{E}_{0}$)
corresponding to $g$ in $G$. Note that $g(W_{n})$ is adjointable although
$u^{\prime}_{g}$ and $u_{g}$ may not. We also note that $g\mapsto g(W_{n})$ is
not necessarily norm-continuous ($W_{n}$ may not be $G$-continuous).
We show that $V_{n}$ satisfies the two desired conditions. Concretely, $V_{n}$
maps $v\in\mathcal{E}_{0}$ to
$(h\to h(\chi)h(W_{n})h(c)v)\in L^{2}(G,H_{X}\otimes B).$
Let $\phi\in C_{c}(X)$. Then, $[V_{n},\phi]$ maps $v\in\mathcal{E}_{0}$ to
$(h\to h(\chi)[h(W_{n}),\phi]h(c)v)\in L^{2}(G,H_{X}\otimes B)$
and we note that $h(\chi)[h(W_{n}),\phi]h(c)=0$ for $h\in G\backslash K$ for
some compact subset $K$ of $G$ which only depends on the support of $\phi$
(and $c$ and $\chi$) but not on $n$. Now, we have
$[V_{n},\phi]^{\ast}[V_{n},\phi]=\int_{h\in
K}h(c)[h(W_{n}),\phi]^{\ast}h(\chi^{2})[h(W_{n}),\phi]h(c)d\mu_{G}(h).$
We note $(h\mapsto[h(W_{n}),\phi])\in
C_{b}(G,\mathfrak{K}(\mathcal{E}_{0},H_{X}\otimes B))$ since we have
$(h\mapsto[W_{n},h^{-1}(\phi)])\in
C_{b}(G,\mathfrak{K}(\mathcal{E}_{0},H_{X}\otimes B))$. We also have
$\lim_{n\to\infty}\lVert[W_{n},h^{-1}(\phi)]\rVert=0$ uniformly in $h\in K$.
From these, we see that
* •
$[V_{n},\phi]^{\ast}[V_{n},\phi]\in\mathfrak{K}(\mathcal{E}_{0})$,
* •
$\lim_{n\to\infty}\lVert[V_{n},\phi]^{\ast}[V_{n},\phi]\rVert=0$.
We are done. ∎
Let $H_{X}$ be an $X$-$G$-module which is ample as an $X$-module. Let
$\pi\colon C_{0}(X)\to\mathfrak{L}(\tilde{H}_{X}\otimes B)$
be the structure map for the $X$-$G$-Hilbert $B$-module $\tilde{H}_{X}\otimes
B$. Note that we may view $\pi$ as a non-degenerate representation
$\pi\colon C_{0}(X)\to M(\mathfrak{K}(\tilde{H}_{X})\otimes B)$
and $B^{\prime}=\mathfrak{K}(\tilde{H}_{X})\otimes B$ is $G$-stable in a sense
that $B^{\prime}\otimes\mathfrak{K}(L^{2}(G))\cong B^{\prime}$ as
$G$-$C^{*}$-algebras.
In general, for any separable $G$-$C^{*}$-algebra $A$ and for any separable
$G$-stable $G$-$C^{*}$-algebra $B^{\prime}$, let us say that a non-degenerate,
$G$-equivariant representation
$\pi\colon A\to M(B^{\prime})$
is non-degenerately $G$-equivariantly absorbing (c.f. [43, Section 2]) if for
any non-degenerate $G$-equivariant representation
$\pi_{0}\colon A\to M(B^{\prime}),$
there is a sequence $u_{n}$ of $G$-equivariant unitaries in
$\mathfrak{L}(B^{\prime}\oplus B^{\prime},B^{\prime})$ such that for any $a\in
A$,
* •
$u_{n}(\pi_{0}(a)\oplus\pi(a))-\pi(a)u_{n}\in\mathfrak{K}(B^{\prime}\oplus
B^{\prime},B^{\prime})$,
* •
$\lim_{n\to\infty}\lVert u_{n}(\pi_{0}(a)\oplus\pi(a))-\pi(a)u_{n}\rVert=0$.
It is routine to deduce from Lemma 12.1 that the structure map $\pi$ for
$\tilde{H}_{X}\otimes B$ is non-degenerately $G$-equivariantly absorbing.
###### Proposition 12.2.
For any $X$-$G$-module $H_{X}$ which is ample as an $X$-module, the structure
map
$\pi\colon C_{0}(X)\to M(B^{\prime})$
for the $X$-$G$-module $\tilde{H}_{X}\otimes B$ where
$B^{\prime}=\mathfrak{K}(\tilde{H}_{X})\otimes B$, is non-degenerately
$G$-equivariantly absorbing.
###### Proof.
We use a standard trick (see the proof of [2, Corollary 2, p341]). Let
$\pi_{0}\colon C_{0}(X)\to M(B^{\prime})$ be a non-degenerate, $G$-equivariant
representation and let $\pi_{0}^{\infty}\colon C_{0}(X)\to M(B^{\prime})$ be
its amplification so that $\pi_{0}\oplus\pi_{0}^{\infty}$ is unitarily
equivalent to $\pi_{0}^{\infty}$. Lemma 12.1 gives a sequence $v_{n}$ of
$G$-equivariant isometries in $\mathfrak{L}(B^{\prime},B^{\prime})$ such that
for any $\phi\in C_{0}(X)$,
* •
$v_{n}\pi^{\infty}_{0}(\phi)-\pi(\phi)v_{n}\in\mathfrak{K}(B^{\prime},B^{\prime})$,
* •
$\lim_{n\to\infty}\lVert v_{n}\pi^{\infty}_{0}(\phi)-\pi(\phi)v_{n}\rVert=0$.
Let $p_{n}=1-v_{n}v_{n}^{\ast}$ and $\sigma_{n}$ be a $G$-equivariant c.c.p.
map from $C_{0}(X)$ to $\mathfrak{L}(p_{n}B^{\prime})$ defined by
$\sigma_{n}(\phi)=p_{n}\pi(\phi)p_{n}$. We set $w_{n}$ to be a unitary in
$\mathfrak{L}(p_{n}B^{\prime}\oplus B^{\prime},B^{\prime})$ given by the
direct sum of the identity map on $p_{n}B^{\prime}$ and $v_{n}$. Then, $w_{n}$
is a $G$-equivariant unitary in $\mathfrak{L}(p_{n}B^{\prime}\oplus
B^{\prime},B^{\prime})$ such that for any $\phi\in C_{0}(X)$,
* •
$w_{n}(\sigma_{n}(\phi)\oplus\pi^{\infty}_{0}(\phi))-\pi(\phi)w_{n}\in\mathfrak{K}(p_{n}B^{\prime}\oplus
B^{\prime},B^{\prime})$,
* •
$\lim_{n\to\infty}\lVert
w_{n}(\sigma_{n}(\phi)\oplus\pi^{\infty}_{0}(\phi))-\pi(\phi)w_{n}\rVert=0$.
Define unitaries $u_{n}$ in $\mathfrak{L}(B^{\prime}\oplus
B^{\prime},B^{\prime})$ by the composition
$\textstyle{B^{\prime}\oplus
B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{B^{\prime}}\oplus
w_{n}^{\ast}}$$\textstyle{B^{\prime}\oplus(p_{n}B^{\prime}\oplus
B^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{p_{n}B^{\prime}\oplus(B^{\prime}\oplus
B^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1_{p_{n}B^{\prime}}\oplus
w}$$\textstyle{p_{n}B^{\prime}\oplus
B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w_{n}}$$\textstyle{B^{\prime}}$
Where $w\in\mathfrak{L}(B^{\prime}\oplus B^{\prime},B^{\prime})$ is a unitary
equivalence from $\pi_{0}\oplus\pi_{0}^{\infty}$ to $\pi_{0}^{\infty}$. Then,
we have for any $\phi\in C_{0}(X)$,
* •
$u_{n}(\pi_{0}(\phi)\oplus\pi(\phi))-\pi(\phi)u_{n}\in\mathfrak{K}(B^{\prime}\oplus
B^{\prime},B^{\prime})$,
* •
$\lim_{n\to\infty}\lVert
u_{n}(\pi_{0}(\phi)\oplus\pi(\phi))-\pi(\phi)u_{n}\rVert=0$.
∎
For any $G$-equivariant representation
$\pi\colon A\to M(B^{\prime}),$
we define $\mathcal{D}(\pi)^{G}$ to be the $C^{*}$-algebra of $G$-equivariant
elements $x$ in $M(B^{\prime})$ such that $[x,\pi(a)]\in B^{\prime}$ for any
$a\in A$ and $\mathcal{C}(\pi)^{G}$ to be the $C^{*}$-algebra of
$G$-equivariant elements $x$ in $M(B^{\prime})$ such that $x\pi(a),\pi(a)x\in
B^{\prime}$ for any $a\in A$.
We have a group homomorphism
$\Theta\colon K_{1}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})\to
KK_{0}^{G}(A,B^{\prime})$
which sends a unitary $u$ in
$M_{n}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})$ to the following even
Kasparov triple for $KK_{0}^{G}(A,B^{\prime})$,
$(B^{\prime\oplus n}\oplus B^{\prime\oplus n},\pi^{\oplus n}\oplus\pi^{\oplus
n},\,\begin{bmatrix}0&v\\\ v^{\ast}&0\end{bmatrix})$
where $v$ is any lift of $u$ in $M_{n}(\mathcal{D}(\pi)^{G})$. Similarly, we
have a group homomorphism
$\Theta\colon K_{0}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})\to
KK_{1}^{G}(A,B^{\prime})$
which sends a projection $p$ in
$M_{n}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})$ to the following odd
Kasparov triple for $KK_{1}^{G}(A,B^{\prime})$,
$(B^{\prime\oplus n},\pi^{\oplus n},2P-1)$
where $P$ is any (self-adjoint) lift of $p$ in
$M_{n}(\mathcal{D}({\pi})^{G})$.
We recall some terminologies from Section 3 [44]. An even Kasparov triple
$\mathcal{E}=(E,\phi,F)$ for $KK_{0}^{G}(A,B^{\prime})$ is called elementary
if $E=B^{\prime}\oplus B^{\prime}$ and it it called essential if $\phi(A)E=E$.
Similarly, an odd Kasparov triple $\mathcal{E}=(E,\phi,P)$ for
$KK_{1}^{G}(A,B^{\prime})$ is called elementary if $E=B^{\prime}$ and it it
called essential if $\phi(A)E=E$.
###### Proposition 12.3.
([44, Theorem 3.9, Theorem 4.2]) Let $A$ be a separable proper
$G$-$C^{*}$-algebra and $B^{\prime}$ be a separable, $G$-stable
$G$-$C^{*}$-algebra.
1. (1a)
Every element of $KK^{G}_{0}(A,B^{\prime})$ is represented by an even Kasparov
triple which is both elementary and essential.
2. (1b)
Two elementary and essential even Kasparov triples, $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ define the same element of $KK^{G}_{0}(A,B^{\prime})$ if and
only if there are degenerate even Kasparov triples, $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$ for $KK^{G}_{0}(A,B^{\prime})$ which are both elementary and
essential, such that $\mathcal{E}_{1}\oplus\mathcal{D}_{1}$ is operator
homotopic to $\mathcal{E}_{2}\oplus\mathcal{D}_{2}$.
3. (2a)
Every element of $KK^{G}_{1}(A,B^{\prime})$ is represented by an odd Kasparov
triple which is both elementary and essential.
4. (2b)
Two elementary and essential odd Kasparov triples, $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ define the same element of $KK^{G}_{1}(A,B^{\prime})$ if and
only if there are degenerate odd Kasparov triples, $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$ for $KK^{G}_{1}(A,B^{\prime})$ which are both elementary and
essential, such that $\mathcal{E}_{1}\oplus\mathcal{D}_{1}$ is operator
homotopic to $\mathcal{E}_{2}\oplus\mathcal{D}_{2}$.
###### Proof.
In [44], the proof was given for $A$ which is $G$-stable but the only property
of $A$ which is used for the proof is that for any $G$-equivariant
representation $\phi\colon A\to\mathfrak{L}(E)$ on a $G$-Hilbert
$B^{\prime}$-module $E$ with $\phi(A)E=E$, we have $E\oplus B^{\prime}\cong
B^{\prime}$ (see [44, Theorem 2.8, Corollary 2.9]). Any proper
$G$-$C^{*}$-algebra has this property (see [35, Proposition 8.6]). ∎
###### Theorem 12.4.
(c.f. [43, Theorem 3.2]) Let $A$ be a separable proper $G$-$C^{*}$-algebra and
$B^{\prime}$ be a separable, $G$-stable $G$-$C^{*}$-algebra. Suppose a non-
degenerate $G$-equivariant representation $\pi\colon A\to M(B^{\prime})$ is
non-degenerately $G$-equivariantly absorbing. Then, the group homomorphisms
$\Theta$ induce isomorphisms
$K_{1}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})\cong
KK_{0}^{G}(A,B^{\prime})\,\,\,\text{and}\,\,\,K_{0}(\mathcal{D}(\pi)/\mathcal{C}(\pi)^{G})\cong
KK_{1}^{G}(A,B^{\prime}).$
###### Proof.
The proof of [43, Theorem 3.2] works with minor changes so we shall be brief.
We give a proof for the isomorphism
$K_{1}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})\cong
KK_{0}^{G}(A,B^{\prime})$. The other one is parallel. Surjectivity: Take any
even elementary, essential, Kasparov triple $\mathcal{E}=(B^{\prime}\oplus
B^{\prime},\pi_{0}\oplus\pi_{1},\begin{bmatrix}0&x\\\
x^{\ast}&0\end{bmatrix})$ for $KK^{G}_{0}(A,B^{\prime})$. By adding an
essential, degenerate Kasparov triple
$\mathcal{D}=(B^{\prime\infty}\oplus
B^{\prime\infty},(\pi^{\infty}_{0}\oplus\pi^{\infty}_{1})\oplus(\pi^{\infty}_{0}\oplus\pi^{\infty}_{1}),\begin{bmatrix}0&1\\\
1&0\end{bmatrix})\oplus(B^{\prime}\oplus
B^{\prime},\pi\oplus\pi,\begin{bmatrix}0&1\\\ 1&0\end{bmatrix})$
to $\mathcal{E}$, where $(\pi^{\infty}_{0}\oplus\pi^{\infty}_{1})$ is the
infinite direct sum of $(\pi_{0}\oplus\pi_{1})$, we see that the triple
$\mathcal{E}\oplus\mathcal{D}$ is isomorphic to the essential triple
$\mathcal{E}_{1}$ of the form $(B^{\prime}\oplus
B^{\prime},\pi^{\prime}\oplus\pi^{\prime},\begin{bmatrix}0&v\\\
v^{\ast}&0\end{bmatrix})$ where $\pi^{\prime}(a)-\pi(a)\in B^{\prime}$ for any
$a\in A$, i.e. $\mathcal{E}_{1}$ is a compact perturbation of the triple
$(B^{\prime}\oplus B^{\prime},\pi\oplus\pi,\begin{bmatrix}0&v\\\
v^{\ast}&0\end{bmatrix})$. Using a cut-off function (as usual), the latter
triple is a compact perturbation of the triple of the same form where $v$ is
$G$-equivariant. Such a triple is in the image of $\Theta$.
Injectivity: We may assume $\pi$ is of infinite multiplicity. We take a
unitary $u\in\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G}$ and let
$v\in\mathcal{D}(\pi)^{G}$ be any lift of $u$ (it is enough to consider
$1\times 1$-matrix since $\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G}$ is
properly infinite.). Suppose $\Theta(u)=0$ in $KK^{G}_{0}(A,B^{\prime})$.
Then, there are essential, degenerate triples
$\mathcal{D}_{1},\mathcal{D}_{2}$ such that $(B^{\prime}\oplus
B^{\prime},\pi\oplus\pi,\begin{bmatrix}0&v\\\
v^{\ast}&0\end{bmatrix})\oplus\mathcal{D}_{1}$ is operator homotopic to
$(B^{\prime}\oplus B^{\prime},\pi\oplus\pi,\begin{bmatrix}0&1\\\
1&0\end{bmatrix})\oplus\mathcal{D}_{2}$. By adding infinite copies of
$\mathcal{D}_{1}\oplus\mathcal{D}_{2}$ to both, we may assume that
$\mathcal{D}_{1}=\mathcal{D}_{2}$. We can further arrange that
$\mathcal{D}_{1}=\mathcal{D}_{2}$ are of the form $(B^{\prime}\oplus
B^{\prime},\lambda\oplus\lambda,\begin{bmatrix}0&1\\\ 1&0\end{bmatrix})$ with
$\lambda$ of infinite multiplicity. By adding $(B^{\prime}\oplus
B^{\prime},\pi\oplus\pi,\begin{bmatrix}0&1\\\ 1&0\end{bmatrix})$, we can
assume that there is a $G$-equivariant unitary $w\in M(B^{\prime})$ such that
$w\lambda(a)w^{\ast}-\pi(a)\in B^{\prime}$ for any $a\in A$. From here, using
the same reasoning of the proof [43, Theorem 3.2], we get a $G$-equivariant
norm-continuous path $G_{t}$ in the matrix algebra $M_{n}(M(B^{\prime}))$ such
that $G_{0}=1_{n}$ and $G_{1}=v\oplus 1_{n-1}$ and
$(G_{t}^{\ast}G_{t}-1_{n})\pi(a),(G_{t}G_{t}^{\ast}-1_{n})\pi(a),[\pi(a),G_{t}]\in
M_{n}(B^{\prime})$
for all $t$ and for all $a\in A$. The $G$-equivariance can be arranged by
perturbing the operator homotopy into $G$-equivariant one using a cut-off
function $c$ for a proper $G$-space $X$ for which $A$ is a
$G$-$C_{0}(X)$-algebra. More precisely, $G_{t}$ would be a homotopy between
$G_{0}=1_{n}$ and $G_{1}=v^{\prime}\oplus 1_{n-1}$ where
$v^{\prime}=\int_{g\in G}g(c)g(v)g(c)d\mu_{G}(g)=\int_{g\in
G}g(c)vg(c)d\mu_{G}(g)$ but $v^{\prime}$ is still a lift of $u$. This gives a
path of unitaries in $M_{n}(\mathcal{D}(\pi)^{G}/\mathcal{C}(\pi)^{G})$
connecting $u\oplus 1_{n-1}$ to $1_{n}$. ∎
###### Corollary 12.5.
Let $G$ be a locally compact group, $X$ be a $G$-compact proper $G$-space and
$B$ be a separable $G$-$C^{*}$-algebra. Let $H_{X}$ be an $X$-$G$-module which
is ample as an $X$-module. Let $\pi\colon
C_{0}(X)\to\mathfrak{L}(\tilde{H}_{X}\otimes B)$ be the structure map for the
$X$-$G$-Hilbert $B$-module $\tilde{H}_{X}\otimes B$. Then, there are canonical
isomorphisms
$\Theta\colon K_{1}(D^{*}(\pi)^{G}/C^{*}(\pi)^{G})\cong
KK_{0}^{G}(C_{0}(X),B),$ $\Theta\colon
K_{0}(D^{*}(\pi)^{G}/C^{*}(\pi)^{G})\cong KK_{1}^{G}(C_{0}(X),B).$
###### Corollary 12.6.
Let $G$ be a locally compact group, $X$ be a $G$-compact proper $G$-space and
$B$ be a separable $G$-$C^{*}$-algebra. Let $H_{X}$ be a universal
$X$-$G$-module. Assume that $H_{X}$ is of infinite multiplicity. Let
$\pi\colon C_{0}(X)\to\mathfrak{L}(\tilde{H}_{X}\otimes B)$ be the structure
map for the $X$-$G$-Hilbert $B$-module $\tilde{H}_{X}\otimes B$. We have the
following sequence of isomorphisms.
$\textstyle{KK_{\ast}^{G}(C_{0}(X),B)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
K_{\ast+1}(D^{*}(\pi)^{G}/C^{*}(\pi)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\Theta}$$\scriptstyle{\cong}$$\scriptstyle{\eta_{0}^{-1}}$$\textstyle{K_{\ast+1}(D^{*}(\tilde{H}_{X}\otimes
B)^{G}/C^{*}(\tilde{H}_{X}\otimes
B)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\iota}$$\textstyle{K_{\ast+1}(\mathcal{D}_{T}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta^{-1}}$$\textstyle{K_{\ast+1}(\mathcal{D}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\partial}$$\textstyle{K_{\ast}(RL^{*}_{u}(H_{X}\otimes
B)\rtimes_{r}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\rho_{\ast}}$$\textstyle{K_{\ast}(\mathcal{C}_{L}(\tilde{H}_{X}\otimes
B)^{G}_{\mathrm{proper}})}$
In particular, for any universal $X$-$G$-module $H_{X}$, there is a canonical
isomorphism
$K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\cong
KK_{\ast}^{G}(C_{0}(X),B).$
###### Proof.
Combine Theorem 10.12, Proposition 11.7 and Corollary 12.5. ∎
## 13\. $\rho_{X}$ is an isomorphism, part IV
Now we are ready to show that the group homomorphism
$\rho_{X}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
E_{\ast}^{G}(C_{0}(X),B)\cong KK_{\ast}^{G}(C_{0}(X),B)$
is an isomorphism for any $G$-compact proper $G$-space $X$ where $H_{X}$ is a
universal $X$-$G$-module.
For any $X$-$G$-module $H_{X}$, we have a group homomorphism
$\iota_{X}\colon K_{\ast}(C_{L,u}^{*}(H_{X}\otimes B)^{G})\to
E_{\ast}^{G}(C_{0}(X),B)\cong KK_{\ast}^{G}(C_{0}(X),B)$
defined in the same way as $\rho_{X}$ using a canonical asymptotic morphism
$\pi_{X}\otimes\iota\colon C_{0}(X)\otimes C_{L,u}^{*}(H_{X}\otimes
B)^{G}\to\mathfrak{A}(\mathfrak{K}(H_{X}\otimes B))$
such that the image of $\phi\otimes T$ is represented by
$\phi T\in C_{b}([1,\infty),\mathfrak{K}(H_{X}\otimes B)).$
It is clear that we have $\rho_{X}=\iota_{X}\circ\rho_{\ast}$, that is
$\rho_{X}$ factors through the map
$\rho_{\ast}\colon K_{\ast}(RL^{*}_{u}(H_{X}\otimes B)\rtimes_{r}G)\to
K_{\ast}(C_{L,u}^{*}(\tilde{H}_{X}\otimes B)^{G})$
induced by the right-regular representation which is an isomorphism whenever
$H_{X}$ is universal.
###### Lemma 13.1.
(c.f. [16, Theorem 5.1]) Let $G$ be a locally compact group, $X$ be a
$G$-compact proper $G$-space and $B$ be a separable $G$-$C^{*}$-algebra. Let
$H_{X}$ be any $X$-$G$-module. Let $\pi\colon
C_{0}(X)\to\mathfrak{L}(H_{X}\otimes B)$ be the structure map for the
$X$-$G$-Hilbert $B$-module $H_{X}\otimes B$. Then, the following diagram
commutes.
$\textstyle{K_{\ast+1}(D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes
B)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta_{0}}$$\scriptstyle{\iota}$$\textstyle{K_{\ast+1}(\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\eta^{-1}}$$\textstyle{K_{\ast+1}(D^{*}(\pi)^{G}/C^{*}(\pi)^{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta}$$\textstyle{K_{\ast+1}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{KK_{\ast}^{G}(C_{0}(X),B))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{K_{\ast}(\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{E_{\ast}^{G}(C_{0}(X),B)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
K_{\ast}(C^{\ast}_{L,u}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}).}$$\scriptstyle{\iota_{X}}$
###### Proof.
The argument of the proof of [16, Theorem 5.1] works with minor changes. We
give a proof for the case $\ast=0$. The case of $\ast=1$ is simpler and this
is the one of the two cases which is explained carefully in [16]. Take a
unitary $\dot{u}$ in $D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes B)^{G}$
(for simplicity we are taking a unitary in the $1\times 1$-matrix algebra). We
first compute the image of $\dot{u}$ in $E_{0}^{G}(C_{0}(X),B)$ by the clock-
wise composition. The functional calculus for $\dot{u}$ gives a
$\ast$-homomorphism
$\Sigma\ni f\mapsto f(\dot{u})\in D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes
B)^{G}$
where $\Sigma\cong C_{0}(0,1)$ is identified as $C_{0}(S^{1}-\\{1\\})$. We let
$\Sigma\ni f\mapsto f(u)\in D^{*}(H_{X}\otimes B)^{G}$
be its (not necessarily linear) continuous, bounded lift to
$D^{*}(H_{X}\otimes B)^{G}$ (which exists by Bartle–Graves Theorem [4, Theorem
4]). This is an abuse of notations and $f(u)$ is not the functional calculus
applied to some element $u$. We may compose this map $f\mapsto f(u)$ with a
c.c.p. map
$x\to\int_{g\in G}g(c)xg(c)d\mu_{G}(g)$
on $D^{*}(H_{X}\otimes B)^{G}$ to assume that the image of $\Sigma$ is
properly supported as a set. Note that such a composition is still a lift of
$f\mapsto f(\dot{u})$. Let $x\in\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ be as given by Lemma 11.2 for
$D=\\{f(u)\in D^{*}(H_{X}\otimes B)^{G}\subset\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}\mid f\in\Sigma\\}.$
Then,
$\Sigma\ni f\mapsto(1-x)f(u)\in\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$
is a lift of the $\ast$-homomorphism from $\Sigma$ to
$\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$ associated to the unitary $\dot{u}$, or more
precisely the unitary $\iota(\dot{u})\in\mathcal{D}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{T}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}=\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}$. The boundary map $\partial$ sends the element
$[\dot{u}]$ in $K_{1}(\mathcal{D}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}}/\mathcal{C}_{L}(H_{X}\otimes
B)^{G}_{\mathrm{proper}})$ to the element in
$K_{0}(\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}})$ represented by
an asymptotic morphism
$h\otimes f\mapsto h(v_{t})(1-x)f(u)$
from $\Sigma^{2}$ to $\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$.
Here, we let $M$ be a separable $C^{*}$-subalgebra of
$\mathcal{D}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$ generated by
$(1-x)f(u)$ and $v_{t}$ is a continuous approximate unit for
$M\cap\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}}$, quasi-central
with respect to $M$. The morphism $\iota_{X}$ sends this element in
$K_{0}(\mathcal{C}_{L}(H_{X}\otimes B)^{G}_{\mathrm{proper}})$ to the element
in $E_{0}^{G}(C_{0}(X),B)$ represented by an asymptotic morphism
$h\otimes f\otimes\phi\mapsto h(v_{t}(s(t)))(1-x(s(t)))f(u)\phi$
from $\Sigma^{2}\otimes C_{0}(X)$ to $\mathfrak{K}(H_{X}\otimes B)$ where
$t\mapsto s(t)$ is a continuous, increasing map on $[1,\infty)$ with
$s(t)\to\infty$ sufficiently fast as $t\to\infty$.
On the other hand, the downward composition sends the element $[\dot{u}]$ in
$K_{1}(D^{*}(H_{X}\otimes B)^{G}/C^{*}(H_{X}\otimes B)^{G})$ to the element in
$E_{0}^{G}(C_{0}(X),B)$ represented by an asymptotic morphism
$h\otimes f\otimes\phi\mapsto h(u_{t})f(u)\phi$
from $\Sigma^{2}\otimes C_{0}(X)$ to $\mathfrak{K}(H_{X}\otimes B)$ where
$u_{t}$ is an asymptotically $G$-equivariant, continuous approximate (see [26,
Definition 6.2]) in $\mathfrak{K}(H_{X}\otimes B)$ which is quasi-central with
respect to $C_{0}(X)$ and $f(u)$ for $f\in\Sigma$. We can take $u_{t}$ so that
we have
$\lVert(1-u_{t})x(s(t))\phi\rVert\to 0$
as $t\to\infty$ for any $\phi\in C_{0}(X)$. Then, the above asymptotic
morphism is asymptotic to the one
$h\otimes f\otimes\phi\mapsto h(u_{t})(1-x(s(t)))f(u)\phi.$
The two asymptotic morphisms
$h\otimes f\otimes\phi\mapsto h(u_{t})(1-x(s(t)))f(u)\phi,\,\,\,h\otimes
f\otimes\phi\mapsto h(v_{t}(s(t)))(1-x(s(t)))f(u)\phi$
are homotopic through an asymptotic morphism
$h\otimes f\otimes\phi\mapsto h(w^{r}_{t})(1-x(s(t)))f(u)\phi,$
from $\Sigma^{2}\otimes C_{0}(X)$ to $\mathfrak{K}(H_{X}\otimes B)\otimes
C[0,1]$ where
$w^{r}_{t}=(1-r)v_{t}(s(t))+ru_{t}$
for $0\leq r\leq 1$. That this is well-defined can be seen from that
$f\mapsto(1-x(s(t)))f(u)$
is asymptotically a $\ast$-homomorphism modulo the product of $\phi\in
C_{0}(X)$ and $h(w^{r}_{t})$ and from that
$\lVert[h(w^{r}_{t})(1-x(s(t)))f(u),\phi]\rVert\to
0,\,\,\,\lVert[h(w^{r}_{t}),(1-x(s(t)))f(u)]\rVert\to 0$
as $t\to\infty$. ∎
###### Theorem 13.2.
Let $G$ be a locally compact group. The natural transformation
$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}E_{\ast}^{G}(C_{0}(Y),B)\cong\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)$
is an isomorphism for any proper $G$-space $X$ and for any separable
$G$-$C^{*}$-algebra $B$.
###### Proof.
The case when $X$ is $G$-compact follows from Proposition 7.5, Corollary 12.6
and Lemma 13.1. The general case follows since the transformation is natural
with respect to $X$ and both sides are representable by $G$-compact subsets. ∎
###### Theorem 13.3.
Let $G$ be a locally compact group, $X$ be a proper $G$-space and $B$ be a
separable $G$-$C^{*}$-algebra. The forget-control map
$\mathcal{F}\colon\mathbb{D}_{\ast}^{B,G}(X)\to K_{\ast}(B\rtimes_{r}G)$
is naturally equivalent to the Baum–Connes assembly map
$\mu_{X}^{B,G}\colon\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(X),B)\to K_{\ast}(B\rtimes_{r}G).$
That is, there is a natural isomorphism
$\rho_{X}\colon\mathbb{D}_{\ast}^{B,G}(X)\to\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B)$
of the functors from $\mathcal{PR}^{G}$ to $\mathcal{GA}$ and the following
diagram commutes
$\textstyle{\mathbb{D}_{\ast}^{B,G}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\cong}$$\scriptstyle{\mathcal{F}}$$\textstyle{K_{\ast}(B\rtimes_{r}G)}$$\textstyle{\varinjlim_{Y\subset
X,\mathrm{Gcpt}}KK_{\ast}^{G}(C_{0}(Y),B).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{B,G}_{X}}$
###### Proof.
Combine Theorem 9.13 and Theorem 13.2. ∎
As before, let $RL^{0}_{c}(H_{X}\otimes B)$ be the kernel of the evaluation
map $\mathrm{ev}_{1}$ on $RL_{c}^{\ast}(H_{X}\otimes B)$ (see the end of
Section 9).
###### Corollary 13.4.
Let $G$ be a locally compact group and $B$ be a separable $G$-$C^{*}$-algebra.
The Baum–Connes assembly map $\mu^{B,G}_{r}$ is an isomorphism if and only if
$K_{\ast}(RL^{0}_{c}(H_{X}\otimes B)\rtimes_{r}G)=0$
for a universal $X$-$G$-module $H_{X}$ for $X=\underline{E}G$.
## 14\. $X$-$G$-localized element in $KK^{G}(\mathbb{C},\mathbb{C})$
We recall some materials on Kasparov’s $G$-equivariant $KK$-theory from [30]
(see also [7]). The graded (minimal) tensor product is denoted by
$\hat{\otimes}$.
Let $G$ be a second countable, locally compact group. For any (not necessarily
separable) graded $G$-$C^{*}$-algebras $A$ and $B$, Kasparov defines the
abelian group $KK^{G}(A,B)=KK_{0}^{G}(A,B)$ ([30, Definition 2.2] ) as the
group of homotopy classes (up to isomorphisms) of even Kasparov triples
$(E,\pi,T)$ where $E$ is a countably generated, graded $G$-Hilbert $B$-module
$E$ which is equipped with a (graded, $G$-equivariant) $\ast$-homomorphism
$\pi\colon A\to\mathfrak{L}(E)$ and where $T$ is an odd, $G$-continuous
operator in $\mathfrak{L}(E)$ such that for any $a\in A$ and $g\in G$, (we
write $a=\pi(a)$)
$a(1-T^{2}),\,\,[a,T],\,\,a(T-T^{\ast}),\,\,a(g(T)-T)\in\mathfrak{K}(\mathcal{E}).$
We often assume that $T$ is self-adjoint without loss of generality. The
homotopy is defined by the even Kasparov triple for $KK^{G}(A,B[0,1])$ for
$B[0,1]=B\otimes C[0,1]$ and the group structure is given by the direct sum
operation. For any $\ast$-homomorphisms $\phi\colon A_{1}\to A_{2}$,
$\psi\colon B_{1}\to B_{2}$, we have canonically defined homomorphisms ([30,
Definition 2.5]
$\phi^{\ast}\colon KK^{G}(A_{2},B)\to KK^{G}(A_{1},B),\,\,\,\psi_{\ast}\colon
KK^{G}(A,B_{1})\to KK^{G}(A,B_{2})$
and the group $KK^{G}(A,B)$ is homotopy invariant in both variables. If $D$ is
$\sigma$-unital, we have a canonically defined homomorphism
$\sigma_{D}\colon KK^{G}(A,B)\to KK^{G}(A\hat{\otimes}D,B\hat{\otimes}D)$
which sends $(E,\pi,T)$ to
$(E\hat{\otimes}D,\pi\hat{\otimes}\mathrm{id}_{D},T\hat{\otimes}1)$.
For $A$, separable, Kasparov defines the well-defined, bilinear, product law
(Kasparov product) (see [30, Definition 2.10, Theorem 2.11])
$KK^{G}(A,B_{1})\otimes KK^{G}(B_{1},B_{2})\to
KK^{G}(A,B_{2}),\,\,\,(x_{1},x_{2})\mapsto x_{1}\otimes_{B_{1}}x_{2}$
for any $B_{1},B_{2}$. The Kasparov product satisfies several functorial
properties, assuming separability or $\sigma$-unital for the relevant slots
(see [30, Theorem 2.14]).
The descent homomorphism [30, Section 3.11]
$j^{G}_{r}\colon KK^{G}(A,B)\to KK(A\rtimes_{r}G,B\rtimes_{r}G)$
is defined for any $A,B$ and it satisfies functorial properties, assuming
separability or $\sigma$-unital for the relevant slots (see [30, Theorem
3.11]).
The abelian group $KK^{G}_{1}(A,B)$ is defined to be
$KK^{G}_{1}(A,B)=KK^{G}(A\hat{\otimes}\mathbb{C}_{1},B)=KK^{G}(A,B\hat{\otimes}\mathbb{C}_{1})$
where $\mathbb{C}_{1}$ is the first Clifford algebra. We define
$K_{\ast}(A)=KK_{\ast}(\mathbb{C},A)$
for any graded $C^{*}$-algebra $A$ and when $A$ is ungraded, this group is
canonically isomorphic to the topological K-theory of $A$.
In particular, a cycle for the commutative ring
$R(G)=KK^{G}(\mathbb{C},\mathbb{C})$ is a pair $(H,T)$ of an odd, self-
adjoint, $G$-continuous operator $T$ on a (separable) graded $G$-Hilbert space
$H$ satisfying for any $g\in G$,
$1-T^{2},g(T)-T\in\mathfrak{K}(H).$
We call such a pair, a Kasparov cycle for $KK^{G}(\mathbb{C},\mathbb{C})$. A
cycle $(\mathbb{C},0)$ defines the unit of the ring $R(G)$, denoted by
$1_{G}$,
For a proper $G$-space $X$, a graded $X$-$G$-module $H_{X}$ is a graded
$G$-Hilbert space equipped with a non-degenerate (graded, $G$-equivariant)
representation of $C_{0}(X)$. It is just a pair $H_{X}=H_{X}^{(0)}\oplus
H_{X}^{(1)}$ of $X$-$G$-modules. For any graded $X$-$G$-module $H_{X}$, the
representable localization algebra $RL^{*}_{c}(H_{X})$ is naturally a graded
$G$-$C^{*}$-algebra.
###### Definition 14.1.
An $X$-$G$-localized Kasparov cycle for $KK^{G}(\mathbb{C},\mathbb{C})$ is a
pair $(H_{X},T)$ of a graded $X$-$G$-module $H_{X}$ and an odd, self-adjoint,
$G$-continuous element $T$ in the multiplier algebra $M(RL_{c}^{*}(H_{X}))$
satisfying for any $g\in G$,
$1-T^{2},\,\,\,g(T)-T\in RL^{*}_{c}(H_{X}).$
###### Remark 14.2.
Although $RL^{*}_{c}(H_{X})$ is not $\sigma$-unital, one may think
$(RL_{c}^{*}(H_{X}),T)$ as a cycle for $KK^{G}(\mathbb{C},RL^{*}_{c}(H_{X}))$
(see [42, Section 3]).
The evaluation
$\mathrm{ev}_{t=1}\colon RL^{*}_{c}(H_{X})\to\mathfrak{K}(H_{X})$
extends to
$\mathrm{ev}_{t=1}\colon M(RL^{*}_{c}(H_{X}))\to\mathfrak{L}(H_{X}).$
For any $T\in M(RL^{*}_{c}(H_{X}))$, we write $T_{1}\in\mathfrak{L}(H_{X})$,
its image by $\mathrm{ev}_{t=1}$.
###### Lemma 14.3.
Let $X$ be a proper $G$-space and $(H_{X},T)$ be an $X$-$G$-localized Kasparov
cycle for $KK^{G}(\mathbb{C},\mathbb{C})$. Then, a pair $(H_{X},T_{1})$ is a
cycle for $KK^{G}(\mathbb{C},\mathbb{C})$.
###### Definition 14.4.
Let $X$ be a proper $G$-space. We say that an element $x$ in
$KK^{G}(\mathbb{C},\mathbb{C})$ is $X$-$G$-localized if there is an
$X$-$G$-localized Kasparov cycle $(H_{X},T)$ for
$KK^{G}(\mathbb{C},\mathbb{C})$ such that
$[H_{X},T_{1}]=x\,\,\text{in}\,\,\,KK^{G}(\mathbb{C},\mathbb{C}).$
We say that $x\in KK^{G}(\mathbb{C},\mathbb{C})$ factors through a
$G$-$C^{*}$-algebra $B$ if there is $y\in KK^{G}(\mathbb{C},B)$ and $z\in
KK^{G}(B,\mathbb{C})$ such that $x=y\otimes_{B}z$. By definition, $x\in
KK^{G}(\mathbb{C},\mathbb{C})$ is the gamma element if it factors through a
(separable) proper $G$-$C^{*}$-algebra $A$ and it satisfies $x=1_{K}$ in
$R(K)$ for any compact subgroup $K$ of $G$.
###### Proposition 14.5.
Suppose that an element $x\in KK^{G}(\mathbb{C},\mathbb{C})$ factors through a
separable $G$-$C_{0}(X)$-algebra $A$ for a proper $G$-space $X$. Then $x$ is
$X$-$G$-localized.
The following is an immediate corollary.
###### Theorem 14.6.
The gamma element $\gamma$ for $G$, if exists, is $X$-$G$-localized for
$X=\underline{E}G$.
Before giving a proof of Proposition 14.5, we prove two lemmas.
Let $A$ be a graded $G$-$C_{0}(X)$ algebra and $(H,\pi,F)$ be a Kasparov
triple for $KK^{G}(\mathbb{C},A)$. If $\pi$ is non-degenerate, $\pi$ induces a
natural non-degenerate representation of $C_{0}(X)$ on $H$. We view $H$
naturally as a (graded) $X$-$G$-module through this representation and set
$H_{X}=H$. Any element $a$ in $M(A)$ commutes with $C_{0}(X)$ and hence $a$
(as a constant function) is naturally an element in
$M(RL^{*}_{c}(H_{X}))\subset\mathfrak{L}(H_{X}\otimes C_{0}[1,\infty))$.
###### Lemma 14.7.
Let $X$ be a proper $G$-space and $A$ be a graded, separable
$G$-$C_{0}(X)$-algebra. Let $(H,\pi,F_{0})$ be a Kasparov triple for
$KK^{G}(A,\mathbb{C})$ with $\pi$ non-degenerate. We view $H$ naturally as a
graded $X$-$G$-module through this representation and set $H_{X}=H$. Then,
there is an odd, $G$-continuous, self-adjoint element $F$ in
$M(RL^{*}_{c}(H_{X}))\subset\mathfrak{L}(H_{X}\otimes C_{0}[1,\infty))$ such
that
1. (I)
$a(F-F_{0})\in C_{b}([1,\infty),\mathfrak{K}(H_{X}))$ for any $a\in A$, where
$F_{0}\in\mathfrak{L}(H_{X})$ is regarded as a constant function in
$\mathfrak{L}(H_{X}\otimes C_{0}[1,\infty))$,
2. (II)
for any $a\in A$ and $g\in G$,
$a(1-F^{2}),[a,F],a(g(F)-F)\in RL^{*}_{c}(H_{X}).$
###### Proof.
We assume $F_{0}$ is self-adjoint. We assume $F_{0}$ is $G$-equivariant using
the standard trick: Let $c\in C_{b}(X)$ be a cut-off function on $X$. Then,
$F_{0}^{\prime}=\int_{g\in G}g(c)g(F_{0})g(c)d\mu_{G}(g)$
is $G$-equivariant and satisfies
$a(F_{0}^{\prime}-F_{0})\in\mathfrak{K}(H_{X})$
for any $a\in A$. Here, for an isometry $V_{c}\colon H_{X}\to H_{X}\otimes
L^{2}(G)$ defined by
$V_{c}\colon v\mapsto(g\mapsto g(c)v),$
we have $F_{0}^{\prime}=V_{c}^{\ast}\tilde{F}V_{c}$ where $\tilde{F}$ on
$H_{X}\otimes L^{2}(G)$ is the diagonal operator $(g\mapsto g(F_{0}))\in
C_{b}(G,\mathfrak{L}(H_{X}))$. That
$a(F_{0}^{\prime}-F_{0})\in\mathfrak{K}(H_{X})$ can be checked by
$a(F_{0}^{\prime}-F_{0})=\int_{g\in
G}ag(c)(g(F_{0})-F_{0})g(c)+ag(c)[F_{0},g(c)]d\mu_{G}(g)$
which is norm convergent in $\mathfrak{K}(H_{X})$ for compactly supported $a$.
We remark that a cut-off function $c$ may not be $G$-continuous in general |
# Are Large Language Models (LLMs) Good Social Predictors?
Kaiqi Yang1, Hang Li1, Hongzhi Wen1, Tai-Quan Peng2, Jiliang Tang1, Hui Liu1
1Department of Computer Science and Engineering, Michigan State University
2Department of Communication, Michigan State University
<EMAIL_ADDRESS>
(Nov 2023)
###### Abstract
The prediction has served as a crucial scientific method in modern social
studies. With the recent advancement of Large Language Models (LLMs), efforts
have been made to leverage LLMs to predict the human features in social life,
such as presidential voting. These works suggest that LLMs are capable of
generating human-like responses. However, we find that the promising
performance achieved by previous studies is because of the existence of input
shortcut features to the response. In fact, by removing these shortcuts, the
performance is reduced dramatically. To further revisit the ability of LLMs,
we introduce a novel social prediction task, Soc-PRF Prediction, which
utilizes general features as input and simulates real-world social study
settings. With the comprehensive investigations on various LLMs, we reveal
that LLMs cannot work as expected on social prediction when given general
input features without shortcuts. We further investigate possible reasons for
this phenomenon that suggest potential ways to enhance LLMs for social
prediction.
Are Large Language Models (LLMs) Good Social Predictors?
Kaiqi Yang1, Hang Li1, Hongzhi Wen1, Tai-Quan Peng2, Jiliang Tang1, Hui Liu1
1Department of Computer Science and Engineering, Michigan State University
2Department of Communication, Michigan State University<EMAIL_ADDRESS>
## 1 Introduction
Prediction is one of the crucial elements of the scientific methods in social
studies Hofman et al. (2017), with a body of literature Liben-Nowell and
Kleinberg (2003); Bakshy et al. (2011); Cheng et al. (2014) devoted to
estimating unobserved or missing data based on observed features.
Historically, social prediction is made by statistical models such as linear
regression Uyanık and Güler (2013). With the development of machine learning,
supervised methods have been adopted, including random forest, Support Vector
Machines (SVM), and neural networks Chen et al. (2021b). However, the classic
machine learning methods notably rely on extensive labeled training datasets,
which is labor-intensive, especially in social studies. Additionally, the
predictive power of machine learning methods is limited Mackenzie (2015);
Athey (2018) and can hardly model the complicated phenomenon in social life.
Meanwhile, Large Language Models (LLMs) have advanced various text-related
tasks, such as question-answering Zhuang et al. (2023); Tan et al. (2023),
code-generation Nijkamp et al. (2022); Chen et al. (2021a), and math word
problems solving Zhou et al. (2022); Wei et al. (2022)). The extensive world
knowledge Zhao et al. (2023) and inference abilities Creswell et al. (2022) of
LLMs have the potential to mitigate the limitations of classic machine
learning methods in predicting features of social datasets. Therefore, there
are recent works leveraging LLMs in predicting and simulating human responses,
such as voting decisions Argyle et al. (2022); von der Heyde et al. (2023) and
political attitude Rosenbusch et al. (2023). They take advantage of LLMs to
augment existing datasets with previously inaccessible features and promising
performance is reported. However, our preliminary investigation revisiting the
case of voting prediction Argyle et al. (2022) with LLMs indicates that their
performance is bolstered by the presence of shortcuts to the desired response
features. Specifically, these shortcuts arise when the input contains features
directly associated with the feature to be predicted, leading to the
exceptional performance of both machine learning models and LLM-based methods.
Unfortunately, this efficiency comes with a downside: it overlooks the
essential task of uncovering authentic relationships between various features
and the label. When these shortcuts are eliminated, we observe a significant
decline in the LLMs’ effectiveness in addressing social study issues, as
detailed in Section 2. This observation leads us to question the true
capability of LLMs in social predictions, challenging the prevailing
perception of their prowess Argyle et al. (2023).
To investigate this problem, we introduce a set of studies that utilize
general social features as input and simulate real-world settings of feature
prediction. In particular, to comprehensively understand the power of LLMs in
social predicting, we introduce a new task, Soc-PRF Prediction (stands for
Social Profile Prediction). It is designed to predict the social features of
individuals while accounting for selected features as explanatory and response
variables. Besides, informed by social studies Bailey (1998), we categorize
social features into two groups that capture individuals’ features from
different perspectives. This enables us to design three distinct settings,
which rigorously categorize features into groups and assess LLMs’ predictive
capacities. In this work, we evaluate various LLMs, including closed-sourced
models GPT 3.5 OpenAI (2022), GPT 4 Achiam et al. (2023), and Gemini Pro Anil
et al. (2023), as well as lighter open-sourced models like Llama-7B,
Llama-7B-chat Touvron et al. (2023) and Mistral-7B Jiang et al. (2023). Our
studies suggest that LLMs cannot work on social prediction with general input
features without shortcuts. We further explore the potential reasons and
future directions to enhance LLMs for social prediction.
## 2 Revisit Voting Prediction with LLMs
Large Language Models (LLMs) have demonstrated impressive performance across
several societal domains, notably in predicting voting decisions in the United
States Argyle et al. (2022); Veselovsky et al. (2023). In this section, we
revisit the voting prediction study with LLMs in Argyle et al. (2022).
The voting prediction study in Argyle et al. (2022) adopts the American
National Election Studies ANES to construct the dataset. ANES is a survey
conducted in every presidential election year, with features about American
public life, especially political views and decisions. To elicit LLMs’
prediction of individual voting decisions, the study selects 10 input
features, i.e., racial/ethnic self-identification, gender, age, ideological
self-identification, party identification, political interest, church
attendance, if discussing politics with family/friends, patriotism feelings,
state of residence. With all the features above, prompts of individual
profiles are constructed with a first-person template and a question to elicit
prediction. Then the prompts are fed into LLMs for completion, and the words
filled in by LLMs serve as the predicted voting decisions. An example of the
prompts is:
> Racially, I am white. I am male. Ideologically, I describe myself as
> conservative. Politically, I am a strong Republican … In 2016, I voted for:
Intuitively two of the input features are likely equivalent to voting
decisions, i.e. ideological self-placement and party identification. It is
evident from political studies Miller (1991); Dalton (2016) that given the
partisan nature of American politics, voting decisions are closely related to
these two features. To validate the intuition, we calculate their Cramer’s V
scores with the voting decisions. Cramer’s V is a measurement of association
between features, ranging from 0 to 1; the score 0 indicates no association
and 1 indicates a perfect association. We find that these two features are
highly correlated with the vote decisions. For example, in ANES 2016 wave
data, these two features have Cramer’s V scores of $0.86$ and $0.76$,
respectively. This indicates their strong associations with voting decisions
and consequently, they can become the shortcuts to make predictions.
Next, we conduct further experiments to study the impact of such shortcuts on
prediction performance. We choose both GPT-based approaches and classic
supervised machine learning models. For GPT-based approaches, we deploy GPT
3.5 as the basis and follow the prompts and zero-shot setting in Argyle et al.
(2022) to make predictions based on individual profiles. For classic
supervised machine learning models, we choose the Random Forest Classifier.
Since the supervised classifier needs labeled data to train, we split the
dataset into 80%/10%/10% as training, validation, and test sets. There are two
settings for each method: (1) Full, taking the full set of input features; (2)
w/o shortcut, taking input features without the shortcut features. To evaluate
the performance, we deploy accuracy as the metric, given the balanced
distribution of the voting decision (51.9% vs. 48.1%). Besides, we adopt
Cohen’s Kappa $\kappa$ as a metric to evaluate the agreement between
prediction and true data. Cohen’s Kappa $\kappa$ has values ranging from 0 to
1, where 1 indicates stronger agreement and 0 indicates almost no agreement.
Figure 1: The performance of voting prediction. In this table, GPT stands for
the LLM-based approach and we choose GPT 3.5 following Argyle et al. (2022).
The Full indicates settings with all input features, and w/o shortcut stands
for settings without the two shortcut features.
As shown in Figure 1, the GPT-based approach with all input features achieved
the accuracy of 90.82% and the Cohen’s Kappa $\kappa$ of 0.83, which aligns
with Argyle et al. (2022). However, when removing the two shortcut features,
the performance of both methods drops dramatically. For example, the
performance of GPT 3.5 drops from the accuracy of 90.82% and $\kappa$ of 0.83
to an accuracy of 61.60% and $\kappa$ of 0.43; similarly, the performance of
Random Forest drops from 90.29%, 0.78 to 69.22%, 0.23. Given the balanced
distribution of the voting decision feature, the performance without shortcut
features is considerably unsatisfactory.
Our preliminary study suggests that the promising performance of social
prediction via LLMs reported by prior works Argyle et al. (2022) could be from
the existence of shortcut features. This finding motivates us to question if
LLMs are really powerful in social prediction. To explore this question, we
design a set of studies that avoid shortcut features as inputs and resemble
realistic scenarios in the following section.
## 3 Social Profile Prediction
In this section, we introduce a social prediction task that evaluates the
predictive power of LLMs without shortcuts. First, we construct a social
prediction dataset based on survey data where we validate that there are no
shortcut features. Then we introduce three settings of evaluations to simulate
real-world scenarios. Finally, we show the performance of LLMs’ prediction in
the proposed settings and discuss the results.
### 3.1 Task and Dataset
As illustrated in Section 2, the inclusion of shortcut features can affect the
evaluation of the power of LLMs in social prediction. Therefore, we design
Soc-PRF Prediction that takes individual social features as input to predict
other missing features in profiles. Next, we introduce the dataset for the
task.
The dataset derives from Gallup World Poll Gallup (2009), one of the most
prestigious social surveys that guarantees reliability and offers various
types of features. Initialized in 2006, the Gallup World Poll has been
conducted in over 150 countries and follows strict random sampling. Questions
in the Gallup World Poll are designed by political scientists, measuring key
indicators of social life, such as law and order, financial life, civic
engagement, etc. Besides, it collects individual demographic data to construct
the survey dataset Tortora et al. (2010).
In this paper, we construct the dataset based on Gallup World Poll Gallup
(2009) and its corresponding questions. We pick the data from the USA and the
data we use in this work is primarily collected between 2016 and 2020. To keep
the basic information complete, we remove all the samples with missing data in
demographic features. After the data cleaning, the dataset includes 4,941
profiles of American individuals (samples). Additionally, we pick a set of
features from the survey to construct the profiles, which encompasses 16
social features reflecting a variety of socio-demographic characteristics,
attitudes, and behaviors.
### 3.2 Task Settings
In social studies, social datasets have predominantly been derived from two
methodologies: traditional surveys and online data collection Couper (2017);
Diaz et al. (2016); Callegaro et al. (2014). As one characteristic of social
features, mutability measures the features’ propensity to change or be
influenced by social context. Following social studies Bailey (1992);
Brensinger and Eyal (2021); Sen and Wasow (2016); Halley (2017), social
features can be roughly divided into two mutability groups: high-mutability
and low-mutability. In most social datasets, features with high mutability and
low mutability can hardly be collected simultaneously. For example, although
survey data collected through in-person interviews is of high quality, it
predominantly captures features of low mutability; the dynamic tracking of
highly mutable features across all topics and times is nearly impractical due
to associated costs. On the other hand, online data collection methods, such
as crawling posts from social networking platforms, have the advantage of
collecting high-mutability features. Facilitated by natural language
processing (NLP) tools Alghamdi and Alfalqi (2015); Vayansky and Kumar (2020);
Hussein (2018); Yue et al. (2019), real-time human attitudes and opinions are
easy to collect. Yet, features of low mutability (e.g. demographic features)
often remain inaccessible due to privacy constraints.
Among 16 social features in our proposed dataset, we assign them to low-
mutability and high-mutability groups, respectively. The low-mutability
features are socio-demographic features, including age, gender, marriage,
education, employment, income, and urbanicity. Here the age, gender, marriage
status, employment status, urbanicity of residence refer to the individual
status when taking the interview, while education refers to the highest
completed level of education, and income is the annual household income of
last year. The high-mutability features are all about attitudes or behaviors
of social life, whose topics include Internet access, social life, economic
confidence, civic engagement, and approval of leadership. For each topic,
there are one to three questions around it. In save of space, we denote the
questions as IA, SL1, SL2, EC1, EC2, CE1, CE2, CE3, AL, respectively. The
details of questions are shown in Appendix A.1.
According to features’ mutability, we design three settings to assess the
capability of LLMs in predicting missing features for the individual profiles,
which simulate real-world scenarios for social data: from low-mutability
features to predicting high-mutability features (for survey data), and from
high-mutability features to predicting high-mutability or low-mutability
features (for online data). Following the prior works especially Argyle et al.
(2022), we set all three settings as zero-shot, without taking any labeled
data as input.
low2high. In this setting, the input features possess low mutability, and the
output features exhibit high mutability. This setting is designed for
traditional survey datasets, where low-mutability features (e.g. demographic
features) are comprehensively collected, but high-mutability features (such as
attitudes) are sparse.
To construct prompts of individual profiles, we adopt the template in Argyle
et al. (2022), designing the prompt as a self-description of individuals and
inducing the LLMs to predict missing features by completing the self-
description. One example of the prompt is:
> I am a male in the USA. I am 42 years old. My current marital status is
> married. My highest completed level of education is middle level. My current
> employment status is employed. My Annual Household Income is $12600. I am
> from a suburb of a large city.
>
> When I’m asked "Do you have access to the Internet in any way, whether on a
> mobile phone, a computer, or some other device?", my answer is
In the provided prompt, the underlined text indicates the values of individual
features, and italicized content presents the question to elicit responses.
For the subsequent settings, we utilize prompts with the same template.
high2low: This setting denotes the prediction from high-mutability to low-
mutability features. To construct the input profiles of individuals, we take
values from all 9 high-mutability features, with a question about one low-
mutability feature. Serving as the inverse setting of low2high, this setting
is designed for profile construction using online data: aided with NLP tools,
the in-time attitudes of individuals can be captured from online posts with
ease, yet their demographic features are inaccessible.
Figure 2: Correlation between features in the dataset. The metric is Cramer’s
V, and values close to 1 indicate strong correlations.
high2high. In this setting, high-mutability features are utilized as input to
predict other high-mutability features. Different from high2low setting, in
order to avoid shortcuts, the features sharing the same topic with the
response feature are excluded from the input profile prompts; rather, we put
all high-mutability features of each topic as inputs. This setting simulates a
specific real-world scenario, where individuals’ attitudes toward one topic
are collected, but their opinions on other topics of interest remain
unexpressed.
Evaluation Metrics. Most features in the dataset have imbalanced
distributions. For example, the feature IA has 91.82% samples with "yes"
labels, while only 8.18% samples with "no". In this situation, accuracy is not
a suitable metric for imbalanced predictions Gu et al. (2009). As a result, we
employ AUC as the metric to evaluate the classification performance.
### 3.3 Feature Analysis
To ensure that there are no shortcut features used as input for predictions,
we first check the correlations (i.e., the Cramer’s V) of all feature pairs.
As shown in Figure 2, most of the Cramer’s V scores are less than 0.5. The
maximum (between CE2 and CE3) is 0.58, which is merely a moderate level of
correlation. This relatively high correlation is because they share a similar
topic (i.e., civic engagement). Therefore, in the following evaluations, we
will not consider features that share the same topic with the response as
inputs.
To evaluate the predictive power of the selected features, we follow the
traditional supervised setting. Take the setting of low2high as the example,
we first chose Random Forest Classifier as the basis model, and split the
dataset by 80%/10%/10% as training, validation, and test sets. The results are
shown in Figure 3, the AUC scores are much higher than those of the random
guessing method. For example, the AUC score of IA is 95.07, while its
corresponding score of random guessing is 48.34. These observations suggest
that though the selected features are not shortcuts, they are still powerful
in predicting the target responses.
Figure 3: Performance of Random Forest and Random Guessing. The metric is AUC.
### 3.4 LLMs as the Predictor
In this section, we leverage LLMs for the Soc-PRF Prediction task with the
aforementioned three settings. The results for the three settings are
illustrated in Table 1, Table 2, and Figure 4, respectively. In the tables,
"Random" indicates the random guessing method. Note that for the settings
high2high, we only partially show the results because the observations are
similar. We note that the performance of LLMs is closely similar to the random
guessing and is far from satisfactory. The poor results appear consistently in
all the settings and with all the LLMs. These observations indicate that LLMs
are struggling to distinguish individual features with the given information
in the proposed settings.
Table 1: Performance of LLMs of setting low2high.
Model | IA | SL1 | SL2 | EC1 | EC2 | CE1 | CE2 | CE3 | AL
---|---|---|---|---|---|---|---|---|---
Random | 48.34 | 52.09 | 48.47 | 52.12 | 50.07 | 49.89 | 49.16 | 49.32 | 48.60
Llama-7B | 50.00 | 50.00 | 50.00 | 48.75 | 55.41 | 50.00 | 50.00 | 50.00 | 50.00
Llama-7B-chat | 50.00 | 50.00 | 50.00 | 50.95 | 51.80 | 50.00 | 50.00 | 50.00 | 50.00
Mistral-7B | 50.00 | 50.00 | 50.00 | 53.12 | 56.89 | 50.00 | 50.00 | 50.00 | 50.00
Gemini Pro | 50.00 | 50.00 | 50.00 | 50.76 | 60.93 | 50.00 | 50.00 | 50.00 | 50.00
GPT-3.5 | 50.00 | 50.00 | 50.00 | 52.63 | 58.20 | 50.00 | 50.00 | 50.00 | 50.00
GPT-4 | 50.00 | 50.00 | 50.00 | 53.82 | 56.57 | 50.00 | 50.00 | 50.00 | 50.00
Table 2: Performance of LLMs (GPT 3.5 and GPT 4) of setting high2low.
Model | age | gender | marriage | education | employment | income | urbanicity
---|---|---|---|---|---|---|---
Random | 49.50 | 49.62 | 49.45 | 49.99 | 50.54 | 48.14 | 50.22
Llama-7B | 33.50 | 49.81 | 50.00 | 55.15 | 50.00 | 50.05 | 49.85
Llama-7B-chat | 40.00 | 50.00 | 50.00 | 35.21 | 50.33 | 51.18 | 50.09
Mistral-7B | 33.55 | 49.81 | 50.00 | 55.15 | 50.00 | 50.05 | 49.85
Gemini Pro | 38.80 | 51.14 | 50.00 | 66.70 | 50.00 | 50.10 | 49.75
GPT-3.5 | 41.35 | 50.00 | 51.29 | 57.76 | 49.59 | 50.95 | 50.94
GPT-4 | 40.75 | 50.00 | 50.88 | 65.65 | 52.01 | 53.80 | 52.09
Figure 4: Performance of GPT 3.5 of setting high2high. The metric is AUC
score. The sign "-" indicates no valid data, either because the input features
(Y-axis) and output features (X-axis) share the same topic, or they are not
conducted simultaneously in the survey.
### 3.5 Discussions
#### 3.5.1 Population v.s. Individual
As shown in the previous subsection, even advanced LLMs like GPT-4 encounter
challenges in accurately predicting social features, often yielding outcomes
similar to random guessing. To explore the underlying reason for such
phenomena, we use the distribution comparisons between predicted features and
true ones under the setting low2high as our case studies; the results are
shown in Figure 5. From the analysis, we have the following observations.
First, LLM’s prediction of less mutable features, such as IA and SL, is prone
to share similar distribution patterns with the true ones. This fact indicates
that LLMs do contain some global knowledge about these social features at the
population level, but they face challenges in building precise connections to
different individual samples during the prediction. Thus, when making
individual-level predictions, LLMs may simply generate random samples from the
feature distribution. Second, the population-level patterns of highly mutable
features, such as different CEs, are seemingly not captured by existing LLMs,
and LLMs always prefer to generate negative responses on these features. This
fact indicates that building accurate predictors with LLMs for those highly
mutable features is more challenging as it not only requires LLMs to establish
the connection to individual samples but also external population-level
knowledge about the features.
Figure 5: Distributions of Social Features. Note that the last two features
(civic engagement behaviors) are more mutable than the first two.
#### 3.5.2 Incorporating Labeled Data
Although the performance of zero-shot LLMs’ prediction is much worse than our
expectation, the strong performance of the random forest classifier in Figure
3 indicates that our designed prediction task is reasonable if sufficient
labeled data is considered. Based on this finding, we explore the
effectiveness of incorporating supervision signals to LLMs based on the
low2high setting as one example. Following the prior studies Brown et al.
(2020); Song et al. (2023), we leverage the in-context learning ability of
LLMs Dong et al. (2022); Zhang et al. (2023) to incorporate a few labeled
samples as demonstrations. To be specific, for each individual profile, we
sample a few other individual profiles from the dataset as a reference, whose
year and marriage features are of the same group. Then, besides constructing
vanilla prompts as introduced in Section 3.2, we add the full information
(including the input and output features) of these reference samples in the
prompts. Finally, the LLMs are asked to make predictions as the original
settings. One example of such prompts is:
> Here are self-descriptions of two people: "I am a male in the USA … my
> answer is yes"; "I am a female in the USA … my answer is no"; …
>
> I am a male in the USA. I am 42 years old …;
>
> When I’m asked "Do you have access to the Internet in any way, whether on a
> mobile phone, a computer, or some other device?", my answer is
With the augmented prompts, we introduce samples with similar input features
and show the true labels of these samples. Like the supervised methods, these
demonstrations allow LLMs to make predictions with the help of supervision
signals. To provide comprehensive information, the positive and negative
labels are balanced within the reference samples. The results of experiments
with 2 and 4 demonstrations are shown in Table 3. With all selected features,
the prompts with demonstrations help LLMs to achieve better prediction
performance. However, when incorporating 4 demonstrations into the prompt, we
observe marginal or no improvement compared with the setting with 2
demonstrations. This observation suggests that how to effectively incorporate
more demonstrations for LLMs for social prediction still faces great
challenges.
#### 3.5.3 Enriching Input Features
As aforementioned, in reality, no matter survey data or online data, we face
constraints to collect sufficient features to describe individuals. In fact,
we find that such constraints may also limit the power of LLMs to discriminate
different individuals in social prediction. For example, in the low2high
setting, we can identify a sub-group from the population characterized by all
low-mutability features: male, married, age between 30 and 60, higher level of
education, fully employed, annual household income within the middle 30%,
living in suburbs of large cities. Within this subgroup, all samples share the
same input; however, their responses are varied significantly. For example, to
the question CE1, 60.53% of samples answered "no" and 29.47% answered "yes".
This variability indicates the lack of discriminative features poses
significant challenges in precisely predicting individual responses. To unlock
the potential of LLMs in predicting social features, enriching input features
is crucial.
Table 3: The performance of LLMs (GPT 3.5) with a few demonstrations.
| Zero-Shot | 2 Demos | 4 Demos
---|---|---|---
IA | 50.00 | 71.61 | 82.67
SL2 | 50.00 | 50.60 | 50.04
EC1 | 52.63 | 50.52 | 53.47
CE1 | 50.00 | 60.17 | 55.34
CE2 | 50.00 | 53.22 | 52.79
AL | 50.00 | 52.03 | 50.80
## 4 Related Work
In this section, we give an overview of works related to classic machine
learning methods and LLMs in social studies.
### 4.1 Machine Learning Methods in Social Prediction
Quantitative social studies have deployed machine learning methods to model
and predict social features in replace of the classical statistical models
like OLS and logistic regression Chen et al. (2021b); Hindman (2015). In
studies of criminology, K-means and neural networks are deployed to predict
criminal behaviors Reier Forradellas et al. (2020). To predict communication
phenomena, the random forest models are utilized to predict the view count of
online posts Hsu et al. (2017). Besides, Dong et al. (2018) uses the SVM
classifier to predict public emotions to social news. In addition, deep
learning methods have also been widely used in social prediction. For example,
CNN models are used to process image data in social studies Messer and Fausser
(2019); Dong et al. (2018); RNN and its variants have been utilized to predict
sequential data like stock changes Wu et al. (2018) and user interests Liu et
al. (2018). However, these methods either suffer from the need for massive
training data or the limited predictive power of models, highlighting the need
for more powerful prediction models.
### 4.2 Using LLMs to Predict Social Features
With the advent of LLMs, predicting social features with LLMs has been studied
by numerous works Ziems et al. (2023); Veselovsky et al. (2023). Among social
studies, LLMs have been deployed to predict the potential responses or
outcomes with ease, especially in scenarios where traditional methods are
constrained by cost or ethical concerns. In economics, Phelps and Russell
(2023) studied game theory by examining cooperative and competitive behaviors
with LLMs. Within political science, Wu et al. (2023) deployed LLMs to predict
the ideological views of politicians. For communication studies, LLMs are used
to simulate and predict the potential outcomes of toxic discourse Törnberg et
al. (2023), the political affiliation of Twitter posts Törnberg (2023), etc.
Additionally, there are growing interests in leveraging LLMs with social
survey and interview, aiming to replicate human-like responses to certain
questions or attributes of individuals. For example, Argyle et al., 2022
proposed "silicon samples" that deploy LLMs to simulate the people in a survey
or interview and predict their partisan views and voting decisions. Dillion et
al., 2023 examined the LLMs response to psychological tests, comparing the
decisions and judgements from LLMs and humans. Aher et al., 2023 proposed sets
of experiments to check LLMs response to interview and games. Besides, fine-
tuning LLMs is a promising method for better prediction of social attitudes
across years of surveys Kim and Lee (2023). At the same time, discussions
Jansen et al. (2023) are hold about the potential and risks of deploying LLMs
in social survey studies.
## 5 Conclusion
In this study, we introduce a survey-based social prediction task to assess
the LLMs’ predictive ability using general features. Through the replication
of experiments and ablation studies of voting prediction tasks, we reveal a
significant performance gap between input prompts with and without shortcut
features. To further study the LLMs’ predictive ability, we propose a real-
world survey dataset with rigorously selected features. Based on it, we
demonstrate the inability of LLMs to predict social features only with general
features. Furthermore, our empirical studies further showcase the potential
reasons that constrain the LLMs’ predictive power. In our future research, we
aim to explore the efficient methods of providing supervision signals and
reference information to improve LLMs prediction performance. Moreover, with
the abundant social survey and online data, we plan to use fine-tuning methods
to fit the LLMs knowledge with social prediction tasks.
## 6 Limitations
In this study, we examine the predictive power of LLMs. We replicate a voting
prediction study and find the shortcuts producing plausible results. However,
we do not investigate how to select informative features that enhance
prediction and what is the upper bound of this task. Then we conduct
comprehensive experiments of social prediction, and contend that incorporating
labeled data and enrich input features could benefit social prediction.
However, we do not provide experiments to validate these suggestions. Lastly,
the LLMs are not further tuned and the prompts are adopted from prior works.
Tailoring LLMs and proper prompts to meet the needs of social prediction tasks
could be a potential direction to explore. Such adjustments could potentially
unlock ability and applicability of LLMs in social prediction, further
improving the performance of LLMs.
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## Appendix A Appendix
### A.1 Questions for Features
We categorize the selected 16 features into two groups, i.e. high-mutability
and low-mutability features. The details of high-mutability features are shown
in Table 4 and those of low-mutability features are shown in Table 5. The
column "Question Abbrev." indicates the abbreviation of the features, which
are broadly used in this work. The column "Question Identifiers" indicates the
identifier labels of the corresponding questions in the original Gallup
survey.
Topic | | Question
---
Abbrev.
| Question
---
Identifiers
Question | Options
Communication Use | IA | WP16056 | Do you have access to the internet in any way, whether on a mobile phone, a computer, or some other device? | yes, no
Social Life | SL1 | WP27 | If you were in trouble, do you have relatives or friends you can count on to help you whenever you need them, or not? | yes, no
| SL2 | WP10248 | In the city or area where you live, are you satisfied or dissatisfied with the opportunities to meet people and make friends? | satisfied, dissatisfied
Economic Confidence | EC1 | WP148 | Right now, do you think that economic conditions in this country, as a whole, are getting better or getting worse? | better, worse
| EC2 | M30 | How would you rate your economic conditions in this country today – as excellent, good, fair, or poor? | excellent, good, fair, poor
Civic Engagement | CE1 | WP108 | Have you donated money to a charity in the past month? | yes, no
| CE2 | WP109 | Have you volunteered your time to an organization in the past month? | yes, no
| CE3 | WP110 | Have you helped a stranger or someone you did not know who needed help? | yes, no
Approval of Leadership | AL | WP150 | Do you approve or disapprove of the job performance of the leadership of this country? | approve, disapprove
Table 4: Questions and Options of High-mutability Features of Gallup Dataset.
Immutable Attribute | | Question
---
Abbrev.
| Question
---
Identifiers.
Options
Age | age | age | -
Gender | gender | WP1219 | 1\. Man, 2. Woman
Marital Status | marriage | WP1223 | 1\. Single/Never been married, 2. Married, 3. Separated, 4. Divorced, 5. Widowed, 6. Domestic Partner;
Highest Completed Level of Education | education | WP3117 | 1\. Completed elementary education or less (up to 8 years of basic education); 2. Secondary - 3 years Tertiary/Secondary education and some education beyond secondary education (9-15 years of education); 3. Completed four years of education beyond high school and/or received a 4-year college degree;
Employment Status | employment | EMP_2010 | 1\. Employed full time for an employer, 2. Out of workforce, 3. Employed part time do not want full time, 4. Employed full time for self, 5. Employed part time want full time, 6. Unemployed;
Annual Household Income | income | INCOME_1 | -
Living of Urbanicity | urbanicity | WP14 | 1\. A suburb of a large city, 2. A small town or village, 3. A large city, 4. A rural area or on a farm;
Table 5: Questions and Options of Low-mutability Features of Gallup Dataset.
#### A.1.1 Feature Convert Methods
In the main experiments, there are features of integer or several classes,
such as income, employment, etc. We convert them into groups (with the number
of groups no larger than four). For income, we calculate the 35% and 65%
percentiles of the annual household income. Based on them, we categorize
income into three classes: lower level, middle level, and higher level. For
features with more than 4 classes, we combine similar classes to make the
number of classes as 2 or 3.
|
*natbibCitation *BibTex
# Data Augmentation for Deep Graph Learning:
A Survey
Kaize Ding, Zhe Xu, Hanghang Tong, and Huan Liu
###### Abstract
Graph neural networks, a powerful deep learning tool to model graph-structured
data, have demonstrated remarkable performance on numerous graph learning
tasks. To address the data noise and data scarcity issues in deep graph
learning, the research on graph data augmentation has intensified lately.
However, conventional data augmentation methods can hardly handle graph-
structured data which is defined in non-Euclidean space with multi-modality.
In this survey, we formally formulate the problem of graph data augmentation
and further review the representative techniques and their applications in
different deep graph learning problems. Specifically, we first propose a
taxonomy for graph data augmentation techniques and then provide a structured
review by categorizing the related work based on the augmented information
modalities. Moreover, we summarize the applications of graph data augmentation
in two representative problems in data-centric deep graph learning: (1)
optimal graph learning which focuses on enhancing the data usability of the
input graph via graph data augmentation; and (2) low-resource graph learning
which targets on enlarging the labeled training data scale through graph data
augmentation. For each problem, we also provide a hierarchical problem
taxonomy and review the existing literature related to graph data
augmentation. Finally, we point out promising research directions and the
challenges in future research.
###### Index Terms:
Graph Neural Networks, Deep Graph Learning, Graph Data Augmentation.
## Acknowledgments
The authors would like to thank…
| Michael Shell Biography text here.
---|---
John Doe Biography text here.
---
Jane Doe Biography text here.
---
|
# Generalized $n$-series and de Rham complexes
S. K. Devalapurkar and M. L. Misterka
###### Abstract.
The goal of this article is to study some basic algebraic and combinatorial
properties of “generalized $n$-series” over a commutative ring $R$, which are
functions $s:\mathbf{Z}_{\geq 0}\to R$ satisfying a mild condition. A special
example of generalized $n$-series is given by the $q$-integers
$\frac{q^{n}-1}{q-1}\in\mathbf{Z}[\\![q-1]\\!]$. Given a generalized
$n$-series $s$, one can define $s$-analogues of factorials (via
$n!_{s}=\prod_{i=1}^{n}s(n)$) and binomial coefficients. We prove that
Pascal’s identity, the binomial identity, Lucas’ theorem, and the Vandermonde
identity admit $s$-analogues; each of these specialize to their appropriate
$q$-analogue in the case of the $q$-integer generalized $n$-series. We also
study the growth rates of generalized $n$-series defined over the integers.
Finally, we define an $s$-analogue of the ($q$-)derivative, and prove
$s$-analogues of the Poincaré lemma and the Cartier isomorphism for the affine
line, as well as a pullback square due to Bhatt-Lurie.
Part of this work was done when the first author was supported by the PD Soros
Fellowship, Inpher, and NSF DGE-2140743
###### Contents
1. 1 Introduction
1. 1.1 Summary
2. 1.2 Table of commonly-used notation
3. 1.3 Acknowledgements
2. 2 The $s$-Binomial Coefficients
1. 2.1 A generalization of binomial coefficients
2. 2.2 Number-theoretic properties of generalized $n$-series
3. 2.3 The $s$-binomial theorem
4. 2.4 The $s$-Lucas theorem
5. 2.5 The $s$-Vandermonde identity
3. 3 Generalized $n$-Series Over $\mathbf{Z}$
1. 3.1 Lexicographically small nonnegative integer generalized $n$-series
2. 3.2 An upper bound on lexicographically small nonnegative integer GNS
3. 3.3 A lower bound on strictly increasing integer GNS
4. 3.4 A more general $s$-Lucas theorem
4. 4 Generalized $n$-Series and de Rham Complexes
1. 4.1 Basic properties of the $s$-de Rham complex
2. 4.2 $s$-analogues of the Poincaré lemma and Cartier isomorphism
3. 4.3 Formal group law $n$-series and the $s$-derivative
4. 4.4 A variant of the Weyl algebra
5. 4.5 An analogue of the Bhatt-Lurie Cartesian square
## 1\. Introduction
### 1.1. Summary
Recent work of Bhatt, Drinfeld, Lurie, Morrow, Scholze, and others (see, e.g.,
[BMS18, BS19, Sch17, Dri21, Dri22, BL22]) has shown that $q$-deformations of
classical number-theoretic and algebro-geometric concepts play a central role
in arithmetic geometry. The basic premise behind the theory of
$q$-deformations is the idea that the $q$-integers
$[n]_{q}=\frac{q^{n}-1}{q-1}$ display many similarities to the ordinary
integers. This idea has a rich history111Any source recounting the history of
$q$-deformations would be most welcome to the authors!: the basic ideas date
back at least to Euler (e.g., [Eul53]) and Jacobi and the study of basic
hypergeometric series. A $q$-analogue of the derivative originates with
Jackson in 1909 (see [Jac09]). We refer the reader to the book [KC02] for an
exposition of $q$-deformed calculus.
The theory of formal group laws supplies a simultaneous generalization of both
ordinary integers and $q$-integers (see 4.3.1 for a quick summary of the
basics of formal group laws, and [Haz78, Rav86] for a detailed treatment).
Namely, every formal group law $F$ over a ring $R$ defines a sequence of power
series $\langle{n}\rangle$ over $R$ for every integer $n\in\mathbf{Z}$. In the
case of the the additive formal group law $x+y$, we have
$\langle{n}\rangle=n$; and in the case of the multiplicative formal group law
$x+y+xy$, one can identify $\langle{n}\rangle=[n]_{q}$. The goal of this
article is to explore whether certain aspects of $q$-deformed mathematics
(such as $q$-analogues of basic combinatorial formulae, and properties of the
$q$-de Rham complex of [Sch17]) admit generalizations to arbitrary formal
group laws. One of the primary motivations behind our investigation is the
unpublished observation of Arpon Raksit that homotopy-theoretic methods
naturally suggest studying “$F$-analogues” of the $q$-de Rham complexes
arising in the aforementioned work of Bhatt-Morrow-Scholze, as well as the
calculation of [Dev23a, Section 3.3].222Since the actual homotopy theory does
not play any role in this paper, we refer the interested reader to Remark
4.3.24 below for more.
Our primary observation is that one does not need the structure of a formal
group law to define and study these “$F$-analogues”. Instead, the following
significantly weaker structure suffices:
###### Definition (Definition 2.1.4).
Fix a ring $R$ (always assumed commutative with unit). A generalized
$n$-series (GNS) over $R$ is a function $s:\mathbf{Z}_{\geq 0}\to R$ such
that:
1. (1)
$s(0)=0$,
2. (2)
$s(n)$ is not a zero-divisor for any $n>0$,
3. (3)
$s(n-k)\mid s(n)-s(k)$ for all $n>k>0$.
For instance, the map $s:\mathbf{Z}_{\geq 0}\to\mathbf{Z}[\\![q-1]\\!]$
sending $n\mapsto[n]_{q}=\frac{q^{n}-1}{q-1}$ defines a GNS.
In the body of this article, we show that this simple definition is sufficient
for proving several analogues of classical combinatorial identities, and is
also enough to study an “$s$-deformation” of the classical algebraic de Rham
complex. The results of this article do not rely on any sophisticated tools:
rather, the purpose is to demonstrate the efficiency of Definition 2.1.4. The
work done in this article seems closely related to Bhargava’s [Bha00], but we
have not attempted to make a comparison.
If $n\geq 0$, let $n!_{s}=\prod_{k=1}^{n}s(k)$, and let
$\binom{n}{j}_{s}=\frac{n!_{s}}{j!_{s}(n-j)!_{s}}$ denote the $s$-analogues of
the factorial and binomial coefficient, respectively. Our main combinatorial
results are the following; for the full statement of some of these results, we
refer the reader to the body of the text.
###### Theorem A.
Fix a GNS $s$ over $R$. The following hold:
1. (1)
Pascal’s identity (Proposition 2.1.3):
$\binom{n}{k}_{s}=\binom{n-1}{k-1}_{s}+\frac{s(n)-s(k)}{s(n-k)}\binom{n-1}{k}_{s}.$
2. (2)
An $s$-analogue of the binomial and $q$-binomial theorems; see Theorem 2.3.7.
3. (3)
Lucas’ theorem (Proposition 2.4.8): suppose that $s(1)=1$ and
$s(a+b)\equiv s(a)+s(b)\pmod{s(a)s(b)}$
for all $a,b\in\mathbf{Z}_{>0}$. Then, for any prime $p$ and any nonnegative
integers $n_{1},n_{0},k_{1},k_{0}$ such that $n_{0},k_{0}<p$, we have
$\binom{n_{1}p+n_{0}}{k_{1}p+k_{0}}_{s}\equiv\binom{n_{1}}{k_{1}}\binom{n_{0}}{k_{0}}_{s}\pmod{s(p)}.$
4. (4)
An analogue of the Vandermonde and $q$-Vandermonde identities; see Theorem
2.5.4.
In Section 3, we study the growth rate of generalized $n$-series over
$\mathbf{Z}$. For instance, we show in Theorem 3.3.1 that if $s(n)$ is a
strictly increasing generalized $n$-series over $\mathbf{Z}$ which is not a
scalar multiple of $n\mapsto[n]_{q}$ for any $q\in\mathbf{Z}_{>0}$, then
$s(n)=\Omega_{a}(a^{n})$ for all $a\geq 0$.
As one might expect given our motivation above, one important class of
examples of generalized $n$-series arises via formal group laws. Recall (see
4.3.1) that a formal group law over a commutative ring $R$ is a two-variable
power series $x+_{F}y\in R[\\![x,y]\\!]$ such that
$(x+_{F}y)+_{F}z=x+_{F}(y+_{F}z)$ and $x+_{F}y\equiv x+y\pmod{(x,y)^{2}}$. If
$n\geq 0$ is an integer, the $n$-series of $F$ is defined via the formula
$[n]_{F}(t)=\overbrace{t+_{F}t+_{F}\cdots+_{F}t}^{n}\in tR[\\![t]\\!].$
Let $\langle{n}\rangle_{F}=\frac{[n]_{F}(t)}{t}$. Suppose (for simplicity)
that $R$ is torsionfree. Then, the function $\mathbf{Z}_{\geq 0}\to
R[\\![t]\\!]$ sending $n\mapsto\langle{n}\rangle_{F}$ defines a GNS over
$R[\\![t]\\!]$ (Proposition 4.3.4).
One can define the $F$-de Rham complex of the affine line
$\mathbf{A}^{1}=\operatorname{Spec}R[x]$ as the cochain complex
$F\Omega_{\square,\mathbf{A}^{1}}=\left(R[\\![t]\\!][x]\xrightarrow{{\nabla_{F}}}R[\\![t]\\!][x]dx\right),\
x^{n}\mapsto\langle{n}\rangle_{F}x^{n-1}dx.$
This was first defined by Arpon Raksit in unpublished work. Many analytic
properties of the usual ($q$-)derivative continue to hold for the
$F$-derivative: for instance, we show (see Corollary 4.3.15) that there is an
explicit power series $F\mathrm{log}(x)$ which recovers the $q$-logarithm when
$F$ is the $q$-integer GNS, and which satisfies the property that
$\nabla_{F}(F\mathrm{log}(x))=1/x$.
Our main results regarding the $F$-de Rham complex can be summarized as
follows:
###### Theorem B (Theorem 4.3.20 and Theorem 4.5.10).
Let $R$ be a torsionfree (say) commutative ring, and let $F$ be a formal group
law over $R$.
1. (1)
Let $R[\\![t]\\!]\langle{x}\rangle_{F}$ denote the ring
$R[\\![t]\\!][x,\frac{x^{n}}{[n]_{F}!}]_{n\geq 0}$. Then the Poincaré lemma
holds: the cohomology of the complex
$F\Omega_{\square,\mathbf{A}^{1}}\otimes_{R[\\![t]\\!][x]}R[\\![t]\\!]\langle{x}\rangle_{F}$
is concentrated in degree zero, where it is isomorphic to $R[\\![t]\\!]$.
2. (2)
The Cartier isomorphism holds: after setting $\langle{p}\rangle_{F}=0$, the
$i$th cohomology of the complex $F\Omega_{\square,\mathbf{A}^{1}}$ is
isomorphic to the $i$th term of a Frobenius twist of
$F\Omega_{\square,\mathbf{A}^{1}}$.
3. (3)
There is an analogue of the décalage isomorphism of [BO78, BS19] for
$F\Omega_{\square,\mathbf{A}^{1}}$.
4. (4)
Using the aforementioned $F$-analogue $F\mathrm{log}(x)$ of the $q$-logarithm,
we prove a generalization of the Cartesian square of [BL22, Lemma 3.5.18].
Except for the final part, the above result in fact admits a generalization to
arbitary GNS (not just ones which arise from formal group laws), but the
statement is slightly more complicated; see Section 4.2. As with the
combinatorial results above, Theorem 4.3.20 is not technically involved;
however, it is supposed to serve as a blueprint for a more general program
that we outline at the end of Section 4.3. In particular, we state the (almost
certainly false) 4.3.22, stating that the assignment
$R[x_{1},\ldots,x_{n}]\mapsto(F\Omega_{\square,\mathbf{A}^{1}})^{\otimes_{R[\\![t]\\!]}n}$
should extend to a functor from the category of commutative $R$-algebras to
the $\infty$-category of ${\mathbf{E}_{\infty}}$-$R[\\![t]\\!]$-algebras.
### 1.2. Table of commonly-used notation
This article will introduce some notation which will be used heavily
throughout. For the reader’s convenience, we have summarized the commonly-used
ones in the table below.
Symbol | Definition | Location in text
---|---|---
$R[1/s]$ | $R[s(1)^{-1},s(2)^{-1},\cdots]$ | 1
$c_{s}(n,k)$ | $\frac{s(n)-s(k)}{s(n-k)}$ | 2
$(x+y)^{n}_{s}$ | Characterized by specific conditions | Definition 2.3.5
$C_{s}(n,k)$ | $\frac{s(n+k)-s(n)-s(k)}{s(n)s(k)}$ | 2.4.4
$\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}$ | $\sum_{m<i_{1}<i_{2}<\cdots<i_{j}\leq m+n}\left(\prod_{\ell=1}^{j}c_{s}(i_{\ell},i_{\ell}-k+j-\ell)\right)$ | Definition 2.5.2
$\langle n\rangle_{F}(t)$ | $\frac{[n]_{F}(t)}{t}$ for a FGL $F(x,y)$ | 4.3.1
$\ell_{F}(t)$, $\mathscr{E}_{F}(t)$ | Logarithm and exponential of a FGL | 4.3.1
$F\mathrm{log}(x)$ | $\frac{t}{\ell_{F}(t)}\mathrm{log}(x)$ | Corollary 4.3.15
$\mathbf{G}_{m}^{\sharp,F}$ | “$F$-divided power hull” of zero section of $\mathbf{G}_{m}$ | Definition 4.5.5
### 1.3. Acknowledgements
The first author is grateful to Ben Antieau, Dimitar Jetchev, and (especially)
Arpon Raksit for discussions on this topic, as well as Michael Kural for his
help in proving the main result of Section 4.5 in the case of the
multiplicative formal group law. The second author is grateful to Pavel
Etingof for suggesting that an $s$-analogue of the $q$-binomial theorem might
exist. Finally, both authors are grateful to the directors and organizers of
PRIMES-USA for the opportunity to collaborate, as well as for comments on this
article!
## 2\. The $s$-Binomial Coefficients
### 2.1. A generalization of binomial coefficients
Recall that the binomial coefficient $\binom{n}{k}$ is defined by
$\binom{n}{k}=\frac{n!}{k!(n-k)!}.$
The $q$-factorial and $q$-binomial coefficients are defined by
$[n]!_{q}=[1]_{q}\cdot[2]_{q}\cdot\cdots\cdot[n]_{q},\qquad\binom{n}{k}_{q}=\frac{[n]!_{q}}{[k]!_{q}[n-k]!_{q}},$
where $[n]_{q}=(q^{n}-1)/(q-1)\in\mathbf{Z}[\\![q-1]\\!]$. The similarity of
these two definitions hints that it might be interesting to study a
simultaneous generalization of the usual and $q$-binomial coefficients, where
the sequence of elements $n\in\mathbf{Z}$ and
$[n]_{q}\in\mathbf{Z}[\\![q-1]\\!]$ are replaced by a sequence of elements in
a commutative ring satisfying certain conditions.
###### Definition 2.1.1.
Let $R$ be a ring, and let $s:\mathbf{Z}_{\geq 0}\to R$ be a function such
that $s(0)=0$ and for all $n>0$, $s(n)$ is not a zero-divisor. For integers
$n\geq k\geq 0$, we define the $s$-factorial $n!_{s}$ by
$0!_{s}=1,\quad n!_{s}=\prod_{k=1}^{n}s(k),$
and the $s$-binomial coefficient $\binom{n}{k}_{s}$ by
$\binom{n}{k}_{s}=\frac{n!_{s}}{k!_{s}(n-k)!_{s}}.$
###### Remark 2.1.2.
In general, this quotient is undefined in $R$; however, it is always defined
in the localization
(1) $R[1/s]:=R[s(1)^{-1},s(2)^{-1},s(3)^{-1},\dots].$
We will soon restrict to the case where the $s$-binomial coefficients are
elements of $R$.
The $s$-binomial coefficient need not satisfy any nice properties, since there
are no restrictions placed on $s$. Our first observation is the following.
###### Proposition 2.1.3.
Let $R$ be a ring and let $s:\mathbf{Z}_{\geq 0}\to R$ be a function that
satisfies the following conditions:
1. (1)
$s(0)=0$,
2. (2)
$s(n)$ is not a zero-divisor for any $n>0$,
3. (3)
$s(n-k)\mid s(n)-s(k)$ for all $n>k>0$.
Then, for all integers $n\geq k\geq 0$, the $s$-binomial coefficient
$\binom{n}{k}_{s}$ is an element of $R$, and the $s$-binomial coefficients
satisfy an “$s$-Pascal identity”: For all $n>k>0$,
$\binom{n}{k}_{s}=\binom{n-1}{k-1}_{s}+\frac{s(n)-s(k)}{s(n-k)}\binom{n-1}{k}_{s}.$
###### Proof.
Indeed, observe that in the localization $R[1/s]$, we have:
$\displaystyle\frac{s(n)-s(k)}{s(n-k)}\binom{n-1}{k}_{s}$
$\displaystyle=\frac{s(n)-s(k)}{s(n-k)}\frac{(n-1)!_{s}}{k!_{s}(n-k-1)!_{s}}$
$\displaystyle=\frac{s(n)\cdot(n-1)!_{s}}{k!_{s}(n-k)!_{s}}-\frac{s(k)\cdot(n-1)!_{s}}{k!_{s}(n-k)!_{s}}$
$\displaystyle=\frac{n!_{s}}{k!_{s}(n-k)!_{s}}-\frac{(n-1)!_{s}}{(k-1)!_{s}(n-k)!_{s}}$
$\displaystyle=\binom{n}{k}_{s}-\binom{n-1}{k-1}_{s}.$
It remains to show that $\binom{n}{k}_{s}\in R$ for all $n\geq k\geq 0$. We
will use induction on $n$.
The base case is clear, since $\binom{0}{0}_{s}=1\in R$. For the inductive
step, assume that for some fixed $n$ and for all $k$ with $n-1\geq k\geq 0$,
$\binom{n-1}{k}_{s}\in R$. Let $k$ be an integer such that $n\geq k\geq 0$. If
$k=0$, then $\binom{n}{k}_{s}=1\in R$. Otherwise, we can apply the $s$-Pascal
identity:
$\binom{n}{k}_{s}=\binom{n-1}{k-1}_{s}+\frac{s(n)-s(k)}{s(n-k)}\binom{n-1}{k}_{s}.$
By the inductive hypothesis, the two $s$-binomial coefficients on the right-
hand side are in $R$, and by condition (c) in the theorem statement,
$\frac{s(n)-s(k)}{s(n-k)}\in R$. Therefore, $\binom{n}{k}_{s}\in R$. This
completes the induction proof. ∎
Motivated by Proposition 2.1.3, we are led to the following:
###### Definition 2.1.4.
Let $R$ be a ring. A generalized $n$-series (GNS) over $R$ is a function
$s:\mathbf{Z}_{\geq 0}\to R$ such that the following conditions are true:
1. (1)
$s(0)=0$,
2. (2)
$s(n)$ is not a zero-divisor for any $n>0$,
3. (3)
$s(n-k)\mid s(n)-s(k)$ for all $n>k>0$.
If $s$ is a generalized $n$-series, we will define
(2) $c_{s}(n,k):=\frac{s(n)-s(k)}{s(n-k)}.$
###### Example 2.1.5 (Integers).
The inclusion $s:\mathbf{Z}_{\geq 0}\to\mathbf{Z}$ is clearly a GNS over
$\mathbf{Z}$.
###### Example 2.1.6 ($q$-integers).
Consider the function $s:\mathbf{Z}_{\geq 0}\to\mathbf{Z}[\\![q-1]\\!]$ given
by $s(n)=[n]_{q}$. This defines a GNS: the first two conditions are satisfied,
since $[0]_{q}=0$, $[n]_{q}\neq 0$ for $n>0$, and $\mathbf{Z}[\\![q-1]\\!]$ is
an integral domain. For the third condition, note that
$\displaystyle[n]_{q}-[k]_{q}$
$\displaystyle=\frac{q^{n}-1}{q-1}-\frac{q^{k}-1}{q-1}=\frac{q^{n}-q^{k}}{q-1}$
$\displaystyle=q^{k}\left(\frac{q^{n-k}-1}{q-1}\right)=q^{k}[n-k]_{q}.$
Therefore, $s$ is a GNS over $\mathbf{Z}[\\![q-1]\\!]$, and we can apply
Proposition 2.1.3 to conclude that
$\binom{n}{k}_{q}\in\mathbf{Z}[\\![q-1]\\!]$ for all $n\geq k\geq 0$. The
$s$-Pascal identity reduces to the well-known $q$-Pascal identity:
$\binom{n}{k}_{q}=\binom{n-1}{k-1}_{q}+q^{k}\binom{n-1}{k}_{q}.$
###### Remark 2.1.7.
One can extend the definition of the $s$-binomial coefficients to allow
arbitrary integers $k$ by defining $\binom{n}{k}_{s}=0$ when $k<0$ or $k>n$.
Using this extended definition, the $s$-Pascal identity remains true when
$k=0$:
$\binom{n-1}{-1}_{s}+\frac{s(n)-s(0)}{s(n-0)}\binom{n-1}{0}_{s}=0+1\cdot
1=1=\binom{n}{0}_{s}.$
This relies on the condition $s(0)=0$. The fact that Pascal’s identity fails
for $k=0$ if $s(0)\neq 0$ is one motivation for including condition (1) in the
definition of GNS.
### 2.2. Number-theoretic properties of generalized $n$-series
In this section, we prove some number-theoretic properties of generalized
$n$-series, which will be useful later in this article. The main result of
this section is the following:
###### Theorem 2.2.1.
Let $s$ be a generalized $n$-series over a ring $R$. Then, for all
$a,b,n\in\mathbf{Z}_{\geq 0}$,
1. (1)
$a\mid b\implies s(a)\mid s(b)$,
2. (2)
$a\equiv b\pmod{n}\implies s(a)\equiv s(b)\pmod{s(n)}$,
3. (3)
the ideals $(s(a),s(b))$ and $(s(\gcd(a,b)))$ are equal.
If $s(1)$ is a unit in $R$, then for all $a,n\in\mathbf{Z}_{\geq 0}$,
1. (4)
$a\text{ is a unit in }\mathbf{Z}/n\implies s(a)\text{ is a unit in }R/s(n)$.
###### Remark 2.2.2.
In the case $R=\mathbf{Z}$, the equivalence of ideals in Theorem 2.2.1 is
equivalent to
$\gcd(s(a),s(b))=\pm s(\gcd(a,b)).$
We will prove Theorem 2.2.1 as a sequence of lemmas. Fix a generalized
$n$-series $s$ over a ring $R$.
###### Lemma 2.2.3.
Let $a,b\in\mathbf{Z}_{\geq 0}$. Then, $a\mid b$ implies $s(a)\mid s(b)$.
###### Proof.
We will use induction. Base case: $s(a)\mid s(0)$ because $s(0)=0$. Inductive
hypothesis: Let $n\in\mathbf{Z}_{>0}$, and assume that $s(a)\mid s(a(n-1))$.
Then, by the divisibility condition in the definition of generalized
$n$-series,
$s(a)\mid s(a(n-1))=s(an-a)\mid s(an)-s(a),$
so $s(a)\mid s(an)$. ∎
###### Lemma 2.2.4.
Let $a,b,n\in\mathbf{Z}_{\geq 0}$. If $a\equiv b\pmod{n}$ then
$s(a)\equiv s(b)\pmod{s(n)}.$
Another way to state this lemma is that $s$ induces a well-defined function
from $\mathbf{Z}/n$ to $R/s(n)$.
###### Proof.
By the definition of congruence, $n\mid a-b$, so $s(n)\mid s(a-b)$ by Lemma
2.2.3. The definition of generalized $n$-series requires that
$s(a-b)\mid s(a)-s(b),$
so $s(n)\mid s(a)-s(b)$, which means that $s(a)\equiv s(b)\pmod{s(n)}$. ∎
###### Lemma 2.2.5.
Suppose that $s(1)$ is a unit in $R$. If $a,n\in\mathbf{Z}_{\geq 0}$ such that
$a$ is a unit in $\mathbf{Z}/n$, then $s(a)$ is a unit in $R/s(n)$.
###### Proof.
Let $b$ be the multiplicative inverse of $a$ modulo $n$. Then, $ab\equiv
1\pmod{n}$. By Lemmas 2.2.3 and 2.2.4,
$s(a)\mid s(ab)\equiv s(1)\pmod{s(n)}.$
So in the ring $R/s(n)$, $s(a)$ divides $s(1)$, which is a unit (because it is
a unit in $R$). Therefore, $s(a)$ is a unit in $R/s(n)$. ∎
###### Lemma 2.2.6.
Let $a,b\in\mathbf{Z}_{\geq 0}$. Then, we have the following equivalence of
ideals:
$\big{(}s(\gcd(a,b))\big{)}=\big{(}s(a),s(b)\big{)}.$
###### Proof.
Let $d=\gcd(a,b)$. By Lemma 2.2.3, $s(d)\mid s(a)$ and $s(d)\mid s(b)$, so
$s(a),s(b)\in\big{(}s(d)\big{)}$. This means that
$\big{(}s(d)\big{)}\supseteq\big{(}s(a),s(b)\big{)}.$
For the other direction, we can use Bézout’s identity to write $d=am+bn$ for
some $m,n\in\mathbf{Z}$. Taking this equation modulo $a$ gives $d\equiv
bn\pmod{a}$. By Lemma 2.2.4, $s(d)\equiv s(bn)\pmod{s(a)}$. Therefore, Lemma
2.2.3 implies that
$s(d)\in s(bn)+\big{(}s(a)\big{)}\subseteq\big{(}s(a),s(b)\big{)},$
and hence $\big{(}s(d)\big{)}\subseteq\big{(}s(a),s(b)\big{)}$. This shows
that the two ideals are equal. ∎
### 2.3. The $s$-binomial theorem
###### Recollection 2.3.1.
The binomial theorem and the $q$-binomial theorem are the following two
identities:
$\displaystyle(x+y)^{n}$
$\displaystyle=\sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k},$
$\displaystyle(x+y)^{n}_{q}$
$\displaystyle=\sum_{k=0}^{n}\binom{n}{k}_{q}q^{k(k-1)/2}x^{n-k}y^{k},$
where $(x+y)^{n}_{q}$ is defined by
$(x+y)^{n}_{q}=\prod_{k=0}^{n-1}(x+q^{k}y)=(x+y)(x+qy)\cdots(x+q^{n-1}y).$
Note that the $x+y$ in the parentheses is part of the notation, and cannot be
treated as a sum; see [KC02] for this notation.
###### Remark 2.3.2.
For readers who are familiar with the $q$-Pochhammer symbol,
$(x+y)^{n}_{q}=x^{n}(-y/x;q)_{n}$, and $(a;q)_{n}=(1+(-a))^{n}_{q}$.
We will now state and prove an analogue of the binomial theorem for the
$s$-binomial coefficients. We begin by defining an analogue of the symbol
$(x+y)^{n}_{q}$. To motivate the definition, recall that the $q$-analogue
$(x+y)^{n}_{q}$ is the unique polynomial in $\mathbf{Z}[\\![q-1]\\!][x,y]$
such that the following properties hold:
* •
The $q$-derivative with respect to $x$ of $(x+y)^{n}_{q}$ is
$[n]_{q}(x+y)^{n-1}_{q}$. This is analogous to the fact that the classical
derivative of $(x+y)^{n}$ with respect to $x$ is $n(x+y)^{n-1}$.
* •
$(x+y)^{0}_{q}=1$.
* •
If $y=-x$, then $(x+y)^{n}_{q}$ is $0$ for all $n>0$.
To define an $s$-analogue $(x+y)^{n}_{q}$ in a similar way, we need an
$s$-derivative; we will greatly expand on this notion in Section 4.
###### Definition 2.3.3.
Let $s:\mathbf{Z}_{\geq 0}\to R$ be a GNS. The $s$-derivative is the
$R$-linear map $\nabla_{s}:R[x]\to R[x]$ given on monomials by
$\nabla_{s}(x^{n})=s(n)x^{n-1}$.
###### Remark 2.3.4.
When $n=0$, we have $\nabla_{s}(x^{0})=s(0)x^{-1}$. This is not defined in
$R[x]$ unless $s(0)=0$, which is always true when $s$ is a GNS. Continuing
Remark 2.1.7, this observation is another reason for requiring $s(0)=0$ in the
definition of GNS.
We can now define $(x+y)^{n}_{s}$:
###### Definition 2.3.5.
Let $s$ be a GNS over $R$, so that $R[1/s]=R[s(1)^{-1},s(2)^{-1},\dots]$.
Define $(x+y)^{n}_{s}$ for $n\in\mathbf{Z}_{\geq 0}$ to be the unique
polynomial in $R[1/s][x,y]$ such that the following three conditions hold:
1. (1)
$(x+y)^{0}_{s}=1$,
2. (2)
$(x+(-x))^{n}_{s}=0$ for all $n>0$,
3. (3)
$\nabla_{s,x}(x+y)^{n}_{s}=s(n)(x+y)^{n-1}_{s}$.
Here, $\nabla_{s,x}:R[1/s][x,y]\to R[1/s][x,y]$ is the operator given by the
“$s$-derivative with respect to $x$”: it is simply the $R[1/s][y]$-linear
extension of the $s$-derivative $\nabla_{s}:R[x]\to R[x]$ to $R[1/s][x,y]$.
###### Lemma 2.3.6.
The symbol $(x+y)^{n}_{s}$ in Definition 2.3.5 is well-defined: it exists and
is unique. Moreover, $(x+y)^{n}_{s}$ is a homogeneous polynomial of degree
$n$.
###### Proof.
We will use induction on $n$. For the base case $n=0$, observe that
$(x+y)^{0}_{s}=1$ by condition (1).
For the inductive step, fix $n>0$, and suppose that for all $k<n$,
$(x+y)^{k}_{s}$ is well-defined and homogeneous of degree $k$. We can
$s$-antidifferentiate $s(n)(x+y)^{n-1}_{s}$ using the $R[1/s][y]$-linear
operator $I_{s,x}:R[1/s][x,y]\to R[1/s][x,y]$ defined on monomials by
$I_{s,x}(x^{k})=s(k+1)^{-1}x^{k+1}$. By definition, this operator produces
polynomials with no term of $x$-degree $0$. Although the $s$-antiderivative
$f(x,y)=I_{s,x}(s(n)(x+y)^{n-1}_{s})$
is homogeneous of degree $n$ (since the operator $I_{s,x}$ increases
$x$-degree by $1$) and satisfies condition (3), it might not equal
$(x+y)^{n}_{s}$ because it does not have to satisfy condition (2). Since
$f(x,y)$ is homogeneous of degree $n$, $f(x,-x)$ is a scalar multiple of
$x^{n}$, say $ax^{n}$. Then, the polynomial
$g(x,y)=f(x,y)-a(-y)^{n}$
satisfies
$\nabla_{s,x}g(x,y)=\nabla_{s,x}f(x,y)=s(n)(x+y)^{n-1}_{s}$
and
$g(x,-x)=f(x,-x)-a(-(-x))^{n}=ax^{n}-ax^{n}=0.$
It follows that $(x+y)^{n}_{s}$ exists, and one possible value for it is
$g(x,y)$, which is homogeneous of degree $n$.
It remains to show that $(x+y)^{n}_{s}$ is unique. We know that any polynomial
$h(x,y)$ that satisfies the conditions of $(x+y)^{n}_{s}$ must match $g(x,y)$
in every term with positive $x$-degree, because their $s$-derivatives with
respect to $x$ are both $s(n)(x+y)^{n-1}_{s}$. Therefore, $h(x,y)-g(x,y)$ is a
scalar multiple of $y^{n}$, say $by^{n}$. Setting $y=-x$ gives
$b(-x)^{n}=h(x,-x)-g(x,-x)$, which is $0$ by condition (2), so $b=0$.
Therefore, $h(x,y)=g(x,y)$. This proves that $(x+y)^{n}_{s}$ is unique and is
equal to $g(x,y)$. ∎
Recall from Section 2 that we originally defined the $s$-binomial coefficients
as elements of the ring $R[1/s]=R[s(1)^{-1},s(2)^{-1},\dots]$, and later
proved (using the $s$-Pascal identity) that if $s$ is a GNS, then all the
$s$-binomial coefficients are elements of $R$. We will do something similar
for $(x+y)^{n}_{s}$ below, and we will use the $s$-binomial theorem as a lemma
in the proof that $(x+y)^{n}_{s}\in R[x,y]$. Here is the $s$-binomial theorem:
###### Theorem 2.3.7 ($s$-binomial theorem).
Let $s$ be a GNS over $R$. Then, as elements of $R[1/s][x,y]$, we have:
$(x+y)^{n}_{s}=\sum_{k=0}^{n}\binom{n}{k}_{s}x^{n-k}y^{k}(0+1)^{k}_{s}.$
###### Proof.
We will use induction on $n$, and the inductive step will mainly consist of
applying the $s$-antidifferentiation operator $I_{s,x}$ from the proof of
Lemma 2.3.6 to both sides. For the base case, observe that if $n=0$, both
sides are $1$.
For the inductive step, fix $n>0$, and assume that the $s$-binomial theorem is
true for $n-1$:
$(x+y)^{n-1}_{s}=\sum_{k=0}^{n-1}\binom{n-1}{k}_{s}x^{n-k-1}y^{k}(0+1)^{k}_{s}.$
Multiplying both sides by $s(n)$, applying $I_{s,x}$, and using
$R[1/s][y]$-linearity gives:
$\displaystyle I_{s,x}(s(n)(x+y)^{n-1}_{s})$
$\displaystyle=s(n)\sum_{k=0}^{n-1}\binom{n-1}{k}_{s}I_{s,x}(x^{n-k-1})y^{k}(0+1)^{k}_{s}$
$\displaystyle=\sum_{k=0}^{n-1}\frac{s(n)}{s(n-k)}\binom{n-1}{k}_{s}x^{n-k}y^{k}(0+1)^{k}_{s}.$
Notice that
$\frac{s(n)}{s(n-k)}\binom{n-1}{k}_{s}=\frac{s(n)(n-1)!_{s}}{s(n-k)k!_{s}(n-k-1)!_{s}}=\frac{n!_{s}}{k!_{s}(n-k)!_{s}}=\binom{n}{k}_{s}.$
This implies that
$I_{s,x}(s(n)(x+y)^{n-1}_{s})=\sum_{k=0}^{n-1}\binom{n}{k}_{s}x^{n-k}y^{k}(0+1)^{k}_{s}.$
The right-hand side almost looks like the right-hand side of the $s$-binomial
theorem that we are trying to prove, but the upper limit of the summation is
$n-1$ instead of $n$. To fix this, add $y^{n}(0+1)^{n}_{s}$ to both sides,
giving
$I_{s,x}(s(n)(x+y)^{n-1}_{s})+y^{n}(0+1)^{n}_{s}=\sum_{k=0}^{n}\binom{n}{k}_{s}x^{n-k}y^{k}(0+1)^{k}_{s}.$
We just have to show that the left-hand side is equal to $(x+y)^{n}_{s}$.
Let $g(x,y)$ be the left-hand side. The $s$-derivative with respect to $x$ of
$g(x,y)$ is $s(n)(x+y)^{n-1}_{s}$, because $I_{s,x}$ is an $s$-antiderivative
operator (a right inverse of $\nabla_{s,x}$) and $y^{n}(0+1)^{n}_{s}$ is
constant with respect to $x$. This is equal to the $s$-derivative of
$(x+y)^{n}_{s}$, so $g(x,y)$ matches $(x+y)^{n}_{s}$ in all terms with
positive $x$-degree. Both $g(x,y)$ and $(x+y)^{n}_{s}$ are homogeneous of
degree $n$, so the only terms with $x$-degree $0$ are the $y^{n}$ terms. The
coefficient of $y^{n}$ in $g(x,y)$ is $(0+1)^{n}_{s}$ because $I_{s,x}$ never
produces terms with $x$-degree $0$. The $y^{n}$ term of $(x+y)^{n}_{s}$ is
$(0+y)^{n}_{s}$, which is $y^{n}(0+1)^{n}_{s}$ by homogeneity, so the
coefficient of $y^{n}$ in $(x+y)^{n}_{s}$ is also $(0+1)^{n}_{s}$. Therefore,
$g(x,y)=(x+y)^{n}_{s}$, so
$(x+y)^{n}_{s}=\sum_{k=0}^{n}\binom{n}{k}_{s}x^{n-k}y^{k}(0+1)^{k}_{s}.$
This is what we needed to show for the inductive step. ∎
To complete this subsection, we will show that the coefficients of
$(x+y)^{n}_{s}$ are elements of $R$ for all GNS $s$ over $R$ and all
$n\in\mathbf{Z}_{\geq 0}$. This means that the $s$-binomial theorem is really
an equivalence of polynomials in $R[x,y]$, and can be stated without using the
larger ring $R[1/s][x,y]$.
###### Proposition 2.3.8.
Let $s$ be a GNS over a ring $R$. For all nonnegative integers $n$, we have
$(x+y)^{n}_{s}\in R[x,y]$.
###### Proof.
It suffices to show that $(0+1)^{n}_{s}\in R$ for all nonnegative integers
$n$, because using the $s$-binomial theorem, we could conclude that
$(x+y)^{n}_{s}=\sum_{k=0}^{n}\binom{n}{k}_{s}x^{n-k}y^{k}(0+1)^{k}_{s}\in
R[x,y].$
Setting $x=-1$ and $y=1$ in the $s$-binomial theorem, we get
$0=((-1)+1)^{n}_{s}=\sum_{k=0}^{n}\binom{n}{k}_{s}(-1)^{n-k}(0+1)^{k}_{s},$
so
(3) $(0+1)^{n}_{s}=\sum_{k=0}^{n-1}\binom{n}{k}_{s}(-1)^{n-k-1}(0+1)^{k}_{s}.$
This is a recurrence relation for $(0+1)^{n}_{s}$.
To prove that $(0+1)^{n}_{s}\in R$, we will use induction on $n$. For the base
case, note that $(0+1)^{0}_{s}=1$ which is an element of $R$. For the
inductive step, note that if $(0+1)^{k}_{s}\in R$ for all $k<n$, then by the
recurrence relation 3 and the fact that the $s$-binomial coefficients are in
$R$ (Proposition 2.1.3),
$(0+1)^{n}_{s}=\sum_{k=0}^{n-1}\binom{n}{k}_{s}(-1)^{n-k-1}(0+1)^{k}_{s}\in
R.$
This completes the induction. ∎
###### Remark 2.3.9.
Using the recurrence relation 3, it can be shown that
$\frac{(-1)^{n}(0+1)^{n}_{s}}{n!_{s}}=\sum_{\ell=1}^{n}\left(\sum_{k_{1}+\cdots+k_{\ell}=n}\left(\prod_{j=1}^{\ell}\frac{(-1)}{k_{\ell}!_{s}}\right)\right),$
where the inner sum is over all ordered $\ell$-tuples
$(k_{1},k_{2},\dots,k_{\ell})$ of positive integers that sum to $n$. Combining
the inner and outer sums, the right-hand side can be viewed as a sum over
compositions (ordered partitions) of $n$. Isolating $(0+1)^{n}_{s}$ gives
$(0+1)^{n}_{s}=\sum_{\pi\text{ composition of
}n}(-1)^{n-|\pi|}\binom{n}{\pi_{1},\pi_{2},\dots,\pi_{|\pi|}}_{s}.$
The summand is an $s$-multinomial coefficient, defined by
$\binom{n}{k_{1},k_{2},\dots,k_{n}}_{s}=\frac{n!_{s}}{k_{1}!_{s}\cdot
k_{2}!_{s}\cdot\cdots\cdot k_{n}!_{s}},$
and $|\pi|$ denotes the length of $\pi$.
### 2.4. The $s$-Lucas theorem
###### Recollection 2.4.1.
Lucas’s theorem says that for all primes $p$ and all integers
$n_{1},n_{0},k_{1},k_{0}\in\mathbf{Z}_{\geq 0}$ such that $n_{0},k_{0}<p$,
$\binom{n_{1}p+n_{0}}{k_{1}p+k_{0}}\equiv\binom{n_{1}}{k_{1}}\binom{n_{0}}{k_{0}}\pmod{p}.$
There is a $q$-analogue of this identity, known as the $q$-Lucas theorem:
$\binom{n_{1}p+n_{0}}{k_{1}p+k_{0}}_{q}\equiv\binom{n_{1}}{k_{1}}\binom{n_{0}}{k_{0}}_{q}\pmod{[p]_{q}}.$
This identity is also true if $p$ is composite, as long as we replace the
modulus $[p]_{q}$ with the cyclotomic polynomial $\Phi_{p}(q)$. Notice that
the first binomial coefficient on the right-hand side of the $q$-Lucas theorem
is not a $q$-binomial coefficient.
Here is an $s$-analogue of Lucas’s theorem:
###### Theorem 2.4.2 ($s$-Lucas theorem).
Let $s$ be a GNS over $R$ such that $s(1)=1$ and
(4) $s(a+b)\equiv s(a)+s(b)\pmod{s(a)s(b)}$
for all $a,b\in\mathbf{Z}_{>0}$. Then, for any prime $p$ and any nonnegative
integers $n_{1},n_{0},k_{1},k_{0}$ such that $n_{0},k_{0}<p$, we have
$\binom{n_{1}p+n_{0}}{k_{1}p+k_{0}}_{s}\equiv\binom{n_{1}}{k_{1}}\binom{n_{0}}{k_{0}}_{s}\pmod{s(p)}.$
###### Remark 2.4.3.
Define $\Phi_{n}(s)=\prod_{d\mid n}s(d)^{\mu(n/d)}$, where $\mu$ denotes the
Möbius function. Note that Möbius inversion implies the identity
$s(n)=\prod_{d\mid n}\Phi_{n}(s)$. One can prove a “composite version” of the
$s$-Lucas theorem where $s(p)$ is replaced by $\Phi_{n}(s)$, but the proof is
more complicated.
For the rest of this section, we fix the GNS $s$ over $R$. To simplify the
statement of the $s$-Lucas theorem, we will use the following:
###### Notation 2.4.4.
For integers $a,b\in\mathbf{Z}_{>0}$, we define
$C_{s}(a,b)=\frac{s(a+b)-s(a)-s(b)}{s(a)s(b)}\in R[1/s].$
The extra condition 4 in Theorem 2.4.2 is therefore equivalent to $C_{s}(a,b)$
being defined in $R$ for all $a,b\in\mathbf{Z}_{>0}$.
###### Lemma 2.4.5.
Suppose that $C_{s}(a,b)$ is defined in $R$ for all $a,b\in\mathbf{Z}_{>0}$.
Then, for all integers $m>0$ and $n>k\geq 0$,
$c_{s}(mn,mk)\equiv 1\pmod{s(m)}.$
###### Proof.
By definition,
$c_{s}(mn,mk)-1=\frac{s(mn)-s(mk)-s(mn-mk)}{s(mn-mk)}=s(mk)C_{s}(mk,mn-mk).$
By Lemma 2.2.3, $s(m)\mid s(mk)$, so the right-hand side is divisible by
$s(m)$. Therefore, $c_{s}(mn,mk)-1\equiv 0\pmod{s(m)}$. ∎
###### Remark 2.4.6.
A consequence of this lemma is the interesting fact that if we define the
“rescaled” GNS $s_{m}(n)=s(mn)$ for each positive integer $m$, then Pascal’s
identity for the $s_{m}$-binomial coefficients is the same as the usual
Pascal’s identity when we reduce modulo $s(m)$. This implies the following
congruence:
###### Lemma 2.4.7.
Suppose that $C_{s}(a,b)$ is defined for all $a,b\in\mathbf{Z}_{>0}$. For all
integers $m>0$ and $0\leq k\leq n$,
$\binom{n}{k}_{s_{m}}\equiv\binom{n}{k}\pmod{s(m)}.$
###### Proof.
We will use induction on $n$. For the base case, observe that if $n=0$ then
$k=0$, so both sides are $1$. For the inductive step, assume that the desired
claim is true when $n$ is replaced by $n-1$. Then:
$\displaystyle\binom{n}{k}_{s_{m}}$
$\displaystyle=\binom{n-1}{k-1}_{s_{m}}+c_{s_{m}}(n,k)\binom{n-1}{k}_{s_{m}}$
$\displaystyle\text{by the $s$-Pascal identity (\lx@cref{creftype~refnum}{thm:
s-Pascal identity})},$
$\displaystyle=\binom{n-1}{k-1}_{s_{m}}+c_{s}(mn,mk)\binom{n-1}{k}_{s_{m}}$
$\displaystyle\text{by definition of $c_{s_{m}}$},$
$\displaystyle\equiv\binom{n-1}{k-1}_{s_{m}}+\binom{n-1}{k}_{s_{m}}$
$\displaystyle\text{by Lemma \ref{lem: C defined implies c(mn, mk) is 1 mod
s(m)}},$ $\displaystyle\equiv\binom{n-1}{k-1}+\binom{n-1}{k}\qquad$
$\displaystyle\text{by inductive hypothesis},$
$\displaystyle=\binom{n}{k}\pmod{s(m)}$ $\displaystyle\text{by Pascal's
identity}.$
Therefore, by induction, this is true for all $m\in\mathbf{Z}_{>0}$. ∎
Theorem 2.4.2 is a consequence of a slight variant.
###### Proposition 2.4.8 (Another $s$-Lucas theorem).
Let $s$ be a GNS such that $s(1)=1$. Let $p$ be prime and let
$n_{1},n_{0},k_{1},k_{0}\in\mathbf{Z}_{\geq 0}$ such that $n_{0},k_{0}<m$.
Then,
$\binom{n_{1}p+n_{0}}{k_{1}p+k_{0}}_{s}\equiv\binom{n_{1}}{k_{1}}_{s_{p}}\binom{n_{0}}{k_{0}}_{s}\pmod{s(p)},$
where $s_{p}$ is the rescaled GNS from Remark 2.4.6.
###### Proof of Theorem 2.4.2.
This is an immediate consequence of Proposition 2.4.8 and Lemma 2.4.7. ∎
###### Remark 2.4.9.
Proposition 2.4.8 is just Theorem 2.4.2 but with $\binom{n_{1}}{k_{1}}$
replaced by $\binom{n_{1}}{k_{1}}_{s_{p}}$, and no requirement that
$C_{s}(a,b)$ be an element of $R$.
Before we prove Proposition 2.4.8 in full generality, we will prove the
special case where $n_{0}=k_{0}=0$. Then, we will use this case as a lemma in
the proof of the general result.
###### Proof of Theorem 2.4.8 in the case $n_{0}=k_{0}=0$.
Writing out the definition of the $s$-binomial coefficient
$\binom{pn}{pk}_{s}$, we get
$\binom{pn}{pk}_{s}=\frac{s(pn)s(pn-1)\cdots s(pn-pk+1)}{s(pk)s(pk-1)\cdots
s(1)}.$
By Lemma 2.2.5, $s(pn-j)$ is a unit modulo $s(p)$ for all $j$ not divisible by
$p$. Together with Lemma 2.2.4, this implies that we can cancel $s(pn-j)$ with
$s(pk-j)$ (since they are congruent units modulo $s(p)$). After all of this
cancellation, we are left with
$\binom{pn}{pk}_{s}=\frac{s(pn)s(p(n-1))\cdots
s(p(n-k+1))}{s(pk)s(p(k-1))\cdots s(p)},$
which is just $\binom{n}{k}_{s_{p}}$. This means that
$\binom{pn}{pk}_{s}\equiv\binom{n}{k}_{s_{p}}\pmod{s(p)}.$
Combining this with the congruence of $\binom{n}{k}_{s_{p}}$ and
$\binom{n}{k}$, we get
$\binom{pn}{pk}_{s}\equiv\binom{n}{k}\pmod{s(p)},$
which is the special case of Theorem 2.4.8 where $n_{0}=k_{0}=0$. ∎
We will now extend this to any value of $n_{0}$ in the valid range $0\leq
n_{0}<p$.
###### Proof of Theorem 2.4.8 in the case $k_{0}=0$.
We proved Proposition 2.4.8 above for $n_{0}=0$, so let $0<n_{0}<p$. Notice
that the definition of $s$-binomial coefficients implies that
$s(n-k)\binom{n}{k}_{s}=\frac{n!_{s}}{k!_{s}(n-k-1)!_{s}}=s(n)\binom{n-1}{k}_{s}.$
Therefore,
$s(p(n_{1}-k)+n_{0})\binom{pn_{1}+n_{0}}{pk}_{s}=s(pn_{1}+n_{0})\binom{pn_{1}+n_{0}-1}{pk}_{s}.$
Since $n_{0}$ is a unit modulo $p$, we see that $s(p(n_{1}-k)+n_{0})$ is a
unit modulo $s(p)$. But $s(p(n_{1}-k)+n_{0})$ is congruent to
$s(pn_{1}+n_{0})$ by Lemma 2.2.4. Since both are units, this implies that
$\binom{pn_{1}+n_{0}}{pk}_{s}\equiv\binom{pn_{1}+n_{0}-1}{pk}_{s}\pmod{s(p)}.$
A simple induction proof gives
$\binom{pn_{1}+n_{0}}{pk}_{s}\equiv\binom{pn_{1}}{pk}_{s}\pmod{s(p)}.$
It follows that
$\binom{pn_{1}+n_{0}}{pk}_{s}\equiv\binom{pn_{1}}{pk}_{s}\equiv\binom{n_{1}}{k}=\binom{n_{1}}{k}\binom{n_{0}}{0}_{s}\pmod{s(p)},$
which is exactly the $s$-Lucas theorem where $k_{0}=0$. ∎
Finally, we will extend this by induction to any value of $k_{0}$ between $0$
and $p-1$.
###### Proof of Theorem 2.4.8 in full generality.
We will use induction on $k_{0}$. We already proved the base case $k_{0}=0$
above. Suppose we have shown that the induction hypothesis
$\binom{pn_{1}+n_{0}}{pk_{1}+k_{0}-1}_{s}\equiv\binom{n_{1}}{k_{1}}\binom{n_{0}}{k_{0}-1}_{s}\pmod{s(p)}$
is true for some $k_{0}-1$ between $0$ and $p-2$ (or $0<k_{0}<p$). We will
show that it is true with $k_{0}-1$ replaced by $k_{0}$. Notice that for all
integers $1\leq k\leq n$,
$s(k)\binom{n}{k}_{s}=\frac{n!_{s}}{(k-1)!_{s}(n-k)!_{s}}=s(n-k+1)\binom{n}{k-1}_{s},$
so
$s(pk_{1}+k_{0})\binom{pn_{1}+n_{0}}{pk_{1}+k_{0}}_{s}=s(p(n_{1}-k_{1})+n_{0}-k_{0}+1)\binom{pn_{1}+n_{0}}{pk_{1}+k_{0}-1}_{s}.$
Reducing modulo $s(p)$ gives
$s(k_{0})\binom{pn_{1}+n_{0}}{pk_{1}+k_{0}}_{s}\equiv
s(n_{0}-k_{0}+1)\binom{pn_{1}+n_{0}}{pk_{1}+k_{0}-1}\pmod{s(p)}.$
So
$\displaystyle s(k_{0})\binom{pn_{1}+n_{0}}{pk_{1}+k_{0}}_{s}$
$\displaystyle\equiv s(n_{0}-k_{0}+1)\binom{pn_{1}+n_{0}}{pk_{1}+k_{0}-1}$
$\displaystyle\equiv\binom{n_{1}}{k_{1}}s(n_{0}-k_{0}+1)\binom{n_{0}}{k_{0}-1}_{s}$
$\displaystyle=\binom{n_{1}}{k_{1}}s(k_{0})\binom{n_{0}}{k_{0}}_{s}\pmod{s(p)}.$
Since $0<k_{0}<p$, $s(k_{0})$ is a unit modulo $s(p)$, so we can cancel it
from both sides, and we get the congruence we wanted. Therefore, the induction
proof is complete. ∎
When the ring $R$ is $\mathbf{Z}$, there is a version of the $s$-Lucas theorem
which allows $p$ to be any natural number, not just a prime; see Theorem
3.4.4.
### 2.5. The $s$-Vandermonde identity
###### Recollection 2.5.1.
The classical Vandermonde identity states that
$\binom{m+n}{k}=\sum_{j}\binom{m}{k-j}\binom{n}{j};$
this admits a $q$-analogue, known as the the $q$-Vandermonde identity:
$\binom{m+n}{k}_{q}=\sum_{j}\binom{m}{k-j}_{q}\binom{n}{j}_{q}q^{j(m-k+j)}.$
To state an $s$-analogue of the Vandermonde identity, we need a definition.
###### Definition 2.5.2.
For $m,n,k,j\in\mathbf{Z}$ with $0\leq j\leq n$ and $0\leq k-j\leq m$, define
$\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}=\sum_{\begin{subarray}{c}I\subseteq(m,m+n]\cap\mathbf{Z}\\\
|I|=j\end{subarray}}\left(\prod_{\ell=1}^{j}c_{s}(i_{\ell},n-(k-j+\ell))\right),$
where $I=\\{i_{1}<\cdots<i_{j}\\}$. We also set
$\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}=0$ for
$j<0$ or $j>n$.
###### Remark 2.5.3.
For $s(n)=n$, the product inside the sum in Definition 2.5.2 is $1$, so the
sum is just the number of subsets of $(m,m+n]\cap\mathbf{Z}$ of size $j$,
which is $\binom{n}{j}$. In that sense,
$\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}$ is an
$s$-analogue of the subset-counting definition of $\binom{n}{j}$ which depends
on two extra parameters $m$ and $k$. The other type of $s$-binomial
coefficient $\binom{n}{j}_{s}$ is an $s$-analogue of the algebraic definition
of $\binom{n}{j}$.
###### Theorem 2.5.4 ($s$-Vandermonde identity).
For all $m,n,k\in\mathbf{Z}_{\geq 0}$ with $k\leq m+n$,
$\binom{m+n}{k}_{s}=\sum_{j}\binom{m}{k-j}_{s}\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s},$
where the sum is taken over all $j\in\mathbf{Z}$ such that $0\leq j\leq n$ and
$0\leq k-j\leq m$.
###### Question 2.5.5.
As explained in [Sas18], the $q$-Vandermonde identity for $\binom{m+n}{k}_{q}$
can be understood as arising via a motivic cellular decomposition of the
Grassmannian $\mathrm{Gr}_{k}(\mathbf{C}^{m+n})$. Is there a motivic
interpretation of Theorem 2.5.4? (A similar question can also be asked for the
$s$-analogues of the other combinatorial identities proved elsewhere in this
article.)
The proof of Theorem 2.5.4 requires a preliminary lemma, which can be viewed
as an analogue of Pascal’s identity for
$\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}$.
###### Lemma 2.5.6.
For all $m,n,k,j\in\mathbf{Z}$ such that $0\leq k-j\leq m$,
$\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}=\genfrac{(}{|}{0.0pt}{}{n-1}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}+c_{s}(m+n,m+n-k)\genfrac{(}{|}{0.0pt}{}{n-1}{j-1}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k-1}_{s}.$
Notice that this includes the cases $j<0$ and $j>n$.
###### Proof.
We will split up the sum in the definition of
$\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}$ into
two sums, based on whether $I$ contains $m+n$ or not. If $m+n\notin I$, then
$I$ ranges over all $j$-element subsets of $(m,m+n-1]\cap\mathbf{Z}$. If
$m+n\in I$, then we remove it, and we get a $(j-1)$-element subset of
$(m,m+n-1]\cap\mathbf{Z}$. Then, we have to pull out the $\ell=j$ term of the
product, since $i_{j}=m+n$. So we have
$\displaystyle\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}$
$\displaystyle=\sum_{\begin{subarray}{c}I\subseteq(m,m+n]\cap\mathbf{Z}\\\
|I|=j\end{subarray}}\left(\prod_{\ell=1}^{j}c_{s}(i_{\ell},i_{\ell}-(k-j+\ell))\right)$
$\displaystyle=\sum_{\begin{subarray}{c}I\subseteq(m,m+n]\cap\mathbf{Z}\\\
|I|=j\\\ m+n\in
I\end{subarray}}\left(\prod_{\ell=1}^{j}c_{s}(i_{\ell},i_{\ell}-(k-j+\ell))\right)+\sum_{\begin{subarray}{c}I\subseteq(m,m+n-1]\cap\mathbf{Z}\\\
|I|=j\end{subarray}}\left(\prod_{\ell=1}^{j}c_{s}(i_{\ell},i_{\ell}-(k-j+\ell))\right)$
$\displaystyle=\sum_{\begin{subarray}{c}I\subseteq(m,m+n-1]\cap\mathbf{Z}\\\
|I|=j-1\end{subarray}}c_{s}(m+n,m+n-(k-j+j))\left(\prod_{\ell=1}^{j-1}c_{s}(i_{\ell},i_{\ell}-(k-j+\ell))\right)+\genfrac{(}{|}{0.0pt}{}{n-1}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}$
$\displaystyle=c_{s}(m+n,m+n-k)\genfrac{(}{|}{0.0pt}{}{n-1}{j-1}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k-1}_{s}+\genfrac{(}{|}{0.0pt}{}{n-1}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}.$
Rearranging this completes the proof. ∎
###### Proof of Theorem 2.5.4.
We will prove this by induction on $n$. For the base case, observe that if
$n=0$, then all $j$ in the range of summation satisfy $0\leq j\leq 0$, so
either the sum is empty (if $k-0>m$) or its only term is $j=0$ (if $k-0\leq
m$). The conditions of the theorem force $k\leq m+n=m$, so the sum has exactly
one term:
$\sum_{j}\binom{m}{k-j}_{s}\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}=\binom{m}{k}_{s}\genfrac{(}{|}{0.0pt}{}{0}{0}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}.$
We want to show that this is $\binom{m}{k}_{s}$. The definition of
$\genfrac{(}{|}{0.0pt}{}{0}{0}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}$ gives
the rather degenerate identity
$\genfrac{(}{|}{0.0pt}{}{0}{0}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}=\sum_{\begin{subarray}{c}I\subseteq(m,m]\cap\mathbf{Z}\\\
|I|=0\end{subarray}}\left(\prod_{\ell=1}^{0}c_{s}(i_{\ell},i_{\ell}-(k+\ell))\right)=\sum_{I\subseteq\emptyset}1=1,$
which completes the base case.
For the inductive step, suppose that the theorem is true with $n$ replaced by
$n-1$. We will first apply a version of the $s$-Pascal identity to
$\binom{m+n}{k}_{s}$ that has been “flipped” using the identity
$\binom{n}{k}_{s}=\binom{n}{n-k}_{s}$:
$\displaystyle\binom{m+n}{k}_{s}$ $\displaystyle=\binom{m+n}{m+n-k}_{s}$
$\displaystyle=\binom{m+n-1}{m+n-k-1}_{s}+c_{s}(m+n,m+n-k)\binom{m+n-1}{m+n-k}_{s}$
$\displaystyle=\binom{m+n-1}{k}_{s}+c_{s}(m+n,m+n-k)\binom{m+n-1}{k-1}_{s}.$
The inductive hypothesis gives:
$\displaystyle\binom{m+n-1}{k}_{s}+c_{s}(m+n,m+n-k)\binom{m+n-1}{k-1}_{s}$
$\displaystyle=\sum_{j}\binom{m}{k-j}_{s}\genfrac{(}{|}{0.0pt}{}{n-1}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}+c_{s}(m+n,m+n-k)\sum_{j}\binom{m}{k-1-j}_{s}\genfrac{(}{|}{0.0pt}{}{n-1}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k-1}_{s}$
$\displaystyle=\sum_{j}\binom{m}{k-j}_{s}\genfrac{(}{|}{0.0pt}{}{n-1}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}+c_{s}(m+n,m+n-k)\sum_{j}\binom{m}{k-j}_{s}\genfrac{(}{|}{0.0pt}{}{n-1}{j-1}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k-1}_{s}$
$\displaystyle=\sum_{j}\binom{m}{k-j}_{s}\left(\genfrac{(}{|}{0.0pt}{}{n-1}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s}+c_{s}(m+n,m+n-k)\genfrac{(}{|}{0.0pt}{}{n-1}{j-1}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k-1}_{s}\right).$
By Lemma 2.5.6, this is equal to
$\sum_{j}\binom{m}{k-j}_{s}\genfrac{(}{|}{0.0pt}{}{n}{j}\\!\\!\genfrac{|}{)}{0.0pt}{}{m}{k}_{s},$
which completes the induction. ∎
## 3\. Generalized $n$-Series Over $\mathbf{Z}$
### 3.1. Lexicographically small nonnegative integer generalized $n$-series
When doing computations with generalized $n$-series, it is useful to have some
examples over $\mathbf{Z}$ that are lexicographically small (close to $0$ for
small values of $n$). We will restrict to integer GNS that are always
nonnegative. We have already seen some examples of relatively small integer
generalized $n$-series: $s(n)=n$ and $s(n)=[n]_{q}$. Before looking at an
algorithm for constructing lexicographically small GNS, we need a definition:
###### Definition 3.1.1.
Let $R$ be a ring, and let $N\in\mathbf{Z}_{\geq 0}$. A partial generalized
$n$-series is a function $s:\\{0,1,\dots,N\\}\to R$ such that $s(0)=0$, $s(n)$
is not a zero-divisor for $0<n\leq N$, and for all $0\leq k\leq n\leq N$,
$s(n-k)\mid s(n)-s(k)$. An extension of a partial GNS $s$ is a GNS which
agrees with $s$ on the domain of $s$.
Given a partial GNS
$s:\\{0,1,\dots,N\\}\to\mathbf{Z}_{\geq 0}$
we can use a greedy algorithm to construct the lexicographically smallest
nonnegative GNS $\widetilde{s}$ which is an extension of $s$.
###### Definition 3.1.2.
Suppose we are given a partial GNS $s:\\{0,1,\dots,N\\}\to\mathbf{Z}_{\geq
0}$. Define a function $\widetilde{s}:\mathbf{Z}_{\geq 0}\to\mathbf{Z}_{\geq
0}$ in the following way: $\widetilde{s}(n)=s(n)$ for $0\leq n\leq N$, and for
each $n>N$ we define $\widetilde{s}(n)$ in terms of the previous values of
$\widetilde{s}(k)$ to be the smallest positive integer that makes the
restriction $\widetilde{s}|_{\\{0,1,\dots,n\\}}$ a partial GNS.
The fact that $\widetilde{s}$ is well-defined is nontrivial. We have to show
that at each step of the algorithm, $\widetilde{s}(n)$ exists. We will
actually prove a stronger theorem:
###### Theorem 3.1.3.
Consider the following recursive construction of an arbitrary nonnegative GNS
over $\mathbf{Z}$: Start with $s(0)=0$, and for each $N$ starting with $1$ in
increasing order, choose $s(N)$ to be an _arbitrary_ integer such that
$s|_{\\{0,1,\dots,N\\}}$ is a partial GNS. No matter what choices are made, it
is always possible to continue (there are never any contradictions).
###### Proof.
Suppose we have a partial GNS $s$ with domain $\\{0,1,\dots,N-1\\}$, and we
want to choose $s(N)$. We have to show that there exists $s(N)$ such that for
all $0<k<N$, $s(N-k)\mid s(N)-s(k)$. This is equivalent to $s(N)\equiv
s(k)\pmod{s(N-k)}$, so we really have a system of linear congruences. This
system has a solution by the generalized Chinese Remainder Theorem as long as
no two congruences contradict each other. We want to show that if we reduce
two of the congruences $s(N)\equiv s(N-k)\pmod{s(k)}$ and $s(N)\equiv
s(N-j)\pmod{s(j)}$ modulo $\gcd(s(k),s(j))$, they become the same congruence.
That is, we want to show that
$s(N-k)\equiv s(N-j)\pmod{\gcd(s(k),s(j))}.$
By Lemma 2.2.6, $\gcd(s(k),s(j))=\pm s(\gcd(k,j))$. And since
$N-k\equiv N\equiv N-j\pmod{\gcd(k,j)},$
the desired congruence must be true by Lemma 2.2.4. ∎
We have seen that many number-theoretic properties of the positive integers
remain true for arbitrary GNS. However, there are some important differences
between $s(n)=n$ and other GNS. For example, if $n=p^{j}m$ with $p\nmid m$,
then $n/p^{j}$ is a unit modulo $p$. This is generally false for other
generalized $n$-series; using Theorem 3.1.3, we can construct a
counterexample.
###### Example 3.1.4.
Let $s$ be an extension of the partial GNS with domain $\\{0,1,\dots,6\\}$
whose values are $0,1,2,3,10,11,12$. One can check manually that this forms a
partial GNS. For this GNS,
$\frac{s(6)}{s(2)}=\frac{12}{2}=6\equiv 0\pmod{s(2)}.$
### 3.2. An upper bound on lexicographically small nonnegative integer GNS
In this section, we will write partial generalized $n$-series as lists of
numbers, where the first item in the list is $s(0)$. For example:
###### Example 3.2.1.
Define a partial generalized $n$-series $0,1,3$ via the function
$s:\\{0,1,2\\}\to\mathbf{Z}_{\geq 0}$ given by $0\mapsto 0$, $1\mapsto 1$,
$2\mapsto 3$. If we apply the algorithm of Definition 3.1.2 to the partial
generalized $n$-series $0,1,k$ where $k\in\mathbf{Z}_{>0}$, we get a
generalized $n$-series $0,1,k,1,k,\dots$, where the $n$-th term is $0$ if
$n=0$, $1$ if $n$ is odd, and $k$ if $n$ is even and nonzero. If we start with
a longer partial generalized $n$-series $s$, say $0,1,3,4$, then the
generalized $n$-series $\widetilde{s}$ starts with
$0,1,3,4,9,19,552,22081,$
and the next few terms are
$219440979,2669857856653708,6558922971496604200448626056129.$
This seems to grow very fast when $n$ increases, almost doubling the number of
digits each term. As we will see, $\widetilde{s}(n)$ is bounded by
$\displaystyle\widetilde{s}(n)$ $\displaystyle<12^{2^{n-4}},\qquad n\geq
4,\text{ and}$ $\displaystyle\widetilde{s}(n)$
$\displaystyle=\Omega_{a}(a^{n}),\qquad\forall a\geq 0.$
The upper bound (a special case of Theorem 3.2.5) is proved below, and the
lower bound (a special case of Theorem 3.3.1) is proved in the next
subsection.
Let us first show that $\widetilde{s}$ is strictly increasing.
###### Lemma 3.2.2.
Let $N\geq 3$, and let $s:\\{0,1,\dots,N\\}\to\mathbf{Z}_{\geq 0}$ be a
strictly increasing partial generalized $n$-series. Then, $\widetilde{s}$ is
also strictly increasing.
###### Remark 3.2.3.
The condition $N\geq 3$ is important, because we already saw that if $s=0,1,3$
(so $N=2$) then $\widetilde{s}=0,1,3,1,3,\dots$, which is not strictly
increasing.
###### Proof.
Notice that $s(2)>s(1)\geq 1$. Let $n>N$. We will show that
$\widetilde{s}(n)>\widetilde{s}(n-1)$. By Lemma 2.2.4, $\widetilde{s}(n)\equiv
s(1)\pmod{s(n-1)}$, so either $\widetilde{s}(n)=s(1)$ or
$\widetilde{s}(n)>s(n-1)$. The first case cannot happen because
$\widetilde{s}(n)\equiv s(2)\pmod{\widetilde{s}(n-2)}$, and
$\widetilde{s}(n-2)\geq s(2)>s(1)$. Therefore,
$\widetilde{s}(n)>\widetilde{s}(n-1)$. ∎
Given the value of $\widetilde{s}(k)$ for all $1\leq k<n$, there is an upper
bound on $\widetilde{s}(n)$, as long as $\widetilde{s}$ is strictly
increasing:
###### Lemma 3.2.4.
Let $s:\\{0,1,\dots,N\\}\to\mathbf{Z}$ be a strictly increasing partial
generalized $n$-series with $N\geq 3$. Then, for all $n>N$,
$\widetilde{s}(n)<\mathrm{lcm}\\{\widetilde{s}(k)\mid 0<k<n\\}.$
###### Proof.
Recall that the proof of the existence of $\widetilde{s}$ (Theorem 3.1.3)
constructs $\widetilde{s}(n)$ from $\widetilde{s}(k)$ for all $0<k<n$ using
the generalized Chinese Remainder Theorem. The congruences are
$x\equiv\widetilde{s}(n-k)\pmod{\widetilde{s}(k)}$ for each $0<k<n$, so the
generalized CRT proves the existence of a unique solution $x$ modulo
$L=\mathrm{lcm}\\{\widetilde{s}(k)\mid 0<k<n\\}$. Therefore, there exists a
solution to the system of congruences with $0<x\leq L$. The smallest positive
solution is $\widetilde{s}(n)$ by Definition 3.1.2, so $\widetilde{s}(n)\leq
L$. We just have to show that $\widetilde{s}(n)\neq L$.
Suppose for contradiction that $\widetilde{s}(n)=L$. Then, by Lemma 2.2.4,
$L\equiv\widetilde{s}(1)\pmod{\widetilde{s}(n-1)}$. But by the definition of
$L$, $L$ is divisible by $\widetilde{s}(n-1)$, so $\widetilde{s}(1)$ is also
divisible by $\widetilde{s}(n-1)$. Since $n-1>N-1>1$,
$\widetilde{s}(1)<\widetilde{s}(n-1)$ by Lemma 3.2.2, which is a
contradiction. ∎
Using the fact that the LCM of a set is at most its product, we can prove the
following upper bound:
###### Theorem 3.2.5.
Let $s:\\{0,1,\dots,N\\}\to\mathbf{Z}$ be a strictly increasing partial
generalized $n$-series with $N\geq 3$. Then, for all $n>N$,
$\widetilde{s}(n)<\Pi^{2^{n-(N+1)}},$
where $\Pi=\prod_{k=1}^{N}s(k)$.
###### Proof.
We will use strong induction. Suppose that $n>N$ and for all $k$ strictly
between $N$ and $n$,
$\widetilde{s}(k)<\Pi^{2^{k-(N+1)}}.$
By Lemma 3.2.4 and the inductive hypothesis,
$\displaystyle\widetilde{s}(n)$
$\displaystyle<\mathrm{lcm}\\{\widetilde{s}(k)\mid
0<k<n\\}\leq\prod_{k=1}^{n-1}\widetilde{s}(k)$
$\displaystyle\leq\left(\prod_{k=1}^{N}s(k)\right)\left(\prod_{k=N+1}^{n-1}\Pi^{2^{k-(N+1)}}\right)$
$\displaystyle=\Pi\cdot\prod_{k=0}^{n-(N+2)}\Pi^{2^{k}}=\Pi\cdot\Pi^{\left(\sum_{k=0}^{n-(N+2)}2^{k}\right)}$
$\displaystyle=\Pi\cdot\Pi^{2^{n-(N+1)}-1}=\Pi^{2^{n-(N+1)}}.$
This completes the induction. ∎
###### Remark 3.2.6.
Another way to think about this theorem is that the recursively defined
sequence
$a_{n}=\begin{cases}s(n)&\text{if }0\leq n\leq N,\\\
\prod_{k=1}^{n-1}a_{k}&\text{if }n>N\end{cases}$
is exactly equal to $\Pi^{2^{n-(N+1)}}$ for all $n>N$, and we know that
$\widetilde{s}(n)$ satisfies this recurrence but with the equality replaced by
$<$ whenever $n>N$, so it should be true that $\widetilde{s}(n)<a_{n}$ for all
$n>N$.
### 3.3. A lower bound on strictly increasing integer GNS
We proved an upper bound on the lexicographically smallest extension of the
partial GNS $0,1,3,4$ in the previous section, but we also stated a lower
bound. The purpose of this subsection is to prove this lower bound by proving
a more general result.
###### Theorem 3.3.1.
Let $s(n)$ be a strictly increasing generalized $n$-series over $\mathbf{Z}$
that is not a scalar multiple of $n\mapsto[n]_{q}$ for any
$q\in\mathbf{Z}_{>0}$. Then, $s(n)=\Omega_{a}(a^{n})$ for all $a\geq 0$.
In this theorem statement, we use the convention that $[n]_{1}=n$. To prove
Theorem 3.3.1, we will use the following lemma:
###### Lemma 3.3.2.
Let $s$ satisfy the conditions of Theorem 3.3.1, and fix a nonnegative integer
$a$. Then, for all sufficiently large $n$,
$s(n+1)\neq as(n)+s(1).$
###### Proof.
Let $k\in\mathbf{Z}_{\geq 0}$ such that $s(k+1)\neq as(k)+s(1)$. Such a $k$
must exist, because otherwise $s(n+1)=as(n)+s(1)$ for all
$n\in\mathbf{Z}_{\geq 0}$. This would imply by induction that
$s(n)=s(1)[n]_{q}$ with $q=a$ for all $n$, which contradicts our assumption
about $s$. Next, choose an integer $N$ large enough so that
$s(N-k)>\max\\{s(k+1),as(k)+s(1)\\}.$
This is always possible, because $s:\mathbf{Z}_{\geq 0}\to\mathbf{Z}$ is
strictly increasing and therefore unbounded.
To prove the lemma, we will show that for all $n\geq N$, we have $s(n+1)\neq
as(n)+s(1)$. Suppose for contradiction that there exists $n\geq N$ with
$s(n+1)=as(n)+s(1).$
Taking this equation modulo $s(n-k)$ and using Lemma 2.2.4 gives
$s(k+1)\equiv as(k)+s(1)\pmod{s(n-k)}.$
But $n\geq N$, so
$s(n-k)\geq s(N-k)>\max\\{s(k+1),as(k)+s(1)\\},$
so the modulus is greater than both sides of the congruence, which means that
it is an equality. This contradicts the fact that $s(k+1)\neq as(k)+s(1)$. ∎
###### Proof of Theorem 3.3.1.
We want to show that $s(n)=\Omega_{a}(a^{n})$ for all $a\geq 0$. It suffices
to prove this for $a\in\mathbf{Z}_{\geq 0}$. For each integer $b$ with
$0<b<a$, apply Lemma 3.3.2 with $a$ replaced by $b$. This gives an integer
$N_{b}$ such that for all $n\geq N_{b}$,
$s(n+1)\neq bs(n)+s(1).$
Let
$N=\max\\{N_{b}\mid 0<b<a\\}.$
We claim that for all $n\geq N$, we have $s(n+1)\geq as(n)+s(1)$.
To see this, let $n\geq N$. By Lemma 2.2.4,
$s(n+1)\equiv s(1)\pmod{s(n)},$
so $s(n+1)=bs(n)+s(1)$ for some $b\in\mathbf{Z}$. Since $s$ is strictly
increasing, $b$ must be positive. We want to show that $b\geq a$. If $b<a$,
then we defined $N_{b}$ above, and $n\geq N\geq N_{b}$. So $s(n+1)\neq
bs(n)+s(1)$, which is a contradiction. Therefore, $b\geq a$, so
$s(n+1)\geq as(n)+s(1).$
To complete the proof, notice that the claim implies (by induction) that for
all $n\geq N$, $s(n)\geq a^{n-N}s(N)$, which means that
$s(n)=\Omega_{a}(a^{n})$ as $n\to\infty$. ∎
### 3.4. A more general $s$-Lucas theorem
The $s$-Lucas theorem (Proposition 2.4.8) is a congruence modulo $s(p)$, where
$s$ is a GNS and $p$ is prime. The $q$-Lucas theorem has a more general form
which allows $p$ to be composite:
$\binom{n_{1}p+n_{0}}{k_{1}p+k_{0}}_{q}\equiv\binom{n_{1}}{k_{1}}\binom{n_{0}}{k_{0}}_{q}\pmod{\Phi_{p}(q)}.$
In this subsection, we will state and prove an $s$-analogue of the more
general $q$-Lucas theorem in the case $n_{0}=k_{0}=0$.
Recall the $s$-analogue $\Phi_{n}(s)$ of the cyclotomic polynomial from Remark
2.4.3.
###### Theorem 3.4.1.
If $s$ is a GNS over $\mathbf{Z}$, then $\Phi_{n}(s)\in\mathbf{Z}$ for all
$n>0$.
We will prove that $\Phi_{n}(s)\in\mathbf{Z}$ by showing that
$\Phi_{n}(s)=\frac{s(n)}{\mathrm{lcm}\\{s(n/p)\mid p\text{ prime factor of
}n\\}}.$
Here is a lemma:
###### Lemma 3.4.2.
Let $S$ be an arbitrary multiset of positive integers. Then,
$\mathrm{lcm}(S)=\prod_{\begin{subarray}{c}\text{\emph{multiset }}A\subseteq
S\\\ A\neq\emptyset\end{subarray}}\gcd(A)^{(-1)^{|A|-1}}.$
This lemma can be proved using the tools of elementary number theory. Notice
that when $s$ is a two-element multiset $\\{a,b\\}$, this formula reduces to
$\mathrm{lcm}(a,b)=ab/\gcd(a,b)$.
###### Proof of Lemma 3.4.2.
We can decompose the right-hand side into its prime-power factors. Let $p$ be
a prime, and let $T$ be the multiset $\\{v_{p}(a)\mid a\in S\\}$, where
$v_{p}(a)$ is the exponent of $p$ in the prime factorization of $a$. The
exponent of $p$ in the right-hand side of the equality we are trying to prove
is
$\sum_{\begin{subarray}{c}\text{multiset }B\subseteq T\\\
B\neq\emptyset\end{subarray}}(-1)^{|B|-1}\mathrm{min}(B),$
because a GCD of powers of $p$ is $p$ to the power of the minimum exponent. We
want to show that this is equal to $v_{p}(\mathrm{lcm}(S))$, which is
$\max(T)$. Write $T=\\{a_{1},a_{2},\dots,a_{k}\\}$, where $k=|T|$ and
$a_{1}\leq a_{2}\leq\cdots\leq a_{n}$. We will count the number of times each
$a_{j}$ is counted in the sum. Since $\max(T)=a_{n}$, we have to show that
$a_{n}$ is counted once and $a_{j}$ is counted $0$ times for all $1\leq j<n$.
Let $1\leq j\leq n$ and let $1\leq k\leq n$. We want to count how many subsets
$B\subseteq T$ have minimum $j$ and cardinality $n$. Such a subset must
include $a_{j}$, but can have any combination of $k-1$ elements $a_{\ell}$
with $\ell>j$. There are $\binom{n-j}{k-1}$ choices for these remaining
elements. Therefore, the number of times $a_{j}$ is counted in the sum is
$\sum_{k=1}^{n}(-1)^{k-1}\binom{n-j}{k-1}=\sum_{k=0}^{n-1}(-1)^{k}\binom{n-j}{k}.$
Since $\binom{n-j}{k}$ is zero for $k>n-j$, this is just the alternating sum
of row $n-j$ of Pascal’s triangle, which is $0$ if $n-j>0$ and $1$ if $n-j=0$.
Therefore, $a_{j}$ is counted $0$ times for $j<n$ and $1$ time for $j=n$ in
the sum
$\sum_{\begin{subarray}{c}B\subseteq P\\\
B\neq\emptyset\end{subarray}}(-1)^{|B|-1}\mathrm{min}(B),$
so the sum is equal to $a_{n}$. This means that the exponents of $p$ in the
left-hand and right-hand sides of the equality in the theorem statement are
both $a_{n}$. Combining this fact for each prime $p$ proves the lemma. ∎
###### Proof of Theorem 3.4.1.
Let $P$ be the set of prime factors of $n$, and let
$r(n)=\frac{s(n)}{\mathrm{lcm}\\{s(n/p)\mid p\in P\\}}.$
Notice that $r(n)\in\mathbf{Z}$ because Lemma 2.2.3 implies that $s(n/p)\mid
s(n)$ for all prime factors $p$ of $n$. We want to show that
$\Phi_{n}(s)=r(n)$.
By the definition of $r(n)$,
$\frac{s(n)}{r(n)}=\mathrm{lcm}\left\\{s\left(\frac{n}{p}\right)\;\middle|\;p\in
P\right\\}.$
By Lemma 3.4.2 with $S=\\{s(n/p)\mid p\in P\\}$, this is equal to
$\prod_{\begin{subarray}{c}A\subseteq P\\\
A\neq\emptyset\end{subarray}}\gcd\left\\{s\left(\frac{n}{p}\right)\;\middle|\;p\in
A\right\\}^{(-1)^{|A|-1}}.$
Lemma 2.2.6 and the fact that $s$ is nonnegative imply that $s$ preserves
GCDs, so
$\gcd\left\\{s\left(\frac{n}{p}\right)\;\middle|\;p\in
A\right\\}=s\left(\gcd\left\\{\frac{n}{p}\;\middle|\;p\in
A\right\\}\right)=s\left(\frac{n}{\mathrm{lcm}(A)}\right).$
This implies that
$\frac{s(n)}{r(n)}=\prod_{\begin{subarray}{c}A\subseteq P\\\
A\neq\emptyset\end{subarray}}s\left(\frac{n}{\mathrm{lcm}(A)}\right)^{(-1)^{|A|-1}}.$
Dividing both sides by $s(n)$ and taking the reciprocal of both sides gives
$r(n)=\prod_{A\subseteq
P}s\left(\frac{n}{\mathrm{lcm}(A)}\right)^{(-1)^{|A|}}.$
(On the right-hand side, we removed the condition $A\neq\emptyset$ and
multiplied the exponent by $-1$.)
We have to count how many ways each divisor $d$ of $n$ can be written as
$\mathrm{lcm}(A)$ for $A\subseteq P$. Since the elements of $A$ are distinct
primes, $\mathrm{lcm}(A)$ is the product of the elements of $A$. Therefore,
$d=\mathrm{lcm}(A)$ is square-free, and given $d$ there is a unique choice of
$A$ (the set of prime factors of $d$). So we get
$r(d)=\prod_{\begin{subarray}{c}d\mid n\\\ d\text{ square-
free}\end{subarray}}s\left(\frac{n}{d}\right)^{\mu(d)},$
using the fact that $\mu(d)$ is $(-1)^{|A|}$ if $A$ is the set of prime
factors of a square-free number $d$. Since $\mu(d)=0$ for $d$ not square-free,
we can remove the restriction that $d$ is square-free:
$r(n)=\prod_{d\mid n}s\left(\frac{n}{d}\right)^{\mu(d)}=\prod_{d\mid
n}s(d)^{\mu(n/d)}=\Phi_{n}(s).$
This is what we wanted to show. ∎
Before proving the $s$-Lucas theorem for composite $p$, we will prove a lemma
about $\Phi_{n}(s)$.
###### Lemma 3.4.3.
Let $s$ be a nonnegative integer GNS, let $m\in\mathbf{Z}_{>0}$, and let
$a,b\in\mathbf{Z}_{\geq 0}$ such that
$a\equiv b\not\equiv 0\pmod{m}.$
Define
$d=\gcd(a,m)=\gcd(b,m).$
Then,
$\frac{s(a)}{s(d)}\equiv\frac{s(b)}{s(d)}\pmod{\Phi_{m}(s)}.$
Additionally, both sides of this congruence are units modulo $\Phi_{m}(s)$.
Equivalently, the function $\mathbf{Z}_{\geq 0}\to\mathbf{Z}_{\geq 0}$ defined
by $a\mapsto s(a)/s(\gcd(a,m))$ induces a well-defined function
$(\mathbf{Z}/m)\setminus\\{0\\}\to(\mathbf{Z}/\Phi_{m}(s))^{\times}.$
###### Proof.
By Lemma 2.2.4,
$s(a)\equiv s(b)\pmod{s(m)}.$
Since $\gcd(s(m),s(d))=s(d)$, this implies that
$\frac{s(a)}{s(d)}\equiv\frac{s(b)}{s(d)}\quad\left(\\!\\!\\!\\!\\!\\!\mod{\frac{s(m)}{s(d)}}\right).$
Notice that
$\displaystyle\gcd\left\\{\frac{s(m)}{s(d)}\;\middle|\;d\text{ proper divisor
of }m\right\\}$ $\displaystyle=\frac{s(m)}{\mathrm{lcm}\\{s(d)\mid d\text{
proper divisor of }m\\}}$ $\displaystyle=\frac{s(m)}{\mathrm{lcm}\\{s(m/p)\mid
p\text{ prime factor of }m\\}}$ $\displaystyle=\Phi_{m}(s).$
In the second step of this chain of equalities, we used the fact that for
every proper divisor $d$ of $m$ there exists a prime factor $p$ of $m$ such
that $d\mid m/p$. We also used Lemma 2.2.3. We can conclude that
$\Phi_{m}(s)\mid s(m)/s(d)$ for all proper divisors $d$ of $m$. In particular,
for $d=\gcd(a,m)=\gcd(b,m)$, we have
$\frac{s(a)}{s(d)}\equiv\frac{s(b)}{s(d)}\pmod{\Phi_{m}(s)}.$
To complete the proof, we just have to show that $s(a)/s(d)$ is a unit modulo
$\Phi_{m}(s)$. This is true because $\Phi_{m}(s)\mid s(m)/s(d)$ implies that
$\gcd\left(\Phi_{m}(s),\frac{s(a)}{s(d)}\right)\mid\gcd\left(\frac{s(m)}{s(d)},\frac{s(a)}{s(d)}\right)=\frac{\gcd(s(m),s(a))}{s(d)}=\frac{s(d)}{s(d)}=1.\qed$
###### Theorem 3.4.4 (“Composite version” of the $s$-Lucas theorem).
Let $m\in\mathbf{Z}_{>0}$ and let $n,k\in\mathbf{Z}_{\geq 0}$. Then,
$\binom{nm}{km}_{s}\equiv\binom{n}{k}_{s_{m}}\pmod{\Phi_{m}(s)}.$
###### Proof.
By definition,
$\binom{nm}{km}_{s}=\frac{s(nm)s(nm-1)\cdots s(nm-km+1)}{s(km)s(km-1)\cdots
s(1)}=\frac{\prod_{j=1}^{km}s(nm-km+j)}{\prod_{j=1}^{km}s(j)}.$
Our goal will be to cancel each $s(nm-km+j)$ with the corresponding $s(j)$
modulo $\Phi_{m}(s)$. The problem is that in general, these are not units
modulo $\Phi_{m}(s)$. To fix this, we will divide $s(nm-km+j)$ and $s(j)$ by
$s(\gcd(j,m))$:
$\binom{nm}{km}_{s}=\frac{\prod_{j=1}^{km}s(nm-
km+j)/s(\gcd(j,m))}{\prod_{j=1}^{km}s(j)/s(\gcd(j,m))}.$
Since $j\equiv nm-km+j\pmod{m}$, we can apply Lemma 3.4.3 for all $j$ not
divisible by $m$, and we get that
$\frac{s(nm-km+j)}{s(\gcd(j,m))}\quad\text{and}\quad\frac{s(j)}{s(\gcd(j,m))}$
are congruent units modulo $\Phi_{m}(s)$.
Therefore, we can cancel $s(nm-km+j)/s(\gcd(j,m))$ with $s(j)/s(\gcd(j,m))$
modulo $\Phi_{m}(s)$ for every $j$ not divisible by $m$. We get
$\displaystyle\binom{nm}{km}_{s}$ $\displaystyle=\frac{\prod_{j=1}^{km}s(nm-
km+j)/s(\gcd(j,m))}{\prod_{j=1}^{km}s(j)/s(\gcd(j,m))}$
$\displaystyle\equiv\frac{\prod_{\ell=1}^{k}s((n-k+\ell)m)/s(m)}{\prod_{\ell=1}^{k}s(\ell
m)/s(m)}$
$\displaystyle=\frac{\prod_{\ell=1}^{k}s((n-k+\ell)m)}{\prod_{\ell=1}^{k}s(\ell
m)}$ $\displaystyle=\binom{n}{k}_{s_{m}}\pmod{\Phi_{m}(s)}.$
We changed product indices from $j$ to $\ell=j/m$, because the terms with
$m\nmid j$ were canceled in the second step. ∎
###### Remark 3.4.5.
When $m$ is prime, Theorem 3.4.4 reduces to the case $n_{0}=k_{0}=0$ of the
$s$-Lucas theorem (Proposition 2.4.8).
## 4\. Generalized $n$-Series and de Rham Complexes
### 4.1. Basic properties of the $s$-de Rham complex
In this subsection, we will define the “$s$-de Rham complex” using the
$s$-derivative of Definition 2.3.3. Recall that this is the $R$-linear map
$\nabla_{s}:R[x]\to R[x]$ given on monomials by
$\nabla_{s}(x^{n})=s(n)x^{n-1}$. Write
$\mathbf{A}^{1}=\operatorname{Spec}R[x]$ to denote the affine line over $R$.
###### Definition 4.1.1.
The $s$-de Rham complex for $R[x]$ is the $2$-term complex
$s\Omega_{\square,\mathbf{A}^{1}}:=(R[x]\xrightarrow{{\nabla_{s}}}R[x]dx).$
Here, the square indicates the dependence of
$s\Omega_{\square,\mathbf{A}^{1}}$ on the choice of coordinate $x$.
###### Remark 4.1.2.
It is easy to generalize the $s$-de Rham complex to several variables (e.g.,
by defining $s\Omega_{\mathbf{A}^{n}}$ to be
$s\Omega_{\square,\mathbf{A}^{1}}^{\otimes_{R}n}$). Since proving
multivariable analogues of the results below is straightforward, we will only
study the case of a single variable.
###### Example 4.1.3.
Let $s:\mathbf{Z}_{\geq 0}\to\mathbf{Z}[\\![q-1]\\!]$ denote the $q$-integer
GNS from Example 2.1.6. Then Definition 4.1.1 is precisely the $q$-de Rham
complex of [Sch17].
The $s$-Pascal identity of Proposition 2.1.3 can be restated in terms of the
$s$-derivative:
###### Lemma 4.1.4.
There is an equality of operators:
$\left[\frac{\nabla_{s}^{k}}{k!_{s}},x\right]=\frac{\nabla_{s}^{k-1}}{(k-1)!_{s}}.$
###### Proof.
Let $n>k\geq 0$ be integers. Since $\nabla_{s}(x^{n})=s(n)x^{n-1}$, we have
$\nabla_{s}^{k}(x^{n})=s(n)s(n-1)\cdots
s(n-k+1)x^{n-k}=\frac{n!_{s}}{(n-k)!_{s}}x^{n-k}.$
This implies that
$\frac{\nabla_{s}^{k}}{k!_{s}}x^{n}=\binom{n}{k}_{s}x^{n-k}.$
The $s$-Pascal identity can therefore be stated as:
$\frac{\nabla_{s}^{k}}{k!_{s}}x^{n}=\frac{\nabla_{s}^{k-1}}{(k-1)!_{s}}x^{n-1}+x\frac{\nabla_{s}^{k}}{k!_{s}}x^{n-1}.$
Rearranging gives
$\frac{\nabla_{s}^{k}}{k!_{s}}x^{n}-x\frac{\nabla_{s}^{k}}{k!_{s}}x^{n-1}=\frac{\nabla_{s}^{k-1}}{(k-1)!_{s}}x^{n-1}.$
Recognizing the left side as a commutator of operators, this can be written as
$\left[\frac{\nabla_{s}^{k}}{k!_{s}},x\right]x^{n-1}=\frac{\nabla_{s}^{k-1}}{(k-1)!_{s}}x^{n-1}.$
This implies the desired equality of operators. ∎
###### Remark 4.1.5.
One can similarly restate the $s$-Lucas theorem (Proposition 2.4.8) via the
$s$-derivative: namely, if the hypotheses of Proposition 2.4.8 are satisfied,
then for any prime $p$ and any nonnegative integers $n_{1},n_{0},k_{1},k_{0}$
such that $n_{0},k_{0}<p$, we have
$\frac{\nabla_{s}^{k_{1}p+k_{0}}}{(k_{1}p+k_{0})!_{s}}(x^{n_{1}p+n_{0}})\equiv\frac{\partial_{x^{p}}^{k_{1}}}{k_{1}!}(x^{n_{1}p})\frac{\nabla_{s}^{k_{0}}}{k_{0}!_{s}}(x^{n_{0}})\pmod{s(p)}.$
###### Proposition 4.1.6 ($s$-product rule).
For a GNS $s$ over $R$, define an $R$-bilinear operator
$\star_{s}:R[x]\otimes_{R}R[x]\to R[x]$ on monomials by
$x^{a}\star_{s}x^{b}=c_{s}(a+b+1,a)x^{a+b}$
for all $a,b\in\mathbf{Z}_{\geq 0}$. Then, for all $f,g\in R[x]$,
$\nabla_{s}(fg)=\nabla_{s}(f)g+f\star_{s}\nabla_{s}(g).$
###### Proof.
Since $\nabla_{s}$ is $R$-linear and $\star_{s}$ is $R$-bilinear, the theorem
follows from the case where $f$ and $g$ are monomials $x^{a}$ and $x^{b}$. In
this case,
$\displaystyle\nabla_{s}(x^{a})x^{b}+x^{a}\star_{s}\nabla_{s}(x^{b})$
$\displaystyle=s(a)x^{a-1}x^{b}+x^{a}\star_{s}s(b)x^{b-1}$
$\displaystyle=s(a)x^{a+b-1}+s(b)c_{s}(a+b,a)x^{a+b-1}$
$\displaystyle=\left(s(a)+s(b)\cdot\frac{s(a+b)-s(a)}{s(b)}\right)x^{a+b-1}$
$\displaystyle=(s(a)+s(a+b)-s(a))x^{a+b-1}$ $\displaystyle=s(a+b)x^{a+b-1}$
$\displaystyle=\nabla_{s}(x^{a+b}).\qed$
###### Example 4.1.7.
Let $s:\mathbf{Z}_{\geq 0}\to\mathbf{Z}[\\![q-1]\\!]$ denote the $q$-integer
GNS from Example 2.1.6. Then
$c_{s}(a+b+1,a)=\frac{[a+b+1]_{q}-[a]_{q}}{[b+1]_{q}}=\frac{q^{a+b+1}-q^{a}}{q^{b+1}-1}=q^{a},$
so that $x^{a}\star_{s}x^{b}=q^{a}x^{a+b}=(qx)^{a}x^{b}$. In particular,
$f(x)\star_{s}\nabla_{s}(g(x))=f(qx)\nabla_{q}(g(x)),$
so that Proposition 4.1.6 reduces to the usual $q$-Leibniz rule.
###### Example 4.1.8.
One can check that the function $s:\mathbf{Z}_{\geq
0}\to\mathbf{Z}[\\![q-1]\\!]$ given by
$s(n)=\frac{1}{q-1}\frac{q^{n}-(2-q)^{n}}{q^{n}+(2-q)^{n}}\in\mathbf{Z}[\\![q-1]\\!]$
defines a GNS; see Example 4.3.2. It follows that
$c_{s}(a+b+1,a)=2q^{a}(2-q)^{a}\frac{q^{b+1}+(2-q)^{b+1}}{(q^{a+b+1}+(2-q)^{a+b+1})(q^{a}+(2-q)^{a})}.$
In particular, unlike for the $q$-integer GNS, there is no simple expression
for $f(x)\star_{s}g(x)$.
###### Corollary 4.1.9.
Fix a GNS $s$ over $R$. Then the complex $s\Omega_{\square,\mathbf{A}^{1}}$
naturally admits the structure of a (noncommutative) differential graded
$R$-algebra.
###### Proof.
Define a left and right $s\Omega_{\square,\mathbf{A}^{1}}^{0}=R[x]$-module
structure on $s\Omega_{\square,\mathbf{A}^{1}}^{1}$ as follows: the right
module structure is the obvious one, and the left module structure is given by
$g(x)\cdot f(x)dx=g\star_{s}f(x)dx$. Then the $s$-Leibniz rule of Proposition
4.1.6 produces a $s\Omega_{\square,\mathbf{A}^{1}}^{0}$-bimodule structure on
$s\Omega_{\square,\mathbf{A}^{1}}^{1}$ such that the $s$-derivative satisfies
the Leibniz rule; this is precisely the structure of a differential graded
$R$-algebra. ∎
### 4.2. $s$-analogues of the Poincaré lemma and Cartier isomorphism
The Poincaré lemma says that over a field $k$ of characteristic zero, the
cohomology of the de Rham complex is concentrated in degree zero (where it is
isomorphic to $k$). A version of this statement is also true over
$\mathbf{Z}$. Namely, if
$\mathbf{Z}\langle{x}\rangle=\mathbf{Z}[x,\frac{x^{n}}{n!}]_{n\geq 0}$ denotes
the divided power envelope of $\mathbf{Z}[x]$, then the cohomology of the
complex
$\Omega^{\bullet}_{\mathbf{Z}[x]/\mathbf{Z}}\otimes_{\mathbf{Z}[x]}\mathbf{Z}\langle{x}\rangle$
is concentrated in degree zero (where it is isomorphic to $\mathbf{Z}$). This
admits a straightforward generalization to the $s$-de Rham complex:
###### Proposition 4.2.1 ($s$-Poincaré lemma).
Let $s$ be a GNS over $R$, and let $R\langle{x}\rangle_{s}$ denote the ring
$R[x,\frac{x^{n}}{n!_{s}}]_{n\geq 0}$. Then the cohomology of the complex
$s\Omega_{\square,\mathbf{A}^{1}}\otimes_{R[x]}R\langle{x}\rangle_{s}$ is
concentrated in degree zero, where it is isomorphic to $R$.
###### Proof.
We need to show that the $R$-linear map
$R\langle{x}\rangle_{s}\xrightarrow{{\nabla_{s}}}R\langle{x}\rangle_{s}dx$
is surjective, and has kernel $R$. Surjectivity follows from the observation
that $\nabla_{s}\frac{x^{n}}{n!_{s}}=\frac{x^{n-1}}{(n-1)!_{s}}$; this also
implies that the kernel of $\nabla_{s}$ is precisely the $R$-submodule of
$R\langle{x}\rangle_{s}$ generated by the constants. ∎
###### Remark 4.2.2.
Note that the ring $R\langle{x}\rangle_{s}$ is nonzero, since the elements
$n!_{s}\in R$ are not zero-divisors; in fact, $R\langle{x}\rangle_{s}$ is a
subring of $R[1/s][x]$.
The $s$-de Rham complex also satisfies an analogue of the Cartier isomorphism.
###### Recollection 4.2.3.
The Cartier isomorphism says that if $A$ is a smooth $\mathbf{F}_{p}$-algebra,
there is a canonical isomorphism
$\Omega^{i}_{A/\mathbf{F}_{p}}\cong\mathrm{H}^{i}(\Omega^{\bullet}_{A/\mathbf{F}_{p}})$.
If $\varphi$ denotes the Frobenius on $R$, this isomorphism is roughly given
by “$\frac{\varphi}{p^{i}}$”. When $A=\mathbf{F}_{p}[x]$, one can interpret
the Cartier isomorphism as giving a canonical isomorphism
$\Omega^{i}_{\mathbf{Z}[x^{p}]/\mathbf{Z}}\otimes_{\mathbf{Z}}\mathbf{F}_{p}\cong\mathrm{H}^{i}(\Omega^{\bullet}_{\mathbf{Z}[x]/\mathbf{Z}}\otimes_{\mathbf{Z}}\mathbf{F}_{p})$
sending $d(x^{p})\mapsto[x^{p-1}dx]$. In [Sch17, Proposition 3.4], Scholze
proves a $q$-analogue of the Cartier isomorphism. Let $\mathbf{Z}[\zeta_{p}]$
denote the quotient $\mathbf{Z}[\\![q-1]\\!]/[p]_{q}$, so that $\zeta_{p}$
denotes a primitive $p$th root of unity. Then there is a canonical isomorphism
$\Omega^{i}_{\mathbf{Z}[x^{p}]/\mathbf{Z}}\otimes_{\mathbf{Z}}\mathbf{Z}[\zeta_{p}]\cong\mathrm{H}^{i}(q\Omega^{\bullet}_{\mathbf{Z}[x]/\mathbf{Z}}\otimes_{\mathbf{Z}[\\![q-1]\\!]}\mathbf{Z}[\zeta_{p}]).$
Both of these results admit an $s$-analogue.
###### Proposition 4.2.4.
Let $s$ be a GNS over $R$ such that $s(1)$ is a unit in $R/s(p)$. Then there
is a canonical isomorphism
$s\Omega_{\square,\mathbf{A}^{1}}^{i}\otimes_{R}R/s(p)\cong\mathrm{H}^{i}(s\Omega_{\square,\mathbf{A}^{1}}\otimes_{R}R/s(p))$
sending $x^{n}\mapsto x^{np}$ in degree zero and
$x^{n}dx\mapsto[x^{np}x^{p-1}dx]$.
###### Proof.
Let us first compute
$\mathrm{H}^{0}(s\Omega_{\square,\mathbf{A}^{1}}\otimes_{R}R/s(p))$, i.e., the
kernel of $\nabla_{s}$. Observe that if $a\in R$, then $ax^{n}\mapsto
as(n)x^{n-1}dx$. If $p\mid n$, then $s(p)\mid s(n)$, so that
$\nabla_{s}(ax^{n})=0\in R[x]/s(p)$. If $p\nmid n$, it follows from Lemma
2.2.5 that $\nabla_{s}(ax^{n})=0\in R[x]/s(p)$ if and only if $s(p)\mid a$.
This implies that
$\mathrm{H}^{0}(s\Omega_{\square,\mathbf{A}^{1}}\otimes_{R}R/s(p))\cong
R[x^{p}]/s(p)$.
To calculate
$\mathrm{H}^{1}(s\Omega_{\square,\mathbf{A}^{1}}\otimes_{R}R/s(p))$, i.e., the
cokernel of $\nabla_{s}$, we need to determine the image of $\nabla_{s}$. If
$ax^{n}dx$ is in the image of $\nabla_{s}$ for some $a\in R$, then there must
be some $b\in R$ such that $bs(n+1)=a$. If $p\mid n+1$, it follows that
$a=0\in R/s(p)$; if $p\nmid n+1$, then $s(n+1)$ is a unit (by the preceding
discussion), so that $b=\frac{a}{s(n+1)}$. It follows that the image of
$\nabla_{s}$ is precisely $\bigoplus_{p\nmid n+1}R/s(p)\\{x^{n}dx\\}$, so that
$\operatorname{coker}(\nabla_{s})=\bigoplus_{n\geq
1}R/s(p)\\{x^{np-1}dx\\}\cong\bigoplus_{n\geq 1}R/s(p)\\{x^{p(n-1)}\cdot
x^{p-1}dx\\}.$
This implies that
$\mathrm{H}^{1}(s\Omega_{\square,\mathbf{A}^{1}}\otimes_{R}R/s(p))\cong
R[x^{p}]/s(p)d(x^{p})$, as desired. ∎
In fact, the classical and $q$-Cartier isomorphisms are special cases of a
more general result due to Berthelot and Ogus [BO78]. Let us now review this
statement; we will then state and prove the analogue for generalized
$n$-series.
###### Recollection 4.2.5.
Let $R$ be a ring, and let $f\in A$ be a non-zero-divisor. If $M^{\bullet}$ is
a cochain complex of $R$-modules which is termwise $f$-torsionfree, the
décalage $\eta_{f}M^{\bullet}$ is the subcomplex of $M^{\bullet}[1/f]$ defined
via
$(\eta_{f}M)^{i}=\\{x\in f^{i}M^{i}\mid dx\in f^{i+1}M^{i+1}\\}.$
See [BMS18, Section 6] and [BO78]. One basic property of the décalage
construction is the following. Let $\mathrm{H}^{\bullet}(M/f)$ denote the
complex whose underlying graded abelian group is
$\bigoplus_{i\in\mathbf{Z}}\mathrm{H}^{i}(M/f)$, and where the differential is
given by the $f$-Bockstein
$\beta:\mathrm{H}^{i}(M/f)\to\mathrm{H}^{i+1}(M/f)$. Then there is a natural
isomorphism of complexes
(5) $\eta_{f}(M)/f\xrightarrow{{\sim}}\mathrm{H}^{\bullet}(M/f).$
We will only need the case when $M^{\bullet}$ is termwise $f$-torsionfree; but
let us mention that $\eta_{f}$ preserves quasi-isomorphisms, and one can
extend $\eta_{f}$ to a (non-exact) functor $L\eta_{f}:D(R)\to D(R)$ on the the
derived category of $R$.
Let $k$ be a perfect field of characteristic $p>0$. A very special case of a
result of Berthelot and Ogus (in [BO78]) says that if $W(k)$ is the ring of
Witt vectors of $k$ and $A$ is a $W(k)$-algebra, there is a Frobenius333The
existence of a Frobenius on $\Omega_{A}$ is not obvious in general, and
depends on the existence of crystalline cohomology. semilinear isomorphism
$\Omega_{A}\xrightarrow{{\sim}}L\eta_{p}\Omega_{A}$. Suppose for simplicity
that $\Omega_{A}$ is $p$-torsionfree; applying 5 with $f=p$ then defines an
isomorphism of complexes
(6)
$\eta_{p}(\Omega_{A})/p\xrightarrow{{\sim}}\mathrm{H}^{\bullet}(\Omega_{A}/p).$
Note that if we write $A_{0}=A/p$, then $\Omega_{A}/p\cong\Omega_{A_{0}/k}$.
The Frobenius semilinear isomorphism
$\Omega_{A}\xrightarrow{{\sim}}L\eta_{p}\Omega_{A}$ gives a Frobenius
semilinear equivalence
$\Omega_{A_{0}/k}\xrightarrow{{\sim}}\eta_{p}(\Omega_{A_{0}})/p$. Comparing
the left and right-hand sides of 6 recovers the Cartier isomorphism for
$A_{0}$.
A similar result was proved in [BS19, Theorem 1.16(4)] for the $q$-de Rham
complex: namely, if $A$ is a smooth $\mathbf{Z}_{p}[\zeta_{p}]$-algebra, then
there is a Frobenius semilinear equivalence
$q\Omega_{A}\xrightarrow{{\sim}}L\eta_{[p]_{q}}q\Omega_{A}$. As above, using 5
with $f=[p]_{q}$ recovers the $q$-analogue of the Cartier isomorphism.
As one might expect, there is a décalage result for the $s$-de Rham complex,
too:
###### Proposition 4.2.6.
Fix a GNS $s$ over $R$ such that:
1. (1)
there is a ring endomorphism $\varphi:R\to R$ which sends
$s(n)\mapsto\frac{s(np)}{s(p)}$.
2. (2)
$s(1)$ is a unit in $R/s(p)$, and $R$ is $s(p)$-adically complete.
Write $\mathbf{A}^{1,(p)}=\operatorname{Spec}R[x^{p}]$, and define a map
$\Phi:s\Omega_{\square,\mathbf{A}^{1,(p)}}\to
s\Omega_{\square,\mathbf{A}^{1}}$ via
$\displaystyle R[x^{p}]$ $\displaystyle\xrightarrow{{\varphi}}R[x],\ \text{in
degree }0,$ $\displaystyle R[x^{p}]d(x^{p})$
$\displaystyle\xrightarrow{{\varphi,d(x^{p})\mapsto s(p)x^{p-1}dx}}R[x]dx,\
\text{in degree }1.$
Then $\Phi$ factors through a quasi-isomorphism
$\varphi^{\ast}s\Omega_{\square,\mathbf{A}^{1,(p)}}\xrightarrow{{\sim}}\eta_{s(p)}s\Omega_{\square,\mathbf{A}^{1}}$.
###### Proof.
Since $s\Omega_{\square,\mathbf{A}^{1}}$ is $s(p)$-torsionfree, we can
directly compute the complex $\eta_{s(p)}s\Omega_{\square,\mathbf{A}^{1}}$. If
$f(x)=\sum_{n\geq 0}a_{n}x^{n}\in R[x]$, then
$\nabla_{s}f(x)=\sum_{n\geq 0}a_{n}s(n)x^{n-1}.$
Since $s(p)\mid s(pj)$ for any $j\geq 0$, we see that $\nabla_{s}f(x)\in
s(p)R[x]$ if and only if $s(p)\mid a_{n}$ for $p\nmid n$. Therefore,
$\displaystyle\eta_{s(p)}s\Omega_{\square,\mathbf{A}^{1}}$
$\displaystyle=\left(R[x^{p}]+s(p)R[x]\xrightarrow{{\nabla_{s}}}s(p)R[x]dx\right).$
The map $\Phi$ clearly factors through the inclusion
$\eta_{s(p)}s\Omega_{\square,\mathbf{A}^{1}}\subseteq
s\Omega_{\square,\mathbf{A}^{1}}$. It remains to check that the map
$\varphi^{\ast}s\Omega_{\square,\mathbf{A}^{1,(p)}}\to\eta_{s(p)}s\Omega_{\square,\mathbf{A}^{1}}$
induces an isomorphism on cohomology. Observe that
$\displaystyle\mathrm{H}^{0}(s\Omega_{\square,\mathbf{A}^{1,(p)}})$
$\displaystyle\cong R,$
$\displaystyle\mathrm{H}^{1}(s\Omega_{\square,\mathbf{A}^{1,(p)}})$
$\displaystyle\cong\bigoplus_{n\geq 1}R/s(n)\\{(x^{p})^{n-1}d(x^{p})\\},$
and similarly
$\displaystyle\mathrm{H}^{0}(\eta_{s(p)}s\Omega_{\square,\mathbf{A}^{1}})$
$\displaystyle\cong R,$
$\displaystyle\mathrm{H}^{1}(\eta_{s(p)}s\Omega_{\square,\mathbf{A}^{1}})$
$\displaystyle\cong\bigoplus_{n\geq
1}(s(p))/(s(np))\\{x^{np-1}dx\\}\oplus\bigoplus_{p\nmid
m}(s(p))/(s(m)s(p))\\{x^{m-1}dx\\}.$
The ring endomorphism $\varphi$ of $R$ sends $s(n)\mapsto\frac{s(np)}{s(p)}$,
and $s(p)$ is a non-zero-divisor in $R$, we see that $\varphi$ descends to an
isomorphism
$\varphi^{\ast}(R/s(n))\xrightarrow{{\sim}}R/\tfrac{s(np)}{s(p)}\xrightarrow{{\sim}}(s(p))/(s(np)).$
If $p\nmid m$, then $s(m)$ is a unit in $R/s(p)$ by Lemma 2.2.5; since $R$ is
$s(p)$-adically complete, this implies that $s(m)$ is a unit in $R$ itself. It
follows that $0\cong R/s(m)\xrightarrow{{\sim}}(s(p))/(s(m)s(p))$. Putting
these together, we see that the map
$\varphi^{\ast}s\Omega_{\square,\mathbf{A}^{1,(p)}}\xrightarrow{{\sim}}\eta_{s(p)}s\Omega_{\square,\mathbf{A}^{1}}$
is a quasi-isomorphism. ∎
In Proposition 4.2.6, the condition that $R$ be $s(p)$-adically complete is
rather powerful. For instance, one of the most basic tools in $p$-adic
mathematics is the Legendre formula; this admits an analogue for generalized
$n$-series, too.
###### Proposition 4.2.7 ($s$-analogue of Legendre formula).
Let $s$ be a GNS over $R$ satisfying the hypotheses of Proposition 4.2.6. For
any $n\geq 0$, we have
$\displaystyle n!_{s}$ $\displaystyle=u\prod_{j\geq
1}\varphi^{j-1}(s(p))^{\lfloor n/p^{j}\rfloor}$
for some unit $u\in R^{\times}$.
###### Proof.
The argument is a straightforward adaptation of [BS19, Lemma 12.6]. Since $R$
is $s(p)$-adically complete, we know that if $p\nmid m$, then $s(m)$ is a unit
in $R$. This implies that
$\displaystyle(np)!_{s}$
$\displaystyle=u\prod_{i=1}^{n}s(np)=u\prod_{i=1}^{n}\left(\frac{s(np)}{s(p)}\cdot
s(p)\right)$
$\displaystyle=us(p)^{n}\prod_{i=1}^{n}\varphi(s(n))=us(p)^{n}\varphi(n!_{s}),$
where $u=\prod_{1\leq j\leq np,p\nmid j}s(j)$ is a unit in $R$. Using the
above identity to inductively strip powers of $p$ off of $n$ produces the
desired claim. ∎
This implies the following analogue of [BS19, Lemma 12.5], which gives a
criterion for admitting “$s$-divided powers”.
###### Corollary 4.2.8.
Let $s$ be a GNS over $R$ satisfying the hypotheses of Proposition 4.2.6, and
suppose that for any $n\geq 0$, $\varphi(n!_{s})$ is a non-zero-divisor in
$R/s(p)$. Let $A$ be an $s(p)$-completely flat $R$-algebra equipped with a
$R$-linear multiplicative map $\phi:\varphi^{\ast}A\to A$ and an element $x\in
A$ such that $\varphi(x)=x^{p}$, and $\varphi(x)$ is divisible by $s(p)$. Then
$n!_{s}\mid x^{n}$, i.e., $\frac{x^{n}}{n!_{s}}$ is well-defined in $A$.
###### Proof.
Because $s(j)$ is a unit in $R$ is $p\nmid j$, it suffices to show: for any
$n\geq 0$, if $n!_{s}\mid x^{n}$, then $(np)!_{s}\mid x^{np}$. By (the proof
of) Proposition 4.2.7, it suffices to show that $\varphi(n!_{s})s(p)^{n}\mid
x^{np}$. Because $\varphi(n!_{s})$ is a non-zero-divisor in $R/s(p)$ (by
assumption), it is also a non-zero-divisor in $A/s(p)$ by flatness of $A$. It
therefore suffices to show that $\varphi(n!_{s})$ and $s(p)^{n}$ each
individually divide $x^{np}$. Since $n!_{s}\mid x^{n}$, it is clear that
$\varphi(n!_{s})\mid\varphi(x^{n})=x^{np}$. Since $s(p)\mid x^{p}$ by
assumption, we also see that $s(p)^{n}\mid x^{np}$, as desired. ∎
### 4.3. Formal group law $n$-series and the $s$-derivative
Some of the most important (and accessible) examples of generalized $n$-series
come from formal group laws. Throughout this section, we will fix a base
commutative ring $R$.
###### Recollection 4.3.1.
A ($1$-dimensional) formal group law over a commutative ring $R$ is a two-
variable power series $F(x,y)\in R[\\![x,y]\\!]$ such that
$F(F(x,y),z)=F(x,F(y,z))$ and $F(x,y)\equiv x+y\pmod{(x,y)^{2}}$. It will
sometimes be convenient to denote $F(x,y)$ by $x+_{F}y$. A morphism $f:F\to G$
of formal group laws is a power series $f(x)\in R[\\![x]\\!]$ such that
$f(F(x,y))=G(f(x),f(y))$; a morphism is called an isomorphism if it admits a
compositional inverse.
If $n\geq 0$ is an integer, the $n$-series of $F$ is defined via the formula
$[n]_{F}(t)=\overbrace{F(t,F(t,F(t,\dots
F(t,t)\dots)))}^{n}=\overbrace{t+_{F}t+_{F}\cdots+_{F}t}^{n}.$
This can be extended to all integers $n\in\mathbf{Z}$ by using the existence
of inverses for the formal group law. The $n$-series $[n]_{F}(t)$ is an
element of $R[\\![t]\\!]$, and it is always divisible by $t$. We will define
$\langle n\rangle_{F}(t):=[n]_{F}(t)/t$ (where we agree that $\langle
0\rangle_{F}(t)=0$). Sometimes, it will be notationally convenient to simply
write these as $[n]_{F}$ and $\langle n\rangle_{F}$ (it being implicit that
these are functions of $t$). If $R$ is an $\mathbf{F}_{p}$-algebra, then
either $[p]_{F}(t)=0$ or $[p]_{F}(t)=\lambda t^{p^{h}}+O(t^{p^{h}+1})$ for
some $h>0$. If $v_{j}$ denotes the coefficient of $t^{p^{j}}$ in $[p]_{F}(t)$,
then $F$ is said to be of height $\geq n$ if $v_{j}=0$ for $j<n$; if $v_{n}$
is a unit, then $F$ is said to be of height $n$.
If $\mathbf{Q}\subseteq R$, then every formal group law $F(x,y)$ is isomorphic
to the additive formal group law via the logarithm. Let
$F_{y}(x,y)=\partial_{y}F(x,y)$; then, the logarithm is given by the integral
$\ell_{F}(x):=\int^{x}_{0}\frac{dt}{F_{y}(t,0)}.$
We will write $\mathscr{E}_{F}(x)$ to denote its compositional inverse, so
that $F(x,y)=\mathscr{E}_{F}(\ell_{F}(x)+\ell_{F}(y))$. Observe that
$[n]_{F}(t)=\mathscr{E}_{F}(n\ell_{F}(t))$ for any $n\in\mathbf{Z}$.
###### Example 4.3.2.
Fix a base commutative ring $R$. The polynomial $F(x,y)=x+y$ is known as the
additive formal group law, and $\langle n\rangle_{F}(t)=n$. The polynomial
$F(x,y)=x+y+xy$ is known as the multiplicative formal group law, and
$\langle n\rangle_{F}(t)=\frac{(1+t)^{n}-1}{t}\in R[\\![t]\\!].$
Note that $\langle{n}\rangle_{F}=[n]_{q}$, where we set $q=t+1$. The power
series $F(x,y)=\frac{x+y}{1+xy}$ is known as the hyperbolic formal group law
(since it describes the addition law for $\tanh$), and a simple induction on
$n$ shows that
$\langle
n\rangle_{F}(t)=\frac{1}{t}\frac{(1+t)^{n}-(1-t)^{n}}{(1+t)^{n}+(1-t)^{n}}=\frac{1}{q-1}\frac{q^{n}-(2-q)^{n}}{q^{n}+(2-q)^{n}}\in
R[\\![q-1]\\!].$
###### Remark 4.3.3.
Given a formal group law $F(x,y)$ over a (torsionfree, say) commutative ring
$R$, one can define a “rescaled” formal group law $\widetilde{F}(x,y)$ over
$R[\\![t]\\!]$ which is characterized by the property that
$\widetilde{F}(x,y)=\frac{1}{t}F(xt,yt)$. Observe that over the special fiber
(i.e., $t=0$), $\widetilde{F}(x,y)$ degenerates to the additive formal group
law. Over $(R\otimes\mathbf{Q})[\\![t]\\!]$, the logarithm
$\widetilde{\ell}_{F}(x)$ is given by $\frac{1}{t}\ell_{F}(tx)$; note that
this power series does not have polar terms in $t$, since $x\mid\ell_{F}(x)$
(and hence $tx\mid\ell_{F}(tx)$).
We begin by showing that the map $n\mapsto[n]_{F}(t)$ is a GNS over
$R[\\![t]\\!]$, as long as $F$ satisfies a mild condition. The existence of
the power series $\ell_{F}$ is the main reason that GNS arising via formal
group laws are particularly well-behaved.
###### Proposition 4.3.4.
Let $F$ be a formal group law over a ring $R$, and suppose that $[n]_{F}(t)\in
R[\\![t]\\!]$ is not a zero-divisor for any $n>0$. Define $s:\mathbf{Z}_{\geq
0}\to R[\\![t]\\!]$ by $s(n)=[n]_{F}(t)$. Then, $s$ is a GNS over
$R[\\![t]\\!]$. Similarly, the function $s_{F}:\mathbf{Z}_{\geq 0}\to
R[\\![t]\\!]$ sending $s_{F}(n)=\langle n\rangle_{F}(t)$ is a GNS over
$R[\\![t]\\!]$.
###### Proof.
It is easy to see that if $s$ is a GNS, the same is true of $s_{F}$. Let us
now show that $s$ is a GNS by checking the conditions of Definition 2.1.4.
Condition (1) is clear, since $[0]_{F}=0$. Condition (2) is already assumed in
the theorem statement.
For condition (3), let $G(x)\in R[\\![t]\\!][\\![x]\\!]$ be the power series
$F(t,x)$. Then,
$[n+1]_{F}-[k+1]_{F}=G([n]_{F})-G([k]_{F}).$
Since $x-y$ divides $x^{j}-y^{j}$ for each $j\geq 0$, and $G$ is a power
series, $x-y$ also divides $G(x)-G(y)$. In particular,
$s(n)-s(k)=[n]_{F}-[k]_{F}$ divides $G([n]_{F})-G([k]_{F})$; but
$G([n]_{F})-G([k]_{F})=[n+1]_{F}-[k+1]_{F}$ is precisely $s(n+1)-s(k+1)$, so
$s(n)-s(k)$ divides $s(n+1)-s(k+1)$. Inducting, we conclude that $s(n)-s(k)$
divides $s(n+j)-s(k+j)$ for all $n,k,j\in\mathbf{Z}_{\geq 0}$. For $k=0$, this
gives $s(n)\mid s(n+j)-s(j)$ for all $n,j\in\mathbf{Z}_{\geq 0}$. After
relabeling the indices, this becomes $s(n-k)\mid s(n)-s(k)$, which proves
condition (3). ∎
###### Lemma 4.3.5.
Let $F$ be a formal group law over a ring $R$. If $n\in\mathbf{Z}_{>0}$ is not
a zero-divisor in $R$ (e.g., $R$ is torsionfree), then $[n]_{F}$ and $\langle
n\rangle_{F}$ are not zero-divisors in $R[\\![t]\\!]$.
###### Proof.
Suppose for the sake of contradiction that $[n]_{F}(t)$ is a zero-divisor for
some $n$ (the same argument works for $\langle n\rangle_{F}$). Then there
exists a power series $f(t)\in R[\\![t]\\!]$ such that
$f(t)\cdot[n]_{F}(t)=0.$
It follows that the product of the coefficients of the lowest-degree terms of
$f(t)$ and $[n]_{F}(t)$ must be $0$. Since the lowest-degree term of
$[n]_{F}(t)$ is $nt$, this implies that $n$ times the lowest-degree
coefficient of $f(t)$ is $0$. In particular, $n$ is a zero-divisor in $R$. ∎
###### Definition 4.3.6.
Let $F$ be a formal group law over $R$ such that $[n]_{F}(t)\in R[\\![t]\\!]$
is not a zero-divisor for any $n>0$.444Many of the results below do not rely
on this assumption; but we keep it nonetheless, since Proposition 4.3.4 allows
to immediately transport many results about generalized $n$-series obtained
above. The interested reader should have no trouble removing this condition as
necessary in the results below. Let $F\Omega_{\square,\mathbf{A}^{1}}$ denote
the differential graded $R[\\![t]\\!]$-algebra given by the $s$-de Rham
complex associated to the GNS $s:\mathbf{Z}_{\geq 0}\to R$ sending
$n\mapsto\langle n\rangle_{F}$. We will refer to
$F\Omega_{\square,\mathbf{A}^{1}}$ as the $F$-de Rham complex of the affine
line $\mathbf{A}^{1}=\operatorname{Spec}R[x]$; we will abusively also refer to
the differential $\nabla_{F}$ as the $F$-derivative.
###### Example 4.3.7.
* •
For the additive formal group law, $\langle n\rangle_{F}=n$; so the resulting
$F$-de Rham complex is simply the usual de Rham complex.
* •
For the multiplicative formal group law, $\langle n\rangle_{F}=[n]_{q}$; so
the resulting $F$-de Rham complex is simply the $q$-de Rham complex. Note that
the $F$-derivative can be defined directly on polynomials (instead of only on
monomials) via $f(x)\mapsto\frac{f(qx)-f(x)}{(q-1)x}$. This can be seen
directly from Proposition 4.3.9: indeed,
$\mathscr{E}_{F}(\ell_{F}(t)z)=\frac{(1+t)^{z}-1}{t}=\frac{q^{z}-1}{q-1},$
and the operator $q^{x\partial_{x}}$ sends $f(x)\mapsto f(qx)$.
###### Remark 4.3.8.
One can consider a slight variant of the $F$-de Rham complex, given by the
complex
$C^{\bullet}:=\left(R[\\![t]\\!][x]\to R[\\![t]\\!][x]dx\right),\
x^{n}\mapsto[n]_{F}x^{n-1}dx.$
This is the $s$-de Rham complex for the function $s:\mathbf{Z}_{\geq 0}\to R$
sending $n\mapsto[n]_{F}$. Then, there is a quasi-isomorphism
$F\Omega_{\square,\mathbf{A}^{1}}\simeq\eta_{t}C^{\bullet}$.
When $\mathbf{Q}\subseteq R$, there is a general formula for the
$F$-derivative.
###### Proposition 4.3.9.
Suppose that $\mathbf{Q}\subseteq R$, and let $F$ be a formal group law over
$R$. Then there is an equality of $R[\\![t]\\!]$-linear operators on
$R[\\![t]\\!][x]$:
$\nabla_{F}=\frac{1}{tx}\mathscr{E}_{F}(\ell_{F}(t)x\partial_{x}).$
In particular, there is a canonical isomorphism
$F\Omega_{\square,\mathbf{A}^{1}}\cong\Omega_{\mathbf{Q}[x]/\mathbf{Q}}\otimes_{\mathbf{Q}}R[\\![t]\\!]$.
###### Proof.
Let $\nabla_{F}^{\prime}$ denote the expression on the right-hand side. By
definition of $\nabla_{F}$, it suffices to check that
$\nabla_{F}^{\prime}(x^{m})=\langle{m}\rangle x^{m-1}$ for every $m\geq 1$.
Write $\mathscr{E}_{F}(t)=\sum_{n}a_{n}t^{n}$; then
$\displaystyle\nabla^{\prime}_{F}(x^{m})$
$\displaystyle=\frac{1}{xt}\sum_{n}a_{n}\ell_{F}(t)^{n}(x\partial_{x})^{n}(x^{m})$
$\displaystyle=\frac{1}{xt}\sum_{n}a_{n}(m\ell_{F}(t))^{n}x^{m}$
$\displaystyle=\frac{1}{t}\mathscr{E}_{F}(m\ell_{F}(t))x^{m-1}=\langle{m}\rangle
x^{m-1},$
as desired. ∎
###### Remark 4.3.10.
Since every formal group law over a $\mathbf{Q}$-algebra is isomorphic to the
additive formal group law, the final statement of Proposition 4.3.9 is a
special case of the following more general observation: if $F_{1}$ and $F_{2}$
are isomorphic formal group laws, then the associated de Rham complexes are
also isomorphic.
###### Example 4.3.11.
Using Proposition 4.3.9, we can make the $F$-derivative explicit for the
hyperbolic formal group law. In this case, $\mathscr{E}_{F}(t)=\tanh(t)$, so
that $\ell_{F}(t)=\tanh^{-1}(t)$, and
$\mathscr{E}_{F}(\ell_{F}(t)z)=\tanh(z\tanh^{-1}(t))=\frac{q^{z}-(2-q)^{z}}{q^{z}+(2-q)^{z}},$
where $q-1=t$. Since the operator $q^{x\partial_{x}}$ sends $f(x)\mapsto
f(qx)$, the $F$-derivative can be expressed as
$\nabla_{F}:f(x)\mapsto\frac{1}{(q-1)x}\frac{f(qx)-f((2-q)x)}{f(qx)+f((2-q)x)}.$
###### Example 4.3.12.
Let $R_{0}=\mathbf{F}_{p}[v_{n}]$. Then, there is a unique formal group law
(known as the Honda formal group law; see [Hon70]) over $R_{0}$ which is
characterized by the property that its $p$-series is given by
$[p]_{F}(t)=v_{n}t^{p^{n}}$. This implies that up to a unit in
$R_{0}[\\![t]\\!]$, we have $[m]_{F}(t)=v_{n}^{v_{p}(m)}t^{m^{n}}$, where
$v_{p}(m)$ denotes the $p$-adic valuation of $m$. The formal group law over
$R_{0}$ lifts to a formal group law over $R=\mathbf{Z}_{p}[v_{n}]$ such that
over $R\otimes\mathbf{Q}\cong\mathbf{Q}_{p}[v_{n}]$, its logarithm is given by
$\ell_{F}(x)=\sum_{j\geq
0}v_{n}^{\frac{p^{jn}-1}{p^{n}-1}}\frac{x^{p^{jn}}}{p^{j}}.$
For example, if $n=1$ and we adjoin a $(p-1)$st root $\beta$ of $v_{1}$, this
is essentially the logarithm of the Artin-Hasse exponential, so that
$\ell_{F}(x)=-\frac{1}{\beta}\sum_{p\nmid
d}\frac{\mu(d)}{d}\mathrm{log}(1-(\beta x)^{d}).$
One can say something similar for general $n$. Recall that the polylogarithm
is defined by $\mathrm{Li}_{s}(x)=\sum_{j\geq 1}\frac{x^{j}}{j^{s}}$, so that
$\mathrm{Li}_{1}(x)=-\mathrm{log}(1-x)$. If $\beta$ denotes a $(p^{n}-1)$st
root of $v_{n}$, then $\ell_{F}(x)$ can be understood as a “$p^{n}$-typical”
version of $\frac{1}{\beta}\mathrm{Li}_{1/n}(\beta x)$.
When $n=1$, the resulting $F$-de Rham complex over $\mathbf{F}_{p}[\\![t]\\!]$
is closely related to the mod $p$ reduction of the $q$-de Rham complex:
indeed, observe that the $p$-series of the multiplicative formal group law is
congruent to $t^{p}\pmod{p}$. The resulting lifted formal group law over
$\mathbf{Z}_{p}[v_{1}]$ is the $p$-typification of the multiplicative formal
group law (see [Rav86, Appendix 2]). For higher $n$, the resulting $F$-de Rham
complex behaves qualitatively similar to the case $n=1$. For instance, we have
$\mathrm{H}^{0}(F\Omega_{\square,\mathbf{A}^{1}})\cong\mathbf{F}_{p^{n}}[v_{n}][\\![t]\\!],\
\mathrm{H}^{1}(F\Omega_{\square,\mathbf{A}^{1}})\cong\bigoplus_{j\geq
1}\mathbf{F}_{p^{n}}[v_{n},t]/(v_{n}^{v_{p}(j)}t^{j^{n}})\\{x^{j-1}dx\\}.$
###### Example 4.3.13.
Let $n\geq 1$, let $k$ be an algebraically closed field of characteristic
$p>0$, and let $R=W(k)[\\![u_{1},\cdots,u_{n-1}]\\!]$ denote the Lubin-Tate
ring. Let $F$ denote the formal group law associated to the universal
deformation of a chosen formal group law of height $n$ over $k$. Then, the
resulting $F$-de Rham complex specializes to the $q$-de Rham complex when
$n=1$. For general $n$, this $F$-de Rham complex is closely related to deep
phenomena in chromatic homotopy theory (see Remark 4.3.24).
###### Lemma 4.3.14 ($F$-Taylor expansion).
Let $F$ be a formal group law over $R$ such that $[n]_{F}(t)\in R[\\![t]\\!]$
is not a zero-divisor for any $n>0$. If
$f(x)\in(R\otimes\mathbf{Q})[\\![t,x-1]\\!]$, there is a Taylor expansion
$f(x)=\sum_{n\geq 0}\nabla_{F}^{n}(f(x))|_{x=1}\frac{(x-1)^{n}_{s}}{n!_{F}}.$
Here, $(x-1)^{n}_{s}$ denotes the symbol from Definition 2.3.5 with $y=-1$.
###### Proof.
This is the same argument as in [AL20, Proposition 4.4]. First, observe that
if $g(x)\in(R\otimes\mathbf{Q})[\\![t,x-1]\\!]$ is a function such that
$\nabla_{F}^{n}(g(x))|_{x=1}=0$ for all $n\geq 0$, then $g=0$. Indeed, since
$\nabla_{F}$ is simply the usual derivative modulo $t$, we see that $g(x)$ is
divisible by $t$. Write $g(x)=tg_{1}(x)$; then,
$\nabla_{F}^{n}(g_{1}(x))|_{x=1}=0$ for all $n\geq 0$, so $t\mid g_{1}(x)$.
Continuing, we see that $g(x)$ is infinitely $t$-divisible, and hence is zero
(since $t$ is topologically nilpotent).
We can now apply the above observation to
$g(x):=f(x)-\sum_{n\geq
0}\nabla_{F}^{n}(f(x))|_{x=1}\frac{(x-1)^{n}_{s}}{n!_{F}}.$
By definition of $(x-1)^{n}_{s}$, we know that
$\nabla_{F}(\frac{(x-1)^{n}_{s}}{n!_{F}})=\frac{(x-1)^{n-1}_{s}}{(n-1)!_{s}}$;
so $\nabla_{F}^{n}(g(x))|_{x=1}=0$ for all $n\geq 0$, and hence $g=0$, as
desired. ∎
###### Corollary 4.3.15 ($F$-logarithm).
Let $F$ be a formal group law over $R$ such that $[n]_{F}(t)\in R[\\![t]\\!]$
is not a zero-divisor for any $n>0$. Consider the function
$F\mathrm{log}(x)\in(R\otimes\mathbf{Q})[\\![t,x-1]\\!]$ given by
$\frac{t}{\ell_{F}(t)}\mathrm{log}(x)$. Then, we have:
1. (1)
$\nabla_{F}(F\mathrm{log}(x))=\frac{1}{x}$.
2. (2)
$F\mathrm{log}(xy)=F\mathrm{log}(x)+F\mathrm{log}(y)$.
3. (3)
There is a series expansion
$F\mathrm{log}(x)=\sum_{n\geq
1}\frac{\langle{-n+1}\rangle_{F}\cdots\langle{-1}\rangle_{F}}{n!_{F}}(x-1)^{n}_{s}.$
###### Proof.
The first statement follows from Proposition 4.3.9. Indeed, write
$\mathscr{E}_{F}(y)=\sum_{n\geq 1}a_{n}y^{n}$; the condition that
$F(x,y)\equiv x+y\pmod{(x,y)^{2}}$ forces $a_{1}=1$. Since
$(x\partial_{x})(F\mathrm{log}(x))=\frac{t}{\ell_{F}(t)}(x\partial_{x})\mathrm{log}(x)=\frac{t}{\ell_{F}(t)},$
we see that
$\displaystyle x\nabla_{F}(F\mathrm{log}(x))$
$\displaystyle=\frac{1}{t}\sum_{n\geq
1}a_{n}\ell_{F}(t)^{n}(x\partial_{x})^{n}(F\mathrm{log}(x))$
$\displaystyle=\frac{1}{t}\left(\ell_{F}(t)\cdot\frac{t}{\ell_{F}(t)}+\sum_{n\geq
2}a_{n}\ell_{F}(t)^{n}(x\partial_{x})^{n}(F\mathrm{log}(x))\right).$
The second sum vanishes, since $(x\partial_{x})^{n}(F\mathrm{log}(x))=0$ for
$n\geq 2$. The first term cancels out to give
$x\nabla_{F}(F\mathrm{log}(x))=1$, as desired.
The second statement is clear. For the third statement, observe that
$\nabla_{F}^{n}(F\mathrm{log}(x))=\nabla_{F}^{n-1}(1/x)=\langle{-n+1}\rangle_{F}\cdots\langle{-1}\rangle_{F}x^{-n}.$
Evaluating at $x=1$ and using Lemma 4.3.14 gives the desired claim. ∎
###### Warning 4.3.16.
The $F$-logarithm $F\mathrm{log}(x)$ is not the same as the logarithm
$\ell_{F}(x)$ associated to the formal group law. This unfortunate terminology
stems from attempting to simultaneously emulate the standard terminology
“$q$-logarithm” and the “logarithm of the multiplicative formal group law”.
###### Remark 4.3.17.
Corollary 4.3.15 implies that $F\mathrm{log}(x)$ is a well-defined class in
the ring $R[\\![t]\\!]\left[x^{\pm 1},\frac{(x-1)^{n}_{s}}{n!_{F}}\right]$;
this is the ring of functions on an $F$-analogue of the divided power
completion of the identity section of $(\mathbf{G}_{m})_{R[\\![t]\\!]}$.
###### Remark 4.3.18.
The formal series in Corollary 4.3.15(3) can be written for arbitry GNS $s$;
when it exists and converges, its $s$-derivative will formally be $1/x$.
However, we have chosen to state Corollary 4.3.15 only in the case of GNS
arising via formal group laws, since it is otherwise difficult to get a
computational grip on the resulting formal series.
###### Example 4.3.19.
When $F$ is the multiplicative formal group law over $\mathbf{Z}$, the
function $F\mathrm{log}(x)$ can be identified with the $q$-logarithm
$\mathrm{log}_{q}(x)=\sum_{n\geq
1}(-1)^{n+1}q^{-\binom{n}{2}}\frac{(x-1)(x-q)\cdots(x-q^{n-1})}{[n]_{q}}\in\mathbf{Q}[\\![q-1,x-1]\\!].$
Indeed, this follows from Corollary 4.3.15(3) and the observation that
$\langle{-j}\rangle_{F}=[-j]_{q}=-q^{-j}[j]_{q}$. See [AL20, Section 4] for
more on the $q$-logarithm.
Let us summarize some of the results from the previous section upon
specialization to the $F$-de Rham complex:
###### Theorem 4.3.20.
Let $R$ be a commutative ring, and let $F$ be a formal group law over $R$ such
that $[n]_{F}(t)\in R[\\![t]\\!]$ is not a zero-divisor for any $n>0$. Let
$\hat{\mathbf{G}}=\operatorname{Spf}R[\\![t]\\!]$ denote the formal group over
$R$. Then:
1. (1)
Let $R[\\![t]\\!]\langle{x}\rangle_{F}$ denote the ring
$R[\\![t]\\!][x,\frac{x^{n}}{[n]_{F}!}]_{n\geq 0}$. Then the Poincaré lemma
holds: the cohomology of the complex
$F\Omega_{\square,\mathbf{A}^{1}}\otimes_{R[\\![t]\\!][x]}R[\\![t]\\!]\langle{x}\rangle_{F}$
is concentrated in degree zero, where it is isomorphic to $R[\\![t]\\!]$.
2. (2)
The Cartier isomorphism holds: there is a canonical isomorphism
$F\Omega_{\square,\mathbf{A}^{1,(p)}}^{i}\otimes_{R[\\![t]\\!]}R[\\![t]\\!]/\langle{p}\rangle_{F}\cong\mathrm{H}^{i}(F\Omega_{\square,\mathbf{A}^{1}}\otimes_{R[\\![t]\\!]}R[\\![t]\\!]/\langle{p}\rangle_{F})$
sending $(x^{p})^{n}\mapsto x^{np}$ in degree zero and
$(x^{p})^{n}d(x^{p})\mapsto[x^{np}x^{p-1}dx]$. Note that
$\operatorname{Spf}R[\\![t]\\!]/[p]_{F}\cong\hat{\mathbf{G}}[p]$.
3. (3)
The décalage isomorphism holds: replace $R[\\![t]\\!]$ with its
$\langle{p}\rangle_{F}$-adic completion. Let $\varphi:R[\\![t]\\!]\to
R[\\![t]\\!]$ denote the $R$-algebra map sending $t\mapsto[p]_{F}(t)$, i.e.,
the map induced on rings by the multiplication-by-$p$ map
$\hat{\mathbf{G}}\to\hat{\mathbf{G}}$. Then, there is a quasi-isomorphism
$\varphi^{\ast}F\Omega_{\square,\mathbf{A}^{1,(p)}}\xrightarrow{{\sim}}\eta_{\langle{p}\rangle_{F}}F\Omega_{\square,\mathbf{A}^{1}}$.
4. (4)
The hypotheses of Corollary 4.2.8 are satisfied, so that there is a criterion
for admitting “$F$-divided powers”. Namely, replace $R[\\![t]\\!]$ by its
$(p,\langle{p}\rangle_{F})$-adic completion, and suppose that for any $n\geq
0$, $\varphi(n!_{F})$ is a non-zero-divisor in
$R[\\![t]\\!]/\langle{p}\rangle_{F}$. Let $A$ be a
$\langle{p}\rangle_{F}$-completely flat $R[\\![t]\\!]$-algebra equipped with a
$R[\\![t]\\!]$-linear multiplicative map $\phi:\varphi^{\ast}A\to A$. If $x\in
A$ is an element such that $\varphi(x)=x^{p}$ and
$\langle{p}\rangle_{F}\mid\varphi(x)$, then $\frac{x^{n}}{n!_{s}}\in A$.
###### Proof.
The first part is Proposition 4.2.1. The second part is Proposition 4.2.4,
where the hypothesis in the proposition holds because
$\langle{1}\rangle_{F}=1$. The third (and fourth) part is an application of
Proposition 4.2.6. Note that the first hypothesis holds by construction of
$\varphi:R[\\![t]\\!]\to R[\\![t]\\!]$: indeed, $\varphi$ sends
$s(n)=\frac{[n]_{F}(t)}{t}\mapsto\frac{[n]_{F}([p]_{F}(t))}{[p]_{F}(t)}=\frac{[np]_{F}(t)}{[p]_{F}(t)}=\frac{s(np)}{s(p)}.$
The second hypothesis follows from the assumption on $R[\\![t]\\!]$. ∎
###### Remark 4.3.21.
Using Remark 4.3.10, one can upgrade Theorem 4.3.20 to the case when $R$
(rather, $\operatorname{Spec}R$) is replaced by the moduli stack of formal
groups. However, we will not discuss this further in this article.
Motivated by [Sch17, Conjecture 3.1], we propose the following conjecture. In
fact, considerations with ring stacks following [Dri22] strongly indicate that
the conjecture is false, but we have stated it nonetheless in the hopes that
it might spur investigation into these sort of questions. Arpon Raksit has
informed the first author that he is currently working on some variant of this
conjecture.
###### Conjecture 4.3.22.
Let $\mathrm{Poly}_{R}$ denote the category of polynomial $R$-algebras and
$R$-algebra maps between them. Let $F$ be a formal group law over $R$ such
that $[n]_{F}(t)\in R[\\![t]\\!]$ is not a zero-divisor for any $n>0$. Then,
there is a functor
$\Gamma_{F\text{-}\mathrm{dR}}({-}):\mathrm{Poly}_{R}\to\mathrm{CAlg}(R[\\![t]\\!])$
landing in the $\infty$-category of
${\mathbf{E}_{\infty}}$-$R[\\![t]\\!]$-algebras which sends
$R[x_{1},\cdots,x_{n}]\mapsto F\Omega_{\square,\mathbf{A}^{n}}$.555In other
words, the assignment $R[x_{1},\cdots,x_{n}]\mapsto
F\Omega_{\square,\mathbf{A}^{n}}$ is functorial in $R$-algebra maps of
polynomial $R$-algebras. Furthermore, each part of Theorem 4.3.20 admits a
generalization to $\Gamma_{F\text{-}\mathrm{dR}}({-})$.
###### Remark 4.3.23.
If the preceding conjecture is true, then the construction
$F\Omega_{\square,-}$ can be extended to all (animated) $R$-schemes by left
Kan extension:
$\textstyle{\mathrm{Poly}_{R}^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma_{F\text{-}\mathrm{dR}}({-})}$$\textstyle{\mathrm{CAlg}(R[\\![t]\\!])^{\mathrm{op}}}$$\textstyle{\mathrm{Sch}_{R}^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma_{F\text{-}\mathrm{dR}}({-})}$
We expect that the resulting functor
$X\mapsto\Gamma_{F\text{-}\mathrm{dR}}({X})$, if it exists, should be rather
interesting. When $F$ is the multiplicative formal group law, 4.3.22 is true,
and the resulting assignment $X\mapsto\Gamma_{F\text{-}\mathrm{dR}}({X})$ is
the $q$-de Rham cohomology of [BS19].
Let us end this section by discussing the motivation behind the construction
of the $F$-de Rham complex. We will necessarily be brief, since this is not
the main subject of the present article.
###### Remark 4.3.24.
Let $A$ be an even-periodic ${\mathbf{E}_{\infty}}$-ring equipped with a
complex orientation. Then, Quillen defined a canonical formal group law
$F(x,y)$ over $R:=\pi_{0}(A)$, whose underlying formal group is
$\operatorname{Spf}\pi_{0}(A^{hS^{1}})$. Let $\tau_{\geq 0}A$ denote the
connective cover of $A$, and let
$\mathrm{F}^{\star}_{\mathrm{ev}}\mathrm{HP}(\tau_{\geq 0}A[x]/\tau_{\geq
0}A)$ denote the even filtration on the periodic cyclic homology of
$\tau_{\geq 0}A[x]$ (defined in [HRW22]). The assignment $A\mapsto(\tau_{\geq
0}A)^{tS^{1}}$ is the homotopical analogue of the construction of the
“rescaled” formal group law from Remark 4.3.3; in other words,
$\operatorname{Spf}\pi_{0}((\tau_{\geq 0}A)^{tS^{1}})$ is the rescaled
analogue of the Quillen formal group $\operatorname{Spf}\pi_{0}(A^{hS^{1}})$.
Unpublished work of Arpon Raksit shows that the $F$-de Rham complex
$F\Omega_{\square,\mathbf{A}^{1}}$ arises as the zeroth associated graded
piece $\mathrm{gr}^{0}_{\mathrm{ev}}\mathrm{HP}(\tau_{\geq 0}A[x]/\tau_{\geq
0}A)$.666Using similar methods, one can also show that the variant of the
$F$-de Rham complex from Remark 4.3.8 arises as the zeroth associated graded
piece $\mathrm{gr}^{0}_{\mathrm{ev}}\mathrm{HC}^{-}(A[x]/A)$ in the negative
cyclic homology of $A[x]$. In particular, in this case, the $F$-de Rham
complex admits the structure of an
${\mathbf{E}_{\infty}}$-$R[\\![t]\\!]$-algebra. There are homotopical
analogues of each part of Theorem 4.3.20: for example, the décalage
isomorphism of Theorem 4.3.20(3) is proved as [Dev23b, Proposition 3.5.3].
When $A=\mathrm{KU}$ is periodic complex K-theory (so $\tau_{\geq
0}A=\mathrm{ku}$ is connective complex K-theory), the formal group law over
$\pi_{0}(A)$ is precisely the multiplicative one; so the $q$-de Rham complex
arises as
$\mathrm{gr}^{0}_{\mathrm{ev}}\mathrm{HP}(\mathrm{ku}[x]/\mathrm{ku})$. As
explained in [DR23], the $p$-completion of
$\mathrm{HP}(\mathrm{ku}[x]/\mathrm{ku})$ can be understood via the
topological negative cyclic homology of $\mathbf{Z}_{p}[\zeta_{p}][x]$; this
is a homotopical analogue of the Bhatt-Scholze construction [BS19] of $q$-de
Rham cohomology via prismatic cohomology.
When $A=E_{n}$ is the Morava E-theory associated to the Lubin-Tate formal
group (see Example 4.3.13) and $\tau_{\geq 0}A=e_{n}$ is its connective cover,
the $F$-de Rham complex of Example 4.3.13 arises as
$\mathrm{gr}^{0}_{\mathrm{ev}}\mathrm{HP}(e_{n}[x]/e_{n})$. The periodic
cyclic homology $\mathrm{HP}(e_{n}[x]/e_{n})$ plays an important role in
higher chromatic analogues of the work of Bhatt-Morrow-Scholze [BMS19], and
will be explored in future work.
Although this does not quite fall into the above framework, the first author
hopes to show in future work that when $A=L^{s}(\mathbf{Z})$ is the symmetric
L-theory of the integers (see [HLN21]), the $F$-de Rham complex associated to
the hyperbolic formal group law is closely related to
$\mathrm{gr}^{0}_{\mathrm{ev}}\mathrm{HP}(L^{s}(\mathbf{Z})[x]/L^{s}(\mathbf{Z}))$.
This is a manifestation of the observation (going back to the Hirzebruch
signature theorem) that the logarithm of the formal group law associated to
the complex orientation on $L^{s}(\mathbf{Z})$ is given by the hyperbolic
tangent function $\tanh(x)$.
### 4.4. A variant of the Weyl algebra
It is well-known that the classical de Rham complex over a base commutative
ring $R$ is Koszul dual to the usual Weyl algebra of differential operators on
the affine line:
$\mathscr{D}_{\mathbf{A}^{1}}=R\langle{x,\partial_{x}}\rangle/([\partial_{x},x]=1).$
The complex of Remark 4.3.8 for the additive formal group is also Koszul dual
to a rescaled analogue $R[\\![t]\\!]\langle{x,D}\rangle/([D,x]=t)$ of this
Weyl algebra; this rescaling amounts to replacing $\partial_{x}$ by
$t\partial_{x}$. This fact has an analogue for arbitrary formal group laws, as
we now explain. It turns out to be significantly more convenient to study the
Weyl algebra of $\mathbf{G}_{m}$ instead, so we will restrict to that case.
Some of the discussion in this section appears briefly in [Dev23a, Section
3.3]. As usual, fix a base commutative ring $R$ and a formal group law
$F(x,y)$ over $R$.
###### Definition 4.4.1.
The $F$-Weyl algebra of $\mathbf{G}_{m}=\operatorname{Spec}R[x^{\pm 1}]$ is
defined to be the associative $R[\\![t]\\!]$-algebra given by
$F\mathscr{D}_{\square,\mathbf{G}_{m}}:=R[\\![t]\\!]\langle{x^{\pm
1},y}\rangle^{\wedge}_{y}/(yx=xF(y,t)).$
###### Example 4.4.2.
For the additive formal group law, the relation imposed in Definition 4.4.1 is
just $yx=x(y+t)$, or equivalently that $[y,x]=tx$. This defines an isomorphism
between $F\mathscr{D}_{\square,\mathbf{G}_{m}}$ and the rescaled Weyl algebra
for $\mathbf{G}_{m}$ by sending $y$ to the rescaled vector field
$tx\partial_{x}$.
###### Example 4.4.3.
For the multiplicative formal group law, let us write $q=1+t$. If we define
$\widetilde{y}=1+y$, the relation imposed in Definition 4.4.1 is just
$(\widetilde{y}-1)x=x(q\widetilde{y}-1),$
or equivalently that $\widetilde{y}x=qx\widetilde{y}$. Observe that
$\widetilde{y}$ acts as the operator $q^{x\partial_{x}}$, so that we obtain an
isomorphism between $F\mathscr{D}_{\square,\mathbf{G}_{m}}$ and the completion
of $R[\\![q-1]\\!]\langle{x^{\pm
1},q^{x\partial_{x}}}\rangle/(q^{x\partial_{x}}x=qxq^{x\partial_{x}})$ at the
ideal $(q^{x\partial_{x}}-1)$. This algebra is essentially the $q$-Weyl
algebra of $\mathbf{G}_{m}$, and is sometimes known as the “quantum torus”, as
well as the algebra of $q$-difference operators on the torus.
###### Lemma 4.4.4.
The algebra $R[\\![t]\\!][x^{\pm 1}]$ is canonically a left
$F\mathscr{D}_{\square,\mathbf{G}_{m}}$-module, where the action of $y$ sends
$x^{n}\mapsto[n]_{F}x^{n}$.
###### Proof.
Observe that
$(yx)(x^{n})=[n+1]_{F}x^{n+1}=xF([n]_{F},t)x^{n}=xF(y,t)x^{n},$
so that the action prescribed above satisfies the relation in Definition
4.4.1. ∎
The following result describing the complex from Remark 4.3.8 as Koszul dual
to the $F$-Weyl algebra is sketched in [Dev23a, Proposition 3.3.9]. We will
not need this result below, so we only state it for completeness.
###### Proposition 4.4.5.
The derived tensor product $R[\\![t]\\!][x^{\pm
1}]\otimes_{F\mathscr{D}_{\square,\mathbf{G}_{m}}}R[\\![t]\\!][x^{\pm 1}]$ can
be identified with the complex
$C^{\bullet}\otimes_{R[x]}R[x^{\pm 1}]\cong\left(R[\\![t]\\!][x^{\pm 1}]\to
R[\\![t]\\!][x^{\pm 1}]d\mathrm{log}(x)\right),\
x^{n}\mapsto[n]_{F}x^{n}d\mathrm{log}(x).$
###### Remark 4.4.6.
In [Dev23a, Section 3.3], we show that if $R$ is a complex-oriented even-
periodic ${\mathbf{E}_{\infty}}$-ring and $F(x,y)$ is the formal group law
over $\pi_{0}(R)$, then $F\mathscr{D}_{\square,\mathbf{G}_{m}}$ arises as the
loop-rotation equivariant homology $\pi_{\ast}(R[\Omega T]^{h(T\times
S^{1}_{\mathrm{rot}})})$ where $T$ is a compact torus of rank $1$ (i.e., a
circle, not to be confused with the loop-rotation circle).777In fact, the
first author initially came across the $F$-Weyl algebra in this manner, and
this article was originally intended to be about this algebra. However, upon
learning of the $F$-de Rham complex from Arpon Raksit, it became clear that
the $F$-de Rham complex was a much simpler object to work with; hence the
present form of the article. We also explained that, under the discussion of
Remark 4.3.24, the Koszul duality of Proposition 4.4.5 is a manifestation of
the Koszul duality between $R[\Omega T]^{h(T\times S^{1}_{\mathrm{rot}})}$ and
the negative cyclic homology
$R[\mathscr{L}T_{+}]^{hS^{1}_{\mathrm{rot}}}=\mathrm{HC}^{-}(R[\Omega T]/R)$.
###### Remark 4.4.7 (Mellin transform).
Let $\mathbf{A}^{1}_{R[\\![t]\\!]}$ denote the affine line over $R[\\![t]\\!]$
with coordinate $y$, so that $\mathbf{Z}$ acts on
$\mathbf{A}^{1}_{R[\\![t]\\!]}$ via the map $y\mapsto F(y,t)$. It follows from
Definition 4.4.1 and Morita theory that the category of
$F\mathscr{D}_{\square,\mathbf{G}_{m}}$-modules is equivalent to the category
of quasicoherent sheaves on $\mathbf{A}^{1}_{R[\\![t]\\!]}/\mathbf{Z}$ which
are $y$-complete. In the case of the additive formal group law, this is a
$t$-deformation of the Mellin transform, which gives an equivalence between
the category of $\mathscr{D}_{\mathbf{G}_{m}}$-modules and quasicoherent
sheaves on $\mathbf{A}^{1}/\mathbf{Z}$ (where $\mathbf{Z}$ acts by $y\mapsto
y+1$).
Let us now describe some special properties of the center of
$F\mathscr{D}_{\square,\mathbf{G}_{m}}$.
###### Recollection 4.4.8.
One property satisfied by the ordinary Weyl algebra in characteristic $p>0$ is
that it has a large center: namely, if $R$ is an $\mathbf{F}_{p}$-algebra,
there is an isomorphism $Z(\mathscr{D}_{\mathbf{G}_{m}})\cong R[x^{\pm
p},x^{p}\partial_{x}^{p}]$ which identifies $Z(\mathscr{D}_{\mathbf{G}_{m}})$
with the ring of functions on the cotangent bundle of the Frobenius twist
$(\mathbf{G}_{m})^{(p)}$. Note that
$x^{p}\partial_{x}^{p}\equiv(x\partial_{x})^{p}-x\partial_{x}\pmod{p}$. Under
the Koszul duality between $\mathscr{D}_{\mathbf{G}_{m}}$ and the de Rham
complex, this identification of $Z(\mathscr{D}_{\mathbf{G}_{m}})$ is in fact
Koszul dual to the Cartier isomorphism
$\mathrm{H}^{\ast}(\Omega^{\bullet}_{\mathbf{G}_{m}/R})\cong\Omega^{\ast}_{(\mathbf{G}_{m})^{(p)}/R}$.
It is therefore natural to ask for a description of the center of
$F\mathscr{D}_{\square,\mathbf{G}_{m}}$; this leads to the following result,
which is Koszul dual to the Cartier isomorphism of Theorem 4.3.20(b).
###### Theorem 4.4.9.
The center of
$F\mathscr{D}_{\square,\mathbf{G}_{m}}\otimes_{R[\\![t]\\!]}R[\\![t]\\!]/\langle{p}\rangle_{F}$
can be identified as follows:
$Z(F\mathscr{D}_{\square,\mathbf{G}_{m}}\otimes_{R[\\![t]\\!]}R[\\![t]\\!]/\langle{p}\rangle_{F})\cong
R[\\![t]\\!]\left[x^{\pm
p},\prod_{j=0}^{p-1}(y+_{F}[j]_{F})\right]/\langle{p}\rangle_{F}.$
###### Proof.
Observe that
$yx^{p}=x^{p}F(y,[p]_{F})\equiv x^{p}y\pmod{\langle{p}\rangle_{F}},$
so that $x^{p}$ is in the center of
$F\mathscr{D}_{\square,\mathbf{G}_{m}}\otimes_{R[\\![t]\\!]}R[\\![t]\\!]/\langle{p}\rangle_{F}$.
Similarly, since
$\prod_{j=0}^{p-1}(y+_{F}[j]_{F})\equiv\prod_{j=0}^{p-1}(y+_{F}[j]_{F})\pmod{\langle{p}\rangle_{F}}$,
we have
$\prod_{j=0}^{p-1}(y+_{F}[j]_{F})x=x\prod_{j=1}^{p}(y+_{F}[j]_{F})\equiv
x\prod_{j=0}^{p-1}(y+_{F}[j]_{F})\pmod{\langle{p}\rangle_{F}},$
and hence $\prod_{j=0}^{p-1}(y+_{F}[j]_{F})$ is in the center of
$F\mathscr{D}_{\square,\mathbf{G}_{m}}\otimes_{R[\\![t]\\!]}R[\\![t]\\!]/\langle{p}\rangle_{F}$.
This defines an inclusion
$R[\\![t]\\!]\left[x^{p},\prod_{j=0}^{p-1}(y+_{F}[j]_{F})\right]/\langle{p}\rangle_{F}\subseteq
Z(F\mathscr{D}_{\square,\mathbf{G}_{m}}\otimes_{R[\\![t]\\!]}R[\\![t]\\!]/\langle{p}\rangle_{F}),$
which can be checked to be an isomorphism. ∎
###### Example 4.4.10.
For the additive formal group law over $\mathbf{Z}$ (say), we have
$\prod_{j=0}^{p-1}(y+jt)\equiv y^{p}-t^{p-1}y\pmod{p},$
so that upon identifying $y=x\partial_{x}$, Theorem 4.4.9 implies the
following isomorphism for the rescaled Weyl algebra:
$Z(F\mathscr{D}_{\square,\mathbf{G}_{m}}/p)\cong\mathbf{F}_{p}[\\![t]\\!][x^{\pm
p},(x\partial_{x})^{p}-t^{p-1}x\partial_{x}].$
Regarding the above algebra as a graded ring with both $t$ and $x\partial_{x}$
in weight $1$ allows us to replace $t$ by a polynomial generator (instead of a
power series variable). Inverting $t$ and taking $\mathbf{G}_{m}$-invariants
(which is to be thought of as setting $t=1$) then produces the ring
$\mathbf{F}_{p}[x^{\pm
p},t^{-p}x^{p}\partial_{x}^{p}]=\mathscr{O}_{T^{\ast}\mathbf{G}_{m}^{(1)}}$.
###### Example 4.4.11.
For the multiplicative formal group law over $\mathbf{Z}$ (say), if we define
$\widetilde{y}=y+1$ as above (so that $\widetilde{y}=q^{x\partial_{x}}$), we
have
$\prod_{j=0}^{p-1}(y+_{F}(q^{j}-1))=\prod_{j=0}^{p-1}(q^{j}\widetilde{y}-1)\equiv
q^{p(p-1)/2}(\widetilde{y}^{p}-1)\pmod{[p]_{q}}.$
Note that $q^{p(p-1)/2}$ is $1\pmod{[p]_{q}}$ for $p>2$, but is
$-1\pmod{[2]_{q}}$. In either case, $q^{p(p-1)/2}$ is a unit, so Theorem 4.4.9
implies the following isomorphism:
$Z(F\mathscr{D}_{\square,\mathbf{G}_{m}}/[p]_{q})\cong\mathbf{Z}[\zeta_{p}][x^{\pm
p},q^{px\partial_{x}}].$
###### Remark 4.4.12.
Since $F\mathscr{D}_{\square,\mathbf{G}_{m}}$ can be recovered from the
equivariant homology $\pi_{\ast}(R[\Omega T]^{h(T\times
S^{1}_{\mathrm{rot}})})$ for a compact torus $T$ of rank $1$ (see Remark
4.4.6), it is natural to wonder whether there is an explanation of Theorem
4.4.9 from the perspective of the affine Grassmannian. In the case when $R$ is
ordinary (integral) homology or K-theory, this has been answered in [Lon17] —
in fact, the methods there are sufficiently geometric that they work even for
more general $R$ (and for $T$ replaced by a more general connected compact Lie
group!), so we refer the reader to loc. cit. for further discussion of this
question.
### 4.5. An analogue of the Bhatt-Lurie Cartesian square
Throughout this section, we will fix a $p$-completely flat
$\mathbf{Z}_{p}$-algebra $R$ and a formal group law $F(x,y)$ over $R$ (so $R$
is torsionfree). The symbol $R[\\![t]\\!]$ will always denote the $p$-adic
completion of the formal power series ring, and all constructions will be done
internal to the category of $(p,t)$-adically $R[\\![t]\\!]$-schemes. (We have
omitted the completion from the notation for readability.) Let
$\hat{\mathbf{G}}$ denote the associated formal group, so that its underlying
formal scheme is $\operatorname{Spf}R[\\![t]\\!]$.
In [BL22, Lemma 3.5.18], Bhatt-Lurie showed that there is a Cartesian square
of group schemes over $R$:
(7)
$\textstyle{\mathbf{G}_{m}^{\sharp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{log}}$$\textstyle{\mathbf{G}_{a}^{\sharp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto\exp(px)}$$\textstyle{\mathbf{G}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\mapsto
x^{p}}$$\textstyle{\mathbf{G}_{m}^{(1)}.}$
Taking vertical quotients, we obtain an isomorphism between
$\mathbf{G}_{m}/\mathbf{G}_{m}^{\sharp}$ and
$\mathbf{G}_{m}^{(1)}/\mathbf{G}_{a}^{\sharp}$; one can identify
$\mathbf{G}_{m}/\mathbf{G}_{m}^{\sharp}$ with the “de Rham stack”
$\mathbf{G}_{m}^{\mathrm{dR}}$ of $\mathbf{G}_{m}$ (see, e.g., [Bha22]), so
that this isomorphism describes $\mathbf{G}_{m}^{\mathrm{dR}}$ in terms of the
group scheme $\mathbf{G}_{a}^{\sharp}$.
The proof of 7 in loc. cit. used the relationship between the group schemes
appearing in the square and the ring scheme of Witt vectors. In this section,
we prove an $F$-analogue of this result (see Theorem 4.5.10); in the case of
the additive formal group law, this reproves 7. Let us state at the outset
that in the case when $F$ is the multiplicative formal group law, this result
was obtained in a discussion between the first author and Michael Kural.
Moreover, the argument in this section rests crucially on 12, the $q$-analogue
of which (Example 4.5.8) was proved by Michael Kural. Any errors below are
solely the fault of the first author!
###### Definition 4.5.1.
Remark 4.3.3 gives a formal group $\hat{\mathbf{G}}_{t}$ over
$\operatorname{Spf}R[\\![t]\\!]$ whose logarithm is
$\widetilde{\ell}_{F}(x)=\frac{1}{t}\ell_{F}(tx)$. Let $x$ denote the
coordinate on $\hat{\mathbf{G}}_{t}$, so that its underlying formal scheme is
$\operatorname{Spf}R[\\![t,x]\\!]$. Let $\hat{\mathbf{G}}_{t}^{\vee}$ denote
the Cartier dual
$\operatorname{Hom}(\hat{\mathbf{G}}_{t},(\mathbf{G}_{m})_{R[\\![t]\\!]})$ of
$\hat{\mathbf{G}}_{t}$; see [Dri21, Section 3] for some generalities on
Cartier duals of formal groups. The element
$x\in\mathscr{O}_{\hat{\mathbf{G}}_{t}}$ defines a homomorphism
$\tau:\hat{\mathbf{G}}_{t}^{\vee}\to(\mathbf{G}_{a})_{R[\\![t]\\!]}$.
###### Observation 4.5.2.
Over $(R\otimes\mathbf{Q})[\\![t]\\!]$, the rescaled logarithm
$\widetilde{\ell}_{F}$ of Remark 4.3.3 defines an isomorphism
$\widetilde{\ell}_{F}:\hat{\mathbf{G}}_{t}\xrightarrow{{\sim}}(\hat{\mathbf{G}}_{a})_{(R\otimes\mathbf{Q})[\\![t]\\!]}$
of formal groups. Therefore, the canonical pairing
$\hat{\mathbf{G}}_{t}\times_{R[\\![t]\\!]}\hat{\mathbf{G}}_{t}^{\vee}\to(\mathbf{G}_{m})_{R[\\![t]\\!]}$
fits into a diagram
$\textstyle{\hat{\mathbf{G}}_{t}\times_{R[\\![t]\\!]}\hat{\mathbf{G}}_{t}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\sim}$$\scriptstyle{\widetilde{\ell}_{F}\times\mathrm{id}}$$\textstyle{(\hat{\mathbf{G}}_{a})_{(R\otimes\mathbf{Q})[\\![t]\\!]}\times_{R[\\![t]\\!]}\hat{\mathbf{G}}_{t}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{(\mathbf{G}_{m})_{(R\otimes\mathbf{Q})[\\![t]\\!]}.}$
Since $R[\\![t]\\!]$ is $(p,t)$-adically complete, the Cartier dual of
$(\hat{\mathbf{G}}_{a})_{R[\\![t]\\!]}$ can be identified with the divided
power completion $(\mathbf{G}_{a}^{\sharp})_{R[\\![t]\\!]}$. The pairing $\nu$
is base-changed from $R[\\![t]\\!]$ itself, where it is given by the formula
$\nu:(x,y)\mapsto\exp(xy).$
It follows that the pairing $\mu$ is given by
$\mu(x,y)=\exp(\widetilde{\ell}_{F}(x)y).$
This can be expanded as a power series in $x$:
$\mu(x,y)=\sum_{n\geq 0}\beta_{n}(y)x^{n}.$
Unwinding the definition of the Cartier dual, and using that $R[\\![t]\\!]$ is
$p$-torsionfree (using our assumption that $R$ is a $p$-completely flat
$\mathbf{Z}_{p}$-algebra), we see that the ring of functions on
$\hat{\mathbf{G}}_{t}^{\vee}$ has underlying $R[\\![t]\\!]$-module given by
(the $(p,t)$-adic completion of)
$\mathscr{O}_{\hat{\mathbf{G}}_{t}^{\vee}}=R[\\![t]\\!]\\{\beta_{n}(y)\\}_{n\geq
0}.$
###### Example 4.5.3.
When $F$ is the multiplicative formal group law, the function $\mu$ is simply
$\mu(x,y)=\exp\left(\frac{y}{q-1}\mathrm{log}(1+(q-1)x)\right)=(1+(q-1)x)^{y/(q-1)};$
its power series expansion is given by
$\mu(x,y)=\sum_{n\geq 0}\frac{\prod_{j=0}^{n-1}(y-j(q-1))}{n!}x^{n}.$
This expression plays an important role in [Dri21].
###### Example 4.5.4.
When $F$ is the hyperbolic formal group law (so that
$\ell_{F}(x)=\tanh^{-1}(x)=\frac{1}{2}\mathrm{log}\left(\frac{1+x}{1-x}\right)$),
the function $\mu$ is
$\mu(x,y)=\exp\left(\frac{y}{2(q-1)}\mathrm{log}\left(\frac{1+(q-1)x}{1-(q-1)x}\right)\right)=\left(\frac{1+(q-1)x}{1-(q-1)x}\right)^{y/2(q-1)}.$
The power series expansion of this function is somewhat complicated: one can
show that upon writing $\mu(x,y)=\sum_{n\geq 0}\beta_{n}(y)x^{n}$, we have
$\beta_{0}=1$, $\beta_{1}=y$, and there is a recurrence
$\beta_{n+2}(y)=\frac{y\beta_{n+1}(y)+n(q-1)^{2}\beta_{n}(y)}{n+2}.$
Using a computer, one can compute that the first few terms of this expansion
are
$\displaystyle\mu(x,y)$
$\displaystyle=1+yx+\frac{y^{2}}{2}x^{2}+\frac{2(q-1)^{2}y+y^{3}}{3!}x^{3}+\frac{8(q-1)^{2}y^{2}+y^{4}}{4!}x^{4}$
$\displaystyle+\frac{24(q-1)^{4}y+20(q-1)^{2}y^{3}+y^{5}}{5!}x^{5}+\cdots.$
Observe that $\beta_{n}(y)\equiv\frac{y^{n}}{n!}\pmod{2}$, reflecting the fact
that the base-change of the hyperbolic formal group to $\mathbf{F}_{2}$ is
isomorphic to the additive formal group.
###### Definition 4.5.5.
Let $\mathbf{G}_{m}^{\sharp,F}$ denote the formal scheme over
$\operatorname{Spf}R[\\![t]\\!]$ given by (the $(p,t)$-adic completion of)
$\mathbf{G}_{m}^{\sharp,F}=\operatorname{Spf}R[\\![t]\\!]\left[y^{\pm
1},\frac{(y-1)^{n}_{s}}{n!_{F}}\right]_{n\geq 0}.$
This can be viewed as the “$F$-divided power hull” of the identity section of
$(\mathbf{G}_{m})_{R[\\![t]\\!]}$. Equip $\mathbf{G}_{m}^{\sharp,F}$ with the
structure of a group scheme where the coproduct sends $y\mapsto y\otimes y$.
It is not immediate that this is well-defined, but we will prove this below in
Corollary 4.5.9. There is a canonical homomorphism
$\mathrm{can}:\mathbf{G}_{m}^{\sharp,F}\to(\mathbf{G}_{m})_{R[\\![t]\\!]}$.
Note that Remark 4.3.17 implies that $F\mathrm{log}(y)$ defines an element of
the coordinate ring of $\mathbf{G}_{m}^{\sharp,F}$, i.e., it defines a map
$F\mathrm{log}:\mathbf{G}_{m}^{\sharp,F}\to(\mathbf{G}_{a})_{R[\\![t]\\!]}$.
This is in fact a homomorphism, since
$F\mathrm{log}(y_{1}y_{2})=F\mathrm{log}(y_{1})+F\mathrm{log}(y_{2})$.
###### Proposition 4.5.6.
Work over the base $(R\otimes\mathbf{Q})[\\![t]\\!]$. Then, the iterated
$F$-derivative of $\mu(x,F\mathrm{log}(y))$ with respect to the variable $y$
is given by
(8)
$\nabla_{F,y}^{n}\mu(x,F\mathrm{log}(y))=\frac{x(x+_{\widetilde{F}}\langle{-1}\rangle_{F}(t))\cdots(x+_{\widetilde{F}}\langle{-n+1}\rangle_{F}(t))}{y^{n}}\mu(x,F\mathrm{log}(y)).$
###### Proof.
Observe that:
$\displaystyle\mu(x,F\mathrm{log}(y))$ $\displaystyle=\sum_{n\geq
0}\beta_{n}(F\mathrm{log}(y))x^{n}=\exp(F\mathrm{log}(y)\widetilde{\ell}_{F}(x))$
$\displaystyle=\exp\left(\frac{t}{\ell_{F}(t)}\mathrm{log}(y)\cdot\frac{\ell_{F}(tx)}{t}\right)=\exp\left(\mathrm{log}(y)\frac{\ell_{F}(tx)}{\ell_{F}(t)}\right)=y^{\frac{\ell_{F}(tx)}{\ell_{F}(t)}};$
the third equality used the definition of $F\mathrm{log}(y)$ via Corollary
4.3.15 and the definition of $\widetilde{\ell}_{F}(x)$ via Remark 4.3.3. One
can deduce 8 from this; let us illustrate this rather inefficiently. Let us
write $a=\frac{\ell_{F}(tx)}{\ell_{F}(t)}$ for notational simplicity, so that
$y^{a}=\sum_{m\geq 0}\frac{a(a-1)\cdots(a-(m-1))}{m!}(y-1)^{m},$
and $\partial_{y}y^{a}=ay^{a-1}$.
We can now inductively compute the iterated $F$-derivative using Proposition
4.3.9. We begin with the base case $n=1$. Note that
$(y\partial_{y})y^{a}=ay^{a}$, so that
(9) $(y\partial_{y})^{m}y^{a}=a^{m}y^{a}$
by an easy induction on $m$. Write $\mathscr{E}_{F}(z)=\sum_{m\geq
0}b_{m}z^{m}$; then using 9, we have:
$\displaystyle\nabla_{F,y}\mu(x,F\mathrm{log}(y))$
$\displaystyle=\frac{1}{yt}\sum_{m\geq
0}b_{m}\ell_{F}(t)^{m}(y\partial_{y})^{m}y^{a}=\frac{1}{yt}\sum_{m\geq |
# Robust Multivariate Functional Control Charts
Christian Capezza Fabio Centofanti Corresponding author. e-mail:
<EMAIL_ADDRESS>Antonio Lepore Biagio Palumbo
###### Abstract
Profile monitoring assesses the stability over time of one or multiple
functional quality characteristics to quickly detect special causes of
variation that act on a process. In modern Industry 4.0 applications, a huge
amount of data is acquired during manufacturing processes that are often
contaminated with anomalous observations in the form of both casewise and
cellwise outliers. Because anomalous observations can seriously affect the
monitoring performance, profile monitoring methods that are able to
successfully deal with outliers are needed. To this aim, we propose a new
framework, referred to as robust multivariate functional control charts
(RoMFCC), that is able to monitor multivariate functional data while being
robust to both functional casewise and cellwise outliers. The RoMFCC relies on
four main elements: (I) a univariate filter to identify functional cellwise
outliers to be replaced by missing components; (II) a robust functional data
imputation method of missing values; (III) a casewise robust dimensionality
reduction; (IV) a monitoring strategy for the multivariate functional quality
characteristic. An extensive Monte Carlo simulation study is performed to
quantify the monitoring performance of the RoMFCC by comparing it with some
competing methods already available in the literature. Finally, a motivating
real-case study is presented where the proposed framework is used to monitor a
resistance spot welding process in the automotive industry.
Keywords: Functional Data Analysis, Profile Monitoring, Statistical Process
Control, Robust Estimation, Casewise and Cellwise Outliers
## 1 Introduction
In modern industrial applications, data acquisition systems allow to collect
massive amounts of data with high frequency. Several examples may be found in
the current Industry 4.0 framework, which is reshaping the variety of signals
and measurements that can be gathered during manufacturing processes. The
focus in many of these applications is statistical process monitoring (SPM),
whose main aim is to quickly detect unusual conditions in a process when
special causes of variation act on it, i.e., the process is out-of-control
(OC). On the contrary, when only common causes are present, the process is
said to be in-control (IC).
In this context, the experimental measurements of the quality characteristic
of interest are often characterized by complex and high dimensional formats
that can be well represented as functional data or profiles (Ramsay and
Silverman, , 2005; Kokoszka and Reimherr, , 2017). The simplest approach for
monitoring one or multiple functional variables is based on the extraction of
scalar features from each profile, e.g., the mean, followed by the application
of classical SPM techniques for multivariate data (Montgomery, , 2012).
However, feature extraction is known to be problem-specific, arbitrary, and
risks compressing useful information. Thus, there is a growing interest in
profile monitoring (Noorossana et al., , 2011), whose aim is to monitor a
process when the quality characteristic is best characterized by one or
multiple profiles. Some recent examples of profile monitoring applications can
be found in Menafoglio et al., (2018); Capezza et al., (2020); Capezza et
al., 2021a ; Capezza et al., 2021b ; Centofanti et al., 2021b .
The main tools for SPM are control charts that are implemented in two phases.
In Phase I, historical process data are used to set control chart limits to be
used in Phase II, i.e., the actual monitoring phase, where observations
falling outside the control limits are signaled as OC. In classical SPM
applications the historical Phase I data are assumed to come from an IC
process. However, this assumption is not always valid. As an example, let
consider the motivating real-world application, detailed in Section 4, that
concerns the SPM of a resistance spot welding (RSW) process in the assembly of
automobiles. RSW is the most common technique employed in joining metal sheets
during body-in-white assembly of automobiles (Zhou and Cai, , 2014), mainly
because of its adaptability for mass production (Martín et al., , 2014). Among
on-line measurements of RSW process parameters, the so-called dynamic
resistance curve (DRC) is recognized as the full technological signature of
the metallurgical development of a spot weld (Dickinson et al., , 1980;
Capezza et al., 2021b, ) and, thus, can be used to characterize the quality of
a finished sub-assembly. Figure 1 shows 100 DRCs corresponding to 10 different
spot welds, measured in $m\Omega$, that are acquired during the RSW process
from the real-case study presented in Section 4. Several outliers are clearly
visible that should be taken into account to set up an effective SPM strategy.
Figure 1: Sample of 100 DRCs, measured in $m\Omega$, that are acquired during
the RSW process from the real-case study in Section 4. The different panels
refer to the corresponding different spot welds, denoted with names from V1 to
V10.
Indeed, control charts are very sensitive to the presence of outlying
observations in Phase I that can lead to inflated control limits and reduced
power to detect process changes in Phase II. To deal with outliers in the
Phase I sample, SPM methods use two common alternatives, namely the diagnostic
and the robust approaches (Kruger and Xie, , 2012; Hubert et al., , 2015). The
diagnostic approach is based on standard estimates after the removal of sample
units identified as outliers that translates into SPM methods where iterative
re-estimation procedures are considered in Phase I. This approach could be
often safely applied to eliminate the effect of a small number of very extreme
observations. However, it will fail to detect more moderate outliers that are
not always as easy to label correctly. On the contrary, the robust approach
accepts all the data points and tries to find a robust estimator which reduces
the impact of outliers on the final results (Maronna et al., , 2019).
Several robust approaches for the SPM of a multivariate scalar quality
characteristic have been proposed in the literature. Alfaro and Ortega,
(2009) show a comparison of robust alternatives to the classical Hotelling’s
control chart. It includes two alternative Hotelling’s $T^{2}$-type control
charts for individual observations, proposed by Vargas, (2003) and Jensen et
al., (2007), which are based on the minimum volume ellipsoide and the minimum
covariance determinant estimators (Rousseeuw, , 1984), respectively. Moreover,
the comparison includes the control chart based on the reweighted minimum
covariance determinant (RMCD) estimators, proposed by Chenouri et al.,
(2009). More recently, Cabana and Lillo, (2021) propose an alternative robust
Hotelling’s $T^{2}$ procedure using the robust shrinkage reweighted estimator.
Although Kordestani et al., (2020) and Moheghi et al., (2020) propose robust
estimators to monitoring simple linear profiles, to the best of authors’
knowledge, a robust approach able to successfully capture the functional
nature of a multivariate functional quality characteristic has not been
devised in the SPM literature so far. Beyond the SPM literature, several works
have been proposed to deal with outlying functional observations. Several
methods extend the classical linear combination type estimators (i.e.,
L-estimator) (Maronna et al., , 2019) to the functional setting to robustly
estimate the center of a functional distribution through trimming (Fraiman and
Muniz, , 2001; Cuesta-Albertos and Fraiman, , 2006) and functional data depths
(Cuevas and Fraiman, , 2009; López-Pintado and Romo, , 2011). Sinova et al.,
(2018) introduce the notion of maximum likelihood type estimators (i.e.,
M-estimators) in the functional data setting. More recently, Centofanti et
al., 2021a propose a robust functional ANOVA method that reduces the weights
of outlying functional data on the results of the analysis.
Regarding functional principal component analysis (FPCA), robust approaches
are classified by Boente and Salibián-Barrera, (2021) in three groups,
depending on the specific property of principal components on which they
focus. Methods in the first group perform the eigenanalysis of a robust
estimator of the scatter operator, as the spherical principal components
method of Locantore et al., (1999) and the indirect approach of Sawant et
al., (2012). The latter performs a robust PCA method, e.g., ROBPCA (Hubert et
al., , 2005), on the matrix of the basis coefficients corresponding to a basis
expansion representation of the functional data. The second group includes
projection-pursuit approaches (Hyndman and Ullah, , 2007), which sequentially
search for the directions that maximize a robust estimator of the spread of
the data projections. Whereas, the third group is composed of methods that
estimate the principal components spaces by minimizing a robust reconstruction
error measure (Lee et al., , 2013). Finally, it is worth mentioning diagnostic
approaches for functional outlier detection, which have been proposed for both
univariate (Hyndman and Shang, , 2010; Arribas-Gil and Romo, , 2014; Febrero
et al., , 2008) and multivariate functional data (Hubert et al., , 2015; Dai
and Genton, , 2018; Alemán-Gómez et al., , 2022).
In presence of many functional variables, the lack of robust approaches that
deal with outliers is exacerbated by the curse of dimensionality. Traditional
multivariate robust estimators assume a casewise contamination model for the
data, which consists in a mixture of two distributions, where the majority of
the cases is free of contamination and the minority mixture component
describes an unspecified outlier generating distribution. Alqallaf et al.,
(2009) show that these traditional estimators are affected by the problem of
propagation of outliers. In situations where only a small number of cases are
contaminated, the traditional robust approaches work well. However, under an
independent contamination model such as cellwise outliers (i.e., contamination
in each variable is independent from the other variables), when the
dimensionality of the data is high, the fraction of perfectly observed cases
can be rather small and the traditional robust estimators may fail. Moreover,
Agostinelli et al., (2015) point out that both types of data contamination,
casewise and cellwise, may occur together. This problem has been addressed in
the multivariate scalar setting by Agostinelli et al., (2015) that propose a
two steps method. In the first step, a univariate filter is used to eliminate
large cellwise outliers, i.e., detection and replacement by missing values,
then, in the second step, a robust estimation, specifically designed to deal
with missing data, is applied to the incomplete data. Leung et al., (2016)
notice that the univariate filter does not handle well moderate-size cellwise
outliers, therefore they introduce for the first step a consistent bivariate
filter to be used in combination with the univariate filter. Rousseeuw and
Bossche, (2018) propose a method for detecting deviating data cells that
takes the correlations between the variables into account, whereas, Tarr et
al., (2016) devise a method for robust estimation of precision matrices under
cellwise contamination. Other methods that consider cellwise outliers have
been developed for regression and classification (Filzmoser et al., , 2020;
Aerts and Wilms, , 2017).
To deal with multivariate functional outliers, in this paper we propose a new
framework, referred to as robust multivariate functional control chart
(RoMFCC), for SPM of multivariate functional data that is robust to both
functional casewise and cellwise outliers. The latter corresponds to a
contamination model where outliers arise in each variable independently from
the other functional variables. Specifically, to deal with functional cellwise
outliers, the proposed framework considers an extension of the filtering
approach proposed by Agostinelli et al., (2015) to univariate functional data
and an imputation method inspired by the robust imputation technique of
Branden and Verboven, (2009). Moreover, it also considers a robust
multivariate functional principal component analysis (RoMFPCA) based on the
ROBPCA method (Hubert et al., , 2005), and a profile monitoring strategy built
on the Hotelling’s $T^{2}$ and the squared prediction error ($SPE$) control
charts (Noorossana et al., , 2011; Grasso et al., , 2016; Centofanti et al.,
2021b, ; Capezza et al., , 2020; Capezza et al., 2021a, ; Capezza et al., ,
2022). A Monte Carlo simulation study is performed to quantify the probability
of signal (i.e., detecting an OC observation) of RoMFCC in identifying mean
shifts in the functional variables in presence of both casewise and cellwise
outliers. That is, the proposed RoMFCC is compared with other control charts
already present in the literature. Finally, the practical applicability of the
proposed control chart is illustrated on the motivating real-case study in the
automotive manufacturing industry. In particular, the RoMFCC is shown to
adequately identify a drift in the manufacturing process due to electrode
wear.
The article is structured as follows. Section 2 introduces the proposed RoMFCC
framework. In Section 3, a simulation study is presented where RoMFCC is
compared to other popular competing methods. The real-case study in the
automotive industry is presented in Section 4. Section 5 concludes the
article. Supplementary materials for this article are available online. All
computations and plots have been obtained using the programming language R (R
Core Team, , 2021).
## 2 The Robust Multivariate Functional Control Chart Framework
The proposed RoMFCC is a new general framework for SPM of multivariate
functional data that is able to deal with both functional casewise and
cellwise outliers. It relies on the following four main elements.
1. (I)
Functional univariate filter, which is used to identify functional cellwise
outliers to be replaced by missing components.
2. (II)
Robust functional data imputation, where a robust imputation method is applied
to the incomplete data to replace missing values.
3. (III)
Casewise robust dimensionality reduction, which reduces the infinite
dimensionality of the multivariate functional data by being robust towards
casewise outliers.
4. (IV)
Monitoring strategy, to appropriately monitor multivariate functional data.
In what follows, we describe a specific implementation of the RoMFCC framework
where (I) an extension of the filtering proposed by Agostinelli et al.,
(2015), referred to as functional univariate filter (FUF), is considered; (II)
a robust functional data imputation method referred to as RoFDI and based on
the robust imputation technique of Branden and Verboven, (2009) is used;
(III) the RoMFPCA is considered as the casewise robust dimensionality
reduction method. Finally, (IV) the multivariate functional data are monitored
through the profile monitoring approach based on the simultaneous application
of the Hotelling’s $T^{2}$ and the squared prediction error ($SPE$) control
charts. For ease of presentation, the RoMFPCA is presented in Section 2.1.
Then, Section 2.2, Section 2.3, and, Section 2.4 describe the FUF, the RoFDI
method, and the monitoring strategy, respectively. Section 2.5 details the
Phase I and Phase II of the proposed implementation of the RoMFCC framework
where the elements (I-IV) are put together.
### 2.1 Robust Multivariate Functional Principal Component Analysis
Let $\bm{X}=\left(X_{1},\dots,X_{p}\right)^{T}$ a random vector with
realization in the Hilbert space $\mathbb{H}$ of $p$-dimensional vectors of
functions defined on the compact set $\mathcal{T}\in\mathbb{R}$ with
realizations in $L^{2}(\mathcal{T})$, i.e., the Hilbert spaces of square
integrable functions defined on $\mathcal{T}$. Accordingly, the inner product
of two functions $f$ and $g$ in $L^{2}\left(\mathcal{T}\right)$ is $\langle
f,g\rangle=\int_{\mathcal{T}}f\left(t\right)g\left(t\right)dt$, and the norm
is $\lVert\cdot\rVert=\sqrt{\langle\cdot,\cdot\rangle}$. The inner product of
two function vectors $\mathbf{f}=\left(f_{1},\dots,f_{p}\right)^{T}$ and
$\mathbf{g}=\left(g_{1},\dots,g_{p}\right)^{T}$ in $\mathbb{H}$ is
$\langle\mathbf{f},\mathbf{g}\rangle_{\mathbb{H}}=\sum_{j=1}^{p}\langle
f_{j},g_{j}\rangle$ and the norm is
$\lVert\cdot\rVert_{\mathbb{H}}=\sqrt{\langle\cdot,\cdot\rangle_{\mathbb{H}}}$.
We assume that $\bm{X}$ has mean
$\bm{\mu}=\left(\mu_{1},\dots,\mu_{p}\right)^{T}$,
$\mu_{i}(t)=\operatorname{E}(X_{i}(t))$, $t\in\mathcal{T}$ and covariance
$\bm{G}=\\{G_{ij}\\}_{1\leq i,j\leq p}$,
$G_{ij}(s,t)=\operatorname{Cov}(X_{j}(s),X_{j}(t))$, $s,t\in\mathcal{T}$. In
what follows, to take into account for differences in degrees of variability
and units of measurements among $X_{1},\dots,X_{p}$, the transformation
approach of Chiou et al., (2014) is considered. Specifically, let consider
the vector of standardized variables
$\bm{Z}=\left(Z_{1},\dots,Z_{p}\right)^{T}$,
$Z_{i}(t)=v_{i}(t)^{-1/2}(X_{i}(t)-\mu_{i}(t))$, with $v_{i}(t)=G_{ii}(t,t)$,
$t\in\mathcal{T}$. Then, from the multivariate Karhunen-Loève’s Theorem (Happ
and Greven, , 2018) follows that
$\bm{Z}(t)=\sum_{l=1}^{\infty}\xi_{l}\bm{\psi}_{l}(t),\quad t\in\mathcal{T},$
where $\xi_{l}=\langle\bm{\psi}_{l},\bm{Z}\rangle_{\mathbb{H}}$ are random
variables, said principal components scores or simply scores such that
$\operatorname{E}\left(\xi_{l}\right)=0$ and
$\operatorname{E}\left(\xi_{l}\xi_{m}\right)=\lambda_{l}\delta_{lm}$, with
$\delta_{lm}$ the Kronecker delta. The elements of the orthonormal set
$\\{\bm{\psi}_{l}\\}$,
$\bm{\psi}_{l}=\left(\psi_{l1},\dots,\psi_{lp}\right)^{T}$, with
$\langle\bm{\psi}_{l},\bm{\psi}_{m}\rangle_{\mathbb{H}}=\delta_{lm}$, are
referred to as principal components, and are the eigenfunctions of the
covariance $\bm{C}$ of $\bm{Z}$ corresponding to the eigenvalues
$\lambda_{1}\geq\lambda_{2}\geq\dots\geq 0$. Following the approach of Ramsay
and Silverman, (2005), the eigenfunctions and eigenvalues of the covariance
$\bm{C}$ are estimated through a basis function expansion approach.
Specifically, we assume that the functions $Z_{j}$ and a generic eigenfunction
$\bm{\psi}$ of $\bm{C}$ with components $\psi_{j}$, for $j=1,\dots,p$, can be
represented as
$Z_{j}(t)\approx\sum_{k=1}^{K}c_{jk}\phi_{jk}(t),\quad\psi_{j}(t)\approx\sum_{k=1}^{K}b_{jk}\phi_{jk}(t),\quad
t\in\mathcal{T}$ (1)
where $\bm{\phi}_{j}=\left(\phi_{j1},\dots,\phi_{jK}\right)^{T}$,
$\bm{c}_{j}=\left(c_{j1},\dots,c_{jK}\right)^{T}$ and
$\bm{b}_{j}=\left(b_{j1},\dots,b_{jK}\right)^{T}$ are the basis functions and
coefficient vectors for the expansion of $Z_{j}$ and $\psi_{j}$, respectively.
With these assumptions, standard multivariate functional principal component
analysis (Ramsay and Silverman, , 2005; Chiou et al., , 2014) estimates
eigenfunctions and eigenvalues of the covariance $\bm{C}$ by performing
standard multivariate principal component analysis on the random vector
$\bm{W}^{1/2}\bm{c}$, where
$\bm{c}=\left(\bm{c}_{1}^{T},\dots,\bm{c}_{p}^{T}\right)^{T}$ and $\bm{W}$ is
a block-diagonal matrix with diagonal blocks $\bm{W}_{j}$, $j=1,\dots,p$,
whose entries are $w_{k_{1}k_{2}}=\langle\phi_{k_{1}},\phi_{k_{2}}\rangle$,
$k_{1},k_{2}=1,\dots,K$. Then, the eigenvalues of $\bm{C}$ are estimated by
those of the covariance matrix of $\bm{W}^{1/2}\bm{c}$, whereas, the
components $\psi_{j}$ of the generic eigenfunction, with corresponding
eigenvalue $\lambda$, are estimated through Equation (1) with
$\bm{b}_{j}=\bm{W}^{-1/2}\bm{u}_{j}$, where
$\bm{u}=\left(\bm{u}_{1}^{T},\dots,\bm{u}_{p}^{T}\right)^{T}$ is the
eigenvector of the covariance matrix of $\bm{W}^{1/2}\bm{c}$ corresponding to
$\lambda$. However, it is well known that standard multivariate principal
component analysis is not robust to outliers (Maronna et al., , 2019), which
obviously reflects on the functional principal component analysis by probably
providing misleading results. Extending the approach of Sawant et al., (2012)
for multivariate functional data, the proposed RoMFPCA applies a robust
principal component analysis alternative to the random vector
$\bm{W}^{1/2}\bm{c}$. Specifically, we consider the ROBPCA approach (Hubert et
al., , 2005), which is a computationally efficient method explicitly conceived
to produce estimates with high breakdown in high dimensional data settings,
which almost always arise in the functional context, and to handle large
percentage of contamination. Thus, given $n$ independent realizations
$\bm{X}_{i}$ of $\bm{X}$, dimensionality reduction is achieved by
approximating $\bm{X}_{i}$ through $\hat{\bm{X}}_{i}$, for $i=1,\dots,n$, as
$\hat{\bm{X}}_{i}(t)=\hat{\bm{\mu}}(t)+\hat{\bm{D}}(t)\sum_{l=1}^{L}\hat{\xi}_{il}\hat{\bm{\psi}}_{l}(t)\quad
t\in\mathcal{T}$ (2)
where $\hat{\bm{D}}$ is a diagonal matrix whose diagonal entries are robust
estimates $\hat{v}_{j}^{1/2}$ of $v_{j}^{1/2}$,
$\hat{\bm{\mu}}=\left(\hat{\mu}_{1},\dots,\hat{\mu}_{p}\right)^{T}$ is a
robust estimate of $\bm{\mu}$, $\hat{\bm{\psi}}_{l}$ are the first $L$
robustly estimated principal components and
$\hat{\xi}_{il}=\langle\hat{\bm{\psi}}_{l},\hat{\bm{Z}}_{i}\rangle_{\mathbb{H}}$
are the estimated scores with robustly estimated variances
$\hat{\lambda}_{l}$. The estimates $\hat{\bm{\psi}}_{l}$ and
$\hat{\lambda}_{l}$ are obtained through the $n$ realizations of $\bm{Z}_{i}$
estimated by using $\hat{\mu}_{j}$ and $\hat{v}_{j}$. The robust estimates
$\hat{\mu}_{j}$ and $\hat{v}_{j}$ are obtained through the scale equivariant
functional $M$-estimator and the functional normalized median absolute
deviation estimator proposed by Centofanti et al., 2021a . As in the
multivariate setting, $L$ is generally chosen such that the retained principal
components $\hat{\bm{\psi}}_{1},\dots,\hat{\bm{\psi}}_{L}$ explain at least a
given percentage of the total variability, which is usually in the range
70-90$\%$, however, more sophisticated methods could be used as well, see
Jolliffe, (2011) for further details.
### 2.2 Functional Univariate Filter
To extend the filter of Agostinelli et al., (2015) and Leung et al., (2016)
to univariate functional data, let consider $n$ independent realizations
$X_{i}$ of a random function $X$ with values in $L^{2}(\mathcal{T})$. The
proposed FUF considers the functional distances $D_{i}^{fil}$, $i=1,\dots,n$,
defined as
$D_{i}^{fil}=\sum_{l=1}^{L^{fil}}\frac{(\hat{\xi}_{il}^{fil})^{2}}{\hat{\lambda}_{l}^{fil}},$
(3)
where the estimated scores
$\hat{\xi}_{il}^{fil}=\langle\hat{{\psi}}_{l}^{fil},\hat{{Z}}_{i}\rangle$, the
estimated eigenvalues $\hat{\lambda}_{l}^{fil}$, the estimated principal
components $\hat{{\psi}}_{j}^{fil}$, and the estimated standardized
observations $\hat{{Z}}_{i}$ of ${X}_{i}$ are obtained by applying, with
$p=1$, the RoMFPCA described in Section 2.1 to the sample $X_{i}$,
$i=1,\dots,n$. In this setting, RoMFPCA is used to appropriately represent
distances among $X_{i}$’s and not to perform dimensionality reduction, this
means that $L^{fil}$ should be sufficiently large to capture a large
percentage of the total variability $\delta^{fil}$. Let $G_{n}$ the empirical
distribution of $D_{i}^{fil}$, that is
$G_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}I(D_{i}^{fil}\leq x),\quad x\geq 0,$
where $I$ is the indicator function. Then, functional observations $X_{i}$ are
labeled as cellwise outliers by comparing $G_{n}(x)$ with $G(x)$, $x\geq 0$,
where $G$ is a reference distribution for $D_{i}^{fil}$. Following Leung et
al., (2016), we consider the chi-squared distribution with $L^{fil}$ degrees
of freedom, i.e., $G=\chi^{2}_{L^{fil}}$. The proportion of flagged cellwise
outliers is defined by
$d_{n}=\sup_{x\geq\eta}\\{G(x)-G_{n}(x)\\}^{+},$
where $\\{a\\}^{+}$ represents the positive part of $a$, and
$\eta=G^{-1}(\alpha)$ is a large quantile of $G$. Following Agostinelli et
al., (2015), in the following we consider $\alpha=0.95$ as the aim is to
detect extreme cellwise outliers, but other choices could be considered as
well. Finally, we flag $\left[nd_{n}\right]$ observations with the largest
functional distances $D_{i}^{fil}$ as functional cellwise outliers (here
$\left[a\right]$ is the largest integer less than or equal to $a$). From the
arguments in Agostinelli et al., (2015) and Leung et al., (2016) the FUF is
a consistent even when the actual distribution of $D_{i}^{fil}$ is unknown,
that is asymptotically the filter will not wrongly flag a cellwise outlier
provided that the tail of the chosen reference distribution $G$ is heavier (or
equal) than that of the actual unknown distribution.
### 2.3 Robust Functional Data Imputation
Let consider $n$ independent realizations
$\bm{X}_{i}=\left(X_{i1},\dots,X_{ip}\right)^{T}$ of a random vector of
functions $\bm{X}$ as defined in Section 2.1. This section considers the
setting where $\bm{X}_{i}$, $i=1,\dots,n$, may presents missing components,
i.e., at least one of $X_{i1},\dots,X_{ip}$ is missing. Thus, for each
realization $\bm{X}_{i}$ we can identify the missing
$\bm{X}^{m}_{i}=\left(X_{im_{i1}},\dots,X_{im_{is_{i}}}\right)^{T}$ and
observed $\bm{X}^{o}_{i}=\left(X_{io_{i1}},\dots,X_{io_{it_{i}}}\right)^{T}$
components with $t_{i}=p-s_{i}$, and where $\\{m_{ij}\\}$ and $\\{o_{ij}\\}$
are disjoint set of indices, whose union coincides with the set of indeces
$\\{1,\dots,p\\}$, that indicate which components of the realization $i$ are
either observed or missing. Moreover, we assume that a set $S_{c}$ of $c$
realizations free of missing components is available. The proposed RoFDI
method extends to the functional setting the robust imputation approach of
Branden and Verboven, (2009) that sequentially estimates the missing part of
an incomplete observation such that the imputed observation has minimum
distance from the space generated by the complete realizations. Analogously,
starting from $S_{c}$, we propose that the missing components of the
observation $\bm{X}_{\underline{i}}\notin S_{c}$, which correspond to the
smallest $s_{i}$, are sequentially imputed by minimizing, with respect to
$\bm{X}^{m}_{\underline{i}}$,
$D(\bm{X}_{\underline{i}}^{m})=\sum_{l=1}^{L^{imp}}\frac{(\hat{\xi}_{il}^{imp})^{2}}{\hat{\lambda}_{l}^{imp}},$
(4)
where the estimated scores
$\hat{\xi}_{il}^{imp}=\langle\hat{\bm{\psi}}_{l}^{imp},\hat{\bm{Z}}_{\underline{i}}\rangle_{\mathbb{H}}$,
eigenvalues $\hat{\lambda}_{l}^{imp}$, principal components
$\hat{\bm{\psi}}_{l}^{imp}$ and standardized observations
$\hat{\bm{Z}}_{\underline{i}}$ of $\bm{X}_{\underline{i}}$ are obtained by
applying the RoMFPCA (Section 2.1) to the free of missing data observations in
$S_{c}$. Analogously to Section 2.2, RoMFPCA is used to define the distance of
$\bm{X}_{\underline{i}}$ from the space generated by the free of missing data
observations, thus, $L^{imp}$ should be sufficiently large to capture a large
percentage of the total variability $\delta^{imp}$. Because
$\hat{\bm{Z}}_{\underline{i}}$ is the standardized version of
$\bm{X}_{\underline{i}}$, we can identify the missing
$\hat{\bm{Z}}^{m}_{\underline{i}}$ and observed
$\hat{\bm{Z}}^{o}_{\underline{i}}$ components of
$\hat{\bm{Z}}_{\underline{i}}$. Thus, the minimization problem in Equation (4)
can be equivalently solved with respect to $\hat{\bm{Z}}_{\underline{i}}^{m}$,
and the resulting solution can be unstandardized to obtain the imputed
components of $\bm{X}_{\underline{i}}^{m}$.
Due to the approximations in Equation (1), $\hat{\bm{Z}}_{\underline{i}}$ is
uniquely identified by the coefficient vectors $\bm{c}_{\underline{i}j}$,
$j=1,\dots,p$, related to the basis expansions of its components. Let
$\bm{c}^{m}_{\underline{i}j}$,
$j=m_{\underline{i}1},\dots,m_{\underline{i}s_{\underline{i}}}$ and
$\bm{c}^{o}_{\underline{i}j}$,
$j=o_{\underline{i}1},\dots,o_{\underline{i}t_{\underline{i}}}$ be the
coefficient vectors corresponding to the missing and observed components of
$\hat{\bm{Z}}_{\underline{i}}$, respectively, and,
$\bm{c}^{m}_{\underline{i}}=\left(\bm{c}_{\underline{i}m_{\underline{i}1}}^{mT},\dots,\bm{c}_{\underline{i}m_{\underline{i}s_{\underline{i}}}}^{mT}\right)^{T}$
and
$\bm{c}_{\underline{i}}^{o}=\left(\bm{c}_{\underline{i}o_{\underline{i}1}}^{oT},\dots,\bm{c}_{\underline{i}o_{\underline{i}t_{\underline{i}}}}^{oT}\right)^{T}$.
Moreover, let indicate with $\hat{\bm{b}}_{lj}$, $l=1,\dots,L^{imp}$,
$j=1,\dots,p$, the coefficient vectors related to the basis expansions of the
components of the estimated principal components $\hat{\bm{\psi}}_{l}^{imp}$,
and with
$\hat{\bm{B}}=\left(\hat{\bm{b}}_{1},\dots,\hat{\bm{b}}_{L^{imp}}\right)$ the
matrix whose columns are
$\hat{\bm{b}}_{l}=\left(\hat{\bm{b}}_{l1}^{T},\dots,\hat{\bm{b}}_{lp}^{T}\right)^{T}$.
Then, the solution of the minimization problem in Equation (4) is
$\hat{\bm{c}}_{\underline{i}}^{m}=-\bm{C}_{mm}^{+}\bm{C}_{mo}\bm{c}_{\underline{i}}^{o}$
(5)
where $\bm{A}^{+}$ is the Moore-Penrose inverse of the matrix $\bm{A}$,
$\bm{C}_{mm}$ and $\bm{C}_{mo}$ are the matrices constructed by taking the
columns $m_{\underline{i}1},\dots,m_{\underline{i}s_{\underline{i}}}$ and
$o_{\underline{i}1},\dots,o_{\underline{i}t_{\underline{i}}}$ of the matrix
composed by the rows
$m_{\underline{i}1},\dots,m_{\underline{i}s_{\underline{i}}}$ of
$\bm{C}=\bm{W}\hat{\bm{B}}\hat{\bm{\Lambda}}^{-1}\hat{\bm{B}}^{T}\bm{W}$
with $\bm{W}$ the block-diagonal matrix defined in Section 2.1, and
$\hat{\bm{\Lambda}}$ the diagonal matrix whose diagonal entries are the
estimated eigenvalues $\hat{\lambda}_{l}^{imp}$, $l=1,\dots,L^{imp}$.
Moreover, to address the correlation bias issue typical of deterministic
imputation approaches (Little and Rubin, , 2019; Van Buuren, , 2018), we
propose to impute $\bm{c}_{\underline{i}}^{m}$ through
$\bm{c}_{\underline{i}}^{m,imp}=\left(\bm{c}_{\underline{i}1}^{m,impT},\dots,\bm{c}_{\underline{i}s_{\underline{i}}}^{m,impT}\right)^{T}$
as follows
$\bm{c}_{\underline{i}}^{m,imp}=\hat{\bm{c}}_{\underline{i}}^{m}+\bm{\varepsilon}_{i}$
(6)
where $\bm{\varepsilon}_{i}$ is a multivariate normal random variable with
mean zero and a residual covariance matrix robustly estimated from the
regression residuals of the coefficient vectors of the missing component on
those of the observed component for the observations in $S_{c}$ through the
model in Equation (5). Thus, the proposed RoFDI approach is a stochastic
imputation method (Little and Rubin, , 2019; Van Buuren, , 2018). Then, the
components of $\hat{\bm{Z}}_{\underline{i}}^{m}$ are imputed, for
$j=1,\dots,s_{\underline{i}}$, as
$\hat{Z}_{\underline{i}j}^{m,imp}(t)=\left(\bm{c}_{\underline{i}j}^{m,imp}\right)^{T}\bm{\phi}_{j}(t),\quad
t\in\mathcal{T},$
where $\bm{\phi}_{j}$ is the vector of basis function corresponding to the
$j$-th component of $\hat{\bm{Z}}$ (Section 2.1), and the imputed missing
components of $\bm{X}_{\underline{i}}$ are obtained by unstandardizing
$\hat{\bm{Z}}_{\underline{i}}^{m}$. Once the missing components of
$\bm{X}_{\underline{i}}$ are imputed, the whole observation is added to
$S_{c}$ and the next observation, which does not belong to $S_{c}$ and
corresponds to the smallest $s_{\underline{i}}$, is considered. Similarly to
Branden and Verboven, (2009), if the cardinality of $S_{c}$ at the first
iteration is sufficiently large, we suggest to not update the RoMFPCA model
each time a new imputed observation is added to $S_{c}$ to avoid infeasible
time complexity of the RoFDI, otherwise, the RoMFPCA model could be updated
each time a given number of observations are added to $S_{c}$. Finally, to
take into account the increased noise due to single imputation, the proposed
RoFDI can be easily included in a multiple imputation framework (Van Buuren, ,
2018; Little and Rubin, , 2019), indeed, differently imputed datasets may be
obtained by performing several times the RoFDI due to the presence of the
stochastic component $\bm{\varepsilon}_{i}$ in Equation (6).
### 2.4 The Monitoring Strategy
The (IV) element of the proposed RoMFCC implementation relies on the
consolidated monitoring strategy for a multivariate functional quality
characteristic $\bm{X}$ based on the Hotelling’s $T^{2}$ and $SPE$ control
charts. The former assesses the stability of $\bm{X}$ in the finite
dimensional space spanned by the first principal components identified through
the RoMFPCA (Section 2.1), whereas, the latter monitors changes along
directions in the complement space. Specifically, the Hotelling’s $T^{2}$
statistic for $\bm{X}$ is defined as
$T^{2}=\sum_{l=1}^{L^{mon}}\frac{(\xi_{l}^{mon})^{2}}{\lambda_{l}^{mon}},$
where $\lambda_{l}^{mon}$ are the variances of the scores
$\xi_{l}^{mon}=\langle\bm{\psi}_{l}^{mon},\bm{Z}\rangle_{\mathbb{H}}$ where
$\bm{Z}$ is the vector of standardized variable of $\bm{X}$ and
$\bm{\psi}_{l}^{mon}$ are the corresponding principal components as defined in
Section 2.1. The number $L^{mon}$ is chosen such that the retained principal
components explain at least a given percentage $\delta^{mon}$ of the total
variability. The statistic $T^{2}$ is the standardized squared distance from
the centre of the orthogonal projection of $\bm{Z}$ onto the principal
component space spanned by
$\bm{\psi}_{1}^{mon},\dots,\bm{\psi}_{L^{mon}}^{mon}$. Whereas, the distance
between $\bm{Z}$ and its orthogonal projection onto the principal component
space is measured through the $SPE$ statistic, defined as
$SPE=||\bm{Z}-\hat{\bm{Z}}||_{\mathbb{H}}^{2},$
where $\hat{\bm{Z}}=\sum_{l=1}^{L^{mon}}\xi_{l}^{mon}\bm{\psi}_{l}^{mon}$.
Under the assumption of multivariate normality of $\xi_{l}^{mon}$, which is
approximately true by the central limit theorem (Nomikos and MacGregor, ,
1995), the control limits of the Hotelling’s $T^{2}$ control chart can be
obtained by considering the $(1-\alpha^{*})$ quantiles of a chi-squared
distribution with $L^{mon}$ degrees of freedom (Johnson et al., , 2014).
Whereas, the control limits for $SPE$ control chart can be computed by using
the following equation (Jackson and Mudholkar, , 1979)
$CL_{SPE,\alpha^{*}}=\theta_{1}\left[\frac{c_{\alpha^{*}}\sqrt{2\theta_{2}h_{0}^{2}}}{\theta_{1}}+1+\frac{\theta_{2}h_{0}(h_{0}-1)}{\theta_{1}^{2}}\right]^{1/h_{0}}$
where $c_{\alpha^{*}}$ is the normal deviate corresponding to the upper
$(1-\alpha^{*})$ quantile, $h_{0}=1-2\theta_{1}\theta_{3}/3\theta_{2}^{2}$,
$\theta_{j}=\sum_{l=L^{mon}+1}^{\infty}(\lambda_{l}^{mon})^{j}$, $j=1,2,3$.
Note that, to control the family wise error rate (FWER), $\alpha^{*}$ should
be chosen appropriately. We propose to use the Šidák correction
$\alpha^{*}=1-\left(1-\alpha\right)^{1/2}$ (Šidák, , 1967), where $\alpha$ is
the overall type I error probability.
### 2.5 The Proposed Method
The proposed RoMFCC implementation collects all the elements introduced in the
previous sections for the Phase II monitoring strategy where a set of Phase I
observations, which can be contaminated with both functional casewise and
cellwise outliers, is used for the design of the control chart. Both phases
are outlined in the scheme of Figure 2 and detailed in the following sections.
Figure 2: Scheme of the RoMFCC approach.
#### 2.5.1 Phase I
Let $\bm{X}_{i}$, $i=1,\dots,n$, the Phase I random sample of the multivariate
functional quality characteristic $\bm{X}$, used to characterize the normal
operating conditions of the process. It can be contaminated with both
functional casewise and cellwise outliers.
In the filtering step, functional cellwise outliers are identified through the
FUF described in Section 2.2, and, then, replaced by missing components. Note
that, if cellwise outliers are identified for each component of a given
observation, then, that observation is removed from the sample because its
imputation does not provide any additional information for the analysis. In
the imputation step, missing components are imputed through the RoFDI method
presented in Section 2.3. Once the imputed Phase I sample is obtained, it used
to to estimate the RoMFPCA model and perform the dimensionality reduction step
as described in Section 2.1. The Hotelling’s $T^{2}$ and $SPE$ statistics are,
then, computed for each observation in the Phase I sample after the imputation
step. Specifically, the values $T^{2}_{i}$ and $SPE_{i}$ of the statistics are
computed as described in Section 2.4 by considering the estimated RoMFPCA
model obtained in the dimensionality reduction step. Finally, control limits
for the Hotelling’s $T^{2}$ and $SPE$ control charts are obtained as described
in Section 2.4. Note that the parameters $\theta_{j}$ to estimate
$CL_{SPE,\alpha^{*}}$ are approximated by considering a finite summation to
the maximum number of estimable principal components, which is finite for a
sample of $n$ observations. If a multiple imputation strategy is employed by
performing the imputation step several times, then the multiple estimated
RoMFPCA models could be combined by averaging the robustly estimated
covariance functions as suggested in Van Ginkel et al., (2007).
Note that, when the sample size $n$ is small compared to the number of process
variables, undesirable effect upon the performance of the RoMFCC could arise
(Ramaker et al., , 2004; Kruger and Xie, , 2012). To reduce possible
overfitting issues and, thus, increase the monitoring performance of the
RoMFCC, a reference sample of Phase I observations, referred to as tuning set,
could be considered, which is different from the one used in the previous
steps, referred to as training set. Specifically, on the line of Kruger and
Xie, (2012), Chapter 6.4, the tuning set is passed through the filtering and
imputation steps and, then, it is projected on the RoMFPCA model, estimated
through the training set observations, to robustly estimate the distribution
of the resulting scores. Finally, Hotelling’s $T^{2}$ and $SPE$ statistics and
control charts limits are calculated by taking into account the estimated
distribution of the tuning set scores.
#### 2.5.2 Phase II
In the actual monitoring phase (Phase II), a new observation $\bm{X}_{new}$ is
projected on the RoMFPCA model to compute the values of $T^{2}_{new}$ and
$SPE_{new}$ statistics accordingly to the score distribution identified in
Phase I. An alarm signal is issued if at least one between $T^{2}_{new}$ and
$SPE_{new}$ violates the control limits.
## 3 Simulation Study
The overall performance of the proposed RoMFCC is evaluated by means of an
extensive Monte Carlo simulation study. The aim of this simulation is to
assess the performance of the RoMFCC in identifying mean shifts of the
multivariate functional quality characteristic when the Phase I sample is
contaminated with both functional cellwise and casewise outliers. The data
generation process (detailed in Supplementary Material A) is inspired by the
real-case study in Section 4 and mimics typical behaviours of DRCs in a RSW
process. Specifically, it considers a multivariate functional quality
characteristic with $p=10$ components. Two main scenarios are considered that
are characterized by different Phase I sample contamination. Specifically, the
Phase I sample is contaminated by functional cellwise outliers in Scenario 1
and by functional casewise outliers in Scenario 2, with a contamination
probability equal to 0.05 in both cases. In the Supplementary Material B,
additional results obtained by considering contamination probability equal to
0.1 are shown. For each scenario, two contamination models, referred to as Out
E and Out P, with three increasing contamination levels, referred to as C1, C2
and C3, are considered. The former mimics a splash weld (expulsion), caused by
excessive welding current, while the latter resembles phase shift of the peak
time caused by an increased electrode force (Xia et al., , 2019). Moreover, we
consider also a scenario, referred to as Scenario 0, representing settings
where the Phase I sample is not contaminated. To generate the Phase II sample,
two types of OC conditions, referred to as OC E and OC P, are considered that
are generated analogously to the two contamination models Out E and Out P,
respectively, at 4 different severity levels $SL=\\{1,2,3,4\\}$.
The proposed RoMFCC implementation is compared with several natural competing
approaches. The first approaches are control charts for multivariate scalar
data, built on the average value of each component of the multivariate
functional data. Among them, we consider the multivariate classical
Hotelling’s $T^{2}$ control chart, referred to as M, the multivariate
iterative variant, referred to as Miter, where outliers detected by the
control chart in Phase I are iteratively removed until all data are assumed to
be IC, and the multivariate robust control chart proposed by Chenouri et al.,
(2009), referred to as MRo. We further consider also two approaches recently
appeared in the profile monitoring literature, i.e., the multivariate
functional control charts, referred to as MFCC, proposed by Capezza et al.,
(2020, 2022), and the multivariate iterative functional control charts
variant, referred to as MFCCiter, where outliers detected by the control chart
in Phase I are iteratively removed until all data are assumed to be IC. The
RoMFCC is implemented as described in Section 2 with
$\delta_{fil}=\delta_{imp}=0.999$ and $\delta_{mon}=0.7$, and to take into
account the increased noise due to single imputation, 5 differently imputed
datasets are generated through RoFDI. While data are observed through noisy
discrete values, each component of the generated quality characteristic
observations is obtained by considering the approximation in Equation (1) with
$K=10$ cubic B-splines estimated through the spline smoothing approach based
on a roughness penalty on the second derivative (Ramsay and Silverman, ,
2005). For each scenario, contamination model, contamination level, OC
condition and severity level, 50 simulation runs are performed. Each run
considers a Phase I sample of 4000 observations, where for MFCC, MFCCIter and
RoMFCC, 1000 are used as training set, and the remaining 3000 are used as
tuning set. The Phase II sample is composed of 4000 i.i.d. observations. The
RoMFCC and the competing methods performances are assessed by means of true
detection rate (TDR), which is the proportion of points outside the control
limits whilst the process is OC, and the false alarm rate (FAR), which is the
proportion of points outside the control limits whilst the process is IC. The
FAR should be as similar as possible to the overall type I error probability
$\alpha$ considered to obtain the control limits and set equal to 0.05,
whereas the TDR should be as close to one as possible.
Figure 3-5 display for Scenario 0, Scenario 1 and Scenario 2, respectively, as
a function of the severity level $SL$, the mean FAR ($SL=0$) or TDR ($SL\neq
0$) for each OC condition OC E and OC P, contamination level C1, C2 and C3 and
contamination model Out E and Out P.
Figure 3: Mean FAR ($SL=0$) or TDR ($SL\neq 0$) achieved by M, Miter, MRo, MFCC, MFCCiter and RoMFCC for each OC condition (OC E and OC P) as a function of the severity level $SL$ in Scenario 0. OC E | OC P
---|---
|
Figure 4: Mean FAR ($SL=0$) or TDR ($SL\neq 0$) achieved by M, Miter, MRo, MFCC, MFCCiter and RoMFCC for each contamination level (C1, C2 and C3), OC condition (OC E and OC P) as a function of the severity level $SL$ with contamination model Out E and Out P in Scenario 1. | Out E | Out P
---|---|---
| OC E | OC P | OC E | OC P
C1 | | | |
C2 | | | |
C3 | | | |
Figure 5: Mean FAR ($SL=0$) or TDR ($SL\neq 0$) achieved by M, Miter, MRo, MFCC, MFCCiter and RoMFCC for each contamination level (C1, C2 and C3), OC condition (OC E and OC P) as a function of the severity level $SL$ with contamination model Out E and Out P in Scenario 2. | Out E | Out P
---|---|---
| OC E | OC P | OC E | OC P
C1 | | | |
C2 | | | |
C3 | | | |
When the Phase I sample is not contaminated by outliers, Figure 3 shows that
all the approaches that take into account the functional nature of the data,
i.e., MFCC, MFCCiter, RoMFCC, achieve the same performance for both OC
conditions. Although this setting should be not favourable to approaches
specifically designed to deal with outliers, MFCCiter and RoMFCC perform equal
to MFCC. The non-functional approaches, i.e., M, Miter, MRo, show worse
performance than the functional counterparts, and there is no significant
performance difference among them as well. Figure 4 show the results for
Scenario 1, where the Phase I sample is contaminated by cellwise outliers. The
proposed RoMFCC largely outperforms the competing methods for each
contmination model, contamination level and OC condition. As expected, as the
contamination level increases the differences in performance between the
RoMFCC and the competing methods increase as well. Indeed, the performance of
RoMFCC is almost insensible to the contamination levels as well as the
contamination models, differently from the competing methods whose performance
decreases as the contamination level increases and the contamination model
changes. Moreover, the MFCCiter, which is representative of the baseline
method in this setting, only slightly improves the performance of the MFCC.
This is probably due to the masking effect that prevents the MFCCiter to
iteratively identify functional cellwise outlier in the Phase I sample and,
thus, makes it equivalent to MFCC. The performance of the non-functional
methods are totally unsatisfactory because they are not able to both capture
the functional nature of the data and successfully deal with functional
cellwise outliers. Specifically, M is the worst method overall readily
followed by Miter and MRo. Furthermore, as the contamination level increases,
the competing functional methods lose their favourable performance and tend to
achieve performance comparable to the non-functional approaches. This shows
that if outliers are not properly dealt with, terrible performance could arise
independently of the suitability of the methods considered.
From Figure 5, also in Scenario 2, RoMFCC is clearly the best method. However,
in this scenario the difference in performance between the RoMFCC and the
competing methods is less pronounced than in Scenario 1. This is expected
because the Phase I sample is now contaminated by functional casewise
outliers, which is the only contamination type against which the competing
methods are able to provide a certain degree of robustness. Note that, the
performance of RoMFCC are almost unaffected by contamination in the Phase I
sample also in this scenario, which proves the ability of the proposed method
to deal with both functional cellwise and casewise outliers.
## 4 Real-Case Study
To demonstrate the potential of the proposed RoMFCC in practical situations, a
real-case study in the automotive industry is presented henceforth. As
introduced in Section 1, it addresses the issue of monitoring the quality of
the RSW process, which guarantees the structural integrity and solidity of
welded assemblies in each vehicle (Martín et al., , 2014). The RSW process
(Zhang and Senkara, , 2011) is an autogenous welding process in which two
overlapping conventional steel galvanized sheets are joint together, without
the use of any filler material. Joints are formed by applying pressure to the
weld area from two opposite sides by means of two copper electrodes. Voltage
applied to the electrodes generates a current flowing between them through the
material. The electrical current flows because the resistance offered by
metals causes significant heat generation (Joule effect) that increases the
metal temperature at the faying surfaces of the work pieces up to the melting
point. Finally, due to the mechanical pressure of the electrodes, the molten
metal of the jointed metal sheets cools and solidifies, forming the so-called
weld nugget (Raoelison et al., , 2012). To monitor the RSW process, the modern
automotive Industry 4.0 framework allows the automatic acquisition of a large
volume of process parameters. The DRC is considered the most important of
these parameters to describe the quality of the RSW process. Further details
on how the typical behaviour of a DRC is related to the physical and
metallurgical development of a spot weld are provided by Capezza et al., 2021b
.
Data analyzed in this study are courtesy of Centro Ricerche Fiat and are
recorded at the Mirafiori Factory during lab tests and are acquired during RSW
processes made on the body of the Fiat 500BEV. A body is characterized by a
large number of spot welds with different characteristics, e.g, the thickness
and material of the sheets to be joined together and the welding time. In this
paper, we focus on monitoring a set of ten spot welds made on the body by one
of the welding machines. Therefore, for each sub-assembly the multivariate
functional quality characteristic is a vector of the ten DRCs, corresponding
to the the second welding pulse of ten spot welds normalized on the time
domain $[0,1]$, for a total number of assemblies equal to 1839. Moreover,
resistance measurements were collected at a regular grid of points equally
spaced by 1 ms.
The RSW process quality is directly affected by electrode wear since the
increase in weld numbers leads to changed electrical, thermal and mechanical
contact conditions at electrode and sheet interfaces (Manladan et al., ,
2017). Thus, to take into account the wear issue, electrodes go through
periodical renovations. In this setting, a paramount issue refers to the swift
identification of DRCs mean shifts caused by electrode wear, which could be
considered as a criterion for electrode life termination and guide the
electrode renovation strategy.
In the light of this, the 919 multivariate profiles corresponding to spot
welds made immediately before electrode renewal are used to form the Phase I
sample, whereas, the remaining 920 observations are used in Phase II to
evaluate the proposed chart performance. We expect that the mean shift of the
Phase II DRCs caused by electrode wear should be effectively captured by the
proposed control chart. The RoMFCC is implemented as in the simulation study
in Section 3 with the training and tuning sets, each composed by 460 Phase I
observations, randomly selected without remittance. As shown in Figure 1 of
Section 1, data are plausibly contaminated by several outliers. This is
further confirmed by Figure 6, which shows the boxplot of the functional
distance square roots $\sqrt{D_{i,fil}}$ (Equation (3)) obtained from the FUF
applied on the training set. Some components clearly show the presence of
functional cellwise outliers that are possibly arranged in groups, while other
components seem less severely contaminated.
Figure 6: Boxplot of the functional distance square roots $\sqrt{D_{i,fil}}$
(Equation (3)) obtained from the FUF applied on the training set.
Figure 7 shows the application of the proposed RoMFCC.
Figure 7: Hotelling’s $T^{2}$ and $SPE$ control charts for the RoMFCC in the
real-case study. The vertical line separates the monitoring statistics
calculated for the tuning set, on the left, and the Phase II data set on the
right, while the horizontal lines define the control limits.
The vertical line separates the monitoring statistics calculated for the
tuning set, on the left, and the Phase II data set on the right, while the
horizontal lines define the control limits. Note that, a significant number of
tuning set observations are signaled as OC, highlighted in red in Figure 7.
This is expected because these points may include functional casewise outlier
not filtered out by the FUF. In the monitoring phase, many points are signaled
as OCs by the RoMFCC. In particular, the RoMFCC signals 72.3% of the
observations in the Phase II data set as OC. This shows that the proposed
method is particularly sensible to mean shifts caused by an increased
electrode wear.
Finally, the proposed method is compared with the competing methods presented
in the simulation study in Section 3. Table 1 shows the estimated TDR values
$\widehat{TDR}$ on the Phase II sample for all the considered competing
methods. Similarly to Centofanti et al., 2021b , the uncertainty of
$\widehat{TDR}$ is quantified through a bootstrap analysis (Efron and
Tibshirani, , 1994). Table 1 reports the mean $\overline{TDR}$ of the
empirical bootstrap distribution of $\widehat{TDR}$, and the corresponding
bootstrap 95% confidence interval (CI) for each monitoring method.
| $\widehat{TDR}$ | $\overline{TDR}$ | CI
---|---|---|---
M | 0.336 | 0.335 | [0.305,0.368]
Miter | 0.462 | 0.461 | [0.428,0.496]
MRo | 0.513 | 0.512 | [0.481,0.547]
MFCC | 0.541 | 0.541 | [0.511,0.574]
MFCCiter | 0.632 | 0.632 | [0.595,0.664]
RoMFCC | 0.723 | 0.723 | [0.695,0.753]
Table 1: Estimated TDR values $\hat{TDR}$ on the Phase II sample, mean
$\overline{T}DR$ of the empirical bootstrap distribution of $\hat{TDR}$, and
the corresponding bootstrap 95% confidence interval (CI) for each monitoring
method in the real-case study.
The bootstrap analysis shows that the RoMFCC outperforms the competing control
charts, indeed bootstrap 95% confidence intervals are strictly above those of
all considered monitoring approaches. As in Section 3, non-functional
approaches, i.e., M, Miter, MRo, show worse performance than the functional
counterparts because they are not able to satisfactorily capture the
functional nature of the data and robust approaches always improve the non-
robust ones. Therefore, the proposed RoMFCC stands out as the best method to
promptly identify OC conditions in the RWS process caused by an increased
electrode wear with a Phase I sample contaminated by functional outliers.
## 5 Conclusions
In this paper, we propose a new robust framework for the statistical process
monitoring of multivariate functional data, referred to as robust multivariate
functional control charts (RoMFCC). The RoMFCC is designed to assess the
presence of assignable causes of variation while being robust to both
functional casewise and cellwise outliers. The proposed method is suitable for
those industrial processes where many functional variables are available and
occasional outliers are produced, such as anomalies in the data acquisition
and data collected during a fault in the process. Specifically, the RoMFCC
framework is based on four main elements, i.e. a functional univariate filter
to identify functional cellwise outliers, to be replaced by missing values, a
robust functional data imputation of these missing values, a casewise robust
dimensionality reduction based on ROBPCA and a monitoring strategy based on
the Hotelling’s $T^{2}$ and $SPE$ control charts. These elements are combined
in a Phase II monitoring strategy where a set of Phase I observations, which
can be contaminated with both functional casewise and cellwise outliers, is
used for the design of the control chart.
To the best of the authors’ knowledge, the RoMFCC framework is the first
monitoring scheme that is able to monitor a multivariate functional quality
characteristic while being robust to functional casewise and cellwise
outliers. Indeed, methods already present in the literature either apply
robust approaches to multivariate scalar features extracted from the profiles
or use diagnostic approaches on the multivariate functional data to
iteratively remove outliers. However, the former are not able to capture the
functional nature of the data, while the latter are not able to deal with
functional cellwise outliers.
The performance of the RoMFCC framework is assessed through an extensive Monte
Carlo simulation study where it is compared with several competing monitoring
methods for multivariate scalar data and multivariate functional data. The
ability of the proposed method to estimate the distribution of the data
without removing observations while being robust to both functional casewise
and cellwise outliers allows the RoMFCC to outperform the competitors in all
the considered scenarios. Lastly, the practical applicability of the proposed
method is illustrated through a motivating real-case study, which addresses
the issue of monitoring the quality of a resistance spot-welding process in
the automotive industry. Also in this case, the RoMFCC shows better
performance than the competitors in the identification of out-of-control
condition of the dynamic resistance curves.
## Supplementary Materials
The Supplementary Materials contain additional details about the data
generation process in the simulation study (A), additional simulation results
(B), as well as the R code to reproduce graphics and results over competing
methods in the simulation study.
## Acknowledgments
The present work was developed within the activities of the project
ARS01_00861 “Integrated collaborative systems for smart factory - ICOSAF”
coordinated by CRF (Centro Ricerche Fiat Scpa - www.crf.it) and financially
supported by MIUR (Ministero dell’Istruzione, dell’Università e della
Ricerca).
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|
11institutetext: Free University of Bozen-Bolzano 11email:
<EMAIL_ADDRESS>22institutetext: University of Trento 22email:
<EMAIL_ADDRESS>33institutetext: University of Oslo & University of
Bergen 33email<EMAIL_ADDRESS>
# Non-Normal Modal Description Logics
(Extended Version)
Tiziano Dalmonte1 Andrea Mazzullo2 Ana Ozaki3 Nicolas Troquard1
###### Abstract
Modal logics are widely used in multi-agent systems to reason about actions,
abilities, norms, or epistemic states. Combined with description logic
languages, they are also a powerful tool to formalise modal aspects of
ontology-based reasoning over an object domain. However, the standard
relational semantics for modalities is known to validate principles deemed
problematic in agency, deontic, or epistemic applications. To overcome these
difficulties, weaker systems of so-called _non-normal_ modal logics, equipped
with _neighbourhood semantics_ that generalise the relational one, have been
investigated both at the propositional and at the description logic level. We
present here a family of _non-normal modal description logics_ , obtained by
extending $\smash{\mathcal{ALC}}$-based languages with non-normal modal
operators. For formulas interpreted on neighbourhood models over varying
domains, we provide a modular framework of terminating, correct, and complete
tableau-based satisfiability checking algorithms in NExpTime. For a subset of
these systems, we also consider a reduction to satisfiability on constant
domain relational models. Moreover, we investigate the satisfiability problem
in fragments obtained by disallowing the application of modal operators to
description logic concepts, providing tight ExpTime complexity results.
## 1 Introduction
_Modal logics_ are powerful tools used to represent and reason about actions
and abilities [10, 15], coalitions [30, 38], knowledge and beliefs [1, 8, 39,
25], obligations and permissions [2, 19, 41], etc. In combination with
_description logics_ , they give rise to _modal description logics_ [42, 17],
knowledge representation formalisms used for modal reasoning over an object
domain and with a good trade-off between expressive power and decidability.
The standard _relational semantics_ for modal operators is given in terms of
_frames_ consisting of a set of _possible worlds_ equipped with binary
_accessibility relations_. The foundations of modal description logics, so
far, have also mostly been studied with relational semantics. However, all the
modal systems interpreted with respect to this semantics, known as _normal_ ,
validate principles that have been considered problematic or debatable for
agency-based, coalitional, epistemic, or deontic applications, in that they
lead to unacceptable conclusions, e.g., _logical omniscience_ in epistemic
settings [39], as well as _agency_ or _deontic paradoxes_ in the
representation of agents’ abilities [15] and obligations [31, 3, 16].
To overcome these problems, a generalisation of relational semantics, known as
_neighbourhood semantics_ , was introduced by Scott [33] and Montague [26].
Since it avoids in general the problematic principles validated by relational
semantics, it has been used to interpret a number of _non-normal_ modal
logics, first studied by C.I. Lewis [24], Lemmon [23], Kripke [22], Segerberg
[34], and Chellas [11], among others. A _neighbourhood frame_ consists of a
set of worlds, each one associated with a “neighbourhood”, i.e., a set of
subsets of worlds. Intuitively, a subset of worlds can be thought of as
representing a fact in a model, namely, those worlds where that fact holds.
Hence, the idea is that every world is assigned to a collection of facts,
those that are brought about, known, obligatory, etc., in that world of the
model.
These are the neighbourhood semantics ingredients for _propositional_ non-
normal modal logics. A further line of research focuses on the behaviour of
modal operators interpreted on neighbourhood frames in combination with
_first-order_ logic. In this direction, completeness results for first-order
non-normal modal logics have been provided [4, 5]. In addition, _non-normal
modal_ _description logics_ , extending standard description logics, with
modal operators interpreted on neighbourhood frames, have been considered for
knowledge representation applications [35, 13, 14], also in multi-agent
coalitional settings [37, 36].
To illustrate the expressivity of non-normal modal description logic
languages, as well some of the limitations of relational frames behind
adoption of neighbourhood semantics, we provide an example based on a classic
multi-agent purchase choreography scenario [27] (see the Appendix for a
detailed version). Our multi-agent setting involves a customer $\mathit{c}$
and a seller $\mathit{s}$, as well as agency operators $\mathbb{D}_{i}$ and
$\mathbb{C}_{i}$, for $i\in\\{c,s\\}$, read as ‘agent $i$ does/makes’ and
‘agent $i$ can do/make’, respectively [15, 20]. The formula
$\mathsf{Ord}\equiv\mathbb{D}_{c}\exists\mathsf{req}.(\mathsf{Prod}\sqcap\mathsf{InCatal})$
defines an order $\mathsf{Ord}$ as a request made by customer $c$ of an in-
catalogue product.
By stating
$\exists\mathsf{req}.(\mathsf{Prod}\sqcap\mathsf{InCatal})\sqsubseteq\mathsf{Confirm}\sqcup\lnot\mathsf{Confirm},$
we can also enforce that any request of an in-catalogue product is either
confirmed or not confirmed. However, relational semantics validates the so-
called _$\mathbf{M}$ -principle_ (often called _monotonicity_) as well,
according to which $C\sqsubseteq D$ always entails
$\mathbb{D}_{c}C\sqsubseteq\mathbb{D}_{c}D$, for any concepts $C,D$. Thus,
from the $\mathbf{M}$-principle and $\mathsf{Ord}$ definition, we obtain
$\mathsf{Ord}\sqsubseteq\mathbb{D}_{c}(\mathsf{Confirm}\sqcup\lnot\mathsf{Confirm}),$
meaning that any order is made confirmed or not confirmed by $c$. This is an
unwanted conclusion in our agency-based scenario, since customers’ actions
should be unrelated to order confirmation aspects.111Other approaches (out of
the scope of this paper) to avoid such consequences would involve rejecting
the principle of _excluded middle_ , as done e.g. in _intuitionistic
description logics_ [29, 9, 32].
Moreover, the formula
$\mathsf{SubmitOrd}\sqsubseteq\mathbb{C}_{s}\mathsf{Confirm}\sqcap\mathbb{C}_{s}\mathsf{PartConf}\sqcap\mathbb{C}_{s}\mathsf{Reject}$
states that a submitted order can be confirmed, can be partially confirmed,
and can be rejected by the seller $s$. On relational frames,
$\mathbb{C}_{s}C\sqcap\mathbb{C}_{s}D\sqsubseteq\mathbb{C}_{s}(C\sqcap D)$ is
a valid formula, for any concepts $C,D$, known as the _$\mathbf{C}$
-principle_ (or _agglomeration_). Therefore, by the $\mathbf{C}$-principle,
under relational semantics we would be forced to conclude that
$\mathsf{SubmitOrd}\sqsubseteq\mathbb{C}_{s}(\mathsf{Confirm}\sqcap\mathsf{PartConf}\sqcap\mathsf{Reject}),$
meaning that any submitted order is such that the seller $s$ has the ability
to make it confirmed, partially confirmed, and rejected, all _at once_ , which
is unreasonable.
Finally, consider the formula
$\top\sqsubseteq\mathsf{Confirm}\sqcup\lnot\mathsf{Confirm},$ i.e., the truism
stating that anything is either confirmed or not confirmed. By the so called
_$\mathbf{N}$ -principle_ (or _necessitation_) of relational semantics, we
have that if $\top\sqsubseteq C$ is valid on relational frames, then
$\top\sqsubseteq\mathbb{D}_{c}C$ holds as well, for any concept $C$. Thus,
from the $\mathbf{N}$-principle it would follow on relational semantics that
$\top\sqsubseteq\mathbb{D}_{c}(\mathsf{Confirm}\sqcup\lnot\mathsf{Confirm}),$
thereby forcing us to the consequence that every object is made by customer
$c$ to be either confirmed or not confirmed, hence leading again to an
unreasonable connection between customer’s actions and confirmation of orders.
The $\mathbb{D}_{i}$ and $\mathbb{C}_{i}$ modalities are axiomatised similarly
to [15], by means of additional principles as well: $\mathbb{D}_{i}$ obeys the
$\mathbf{C}$\- (seen above) and _$\mathbf{T}$ -principle_
($\mathbb{D}_{w}C\sqsubseteq C$), stating a _factivity of actions_ principle,
well-known also in epistemic logic; and both satisfy the _$\mathbf{Q}$
-principle_ ($\top\sqsubseteq\lnot\mathbb{D}_{c}\top$), asserting a principle
of _impotence towards tautologies_ that is unsatisfiable in relational frames,
but admissible over neighbourhood ones, and the _$\mathbf{E}$ -principle_
($C\equiv D$ entails $\mathbb{D}_{i}C\equiv\mathbb{D}_{i}D$ and
$\mathbb{C}_{i}C\equiv\mathbb{C}_{i}D$), valid both on relational and
neighbourhood frames.
In this paper, which is an extension of [13, 14], we investigate reasoning in
a family of non-normal modal description logics, providing terminating, sound,
and complete tableau algorithms for checking formula satisfiability on
neighbourhood models based on _varying domains_ of objects. Moreover, we study
the complexity of reasoning in a restricted fragment that disallows modalities
on description logic concepts. Finally, for two modal description logics
interpreted on _constant domain_ neighbourhood models, we adjust a reduction
(known from the propositional case) to satisfiability with respect to standard
relational semantics.
The paper is structured as follows. Section 2 provides the necessary
definitions and the preliminary results on non-normal modal description
logics. In Section 3 we present the tableau algorithms for the family of
logics here considered. The case of fragments without modalised concepts is
then studied in Section 4. Section 5 contains the results for the constant
domain case. Finally, Section 6 concludes the paper, discussing related work
and possible future research directions.
## 2 Preliminaries
Here we introduce modal description logics, first presenting their syntax, and
then their semantics based on neighbourhood and relational models,
respectively. Finally, we introduce the family of frame conditions here
considered.
### 2.1 Syntax
Let ${\sf N_{C}}$, ${\sf N_{R}}$ and ${\sf N_{I}}$ be countably infinite and
pairwise disjoint sets of _concept_ , _role_ , and _individual names_ ,
respectively. An $\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ _concept_ is an
expression of the form $C::=A\mid\lnot C\mid C\sqcap C\mid\exists
r.C\mid\Box_{i}C,$ where $A\in{\sf N_{C}}$, $r\in{\sf N_{R}}$, and $\Box_{i}$
such that $i\in J=\\{1,\ldots,n\\}$. An
_$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ atom_ is a _concept inclusion_
(_CI_) of the form $(C\sqsubseteq D)$, or an _assertion_ of the form $C(a)$ or
$r(a,b)$, with $C,D$ $\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ concepts,
$r\in{\sf N_{R}}$, and $a,b\in{\sf N_{I}}$. An
_$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ formula_ has the form
$\varphi::=\pi\mid\neg\varphi\mid\varphi\land\varphi\mid\Box_{i}\varphi,$
where $\pi$ is an $\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ atom and $i\in
J$. We use the following standard definitions for concepts: $\forall
r.C:=\lnot\exists r.\lnot C$; $(C\sqcup D):=\lnot(\lnot C\sqcap\lnot D)$;
$\bot:=A\sqcap\lnot A$, $\top:=A\sqcup\lnot A$ (for an arbitrarily fixed
$A\in{\sf N_{C}}$); $\Diamond_{i}C:=\lnot\Box_{i}\lnot C$. Concepts of the
form $\Box_{i}C$, $\Diamond_{i}C$ are _modalised concepts_. Analogous
conventions hold for formulas, writing $C\equiv D$ for $(C\sqsubseteq
D)\land(D\sqsubseteq C)$ and setting $\mathsf{false}:=(\top\sqsubseteq\bot)$,
$\mathsf{true}:=(\bot\sqsubseteq\top)$.
### 2.2 Semantics
We now define neighbourhood semantics, which (as already mentioned) can be
seen as a generalisation of the relational semantics, introduced immediately
after.
#### 2.2.1 Neighbourhood Semantics
A _neighbourhood frame_ , or simply _frame_ , is a pair
$\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$, where
$\mathcal{W}$ is a non-empty set of _worlds_ and
$\mathcal{N}_{i}\colon\mathcal{W}\rightarrow 2^{2^{\mathcal{W}}}$ is a
_neighbourhood function_ , for each _agent_ $i\in J=\\{1,\ldots,n\\}$. An
_$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ varying domain neighbourhood
model_, or simply _model_ , based on a neighbourhood frame $\mathcal{F}$ is a
pair $\mathcal{M}=(\mathcal{F},\mathcal{I})$, where
$\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$ is a neighbourhood
frame and $\mathcal{I}$ is a function associating with every $w\in\mathcal{W}$
an _$\smash{\mathcal{ALC}}$ interpretation_
$\mathcal{I}_{w}=(\Delta_{w},\cdot^{\mathcal{I}_{w}})$, with non-empty
_domain_ $\Delta_{w}$, and where $\cdot^{\mathcal{I}_{w}}$ is a function such
that: for all $A\in{\sf N_{C}}$, $A^{\mathcal{I}_{w}}\subseteq\Delta_{w}$; for
all $r\in{\sf N_{R}}$,
$r^{\mathcal{I}_{w}}\subseteq\Delta_{w}{\times}\Delta_{w}$; for all $a\in{\sf
N_{I}}$, $a^{\mathcal{I}_{w}}\in\Delta_{w}$. An
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ _constant domain neighbourhood
model_ is defined in the same way, except that, for all
$w,w^{\prime}\in\mathcal{W}$, we have that $\Delta_{w}=\Delta_{w^{\prime}}$
and, for all $u,v\in\mathcal{W}$, we require
$a^{\mathcal{I}_{u}}=a^{\mathcal{I}_{v}}$ (denoted by $a^{\mathcal{I}}$), that
is, individual names are _rigid designators_. We often write
$\mathcal{M}=(\mathcal{F},\Delta,\mathcal{I})$ to denote a constant domain
neighbourhood model $\mathcal{M}=(\mathcal{F},\mathcal{I})$ with domain
$\Delta=\Delta_{w}$, for every $w\in\mathcal{W}$. Given a model
$\mathcal{M}=(\mathcal{F},\mathcal{I})$ and a world $w\in\mathcal{W}$ of
$\mathcal{F}$ (or simply _$w$ in $\mathcal{F}$_), the _interpretation
$C^{\mathcal{I}_{w}}$ of a concept $C$ in $w$_ is defined as: $(\neg
D)^{\mathcal{I}_{w}}=\Delta_{w}\setminus D^{\mathcal{I}_{w}},\quad(D\sqcap
E)^{\mathcal{I}_{w}}=D^{\mathcal{I}_{w}}\cap E^{\mathcal{I}_{w}},$ $(\exists
r.D)^{\mathcal{I}_{w}}=\\{d\in\Delta_{w}\mid\exists e\in
D^{\mathcal{I}_{w}}{:}(d,e)\in r^{\mathcal{I}_{w}}\\},$
$(\Box_{i}D)^{\mathcal{I}_{w}}=\\{d\in\Delta_{w}\mid\llbracket
D\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(w)\\},$ where, for all
$d\in\bigcup_{w\in\mathcal{W}}\Delta_{w}$, the set $\llbracket
D\rrbracket^{\mathcal{M}}_{d}=\\{v\in\mathcal{W}\mid d\in
D^{\mathcal{I}_{v}}\\}$ is called the _truth set of $D$ with respect to
$\mathcal{M}$ and $d$_. We say that a concept $C$ is _satisfied in
$\mathcal{M}$_ if there is $w$ in $\mathcal{F}$ such that
$C^{\mathcal{I}_{w}}\neq\emptyset$, and that $C$ is _satisfiable_ (over
varying or constant neighbourhood models, respectively) if there is a (varying
or constant domain, respectively) neighbourhood model in which it is
satisfied. The _satisfaction of an $\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$
formula $\varphi$ in $w$ of $\mathcal{M}$_, written
$\mathcal{M},w\models\varphi$, is defined as follows:
$\displaystyle\mathcal{M},w\models C\sqsubseteq D$ iff $\displaystyle
C^{\mathcal{I}_{w}}\subseteq D^{\mathcal{I}_{w}},$ $\displaystyle\ \
\mathcal{M},w\models C(a)$ iff $\displaystyle a^{\mathcal{I}_{w}}\in
C^{\mathcal{I}_{w}},$ $\displaystyle\mathcal{M},w\models r(a,b)$ iff
$\displaystyle(a^{\mathcal{I}_{w}},b^{\mathcal{I}_{w}})\in
r^{\mathcal{I}_{w}},$ $\displaystyle\ \ \mathcal{M},w\models\neg\psi$ iff
$\displaystyle\mathcal{M},w\not\models\psi,$
$\displaystyle\mathcal{M},w\models\psi\land\chi$ iff
$\displaystyle\mathcal{M},w\models\psi\text{ and }\mathcal{M},w\models\chi,$
$\displaystyle\ \ \mathcal{M},w\models\Box_{i}\psi$ iff
$\displaystyle\llbracket\psi\rrbracket^{\mathcal{M}}\in\mathcal{N}_{i}(w),$
where
$\llbracket\psi\rrbracket^{\mathcal{M}}=\\{v\in\mathcal{W}\mid\mathcal{M},v\models\psi\\}$
is the _truth set of $\psi$_. As a consequence of the above definition, we
obtain the following condition for $\Diamond_{i}$ formulas:
$\mathcal{M},w\models\Diamond_{i}\psi$ iff
$\llbracket\neg\psi\rrbracket^{\mathcal{M}}\notin\mathcal{N}_{i}(w)$. Given a
neighbourhood frame $\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$
and a neighbourhood model $\mathcal{M}=(\mathcal{F},\mathcal{I})$, we say that
$\varphi$ is _satisfied in $\mathcal{M}$_ if there is $w\in\mathcal{W}$ such
that $\mathcal{M},w\models\varphi$, and that $\varphi$ is _satisfiable_ (over
varying or constant domain neighbourhood models, respectively) if it is
satisfied in some (varying or constant domain, respectively) neighbourhood
model. Also, $\varphi$ is _valid in $\mathcal{M}$_,
$\mathcal{M}\models\varphi$, if it is satisfied in all $w$ of $\mathcal{M}$,
and it is _valid on $\mathcal{F}$_ if, for all $\mathcal{M}$ based on
$\mathcal{F}$, $\varphi$ is valid in $\mathcal{M}$, writing
$\mathcal{F}\models\varphi$.
#### 2.2.2 Relational Semantics
A _relational frame_ is a pair $F=(W,\\{R_{i}\\}_{i\in J})$, with $W$ non-
empty set and $R_{i}$ binary relation on $W$, for $i\in J=\\{1,\ldots,n\\}$.
An _$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ (constant domain) relational
model_ based on a relational frame $F=(W,\\{R_{i}\\}_{i\in J})$ is a pair
$M=(F,I)$, where $I$ is a function associating with every $w\in W$ an
$\smash{\mathcal{ALC}}$ _interpretation_ $I_{w}=(\Delta,\cdot^{I_{w}})$,
having non-empty _constant domain_ $\Delta$, and where $\cdot^{I_{w}}$ is a
function such that: for all $A\in{\sf N_{C}}$, $A^{I_{w}}\subseteq\Delta$; for
all $r\in{\sf N_{R}}$, $r^{I_{w}}\subseteq\Delta{\times}\Delta$; for all
$a\in{\sf N_{I}}$, $a^{I_{w}}\in\Delta$, and for all $u,v\in W$,
$a^{I_{u}}=a^{I_{v}}$(denoted by $a^{I}$). Given a relational model $M=(F,I)$
and a world $w\in W$ of $F$ (or simply $w$ in $F$), the _interpretation of a
concept $C$ in $w$_, written $C^{I_{w}}$, is defined by taking: $(\neg
C)^{I_{w}}=\Delta\setminus C^{I_{w}},$ $(C\sqcap D)^{I_{w}}=C^{I_{w}}\cap
D^{I_{w}},$ $(\exists r.C)^{I_{w}}=\\{d\in\Delta\mid\exists e\in
C^{I_{w}}{:}(d,e)\in r^{I_{w}}\\},$ $(\Box_{i}C)^{I_{w}}=\\{d\in\Delta\mid\
\forall v\in W:wR_{i}v\Rightarrow d\in C^{I_{v}}\\}.$
A concept $C$ is _satisfied in $M$_ if there is $w$ in $F$ such that
$C^{I_{w}}\neq\emptyset$, and that $C$ is _satisfiable on relational models_
if there is a relational model in which it is satisfied. The _satisfaction of
a $\smash{\mathcal{ML}_{\mathcal{ALC}}}$ formula $\varphi$ in $w$ of $M$_,
written $M,w\models\varphi$, is defined, for atoms, negation and conjunction,
similarly to the previous case, and as follows for the $\Box_{i}$ case:
$M,w\models\Box_{i}\varphi\text{ \ iff \ }\forall v\in W:wR_{i}v\Rightarrow
M,v\models\varphi.$ Given a relational frame $F=(W,\\{R_{i}\\}_{i\in J})$ and
a relational model $M=(F,\Delta,I)$, we say that $\varphi$ is _satisfied in
$M$_ if there is $w\in W$ such that $M,w\models\varphi$, and that $\varphi$ is
_satisfiable on relational models_ if it is satisfied in some relational
model. Also, $\varphi$ is said to be _valid in $M$_, $M\models\varphi$, if it
is satisfied in all $w$ of $M$, and it is _valid on $F$_ if, for all $M$ based
on $F$, $\varphi$ is valid in $M$, writing $F\models\varphi$.
### 2.3 Frame Conditions and Formula Satisfiability
We consider the following conditions on neighbourhood frames
$\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$. We say that
_$\mathcal{F}$ satisfies the_:
$\mathbf{E}$-condition iff $\mathcal{N}_{i}$ is a neighbourhood function;
$\mathbf{M}$-condition iff $\alpha\in\mathcal{N}_{i}(w)$ and
$\alpha\subseteq\beta$ implies $\beta\in\mathcal{N}_{i}(w)$;
$\mathbf{C}$-condition iff $\alpha\in\mathcal{N}_{i}(w)$ and
$\beta\in\mathcal{N}_{i}(w)$ implies $\alpha\cap\beta\in\mathcal{N}_{i}(w)$;
$\mathbf{N}$-condition iff $\mathcal{W}\in\mathcal{N}_{i}(w)$;
$\mathbf{T}$-condition iff $\alpha\in\mathcal{N}_{i}(w)$ implies $w\in\alpha$;
$\mathbf{D}$-condition iff $\alpha\in\mathcal{N}_{i}(w)$ implies
$\mathcal{W}\setminus\alpha\not\in\mathcal{N}_{i}(w)$; $\mathbf{P}$-condition
iff $\emptyset\not\in\mathcal{N}_{i}(w)$; $\mathbf{Q}$-condition iff
$\mathcal{W}\not\in\mathcal{N}_{i}(w)$;
for every $w\in\mathcal{W}$, $\alpha,\beta\subseteq\mathcal{W}$. Combinations
of conditions, such as the $\mathbf{EMCN}$-condition, are obtained by suitably
joining the ones above. Moreover, since the $\mathbf{E}$-condition is always
satisfied by any neighbourhood frame, we often omit the letter $\mathbf{E}$
from this naming scheme, writing for instance ‘$\mathbf{MCN}$’ in place of
‘$\mathbf{EMCN}$’.
On the relationships among (combinations of) neighbourhood frame conditions,
we make the following observations.
###### Theorem 2.1 ().
Given a neighbourhood frame
$\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$, the following
statements hold, for $i\in J$.
1. 1.
If $\mathcal{N}_{i}$ satisfies the $\mathbf{MQ}$-condition then, for every
$w\in\mathcal{W}$, $\mathcal{N}_{i}(w)=\emptyset$. Hence, $\mathcal{N}_{i}$
satisfies all but the $\mathbf{N}$-condition.
2. 2.
$\mathcal{N}_{i}$ satisfies the $\mathbf{P}$-condition, if $\mathcal{N}_{i}$
satisfies one of the following:
(i) $\mathbf{MD}$-condition; (ii) $\mathbf{ND}$-condition; or (iii)
$\mathbf{T}$-condition.
3. 3.
$\mathcal{N}_{i}$ satisfies the $\mathbf{D}$-condition, if $\mathcal{N}_{i}$
satisfies one of the following:
(i) $\mathbf{CP}$-condition; or (ii) $\mathbf{T}$-condition.
4. 4.
$\mathcal{N}_{i}$ does not satisfy the $\mathbf{NQ}$-condition.
$\smash{\mathbf{EMCNT}}$$\smash{\mathbf{EMCNTD}}$$\smash{\mathbf{EMCNTP}}$$\smash{\mathbf{EMCTDP}}$$\smash{\mathbf{EMCND}}$$\smash{\mathbf{EMCNP}}$$\smash{\mathbf{EMCNDP}}$$\smash{\mathbf{EMCN}}$$\smash{\mathbf{EMCT}}$$\smash{\mathbf{EMCTD}}$$\smash{\mathbf{EMCTP}}$$\smash{\mathbf{EMCTDP}}$$\smash{\mathbf{EMCD}}$$\smash{\mathbf{EMCP}}$$\smash{\mathbf{EMCDP}}$$\smash{\mathbf{EMNT}}$$\smash{\mathbf{EMNTD}}$$\smash{\mathbf{EMNTP}}$$\smash{\mathbf{EMNTDP}}$$\smash{\mathbf{EMND}}$$\smash{\mathbf{EMNDP}}$$\smash{\mathbf{EMNP}}$$\smash{\mathbf{ECNT}}$$\smash{\mathbf{ECNTD}}$$\smash{\mathbf{ECNTP}}$$\smash{\mathbf{ECNTDP}}$$\smash{\mathbf{ECND}}$$\smash{\mathbf{ECNP}}$$\smash{\mathbf{ECNDP}}$$\smash{\mathbf{ECTQ}}$$\smash{\mathbf{ECTPQ}}$$\smash{\mathbf{ECTDQ}}$$\smash{\mathbf{ECTDPQ}}$$\smash{\mathbf{ECDQ}}$$\smash{\mathbf{ECPQ}}$$\smash{\mathbf{ECDPQ}}$$\smash{\mathbf{EDPQ}}$$\smash{\mathbf{EMC}}$$\smash{\mathbf{EMN}}$$\smash{\mathbf{EMT}}$$\smash{\mathbf{EMTD}}$$\smash{\mathbf{EMTP}}$$\smash{\mathbf{EMTDP}}$$\smash{\mathbf{EMD}}$$\smash{\mathbf{EMDP}}$$\smash{\mathbf{EMP}}$$\smash{\mathbf{ECN}}$$\smash{\mathbf{ECT}}$$\smash{\mathbf{ECTD}}$$\smash{\mathbf{ECTP}}$$\smash{\mathbf{ECTDP}}$$\smash{\mathbf{ECD}}$$\smash{\mathbf{ECP}}$$\smash{\mathbf{ECQ}}$$\smash{\mathbf{ENT}}$$\smash{\mathbf{ENTD}}$$\smash{\mathbf{ENTP}}$$\smash{\mathbf{EMTDP}}$$\smash{\mathbf{END}}$$\smash{\mathbf{ENP}}$$\smash{\mathbf{ETQ}}$$\smash{\mathbf{ETDQ}}$$\smash{\mathbf{ETDQ}}$$\smash{\mathbf{ETDPQ}}$$\smash{\mathbf{EDQ}}$$\smash{\mathbf{EDP}}$$\smash{\mathbf{EPQ}}$$\smash{\mathbf{EM}}$$\smash{\mathbf{EC}}$$\smash{\mathbf{EN}}$$\smash{\mathbf{ET}}$$\smash{\mathbf{ETD}}$$\smash{\mathbf{ETP}}$$\smash{\mathbf{ETDP}}$$\smash{\mathbf{ED}}$$\smash{\mathbf{EP}}$$\smash{\mathbf{EQ}}$$\smash{\mathbf{E}}$
Figure 1: Implications among $L$-conditions in $\smash{\mathsf{Pantheon}}$
(equivalent ones are listed in the same nodes, with underlined
representatives).
Based on these results, Figure 1 depicts the relations between combinations of
frame conditions: nodes are (groups of equivalent) conditions (with the
canonical representative underlined), and arrows represent logical
implications. Any combination containing the $\mathbf{NQ}$-condition has been
omitted, as it leads to inconsistency (Theorem 2.1, Point 4). Moreover, due to
Theorem 2.1, Point 1, any combination that includes the
$\mathbf{MQ}$-condition is not considered, since for any neighbourhood frame
$\mathcal{F}$ satisfying such condition and any
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ concept $C$, we have
$\mathcal{F}\models\Box_{i}C\equiv\bot$, and similarly for formulas, hence
trivialising the modal operators. Thus, we consider in the remainder the set
$\smash{\mathsf{Pantheon}}$ of 39 non-equivalent combinations shown (as nodes
or canonical representatives) in Figure 1.
For $\mathit{L}\in\smash{\mathsf{Pantheon}}$, we say that a neighbourhood
frame $\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$, with
$J=\\{1,\ldots,n\\}$, is an _$L^{n}$ frame_ iff its neighbourhood functions
$\mathcal{N}_{i}$, for $i\in J$, satisfy the _$\mathit{L}$ -condition_,
obtained by combining the conditions associated with letters in $\mathit{L}$.
For a class of neighbourhood frames $\mathcal{C}$, the _satisfiability in
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ on_ (_varying_ or _constant domain_
, resp.) _neighbourhood models based on a frame in $\mathcal{C}$_ is the
problem of deciding whether an $\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$
formula is satisfied in a (varying or constant domain, resp.) neighbourhood
model based on a frame in $\mathcal{C}$. Satisfiability in
_$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ on_ (_varying_ or _constant domain_
, respectively) _neighbourhood models_ is satisfiability in
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ on (varying or constant domain,
resp.) neighbourhood models based on a frame in the class of $L^{n}$ frames.
Finally, _satisfiability in $\smash{\mathbf{K}^{n}_{\mathcal{ALC}}}$ on_
(_constant domain_) _relational models_ is satisfiability in
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ on relational models based on any
relational frame.
_${\mathbf{E}}$ -principle_ | $S\models C\equiv D$ implies $S\models\Box_{i}C\equiv\Box_{i}D$.
---|---
$S\models\varphi\leftrightarrow\psi$ implies
$S\models\Box_{i}\varphi\leftrightarrow\Box_{i}\psi$.
_${\mathbf{M}}$ -principle_ | $S\models C\sqsubseteq D$ implies $S\models\Box_{i}C\sqsubseteq\Box_{i}D$.
$S\models\varphi\to\psi$ implies $S\models\Box_{i}\varphi\to\Box_{i}\psi$.
_${\mathbf{C}}$ -principle_ | $S\models\Box_{i}C\sqcap\Box_{i}D\sqsubseteq\Box_{i}(C\sqcap D)$.
$S\models\Box_{i}\varphi\land\Box_{i}\psi\to\Box_{i}(\varphi\land\psi)$.
_${\mathbf{N}}$ -principle_ | $S\models\top\sqsubseteq C$ implies $S\models\top\sqsubseteq\Box_{i}C$.
$S\models\varphi$ implies $S\models\Box_{i}\varphi$.
_${\mathbf{T}}$ -principle_ | $S\models\Box_{i}C\sqsubseteq C$.
---|---
$S\models\Box_{i}\varphi\to\varphi$.
_${\mathbf{D}}$ -principle_ | $S\models\Box_{i}C\sqsubseteq\Diamond_{i}C$.
$S\models\Box_{i}\varphi\to\Diamond_{i}\varphi$.
_${\mathbf{P}}$ -principle_ | $S\models\top\sqsubseteq\lnot\Box_{i}\bot$.
$S\models\lnot\Box_{i}\mathsf{false}$.
_${\mathbf{Q}}$ -principle_ | $S\models\top\sqsubseteq\lnot\Box_{i}\top$.
$S\models\lnot\Box_{i}\mathsf{true}$.
Table 1: Principles over neighbourhood or relational frames and models $S$.
We now study the correspondence between conditions presented in Section 2.2
and the principles in Table 1, where $S$ is either a (neighbourhood or
relational) frame or a (neighbourhood or relational) model and the
$L$-principle is obtained by suitably combining the basic principles. We say
that the $L$-principle holds in $S$ if the corresponding expressions in Table
1 are satisfied. On the correspondence between the principles in Table 1 and
conditions over frames and models, we have the following results (see e.g.
[28] for the propositional case).
###### Proposition 1 ().
Given a neighbourhood frame $\mathcal{F}$, the $\mathit{L}$-principle holds in
$\mathcal{F}$ iff $\mathcal{F}$ satisfies the $\mathit{L}$-condition.
###### Proposition 2 ().
The following statements hold.
1. 1.
For a (varying or constant domain) neighbourhood model $\mathcal{M}$, we have
that if $\mathcal{M}$ satisfies the $\mathit{L}$-condition, then the
$\mathit{L}$-principle holds in $\mathcal{M}$. However, in general, the
converse is not true.
2. 2.
For a relational frame $F$ and a relational model $M$ based on $F$, the
$\mathbf{EMCN}$-principle holds in $M$, hence in $F$. Moreover, in $M$, hence
in $F$, the $\mathbf{D}$-principle holds iff the $\mathbf{P}$-principle holds,
and the $\mathbf{Q}$-principle does not hold.
## 3 Tableaux for Formula Satisfiability
We provide terminating, sound, and complete tableau algorithms to check
satisfiability of formulas in varying domain neighbourhood models. The
notation partly adheres to that of [17], while the model construction in the
soundness proof is based on the strategy of [12]. In this section, we use
concepts and formulas in _negation normal form_ (_NNF_) and, for this reason,
we consider all the logical connectives $\sqcup,\lor,\forall,\Diamond$ as
primitive, rather than defined. For a concept or formula $\gamma$, we denote
by $\dot{\lnot}\gamma$ the negation of $\gamma$ put in NNF, defined as
follows: a concept is in _NNF_ if negation occurs in it only in front of
concept names; a formula is in _NNF_ if all concepts in it are in NNF and
negation occurs in the formula only in front of CIs or assertions of the form
$r(a,b)$ (regarding assertions of the form $A(a)$, we recall that a formula
$\lnot\psi$, with $\psi=C(a)$, is equivalent to the assertion $D(a)$, with
$D=\lnot C$). Given an $\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ formula
$\varphi$, we assume without loss of generality that $\varphi$ is in NNF
(using De Morgan laws) and it contains CIs only of the form $\top\sqsubseteq
C$, since $C\sqsubseteq D$ is equivalent to $\top\sqsubseteq\lnot C\sqcup D$).
We denote by $\mathsf{con}(\varphi)$ and $\mathsf{for}(\varphi)$ the set of
subconcepts and subformulas of $\varphi$, respectively, and then we set
$\mathsf{con}_{\dot{\lnot}}(\varphi)=\mathsf{con}(\varphi)\cup\\{\dot{\lnot}C\mid
C\in\mathsf{con}(\varphi)\\}\cup\\{\top\\}$ and
$\mathsf{for}_{\dot{\lnot}}(\varphi)=\mathsf{for}(\varphi)\cup\\{\dot{\lnot}\psi\mid\psi\in\mathsf{for}(\varphi)\\}$.
The sets $\mathsf{rol}(\varphi)$ and $\mathsf{ind}(\varphi)$ are,
respectively, the sets of role names and of individual names occurring in
$\varphi$. Let
$\mathsf{Fg}(\varphi)=\mathsf{for}_{\dot{\lnot}}(\varphi)\cup\mathsf{con}_{\dot{\lnot}}(\varphi)\cup\mathsf{rol}(\varphi)\cup\mathsf{ind}(\varphi)$
be the _fragment induced by $\varphi$_.
Moreover, let ${\sf N_{V}}$ be a countable set of _variables_ , denoted by the
letters $u,v$. The _terms for $\varphi$_, denoted by the letters $x,y$, are
either individual names in $\mathsf{ind}(\varphi)$ or variables in ${\sf
N_{V}}$. We assume that the set of terms for $\varphi$ is strictly well-
ordered by the relation $<$. In addition, let $\mathsf{N_{L}}$ be a countable
set of _labels_. Given an $\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ formula
$\varphi$ and a label $n\in\mathsf{N_{L}}$, an _$n$ -labelled constraint for
$\varphi$_ takes the form $n:\psi$, or $n:C(x)$, or $n:r(x,y)$, where
$\psi\in\mathsf{for}_{\dot{\lnot}}(\varphi)$, $x,y$ are terms for $\varphi$,
$C\in\mathsf{con}_{\dot{\lnot}}(\varphi)$, and $r\in\mathsf{rol}(\varphi)$.
For every $n\in\mathsf{N_{L}}$, an _$n$ -labelled constraint system for
$\varphi$_ is a set $S_{n}$ of $n$-labelled constraints for $\varphi$. A
_labelled constraint for $\varphi$_ is an $n$-labelled constraint for
$\varphi$, for some $n\in\mathsf{N_{L}}$, and similarly for a _labelled
constraint system for $\varphi$_. A _completion set $\mathcal{T}$ for
$\varphi$_ is a non-empty union of labelled constraint systems for $\varphi$,
and we set $\mathsf{L}_{\mathcal{T}}=\\{n\in\mathsf{N_{L}}\mid
S_{n}\subseteq\mathcal{T}\\}$.
About terms, we adopt the following terminology. A term $x$ _occurs in
$S_{n}$_ if $S_{n}$ contains $n$-labelled constraints of the form $n:C(x)$ or
$n:r(\tau,\tau^{\prime})$, where $\tau=x$, or $\tau^{\prime}=x$, and
$n\in\mathsf{N_{L}}$. In addition, a variable $u$ is said to be _fresh for
$S_{n}$_ if $u$ does not occur in $S_{n}$. (These notions can be used with
respect to $\mathcal{T}$, whenever $S_{n}\subseteq\mathcal{T}$). Finally,
given variables $u,v$ in an $n$-labelled constraint system $S_{n}$, we say
that $u$ is _blocked by $v$ in $S_{n}$_ if $u>v$ and $\\{C\mid n:C(u)\in
S_{n}\\}\subseteq\\{C\mid n:C(v)\in S_{n}\\}$.
A completion set $\mathcal{T}$ contains a _clash_ if, for some
$m\in\mathsf{N_{L}}$, concept $C$, role $r$, and terms $x,y$, one of the
following holds: $\\{m:(\top\sqsubseteq C),m:\lnot(\top\sqsubseteq
C)\\}\subseteq\mathcal{T}$; or $\\{m:A(x),m:\lnot
A(x)\\}\subseteq\mathcal{T}$; or $\\{m:r(x,y),m:\lnot
r(x,y)\\}\subseteq\mathcal{T}$. A completion set that does not contain a clash
is _clash-free_.
For every $\mathit{L}\in\smash{\mathsf{Pantheon}}$, we associate to
$\mathit{L}$ the set of _$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ -rules_ from
Figure 2 (bottom part) containing $\mathsf{R}_{\land}$, $\mathsf{R}_{\sqcap}$,
$\mathsf{R}_{\lor}$, $\mathsf{R}_{\sqcup}$, $\mathsf{R}_{\exists}$,
$\mathsf{R}_{\forall}$, $\mathsf{R}_{\sqsubseteq}$,
$\mathsf{R}_{\not\sqsubseteq}$, $\mathsf{R}_{\mathit{L}}$, and
$\mathsf{R}_{\mathit{L}\mathbf{X}}$, for every
$\mathbf{X}\in\\{\mathbf{N,T,P,Q,D}\\}$ such that $\mathbf{X}\in\mathit{L}$.
Given $\mathit{L}\in\smash{\mathsf{Pantheon}}$, a completion set $\mathcal{T}$
is $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-_complete_ if no
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-rule is applicable to $\mathcal{T}$,
where $\gamma_{j}$ is either $\psi_{j}\in\mathsf{for}_{\dot{\lnot}}(\varphi)$
or $C_{j}(x_{j})$, with $C_{j}\in\mathsf{con}_{\dot{\lnot}}(\varphi)$, for
$j=1,\ldots,k$, and $\delta$ is either
$\chi\in\mathsf{for}_{\dot{\lnot}}(\varphi)$ or $D(y)$, with
$D\in\mathsf{con}_{\dot{\lnot}}(\varphi)$, with respect to the _application
conditions_ associated to each $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-rule
from Figure 2 (top part).
Application conditions
1. $(\mathsf{R}_{\land})$
$\\{n:\psi,n:\chi\\}\not\subseteq\mathcal{T}$;
2. $(\mathsf{R}_{\sqcap})$
$\\{n:C(x),n:D(x)\\}\not\subseteq\mathcal{T}$;
3. $(\mathsf{R}_{\lor})$
$\\{n:\psi,n:\chi\\}\cap\mathcal{T}=\emptyset$;
4. $(\mathsf{R}_{\sqcup})$
$\\{n:C(x),n:D(x)\\}\cap\mathcal{T}=\emptyset$;
5. $(\mathsf{R}_{\exists})$
$x$ is not blocked by any variable in $S_{n}$, there is no $y$ such that
$\\{n:r(x,y),n:C(y)\\}\subseteq\mathcal{T}$, and $v$ is the $<$-minimal
variable fresh for $S_{n}$;
6. $(\mathsf{R}_{\forall})$
$n:C(y)\notin\mathcal{T}$;
7. $(\mathsf{R}_{\sqsubseteq})$
either $x$ occurs in $S_{n}$ and $n:C(x)\notin\mathcal{T}$; or no term occurs
in $S_{n}$ and $x$ is the $<$-minimal variable fresh for $S_{n}$;
8. $(\mathsf{R}_{\not\sqsubseteq})$
there is no $x$ such that $n:\dot{\lnot}C(x)\in\mathcal{T}$ and $v$ is the
$<$-minimal variable fresh for $S_{n}$;
9. $(\mathsf{R}_{\mathit{L}})$
$m$ is fresh for $\mathcal{T}$, and there is no $o\in\mathsf{N_{L}}$ such that
$\\{o:\gamma_{1},\ldots,o:\gamma_{k},o:\delta\\}\subseteq\mathcal{T}$, or
$\\{o:\dot{\lnot}\gamma_{j},o:\dot{\lnot}\delta\\}\subseteq\mathcal{T}$, for
some $j\leq k$;
10. $(\mathsf{R}_{\mathit{L}\mathbf{N}})$
$m$ is fresh for $\mathcal{T}$, and there is no $o\in\mathsf{N_{L}}$ such that
$o:\gamma\in\mathcal{T}$;
11. $(\mathsf{R}_{\mathit{L}\mathbf{T}})$
$n:\gamma\not\in\mathcal{T}$;
12. $(\mathsf{R}_{\mathit{L}\mathbf{P}})$
$m$ is fresh for $\mathcal{T}$, and there is no $o\in\mathsf{N_{L}}$ such that
$\\{o:\gamma_{1},\ldots,o:\gamma_{k}\\}\subseteq\mathcal{T}$;
13. $(\mathsf{R}_{\mathit{L}\mathbf{Q}})$
$m$ is fresh for $\mathcal{T}$, and there is no $o\in\mathsf{N_{L}}$ such that
$o:\dot{\lnot}\gamma_{j}\in\mathcal{T}$, for some $j\leq k$;
14. $(\mathsf{R}_{\mathit{L}\mathbf{D}})$
$m$ is fresh for $\mathcal{T}$, and there is no $o\in\mathsf{N_{L}}$ such that
$\\{o:\gamma_{1},\ldots,o:\gamma_{k},o:\delta_{1},\ldots,o:\delta_{k}\\}\subseteq\mathcal{T}$,
or
$\\{o:\dot{\lnot}\gamma_{j},o:\dot{\lnot}\delta_{\ell}\\}\subseteq\mathcal{T}$,
for some $j\leq k$, $\ell\leq h$.
Rules
$n:\psi\land\chi$ ($\mathsf{R}_{\land}$) $n:\psi$ , $n:\chi$ $n:\psi\lor\chi$
($\mathsf{R}_{\lor}$) $n:\psi$ $n:\chi$ $n:C\sqcap D(x)$
($\mathsf{R}_{\sqcap}$) $n:C(x)$, $n:D(x)$ $n:C\sqcup D(x)$
($\mathsf{R}_{\sqcup}$) $n:C(x)$ $n:D(x)$
$n:\exists r.C(x)$ ($\mathsf{R}_{\exists}$) $n:r(x,v)$, $n:C(v)$ $n:\forall
r.C(x)$, $n:r(x,y)$ ($\mathsf{R}_{\forall}$) $n:C(y)$
$n:\top\sqsubseteq C$ ($\mathsf{R}_{\sqsubseteq}$) $n:C(x)$
$n:\lnot(\top\sqsubseteq C)$ ($\mathsf{R}_{\not\sqsubseteq}$)
$n:\dot{\lnot}C(v)$
$n:\Box_{i}\gamma_{1}$, $\ldots$, $n:\Box_{i}\gamma_{k}$,
$n:\Diamond_{i}\delta$ ($\mathsf{R}_{\mathit{L}}$) $m:\gamma_{1}$, $\ldots$,
$m:\gamma_{k}$ , $m:\delta$ $m:\dot{\lnot}\gamma_{1}$, $m:\dot{\lnot}\delta$
$\ \dots$ $m:\dot{\lnot}\gamma_{k}$, $m:\dot{\lnot}\delta$
$\underbrace{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{\qquad\qquad\qquad\qquad--\qquad\qquad\qquad\quad\
\ }}}_{\text{if }\mathbf{M}\not\in L}$ $n:\Diamond_{i}\gamma$
($\mathsf{R}_{\mathit{L}\mathbf{N}}$) $m:\gamma$
$n:\Box_{i}\gamma$ ($\mathsf{R}_{\mathit{L}\mathbf{T}}$) $n:\gamma$
$n:\Box_{i}\gamma_{1}$, $\ldots$, $n:\Box_{i}\gamma_{k}$
($\mathsf{R}_{\mathit{L}\mathbf{P}}$) $m:\gamma_{1}$, $\ldots$,
$m:\gamma_{k}$ $n:\Box_{i}\gamma_{1}$, $\ldots$, $n:\Box_{i}\gamma_{k}$
($\mathsf{R}_{\mathit{L}\mathbf{Q}}$) $m:\dot{\lnot}\gamma_{1}$ $\ \dots$
$m:\dot{\lnot}\gamma_{k}$
$n:\Box_{i}\gamma_{1}$ , $\ldots$ , $n:\Box_{i}\gamma_{k}$ ,
$n:\Box_{i}\delta_{1}$ , $\ldots$ , $n:\Box_{i}\delta_{h}$
($\mathsf{R}_{\mathit{L}\mathbf{D}}$) $m:\gamma_{1}$, $\ldots$,
$m:\gamma_{k}$, $m:\dot{\lnot}\gamma_{1}$, $\ \dots$
$m:\dot{\lnot}\gamma_{k}$, $\ \dots$ $m:\dot{\lnot}\gamma_{1}$, $\ \dots$
$m:\dot{\lnot}\gamma_{k}$, $m:\delta_{1}$, $\ldots$, $m:\delta_{h}$
$m:\dot{\lnot}\delta_{1}$ $m:\dot{\lnot}\delta_{1}$ $m:\dot{\lnot}\delta_{h}$
$m:\dot{\lnot}\delta_{h}$
$\underbrace{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}{\qquad\qquad\qquad\qquad--\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}}_{\text{if
}\mathbf{M}\not\in L}$
Figure 2: $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-rules, where $k,h\geq 1$
if $\mathbf{C}\in\mathit{L}$ and $k=h=1$ if $\mathbf{C}\not\in\mathit{L}$. In
the rules $\mathsf{R}_{\mathit{L}}$ and $\mathsf{R}_{\mathit{L}\mathbf{D}}$,
the number of possible expansions depend on whether $\mathbf{M}\in\mathit{L}$:
if $\mathbf{M}\in\mathit{L}$ only the first expansion is possible, if
$\mathbf{M}\notin\mathit{L}$ all other expansions are also possible.
The $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-rules essentially state how to
extend a completion set on the basis of the information contained in it.
Branching rules entail a _non-deterministic choice_ in the completion set
expansion.
For each $\mathit{L}\in\smash{\mathsf{Pantheon}}$, we now present a tableau-
based non-deterministic decision procedure for the
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ formula satisfiability problem on
varying domain neighbourhood models, based on Algorithm 1 (simply referred to
as _$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm_). We have that
a formula $\varphi$ is $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ satisfiable if
and only if there exists at least one execution of the
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm that constructs an
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-complete and clash-free completion
set for $\varphi$. This non-deterministic algorithm gives priority to non-
generating $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-rules, i.e., those that do
not introduce new variables or labels, with respect to generating ones, so to
minimise the size of the completion set constructed by its application, and
terminates in exponential time for every formula $\varphi$. Thus, we obtain
the following.
###### Theorem 3.1.
Satisfiability in $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ on varying domain
neighbourhood models is decidable in NExpTime.
Input: the initial completion set $\mathcal{T}:=\\{0:\varphi\\}$ of an
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ formula $\varphi$ in NNF.
Output: a completion set for $\varphi$, extending the initial one, that either
contains a clash, or is complete and clash-free.
1 while _$\mathcal{T}$ is clash-free and not
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-complete_ do
2 if _a rule
$\mathsf{R}\in\\{\mathsf{R}_{\land},\mathsf{R}_{\lor},\mathsf{R}_{\sqcap},\mathsf{R}_{\sqcup},\mathsf{R}_{\forall},\mathsf{R}_{\sqsubseteq},\mathsf{R}_{\mathit{L}\mathbf{T}}\\}$
is applicable to $\mathcal{T}$_ then
3 apply $\mathsf{R}$ to $\mathcal{T}$;
4 else if _a rule
$\mathsf{R}\in\\{\mathsf{R}_{\exists},\mathsf{R}_{\mathit{L}},\mathsf{R}_{\mathit{L}\mathbf{N}},\mathsf{R}_{\mathit{L}\mathbf{P}},\mathsf{R}_{\mathit{L}\mathbf{Q}},\mathsf{R}_{\mathit{L}\mathbf{D}}\\}$
is applicable to $\mathcal{T}$_ then
5 apply $\mathsf{R}$ to $\mathcal{T}$;
6
7 end while
Algorithm 1 $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm on
varying domain neighbourhood models for $\varphi$
As an immediate consequence of the correctness of the tableau we also obtain a
(constructive) proof of the following kind of _exponential model property_.
###### Corollary 1 ().
Every $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ satisfiable formula $\varphi$
has a model with at most $p(|\mathsf{Fg}(\varphi)|)$ worlds, if
$\mathbf{C}\notin\mathit{L}$, and at most $2^{q(|\mathsf{Fg}(\varphi)|)}$)
worlds, if $\mathbf{C}\in\mathit{L}$, each of them having a domain with at
most $2^{r(|\mathsf{Fg}(\varphi)|)}$ elements, with $p$, $q$, $r$ polynomial
functions.
## 4 Fragments without Modalised Concepts
Here we study fragments of $\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ without
modalised concepts. An _$\smash{\mathcal{ALC}\textnormal{-}\mathcal{ML}^{n}}$
formula_ is defined similarly to the
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ case, by disallowing modalised
concepts. Given $\mathit{L}\in\smash{\mathsf{Pantheon}}$, _satisfiability in
$\smash{{\mathcal{ALC}}\textnormal{-}{\mathit{L}^{n}}}$ on varying_ _domain
neighbourhood models_ is $\smash{\mathcal{ALC}\textnormal{-}\mathcal{ML}^{n}}$
satisfiability on varying domain neighbourhood models based on neighbourhood
frames in the respective class for $\mathit{L}$. An
_$\smash{\mathcal{ML}^{n}}$ formula_, instead, is defined analogously to
$\smash{\mathcal{ALC}\textnormal{-}\mathcal{ML}^{n}}$, except that we build it
from the standard propositional (rather than $\smash{\mathcal{ALC}}$) language
over a countable set of _propositional letters_ $\mathsf{N_{P}}$, disjoint
from ${\sf N_{C}}$, ${\sf N_{R}}$, and ${\sf N_{I}}$. The semantics of
$\smash{\mathcal{ML}^{n}}$ formulas is given in terms of _propositional
neighbourhood models_ (or simply _models_) $\mathcal{M}^{\sf
P}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J},\mathcal{V})$, where
$(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$ is a neighbourhood frame, with
$J=\\{1,\ldots,n\\}$ in the following, and $\mathcal{V}:{\sf N_{P}}\rightarrow
2^{\mathcal{W}}$ is a function mapping propositional letters to sets of worlds
(see [11, 40]). _Satisfiability in $\smash{\mathit{L}^{n}}$_ is satisfiability
in $\smash{\mathcal{ML}^{n}}$ on propositional neighbourhood models based on
neighbourhood frames in the respective class for $\mathit{L}$. A propositional
neighbourhood model based on a neighbourhood frame in the respective class for
$\mathit{L}$ is called _$\mathit{L}^{n}$ model_.
We prove tight complexity results for
$\smash{{\mathcal{ALC}}\textnormal{-}{\mathit{L}^{n}}}$ satisfiability on
varying domain neighbourhood models, where $L\in\smash{\mathsf{Pantheon}}$,
using the notion of a propositional abstraction of a formula (as in, e.g.,
[6]). Here, one can separate the satisfiability test into two parts, one for
the description logic dimension and one for the neighbourhood frame dimension.
For an $\smash{\mathcal{ALC}\textnormal{-}\mathcal{ML}^{n}}$ formula
$\varphi$, the _propositional abstraction_ $\varphi_{\sf prop}$ is the result
of replacing each $\smash{\mathcal{ALC}}$ atom $\pi$ in $\varphi$ by a
propositional variable $p_{\pi}\in{\sf N_{P}}$. Define the set
$\Sigma_{\varphi}=\\{p_{\pi}\in{\sf N_{P}}\mid\pi\text{ is an
$\smash{\mathcal{ALC}}$ atom in }\varphi\\}$. A (propositional neighbourhood)
$L^{n}$ model $\mathcal{M}^{\sf P}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in
J},\mathcal{V})$ is _$\Sigma_{\varphi}$ -consistent_ if, for all
$w\in\mathcal{W}$, the following $\smash{\mathcal{ALC}}$ formula is
satisfiable:
$\hat{\varphi}_{\mathcal{V},w}=\bigwedge_{p_{\pi}\in{\boldsymbol{f}}^{\mathcal{V},w}_{\varphi}}{\pi}\
\wedge\bigwedge_{p_{\pi}\in\Sigma_{\varphi}\setminus{\boldsymbol{f}}^{\mathcal{V},w}_{\varphi}}\neg{\pi},$
where
${\boldsymbol{f}}^{\mathcal{V},w}_{\varphi}=\\{p_{\pi}\in\Sigma_{\varphi}\mid
w\in\mathcal{V}(p_{\pi})\\}$. We formalise the connection between
$\smash{{\mathcal{ALC}}\textnormal{-}{\mathit{L}^{n}}}$ satisfiable formulas
and their propositional abstractions with the following lemma.
###### Lemma 1 ().
A formula $\varphi$ is $\smash{{\mathcal{ALC}}\textnormal{-}{\mathit{L}^{n}}}$
satisfiable on varying domain neighbourhood models iff $\varphi_{\sf prop}$ is
satisfied in a $\Sigma_{\varphi}$-consistent $L^{n}$ model.
We now introduce definitions and notation used to prove our complexity result
on fragments without modalised concepts. Let $\Sigma=\\{p_{\pi}\in{\sf
N_{P}}\mid\pi\text{ is an $\smash{\mathcal{ALC}}$}$ $\text{atom in
}\varphi\\}$, for a fixed but arbitrary
$\smash{\mathcal{ALC}\textnormal{-}\mathcal{ML}^{n}}$ formula $\varphi$, and
let $\phi$ be an $\smash{\mathcal{ML}^{n}}$ formula built from symbols in
$\Sigma$. We denote by ${\sf sub}(\phi)$ the set of subformulas of $\phi$
closed under single negation. A _valuation_ for $\phi$ is a function $\nu:{\sf
sub}(\phi)\rightarrow\\{0,1\\}$ that satisfies the conditions: (1) for all
$\neg\psi\in{\sf sub}(\phi)$, $\nu(\psi)=1$ iff $\nu(\neg\psi)=0$; (2) for all
$\psi_{1}\wedge\psi_{2}\in{\sf sub}(\phi)$, $\nu(\psi_{1}\wedge\psi_{2})=1$
iff $\nu(\psi_{1})=1$ and $\nu(\psi_{2})=1$; and (3) $\nu(\phi)=1$. A
valuation $\nu$ for $\phi$ is _$\Sigma$ -consistent_ if the following
$\smash{\mathcal{ALC}}$ formula is satisfiable:
$\textstyle\bigwedge_{\nu(p_{\pi})=1}\ {\pi}\ \wedge\
\bigwedge_{\nu(p_{\pi})=0}\ \neg{\pi},$ where $p_{\pi}\in\Sigma$. Lemma 2
establishes that satisfiability of $\phi$ in a $\Sigma$-consistent model is
characterised by the existence of a $\Sigma$-consistent valuation satisfying
suitable properties. In the following, we use $\mathsf{ff}$ as an abbreviation
for $p\land\neg p$, for a fixed but arbitrary $p\in\mathsf{N_{P}}$.
###### Lemma 2 ().
Given $\mathit{L}$ and an $\smash{\mathcal{ML}^{n}}$ formula $\phi$ built from
symbols in $\Sigma$ (defined as above), let:
$\boldsymbol{\kappa}=\begin{cases}|{\sf sub}({\phi})|,&\text{if
$\mathbf{C}\in\mathit{L}$}\\\ 1,&\text{if
$\mathbf{C}\not\in\mathit{L}$}\end{cases}.$
A formula $\phi$ is satisfied in a $\Sigma$-consistent $\mathit{L}^{n}$ model
iff there is a $\Sigma$-consistent valuation $\nu$ for $\phi$ such that, for
every $1\leq k\leq\boldsymbol{\kappa}$, if
$\Box_{i}\psi_{1},\dots,\Box_{i}\psi_{k},\Box_{i}\chi\in{\sf sub}(\phi)$,
$\nu(\Box_{i}\psi_{j})=1$ for all $1\leq j\leq k$, and $\nu(\Box_{i}\chi)=0$,
then
1. 1.
$(\bigwedge^{k}_{j=1}\psi_{j}\wedge\neg\chi)\vee\boldsymbol{\vartheta}$ is
satisfied in a $\Sigma$-consistent $\mathit{L}^{n}$ model, where:
$\boldsymbol{\vartheta}=\mathsf{ff}$, if $\mathbf{M}\in\mathit{L}$;
$\boldsymbol{\vartheta}=\bigvee^{k}_{j=1}(\neg\psi_{j}\wedge\chi)$, if
$\mathbf{M}\not\in\mathit{L}$; and
2. 2.
for $\mathbf{X}\in\\{\mathbf{N,T,P,Q,D}\\}$, if $\mathbf{X}\in\mathit{L}$,
then $\nu$ satisfies the condition $(\mathbf{X})$ below, for every $1\leq
k,h\leq\boldsymbol{\kappa}$:
* ($\mathbf{N}$)
if $\Box_{i}\psi\in{\sf sub}(\phi)$ and $\nu(\Box_{i}\psi)=0$, then $\neg\psi$
is satisfied in a $\Sigma$-consistent $\mathit{L}^{n}$ model;
* ($\mathbf{T}$)
if $\Box_{i}\psi\in{\sf sub}(\phi)$ and $\nu(\Box_{i}\psi)=1$ then
$\nu(\psi)=1$;
* ($\mathbf{P}$)
if $\Box_{i}\psi_{1},\dots,\Box_{i}\psi_{k}\in{\sf sub}(\phi)$ and
$\nu(\Box_{i}\psi_{j})=1$ for all $1\leq j\leq k$, then
$\bigwedge^{k}_{j=1}\psi_{j}$ is satisfied in a $\Sigma$-consistent
$\mathit{L}^{n}$ model;
* ($\mathbf{Q}$)
if $\Box_{i}\psi_{1},\dots,\Box_{i}\psi_{k}\in{\sf sub}(\phi)$ and
$\nu(\Box_{i}\psi_{j})=1$ for all $1\leq j\leq k$, then
$\bigvee^{k}_{j=1}\neg\psi_{j}$ is satisfied in a $\Sigma$-consistent
$\mathit{L}^{n}$ model;
* ($\mathbf{D}$)
if
$\Box_{i}\psi_{1},\dots,\Box_{i}\psi_{k},\Box_{i}\chi_{1},\dots,\Box_{i}\chi_{h}\in{\sf
sub}(\phi)$, $\nu(\Box_{i}\psi_{j})=1$ for all $1\leq j\leq k$, and
$\nu(\Box_{i}\chi_{\ell})=1$ for all $1\leq\ell\leq h$, then
$(\bigwedge^{k}_{j=1}\psi_{j}\land\bigwedge^{h}_{\ell=1}\chi_{\ell})\vee\boldsymbol{\eta}$
is satisfied in a $\Sigma$-consistent $\mathit{L}^{n}$ model, where:
$\boldsymbol{\eta}=\begin{cases}\mathsf{ff},&\text{if
$\mathbf{M}\in\mathit{L}$}\\\
\neg(\bigwedge^{k}_{j=1}\psi_{j})\land\neg(\bigwedge^{h}_{\ell=1}\chi_{\ell}),&\text{if
$\mathbf{M}\not\in\mathit{L}$}\end{cases}.$
By using Lemmas 1-2, the following theorem provides a procedure that runs in
exponential time to check
$\smash{{\mathcal{ALC}}\textnormal{-}{\mathit{L}^{n}}}$ satisfiability on
varying domains. Since $\smash{\mathcal{ALC}}$ formula satisfiability is
already ExpTime-hard, our upper bound is tight.
###### Theorem 4.1 ().
Satisfiability in $\smash{{\mathcal{ALC}}\textnormal{-}{\mathit{L}^{n}}}$ on
varying domain neighbourhood models is ExpTime-complete.
## 5 Reasoning on Constant Domain
We now study the complexity of the formula satisfiability problem in
$\smash{\mathbf{E}^{n}_{\mathcal{ALC}}}$ and
$\smash{\mathbf{EM}^{n}_{\mathcal{ALC}}}$ on constant domain neighbourhood
models. We provide a NExpTime upper bound for satisfiability in
$\smash{\mathbf{E}^{n}_{\mathcal{ALC}}}$ and
$\smash{\mathbf{EM}^{n}_{\mathcal{ALC}}}$ by using a reduction, lifted from
the propositional case, to multi-modal
$\smash{\mathbf{K}^{m}_{\mathcal{ALC}}}$. The translation $\cdot^{\dagger}$
from $\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ to
$\smash{\mathcal{ML}^{3n}_{\mathcal{ALC}}}$ is defined as [21, 18]:
$A^{\dagger}=A$, $(\lnot C)^{\dagger}=\lnot C^{\dagger}$, $(C\sqcap
D)^{\dagger}=C^{\dagger}\sqcap D^{\dagger}$, $(\exists r.C)^{\dagger}=\exists
r.C^{\dagger}$; $(C(a))^{\dagger}=C^{\dagger}(a)$,
$(r(a,b))^{\dagger}=r(a,b)$, $(C\sqsubseteq
D)^{\dagger}=C^{\dagger}\sqsubseteq D^{\dagger}$,
$(\lnot\psi)^{\dagger}=\lnot\psi^{\dagger}$,
$(\psi\land\chi)^{\dagger}=\psi^{\dagger}\land\chi^{\dagger}$;
$(\Box_{i}\gamma)^{\dagger}=\Diamond_{i_{1}}(\Box_{i_{2}}\gamma^{\dagger}\circ\Box_{i_{3}}\lnot\gamma^{\dagger})$;
where $A\in{\sf N_{C}}$, $r\in{\sf N_{R}}$, $\gamma$ is either an
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ concept or formula, and
$\circ\in\\{\sqcap,\land\\}$ accordingly. Using this translation, one can show
that satisfiability on neighbourhood models is reducible to satisfiability on
the relational models [21, 18]. Since satisfiability in
$\smash{\mathbf{K}^{3n}_{\mathcal{ALC}}}$ constant domain relational models is
NExpTime-complete [17, Theorem 15.15], we obtain the following complexity
result.
###### Theorem 5.1 ().
Satisfiability in $\smash{\mathbf{E}^{n}_{\mathcal{ALC}}}$ on constant domain
neighbourhood models is decidable in NExpTime.
The translation $\cdot^{\ddagger}$ from
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ to
$\smash{\mathcal{ML}^{2n}_{\mathcal{ALC}}}$ is defined as $\cdot^{\dagger}$ on
all concepts and formulas, except for the modalised concepts or formulas
$\gamma$:
$(\Box_{i}\gamma)^{\ddagger}=\Diamond_{i_{1}}\Box_{i_{2}}\gamma^{\ddagger}$.
We obtain an upper bound analogous to the one for
$\smash{\mathbf{E}^{n}_{\mathcal{ALC}}}$ by a reduction of the formula
satisfiability problem for $\smash{\mathbf{EM}^{n}_{\mathcal{ALC}}}$ to the
$\smash{\mathbf{K}^{2n}_{\mathcal{ALC}}}$ one [21, 18, 17].
###### Theorem 5.2 ().
Satisfiability in $\smash{\mathbf{EM}^{n}_{\mathcal{ALC}}}$ on constant domain
neighbourhood models is decidable in NExpTime.
## 6 Discussion
We investigated reasoning in non-normal modal description logics, focussing
on: $(i)$ tableaux algorithms to check satisfiability of multi-modal
description logics formulas in varying domain neighbourhood models based on
classes of frames for 39 different non-normal systems; $(ii)$ complexity of
satisfiability restricted to fragments with modal operators applied only over
formulas, and interpreted on varying domain models; $(iii)$ preliminary
reduction of formula satisfiability for two non-normal modal description
logics to satisfiability in the standard relational semantics on a constant
domain. We now discuss possible future work.
First, we intend to devise tableaux for formula satisfiability on
neighbourhood models with constant domain, by solving the problem of newly
introduced variables that do not occur in other previously expanded labelled
constraints systems. For instance, by applying the
$\mathbf{M}^{n}_{\smash{\mathcal{ALC}}}$-rules to the $n$-labelled constraint
system $S_{n}=\\{n:\Diamond_{i}\exists r.A(x),\Box_{i}\lnot A(x)\\}$, we get
the $m$-labelled constraint system $S_{m}=\\{m:\exists r.A(x),m:\lnot
A(x),m:r(x,y),m:A(y)\\}$. The fresh variable $y$ in $S_{m}$ does not allow for
the direct extraction of a constant domain model, as no object in the domain
of the world associated with $S_{n}$ would be capable of representing $y$
correctly. An alternative approach involves _quasimodels_ [17], to
characterise satisfiability on constant domain models in terms of structures
representing “abstractions” of the actual models of a formula. Objects across
worlds can be represented by means of _runs_ , i.e., functions to guarantee
their modal properties and the constant domain assumption. A similar strategy
is presented in [35, 37, 36], where the definition of runs (which is not
carried out in detail) involves the introduction of suitable world “copies”.
We conjecture that a quasimodel-based approach with _marked variables_ , as
illustrated in [17], can also be adopted to solve the constant domain model
extraction issue.
Moreover, we aim at tight complexities for
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ satisfiability, both in varying and
in constant domain models. While $\smash{\mathcal{ALC}}$ formula
satisfiability is ExpTime-complete, it is unclear whether the upper bound for
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ on varying or constant domain
neighbourhood models can be improved to ExpTime-membership, for any
$\mathit{L}\in\smash{\mathsf{Pantheon}}$. At the propositional level, the
formula satisfiability problem for the systems based on the $L$-condition,
with $\mathbf{C}\not\in L$, is NP-complete, rising to PSpace if the
$\mathbf{C}$-condition is included [40]. For normal modal description logics,
instead, the (tight) NExpTime-hardness results are based on complexity proofs
of _product logics_ over relational product frames [17], and cannot be
immediately adapted to neighbourhood semantics, where an analogous notion of
product is not yet well understood. Nonetheless, we conjecture that the
NExpTime-hardness known for, e.g., $\mathbf{K}_{\mathcal{ALC}}$ on constant
domain relational models, also holds in the neighbourhood case, at least in
presence of the $\mathbf{C}$-condition.
Finally, we plan to study: non-normal modal description logics in
_coalitional_ and _strategic_ settings [30, 38, 37], with an interplay between
abilities and powers of _groups_ of agents, rather than single ones;
additional description logics constructs (e.g. _nominals_ , _inverse roles_ ,
or _number restrictions_ [7]); and _interactions between modalities_ , with
axioms expressing e.g. that an agent _can do_ anything they _actually do_ , by
means of formulas of the form $\mathbb{D}_{i}C\sqsubseteq\mathbb{C}_{i}C$ or
$\mathbb{D}_{i}\varphi\to\mathbb{C}_{i}\varphi$.
## Acknowledgements
This research has been partially supported by the Province of Bolzano and DFG
through the project D2G2 (DFG grant n. 500249124). Andrea Mazzullo
acknowledges the support of the MUR PNRR project FAIR - Future AI Research
(PE00000013) funded by the NextGenerationEU. Ana Ozaki is supported by the
Research Council of Norway, project number 316022.
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## Appendix 0.A Modelling Scenario
In the following, we present the modelling of an example scenario in the
classic domain of multi-agent purchase choreography [27]. The aim is twofold.
First, it displays some of the limitations of modalities defined on relational
frames, motivating the adoption of neighbourhood semantics in modal extensions
of description logics. Second, it illustrates the expressivity of (non-normal)
modal description logic languages, showing interactions between modalities and
the constructs of the standard description logic $\smash{\mathcal{ALC}}$.
Our multi-agent setting involves a customer $\mathit{c}$, a marketplace
$\mathit{m}$, a seller $\mathit{s}$, and a warehouse $\mathit{w}$, with agency
operators $\mathbb{D}_{i}$ and $\mathbb{C}_{i}$, for $i\in\\{c,m,s,w\\}$, read
as ‘agent $i$ does/makes’ and ‘agent $i$ can do/make’, respectively [15, 20].
Concept names $\mathsf{Ord}$, $\mathsf{Prod}$, and $\mathsf{InCatal}$ are used
to represent, respectively, orders, products, and the class of objects
displayed as in-catalogue, while $\mathsf{req}$ is a role name for the request
relation. The formula
$\mathsf{Ord}\equiv\mathbb{D}_{c}\exists\mathsf{req}.(\mathsf{Prod}\sqcap\mathsf{InCatal})$
defines an order $\mathsf{Ord}$ as a request made by customer $c$ of an in-
catalogue product. Using the concept name $\mathsf{Confirm}$ to represent the
class of objects that are confirmed, we also have that
$\exists\mathsf{req}.(\mathsf{Prod}\sqcap\mathsf{InCatal})\sqsubseteq\mathsf{Confirm}\sqcup\lnot\mathsf{Confirm},$
meaning that any request of an in-catalogue product is either confirmed or not
confirmed. However, relational semantics validates the so-called _$\mathbf{M}$
-principle_ (often called _monotonicity_) as well, according to which
$C\sqsubseteq D$ always entails $\mathbb{D}_{c}C\sqsubseteq\mathbb{D}_{c}D$,
for any concepts $C,D$. Thus, from the $\mathbf{M}$-principle we would obtain
$\mathsf{Ord}\sqsubseteq\mathbb{D}_{c}(\mathsf{Confirm}\sqcup\lnot\mathsf{Confirm}),$
meaning that any order is made confirmed or not confirmed by $c$. This is an
unwanted conclusion in our agency-based scenario, since customers’ actions
should be unrelated to any aspect of order confirmation.
Moreover, assume that the concept name $\mathsf{SubmitOrd}$ stands for the
class of submitted orders, and that $\mathsf{PartConf}$ and $\mathsf{Reject}$
are used in our knowledge base to represent, respectively, the partially
confirmed and the rejected entities. Now consider the formula
$\mathsf{SubmitOrd}\sqsubseteq\mathbb{C}_{s}\mathsf{Confirm}\sqcap\mathbb{C}_{s}\mathsf{PartConf}\sqcap\mathbb{C}_{s}\mathsf{Reject},$
stating that a submitted order can be confirmed, can be partially confirmed,
and can be rejected by the seller $s$. On relational frames, however, we have
that $\mathbb{C}_{s}C\sqcap\mathbb{C}_{s}D\sqsubseteq\mathbb{C}_{s}(C\sqcap
D)$ is a valid formula, for any concepts $C,D$, known as the _$\mathbf{C}$
-principle_ (or _agglomeration_). Therefore, by the $\mathbf{C}$-principle,
under relational semantics we would be forced to conclude that
$\mathsf{SubmitOrd}\sqsubseteq\mathbb{C}_{s}(\mathsf{Confirm}\sqcap\mathsf{PartConf}\sqcap\mathsf{Reject}),$
meaning that any submitted order is such that the seller $s$ has the ability
to make it confirmed, partially confirmed, and rejected, all _at once_ , which
is unreasonable.
Consider now the formula
$\top\sqsubseteq\mathsf{Confirm}\sqcup\lnot\mathsf{Confirm},$ i.e., the truism
stating that anything is either confirmed or not confirmed. By the so called
_$\mathbf{N}$ -principle_ (or _necessitation_) of relational semantics, we
have that if $\top\sqsubseteq C$ is valid on relational frames, then
$\top\sqsubseteq\mathbb{D}_{c}C$ holds as well, for any concept $C$. Thus,
from the $\mathbf{N}$-principle of relational semantics it would follow that
$\top\sqsubseteq\mathbb{D}_{c}(\mathsf{Confirm}\sqcup\lnot\mathsf{Confirm}),$
thereby forcing us to the consequence that every object is made by customer
$c$ to be either confirmed or not confirmed, and hence leading again to an
unreasonable connection between customer’s actions and confirmation of orders.
In fact, since customer $c$ plays no role in confirmation actions, it is
sensible to assume that, for any object of the domain, it is not the case that
$c$ makes it confirmed or not confirmed. This can be achieved by the formula
$\top\sqsubseteq\lnot\mathbb{D}_{c}(\mathsf{Confirm}\sqcup\lnot\mathsf{Confirm}),$
an instance of a principle sometimes known as the _$\mathbf{Q}$ -principle_,
which is _unsatisfiable_ in relational frames, while admissible over
neighbourhood ones.
Finally, we consider additional principles that can be adopted both in
relational and neighbourhood semantics. The formula
$\mathbb{D}_{w}\mathsf{Avail}\sqsubseteq\mathsf{Avail}$ states that anything
that is _made_ available by the warehouse $w$ is _actually_ available. This is
an instance of the so-called _$\mathbf{T}$ -principle_ (also _factivity
principle_), well-known in modal logic, particularly for its epistemic
applications (if an agent _knows_ something, it has to be true). Both in
relational and neighbourhood semantics, the $\mathbf{T}$-principle entails the
so-called _$\mathbf{D}$ -principle_. This is instantiated by
$\mathbb{D}_{w}\mathsf{Avail}\sqsubseteq\lnot\mathbb{D}_{w}\lnot\mathsf{Avail},$
a formula asserting that anything that is _made available_ by the warehouse
$w$ is _not made unavailable_ by $w$. This principle, also well-known for its
epistemic implications (anything that is _known_ by an agent is _compatible_
with their knowledge), in relational semantics is _equivalent_ to a much
lesser known principle, sometimes called _$\mathbf{P}$ -principle_. An example
of it is given by the formula
$\top\sqsubseteq\lnot\mathbb{D}_{w}(\mathsf{Avail}\sqcap\lnot\mathsf{Avail}),$
which states that, for any object, it is not the case that the warehouse $w$
makes it both available _and_ unavailable. Neighbourhood semantics, under
which the $\mathbf{D}$\- and the $\mathbf{P}$-principle are _not equivalent_ ,
allows for distinctions between modal constraints that are hence more fine-
grained than in relational semantics.
$\begin{array}[]{@{}r@{~~}c@{~~}l@{\qquad}r@{~~}c<EMAIL_ADDRESS>\mathsf{Confirm}~{}~{}&\sqsubseteq\hfil~{}~{}&\lnot\mathsf{PartConf}&\mathsf{SubmitOrd}~{}~{}&\sqsubseteq\hfil~{}~{}&\mathbb{C}_{s}\forall\mathsf{req}.(\mathbb{D}_{w}\mathsf{Avail}\sqcup\mathbb{D}_{w}\lnot\mathsf{Avail})\\\
\mathsf{Confirm}~{}~{}&\sqsubseteq\hfil~{}~{}&\lnot\mathsf{Reject}&\mathsf{SubmitOrd}~{}~{}&\sqsubseteq\hfil~{}~{}&\mathbb{C}_{s}\mathsf{Confirm}\sqcap\mathbb{C}_{s}\mathsf{PartConf}\sqcap\mathbb{C}_{s}\mathsf{Reject}\\\
\mathsf{PartConf}~{}~{}&\sqsubseteq\hfil~{}~{}&\lnot\mathsf{Reject}&\mathbb{D}_{s}\mathsf{Confirm}~{}~{}&\sqsubseteq\hfil~{}~{}&\forall\mathsf{req}.\mathbb{D}_{w}\mathsf{Avail}\\\
\mathsf{Ord}~{}~{}&\equiv\hfil~{}~{}&\mathbb{D}_{c}\exists\mathsf{req}.(\mathsf{Prod}\sqcap\mathsf{InCatal})&\mathbb{C}_{s}\mathsf{PartConf}~{}~{}&\sqsupseteq\hfil~{}~{}&\exists\mathsf{req}.\mathbb{D}_{w}\mathsf{Avail}\
\sqcap\exists\mathsf{req}.\mathbb{D}_{w}\lnot\mathsf{Avail}\\\
\mathsf{SubmitOrd}~{}~{}&\equiv\hfil~{}~{}&\mathsf{Ord}\sqcap\mathbb{D}_{c}\mathsf{Submit}&\mathbb{C}_{s}\mathsf{Reject}~{}~{}&\sqsupseteq\hfil~{}~{}&\forall\mathsf{req}.\mathbb{D}_{w}\lnot\mathsf{Avail}\\\
\mathsf{IncomplOrd}~{}~{}&\equiv\hfil~{}~{}&\mathsf{Ord}\sqcap\lnot\mathbb{D}_{c}\mathsf{Submit}&\mathbb{D}_{m}(\mathsf{ConfirmOrd}~{}~{}&\sqsubseteq\hfil~{}~{}&\lnot\mathbb{C}_{s}\mathsf{Reject})\\\
\mathsf{InvalidOrd}~{}~{}&\equiv\hfil~{}~{}&\mathsf{Ord}\sqcap\lnot\mathbb{C}_{c}\mathsf{Submit}&\mathsf{DispatchOrd}~{}~{}&\equiv\hfil~{}~{}&\mathsf{Ord}\sqcap\mathbb{D}_{s}\mathbb{D}_{w}\mathsf{Dispatch}\\\
\mathsf{ConfirmOrd}~{}~{}&\equiv\hfil~{}~{}&\mathsf{Ord}\sqcap\mathbb{D}_{s}\mathsf{Confirm}&\mathbb{D}_{m}(\mathsf{ConfirmOrd}~{}~{}&\sqsubseteq\hfil~{}~{}&\mathsf{DispatchOrd})\end{array}$
Figure 3: Purchase choreography knowledge base.
The full knowledge base describing the purchase choreography scenario is
reported in Figure 3. Notably, it displays a great range of interactions
between modal and description logics constructs. The modal operators
$\mathbb{D}_{i}$ and $\mathbb{C}_{i}$ are axiomatized similarly to [15]:
$\mathbb{D}_{i}$ obeys the $\mathbf{C}$\- and $\mathbf{T}$-principles; and
both satisfy the $\mathbf{Q}$-principle, and the $\mathbf{E}$-principle:
$C\equiv D$ entails $\mathbb{D}_{i}C\equiv\mathbb{D}_{i}D$ and
$\mathbb{C}_{i}C\equiv\mathbb{C}_{i}D$.
On the left column, the first four axioms impose that the range of the request
relation is a product, and that the classes of confirmed, partially confirmed,
and rejected objects are all disjoint. We have already discussed the fifth
axiom, which is the definition of an order as a request, made by a customer,
for some in-catalogue product. The last four axioms on the left column define,
respectively: a submitted order as an order that is actually submitted by a
customer; an incomplete order as an order that is not submitted by the
customer; an invalid order as an order that cannot be submitted by the
customer; and a confirmed order as an order that is confirmed by the seller.
On the right column, the first formula states that an invalid order can be
cancelled by the seller. The second axiom asserts that, given a submitted
order, the seller can check at the warehouse for availability of all of its
requested products. The third formula, as already discussed, requires that a
submitted order can be confirmed, can be partially confirmed, and can be
rejected by the seller. The subsequent three axioms impose, respectively, the
following constraints: anything that is confirmed by the seller has to request
only products made available by the warehouse; anything that requests products
that are made available, as well as products that are brought about to be
unavailable by the warehouse, can be partially confirmed by the seller;
finally, any request of products that are all unavailable at the warehouse can
be rejected by the seller. The third formula from the bottom of the right
column states that the marketplace enforces that a confirmed order cannot be
rejected by the seller. The second last axiom defines a dispatched order as an
order that the seller makes the warehouse dispatch. Finally, the last formula
imposes that the marketplace brings it about that any confirmed order is also
dispatched.
## Appendix 0.B Proofs for Section 2
See 2.1
###### Proof.
Point 1. Suppose that, for every $w\in\mathcal{W}$,
$\alpha,\beta\subseteq\mathcal{W}$, we have: ($\mathbf{M}$-condition)
$\alpha\in\mathcal{N}_{i}(w)$ and $\alpha\subseteq\beta$ implies
$\beta\in\mathcal{N}_{i}(w)$; and ($\mathbf{Q}$-condition)
$\mathcal{W}\not\in\mathcal{N}_{i}(w)$. This means that
$\alpha\not\in\mathcal{N}_{i}(w)$, for every $\alpha\subseteq\mathcal{W}$,
i.e., $\mathcal{N}_{i}(w)=\emptyset$. Thus, every condition, except for the
$\mathbf{N}$-condition, is satisfied by $\mathcal{N}_{i}$.
Point 2. Suppose $(i)$ that, for every $w\in\mathcal{W}$,
$\alpha,\beta\subseteq\mathcal{W}$, we have: ($\mathbf{M}$-condition)
$\alpha\in\mathcal{N}_{i}(w)$ and $\alpha\subseteq\beta$ implies
$\beta\in\mathcal{N}_{i}(w)$; and ($\mathbf{D}$-condition)
$\alpha\in\mathcal{N}_{i}(w)$ implies
$\mathcal{W}\setminus\alpha\not\in\mathcal{N}_{i}(w)$. Towards a
contradiction, suppose that $\emptyset\in\mathcal{N}_{i}(w)$. Then, we have
$\mathcal{W}=\mathcal{W}\setminus\emptyset\not\in\mathcal{N}_{i}(w)$ as well.
By contraposition, this implies in particular that
$\emptyset\not\in\mathcal{N}_{i}(w)$, a contradiction. Thus,
($\mathbf{P}$-condition) $\emptyset\not\in\mathcal{N}_{i}(w)$
Moreover, suppose $(ii)$ that, for every $\alpha\subseteq\mathcal{W}$, we
have: ($\mathbf{N}$-condition) $\mathcal{W}\in\mathcal{N}_{i}(w)$, i.e.,
$\mathcal{W}\setminus\emptyset\in\mathcal{N}_{i}(w)$; and
($\mathbf{D}$-condition) $\alpha\in\mathcal{N}_{i}(w)$ implies
$\mathcal{W}\setminus\alpha\not\in\mathcal{N}_{i}(w)$. By contraposition, we
obtain that ($\mathbf{P}$-condition) $\emptyset\not\in\mathcal{N}_{i}(w)$.
Finally, suppose $(iii)$ that, for every $w\in\mathcal{W}$,
$\alpha,\beta\subseteq\mathcal{W}$, we have: ($\mathbf{T}$-condition)
$\alpha\in\mathcal{N}_{i}(w)$ implies $w\in\alpha$. Then, we have
($\mathbf{P}$-condition) $\emptyset\not\in\mathcal{N}_{i}(w)$, for otherwise
we would get a contradiction.
Point 3. Suppose $(i)$ that, for every $w\in\mathcal{W}$,
$\alpha,\beta\subseteq\mathcal{W}$, we have: ($\mathbf{C}$-condition)
$\alpha\in\mathcal{N}_{i}(w)$ and $\beta\in\mathcal{N}_{i}(w)$ implies
$\alpha\cap\beta\in\mathcal{N}_{i}(w)$; and ($\mathbf{P}$-condition)
$\emptyset\not\in\mathcal{N}_{i}(w)$. Given $\alpha\in\mathcal{N}_{i}(w)$,
suppose towards a contradiction that
$\mathcal{W}\setminus\alpha\in\mathcal{N}_{i}(w)$. From this, we obtain that
$\emptyset=\alpha\cap(\mathcal{W}\setminus\alpha)\in\mathcal{N}_{i}(w)$, a
contradiction. Thus, we have that ($\mathbf{D}$-condition)
$\alpha\in\mathcal{N}_{i}(w)$ implies
$\mathcal{W}\setminus\alpha\not\in\mathcal{N}_{i}(w)$.
Moreover, suppose $(ii)$ that, for every $w\in\mathcal{W}$,
$\alpha,\beta\subseteq\mathcal{W}$, we have: ($\mathbf{T}$-condition)
$\alpha\in\mathcal{N}_{i}(w)$ implies $w\in\alpha$. Consider
$\alpha\in\mathcal{N}_{i}(w)$ and suppose, towards a contradiction, that also
$\mathcal{W}\setminus\alpha\in\mathcal{N}_{i}(w)$. We obtain that $w\in\alpha$
and $w\in\mathcal{W}\setminus\alpha$ as well, a contradiction. Hence, we have
that ($\mathbf{D}$-condition) $\alpha\in\mathcal{N}_{i}(w)$ implies
$\mathcal{W}\setminus\alpha\not\in\mathcal{N}_{i}(w)$.
Point 4. Straightforward, because otherwise we immediately have a
contradiction. ∎
See 1
###### Proof.
Here we present a proof only for $\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$
concept inclusions and only for the basic principles
$L\in\\{\mathbf{E,M,C,N,P,Q,D,T}\\}$. For
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ formulas, the proof is similar to
the case of propositional non-normal modal logics (see e.g. [28]). More
complex principles (e.g. $\mathbf{EMCN}$) can be obtained by suitably
combining the basic principles. Let
$\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$ be a neighbourhood
frame and let $L$ be as above.
$\mathit{L=\mathbf{E}}$.
If $C\equiv D$ is valid on $\mathcal{F}$, then for all
$\mathcal{M}=(\mathcal{F},\mathcal{I})$ based on $\mathcal{F}$, and all $w$ in
$\mathcal{M}$, we have $C^{\mathcal{I}_{w}}=D^{\mathcal{I}_{w}}$. Thus, for
all $d\in\Delta_{w}$, $\llbracket C\rrbracket^{\mathcal{M}}_{d}=\llbracket
D\rrbracket^{\mathcal{M}}_{d}$. So for all $v\in\mathcal{W}$, $\llbracket
C\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(v)$ iff $\llbracket
D\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(v)$, which implies
$d\in(\Box_{i}C)^{\mathcal{I}_{v}}$ iff $d\in(\Box_{i}D)^{\mathcal{I}_{v}}$,
that is $(\Box_{i}C)^{\mathcal{I}_{v}}=(\Box_{i}D)^{\mathcal{I}_{v}}$. Then,
$\Box_{i}C\equiv\Box_{i}D$ is valid on $\mathcal{F}$, for all $i\in J$.
$\mathit{L=\mathbf{M}}$.
From right to left, assume that $\mathcal{F}$ is supplemented and
$C\sqsubseteq D$ is valid on $\mathcal{F}$. Then, for all
$\mathcal{M}=(\mathcal{F},\mathcal{I})$ based on $\mathcal{F}$, and all $w$ in
$\mathcal{M}$, we have $C^{\mathcal{I}_{w}}\subseteq D^{\mathcal{I}_{w}}$.
Thus, for all $d\in\Delta_{w}$, $\llbracket
C\rrbracket^{\mathcal{M}}_{d}\subseteq\llbracket
D\rrbracket^{\mathcal{M}}_{d}$. By supplementation we have that, for all
$v\in\mathcal{W}$, $\llbracket
C\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(v)$ implies $\llbracket
D\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(v)$. So
$d\in(\Box_{i}C)^{\mathcal{I}_{v}}$ implies
$d\in(\Box_{i}D)^{\mathcal{I}_{v}}$, that is
$(\Box_{i}C)^{\mathcal{I}_{v}}\subseteq(\Box_{i}D)^{\mathcal{I}_{v}}$. Then
$\Box_{i}C\sqsubseteq\Box_{i}D$ is valid on $\mathcal{F}$. For the left-to-
right direction, assume that $\mathcal{F}$ is not supplemented. Then there are
$w\in\mathcal{W}$, $\alpha,\beta\subseteq\mathcal{W}$ such that
$\alpha\subseteq\beta$, $\alpha\in\mathcal{N}_{i}(w)$ and
$\beta\notin\mathcal{N}_{i}(w)$. We define over $\mathcal{F}$ the model
$\mathcal{M}=(\mathcal{F},\mathcal{I})$, where $\Delta_{w}=\\{d\\}$, for every
$w\in\mathcal{W}$, and the interpretation of two concept names $A,B\in{\sf
N_{C}}$ is defined as follows: $d\in A^{\mathcal{I}_{v}}$ iff $v\in\alpha$,
and $d\in B^{\mathcal{I}_{v}}$ iff $v\in\beta$ (and defined arbitrarily on all
other symbols in $({\sf N_{C}}\cup{\sf N_{R}})\setminus\\{A,B\\}$). As a
consequence, we have that $\llbracket A\rrbracket_{d}^{\mathcal{M}}=\alpha$
and $\llbracket B\rrbracket_{d}^{\mathcal{M}}=\beta$, which implies
$\llbracket A\rrbracket_{d}^{\mathcal{M}}=\llbracket
A\rrbracket_{d}^{\mathcal{M}}\cap\llbracket
B\rrbracket_{d}^{\mathcal{M}}=\llbracket A\sqcap
B\rrbracket_{d}^{\mathcal{M}}$. Thus $\llbracket A\sqcap
B\rrbracket_{d}^{\mathcal{M}}\in\mathcal{N}_{i}(w)$ and $\llbracket
B\rrbracket_{d}^{\mathcal{M}}\notin\mathcal{N}_{i}(w)$. By definition we have
$d\in(\Box_{i}(A\sqcap B))^{\mathcal{I}_{w}}$ and
$d\notin(\Box_{i}B)^{\mathcal{I}_{w}}$.
$\mathit{L=\mathbf{C}}$.
From right-to-left, assume that $\mathcal{F}$ is closed under intersection.
Moreover, let $\mathcal{M}=(\mathcal{F},\mathcal{I})$ be a model based on
$\mathcal{F}$, with $w$ world of $\mathcal{M}$, and $d\in\Delta$ such that
$d\in(\Box_{i}C\sqcap\Box_{i}D)^{\mathcal{I}_{w}}$. Thus
$d\in(\Box_{i}C)^{\mathcal{I}_{w}}$ and $d\in(\Box_{i}D)^{\mathcal{I}_{w}}$,
that is $[C]_{d}^{\mathcal{M}},[D]_{d}^{\mathcal{M}}\in\mathcal{N}_{i}(w)$. By
closure under intersection,
$[C]_{d}^{\mathcal{M}}\cap[D]_{d}^{\mathcal{M}}=[C\sqcap
D]_{d}^{\mathcal{M}}\in\mathcal{N}_{i}(w)$. Then $d\in(\Box_{i}(C\sqcap
D))^{\mathcal{I}_{w}}$. For the left-to-right direction, assume that
$\mathcal{F}$ is not closed under intersection. Then, there are
$w\in\mathcal{W}$, $\alpha,\beta\subseteq\mathcal{W}$ such that
$\alpha,\beta\in\mathcal{N}_{i}(w)$ and
$\alpha\cap\beta\notin\mathcal{N}_{i}(w)$. We define over $\mathcal{F}$ the
model $\mathcal{M}=(\mathcal{F},\mathcal{I})$, where $\Delta_{w}=\\{d\\}$, for
all $w\in\mathcal{W}$, and the interpretation of two concept names $A,B\in{\sf
N_{C}}$ is defined as follows: $d\in A^{\mathcal{I}_{v}}$ iff $v\in\alpha$,
and $d\in B^{\mathcal{I}_{v}}$ iff $v\in\beta$ (and defined arbitrarily on all
other symbols in $({\sf N_{C}}\cup{\sf N_{R}})\setminus\\{A,B\\}$). We have
$\llbracket A\rrbracket_{d}^{\mathcal{M}}=\alpha$ and $\llbracket
B\rrbracket_{d}^{\mathcal{M}}=\beta$, which implies
$d\in(\Box_{i}A)^{\mathcal{I}_{w}}$ and $d\in(\Box_{i}B)^{\mathcal{I}_{w}}$.
Moreover, $\llbracket A\sqcap B\rrbracket_{d}^{\mathcal{M}}=\llbracket
A\rrbracket_{d}^{\mathcal{M}}\cap\llbracket
B\rrbracket_{d}^{\mathcal{M}}=\alpha\cap\beta\notin\mathcal{N}_{i}(w)$. Thus
$d\notin(\Box_{i}(A\sqcap B))^{\mathcal{I}_{w}}$.
$\mathit{L=\mathbf{N}}$.
From right-to-left, assume that $\mathcal{F}$ contains the unit and
$\top\sqsubseteq C$ is valid on $\mathcal{F}$. Then for all
$\mathcal{M}=(\mathcal{F},\mathcal{I})$ based on $\mathcal{F}$, and all $w$ in
$\mathcal{M}$, we have $C^{\mathcal{I}_{w}}=\Delta_{w}$. As a consequence, for
all $d\in\Delta_{w}$, $\llbracket C\rrbracket^{\mathcal{M}}_{d}=\mathcal{W}$.
By the property of containing the unit we have that, for all
$v\in\mathcal{W}$, $\llbracket
C\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(v)$. So
$d\in(\Box_{i}C)^{\mathcal{I}_{v}}$ for all $d\in\Delta_{v}$, that is,
$\top\sqsubseteq\Box_{i}C$ is valid on $\mathcal{F}$. For the left-to-right
direction, assume that $\mathcal{F}$ does not contain the unit, i.e., there is
$w\in\mathcal{W}$ such that $\mathcal{W}\notin\mathcal{N}_{i}(w)$. Then, for
all models $\mathcal{M}=(\mathcal{F},\mathcal{I})$ based on $\mathcal{F}$, all
$w\in\mathcal{W}$, and all $d\in\Delta_{w}$, we have
$d\in\top^{\mathcal{I}_{w}}$. Moreover, since
$d\in(\Box_{i}\top)^{\mathcal{I}_{w}}$ iff
$\llbracket\top\rrbracket_{d}^{\mathcal{M}}\in\mathcal{N}_{i}(w)$ iff
$\mathcal{W}\in\mathcal{N}_{i}(w)$, we also have
$d\notin(\Box_{i}\top)^{\mathcal{I}_{w}}$.
$\mathit{L=\mathbf{P}}$.
From right-to-left, assume that
$\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$ satisfies the
$\mathbf{P}$-condition. I.e., for all $w\in\mathcal{W}$ and $i\in J$,
$\emptyset\not\in\mathcal{N}_{i}(w)$. Then, for all models
$\mathcal{M}=(\mathcal{F},\mathcal{I})$ based on $\mathcal{F}$, all
$w\in\mathcal{W}$, and all $d\in\Delta_{w}$, we have
$\emptyset=\llbracket\bot\rrbracket^{\mathcal{M}}_{d}\not\in\mathcal{N}_{i}(w)$.
So $d\not\in(\Box_{i}\bot)^{\mathcal{I}_{w}}$, or equivalently
$d\in(\lnot\Box_{i}\bot)^{\mathcal{I}_{w}}$. Also,
$d\in\Delta_{w}=\top^{\mathcal{I}_{w}}$. So
$\top^{\mathcal{I}_{w}}\subseteq(\lnot\Box_{i}\bot)^{\mathcal{I}_{w}}$. Then
$\mathcal{M},w\models\top\sqsubseteq\lnot\Box_{i}\bot$. Hence
$\top\sqsubseteq\lnot\Box_{i}\bot$ is valid on $\mathcal{F}$. For the left-to-
right direction, assume that $\mathcal{F}$ does not satisfy the $P$-condition
for some $i\in J$. This means that there is $w\in\mathcal{W}$ such that
$\emptyset\in\mathcal{N}_{i}(w)$. So there exists a model
$\mathcal{M}=(\mathcal{F},\mathcal{I})$ based on $\mathcal{F}$, a
$w\in\mathcal{W}$, and a $d\in\Delta_{w}$, such that
$\emptyset=\llbracket\bot\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(w)$.
So $d\in(\Box_{i}\bot)^{\mathcal{I}_{w}}$, or equivalently
$d\not\in(\lnot\Box_{i}\bot)^{\mathcal{I}_{w}}$. But
$d\in\Delta_{w}=\top^{\mathcal{I}_{w}}$. So
$\top^{\mathcal{I}_{w}}\not\subseteq(\lnot\Box_{i}\bot)^{\mathcal{I}_{w}}$.
Then $\mathcal{M},w\not\models\top\sqsubseteq\lnot\Box_{i}\bot$. Hence
$\top\sqsubseteq\lnot\Box_{i}\bot$ is not valid on $\mathcal{F}$.
$\mathit{L=\mathbf{Q}}$.
From right to left, assume that
$\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$ satisfies the
$\mathbf{Q}$-condition. I.e., for all $w\in\mathcal{W}$ and $i\in J$,
$\mathcal{W}\not\in\mathcal{N}_{i}(w)$. Then, for all models
$\mathcal{M}=(\mathcal{F},\mathcal{I})$ based on $\mathcal{F}$, all
$w\in\mathcal{W}$, and all $d\in\Delta_{w}$, we have
$\mathcal{W}=\llbracket\top\rrbracket^{\mathcal{M}}_{d}\not\in\mathcal{N}_{i}(w)$.
So $d\not\in(\Box_{i}\top)^{\mathcal{I}_{w}}$, or equivalently
$d\in(\lnot\Box_{i}\top)^{\mathcal{I}_{w}}$. Also
$d\in\Delta=\top^{\mathcal{I}_{w}}$. So
$\top^{\mathcal{I}_{w}}\subseteq(\lnot\Box_{i}\top)^{\mathcal{I}_{w}}$. Then
$\mathcal{M},w\models\top\sqsubseteq\lnot\Box_{i}\top$. Hence
$\top\sqsubseteq\lnot\Box_{i}\top$ is valid on $\mathcal{F}$. For the other
direction, assume that $\mathcal{F}$ does not satisfy the
$\mathbf{Q}$-condition for some $i\in J$. This means that there is
$w\in\mathcal{W}$ such that $\mathcal{W}\in\mathcal{N}_{i}(w)$. So there
exists a model $\mathcal{M}=(\mathcal{F},\mathcal{I})$ based on $\mathcal{F}$,
a $w\in\mathcal{W}$, and a $d\in\Delta_{w}$, such that
$\mathcal{W}=\llbracket\top\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(w)$.
So $d\in(\Box_{i}\top)^{\mathcal{I}_{w}}$, or equivalently
$d\not\in(\lnot\Box_{i}\top)^{\mathcal{I}_{w}}$. But
$d\in\Delta=\top^{\mathcal{I}_{w}}$. So
$\top^{\mathcal{I}_{w}}\not\subseteq(\lnot\Box_{i}\top)^{\mathcal{I}_{w}}$.
Then $\mathcal{M},w\not\models\top\sqsubseteq\lnot\Box_{i}\top$. Hence
$\top\sqsubseteq\lnot\Box_{i}\top$ is not valid on $\mathcal{F}$.
$\mathit{L=\mathbf{D}}$.
From right-to-left, assume that
$\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$ satisfies the
$\mathbf{D}$-condition. Moreover, let $\mathcal{M}=(\mathcal{F},\mathcal{I})$
based on $\mathcal{F}$, with a world $w\in\mathcal{W}$, and $d\in\Delta_{w}$,
a concept $C$, and $i\in J$, all arbitrarily chosen. Suppose that
$d\in(\Box_{i}C)^{\mathcal{I}_{w}}$. It means that
$[C]^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(w)$. By the $\mathbf{D}$-condition,
$\mathcal{W}\setminus[C]^{\mathcal{M}}_{d}\not\in\mathcal{N}_{i}(w)$.
Equivalently, $[\lnot C]^{\mathcal{M}}_{d}\not\in\mathcal{N}_{i}(w)$. This
means that $d\not\in(\Box_{i}\lnot C)^{\mathcal{I}_{w}}$, or equivalently,
that $d\in(\lnot\Box_{i}\lnot C)^{\mathcal{I}_{w}}$. So
$(\Box_{i}C)^{\mathcal{I}_{w}}\subseteq(\lnot\Box_{i}\lnot
C)^{\mathcal{I}_{w}}$. Then
$\mathcal{M},w\models\Box_{i}C\sqsubseteq\Diamond_{i}C$. Hence,
$\Box_{i}C\sqsubseteq\Diamond_{i}C$ is valid on $\mathcal{F}$. For the left-
to-right direction, assume that $\mathcal{F}$ does not satisfy the
$\mathbf{D}$-condition for $i\in J$. So there is a $w\in\mathcal{W}$, such
that, for some $\alpha\subseteq\mathcal{W}$, we have
$\alpha\in\mathcal{N}_{i}(w)$ and
$\mathcal{W}\setminus\alpha\in\mathcal{N}_{i}(w)$. We define
$\mathcal{M}=(\mathcal{F},\mathcal{I})$ based on $\mathcal{F}$, where, for any
$v\in\mathcal{W}$, $\Delta_{v}=\\{d\\}$, and $A^{\mathcal{I}_{v}}=\\{d\\}$ iff
$v\in\alpha$ (and defined arbitrarily on all other symbols in $({\sf
N_{C}}\cup{\sf N_{R}})\setminus\\{A\\}$). We have $\llbracket
A\rrbracket^{\mathcal{M}}_{d}=\\{v\in\mathcal{W}\mid d\in
A^{\mathcal{I}_{w}}\\}=\alpha$, and $\llbracket\lnot
A\rrbracket^{\mathcal{M}}_{d}=\mathcal{W}\setminus\alpha$. So
$d\in(\Box_{i}A)^{\mathcal{I}_{w}}$ and $d\in(\Box_{i}\lnot
A)^{\mathcal{I}_{w}}$, and then $d\not\in(\lnot\Box_{i}\lnot
A)^{\mathcal{I}_{w}}$. So
$(\Box_{i}A)^{\mathcal{I}_{w}}\not\subseteq(\lnot\Box_{i}\lnot
A)^{\mathcal{I}_{w}}$. This means that,
$\mathcal{M},w\not\models\Box_{i}A\sqsubseteq\Diamond_{i}A$. Hence,
$\Box_{i}C\sqsubseteq\Diamond_{i}C$ is not valid on $\mathcal{F}$.
$\mathit{L=\mathbf{T}}$.
From right-to-left, assume that
$\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$ satisfies the
$\mathbf{T}$-condition. Moreover, let $\mathcal{M}=(\mathcal{F},\mathcal{I})$
based on $\mathcal{F}$, with a world $w\in\mathcal{W}$, and $d\in\Delta_{w}$,
a concept $C$, and $i\in J$, all arbitrarily chosen. Suppose that
$d\in(\Box_{i}C)^{\mathcal{I}_{w}}$. It means that $\llbracket
C\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(w)$. By the
$\mathbf{T}$-condition, $w\in\llbracket C\rrbracket^{\mathcal{M}}_{d}$. So
$d\in C^{\mathcal{I}_{w}}$. So $(\Box_{i}C)^{\mathcal{I}_{w}}\subseteq
C^{\mathcal{I}_{w}}$. Then $\mathcal{M},w\models\Box_{i}C\sqsubseteq C$.
Hence, $\Box_{i}C\sqsubseteq C$ is valid on $\mathcal{F}$. For the left-to-
right direction, assume that $\mathcal{F}$ does not satisfy the
$\mathbf{T}$-condition for $i\in J$. So there is a $w\in\mathcal{W}$, such
that for some $\alpha\subseteq\mathcal{W}$ we have
$\alpha\in\mathcal{N}_{i}(w)$ and $w\not\in\alpha$. We define
$\mathcal{M}=(\mathcal{F},\mathcal{I})$ based on $\mathcal{F}$, where
$\Delta_{v}=\\{d\\}$, for any $v\in\mathcal{W}$, and
$A^{\mathcal{I}_{v}}=\\{d\\}$ iff $v\in\alpha$ (and defined arbitrarily on all
other symbols in $({\sf N_{C}}\cup{\sf N_{R}})\setminus\\{A\\}$). We have
$\llbracket A\rrbracket^{\mathcal{M}}_{d}=\\{v\in\mathcal{W}\mid d\in
A^{\mathcal{I}_{w}}\\}=\alpha$. So $d\in(\Box_{i}A)^{\mathcal{I}_{w}}$. Since
$w\not\in\alpha$, $w\in\mathcal{W}\setminus\alpha$. That is,
$w\in\mathcal{W}\setminus\llbracket A\rrbracket^{\mathcal{M}}_{d}$, or
equivalently $w\in\llbracket\lnot A\rrbracket^{\mathcal{M}}_{d}$. Then,
$d\in(\lnot A)^{\mathcal{I}_{w}}$, or equivalently
$d\not\in(A)^{\mathcal{I}_{w}}$. So,
$(\Box_{i}A)^{\mathcal{I}_{w}}\not\subseteq A^{\mathcal{I}_{w}}$, meaning that
$\mathcal{M},w\not\models\Box_{i}A\sqsubseteq A$. Hence, $\Box_{i}C\sqsubseteq
C$ is not valid on $\mathcal{F}$, as required. ∎
See 2
###### Proof.
_Point 1._ The $(\Rightarrow)$ direction follows from the proof of Proposition
1. To see that the $(\Leftarrow)$ direction does not hold in general, we
provide the following counterexample showing that the $\mathbf{T}$-principle
holds in a model that does not satisfy the $\mathbf{T}$-condition. Consider
$\mathcal{M}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J},\mathcal{I})$, where
* •
$\mathcal{W}=\\{w,v\\}$;
* •
$\mathcal{N}_{i}(w)=\\{\\{v\\},\mathcal{W}\\}$ and
$\mathcal{N}_{i}(v)=\\{\\{w\\},\mathcal{W}\\}$, for $i\in J$;
* •
$\mathcal{I}_{w}=\mathcal{I}_{v}$, with $\Delta_{w}=\Delta_{v}=\\{d\\}$.
$\mathcal{M}$ does not satisfy the $\mathbf{T}$-condition, since
$\\{v\\}\in\mathcal{N}_{i}(w)$ but $w\notin\\{v\\}$. We show that the
$\mathbf{T}$-principle holds in $\mathcal{M}$.
###### Claim 1.
For all concepts $C$, $\llbracket C\rrbracket^{\mathcal{M}}_{d}=\emptyset$ or
$\llbracket C\rrbracket^{\mathcal{M}}_{d}=\mathcal{W}$.
###### Proof of Claim.
By induction on the construction of $C$.
For the base case $C=A\in{\sf N_{C}}$, it follows from the definition that
either $d\in A^{\mathcal{I}_{w}}$ and $d\in A^{\mathcal{I}_{v}}$, hence
$\llbracket A\rrbracket^{\mathcal{M}}_{d}=\mathcal{W}$, or $d\notin
A^{\mathcal{I}_{w}}$ and $d\notin A^{\mathcal{I}_{v}}$, hence $\llbracket
A\rrbracket^{\mathcal{M}}_{d}=\emptyset$.
We now show the inductive cases. For $C=\lnot D,D\sqcap E$, the proof is
immediate by applying the induction hypothesis.
For $C=\exists r.D$, the proof follows by the application of the induction
hypothesis and the fact that $r^{\mathcal{I}_{w}}=r^{\mathcal{I}_{v}}$.
For $C=\Box_{i}D$, by induction hypothesis $\llbracket
D\rrbracket^{\mathcal{M}}_{d}=\emptyset$ or $\llbracket
D\rrbracket^{\mathcal{M}}_{d}=\mathcal{W}$. In the first case, $\llbracket
D\rrbracket^{\mathcal{M}}_{d}\notin\mathcal{N}_{i}(w)$ and $\llbracket
D\rrbracket^{\mathcal{M}}_{d}\notin\mathcal{N}_{i}(v)$, hence
$w\notin\llbracket\Box_{i}D\rrbracket^{\mathcal{M}}_{d}$ and
$v\notin\llbracket\Box_{i}D\rrbracket^{\mathcal{M}}_{d}$, thus
$\llbracket\Box_{i}D\rrbracket^{\mathcal{M}}_{d}=\emptyset$. In the second
case, $\llbracket D\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(w)$ and
$\llbracket D\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(v)$, hence
$w\in\llbracket\Box_{i}D\rrbracket^{\mathcal{M}}_{d}$ and
$v\in\llbracket\Box_{i}D\rrbracket^{\mathcal{M}}_{d}$, thus
$\llbracket\Box_{i}D\rrbracket^{\mathcal{M}}_{d}=\mathcal{W}$. ∎
Now, given a concept $C$, suppose that $d\in(\Box_{i}C)^{\mathcal{I}_{w}}$,
that is, $\llbracket C\rrbracket^{\mathcal{M}}_{d}\in\mathcal{N}_{i}(w)$. By
the claim, $\llbracket C\rrbracket^{\mathcal{M}}_{d}=\emptyset$ or $\llbracket
C\rrbracket^{\mathcal{M}}_{d}=\mathcal{W}$. Since
$\emptyset\notin\mathcal{N}_{i}(w)$, we have that $\llbracket
C\rrbracket^{\mathcal{M}}_{d}=\mathcal{W}$, and thus $w\in\llbracket
C\rrbracket^{\mathcal{M}}_{d}$. This means that $d\in C^{\mathcal{I}_{w}}$. By
the same argument, we can show that $d\in(\Box_{i}C)^{\mathcal{I}_{v}}$
implies $d\in C^{\mathcal{I}_{v}}$. Therefore
$\mathcal{M}\models\Box_{i}C\sqsubseteq C$. Similarly we can prove that
$\mathcal{M}\models\Box_{i}\varphi\to\varphi$, for any formula $\varphi$.
_Point 2._ Let $F=(W,\\{R_{i}\\}_{i\in J})$ be a relational frame, and let
$M=(F,\Delta,I)$ be a relational model based on $F$.
($\mathbf{E}$-principle) Follows directly from the ($\mathbf{M}$-principle)
case below.
($\mathbf{M}$-principle) Assume $C\sqsubseteq D$ valid in $M$. Then
$C^{I_{w}}\subseteq D^{I_{w}}$, for all $w\in W$. Now, suppose that
$d\in(\Box_{i}C)^{I_{w}}$, for $d\in\Delta$ and $w\in W$. For all $v\in W$,
$wR_{i}v$ implies $d\in C^{I_{v}}$, hence $d\in D^{I_{v}}$. Therefore,
$d\in(\Box_{i}D)^{I_{w}}$.
($\mathbf{C}$-principle) Assume $d\in(\Box_{i}C\sqcap\Box_{i}D)^{I_{w}}$, that
is, $d\in(\Box_{i}C)^{I_{w}}$ and $d\in(\Box_{i}D)^{I_{w}}$. Then, for all
$v\in W$, $wR_{i}v$ implies $d\in C^{I_{v}}$ and $d\in D^{I_{v}}$, that is
$d\in(C\sqcap D)^{I_{v}}$. Therefore, $d\in(\Box_{i}(C\sqcap D))^{I_{w}}$.
($\mathbf{N}$-principle) Assume $\top\sqsubseteq C$ valid in $M$. Then, for
all $w\in W$, $C^{I_{w}}=\Delta$. Thus, for all $d\in\Delta$ and all $v\in W$,
$wR_{i}v$ implies $d\in C^{I_{v}}$. In conclusion, for all $d\in\Delta$,
$d\in(\Box_{i}C)^{I_{w}}$.
In conclusion, the $\mathbf{E}$-, $\mathbf{M}$-, $\mathbf{C}$-, and
$\mathbf{N}$-principle hold in $M$, and hence in $F$.
($\mathbf{D}$-principle and $\mathbf{P}$-principle are equivalent) It is easy
to see that both the $\mathbf{D}$\- and the $\mathbf{P}$-principle hold if and
only if $R_{i}$ is serial, for all $i\in J$ (that is, for all $w\in W$, there
is $v\in W$ such that $wR_{i}v$).
($\mathbf{Q}$-principle does not hold) The $\mathbf{Q}$-principle is
incompatible with the $\mathbf{N}$-principle, whcih holds in relational
models. Indeed, if both principles hold in the relational model $M$, then
$M\models\top\sqsubseteq\Box_{i}\top\sqcap\lnot\Box_{i}\top$, against the fact
that $\smash{\mathcal{ALC}}$ domains are non-empty. ∎
## Appendix 0.C Proofs for Section 3
Here is an $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm example
application.
###### Example 1.
Consider the formula
$\varphi=\lnot(\mathbb{D}_{c}\exists\mathsf{req}.(\mathsf{Prod}\sqcap\mathsf{InCatal})\sqsubseteq\mathbb{D}_{c}(\mathsf{Conf}\sqcup\lnot\mathsf{Conf})),$
related to the discussion in Section 0.A. We recall that the formula is
unsatisfiable in models validating the $\mathbf{M}$-condition, and it is
satisfiable otherwise. Here we show that the algorithm provides different
answers depending whether $\mathbf{M}\in\mathit{L}$. First, we rewrite
$\varphi$ in NNF, using $\widehat{\mathbb{D}}_{c}$ as the dual operator of
$\mathbb{D}_{c}$, thus obtaining
$\lnot(\top\sqsubseteq\widehat{\mathbb{D}}_{c}\forall\mathsf{req}.(\lnot\mathsf{Prod}\sqcup\lnot\mathsf{InCatal})\sqcup\mathbb{D}_{c}(\mathsf{Conf}\sqcup\lnot\mathsf{Conf})).$
We then consider the following applications of the tableau algorithm. In the
first case we assume $\mathbf{M}\in\mathit{L}$:
$0:\varphi$
$0:\mathbb{D}_{c}\exists\mathsf{req}.(\mathsf{Prod}\sqcap\mathsf{InCatal})\sqcap\widehat{\mathbb{D}}_{c}(\lnot\mathsf{Conf}\sqcap\mathsf{Conf})(v)$
($\mathsf{R}_{\not\sqsubseteq}$)
$1:\exists\mathsf{req}.(\mathsf{Prod}\sqcap\mathsf{InCatal})(v)$
($\mathsf{R}_{\mathit{L}}$)
$1:\lnot\mathsf{Conf}\sqcap\mathsf{Conf}(v)$ ($\mathsf{R}_{\mathit{L}}$)
$1:\lnot\mathsf{Conf}(v)$ ($\mathsf{R}_{\sqcap}$)
$1:\mathsf{Conf}(v)$ ($\mathsf{R}_{\sqcap}$)
The completion set constructed by the application of the
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm contains a clash,
hence the algorithm returns $\mathsf{unsatisfiable}$ on input $\varphi$. Now,
assume $\mathbf{M}\notin\mathit{L}$:
$0:\varphi$
$0:\mathbb{D}_{c}\exists\mathsf{req}.(\mathsf{Prod}\sqcap\mathsf{InCatal})\sqcap\widehat{\mathbb{D}}_{c}(\lnot\mathsf{Conf}\sqcap\mathsf{Conf})(v)$
($\mathsf{R}_{\not\sqsubseteq}$)
$1:\forall\mathsf{req}.(\lnot\mathsf{Prod}\sqcup\lnot\mathsf{InCatal})(v)$
($\mathsf{R}_{\mathit{L}}$)
$1:\mathsf{Conf}\sqcup\lnot\mathsf{Conf}(v)$ ($\mathsf{R}_{\mathit{L}}$)
$1:\mathsf{Conf}(v)$ ($\mathsf{R}_{\sqcup}$)
The completion set is clash-free and
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-complete, hence the algorithm returns
$\mathsf{satisfiable}$ on input $\varphi$. Note that the latter applications
of $\mathsf{R}_{\mathit{L}}$ are only possible if
$\mathbf{M}\notin\mathit{L}$.
In this appendix we prove termination, soundness, and completeness of the
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm. We define the
_weight_ $|C|$ of a concept $C$ in NNF as follows: $|A|=|\lnot A|=0$;
$|\exists r.D|=|\forall r.D|=|\Box_{i}D|=|\Diamond_{i}D|=|D|+1$; $|D\sqcap
E|=|D\sqcup E|=|D|+|E|+1$. The _weight $|\varphi|$_ of a formula $\varphi$ in
NNF is defined as: $|C(a)|=|r(a,b)|=|\lnot r(a,b)|=|(C\sqsubseteq
D)|=|\lnot(C\sqsubseteq D)|=0$; $|\Box_{i}\psi|=|\Diamond_{i}\psi|=|\psi|+1$;
$|\psi\land\chi|=|\psi\lor\chi|=|\psi|+|\chi|+1$. Observe that, for a concept
or formula $\gamma$, we have that $|\gamma|=|\dot{\lnot}\gamma|$.
###### Theorem 0.C.1 (Termination).
The $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm for $\varphi$
terminates after at most $2^{p(|\mathsf{Fg}(\varphi)|)}$ steps, where $p$ is a
polynomial function.
###### Proof.
We first require the following claims.
###### Claim 2.
Let $\mathcal{T}$ be a completion set obtained by applying the
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm for $\varphi$. For
each $n\in\mathsf{L}_{\mathcal{T}}$, the number of $n$-labelled constraints
for $\varphi$ in $\mathcal{T}$ does not exceed
$2^{q(|\mathsf{Fg}(\varphi)|)}$, where $q$ is a polynomial function.
###### Proof of Claim.
We remark that, for each $S_{n}\subseteq\mathcal{T}$, the
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableaux algorithm behaves exactly
like a standard (non-modal) $\smash{\mathcal{ALC}}$ tableaux algorithm (cf.
e.g. [17, Theorem 15.4]), except possibly for the additional rule
$\mathsf{R}_{\mathit{L}\mathbf{T}}$ which introduces at most
$|\mathsf{Fg}(\varphi)|$ $n$-labelled contraints. ∎
###### Claim 3.
Let $\mathcal{T}$ be a completion set obtained by applying the
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm for $\varphi$. Then
$|\mathsf{L}_{\mathcal{T}}|\leq r(|\mathsf{Fg}(\varphi)|)$ if
$\mathbf{C}\notin\mathit{L}$, and $|\mathsf{L}_{\mathcal{T}}|\leq
2^{r^{\prime}(|\mathsf{Fg}(\varphi)|)}$ if $\mathbf{C}\in\mathit{L}$, for some
for some polynomial functions $r$ and $r^{\prime}$.
###### Proof of Claim.
Labels $n$ are generated in $\mathcal{T}$ by means of the application of the
rules $\mathsf{R}_{\mathit{L}}$, $\mathsf{R}_{\mathit{L}\mathbf{N}}$,
$\mathsf{R}_{\mathit{L}\mathbf{P}}$, $\mathsf{R}_{\mathit{L}\mathbf{Q}}$,
$\mathsf{R}_{\mathit{L}\mathbf{D}}$. If $\mathbf{C}\notin\mathit{L}$, these
rules are applied to either one or two $n$-labelled contraints, while if
$\mathbf{C}\in\mathit{L}$, they are applied to $k$, $k+1$ or $k+h$
$n$-labelled contraints. By the application conditions of the rules, each such
combination of constraints generates at most one label $m$. Therefore, the
number of labels that can be generated in $\mathcal{T}$ is bounded by the
number of possible such combinations, which is at most
$2\cdot|\mathsf{Fg}(\varphi)|^{2}+3\cdot|\mathsf{Fg}(\varphi)|$, if
$\mathbf{C}\not\in\mathit{L}$, and at most
$2^{|\mathsf{Fg}(\varphi)|}\cdot|\mathsf{Fg}(\varphi)|+|\mathsf{Fg}(\varphi)|+2^{|\mathsf{Fg}(\varphi)|+1}+2^{2|\mathsf{Fg}(\varphi)|}$,
if $\mathbf{C}\in\mathit{L}$. ∎
The theorem is then a consequence of the following observations. Given a
completion set $\mathcal{T}$ constructed by the
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm for $\varphi$, we
have by Claim 3 that the number of applications of the rules
$\mathsf{R}_{\mathit{L}}$, $\mathsf{R}_{\mathit{L}\mathbf{N}}$,
$\mathsf{R}_{\mathit{L}\mathbf{P}}$, $\mathsf{R}_{\mathit{L}\mathbf{Q}}$, and
$\mathsf{R}_{\mathit{L}\mathbf{D}}$ is bounded by
$|\mathsf{L}_{\mathcal{T}}|$, which is at most $r(|\mathsf{Fg}(\varphi)|)$,
for $\mathit{L}$ such that $\mathbf{C}\notin\mathit{L}$, and at most
$2^{r^{\prime}(|\mathsf{Fg}(\varphi)|)}$, for $\mathit{L}$ such that
$\mathbf{C}\in\mathit{L}$, where $r$ and $r^{\prime}$ are polynomial
functions. Moreover, for a given label $n$, the number of possible
applications of the rules $\mathsf{R}_{\land}$, $\mathsf{R}_{\lor}$ and
$\mathsf{R}_{\mathit{L}\mathbf{T}}$ to constraints of the form $n:\psi$ is
linearly bounded by $\mathsf{Fg}(\varphi)$, hence there are at most
$|\mathsf{L}_{\mathcal{T}}|\cdot q^{\prime}(|\mathsf{Fg}(\varphi)|)$ such rule
applications, where $q^{\prime}$ is a polynominal function. Finally, by Claim
2, for each label $n$, the number of applications of the rules
$\mathsf{R}_{\sqcap},\mathsf{R}_{\sqcup},\mathsf{R}_{\forall},\mathsf{R}_{\exists},\mathsf{R}_{\sqsubseteq},\mathsf{R}_{\not\sqsubseteq}$
and $\mathsf{R}_{\mathit{L}\mathbf{T}}$ to constraints of the form $n:C(x)$ or
$n:r(x,y)$ is bounded by $2^{q(|\mathsf{Fg}(\varphi)|)}$, where $q$ is a
polynomial function, hence there are at most $|\mathsf{L}_{\mathcal{T}}|\cdot
2^{q(|\mathsf{Fg}(\varphi)|)}$ such rule applications. It follows that the
overall number of rule applications is bounded by
$2^{p(|\mathsf{Fg}(\varphi)|)}$ for some polynomial function $p$. ∎
We now proceed to prove that the $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$
tableau algorithm is sound.
###### Theorem 0.C.2 (Soundness).
If there exists an execution of the $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$
tableau algorithm for $\varphi$ that constructs a complete and clash-free
completion set, then $\varphi$ is $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$
satisfiable.
###### Proof.
Suppose that the $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ tableau algorithm
for $\varphi$ constructs an $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-complete
and clash-free completion set $\mathcal{T}$ for $\varphi$. We define, for
$n\in\mathsf{L}_{\mathcal{T}}$, $\psi\in\mathsf{for}_{\dot{\lnot}}(\varphi)$,
$C\in\mathsf{con}_{\dot{\lnot}}(\varphi)$, and $x$ occurring in $\mathcal{T}$,
$\displaystyle\lfloor C\rfloor_{x}$
$\displaystyle=\\{n\in\mathsf{L}_{\mathcal{T}}\mid n:C(x)\in S_{n}\\},$
$\displaystyle\lceil C\rceil_{x}$
$\displaystyle=\mathsf{L}_{\mathcal{T}}\setminus\\{n\in\mathsf{L}_{\mathcal{T}}\mid
n:\dot{\lnot}C(x)\in S_{n}\\},$ $\displaystyle\lfloor\psi\rfloor$
$\displaystyle=\\{n\in\mathsf{L}_{\mathcal{T}}\mid n:\psi\in S_{n}\\},$
$\displaystyle\lceil\psi\rceil$
$\displaystyle=\mathsf{L}_{\mathcal{T}}\setminus\\{n\in\mathsf{L}_{\mathcal{T}}\mid
n:\dot{\lnot}\psi\in S_{n}\\}.$
Moreover, define $\Gamma^{x}_{n}=\\{\psi\mid n:\psi\in S_{n}\\}\cup\\{C\mid
n:C(x)\in S_{n}\\}$ and let $\gamma,\delta$ range over
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ formulas or concepts, where:
$\lfloor\gamma\rfloor_{x}=\lfloor\psi\rfloor$, if $\gamma=\psi$, and
$\lfloor\gamma\rfloor_{x}=\lfloor C\rfloor_{x}$, if $\gamma=C$; and similarly
for $\lceil\gamma\rceil_{x}$. We set $\mathcal{M}=(\mathcal{F},\mathcal{I})$,
with $\mathcal{F}=(\mathcal{W},\\{\mathcal{N}_{i}\\}_{i\in J})$ and
$\mathcal{I}_{n}=(\Delta_{n},\cdot^{\mathcal{I}_{n}})$, for $n\in\mathcal{W}$,
defined as follows:
* •
$\mathcal{W}=\mathsf{L}_{\mathcal{T}}$;
* •
for every $i\in J=\\{1,\ldots,n\\}$, we set
$\mathcal{N}_{i}\colon\mathcal{W}\rightarrow 2^{2^{\mathcal{W}}}$ such that:
$\displaystyle\mathcal{N}_{i}(n)=\big{\\{}\alpha\mid$
$\displaystyle\textnormal{\ for some \
}\Box_{i}\gamma_{1}\in\Gamma^{{x}_{1}}_{n},\ldots,\Box_{i}\gamma_{\mathsf{k}}\in\Gamma^{{x}_{\mathsf{k}}}_{n}\colon$
$\displaystyle\mathsf{LB}(\overline{\gamma_{\mathsf{k}}})\subseteq\alpha\subseteq\mathsf{UB}(\overline{\gamma_{\mathsf{k}}})\big{\\}}\cup\mathsf{S};$
where:
* –
$\mathsf{LB}(\overline{\gamma_{\mathsf{k}}})=\bigcap^{\mathsf{k}}_{j=1}\lfloor\gamma_{j}\rfloor_{{{x}_{j}}}$;
* –
$\mathsf{UB}(\overline{\gamma_{\mathsf{k}}})=\begin{cases}\mathcal{W},&\text{if
$\mathbf{M}\in L$}\\\
\bigcap^{\mathsf{k}}_{j=1}\lceil\gamma_{j}\rceil_{{{x}_{j}}},&\text{if
$\mathbf{M}\not\in L$}\end{cases};$
* –
$\mathsf{k}\begin{cases}\geq 1,&\text{if $\mathbf{C}\in L$}\\\ =1,&\text{if
$\mathbf{C}\not\in L$}\end{cases};$
* –
$\mathsf{S}=\begin{cases}\\{\mathcal{W}\\},&\text{if $\mathbf{N}\in L$}\\\
\emptyset,&\text{if $\mathbf{N}\not\in L$}\end{cases};$
* •
$\Delta_{n}=\\{x\mid x\ \text{is a term occurring in}\ S_{n}\\}$;
* •
$A^{\mathcal{I}_{n}}=\\{x\in\Delta_{n}\mid n:A(x)\in S_{n}\\}$;
* •
$a^{\mathcal{I}_{n}}=\begin{cases}a,&\text{if $a$ occurs in $S_{n}$}\\\
\text{arbitrary},&\text{otherwise}\end{cases}$ ;
* •
$r^{\mathcal{I}_{n}}=\\{(x,y)\in\Delta_{n}\times\Delta_{n}\mid n:r(x,y)\in
S_{n}\ \text{or}\ n:r(z,y)\in S_{n},$ for some $z$ blocking $x$ in $S_{n}\\}$.
We require the following claims.
###### Claim 4.
For $\mathbf{X}\in\\{\mathbf{M,C,N,T,P,Q,D}\\}$, if $\mathbf{X}\in\mathit{L}$,
then $\mathcal{M}$ satisfies the $X$-condition.
###### Proof of Claim.
1. $\mathbf{M}\in\mathit{L}$.
Suppose that $\alpha\in\mathcal{N}_{i}(n)$ and
$\alpha\subseteq\beta\subseteq\mathcal{W}$. By definition, there are
$\Box_{i}\gamma_{1}\in\Gamma^{{x}_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{{x}_{k}}_{n}$
such that $\mathsf{LB}(\overline{\gamma_{k}})\subseteq\alpha$. Then
$\mathsf{LB}(\overline{\gamma_{k}})\subseteq\beta$, hence
$\beta\in\mathcal{N}_{i}(n)$.
2. $\mathbf{C}\in\mathit{L}$.
Suppose that $\alpha,\beta\in\mathcal{N}_{i}(n)$. Then there are
$\Box_{i}\gamma_{1}\in\Gamma^{{x}_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{{x}_{k}}_{n}$
such that
$\mathsf{LB}(\overline{\gamma_{k}})\subseteq\alpha\subseteq\mathsf{UB}(\overline{\gamma_{k}})$,
and there are
$\Box_{i}\delta_{1}\in\Gamma^{{x}_{1}}_{n},\ldots,\Box_{i}\delta_{h}\in\Gamma^{{x}_{h}}_{n}$
such that
$\mathsf{LB}(\overline{\delta_{h}})\subseteq\beta\subseteq\mathsf{UB}(\overline{\delta_{h}})$.
Then
$\mathsf{LB}(\overline{\gamma_{k}})\cap\mathsf{LB}(\overline{\delta_{h}})=\mathsf{LB}(\overline{\gamma_{k},\delta_{h}})\subseteq\alpha\cap\beta\subseteq\mathsf{UB}(\overline{\gamma_{k}})\cap\mathsf{UB}(\overline{\delta_{h}})=\mathsf{UB}(\overline{\gamma_{k},\delta_{h}})$,
which implies $\alpha\cap\beta\in\mathcal{N}_{i}(n)$.
3. $\mathbf{N}\in\mathit{L}$.
By construction, $\mathcal{W}\in\mathcal{N}_{i}(n)$ for all $n\in\mathcal{W}$.
4. $\mathbf{P}\in\mathit{L}$.
Suppose that $\alpha\in\mathcal{N}_{i}(n)$. Then there are
$\Box_{i}\gamma_{1}\in\Gamma^{{x}_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{{x}_{k}}_{n}$
such that
$\mathsf{LB}(\overline{\gamma_{k}})\subseteq\alpha\subseteq\mathsf{UB}(\overline{\gamma_{k}})$.
Since $\mathcal{T}$ is $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-complete, by
the rule $\mathsf{R}_{\mathit{L}\mathbf{P}}$, there is $m$ such that
$m:\gamma_{j}\in\mathcal{T}$ for all $1\leq j\leq k$, that is
$m\in\mathsf{LB}(\overline{\gamma_{k}})$. Then $\alpha\not=\emptyset$.
5. $\mathbf{Q}\in\mathit{L}$.
Suppose that $\alpha\in\mathcal{N}_{i}(n)$. Then there are
$\Box_{i}\gamma_{1}\in\Gamma^{{x}_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{{x}_{k}}_{n}$
such that
$\mathsf{LB}(\overline{\gamma_{k}})\subseteq\alpha\subseteq\mathsf{UB}(\overline{\gamma_{k}})$.
Since $\mathcal{T}$ is $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-complete, by
the rule $\mathsf{R}_{\mathit{L}\mathbf{Q}}$, there is $m$ such that
$m:\dot{\lnot}\gamma_{j}\in\mathcal{T}$ for some $1\leq j\leq k$, that is
$m\in\mathcal{W}$ and $m\notin\lceil\gamma_{j}\rceil_{{{x}_{j}}}$, hence
$m\notin\mathsf{UB}(\overline{\gamma_{k}})$. Then $\alpha\not=\mathcal{W}$.
6. $\mathbf{D}\in\mathit{L}$.
Suppose that $\alpha,\beta\in\mathcal{N}_{i}(n)$. Then there are
$\Box_{i}\gamma_{1}\in\Gamma^{{x}_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{{x}_{k}}_{n}$
such that
$\mathsf{LB}(\overline{\gamma_{k}})\subseteq\alpha\subseteq\mathsf{UB}(\overline{\gamma_{k}})$,
and there are
$\Box_{i}\delta_{1}\in\Gamma^{{x}_{1}}_{n},\ldots,\Box_{i}\delta_{h}\in\Gamma^{{x}_{h}}_{n}$
such that
$\mathsf{LB}(\overline{\delta_{h}})\subseteq\beta\subseteq\mathsf{UB}(\overline{\delta_{h}})$.
Since $\mathcal{T}$ is $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-complete, b
the rule $\mathsf{R}_{\mathit{L}\mathbf{D}}$, there is $m$ such that
$m:\gamma_{j},m:\delta_{\ell}\in\mathcal{T}$ for all $1\leq j\leq k$,
$1\leq\ell\leq h$; or
$m:\dot{\lnot}\gamma_{j},m:\dot{\lnot}\delta_{\ell}\in\mathcal{T}$ for some
$1\leq j\leq k$, $1\leq\ell\leq h$. In the first case,
$m\in\mathsf{LB}(\overline{\gamma_{k}})\cap\mathsf{LB}(\overline{\delta_{h}})$,
hence $m\in\alpha\cap\beta$. In the second case,
$m\in(\mathcal{W}\setminus\mathsf{UB}(\overline{\gamma_{k}}))\cap(\mathcal{W}\setminus\mathsf{UB}(\overline{\delta_{h}}))$,
hence $m\in(\mathcal{W}\setminus\alpha)\cap(\mathcal{W}\setminus\beta)$. In
either case $\beta\neq\mathcal{W}\setminus\alpha$.
7. $\mathbf{T}\in\mathit{L}$.
Suppose that $\alpha\in\mathcal{N}_{i}(n)$. Then there are
$\Box_{i}\gamma_{1}\in\Gamma^{{x}_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{{x}_{k}}_{n}$
such that
$\mathsf{LB}(\overline{\gamma_{k}})\subseteq\alpha\subseteq\mathsf{UB}(\overline{\gamma_{k}})$.
Since $\mathcal{T}$ is $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-complete, by
the rule $\mathsf{R}_{\mathit{L}\mathbf{T}}$, $n:\gamma_{j}\in\mathcal{T}$ for
all $1\leq j\leq k$, then $n\in\mathsf{LB}(\overline{\gamma_{k}})$, thus
$n\in\alpha$.∎
###### Claim 5.
For every $n\in\mathcal{W}$, $C\in\mathsf{con}_{\dot{\lnot}}(\varphi)$, and
$x\in\Delta_{n}$: if $n:C(x)\in S_{n}$, then $x\in C^{\mathcal{I}_{n}}$.
###### Proof of Claim.
We show the claim by induction on the weight of $C$ (in NNF). The base case of
$C=A$ comes immediately from the definitions. For the base case of $C=\lnot
A$, suppose that $n:\lnot A(x)\in S_{n}$. Since $\mathcal{T}$ is clash-free,
we have that $n:A(x)\not\in S_{n}$, and thus $x\not\in A^{\mathcal{I}_{n}}$ by
definition of $A^{\mathcal{I}_{n}}$, meaning $x\in(\lnot
A)^{\mathcal{I}_{n}}$. The inductive cases of $C=D\sqcap E$ and $C=D\sqcup E$
come from the fact that $S_{n}$ is closed under $\mathsf{R}_{\sqcap}$ and
$\mathsf{R}_{\sqcup}$, respectively, and straightforward applications of the
inductive hypothesis. We show the remaining cases (cf. also [17, Claim 15.2]).
1. $C=\exists r.D$.
Let $n:\exists r.D(x)\in S_{n}$, meaning that $\exists r.D\in\Gamma^{x}_{n}$.
We distinguish two cases.
* $(i)$
$x$ is not blocked by any variable in $S_{n}$. Since $S_{n}$ is closed under
$\mathsf{R}_{\exists}$, there exists $y$ occurring in $S_{n}$ such that
$n:r(x,y)\in S_{n}$ and $n:D(y)\in S_{n}$. Thus, by definition, $(x,y)\in
r^{\mathcal{I}_{n}}$ and $n:D(y)\in S_{n}$. By inductive hypothesis, we obtain
that $x\in(\exists r.D)^{\mathcal{I}_{n}}$.
* $(ii)$
$x$ is blocked by a variable in $S_{n}$, implying that there exists a
$<$-minimal (since $<$ is a well-ordering) $y$ occurring in $S_{n}$ such that
$y<x$ and $\\{E\mid n:E(x)\in S_{n}\\}\subseteq\\{E\mid n:E(y)\in S_{n}\\}$.
In turn, this implies that $y$ is not blocked by any other variable $z$ in
$S_{n}$ (for otherwise $z$ would block $x$, with $z<y$, against the fact that
$y$ is $<$-minimal). By reasoning as in the case above, since $y$ is not
blocked and $S_{n}$ is closed under $\mathsf{R}_{\exists}$, we have a variable
$z$ occurring in $S_{n}$ such that $n:r(y,z)\in S_{n}$ and $n:D(z)\in S_{n}$.
Since $y$ blocks $x$, by definition we have that $(x,z)\in
r^{\mathcal{I}_{n}}$, and by inductive hypothesis we get from $n:D(z)$ that
$z\in D^{\mathcal{I}_{n}}$. Thus, $x\in(\exists r.D)^{\mathcal{I}_{n}}$.
2. $C=\forall r.D$.
Let $n:\forall r.D(x)\in S_{n}$, meaning that $\forall r.D\in\Gamma^{x}_{n}$,
and suppose that $(x,y)\in r^{\mathcal{I}_{n}}$. By definition, either
$n:r(x,y)\in S_{n}$ or $n:r(z,y)\in S_{n}$, for some $z$ blocking $x$ in
$S_{n}$. In the former case, since $S_{n}$ is closed under
$\mathsf{R}_{\forall}$, we get that $n:D(y)\in S_{n}$. In the latter case,
since $z$ blocks $x$ in $S_{n}$, we obtain $n:\forall r.D(z)\in S_{n}$; again,
since $S_{n}$ is closed under $\mathsf{R}_{\forall}$, this implies that
$n:D(y)\in S_{n}$. Hence, in both cases, we have $n:D(y)\in S_{n}$. By
inductive hypothesis, this means that $y\in D^{\mathcal{I}_{n}}$. Since $y$
was arbitrary, we conclude that $x\in(\forall r.D)^{\mathcal{I}_{n}}$.
3. $C=\Box_{i}D$.
Let $n:\Box_{i}D(x)\in S_{n}$, meaning that $\Box_{i}D\in\Gamma^{x}_{n}$. We
have by inductive hypothesis that $\lfloor D\rfloor_{x}=\\{n\in\mathcal{W}\mid
n:D(x)\in S_{n}\\}\subseteq\\{n\in\mathcal{W}\mid x\in
D^{\mathcal{I}_{n}}\\}=\llbracket D\rrbracket^{\mathcal{M}}_{x}$. By inductive
hypothesis (since $|\dot{\lnot}D|=|D|$), we also have that
$\\{n\in\mathcal{W}\mid n:\dot{\lnot}D(x)\in
S_{n}\\}\subseteq\\{n\in\mathcal{W}\mid
x\in(\dot{\lnot}D)^{\mathcal{I}_{n}}\\}=\llbracket\dot{\lnot}D\rrbracket^{\mathcal{M}}_{x}=\mathcal{W}\setminus\llbracket
D\rrbracket^{\mathcal{M}}_{x}$. Hence, $\llbracket
D\rrbracket^{\mathcal{M}}_{x}\subseteq\mathcal{W}\setminus\\{w\in\mathcal{W}\mid
n:\dot{\lnot}D(x)\in S_{n}\\}=\lceil D\rceil_{x}$. In conclusion, we have
$\Box_{i}D\in\Gamma^{x}_{n}$ such that $\lfloor
D\rfloor_{x}\subseteq\llbracket D\rrbracket^{\mathcal{M}}_{x}\subseteq\lceil
D\rceil_{x}$. Thus, by definition, $\llbracket
D\rrbracket^{\mathcal{M}}_{x}\in\mathcal{N}_{i}(n)$, as required. (If
$\mathbf{M}\in\mathit{L}$, $\lfloor D\rfloor_{x}\subseteq\llbracket
D\rrbracket^{\mathcal{M}}_{x}$, and by definition this means $\llbracket
D\rrbracket^{\mathcal{M}}_{x}\in\mathcal{N}_{i}(n)$, as required.)
4. $C=\Diamond_{i}D$.
Let $n:\Diamond_{i}D(x)\in S_{n}$. We distinguish two cases.
* $(i)$
There exists no $\Box_{i}\gamma\in\Gamma^{y}_{n}$. Then if
$\mathbf{N}\notin\mathit{L}$, $\mathcal{N}_{i}(n)=\emptyset$, thus
$\mathcal{W}\setminus\llbracket
D\rrbracket^{\mathcal{M}}_{x}\not\in\mathcal{N}_{i}(n)$, meaning that
$x\in(\Diamond_{i}D)^{\mathcal{I}_{n}}$. If instead $\mathbf{N}\in\mathit{L}$,
then $\mathcal{N}_{i}(n)=\mathcal{W}$. Moreover, since $\mathcal{T}$ is
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-complete, by the rule
$\mathsf{R}_{\mathit{L}\mathbf{N}}$, there is $m$ such that $m:D(x)\in S_{m}$.
By inductive hypothesis, this implies $x\in D^{\mathcal{I}_{m}}$, that is,
$\llbracket D\rrbracket^{\mathcal{M}}_{x}\neq\emptyset$. Then we have
$\mathcal{W}\setminus\llbracket D\rrbracket^{\mathcal{M}}_{x}\neq\mathcal{W}$,
and thus $\mathcal{W}\setminus\llbracket
D\rrbracket^{\mathcal{M}}_{x}\not\in\mathcal{N}_{i}(n)$. Hence,
$x\in(\Diamond_{i}D)^{\mathcal{I}_{n}}$.
* $(ii)$
There exist
$\Box_{i}\gamma_{1}\in\Gamma^{y_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{y_{k}}_{n}$.
Since $\mathcal{T}$ is $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-complete,
there exists $m\in\mathcal{W}$ such that:
$\gamma_{1}\in\Gamma^{y_{1}}_{m},\ldots,\gamma_{k}\in\Gamma^{y_{k}}_{m}$ and
$D\in\Gamma^{x}_{m}$; or $\dot{\lnot}\gamma_{j}\in\Gamma^{y_{j}}_{m}$ and
$\dot{\lnot}D\in\Gamma^{x}_{m}$, for some $j\leq k$. By inductive hypothesis,
the previous step implies that there exists $m\in\mathcal{W}$ such that:
$\gamma_{1}\in\Gamma^{y_{1}}_{m},\ldots,\gamma_{k}\in\Gamma^{y_{k}}_{m}$ and
$x\in D^{\mathcal{I}_{m}}$; or $\dot{\lnot}\gamma_{j}\in\Gamma^{y_{j}}_{m}$
and $x\in\dot{\lnot}D^{\mathcal{I}_{m}}$, for some $j\leq k$. Thus
$\bigcap_{j=1}^{k}\lfloor\gamma_{j}\rfloor_{y_{j}}\not\subseteq\mathcal{W}\setminus\llbracket
D\rrbracket^{\mathcal{M}}_{x}$; or $\mathcal{W}\setminus\llbracket
D\rrbracket^{\mathcal{M}}_{x}\not\subseteq\bigcap_{j=1}^{k}\lceil\gamma_{l}\rceil_{y_{l}}$.
Since this holds for every
$\Box_{i}\gamma_{1}\in\Gamma^{y_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{y_{k}}_{n}$,
we conclude that $\mathcal{W}\setminus\llbracket
D\rrbracket^{\mathcal{M}}_{x}\not\in\mathcal{N}_{i}(n)$, i.e.,
$x\in(\Diamond_{i}D)^{\mathcal{I}_{n}}$, as required. (If
$\mathbf{M}\in\mathit{L}$, there exists $m\in\mathcal{W}$ such that
$\gamma_{1}\in\Gamma^{y_{1}}_{m},\ldots,\gamma_{k}\in\Gamma^{y_{k}}_{m}$ and
$D\in\Gamma^{x}_{m}$, thus $x\in D^{\mathcal{I}_{m}}$, hence
$\bigcap_{j=1}^{k}\lfloor\gamma_{j}\rfloor_{y_{j}}\not\subseteq\mathcal{W}\setminus\llbracket
D\rrbracket^{\mathcal{M}}_{x}$, therefore $\mathcal{W}\setminus\llbracket
D\rrbracket^{\mathcal{M}}_{x}\not\in\mathcal{N}_{i}(n)$.)∎
###### Claim 6.
For every $n\in\mathcal{W}$ and $\psi\in\mathsf{con}_{\dot{\lnot}}(\varphi)$:
if $n:\psi\in S_{n}$, then $\mathcal{M},n\models\psi$.
###### Proof of Claim.
We prove the claim by induction on the weight of $\varphi$ (in NNF).
1. $\psi=C(a)$.
Let $n:C(a)\in S_{n}$. By definition of $\mathcal{I}_{n}$ and Claim 5, we have
that $a^{\mathcal{I}_{n}}\in C^{\mathcal{I}_{n}}$, hence $\mathcal{M},n\models
C(a)$. (For $\psi=\lnot C(a)$, recall that $\lnot C(a)$ is equivalent to
$D(a)$ with $D=\lnot C$).
2. $\psi=r(a,b)$.
Let $n:r(a,b)\in S_{n}$. By definition of $\mathcal{I}_{n}$, this implies
$(a^{\mathcal{I}_{n}},b^{\mathcal{I}_{n}})\in r^{\mathcal{I}_{n}}$, hence
$\mathcal{M},n\models r(a,b)$.
3. $\psi=\lnot r(a,b)$.
Let $n:\lnot r(a,b)\in S_{n}$. Since $\mathbf{T}$ is clash-free, we have that
$n:r(a,b)\not\in S_{n}$. Thus, by definition of $\mathcal{I}_{n}$ we have
$(a^{\mathcal{I}_{n}},b^{\mathcal{I}_{n}})\not\in r^{\mathcal{I}_{n}}$,
meaning that $\mathcal{M},n\not\models r(a,b)$.
4. $\psi=(\top\sqsubseteq C)$.
Let $n:\top\sqsubseteq C\in S_{n}$ and let $x\in\Delta_{n}$. Since $S_{n}$ is
closed under $(\mathsf{R}_{\sqsubseteq})$ and $x$ occurs in $S_{n}$, we have
that $n:C(x)\in S_{n}$. By Claim 5, we have that $x\in C^{\mathcal{I}_{n}}$.
Given that $x$ is arbitrary, we conclude that
$\mathcal{M},n\models\top\sqsubseteq C$.
5. $\psi=\lnot(\top\sqsubseteq C)$.
Let $n:\lnot(\top\sqsubseteq C)\in S_{n}$. Since $S_{n}$ is closed under
$(\mathsf{R}_{\not\sqsubseteq})$, there exists $x$ occurring in $S_{n}$ such
that $n:\dot{\lnot}C(x)\in S_{n}$. By Claim 5, we obtain that
$x\in(\dot{\lnot}C)^{\mathcal{I}_{n}}$, for some $x\in\Delta_{w}$. Hence,
$\mathcal{M},n\models\lnot(\top\sqsubseteq C)$.
The inductive cases of $\psi=\chi\land\vartheta$ and $\psi=\chi\lor\vartheta$
follow from the definitions and straighforward applications of the inductive
hypothesis. Moreover the inductive cases of $\psi=\Box_{i}\chi$ and
$\psi=\Diamond_{i}\chi$ can be proved analogously to Claim 5. ∎
Since, by definition, we have $0:\varphi\in S_{0}\subseteq\mathbf{T}$, thanks
to Claim 6 we obtain $\mathcal{M},0\models\varphi$. Moreover, by Claim 4,
$\mathcal{M}$ is a $\mathit{L}^{n}$ model. Therefore $\varphi$ is
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ satisfiable. ∎
We finally show completeness of the $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$
tableau algorithm.
###### Theorem 0.C.3 (Completeness).
If $\varphi$ is $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$ satisfiable, then
there exists an execution of the $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$
tableau algorithm for $\varphi$ that constructs a complete and clash-free
completion set.
###### Proof.
In the proof we assume $\mathbf{C}\in\mathit{L}$, for the case
$\mathbf{C}\notin\mathit{L}$ consider $k=h=1$. Let
$\mathcal{M}=(\mathcal{F},\mathcal{I})$ be an
$\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-model satisfying $\varphi$, with
$\mathcal{F}=(\mathcal{W},\\{\mathcal{N}\\}_{i\in J})$, i.e.,
$\mathcal{M},w_{\varphi}\models\varphi$, for some $w_{\varphi}\in\mathcal{W}$.
We require the following definitions and technical results. First, we let
$\gamma,\delta$ (possibly indexed) range over
$\smash{\mathcal{ML}^{n}_{\mathcal{ALC}}}$ concepts and formulas, with
$\llbracket\gamma\rrbracket^{\mathcal{M}}_{d}=\llbracket\psi\rrbracket^{\mathcal{M}}$,
if $\gamma=\psi$, and $\llbracket\gamma\rrbracket^{\mathcal{M}}_{d}=\llbracket
C\rrbracket^{\mathcal{M}}_{d}$, if $\gamma=C$. Then, for $w\in\mathcal{W}$ and
$d\in\bigcup_{v\in\mathcal{W}}\Delta_{v}$, define
$\Phi^{d}_{w}=\\{\psi\in\mathsf{for}_{\dot{\lnot}}(\varphi)\mid\mathcal{M},w\models\psi\\}\cup\\{C\in\mathsf{con}_{\dot{\lnot}}(\varphi)\mid
d\in C^{\mathcal{I}_{w}}\\}$. Observe that, if $C\in\Phi^{d}_{w}$, then
$d\in\Delta_{w}$. Moreover, given a completion set $\mathcal{T}$ for $\varphi$
and $S_{n}\subseteq\mathcal{T}$, let $\Gamma^{x}_{n}=\\{\psi\mid n:\psi\in
S_{n}\\}\cup\\{C\mid n:C(x)\in S_{n}\\}$. We say that a completion set
$\mathcal{T}$ for $\varphi$ is _$\mathcal{M}$ -compatible_ if there exists a
function $\pi$ from $\mathsf{L}_{\mathcal{T}}$ to $\mathcal{W}$, and, for
every $n\in\mathsf{L}_{\mathcal{T}}$, there exists a function $\pi_{n}$ from
the set of terms occurring in $S_{n}$ to $\Delta_{\pi(n)}$, such that
$\gamma\in\Gamma^{x}_{n}$ implies $\gamma\in\Phi^{\pi_{n}(x)}_{\pi(n)}$. We
require the following claim.
###### Claim 7.
If a completion set $\mathcal{T}$ for $\varphi$ is $\mathcal{M}$-compatible
and $\mathcal{M}$ is an $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-model, then
for every $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-rule $\mathsf{R}$
applicable to $\mathcal{T}$, there exists a completion set
$\mathcal{T}^{\prime}$ obtained from $\mathcal{T}$ by an application of
$\mathsf{R}$ such that $\mathcal{T}^{\prime}$ is $\mathcal{M}$-compatible.
###### Proof.
Given an $\mathcal{M}$-compatible completion set $\mathcal{T}$ for $\varphi$
and a label $n\in\mathsf{L}_{\mathcal{T}}$, let $\pi$ and $\pi_{n}$ be the
functions provided by the definition of $\mathcal{M}$-compatibility. We need
to consider each $\mathit{L}^{n}_{\smash{\mathcal{ALC}}}$-rule $\mathsf{R}$.
For
$\mathsf{R}\in\\{\mathsf{R}_{\land},\mathsf{R}_{\lor},\mathsf{R}_{\sqcap},\mathsf{R}_{\sqcup},\mathsf{R}_{\forall},\mathsf{R}_{\exists},\mathsf{R}_{\sqsubseteq},\mathsf{R}_{\not\sqsubseteq}\\}$,
we proceed similarly to [17, Claim 15.14]. Here we consider the modal rules.
1. ($\mathsf{R}_{\mathit{L}}$)
Suppose that $\mathsf{R}_{\mathit{L}}$ is applicable to $\mathcal{T}$. Then
there are
$\Box_{i}\gamma_{1}\in\Gamma^{x_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{x_{k}}_{n},\Diamond_{i}\delta\in\Gamma^{y}_{n}$.
Since $\mathcal{T}$ is $\mathcal{M}$-compatible, we have that
$\Box_{i}\gamma_{1}\in\Phi^{\pi_{n}(x_{1})}_{\pi(n)},\ldots,\Box_{i}\gamma_{k}\in\Phi^{\pi_{n}(x_{k})}_{\pi(n)}$
and $\Diamond_{i}\delta\in\Phi^{\pi_{n}(y)}_{\pi(n)}$, meaning that
$\llbracket\gamma_{j}\rrbracket^{\mathcal{M}}_{d_{j}}\in\mathcal{N}_{i}(\pi(n))$,
for $j=1,\ldots,k$, hence by the $C$-condition
$\bigcap_{j=1}^{k}\llbracket\gamma_{j}\rrbracket^{\mathcal{M}}_{d_{j}}\in\mathcal{N}_{i}(\pi(n))$,
and
$\mathcal{W}\setminus\llbracket\delta\rrbracket^{\mathcal{M}}_{e}\not\in\mathcal{N}_{i}(\pi(n))$,
i.e.,
$\llbracket\dot{\lnot}\delta\rrbracket^{\mathcal{M}}_{e}\not\in\mathcal{N}_{i}(\pi(n))$.
Then
$\bigcap_{j=1}^{k}\llbracket\gamma_{j}\rrbracket^{\mathcal{M}}_{d_{j}}\neq\llbracket\dot{\lnot}\delta\rrbracket^{\mathcal{M}}_{e}$
(if $\mathbf{M}\in\mathit{L}$,
$\bigcap_{j=1}^{k}\llbracket\gamma_{j}\rrbracket^{\mathcal{M}}_{d_{j}}\not\subseteq\llbracket\dot{\lnot}\delta\rrbracket^{\mathcal{M}}_{e}$).
It follows that there exists $v\in\mathcal{W}$ such that
$\gamma_{1}\in\Phi^{\pi_{n}(x_{1})}_{v},\ldots,\gamma_{k}\in\Phi^{\pi_{n}(x_{k})}_{v}$
and $\delta\in\Phi^{\pi_{n}(y)}_{v}$; or
$\dot{\lnot}\gamma_{j}\in\Phi^{\pi_{n}(x_{j})}_{v}$ and
$\dot{\lnot}\delta\in\Phi^{\pi_{n}(y)}_{v}$, for some $j\leq k$. Then by
applying the rule $\mathsf{R}_{\mathit{L}}$ accordingly, we expand
$\mathcal{T}$ to $\mathcal{T}^{\prime}$ with
$m:\gamma_{1},\ldots,m:\gamma_{k},m:\delta$, or with
$m:\dot{\lnot}\gamma_{j},m:\dot{\lnot}\delta$, for some $j\leq k$, for some
$m$ satisfying the application condition of $\mathsf{R}_{\mathit{L}}$. Since
$m$ is fresh, we can extend $\pi$ with $\pi(m)=v$, and $\pi_{m}$ with
$\pi_{m}(x_{1})=\pi_{n}(x_{1})$, …, $\pi_{m}(x_{k})=\pi_{n}(x_{k})$,
$\pi_{m}(y)=\pi_{n}(y)$, thus obtaining that $\mathcal{T}^{\prime}$ is
$\mathcal{M}$-compatible.
2. ($\mathsf{R}_{\mathit{L}\mathbf{N}}$)
Suppose that $\mathsf{R}_{\mathit{L}\mathbf{N}}$ is applicable to
$\mathcal{T}$. Then there is $\Diamond_{i}\delta\in\Gamma^{y}_{n}$. Since
$\mathcal{T}$ is $\mathcal{M}$-compatible, we have that
$\Diamond_{i}\delta\in\Phi^{\pi_{n}(y)}_{\pi(n)}$, meaning that
$\mathcal{W}\setminus\llbracket\delta\rrbracket^{\mathcal{M}}_{e}\not\in\mathcal{N}_{i}(\pi(n))$.
At the same time, by the $N$-condition,
$\mathcal{W}\in\mathcal{N}_{i}(\pi(n))$, hence
$\llbracket\delta\rrbracket^{\mathcal{M}}_{e}\not=\emptyset$, that is there
exists $v\in\mathcal{W}$ such that $\delta\in\Phi^{\pi_{n}(y)}_{v}$. Then we
expand $\mathcal{T}$ with $m:\delta$, for some $m$ satisfying the application
condition of $\mathsf{R}_{\mathit{L}\mathbf{N}}$. Since $m$ is fresh, we can
extend $\pi$ with $\pi(m)=v$, and $\pi_{m}$ with $\pi_{m}(y)=\pi_{n}(y)$, thus
obtaining that $\mathcal{T}^{\prime}$ is $\mathcal{M}$-compatible.
3. ($\mathsf{R}_{\mathit{L}\mathbf{P}}$)
Suppose that $\mathsf{R}_{\mathit{L}\mathbf{P}}$ is applicable to
$\mathcal{T}$. Then there are
$\Box_{i}\gamma_{1}\in\Gamma^{x_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{x_{k}}_{n}$.
Since $\mathcal{T}$ is $\mathcal{M}$-compatible, we have that
$\Box_{i}\gamma_{1}\in\Phi^{\pi_{n}(x_{1})}_{\pi(n)},\ldots,\Box_{i}\gamma_{k}\in\Phi^{\pi_{n}(x_{k})}_{\pi(n)}$,
meaning that
$\llbracket\gamma_{j}\rrbracket^{\mathcal{M}}_{d_{j}}\in\mathcal{N}_{i}(\pi(n))$,
for $j=1,\ldots,k$, hence
$\bigcap_{j=1}^{k}\llbracket\gamma_{j}\rrbracket^{\mathcal{M}}_{d_{j}}\in\mathcal{N}_{i}(\pi(n))$.
At the same time, by the $P$-condition,
$\bigcap_{j=1}^{k}\llbracket\gamma_{j}\rrbracket^{\mathcal{M}}_{d_{j}}\neq\emptyset$,
that is there exists $v\in\mathcal{W}$ such that
$\gamma_{1}\in\Phi^{\pi_{n}(x_{1})}_{v},\ldots,\gamma_{k}\in\Phi^{\pi_{n}(x_{k})}_{v}$.
Then we expand $\mathcal{T}$ with $m:\gamma_{1},\ldots,m:\gamma_{k}$, for some
$m$ satisfying the application condition of
$\mathsf{R}_{\mathit{L}\mathbf{P}}$. Since $m$ is fresh, we can extend $\pi$
with $\pi(m)=v$, and $\pi_{m}$ with $\pi_{m}(x_{1})=\pi_{n}(x_{1})$, …,
$\pi_{m}(x_{k})=\pi_{n}(x_{k})$, thus obtaining that $\mathcal{T}^{\prime}$ is
$\mathcal{M}$-compatible.
4. ($\mathsf{R}_{\mathit{L}\mathbf{Q}}$)
Suppose that $\mathsf{R}_{\mathit{L}\mathbf{Q}}$ is applicable to
$\mathcal{T}$. Then there are
$\Box_{i}\gamma_{1}\in\Gamma^{x_{1}}_{n},\ldots,\Box_{i}\gamma_{k}\in\Gamma^{x_{k}}_{n}$.
Since $\mathcal{T}$ is $\mathcal{M}$-compatible, we have that
$\Box_{i}\gamma_{1}\in\Phi^{\pi_{n}(x_{1})}_{\pi(n)},\ldots,\Box_{i}\gamma_{k}\in\Phi^{\pi_{n}(x_{k})}_{\pi(n)}$,
meaning that
$\llbracket\gamma_{j}\rrbracket^{\mathcal{M}}_{d_{j}}\in\mathcal{N}_{i}(\pi(n))$,
for $j=1,\ldots,k$, hence
$\bigcap_{j=1}^{k}\llbracket\gamma_{j}\rrbracket^{\mathcal{M}}_{d_{j}}\in\mathcal{N}_{i}(\pi(n))$.
At the same time, by the $Q$-condition,
$\bigcap_{j=1}^{k}\llbracket\gamma_{j}\rrbracket^{\mathcal{M}}_{d_{j}}\neq\mathcal{W}$,
that is there exists $v\in\mathcal{W}$ such that
$\gamma_{j}\notin\Phi^{\pi_{n}(x_{j})}_{v}$ for some $j\leq k$. Then by
applying $\mathsf{R}_{\mathit{L}\mathbf{Q}}$ accordingly, we expand
$\mathcal{T}$ with $m:\dot{\lnot}\gamma_{j}$, for some $m$ satisfying the
application condition of $\mathsf{R}_{\mathit{L}\mathbf{Q}}$. Since $m$ is
fresh, we can extend $\pi$ with $\pi(m)=v$, and $\pi_{m}$ with
$\pi_{m}(x_{j})=\pi_{n}(x_{j})$, thus obtaining that $\mathcal{T}^{\prime}$ is
$\mathcal{M}$-compatible.
5. ($\mathsf{R}_{\mathit{L}\mathbf{D}}$)
Suppose that $\mathsf{R}_{\mathit{L}\mathbf{D}}$ is applicable to
$\mathcal{T}$. Then there are |
# RoboPianist: A Benchmark for
High-Dimensional Robot Control
Kevin Zakka1,2 Laura Smith1 Nimrod Gileadi3 Taylor Howell4 Xue Bin Peng5
Sumeet Singh2 Yuval Tassa3 Pete Florence2 Andy Zeng2 Pieter Abbeel1
https://kzakka.com/robopianist/ 1UC Berkeley 2Robotics at Google 3DeepMind
4Stanford University 5Simon Fraser University
###### Abstract
We introduce a new benchmarking suite for high-dimensional control, targeted
at testing high spatial and temporal precision, coordination, and planning,
all with an underactuated system frequently making-and-breaking contacts. The
proposed challenge is mastering the piano through bi-manual dexterity, using a
pair of simulated anthropomorphic robot hands. We call it RoboPianist, and the
initial version covers a broad set of 150 variable-difficulty songs. We
investigate both model-free and model-based methods on the benchmark,
characterizing their performance envelopes. We observe that while certain
existing methods, when well-tuned, can achieve impressive levels of
performance in certain aspects, there is significant room for improvement.
RoboPianist provides a rich quantitative benchmarking environment, with human-
interpretable results, high ease of expansion by simply augmenting the
repertoire with new songs, and opportunities for further research, including
in multi-task learning, zero-shot generalization, multimodal (sound, vision,
touch) learning, and imitation. Supplementary information, including videos of
our control policies, can be found at https://kzakka.com/robopianist/.
## I Introduction
As the fields of control and reinforcement learning continue to develop, a key
question is how will progress be measured. In fields like computer vision and
natural language processing, progress has been fueled by robust, quantifiable,
and interpretable benchmarks – which in aggregate achieve breadth [1], and in
each explore complementary focuses in depth: take for example the famous
challenge of “Winograd schema” [2], first proposed in 1972 and later developed
in 2012 [3] into an influential benchmark. While in control and reinforcement
learning, certain benchmarking efforts have begun both to aggregate [4] and
explore different aspects of depth, a particularly under-served area has been
robust benchmarks which focus on high-dimensional control, including in
particular the perhaps ultimate “challenge problem” of high-dimensional
robotics: mastering bi-manual (two-handed) multi-fingered control.
Figure 2: A full example set of F1 scores on all tasks in RoboPianist-
repertoire-150, highlighting the breadth of difficulty offered by the
benchmark. This example set of scores is from the model-free method discussed
in Section IV (errors bars represent standard deviations over 3 seeds). Both
model-free and model-based implementations as well as evaluations are
discussed in Subsection V-B.
In fact, despite decades-long research into replicating the dexterity of the
human hand, high-dimensional control remains a grand challenge in robotics.
This topic has inspired considerable research from both a mechanical design
[5, 6, 7] and control theoretic point of view [8, 9, 10, 11, 12]. While
learning-based approaches [13, 14, 15, 16, 17, 18, 19] have dominated the
recent literature, the class of problems typically considered corresponds to a
limited definition of dexterity. In particular, most all such tasks are well-
specified using a single goal-state or termination condition, limiting the
complexity of the solution space and oftentimes yielding unnatural-looking
behaviors so long as the desired terminal state is reached.
In this work, we introduce a new benchmark suite for high-dimensional control,
RoboPianist, where bi-manual simulated anthropomorphic robot hands are tasked
with playing a variety of songs (i.e., correctly pressing sequences of keys on
a keyboard) conditioned on sheet music, in the form of a Musical Instrument
Digital Interface (MIDI) transcription. The robot hands themselves exhibit
high degrees of freedom (22 actuators per hand, for a total of 44), and are
partially underactuated, akin to human hands. We specifically chose this
domain because playing a song successfully means being able to sequence
actions in ways that exemplify many of the properties that we look for in
high-dimensional control policies, including (i) spatial and temporal
precision (hitting the right notes, at the right time), (ii) coordination
(simultaneously achieving multiple different goals, in this case, fingers on
each hand hitting different notes, without colliding), and (iii) planning (how
a key is pressed should be conditioned on the expectation of how it would
enable the policy to reach future notes). Additionally, piano playing also
presents an immediately interpretable task success signal (i.e., “does it
sound good?”), and this can e.g., be disentangled from the particular
reward/cost functions used.
The initial RoboPianist-repertoire-150 benchmark covers 150 songs, where each
song is effectively a different task. Through extensive experiments, we
investigate the performance envelope of both model-free and model-based
methods and demonstrate that, while there is ample room for improvement, our
policies are able to produce compelling performances. We encourage the reader
to watch and listen to the performances at https://kzakka.com/robopianist/.
For expanding further on the benchmarking suite, since song data is also
freely available on the Internet (either by parsing MIDI files, or YouTube
data), one can also effectively scale up the number of tasks presented in this
domain over time. Figure 2 also shows that we can sort songs (i.e., tasks) by
difficulty, reflected in how well a policy can learn the song. Being able to
sort tasks via such metrics can foster additional research across a multitude
of topics in robot learning, including curriculum learning and transfer
learning. Overall, RoboPianist exhibits a straightforward task, easy to
simulate environment, clear evaluation metrics, and is amenable to a variety
of expansion opportunities in the future. Code for simulation, control and
learning, baselines and datasets will be made available at
https://github.com/google-research/robopianist/.
To summarize, we believe RoboPianist provides many distinct and complementary
aspects relative to previous benchmarks for high-dimensional control and is a
strong addition to the community for the following reasons:
* •
Challenging. As we show in Subsection V-B, RoboPianist is challenging for
model-based and model-free methods. Without a shaped reward in the form of
fingering information, they perform poorly, and even with extensive reward
shaping, there is plenty room for improvement.
* •
Interpretable. One can simply look at and listen to a policy to gauge its
performance.
* •
Multimodal. Playing the piano is sensory-rich: policies can make use of sound,
vision and touch.
* •
Data rich and extendable. On top of just adding more MIDI files to expand the
number of tasks, one can incorporate other abundant sources of data such as
YouTube videos to create third-person demonstrations.
* •
Enables knowledge reuse. Similar songs share note and pattern structures which
can be reused to solve new songs zero-shot. More broadly, RoboPianist is a
natural playground for studying multi-task and meta-learning.
* •
Open source. We have fully open-sourced the benchmark and dataset at
https://github.com/google-research/robopianist.
## II Related Work
We address related work within two primary areas: dexterous high-dimensional
control, and robotic pianists.
Dexterous Manipulation and High-Dimensional Control The vast majority of the
control literature uses much lower-dimensional systems (i.e., single-arm,
simple end-effectors) than high-dimensional dexterous hands. Specifically,
only a handful of general-purpose policy optimization methods have been shown
to work on high-dimensional hands, even for a single hand [14, 13, 16, 15, 20,
18, 21, 11], and of these, only a subset has demonstrated results in the real
world [14, 13, 16, 15, 20]. Results with bi-manual hands are even rarer, even
in simulation only [19, 22].
As a benchmark, perhaps the most distinguishing aspect of RoboPianist is in
the definition of “task success”. As an example, general manipulation tasks
are commonly framed as the continual application of force/torque on an object
for the purpose of a desired change in state (e.g., SE(3) pose and velocity).
Gradations of dexterity are predominantly centered around the kinematic
redundancy of the arm or the complexity of the end-effector, ranging from
parallel jaw-grippers to anthropomorphic hands [23, 19]. A gamut of methods
have been developed to accomplish such tasks, ranging from various
combinations of model-based and model-free RL, imitation learning,
hierarchical control, etc. [24, 14, 17, 16, 25, 26]. However, the class of
problems generally tackled corresponds to a definition of dexterity pertaining
to traditional manipulation skills [27], such as re-orientation, relocation,
manipulating simply-articulated objects (e.g., door opening, ball throwing and
catching), and using simple tools (e.g., hammer) [4, 28, 19, 15, 29]. The only
other task suite that we know of that presents bi-manual tasks, the recent Bi-
Dex [19] suite, presents a broad collection of tasks that fall under this
category.
While these works represent an important class of problems, we explore an
alternative notion of dexterity and success. In particular, for most all the
aforementioned suite of manipulation tasks, the “goal” state is some explicit,
specific geometric function of the final states; for instance, an open/closed
door, object re-oriented, nail hammered, etc. This effectively reduces the
search space for controls to predominantly a single “basin-of-attraction" in
behavior space per task. In contrast, the RoboPianist suite of tasks
encompasses a more complex notion of a goal, which is encoded through a
musical performance. In effect, this becomes a highly combinatorially variable
sequence of goal states, extendable to arbitrary difficulty by only varying
the musical score. “Success” is graded on accuracy over an entire episode;
concretely, via a time-varying non-analytic output of the environment, i.e.,
the music. Thus, it is not a matter of the “final-state” that needs to satisfy
certain termination/goal conditions, a criterion which is generally permissive
of less robust execution through the rest of the episode, but rather the
behavior of the policy _throughout the episode_ needs to be precise and
musical.
Similarly, the literature on humanoid locomotion and more broadly, “character
control", another important area of high-dimensional control, primarily
features tasks involving the discovery of stable walking/running gaits [30,
31, 32], or the distillation of a finite set of whole-body movement priors
[33, 34, 35], to use downstream for training a task-level policy. Task success
is typically encoded via rewards for motion progress and/or reaching a
terminal goal condition. It is well-documented that the endless pursuit of
optimizing for these rewards can yield unrealistic yet “high-reward"
behaviors. While works such as [33, 36] attempt to capture _stylistic_
objectives via leveraging demonstration data, these reward functions are
simply appended to the primary task objective. This scalarization of multiple
objectives yields an arbitrarily subjective Pareto curve of optimal policies.
In contrast, performing a piece of music entails both objectively measurable
precision with regards to melodic and rhythmic accuracy, as well as a
subjective measure of musicality. Mathematically, this translates as
_stylistic_ constraint satisfaction, paving the way for innovative algorithmic
advances.
Robotic Piano Playing Robotic pianists have a rich history within the
literature, with several works dedicated to the design of specialized hardware
[37, 38, 39, 40, 41, 42], and/or customized controllers for playing back a
song using pre-programmed commands (open-loop) [43, 44]. The work in [45]
leverages a combination of inverse kinematics and trajectory stitching to play
single keys and playback simple patterns and a song with a Shadow hand [46].
More recently, in [47], the author simulated robotic piano playing using
offline motion planning with inverse kinematics for a 7-DoF robotic arm, along
with an Iterative Closest Point-based heuristic for selecting fingering for a
four-fingered Allegro hand. Each hand is simulated separately, and the audio
results are combined post-hoc. Finally, in [48], the authors formulate piano
playing as an RL problem for a single Allegro hand (four fingers) on a
miniature piano, and additionally leverage tactile sensor feedback. However,
the tasks considered are rather simplistic (e.g., play up to six successive
notes, or three successive chords with only two simultaneous keys pressed for
each chord). The RoboPianist benchmark suite is designed to allow a general
bi-manual controllable agent to emulate a pianist’s growing proficiency on the
instrument by providing a curriculum of musical pieces, graded in difficulty.
Leveraging two underactuated anthropomorphic hands as actuators provides a
level of realism and exposes the challenge of mastering this suite of high-
dimensional control problems.
## III The RoboPianist Benchmark
(a)
(b)
(c)
Figure 3: From left to right, (a): 24 DoF, under-actuated right Shadow Hand
model, (b): 88 DoF digital piano model, and (c): RoboPianist task setup with
left and right Shadow Hands placed above the piano on an invisible gantry.
In this section, we introduce the main elements of the RoboPianist benchmark.
We begin by describing the general setup of our benchmark, walk through its
practical implementation in simulation, and finally, detail our MIDI dataset
which defines the task distribution available in the benchmark.
### III-A Benchmark setup
States | Unit | Size
---|---|---
Hand joint angles | $\text{\,}\mathrm{rad}$ | 48
Hand joint velocities | $\text{\,}\mathrm{rad}\mathrm{/}\mathrm{s}$ | 48
Forearm positions | $\text{\,}\mathrm{m}$ | 4
Forearm velocities | $\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$ | 4
Piano key angles | $\text{\,}\mathrm{rad}$ | 88
Piano key velocities | $\text{\,}\mathrm{rad}\mathrm{/}\mathrm{s}$ | 88
Fingering (goal) | discrete | 10
Piano key press (goal) | discrete | 88
Total | | 378
Actions | Unit | Size
Desired hand joint angles | $\text{\,}\mathrm{rad}$ | 40
Desired forearm positions | $\text{\,}\mathrm{m}$ | 4
Total | | 44
Observations | Unit | Size
Hand and forearm joints | $\text{\,}\mathrm{rad}$ | 52
Cartesian forearm position | $\text{\,}\mathrm{m}$ | 6
Piano key angles | $\text{\,}\mathrm{rad}$ | 88
Fingering | discrete | 10
Piano key press | discrete | 88
Previous reward | — | 1
Previous action | $\text{\,}\mathrm{rad}$ | 44
Total | | 289
TABLE I: The state, action, and observation spaces of the RoboPianist MDP.
Fingering is a discrete boolean vector with size equal to the number of
fingers, with each component indicating if the corresponding finger is
expected to be in contact with a key. This information is extracted from the
fingering annotations in the MIDI file for each piece/task.
Learning to play the piano can be formulated as a finite-horizon Markov
Decision Process (MDP) defined by a tuple
$(\mathcal{S},\mathcal{A},\rho,p,r,\gamma,H)$ where
$\mathcal{S}\subset\mathbb{R}^{n}$ is the state space,
$\mathcal{A}\subset\mathbb{R}^{m}$ is the action space, $\rho(\cdot)$ is the
initial state distribution, $p(\cdot|s,a)$ governs the dynamics,
$r:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}$ defines the rewards,
$\gamma\in[0,1)$ is the discount factor, and $H$ is the horizon. The goal of
an agent is to maximize its total expected reward over the horizon
$\operatorname*{\mathbb{E}}\left[\sum_{t=0}^{H}\gamma^{t}r(s_{t},a_{t})\right]$.
In the RoboPianist benchmark, each song is a separate task with a different
note trajectory and horizon $H$.
All tasks in RoboPianist share the same underlying state, action, and
observation spaces (Table I)111The MDP is fully observable, but in practice we
use an observation space that is not exactly the state space – for example, to
better align with hardware in that there are no direct velocity sensors. Any
lack of observability on joint velocities can be remedied simply by
incorporating the last two measurements for the agent.. The action space
$\mathcal{A}$ is a 44 dimensional bounded, continuous action corresponding to
desired joint positions for the hands and forearms. The desired positions are
converted to torques at the 52 joints using proportional-position
actuators222The control signal for the distal actuator of each non-thumb
finger ($4\cdot 2=8$) is split amongst the middle and distal joints.. In
particular, we note that both proprioceptive states (i.e., joint and forearm
positions) and exteroceptive states (i.e., key joint positions) are
measurements that are readily available in the real world.
A RoboPianist task is illustrated in Figure 3. At episode initialization, the
hands are placed in the middle of the keyboard and the MIDI file is converted
into a piano roll, i.e., a binary matrix in $\mathbb{Z}^{T\times 88}$, where
$T$ is the number of time steps. The piano roll is a time-indexed trajectory
telling us which of the 88 notes should be active at every time step,
essentially defining a time-varying “goal trajectory." Each task has a
variable length $T$ which is a function of both the MIDI length and the
control frequency. At an implementation level, all tasks in RoboPianist are
exposed as distinct dm_env [49] or gym [50] environments, which makes it
straightforward to plug them directly into existing RL or imitation learning
codebases.
### III-B Simulation details
Reward | Formula | Weight | Explanation
---|---|---|---
Key Press | $0.5\cdot g(||k_{s}-k_{g}||_{2})+0.5\cdot(1-\mathbf{1}_{\\{\mathsf{false\;positive}\\}})$ | 1 | Press the right keys and only the right keys
Forearm Penalty | $1-\mathbf{1}_{\\{\mathsf{collision}\\}}$ | 0.4 | Minimize forearm collisions
Energy Penalty | $|\mathsf{\tau_{joints}}|^{\top}|\mathsf{v_{joints}}|$ | -5e-3 | Minimize energy expenditure
Finger Close to Key | $g(||p_{f}-p_{k}||_{2})$ | 1 | Shaped reward to bring fingers to key
TABLE II: The reward structure used for the model-free baseline. $\tau$
represents the joint torque, $v$ is the joint velocity, $p_{f}$ and $p_{k}$
represent the position of the finger and key in the world frame respectively,
$k_{s}$ and $k_{g}$ represent the current and the goal states of the key
respectively, and $g$ is a function that transforms the distances to rewards
in the $[0,1]$ range. See the appendix for a detailed description of each
term.
We use the open-source MuJoCo [51] simulator with the dm_control Python
bindings [52]. We chose MuJoCo for a few reasons: 1) faster-than-realtime
rigid body simulation in the presence of contacts, 2) MJCF model definition
combined with ease of task creation with the Composer module, 3) the feature-
rich interactive viewer allowing for visual debugging, playback and
interaction with the physical models using the mouse input, 4) recently, the
addition of a high-quality robot models [53] maintained by the creators of the
simulator.
Piano model. We create a full-size standard (“88-key”) digital piano in
simulation, which consists of 52 white keys and 36 black keys, spanning 12
major scales. We use a Kawai reference manual [54] to closely match the
dimensions, shape, positioning and spacing of the keys on the keyboard. Each
key is modeled using a linear spring. To speed up the simulation, we
explicitly disable collision checking between the white keys since they can’t
possibly ever be in contact. Additionally, rather than creating a custom key
mesh for the white keys (i.e., a box with a crevice in which the black key
rests), we disable collision checking between white and black keys. The result
is practically equivalent but faster to simulate since we exclusively deal
with primitive box geometries.
Hand model. We use the left and right Shadow Dexterous Hand [46] models from
MuJoCo Menagerie [53]. This anthropomorphic hand has been designed to closely
reproduce the kinematics and dexterity of the human hand. The Shadow Hand is
underactuated: it has 24 degrees of freedom, but only 20 actuators. This is
because the non-thumb finger distal joints are coupled. We add two degrees of
freedom to the base of the hand forearms to simulate the equivalent of a
planar gantry: a prismatic joint to translate laterally along the piano
length, and a prismatic joint to translate longitudinally along the piano
depth. Note that the Shadow Hand itself provides two rotational degrees of
freedom in each wrist (similar to human wrist joint). Since [19] used two
copies of a right hand, RoboPianist is the first benchmarking suite we know of
that uses both left and right hands, and also does so in a way in which the
handedness is relevant (piano pieces are designed for a pair of left and right
human hands).
Sound representation. We use the Musical Instrument Digital Interface (MIDI)
standard to represent piano pieces and synthesize sounds. Very briefly, a MIDI
file stores note_on and note_off messages. Each such message stores a note
number, a note velocity and a timestamp. The MIDI number is an integer between
0 and 127 and encodes a note’s pitch. The velocity is also an integer from 0
to 127 and controls the intensity of the sound. The timestamp specifies when
to execute the message. The moment a key is pressed on a piano, it generates a
note_on event, and the moment it is released, it generates a note_off event.
The intensity of the generated sound is controlled by the velocity of the
keystroke. In our simulation, a key is considered active when its joint
position exceeds the halfway point of its range. When this occurs, we emit the
note_on message. We currently do not model the intensity of each keynote,
captured by the velocity dimension in the MIDI representation.
### III-C MIDI dataset
A contribution of RoboPianist is providing a rich initial song corpora with
which we can evaluate high-dimensional control policies. To do this, we
identify a dataset developed for research into estimating human fingering for
piano, called the PIG dataset [55], and transform it into a corpus of MIDI
files that can be played in our simulated environment. This dataset contains
piano pieces from twenty-four Western composers spanning the baroque,
classical and romantic periods. The pieces vary in difficulty, ranging from
relatively easy (e.g., Mozart’s Piano Sonata K 545 in C major) to
significantly harder (e.g., Scriabin’s Piano Sonata No. 5). We refer the
reader to [55] for a detailed list of the pieces.
The fingering labels are encoded as a 10-dimensional boolean vector, with each
component indicating if the corresponding finger is expected to be in contact
with a key at that timestep. Note that fingering information is generally
quite sparse for most pieces of music, and is usually reserved for
particularly tricky sections of the piece. Pianists are expected to rely on
their skill and experience with the instrument to discover the most efficient
fingering. Further, most fingering markings on sheet music are a general
guide, and it is up to the pianist to gracefully incorporate this assistive
information and ensure a smooth musical flow. All these factors provide
enticing opportunities for the design of a controller’s combinatorial
reasoning. In our presented solution methodologies, we incorporate the boolean
fingering information within the reward formulation (Table II). For future
research enabled by the benchmark, having a policy discover the fingering
would be an impressive additional challenge.
### III-D Evaluation criteria
Agents in RoboPianist are evaluated based on precision, recall and F1 scores.
These metrics are computed by comparing the state of the piano keys at every
time step with the corresponding ground-truth state stored in the MIDI file,
averaged across all time steps. Intuitively, precision measures how good a
policy is at not hitting the wrong keys (i.e., low false positive rate) and
recall measures how good it is at hitting the right keys (i.e., low false
negative). Thus, a policy that hovers above the keyboard without hitting any
keys has high precision but low recall, and conversely, a policy that hits all
the keys simultaneously (assuming it were physically possible) has high recall
but low precision.
## IV High-Dimensional Robot Control
High-dimensional robot control requires amongst other things, precision in
space and time, coordination, planning, and contact. The RoboPianist task
suite and dataset stresses precisely these properties in a piano playing
environment (Section III). In this section, we describe initial steps towards
tackling the benchmark by introducing two popular policy optimization methods,
as well as various design considerations we incorporated to improve their
performance. These may serve as both (i) expert data generators (e.g., for
imitation learning), or (ii) baselines for future work on the benchmark.
### IV-A Baseline methods
There are a variety of method-agnostic criteria and solution regimes that one
may be interested in applying to the benchmark. For one, does the policy
method itself have access to the true dynamics of the environment (model-
based), or must it only learn through trial-and-error interactions with the
environment (model-free)? If it needs environment interactions, how many are
allowed? Additionally, although the exact cutoff limit of what may be
considered “real-time” is dependent on specific compute hardware, the ability
of a method to be approximately-fast-enough on common compute hardware in
order to synthesize behaviors in real-time, whether through a “direct”
feedback policy or through online model predictive control (MPC), is another
primary consideration that impacts the eventual deployable regimes on real-
world hardware.
As baseline methods, we present both model-free and model-based variants. By
their nature, the comparison is somewhat inherently “apples vs. oranges”. For
one, the model-based method uses the ground truth dynamics which are not used
by the model-free variant. Meanwhile, the model-free variant requires a large
number (order of 1 million) environment interactions, whereas the model-based
variant requires none. Additionally, the way in which computational resources
are consumed differs greatly. As is common for model-free RL, considerable
compute is used at training time, but the learned policy is encoded into a
learned neural network which runs quickly at inference time (we present
results with sub-millisecond feed-forward time on an M1 Max CPU). Meanwhile
for model-based MPC with a ground-truth model, it is common for no “training”
compute to be required, but instead inference may be computationally
intensive, and better results can be obtained simply by expending more
inference-time compute. Due to this “apples vs. oranges” nature, our goal is
not to proclaim which is “better”, but rather to characterize the performance
envelopes of both methods. We choose both to be in the regime of near-real-
time policy inference, where we allow the model-based method to slow down
simulation time as much as $10\%$ real-time. Model-based methods with
additional offline pre-computation are left for future work.
Model-free. The first baseline approach we consider is in the category of
model-free reinforcement learning. We present results with the off-policy
algorithm DroQ [56], one of several regularized variants of the widely-used
Soft-Actor-Critic [57] algorithm, as it is state-of-the-art in terms of
performance and sample efficiency. We train a separate policy per-song (multi-
task is left for future work) for one million environment interactions using
the reward structure in Table II. We found that even with $10\times$ the
number of samples, PPO [58] with default Stable Baselines3 [59]
hyperparameters was not able to achieve reasonable performance.
Figure 4: RoboPianist-prelude: Peformance of model-free RL with different MDP
design considerations. From top to bottom, in the order specified by the
legend, each curve inherits all MDP attributes of the curve before it, but
makes one additional modification as specified by its label. The PPO run,
which also inherits all the MDP tweaks, starts making progress at 10M
environment interactions, see the appendix for more information.
Model-based. The second baseline we present uses MPC, and specifically uses
the implementation from [60] which was shown to solve the previously-
considered-challenging [13, 17, 15] dexterous task of one-handed cube re-
orientation in simulation. Specifically amongst the various implementation
options presented in [60], we found the most success with the derivative-free
sampling-based method (“Predictive Sampling”). The cost formulation for the
MPC baseline is detailed in the appendix.
Note that while the method with which we received the best success was the
derivative-free “Predictive Sampling”, we also tried the optimized derivative-
based implementation of iLQG [61] also provided by [60], but this was not able
to make substantial progress even at significantly slower than real-time
speeds. We expect that it should be possible to acquire strong results with
derivative-based methods especially with sufficient reward shaping, both in
the near-real-time regime and also with offline model-based trajectory
optimization. We note, however, that the (i) high dimensionality, (ii) complex
sequence of goals adding many constraints, and (iii) overall temporal length
(tens of seconds) of the trajectories pose challenges for some methods that
are typically applied to significantly smaller problems.
### IV-B System design
We found that simple modifications to the MDP had a large impact on policy
performance. We detail these design considerations and analyze their effect in
the following sections.
* •
Fingering reward: We use the fingering annotations from the MIDI dataset to
encourage the fingers of the hand to reach the corresponding keys.
* •
Lookahead horizon: Instead of just goal-conditioning the policy for the
current timestep, we additionally include the “goal-trajectory" up to some
lookahead horizon $H$ into the future.
* •
Energy penalty: We add an extra reward term that penalizes the energy output
of the hand actuators, see Table II.
* •
Action space reduction: We reduced the dimensionality of the action space by
disabling some DoFs in the hand. We also restricted the range of some
actuators.
* •
Action-reward: We appended the action and reward obtained at the previous
timestep to the state, see Table II.
## V Experiments
(a)
(b)
Figure 5: RoboPianist-repertoire-150: (a) Performance of the model-free RL
baseline evaluated at checkpoints trained on increasing amounts of environment
interactions. Each point corresponds to an average over 150 MIDI files, 3
seeds each, with a standard deviation shading computed over all MIDI files.
(b) Performance of the MPC baseline evaluated with an increasing compute
budget (e.g., 2 = double the compute used by the same method running
realtime). Each point corresponds to an average over 150 songs, with a
standard deviation shading computed over all MIDI files.
The goal of our experiments is twofold. First, to characterize the performance
envelope of the algorithms in Section IV and gain insight into the key
challenges of the RoboPianist benchmark. Second, to investigate how the MDP
structure influences overall policy performance. We study these questions in
the context of three experimental setups: RoboPianist-prelude, RoboPianist-
etude-12 and RoboPianist-repertoire-150.
### V-A Experimental setup
RoboPianist-prelude consists of one piece with which one can extensively
compare the effects of the design decisions introduced in Subsection IV-B on
the model-free RL baseline. For these experiments, we use a 10-second snippet
of Mozart’s Twelve Variations on “Ah Vous Dirai-Je, Maman” (a.k.a. Twinkle
Twinkle Little Star) – a piece that (i) strikes a good balance in difficulty:
it is not too simple (e.g., requires both hands, there exists an ornament,
etc.) and not too hard (e.g., there are no chords), and (ii) is universally
recognizable and, thus, intuitive for humans to quickly listen to and evaluate
qualitative success.
RoboPianist-repertoire-150 considers a dataset of 150 MIDI files and evaluates
both algorithms mentioned in Section IV (including the best performing variant
of the model-free baseline from RoboPianist-prelude).
RoboPianist-etude-12 considers 12 randomly sampled songs from the full
repertoire of 150 MIDI files and again, evaluates both of the algorithms
mentioned in Section IV. This etude serves as a proxy for the full repertoire,
for smaller scale experiments and/or more modest compute budgets. For results
and analysis on this etude, we refer the reader to the appendix.
Training details. Training the model-free RL baseline takes approximately 5
hours per song on an Intel Xeon E5-2696V3 Processor hardware with 32 cores
(2.3 GHz base clock), 416 GB RAM and 4 Tesla K80 GPUs (n1-highmem-64 machine
type on Google Cloud). Inference for both model-free and model-based variants
is done on an M1 Max, 64GB RAM processor. All model-free policies are trained
and evaluated three times on each task with different random seeds. Model-
based policies are evaluated once per task. We implement our model-free
baseline using JAX [62], and our MPC baseline using MJPC [60]. Complete
hyperparameters and experimental details are listed in the appendix.
### V-B Overall evaluation
Heuristic | Pearson Correlation
---|---
Notes Per Second | -0.266
Pitch Class Entropy | -0.312
Total Time | -0.383
Max Polyphony | -0.458
Mean Polyphony | -0.447
Unique Pitches | -0.560
Unique Pitch Classes | -0.238
Pitch Range | -0.486
TABLE III: Correlation of musical properties of the MIDI dataset with the
performance of the model-free RL baseline as measured by its F1 score on
RoboPianist-repertoire-150. This shows that for example, more unique pitches
(different keys pressed) in a song is especially correlated with it being
harder.
The performance of the model-free RL baseline as well as the MPC baseline on
RoboPianist-repertoire-150 on all 150 tasks is illustrated in Figure 5. See
Figure 2 for individual results on all songs for model-free, and the appendix
for full results for model-based.
(a)
(b)
(c)
Figure 6: RoboPianist-prelude: (a) Performance of the model-free RL baseline
trained with different control frequencies, (b) Performance of the model-free
RL baseline trained with different values of the goal lookahead horizon, and
(c) Performance of the model-free RL baseline trained with and without a
fingering reward.
What makes a task hard? We can sort the songs in descending order based on the
F1 scores of the model-free RL policies, which presents a birds-eye view of
the difficulty between songs (i.e., lower is harder). By measuring the Pearson
correlation between the F1 scores against heuristics calculated per song from
the 150 MIDI files (shown in Table III), we can observe which task attributes
contribute the most to the difficulty.
Interestingly, we observe that the strongest negatively correlated variable is
the number of unique pitches (i.e., different keys on the keyboard) – the more
there are, the harder the task. This could suggest that the number of unique
notes to hit for a task contributes to the diversity of control trajectories
that need to be addressed by the policy. This attribute is followed by the
second-most negatively correlated variable maximum polyphony, i.e., the
average maximum number of notes active at any point in time. Successfully
playing multiple notes imposes additional constraints on precision. For high-
dimensional control, this could be akin to threading the needle through a more
narrow solution space, which can be more challenging to discover with RL
through trial and error. These correlations allow us to project difficulty
onto new prospective tasks as the dataset expands, and point to interesting
areas for future research.
### V-C Results and baseline studies
The following subsection discusses observations from varying the design
decisions detailed in Section IV of the model-free baseline policy on
RoboPianist-prelude. These observations may continue to be prevalent for
related future work in high-dimensional control.
Energy penalty leads to more spatially precise control. High-dimensional
control policies trained with RL and random exploration are subject to
learning more ecstatic policies (visible with non-smooth trajectories). In
particular, these movements deteriorate performance on a task like piano
playing, which requires both spatial precision and timing. As shown in Figure
4, we observe substantial improvements in quantitative performance (and much
less variance) by adding an energy penalty to the policy, which reduces the
ecstatic movements and bang-bang-like control during the initial stages of
training.
Looking into the future leads to more temporally precise control. We observe
additional improvements, both in performance and variance, from appending
future goal states (up to a certain horizon) to the goal vector. Intuitively,
this allows the policy to plan better for future notes – for example by
placing the non-finger joints (e.g., the wrist) in a manner that allows more
timely reaching of notes at the next timestep. However, as we show in 6(b),
increasing the goal lookahead horizon above 0.5 seconds leads to reduced
performance.
Reducing the action space trains faster and reduces un-human-like thumb
movement. To alleviate exploration even further, we explore the effect of
disabling degrees of freedom in the Shadow Hand that either do not exist in
the human hand (e.g., the little finger being opposable) or are not strictly
necessary for most songs. We additionally reduce the joint range of the thumb,
which on the Shadow Hand, can reach backward in a manner not possible on a
human hand. As shown in Figure 4, this change leads to faster training.
Qualitatively, the change is even more pronounced with the policies trained
with the reduced action space exhibiting more natural-looking thumb behaviors
(see the project website for a side-by-side comparison).
Fingering information alleviates exploration. Without the fingering reward,
the policy struggles to learn meaningful behaviors as shown in 6(c).
The control frequency matters. As shown in 6(a), the control frequency has a
substantial effect on learning. We find that 20Hz is a sweet spot, with
notably 100Hz drastically reducing the final score, and 10Hz converging a bit
slower. At 100Hz, the MDP becomes too long-horizon, which complicates
exploration, and at 10Hz, the discretization of the MIDI file becomes too
coarse, which negatively impacts the timing of the notes.
Adding the action and reward to the state helps. As shown in Figure 4, adding
the action and reward from the previous timestep, while marginal, also helps
improve the convergence speed of the policy. The previous action helps the
policy reason about velocities, which can enable more spatially and temporally
precise control. Note we could have obtained a similar effect by stacking a
history of past observations – we chose the option that has the smallest
increase on its dimension.
## VI Discussion and Conclusion
In this paper, we introduced the RoboPianist benchmark, which provides a
simulation framework and suite of tasks in the form of a corpora of songs,
together with baseline methods and evaluations, for studying the challenging
high-dimensional control problem of mastering piano-playing with bi-manual
hands. We showed that both well-tuned model-free and model-based baselines
struggle on this benchmark and explored different ways in which to improve
them. There is an array of exciting future directions to explore with
RoboPianist, including for example: pushing both model-based and model-free
methods, using human priors to accelerate learning, studying zero shot
generalization to new songs, and using multimodal data like sound and touch.
## VII Acknowledgments
This project was supported in part by ONR #N00014-22-1-2121 under the Science
of Autonomy program.
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## Appendix A Experiment details
### A-A Model-free reinforcement learning
Computing infrastructure and experiment running time
Our model-free RL codebase is implemented in JAX [62]. Experiments were
performed on a Google Cloud n1-highmem-64 machine with an Intel Xeon E5-2696V3
Processor hardware with 32 cores (2.3 GHz base clock), 416 GB RAM and 4 Tesla
K80 GPUs. Each “run”, i.e., the training and evaluation of a policy on one
task with one seed, took an average of 5 hrs wall clock time. These run times
are recorded while performing up to 8 runs in parallel.
Network architecture
We use a regularized variant of clipped double Q-learning [63, 64],
specifically DroQ [56], for the critic. Each $Q$-function is parameterized by
a 3-layer multi-layer perceptron (MLP) with ReLU activations. Each linear
layer is followed by dropout [65] with a rate of $0.01$ and layer
normalization [66]. The actor is implemented as a tanh-diagonal-Gaussian, and
is also parameterized by a 3-layer MLP that outputs a mean and covariance.
Both actor and critic MLPs have hidden layers with $256$ neurons and their
weights are initialized with Xavier initialization [67], while their biases
are initialized to zero.
Training and evaluation
We first collect $5000$ seed observations with a uniform random policy, after
which we sample actions using the RL policy. We then perform one gradient
update every time we receive a new environment observation. We use the Adam
[68] optimizer for neural network optimization. Evaluation happens in parallel
in a background thread every $10000$ steps. The latest policy checkpoint is
rolled out by taking the mean of the output (i.e., no sampling). Since our
environment is “fixed”, we perform only one rollout per evaluation.
Reward formulation
The reward function for training the model-free RL baseline consists of four
terms: 1) a key press term $r_{\text{key}}$, 2) a move finger to key term
$r_{\text{finger}}$, 3) a forearm collision term $r_{\text{forearm}}$ and 4)
an energy penalty term $r_{\text{energy}}$.
$r_{\text{key}}$ encourages the policy to press the keys that need to be
pressed and discourages it from pressing keys that shouldn’t be pressed. It is
implemented as:
$r_{\text{key}}=0.5\cdot\left(\dfrac{1}{K}\sum_{i}^{K}g(||k^{i}_{s}-1||_{2})\right)+0.5\cdot(1-\mathbf{1}_{\\{\mathsf{false\;positive}\\}}),$
where K is the number of keys that need to be pressed at the current timestep,
$k_{s}$ is the normalized joint position of the key between 0 and 1, and
$\mathbf{1}_{\\{\mathsf{false\;positive}\\}}$ is an indicator function that is
1 if any key that should not be pressed creates a sound (i.e., its normalized
joint position crossed 0.5). $g$ is the tolerance function from the dm_control
[32] library: it takes the L2 distance of $k_{s}$ and $1$ and converts it into
a bounded positive number between 0 and 1. We use the parameters bounds=0.05
and margin=0.5.
$r_{\text{finger}}$ encourages the fingers that are active at the current
timestep to move as close as possible to the keys they need to press. It is
implemented as:
$r_{\text{finger}}=\dfrac{1}{K}\sum_{i}^{K}g(||p^{i}_{f}-p^{i}_{k}||_{2}),$
where $p_{f}$ is the Cartesian position of the finger and $p_{i}$ is the
Cartesian position of a point centered at the surface of the key. $g$ for this
reward is parameterized by bounds=0.01 and margin=0.1.
$r_{\text{forearm}}$ encourages the shadow hand forearms not to collide. It is
implemented as:
$r_{\text{forearm}}=1-\mathbf{1}_{\\{\mathsf{collision}\\}},$
where $\mathbf{1}_{\\{\mathsf{collision}\\}}$ is 1 if the forearms are in
collision and 0 otherwise.
Finally, $r_{\text{energy}}$ penalizes high energy expenditure and is
implemented as:
$r_{\text{energy}}=|\mathsf{\tau_{joints}}|^{\top}|\mathsf{v_{joints}}|,$
where $\mathsf{\tau_{joints}}$ is a vector of joint torques and
$\mathsf{v_{joints}}$ is a vector of joint velocities.
The final reward function sums up the aforementioned terms as follows:
$r_{\text{total}}=r_{\text{key}}+r_{\text{finger}}+0.4\cdot
r_{\text{forearm}}-0.005\cdot r_{\text{energy}}$
Other hyperparameters
For a comprehensive list of hyperparameters used for training the model-free
RL policy, see Table IV.
Hyperparameter | Value
---|---
Total train steps | 1M
Optimizer |
Type | ADAM
Learning rate | $3\times 10^{-4}$
$\beta_{1}$ | $0.9$
$\beta_{2}$ | $0.999$
Critic |
Hidden units | $256$
Hidden layers | $3$
Non-linearity | ReLU
Dropout rate | 0.01
Actor |
Hidden units | $256$
Hidden layers | $3$
Non-linearity | ReLU
Misc. |
Discount factor | $0.99$
Minibatch size | $256$
Replay period every | $1$ step
Eval period every | $10000$ step
Number of eval episodes | $1$
Replay buffer capacity | 1M
Seed steps | $5000$
Critic target update frequency | $1$
Actor update frequency | $1$
Critic target EMA momentum ($\tau_{Q}$) | $0.005$
Actor log std dev. bounds | $[-20,2]$
Entropy temperature | $1.0$
Learnable temperature | True
TABLE IV: Hyperparameters for all model-free RL experiments.
### A-B Model predictive control
Computing infrastructure and experiment running time
Our model-based codebase is implemented in C++ with MJPC [60]. Experiments
were performed on a 2021 M1 Max Macbook Pro with 64 GB of RAM.
Algorithm
We use MPC with Predictive Sampling (PS) as the planner. PS is a derivative-
free sampling-based algorithm that iteratively improves a nominal sequence of
actions using random search. Concretely, $N$ candidates are created at every
iteration by sampling from a Gaussian with the nominal as the mean and a fixed
standard deviation $\sigma$. The returns from the candidates are evaluated,
after which the highest scoring candidate is set as the new nominal. The
action sequences are represented with cubic splines to reduce the search space
and smooth the trajectory. In our experiments, we used $N=10$, $\sigma=0.05$,
and a spline dimension of 2. We plan over a horizon of $0.2$ seconds, use a
planning time step of $0.01$ seconds and a physics time step of $0.005$
seconds.
Cost formulation
The cost function for the MPC baseline consists of 2 terms: 1) a key press
term $c_{\text{key}}$, 2) and a move finger to key term $c_{\text{finger}}$.
The costs are implemented similarly to the model-free baseline, but don’t make
use of the $g$ function, i.e., they solely consist in unbounded l2 distances.
The total cost is thus:
$c_{\text{total}}=c_{\text{key}}+c_{\text{finger}}$
Note that we experimented with a control cost and an energy cost but they
decreased performance so we disabled them.
## Appendix B Supplementary results
### B-A RoboPianist-prelude
PPO baseline
As mentioned in Section IV, we experimented with training a model-free policy
with PPO [58], but found that it performed much worse than DroQ both in terms
of sample efficiency and wall clock time. In Figure 7, we report the results
of the full training run on RoboPianist-prelude.
Figure 7: Performance of DroQ vs PPO on RoboPianist-prelude.
### B-B RoboPianist-repertoire-150
Figure 8: Full set of F1 scores achieved by the model-based MPC baseline on
all tasks in RoboPianist-repertoire-150.
The per-MIDI performance of the MPC baseline is illustrated in Figure 8.
### B-C RoboPianist-etude-12
Baseline | Avg. F1 | Avg. Precision | Avg. Recall
---|---|---|---
Model-free RL | 0.538 $\pm$ 0.122 | 0.990 $\pm$ 0.006 | 0.462 $\pm$ 0.109
MPC (100% realtime) | 0.266 $\pm$ 0.079 | 0.816 $\pm$ 0.073 | 0.225 $\pm$ 0.084
MPC (80% realtime) | 0.276 $\pm$ 0.069 | 0.813 $\pm$ 0.076 | 0.233 $\pm$ 0.081
MPC (20% realtime) | 0.402 $\pm$ 0.056 | 0.786 $\pm$ 0.060 | 0.351 $\pm$ 0.064
MPC (10% realtime) | 0.433 $\pm$ 0.072 | 0.775 $\pm$ 0.059 | 0.385 $\pm$ 0.077
TABLE V: robopianist-etude-12: Aggregate scores for model-free and model-based
baselines averaged over all 12 MIDI files in the etude. The scores for the
model-free baseline are obtained from the final checkpoint at 1M steps.
As mentioned in Section V, we consider a smaller subset of the full repertoire
(12 randomly sampled MIDI files) and report the performance of the model-free
and model-based baselines. These results are summarized in Figure 9 and Table
V.
(a)
(b)
Figure 9: RoboPianist-etude-12: (a) Performance of the model-free RL baseline
evaluated at checkpoints trained on increasing amounts of environment
interactions. Each point corresponds to an average over 12 MIDI files, 3 seeds
each, with a standard deviation shading computed over all MIDI files. (b)
Performance of the MPC baseline evaluated with an increasing compute budget
(e.g., 2 = double the compute used by the same method running realtime). Each
point corresponds to an average over 12 songs, with a standard deviation
shading computed over all MIDI files.
|
# Proximal Policy Optimization-based Transmit Beamforming and Phase-shift
Design in an IRS-aided ISAC System for the THz Band
Xiangnan Liu, Haijun Zhang, , Keping Long, , Mingyu Zhou, Yonghui Li, , and H.
Vincent Poor This work is supported in part by the National Key Research and
Development Program of China (Grant No. 2020YFB1708800), in part by Beijing
Natural Science Foundation (L212004), in part by the National Natural Science
Foundation of China under Grants 61822104 and 61771044, in part by the State
Key Laboratory of Advanced Metallurgy under Grant KF20-04, in part by the
Fundamental Research Funds for the Central Universities under Grants FRFTP-
19-002C1 and RC1631, and in part by Beijing Top Discipline for Artificial
Intelligent Science and Engineering, University of Science and Technology
Beijing (_Corresponding author: Haijun Zhang_.) Xiangnan Liu, Haijun Zhang,
and Keping Long are with Institute of Artificial Intelligence, Beijing
Advanced Innovation Center for Materials Genome Engineering, Beijing
Engineering and Technology Research Center for Convergence Networks and
Ubiquitous Services, University of Science and Technology Beijing, Beijing
100083, China (email<EMAIL_ADDRESS><EMAIL_ADDRESS>longkeping@ustb.edu.cn). Mingyu Zhou is with BaiCells Technologies Company,
Beijing, China, 100094 (email: Zhoumingyu@baicells.com). Yonghui Li is with
the School of Electrical and Information Engineering, The University of
Sydney, Sydney, NSW 2006, Australia (email: yonghui.li@sydney.edu.au). H.
Vincent Poor is with the Department of Electrical and Computer, Princeton
University, NJ 08544 USA (e-mail: poor@princeton.edu).
###### Abstract
In this paper, an IRS-aided integrated sensing and communications (ISAC)
system operating in the terahertz (THz) band is proposed to maximize the
system capacity. Transmit beamforming and phase-shift design are transformed
into a universal optimization problem with ergodic constraints. Then the joint
optimization of transmit beamforming and phase-shift design is achieved by
gradient-based, primal-dual proximal policy optimization (PPO) in the multi-
user multiple-input single-output (MISO) scenario. Specifically, the actor
part generates continuous transmit beamforming and the critic part takes
charge of discrete phase shift design. Based on the MISO scenario, we
investigate a distributed PPO (DPPO) framework with the concept of multi-
threading learning in the multi-user multiple-input multiple-output (MIMO)
scenario. Simulation results demonstrate the effectiveness of the primal-dual
PPO algorithm and its multi-threading version in terms of transmit beamforming
and phase-shift design.
###### Index Terms:
Integrated sensing and communications, transmit beamforming, phase shift
design, intelligent reflecting surface, distributed reinforcement learning.
## I Introduction
Rrecently, the integrated sensing and communications (ISAC) has been proposed
to enhance the sensing capability in the location/environment-aware scenarios.
Many scenarios require ISAC to obtain high-rate transmission and high-
resolution target detection, such as autonomous vehicles, indoor localization,
and extended reality (XR) [1]. ISAC is considered to be a promising technique
for the next generation wireless systems to support high-accuracy sensing
services [2]. Following the communication signal processing, the millimeter
wave (mmWave) band is considered in the ISAC system. Previous discussions of
ISAC systems have considered the use of the mmWave bands for use in automotive
radars and high-resolution imaging radars [3]. Therefore, higher bandwidth and
transmission rate is a technique development in the envisioned sixth
generation (6G) networks.
In this context, terahertz (THz) communication is expected to enable ultra-
high-speed communications in the era of 6G. The 6G networks will require
higher data rates and capacity to maintain quality of service, economical
operation, and flexible resource management [4]. The THz band can provide
wider communication bandwidth than the current wireless communication band.
THz communication can obtain Gbps wireless transmission rate, which will
enable new sensing applications such as miniaturized radars for gesture
detection and touchless smartphones, spectrometers for explosive detection and
gas sensing. THz communication can provide higher transmission rates and wider
bandwidths than communication in lower-frequency bands. However, the molecular
absorption in the THz band is severe. And the THz waves with strong
directivity and poor diffraction are blocked by many obstacles more easily.
To tackle these challenges, the intelligent reflecting surface (IRS) has
recently emerged as a promising technique to address the issue of excessive
pathloss by creating better propagation environments [5]. IRSs are composed of
passive antenna elements with adjustable phase shifts. Combined with a phase-
shift design in an IRS, transmit beamforming can realize better ISAC
performance. On the one hand, the transmit beamforming can be utilized to
synthesize multiple beams towards existing users and targets [6]. On the other
hand, the IRS can boost the ISAC signal by changing the phase optimization
dynamically [7]. Compared with a conventional relay, the IRS does not employ a
radio frequency unit and so can save energy. Thus an IRS-aided wireless
communication system can obtain higher spectrum and energy efficiencies.
By utilizing the IRS technique, the communication performance of the ISAC
system can be enhanced. Meanwhile, high-data and high-resolution sensing
performance can be achieved in the THz band. While IRS-aided systems open new
possibilities, they also bring new design challenges, such as passive
beamforming, channel realization, and deployment of the IRS. Moreover, there
have been few studies of transmit beamforming and phase shift design for the
IRS-aided ISAC system. Therefore. we focus on the joint optimization of
transmit beamforming and phase-shift design in this paper.
### I-A Related Works
Transmit beamforming and phase-shift design in general IRS-aided system have
attracted extensive attention in recent years. They have been studied in
various systems, such as multi-user multiple-input single-output (MISO) [8],
multiple-input multiple-output (MIMO) communication [9], and simultaneous
wireless information and power transfer (SWIPT) systems [10]. To reduce the
power consumption, Abeywickrama et.al. proposed a practical method by jointly
designing the access point transmit beamforming and IRS passive phase
optimization, subject to constraints on users’ individual signal-to-
interference-plus-noise ratios (SINRs) [8]. Transmit beamforming was studied
in an IRS-assisted SWIPT system, with the constraints of quality-of-service
(QoS) constraints at all users, by applying a penalty-based optimization
method [10]. In [11], Shen et al. first derived a closed-form solution for the
base station (BS)’s transmit beamforming to maximize the signal-to-noise ratio
(SNR) of a radar signal. Furthermore, transmit beamforming has been designed
to optimize various objective functions, such as power consumption [12],
achievable rate [13], and energy efficiency [14]. On the one hand, THz will
enable new sensing applications such as miniaturized radars for gesture
detection and touchless smartphones, spectrometers for explosive detection and
gas sensing. On the other hand, IRS is a powerful relay with unique properties
to enhance the radar and communication signal strength in the THz band.
Therefore, the joint optimization of transmit beamforming and phase-shift
design is meaningful, in terms of the communication and radar performance.
Deep reinforcement learning (DRL) is a state-of-the-art method that can be
used to optimize radio resource allocation [15]. Despite the fact that
excellent performances of many optimization algorithms have been observed
through numerical simulations and theoretical analysis, implementing them in
real systems still faces many serious difficulties. In particular, the high
computational cost incurred by these algorithms has been one of the most
challenging issues. Deep learning can reduce the computational cost but it
acquires the training set, resulting in poor flexibility of deep learning.
Furthermore, DRL-based methods do not need training labels and possess the
property of online learning and sample generation, which is more storage-
efficient. Recently, it has been leveraged to optimize performance in wireless
systems extensively, for problems such as power control [16], bandwidth
scheduling [17], computation offloading [18], and caching deployment [19].
Mismar et al. proposed joint optimization of analog beamforming, power
allocation, and interference coordination in the sub-6 GHz and mmWave band. A
deep Q-learning network (DQN) was applied to tackle this joint optimization
[20]. Recently DRL has been applied to solve optimization problems in IRS
assisted communications [21, 22, 23]. In [21], a DRL framework is designed to
investigate the passive phase optimization for the IRS-aided MISO scenario to
maximize the SNR. The results demonstrated a considerable gain compared to
existing schemes. In [22], the transmit beamforming and passive phase were
jointly optimized by using a DRL framework. The developed framework can
support large-dimensional optimization problems in massive MIMO systems. The
transmit beamforming in the BS and passive beamforming in the IRS were jointly
designed to optimize the system transmission rate in [23] by using
prioritized. Towards the complicated practical scenario, the DRL with a single
thread is intractable because of its lower utilization of computing process.
Furthermore, distributed DRL is a promising technique to solve more
complicated wireless networks problems. Distributed DRL is composed of a
central controller and a group of learners. It can be classified as multi-
agent DRL or multi-threading DRL. The latter uses the multi-threading method
to train agents. Interactive learning takes place simultaneously in multiple
threads and each thread summarizes the learning results, sorting and saving
them in a public place. Compared with multi-agent DRL, multi-threading DRL can
save power consumption. Additionally, multi-threading DRL can save training
time and keep the training process stable [24]. Du et al. proposed an
asynchronous advantage actor-critic (A3C) algorithm to solve joint
optimization of viewport rendering offloading and power allocation [25]. Zhang
et al. solved the optimization problem of power control in the cognitive radio
network by utilizing distributed DRL methods, such as A3C and distributed
proximal policy optimization (DPPO) [26]. Dinh et al. designed a distributed
model-free algorithm DeepPool to optimize ride-sharing platforms [27].
Compared with multi-agent reinforcement learning, there are relatively few
studies of multi-threaded DRL, especially for transmit beamforming and passive
phase optimization.
On the other hand, ergodic stochastic optimization, which uses term averages
to reflect system performance, is suitable for wireless resource allocation.
This idea was originated by Ribeiro [28], who proposed the average variable
scheme for radio resource management. This ergodic system performance metric
is valid for both independent channels and correlated channels. A further
study [29] considered optimizing primal and dual variables by learning
resource allocation policy gradients. Lee et al. in [30] proposed a
distributed scheme involving average variables to capture system performance,
and two deep neural networks (DNNs) were introduced to approximate the
allocation policy. Combining this scheme of average variables with a DRL
framework is a promising approach, which also becomes one of motivations of
this paper.
### I-B Contributions
In this context, we study an IRS-aided ISAC system in the THz band. In
particular, transmit beamforming and phase-shift design in an IRS-aided
device-based ISAC system with multiple users are studied. The capacity with
ergodic constraints is set as the utility function to evaluate the IRS-aided
ISAC system performance. This problem is challenging because of the joint
optimization of transmit beamforming and phase-shift design. To tackle this
problem, proximal policy optimization (PPO) and its distributed forms are
developed in different scenarios, including MISO and MIMO. The main
contributions of this paper are summarized below.
* •
_The IRS-aided ISAC system in the THz band:_ The IRS is explored to compensate
for the high path loss caused by molecular absorption in the THz band. We
consider an IRS-aided ISAC system consisting one BS equipped with several
antennas, several users and an IRS. The capacity of the ISAC system is
optimized through transmit beamforming and passive phase-shift design of the
IRS.
* •
_The primal-dual model-free PPO with ergodic constraints:_ The optimization
problem of capacity maximization is formulated as a universal optimization
problem with ergodic constraints and is transformed into the dual domain. A
gradient-based, primal-dual PPO algorithm is designed to solve the problem of
capacity maximization with ergodic constraints.
* •
_The joint optimization via the actor-critic structure:_ PPO is leveraged to
achieve joint optimization on continuous transmit beamforming of the ISAC BS
and discrete phase design of the IRS. The actor part optimizes transmit
beamforming and the critic part designs the phase shifts, to maximize the
total capacity in the multi-user MISO scenario.
* •
_The optimization of multi-user MIMO scenario through distributed DRL:_
Distributed PPO (DPPO) is introduced in the established model to solve this
joint optimization. The concept of multi-threading applied to the more
complicated matrix processing in the MIMO scenario. Each worker docks with one
user to collect observations and transmit them to the global PPO. The global
PPO then broadcasts the optimized gradients to each worker.
The DRL-based method to achieve the joint optimization of transmit beamforming
and phase-shift design is different from the current studies of transmit
beamforming in ISAC system. Utilizing the IRS to counteract the high pathloss
in the THz band, this emerging technique still lacks in-depth studies studies
for ISAC system. Towards the joint optimization in the multi-user MISO
scenario, the proposed primal-dual PPO algorithm is novel and effective.
Beyond the multi-user MISO scenario, the primal-dual PPO algorithm’s
distributed version can be applied to the multi-user MIMO scenario. Simulation
results will demonstrate the effectiveness of primal-dual PPO-based algorithm
in both the multi-user MISO scenario and multi-user MIMO scenario.
The rest of this paper is organized as follows. Section II introduces the
system model and formulates the optimization problem of transmit beamforming
and phase-shift design. Section III presents the primal-dual PPO algorithm to
realize joint optimization of transmit beamforming and passive phase
optimization in the multi-user MISO scenario. In Section IV, the primal-dual
DPPO is extended to the MIMO scenario. Section V shows the pseudo codes and
implementations of these two algorithms. The proposed algorithms are verified
by simulations results in VI, and Section VII concludes the paper.
## II System Model and Problem Formulation
### II-A System Model
Figure 1: Primal-dual PPO learning beamforming and phase-shift design in the
IRS-aided ISAC system.
As shown in Fig. 1, a multi-user MISO model is considered in the IRS-aided
ISAC system. It consists of one ISAC BS with $M$ antennas and $K$ users
equipped with single antenna. Let ${\cal M}=\left\\{{1,2,...,M}\right\\}$ and
${\cal K}=\left\\{{1,2,...,K}\right\\}$ denote the set of the BS’s antennas
and served users.
${\bf{s}}={\left[{{s_{1}},s_{2},...,s_{k}}\right]^{T}}\in{{\mathbb{C}}^{K\times
1}}$ and ${\bf{W}}\in{{\mathbb{C}}^{M\times K}}$ are the information-bearing
symbol and the transmit beamforming matrix, respectively. The transmit signal
${\bf{X}}\in{{\mathbb{C}}^{M\times 1}}$ after transmit beamforming is given by
${\bf{X}}={\bf{W}}\cdot{\bf{s}}.$ (1)
_1) Communication Signal:_
Let ${{\bf{H}}_{1}}\in{{\mathbb{C}}^{N\times M}}$,
${{\bf{H}}_{0}}\in{\mathbb{C}}^{{M\times K}}$, and
${{\bf{H}}_{2}}\in{\mathbb{C}}^{{N\times K}}$ represent the channel gain from
the ISAC BS to the IRS, and the channel gain from the ISAC BS to user $k$, the
channel gain from the IRS’s element $n$ to user $k$, respectively. The
communication signal ${{\bf{y}}_{c}}\in{{\mathbb{C}}^{K\times 1}}$ received by
users can be written as
${\bf{y}}_{c}=\left({{{\bf{H}}_{0}^{H}}+{{\bf{H}}_{2}^{H}}{\bf{V}}{{\bf{H}}_{1}}}\right){{\bf{X}}}+{\bf{n}}_{c},$
(2)
${\bf{n}}_{c}\in{{\mathbb{C}}^{K\times 1}}$ is the independent identically
distributed complex Gaussian stochastic process. The effective diagonal phase
matrix $\bf V$ is
$diag\left\\{{{\gamma_{1}}{e^{j{\beta_{1}}}},{\gamma_{2}}{e^{j{\beta_{2}}}},...,{\gamma_{N}}{e^{j{\beta_{N}}}}}\right\\}$,
where ${\gamma_{n}}\in\left[{0,1}\right]$ and
${\beta_{n}}\in\left[{0,2\pi}\right]$ are amplitude and phase, respectively.
Let ${\cal N}=\left\\{{1,2,...,N}\right\\}$ denote the set of the IRS’s
elements. Define $\gamma_{n}$ as the amplitude reflection coefficient of the
IRS and it can be calculated as [8]
${\gamma_{n}}\left({{\beta_{n}}}\right)=\left({1-{\gamma_{\min}}}\right){\left({\frac{{\sin\left({{\beta_{n}}-\varphi}\right)+1}}{2}}\right)^{\varepsilon}}+{\gamma_{\min}}.$
(3)
${\gamma_{\min}}$ is the minimum amplitude, $\varphi$ is the horizontal
distance between $-\pi/2$ and ${\gamma_{\min}}$. $\varphi\geq 0$ and
$\varepsilon\geq 0$ are the constants depending on the IRS’s circuit.
All of them are modeled as the Rician channel [31],
${{\bf{H}}_{i}}=\sqrt{\frac{\gamma_{AI}}{{1+\gamma_{AI}}}}{{\bf{H}}_{LOS}}+\sqrt{\frac{1}{{1+\gamma_{AI}}}}{{\bf{H}}_{NLOS}},i=0,1,2,$
(4)
where $\gamma_{AI}$ is the Rician factor, ${\bf{H}}_{LOS}$ is the line-of-
sight (LoS) component and it can be expressed as
${{\bf{H}}_{LOS}}={{\alpha}_{LOS}\left({f,l}\right){G_{t}}{G_{r}}{{\bf{b}}_{r}{\left({{\theta_{AoA}}}\right)}}{\bf{a}}_{t}^{H}\left({{\theta_{AoD}}}\right)}$,
where ${{\bf{b}}_{r}}\left(\theta_{AoA}\right)$ and
${{\bf{a}}_{t}}\left(\theta_{AoD}\right)$ is the receive response vector and
transmit response vector, respectively.
${{\mathbf{a}}_{t}^{H}}\left(\cdot\right)$ denotes the conjugate array
steering vector. Additionally, $\theta_{AoD}$ and $\theta_{AoA}$ are the angle
of departure (AoD) and the angle of arrival (AoA), respectively.
We consider a uniform linear antenna (ULA) to be the structure of the ISAC
BS’s antennas, so the steering vector
${{\bf{a}}}\left(\theta\right)\in{{\mathbb{C}}^{M\times 1}}$ is given by
${{{\bf{a}}}\left(\theta\right)=\sqrt{\frac{1}{M}}{\left[{1,{e^{-j{{2\pi
d}\mathord{\left/{\vphantom{{2\pi
d}\lambda}}\right.\kern-1.2pt}\lambda}\cos\theta}},...,{e^{-j{{2\pi
d}\mathord{\left/{\vphantom{{2\pi
d}\lambda}}\right.\kern-1.2pt}\lambda}\left({M-1}\right)\cos\theta}}}\right]^{T}}.}$
(5)
$d$ is the antenna element spacing and $\lambda$ is the wavelength.
For the IRS, an $N\times N$ element uniform planar antenna (UPA) is
considered. The array steering vector
${{\mathbf{v}}}\left({\phi,{\mathbf{\theta}}}\right)\in{{\mathbb{C}}^{N\times
1}}$ is given by
$\begin{array}[]{*{20}{l}}{{{\mathbf{v}}}\left({\phi,{\mathbf{\theta}}}\right)=\frac{1}{N}\left[{1,...,{e^{j\frac{{2\pi}}{\lambda}d\left({n\cos{\phi}\cos{\theta}+n\sin{\phi}\sin{\theta}}\right)}},...{\kern
1.0pt}{\kern 1.0pt},}\right.}\\\ {{{\left.{{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}{e^{j\frac{{2\pi}}{\lambda}d\left({\left({\sqrt{N}-1}\right)\cos{\phi}\cos{\theta}+\left({\sqrt{N}-1}\right)\sin{\phi}\sin{\theta}}\right)}}}\right]}^{T}}}.\end{array}$
(6)
$n$ is the antenna element index with $0\leq n\leq N-1$.
In the THz band, there is not only free space pathloss $L_{spread}$, but also
the module pathloss $L_{medium}$. The pathloss of LoS
${\left|{{\alpha_{L}}\left({f,l}\right)}\right|^{2}}$ is expressed by
${\left|{{\alpha_{L}}\left({f,l}\right)}\right|^{2}}={L_{spread}}\left({f,l}\right){L_{medium}}\left({f,l}\right).$
(7)
$f$ is the carrier frequency and $l$ is the transmission distance. The free
space pathoss $L_{spread}$ can be calculated by
${L_{spread}}\left({f,l}\right){\text{ = }}{\left({\frac{c}{{4\pi
fl}}}\right)^{2}}.$ (8)
Similar to the channel state of the LoS path ${{\bf{H}}_{LOS}}$,
${{\bf{H}}_{NLOS}}={\sum\limits_{i=1}^{{n_{NL}}}{{\alpha_{i}}\left({f,l}\right){G_{t}}{G_{r}}{{\bf{b}}_{r}}\left({{\theta_{AoA,i}}}\right){\bf{a}}_{t}^{H}\left({{\theta_{AoD,i}}}\right)}}$
is set as the non-line-of-sight (NLoS) component. Let $G_{t}$ and $G_{r}$
represent channel gain from transmit antenna and received antenna,
respectively. ${n_{NL}}$ is the number of NLoS rays and
$\alpha\left({f,l}\right)$ is the complex gain of the path.
The large-scale channel gain $\alpha_{LOS}$ and $\alpha_{NLOS}$ can be given
by
$\displaystyle\begin{array}[]{*{20}{l}}{\alpha_{LOS}\left({f,l}\right)}={L_{spread}}\left({f,l}\right){L_{abs}}\left({f,l}\right)\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}={\left({\frac{c}{{4\pi
fl}}}\right)^{2}}{e^{-{k_{abs}}\left(f\right)l}},\end{array}$ (9c)
$\displaystyle\begin{array}[]{*{20}{l}}{\alpha_{NLOS}\left({f,l}\right)}={\Gamma^{2}\left(f\right)}{L_{spread}}\left({f,l}\right){L_{abs}}\left({f,l}\right)\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}={\left({\frac{c}{{4\pi
fl}}}\right)^{2}}{e^{-{k_{abs}}\left(f\right)l}},\end{array}$ (9f)
the speed of light is
$c=3\times{10^{8}}\;{m\mathord{\left/{\vphantom{ms}}\right.\kern-1.2pt}s}$.
Let $l$ denote the transmission distance in free space. The absorption
coefficient $k_{abs}\left(f\right)$ can be calculated by the medium available
at a molecular level [25]. The reflecting coefficient ${\Gamma}\left(f\right)$
is the product of the Fresnel reflection coefficient $\iota$ and the Rayleigh
roughness factor $\xi$. The high reflection loss (i.e. up to second order
reflections) is only considered in the THz band. The Fresnel reflection
coefficient $\iota$ can describe the behavior of light in medium with
different refractive indices. It can be calculated by
$\iota\left(f\right)=\frac{{Z\left(f\right)\cos{\varphi_{in}}-{Z_{0}}\cos{\varphi_{ref}}}}{{Z\left(f\right)\cos{\varphi_{in}}+{Z_{0}}\cos{\varphi_{ref}}}},$
(10)
where $Z\left(f\right)$ is the reflecting material ’s frequency-dependent wave
impedance and ${{\rm{Z}}_{0}}{\rm{=377}}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}\Omega{\rm{}}$ is the free space wave impedance. Let
$\varphi_{in}$ denote the angle of incidence and reflection. Additionally,
${\varphi_{ref}}=\arcsin\left({{\textstyle{Z\over{{Z_{0}}}}}\sin{\varphi_{in}}}\right)$
represents the angle of refraction.
The Rayleigh roughness factor $\xi$ is calculated by
$\zeta\left(f\right)={e^{-{\textstyle{1\over 2}}{{\left({{\textstyle{{4\pi
f\sigma\cos{\varphi_{in}}}\over c}}}\right)}^{2}}}}.$ (11)
They are composed of reflection and scattering, so each user $k$’s channel
state can be calculated by [32]
${\bf{y}}_{c,k}=\left({{{\bf{H}}_{0,k}^{H}}+{{\bf{H}}_{2,k}^{H}}{\bf{V}}{{\bf{H}}_{1,k}}}\right){{\bf{X}}}+{\bf{n}}_{c,k}={\bf
H}_{k}{\bf X}_{k}+{\bf{n}}_{c,k},$ (12)
Furthermore, the user $k$ ’s signal-to-interference-to-noise ratio (SINR) can
be calculated by
${SINR}_{k}=\frac{{{{\left|{{\bf{H}}_{k}^{H}{{\bf{W}}_{k}}}\right|}^{2}}}}{{\sum\limits_{l\neq
k}{{{\left|{{\bf{H}}_{k}^{H}{{\bf{W}}_{l}}}\right|}^{2}}}+\sigma_{c}^{2}}}.$
(13)
_2) Radar Signal:_
The radar signal received at the ISAC receiver is given by [33]
${\bf{y}}_{r}=\left({{\bf{H}}_{1}^{H}{\bf{VB}}{{\bf{V}}^{H}}{{\bf{H}}_{1}}{\bf{+A}}}\right){\bf{X}}+{\bf{n}}_{r},$
(14)
It is assumed that IRS is a mono-static MIMO radar[2]. Thus,
${\bf{A}}\in{{\mathbb{C}}^{M\times M}}$ is the target response matrix of the
ISAC BS and ${\bf{B}}\in{{\mathbb{C}}^{N\times N}}$ is the target response
matrix based on parameters of the IRS’s elements in the device-based ISAC
system. They can be calculated by,
$\displaystyle{\bf{A}}=\sum\limits_{k=1}^{K}{{\alpha_{k}}{{\bf{a}}_{r}}\left({{\theta_{k}}}\right){{\bf{a}}_{t}^{H}}\left({{\theta_{k}}}\right)},$
(15a)
$\displaystyle{\bf{B}}=\sum\limits_{k=1}^{K}{{\alpha_{k}}{{\bf{v}}_{r}}\left({{\upsilon_{k},\theta_{k}}}\right){\bf{v}}_{t}^{H}\left({{\upsilon_{k},\theta_{k}}}\right)}.$
(15b)
$\alpha_{k}$ is the complex pathloss that include the coefficient of the
pathloss, reflection, and complex radar cross of the target. Vector
${{\mathbf{a}}_{t}}\left(\theta\right)$ and
${{\mathbf{a}}_{r}}\left(\theta\right)$ are array steering vectors from
transmit and receive antennas. For radar signals, beam pattern error is
considered in the proposed system [6]. MIMO radar transmit beamforming design
aims to optimize the transmit power at given directions, or generally match a
desired beam pattern. The beam pattern error is the mean square error (MSE)
between the obtained beam pattern and the ideal beam pattern. It can be
calculated by the following formula
${L_{r}}\left({\bf{R}}\right)=\frac{1}{L}\sum\limits_{l=1}^{L}{{{\left|{d\left({{\psi_{l}}}\right)-P\left({{\psi_{l}}}\right)}\right|}^{2}}},$
(16)
where ${\bf{R}}={\bf{W}}{{\bf{W}}^{H}}$ is covariance matrix. The direction
grids $\left\\{{{\psi_{l}}}\right\\}_{l=1}^{L}$ are sampled in the range of
$-90^{\circ}$ to $90^{\circ}$ with resolution of $0.1^{\circ}$.
$P\left(\psi\right)={{\bf{a}}^{H}}\left(\psi\right){\bf{Ra}}\left(\psi\right)$
is power consumption in direction $\psi$ and $d\left({{\psi}}\right)$ is
desired beam pattern, its calculating principle is
$d\left(\psi\right)=\left\\{{\begin{array}[]{*{20}{l}}{1,}&{{\psi_{p}}-{\Delta\mathord{\left/{\vphantom{\Delta
2}}\right.\kern-1.2pt}2}\leq\psi\leq{\psi_{p}}+{\Delta\mathord{\left/{\vphantom{\Delta
2}}\right.\kern-1.2pt}2},p=1,2,3,}\\\ {0,}&{otherwise},\end{array}}\right.$
(17)
where the ideal beam pattern $\psi_{p}$ consists of three main beams and
$\Delta=10^{\circ}$ is each ideal beam width.
The beam pattern error is set as a constraint and it can be expressed as
${L_{r}}\left({\bf{R}}\right)\leq\ell,$ (18)
The above constraint implies that the beam pattern error is not allowed to
exceed the threshold $\ell$. The threshold symbolizes the accuracy of the
sensing performance.
### II-B Problem Formulation
The problem of transmit beamforming and phase shift design can be transformed
into a long-term instantaneous performance function using the ergodic average
value $\bf x$ to reflect the system performance [28], and this solution can be
applied to optimize other systems, including optimization of frequency
division multiplexing, power control, and random access. It can be calculated
by
${\bf{x}}\leq{\mathbb{E}}\left[{{{\bf
f}_{1}}\left({{\bf{h}},{\bf{p}}\left({\bf{h}}\right)}\right)}\right],$ (19)
where ${{\bf f}_{1}}\left({{\bf{h}},{\bf{p}}\left({\bf{h}}\right)}\right)$ is
an instantaneous performance function. The design is to choose a resource
allocation ${\bf{p}}\left({\bf{h}}\right)$ to maximize the ergodic variable
${\bf{x}}$. The average variable ${\bf{x}}$ reflects the performance of
wireless communication systems in a considerably long period and is influenced
by instantaneous resource allocation ${\bf{p}}\left({\bf{h}}\right)$.
$\begin{array}[]{*{20}{l}}{\max}&{{f_{0}}\left({\mathbf{x}}\right)}\\\
{s.t.}&{{\bf
x}\leqslant\mathbb{E}\left[{{{\mathbf{f}}_{1}}\left({{\mathbf{h}},{\mathbf{p}}\left({\mathbf{h}}\right)}\right)}\right]}\\\
{}&{{{\mathbf{f}}_{2}}\left({\mathbf{x}}\right)\geqslant
0,{\mathbf{x}}\in\chi,{\mathbf{p}}\in\mathcal{P}.}\end{array}$ (20)
In the proposed system model, ${\bf{h}}$ is the channel state $\bf H$,
${\bf{p}}\left({\bf{h}}\right)$ is the transmit beamforming matrix $\bf W$ and
phase optimization $\bf V$.
${{\bf{f}}_{1}}\left({{\bf{h}},{\bf{p}}\left({\bf{h}}\right)}\right)$ is the
corresponding data transmission rate. ${{\bf{f}}_{2}}\left(\bf x\right)$
indicates the vector utility function. The design goal is to maximize the
average vector $\bf{x}$ for transmit beamforming
${\bf{p}}\left({\bf{h}}\right)$ under constraints,
${x_{k}}\leqslant{{\mathbb{E}}_{\bf{H}}}\left[{\log\left({1+SIN{R_{k}}}\right)}\right].$
(21)
Its design goal is to minimize radar signal loss performance with long-term
constraints for transmit beamforming and phase-shift design. It is assumed
that the element of covariance matrix ${\left[\bf{R}\right]_{mm}}$ has power
budget:
${{\left[\bf{R}\right]_{mm}}}\leqslant{{{P_{max}}}\mathord{\left/{\vphantom{{{P_{max}}}M}}\right.\kern-1.2pt}M},m=1,2,...,M,$
(22)
where $P_{max}$ is the total transmit power. The utility
${f_{0}}\left({\bf{x}}\right)$ is set as the sum rate function.
${f_{0}}\left({\bf{x}}\right)=\sum\limits_{k=1}^{K}{{x_{k}}}.$ (23)
Finally, the optimization problem of MISO scenario in the IRS-aided ISAC
system can be formulated as
$\begin{array}[]{*{20}{l}}{\mathop{\max}\limits_{{\bf{V,W}}}{f_{0}}\left({\bf{x}}\right)}\\\
{\begin{array}[]{*{20}{l}}{s.t.{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{x_{k}}\leqslant{{\mathbb{E}}_{\bf{H}}}\left[{\log\left({1+SIN{R_{k}}}\right)}\right]},\\\
{{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}{{\left[\bf{R}\right]_{mm}}}\leq{{{P_{max}}}\mathord{\left/{\vphantom{{{P_{max}}}M}}\right.\kern-1.2pt}M},m=1,2,...,M,}\\\
{{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}{L_{r}}\left({\bf{R}}\right)\leq\ell}.\end{array}}\end{array}$ (24)
## III Primal-dual Proximal Policy Optimization Scheme in Multi-user MISO
Scenario
In this section, we investigate multi-user MISO scenario in the IRS-aided ISAC
system in the THz band, and use the primal-dual optimization to solve the
problem (24).
### III-A Transmit Beamforming and Phase-shift Design via PPO
For the problem (24), the value-based method in DRL is not able to deal with
continuous actions. The value-based method is unable to obtain the optimal
solution with the constraints. Therefore, the research is mainly based on
policy optimization. The traditional policy-based solution is expressed by
[29]:
${L^{PG}}\left({\mbox{\boldmath{$\omega$}}}\right)={\mathbb{E}_{t}}\left[{\log{\mbox{\boldmath{$\pi$}}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right){\mathbf{A}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)}\right].$
(25)
The advantage function can be calculated by
${\bf{A}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)=\sum\nolimits_{t^{\prime}>t}{{\gamma^{t^{\prime}-t}}{r^{t^{\prime}}}}-{v_{\upsilon}}\left({{{\bf{H}}^{t}},{\mbox{\boldmath{$\upsilon$}}}^{t}}\right)$.
The shortcoming of this conventional method is the need to update the step
size. Inappropriate setting of the step size will lead to a degraded
performance. We need find a step size to ensure that the reward function
increases monotonically for each iteration. On the other hand, the
conventional policy gradient methods are sensitive to their hyper parameters
and have high variance. Employing a trust region constraint can be regarded as
an effective choice. The popular solution is trust region policy optimization
(TRPO). It can be given by
$\begin{array}[]{*{20}{l}}{\mathop{\max}\limits_{\mbox{\boldmath{$\omega$}}}}&{{\mathbb{E}_{{\rho_{\omega}}\left(\tau\right)}}\left[{\sum\nolimits_{t}{{\gamma^{t-1}}\frac{{{\mbox{\boldmath{$\pi$}}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}}{{{{\mbox{\boldmath{$\pi$}}}_{old}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}}{{\mathbf{A}}_{old}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}}\right]},\\\
{s.t.}&{{D_{KL}}\left[{\frac{{{\mbox{\boldmath{$\pi$}}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}}{{{{\mbox{\boldmath{$\pi$}}}_{old}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}}}\right]\leqslant\delta}.\end{array}$
(26)
More detailed classifications on TRPO are presented in Appendix A. The PPO
algorithm is an approximate version of TRPO based on the first order
gradients, utilizing DNNs to a large-scale distributed setting. PPO solves
this problem by introducing relative entropy in policy update [34].
$\begin{array}[]{*{20}{l}}{{L^{PPO}}\left({\mbox{\boldmath{$\omega$}}}\right)=\sum\nolimits_{t}{\left({{{{\mbox{\boldmath{$\pi$}}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)}\mathord{\left/{\vphantom{{{\mathbf{\pi}}\left({{\mathbf{H,\omega}}}\right)}{{{\mathbf{\pi}}_{old}}\left({{\mathbf{H,\omega}}}\right)}}}\right.\kern-1.2pt}{{{\mbox{\boldmath{$\pi$}}}_{old}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)}}}\right){\mathbf{A}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)}}\\\
{{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}-\lambda{D_{KL}}\left[{{{{\mbox{\boldmath{$\pi$}}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)}\mathord{\left/{\vphantom{{{\mbox{\boldmath{$\pi$}}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)}{{{\mbox{\boldmath{$\pi$}}}_{old}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)}}}\right.\kern-1.2pt}{{{\mbox{\boldmath{$\pi$}}}_{old}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)}}}\right]}.\end{array}$
(27)
Generally speaking, PPO is based on the actor-critic structure, and the actor
part’s goal is to maximize ${L^{PPO}}\left(\mbox{\boldmath{$\omega$}}\right)$,
which can be divided into two types, the one is
${L^{KLPEN}}\left(\mbox{\boldmath{$\omega$}}\right)$ in PPO1 and the other is
${L^{CLIP}}\left(\mbox{\boldmath{$\omega$}}\right)$ in PPO2. A penalty on
Kullback-Leibler divergence (KL divergence) is used in PPO1, and to adapt the
penalty coefficient so that we achieve some target values of the KL divergence
target each policy update. PPO2 has no KL divergence term in the target, nor
any constraints. Instead, it relies on tailoring the objective function to
eliminate the incentives of the new policy and the old policy.
$\begin{array}[]{*{20}{l}}{{L^{CLIP}}\left({\mbox{\boldmath{$\omega$}}}\right)=}&{{\mathbb{E}_{t}}\left[{\min\left({{\textstyle{{\pi\left({H,{\mbox{\boldmath{$\omega$}}}}\right)}\over{{\pi_{old}}\left({\bf
H,{\mbox{\boldmath{$\omega$}}}}\right)}}}{\bf A}\left({{\bf
H},{\mbox{\boldmath{$\omega$}}}}\right),}\right.}\right.}\\\
{}&{\left.clip\left({{\textstyle{{\pi\left({{\bf
H},{\mbox{\boldmath{$\omega$}}}}\right)}\over{{\pi_{old}}\left({{\bf
H},\omega}\right)}}},1-\epsilon,1+\epsilon}\right){\bf A}\left({{\bf
H},{\mbox{\boldmath{$\omega$}}}}\right)\right]}.\end{array}$ (28)
To get the similarity degree of the probability distribution of actions, KL
divergence $D_{KL}$ can be used to calculate. Under the setting of
hyperparameter is $\epsilon=2$, the second term
$clip\left({\frac{{\pi\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)}}{{{\pi_{old}}\left({{\bf{H,\mbox{\boldmath{$\omega$}}}}}\right)}},1-\epsilon,1+\epsilon}\right)$
modifies the surrogate objective by clipping the probability ratio. In the
formula (28),
$\frac{{\pi\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)}}{{{\pi_{old}}\left({{\bf{H,\mbox{\boldmath{$\omega$}}}}}\right)}}$
can ensure that the distribution gap between the two updates is small. This
clipped operation can avoid unnecessary samples to save training time.
To obtain optimization of transmit beamforming and phase-shift design, the
actor part takes charge of continuous transmit beamforming in the established
PPO framework. Meanwhile, the critic part undertakes the discrete phase-shift
design.
The policy
${\mbox{\boldmath{$\pi$}}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)$
is introduced to act as the transmit beamforming $\bf W$. For trained
parameters $\omega$, we make
${\mbox{\boldmath{$\pi$}}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)={\bf
W}$. And the problem (20) is transformed into the following problem.
$\begin{array}[]{*{20}{l}}{\mathop{\max}\limits_{{{\mathbf{H}}},{{\mbox{\boldmath{$\omega$}}}}}{f_{0}}\left({{{\mathbf{x}}}}\right)}\\\
{\begin{array}[]{*{20}{l}}{s.t.{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{{\mathbf{x}}}\leqslant\mathbb{E}\left[{{{{\bf
f}_{1}}}\left({{\bf{H}},{\mbox{\boldmath{$\pi$}}}\left({{\bf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}\right)}\right]},\\\
{{\mathbf{f}}_{2}}\left({\mathbf{x}}\right)\geqslant
0.\par\end{array}}\end{array}$ (29)
The problem (29) is the general form of the problem (24). The constraint
${{\left[\bf{R}\right]_{mm}}}\leq{{{P_{max}}}\mathord{\left/{\vphantom{{{P_{max}}}M}}\right.\kern-1.2pt}M}$
is clipped by the action output of policy
${\mbox{\boldmath{$\pi$}}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)$.
We assume that
${{\mathbf{f}}_{2}}\left({\mathbf{x}}\right)=\ell-{L_{r}}\left({\bf{R}}\right)$.
To convenience of understanding, the following formulation is based on the
general form (29).
For the critic part, the phase shift of the instantaneous performance function
can be calculated by the output action. The critic part adopts the design
concept of DQN, it needs to be discrete,
${i_{n}}\in\left\\{{0,1,...,{2^{b}}-1}\right\\}$, the corresponding phase
degree ${\beta_{n}}$ is given by
${\beta_{n}}=\frac{{{i_{n}}2\pi}}{{{2^{b}}-1}}.$ (30)
Accordingly, its Q-value function produces a discrete action ${i_{n}}$ as
${i_{n}}=\mathop{\arg\max}\limits_{{{\bf{i}}_{n}}}{\bf Q}^{t}\left({{{{\bf
H}_{k}^{t}}},{{\bf{i}}_{n}}},{{\mbox{\boldmath{$\upsilon$}}}_{k}^{t}}\right).$
(31)
And the amplitude $\gamma_{n}$ also can be calculated through the formula (3).
Therefore, the phase shift design of IRS can be obtained by the critic part.
### III-B Primal-dual Learning Optimization
The primal-dual optimization is introduced to tackle the problem (29). The
Lagrangian of the problem (29) is utility and constraints weighted by their
multipliers. For convenience, its Lagrangian function
$\mathcal{L}\left(\cdot\right)$ for the problem (29) is given by
$\begin{array}[]{*{20}{l}}{\mathcal{L}\left({{\mbox{\boldmath{$\omega$}}},{\bf{x}},{\mbox{\boldmath{$\lambda$}}},{\mbox{\boldmath{$\mu$}}}}\right)={f_{0}}\left({\bf{x}}\right)+{{\mbox{\boldmath{$\mu$}}}^{T}}{{{\bf
f}_{2}}}\left({\bf{x}}\right)}\\\ {{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+{{\mbox{\boldmath{$\lambda$}}}^{T}}\left({{\mathbb{E}}\left[{{{{\bf
f}_{1}}}\left({{\bf{H}},{\mbox{\boldmath{$\pi$}}}\left({{\bf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}\right)}\right]-{\bf{x}}}\right)}.\end{array}$
(32)
Find the gradient of the four parameters in sequence,
${{\mbox{\boldmath{$\omega$}}}^{t+1}}={{\mbox{\boldmath{$\omega$}}}^{t}}+{\tau_{1}}{\nabla_{\omega}}{\mathbb{E}}\left[{{{\bf{f}}_{1}}\left({{\bf{H}},{\mbox{\boldmath{$\pi$}}}\left({{\bf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}\right){{\mbox{\boldmath{$\lambda$}}}^{t}}}\right],$
(33)
${{\bf{x}}^{t+1}}={{\bf{x}}^{t}}+{\tau_{2}}\left({\nabla{f_{0}}\left({{{\bf{x}}^{t}}}\right)+\nabla{{\bf{f}}_{2}}\left({{{\bf{x}}^{t}}}\right){{\mbox{\boldmath{$\mu$}}}^{t}}-1}\right),$
(34)
where $\tau_{1},\tau_{2}>0$ denote scalar step sizes. A gradient update is
performed on current dual iterates $\lambda_{k},\mu_{k}$ in a similar manner
by subtracting the partial stochastic gradients $\nabla_{\lambda}L$,
$\nabla_{\mu}L$ and projecting onto the positive orthant to obtain
${{\mbox{\boldmath{$\lambda$}}}^{t+1}}={{\mbox{\boldmath{$\lambda$}}}^{t}}-{\tau_{3}}\left({{\mathbb{E}_{\bf
H}}{{\bf{f}}_{1}}\left({\bf{H}},{{\mbox{\boldmath{$\pi$}}}\left({{\bf{H}},{{\mbox{\boldmath{$\omega$}}}^{t+1}}}\right)}\right)-{{\bf{x}}^{t+1}}}\right).$
(35)
${{\mbox{\boldmath{$\mu$}}}^{t+1}}={{\mbox{\boldmath{$\mu$}}}^{t}}-{\tau_{4}}{{\bf{f}}_{2}}\left({{{\bf
x}^{t+1}}}\right).$ (36)
with associated step sizes $\tau_{3},\tau_{4}>0$. The gradient primal-dual
updates in (33)–(36) successively optimize the primal and dual variables to
the maximum and minimum points of the Lagrangian function, respectively.
The updates in the formula (33) and (34) can be replaced with the zeroth-
ordered gradient updates. The gradients estimates with finite difference can
be established using function observations in the given initial variables
${{\mbox{\boldmath{$x$}}}_{0},{\mbox{\boldmath{$\omega$}}}_{0}}$ and the
sampled points
${{\bf\widehat{x}}_{1},{\bf\widehat{x}}_{2},{\mbox{\boldmath{$\widehat{\omega}$}}}}$
as,
$\widehat{\nabla}{{{f}}_{0}}\left({{{\bf{x}}_{0}}}\right):=\frac{{{{\widehat{f}}_{0}}\left({{{\bf{x}}_{0}}+{\alpha_{1}}{{{\bf{\hat{x}}}}_{1}}}\right)-{{\widehat{f}}_{0}}\left({{{\bf{x}}_{0}}}\right)}}{{{\alpha_{1}}}}{{\bf{\hat{x}}}_{1}},$
(37)
$\widehat{\nabla}{\bf{f}_{2}}\left({{{\bf{x}}_{0}}}\right):=\frac{{{{\widehat{\bf{f}}}_{2}}\left({{{\bf{x}}_{0}}+{\alpha_{2}}{{{\bf{\hat{x}}}}_{2}}}\right)-{{\widehat{\bf{f}}}_{2}}\left({{{\bf{x}}_{0}}}\right)}}{{{\alpha_{2}}}}\widehat{\bf{x}}_{2}^{T},$
(38)
$\begin{array}[]{*{20}{l}}{{{\widehat{\nabla}}_{\mbox{\boldmath{$\omega$}}}}\mathbb{E}\left[{{\bf{f}_{1}}\left({{\mbox{\boldmath{$\pi$}}}\left({{\bf{H}},{\mbox{\boldmath{$\omega$}}}}\right),{\bf{H}}}\right)}\right]}\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{:=\frac{{\widehat{\bf{f}_{1}}\left({{\bf{H}},{\mbox{\boldmath{$\pi$}}}\left({{\bf{H}},{{\mbox{\boldmath{$\omega$}}}_{0}}+{\alpha_{3}}\widehat{\mbox{\boldmath{$\omega$}}}}\right)}\right)-\widehat{{\bf
f}_{1}}\left({{\bf{H}},{\mbox{\boldmath{$\pi$}}}\left({{\bf{H}},{{\mbox{\boldmath{$\omega$}}_{0}}}}\right)}\right)}}{{{\alpha_{3}}}}{{\widehat{\mbox{\boldmath{$\omega$}}}}^{T}}},\end{array}$
(39)
where $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ are the gradient estimated
upgrading step factors. Furthermore, the $\tau_{1}$, $\tau_{2}$, $\tau_{3}$,
$\tau_{4}$ are the primal-dual upgrading steps.
## IV Distributed Proximal Policy Optimization Scheme for Multi-user MIMO
Scenario
In this section, we will explore how to obtain joint optimization of transmit
beamforming and phase shift design in the multi-user MIMO scenario.
Distributed DRL will be introduced in this section.
### IV-A System Model and Formulation in Multi-user MIMO Scenario
In the multi-user MIMO ISAC system, it is assumed that there exists a BS with
$M$ transmit antennas and $K$ users with $R$ receive antennas. The transmit
signal to the user $k$ through beamforming
${{\mathbf{X}}_{k}}\in{\mathbb{C}^{M\times R}}$ is
${{\mathbf{X}}_{k}}={{\mathbf{W}}_{k}}\cdot{{\mathbf{s}}_{k}},$ (40)
where
${{\mathbf{s}}_{k}}={\left[{{s_{1}},{s_{2}},...,{s_{R}}}\right]^{T}}\in{\mathbb{C}^{R\times
1}}$. The responding transmit beamforming matrix is
${{\mathbf{W}}_{k}}\in{\mathbb{C}^{M\times R}}$ and radar signal is expressed
by:
${{\mathbf{y}}_{r,k}}=\left({\mathbf{H}}_{1}^{H}{{\mathbf{V}}}{\mathbf{B}}{{\mathbf{V}}^{H}}{{\mathbf{H}}_{1}+{\mathbf{A}_{k}}}\right){{\mathbf{X}}_{k}}+{\mathbf{n}}_{r,k}.$
(41)
Similar to the formula (2), the communication signal ${\mathbf{y}}_{c,k}$ is
derived from
${\mathbf{y}}_{c,k}=\left({{{\mathbf{H}}_{0,k}}+{{\mathbf{H}}_{2,k}}{\mathbf{V}}{{\mathbf{H}}_{1}}}\right){{\mathbf{X}}_{k}}+{\mathbf{n}}_{c,k}.$
(42)
Compared with ${{\bf{H}}_{0}}\in{\mathbb{C}}^{{M\times K}}$ and
${{\bf{H}}_{2}}\in{\mathbb{C}}^{{N\times K}}$ in multi-user MISO scenario, the
channel gains are transformed into
${{\mathbf{H}}_{0,k}}\in{\mathbb{C}^{M\times R}}$ and
${{\mathbf{H}}_{2,k}}\in{\mathbb{C}^{N\times R}}$, respectively. And the
corresponding parameterization problem is expressed as follows:
$\begin{array}[]{*{20}{l}}{\mathop{\max}\limits_{{{\mathbf{H}}_{k}},{{\mathbf{\omega}}_{k}}}{f_{0}}\left({{{\mathbf{x}}_{k}}}\right)}\\\
{\begin{array}[]{*{20}{l}}{s.t.{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{{\mathbf{x}}_{k}}\leqslant\mathbb{E}\left[{{\text{log}}\left({1+SIN{R_{k}}\left({{{\mathbf{H}}_{k}},{{{\mbox{\boldmath{$\pi$}}}}_{k}}\left({{{\mathbf{H}}_{k}},{{{\mbox{\boldmath{$\omega$}}}}_{k}}}\right)}\right)}\right)}\right]},\\\
{{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{{\left[{\bf
R}_{k}\right]_{mm}}}\leqslant{{{P_{max}}}\mathord{\left/{\vphantom{{{P_{max}}}M}}\right.\kern-1.2pt}M},m=1,2,...,M},\\\
{{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}{L_{r}}\left({{\mathbf{R}}_{k}}\right)\leqslant\ell.}\end{array}}\end{array}$
(43)
Similar to the problem (29), ${f_{0}}\left({{{\mathbf{x}}_{k}}}\right)$ in the
problem (43) is set as the sum rate function. Typically, the instantaneous
system performance in practical MIMO transmission designs can not reflect the
system performance well. One of solutions adopts the long term average such as
ergodic capacity for transmission optimization [35].
The channel gain ${\bf H}_{k}$ is the state in the environment, what we
optimize is the transmit beamforming and phase-shift design. In this paper,
the transmit beamforming ${\bf W}_{k}$ and phase-shift design ${\bf V}$ is
policy derived form the state, i.e. channel gain ${\bf H}_{k}$. Similar to the
multi-user MISO scenario, the ${{\mbox{\boldmath{$\pi$}}}_{k}\left({\bf
H}_{k},{\mbox{\boldmath{$\omega$}}}_{k}\right)}$ form actor part realizes the
transmit beamforming and the critic part takes charge of the phase-shift
design. When the scenario extended to the MIMO one, the number of receive
antennas $R$ takes place of the number of users $K$. The channel ${\bf
H}\in{\mathbb{C}}^{{M\times R\times K}}$ is 3 dimensions, compared with
2-dimension ${\bf H}$ in MISO scenario. How to solve higher dimension in
multi-user MIMO IRS-aided ISAC system is worthy of to be studied, so DPPO,
this multi-threading DRL technique is introduced in the following subsection.
### IV-B DPPO Optimization via multi-threading DRL
DPPO with multiple threads can be adapted to more complex environments. DPPO
is not a specific algorithm like PG or DQN, it conveys the concept of multi-
threading learning. Its core is that in the training process of DRL, we can
train multiple agents in parallel threading. In the training process, each
worker takes charge of data collection locally. Multiple workers can enrich
the experiences form the environment. After the same episodes, distributed
learning can accelerate the learning speed further because of its richer
experience replay and its higher utilization rate of CPU.
The DPPO is suitable for the proposed multi-user MIMO scenario in IRS-aided
ISAC system. Every user can act as each worker in multi-threading DRL. It can
collect different channel realization from its own situation. Accordingly,
each worker (i.e. user) takes the corresponding policy to optimize its
capacity.
Through the total rewards from different workers (i.e. users), the original
PPO algorithm can be achieved by estimated advantage functions ${{\bf
A}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)$. To facilitate the use
of DNNs with batch updates while also supporting variable length episodes,
L-step returns is proposed to estimate the advantage, i.e. we sum the rewards
over the same L-step windows and bootstrap from the value function after
L-steps:
${{\bf
A}^{t}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)=\sum\nolimits_{l=1}^{L}{{\gamma^{l-1}}{r^{t}}+}{\gamma^{L-1}}{v}\left({{{\bf{H}}^{t+L}}},{{\mbox{\boldmath{$\upsilon$}}}^{t+L}}\right)-{v}\left({{\bf{H}}^{t}},{{\mbox{\boldmath{$\upsilon$}}}^{t}}\right).$
(44)
Workers share a global PPO and the gradients of PPO would not be calculated by
workers. Gradients will not be transformed to the chief like A3C. Workers only
push the data collection by themselves to the global PPO. Each worker
calculates the local $L^{CLIP}_{k}\left(\mbox{\boldmath{$\omega$}}\right)$ by
the formula (29), computes gradient ${\nabla_{\omega}}{L^{CLIP}}$ and sends to
the global PPO. The global PPO updates a certain batch of data from multiple
workers (workers stop collecting when the chief is updating). After updating,
workers collect data with the latest policy ${\mbox{\boldmath{$\pi$}}}^{t}$.
### IV-C Transmit Beamforming and Phase-shift Design in Multi-user MIMO
Scenario
Figure 2: Primal-dual DPPO beamforming and phase-shift design in the multi-
user MIMO scenario.
In the design of the proposed multi-user MIMO scenario, the dimension of
channel state between the ISAC BS and users consists of $M\times R\times K$.
It is quite difficult to realize transmit beamforming and passive phase
optimization directly. With the help of DPPO’s multi-threading training, each
user will be allocated to each worker. The workers can collect observation and
choose actions in parallel. Compared with multi-user MISO scenario, each
worker processes the channel state ${\bf H}_{k}\in{\mathbb{C}}^{{M\times R}}$,
which is similar to ${\bf H}\in{\mathbb{C}}^{{M\times K}}$. The global PPO
will allocate actions to different workers according to the experience memory.
## V Transmit Beamforming and Phase-Shift Design Optimization Algorithm
Design
### V-A Primal-dual PPO Algorithm for Multi-user MISO Scenario
In the proposed algorithm, we utilize the PPO’s architecture to learn two
kinds of actions. Firstly, we allocate the equal initial beamforming to each
user, and initial policy parameters including mean
$\mbox{\boldmath{$\omega$}}^{0}$ and Langrange multipliers
$\mbox{\boldmath{$\lambda$}}^{0}$ and $\mbox{\boldmath{$\mu$}}^{0}$. And then,
each of the ISAC BS’s antennas deploys PPO method in turn, and the sum of
antennas’ iteration is $M$. Next, beginning with the actor part’s and the
critic part’s training, we compute the gradient estimate from replay
experience. The estimated gradient integrates into the primal-dual
optimization. The primal-dual PPO algorithm is used to iteratively calculate
the transmit beamforming matrix ${\bf W}^{\rm*}$ and the phase-shift design
matrix ${\bf V}^{\rm*}$.
In Algorithm 1, episode begins with channel state ${\bf
H}^{0}\in{\mathbb{C}}^{{M\times K}}$ between the ISAC BS and users. It can be
calculated by the formula (4) and the first phase shift ${\bf V}^{0}$ in IRS
is sampled in domain between $\left[{-\pi/2,\pi/2}\right]$. Then, the
iteration begins and the channel state ${\bf
H}_{k}^{0}\in{\mathbb{C}}^{{M\times 1}}$ is transited into the DNN of policy.
Accordingly, the actor part outputs the policy
${\mbox{\boldmath{$\pi$}}}_{k}^{t}\left({\bf
H}_{k}^{t},{\mbox{\boldmath{$\omega$}}}_{k}^{t}\right)$ to realize transmit
beamforming. Simultaneously, the critic part outputs Q-value function ${\bf
Q}^{t}\left({\bf H}_{k}^{t},{\mbox{\boldmath{$\upsilon$}}}_{k}^{t}\right)$.
The choice of ${i_{n}^{t}}$ follows the greedy policy. The algorithm selects a
random discrete action $i_{n}$ with the probability $\varepsilon$ and selects
${i_{n}}=\mathop{\arg\max}\limits_{{{\bf{i}}_{n}}}{\bf Q}^{t}\left({{{{\bf
H}_{k}^{t}}},{{\bf{i}}_{n}}},{{\mbox{\boldmath{$\upsilon$}}}_{k}^{t}}\right)$
with probability $1-\varepsilon$. The reward ${\bf
r}^{t}_{k}={{{f}}_{0}^{t}}\left({{{\bf{x}}}}\right)$ can be calculated by the
obtained channel state ${\bf H}_{k}^{t}$ and transmit beamforming ${\bf
W}_{k}^{t}$. At the same time, the index $i_{n}^{t}$ of the IRS derived from
the critic part affects the next state ${\bf H}_{k}^{t+1}$ through new ${\bf
V}^{t+1}$. Therefore, the batch of
$\left\\{{{\bf{H}}_{k}^{t},{{\mbox{\boldmath{$\pi$}}}}_{k}^{t},{\bf{r}}_{k}^{t}},{{\bf
H}_{k}^{t+1}}\right\\}$ is obtained. Then batches are collected into the
experience memory and parameters of the actor part $\omega$ and the critic
part $\upsilon$ can be updated by the experience memory.
Algorithm 1 Transmit beamforming and phase-shift design via primal-dual PPO
algorithm
Input Initial transmit beamforming ${\bf W}^{0}$, policy parameters
${\mbox{\boldmath{$\omega$}}^{0}}$, Lagrange multipliers
${{\mbox{\boldmath{$\lambda$}}}^{0}},{{\mbox{\boldmath{$\upsilon$}}}^{0}}$ can
be allocated for users. channel states ${\bf H}_{0}$, ${\bf H}_{1}$, and ${\bf
H}_{2}$ are fixed in each episode.
Episode begins
For $i=1,2,...,I$ do:
For users $k=1,2,...,K$ do:
a) Draw samples ${{{\bf\hat{x}}_{1}}}$ and ${{{\bf\hat{x}}_{2}}}$ from a
truncated standard normal distribution;
b) Calculate the channel state ${{\bf H}_{k}^{t}}\in{{\mathbb{C}}^{M\times
1}}$, ${{\bf H}^{t}=}{{\bf{H}}_{0}}{\bf{+}}{{\bf{H}}_{2}}{{\bf
V}^{t}}{{\bf{H}}_{1}}$;
c) ${{\bf H}_{k}^{t}}$ is fed into the actor part, then the actor part outputs
strategy function ${\bf
W}_{k}^{t}={\mbox{\boldmath{$\pi$}}_{k}^{t}}\left({{{\bf
H}_{k}^{t}},{\mbox{\boldmath{$\omega$}}_{k}^{t}}}\right)$; fed into the critic
part, then the critic part outputs Q-value function ${\bf Q}^{t}\left({{{\bf
H}_{k}^{t}},{{\mbox{\boldmath{$\upsilon$}}}_{k}^{t}}}\right)$;
d) Calculate the reward ${\bf
r}^{t}_{k}={{{f}}_{0}^{t}}\left({{{\bf{x}}}}\right)$ via the formula (23);
e) With the probability $\varepsilon$ select a random discrete action $i_{r}$
otherwise select ${i_{r}}=\mathop{\arg\max}\limits_{{{\bf{i}}_{r}}}{\bf
Q}^{t}\left({{{{\bf
H}_{k}^{t}}},{{\bf{i}}_{r}}},{{\mbox{\boldmath{$\upsilon$}}}_{k}^{t}}\right)$;
f) Calculate the amplitude $\gamma_{n}$ and phase $\beta_{n}$ through the
formula (3) and (29). Subsequently, the phase-shift design ${\bf V}^{t+1}$ can
be obtained;
g) The next state ${{\bf H}_{k}^{t+1}}$ can be obtained by ${{\bf
H}^{t+1}=}{{\bf{H}}_{0}}{\bf{+}}{{\bf{H}}_{2}}{{\bf V}^{t+1}}{{\bf{H}}_{1}}$;
h) Collect
$\left\\{{{\bf{H}}_{k}^{t},{{\mbox{\boldmath{$\pi$}}}}_{k}^{t},{\bf{r}}_{k}^{t}},{{\bf
H}_{k}^{t+1}}\right\\}$ into the experience memory to estimate discounted
reward and advantages, then update the actor part and the critic part;
i) Compute the gradient estimate
$\widehat{\nabla}{{{f}}_{0}}\left({{{\bf{x}}}}\right)$, $\widehat{\nabla}{{\bf
f}_{2}}\left({{{\bf{x}}}}\right)$,
${{{\widehat{\nabla}}_{\mbox{\boldmath{$\omega$}}}}\mathbb{E}\left[{{\bf{f}_{1}}\left({{\mbox{\boldmath{$\pi$}}}\left({{\bf{H}},{\mbox{\boldmath{$\omega$}}}}\right),{\bf{h}}}\right)}\right]}$:
Execute the formulas (37)–(39);
j) Update the primal-dual variable ${\mbox{\boldmath{$\omega$}}}^{t+1}_{k}$,
${{\bf{x}}^{t+1}_{k}}$, ${{\mbox{\boldmath{$\lambda$}}}^{t+1}_{k}}$,
${{\mbox{\boldmath{$\mu$}}}^{t+1}_{k}}$: Execute the formulas (33)–(36);
Execute step b) to h) until convergence, and an optimized strategy is obtained
after the PPO training ${{\mbox{\boldmath{$\pi$}}}_{k}^{\rm{*}}}\left({{{\bf
H}_{k}},{{\mbox{\boldmath{$\omega$}}}_{k}}}\right)$;
End for
End for
Episode ends
Output Transmit beamforming ${\bf{W}^{\rm{*}}}$, phase-shift design
$\bf{V^{\rm{*}}}$, and system performances
${{f}_{0}^{\rm{*}}}\left({{{\bf{x}}}^{\rm{*}}}\right)$.
Subsequently, the gradient estimate
$\widehat{\nabla}{{{f}}_{0}}\left({{{\bf{x}}}}\right)$, $\widehat{\nabla}{{\bf
f}_{2}}\left({{{\bf{x}}}}\right)$,
${{{\widehat{\nabla}}_{\mbox{\boldmath{$\omega$}}}}\mathbb{E}\left[{{\bf{f}_{1}}\left({{\mbox{\boldmath{$\pi$}}}\left({{\bf{H}},{\mbox{\boldmath{$\omega$}}}}\right),{\bf{h}}}\right)}\right]}$
can be obtained from the formulas (37)–(39). And primal-dual variables can be
obtained through the formulas (33)–(36). After primal-dual PPO training
performance converges, the optimized transmit beamforming ${\bf{W}^{\rm{*}}}$,
the phase-shift design $\bf{V^{\rm{*}}}$, and the system performances
${{f}_{0}^{\rm{*}}}\left({{{\bf{x}}}^{\rm{*}}}\right)$ can be achieved. For
the sake of clarity, a full notation list is included in TABLE I.
TABLE I: List of Notations. Notations | Explanations
---|---
$\cal M$ | The set of ISAC BS’s antennas
$\cal K$ | The set of served users
$\cal N$ | The set of IRSs’ elements
$\cal R$ | The set of user’s antennas
$\bf A$ | The target response matrix of the ISAC BS
$\bf B$ | The target response matrix of the IRS’s elements
${\bf V}$ | The effective diagonal phase matrix
$i_{n}$ | The discrete action of the IRS’s element index
$\gamma$ | The amplitude reflection coefficient of the IRS
$\beta$ | The phase-shift coefficient of the IRS
$\theta$ | The azimuth angle
$\theta_{AoD}$ | The angle of departure
$\theta_{AoA}$ | The angle of arrival
$\psi_{l}$ | The sampled angle
$d\left({{\psi}}\right)$ | The desired beam pattern in direction $\psi$
$d\left({{\psi_{p}}}\right)$ | The ideal beam pattern in direction $\psi_{p}$
$\phi$ | The elevation angle
${\bf{a}}\left({{\theta}}\right)$ | Steering vector at the ISAC BS
${{\mathbf{v}}}\left({\phi,{\mathbf{\theta}}}\right)$ | Steering vector at the IRSs
${\bf{b}}\left({{\theta}}\right)$ | Steering vector at receive users
$P\left(\psi\right)$ | The power consumption in direction $\psi$
$P_{max}$ | The total transmit power
$\bf R$ | The covariance matrix
${\bf H}_{0}$ | The channel gain from the ISAC BS to user $k$
${\bf H}_{1}$ | The channel gain from the ISAC BS to the IRS
${\bf H}_{2}$ | The channel gain from the IRS’s element $n$ to user $k$
${\bf H}_{LOS}$ | The LoS component
${\bf H}_{NLOS}$ | The NLoS component
$\bf x$ | The ergodic average value
${{f}_{0}}\left(\cdot\right)$ | The sum function
${{\bf f}_{1}}\left(\cdot\right)$ | The instantaneous performance function
${{\bf f}_{2}}\left(\cdot\right)$ | The constraint function
$\omega$ | The parameters of the actor neural network
$\upsilon$ | The parameters of the critic neural network
$\ell$ | The threshold at the target
${\bf{A}}\left({{\bf{H},\mbox{\boldmath{$\omega$}}}}\right)$ | The advantage function
${v}\left({{{\bf{H}},{\mbox{\boldmath{$\upsilon$}}}}}\right)$ | The value function of critic part
${\bf Q}\left({{{{\bf H}}},{{\bf{i}}}},{{\mbox{\boldmath{$\upsilon$}}}}\right)$ | The Q-value function
${\mbox{\boldmath{$\pi$}}}\left({\bf H},{\mbox{\boldmath{$\omega$}}}\right)$ | The policy function in the current episode
${\mbox{\boldmath{$\pi$}}}_{old}\left({\bf H},{\mbox{\boldmath{$\omega$}}}\right)$ | The policy function in the former episode
${L^{PG}}\left({\mbox{\boldmath{$\omega$}}}\right)$ | The loss function of policy gradient method
${L^{PPO}}\left({\mbox{\boldmath{$\omega$}}}\right)$ | The loss function of PPO method
${L^{CLIP}}\left({\mbox{\boldmath{$\omega$}}}\right)$ | The loss function of simplified PPO method
$\lambda$, $\mu$ | The Lagrange multipliers
$\tau$ | The size of updating step length
$\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ | The gradient estumated upgrading step factors
$\epsilon$ | The hyperparameter of PPO
$\varepsilon$ | The exploring rate of greedy policy of DQN
### V-B Primal-dual DPPO Algorithm for Multi-user MIMO Scenario
Algorithm 2 Transmit beamforming and phase-shift design via primal-dual DPPO
algorithm
Input Initial beamforming $\bf{W}^{0}$, Lagrange multipliers
${\mbox{\boldmath{$\lambda$}}}^{0},{\mbox{\boldmath{$\mu$}}}^{0}$, policy
parameters ${{\mbox{\boldmath{$\omega$}}}^{0}}$ and DQN’s parameters
${{\mbox{\boldmath{$\upsilon$}}}^{0}}$. Channel states ${\bf H}_{0,k}$, ${\bf
H}_{1,k}$, and ${\bf H}_{2,k}$ are fixed in each episode.
Episode begins
For $i=1,2,...,I$ do:
Wait until a certain amount of gradients $\omega$ are available average
gradients and update global $\omega$. Wait until a certain amount of
$\upsilon$ are available average gradients and update global $\upsilon$
(workers stop collecting data when the chief is updating).
For the worker (user) $k=1,2,...,K$ do:
For received antennas $n=1,2,...,N$ do:
a) Draw samples ${{{\bf\hat{x}}_{1}}}$ and ${{{\bf\hat{x}}_{2}}}$ from a
truncated standard normal distribution;
b) Channel state ${{\bf H}_{n}^{t}}\in{{\mathbb{C}}^{M\times 1}}$, obtained
from the matrix of ${{\bf H}^{t}=}{{\bf{H}}_{0}}{\bf{+}}{{\bf{H}}_{2}}{{\bf
V}^{t}}{{\bf{H}}_{1}}$;
c) Calculate the reward ${\bf
r}^{t}_{k}={{{f}}_{0}^{t}}\left({{{\bf{x}}}}\right)$ via the formula (23);
d) Each worker $k$ repeats step d) to step j) in Algorithm 1;
End for
Worker (user) ends
End for
Episode ends
Output Transmit beamforming ${\bf{W}^{\rm{*}}}$, phase-shift design
$\bf{V^{\rm{*}}}$, and the system performances
${{f}_{0}^{\rm{*}}}\left({{{\bf{x}}}^{\rm{*}}}\right)$.
For multi-user MIMO scenario, the pseudo code of the proposed primal-dual DPPO
is concluded in Algorithm 2. As discussed in Section IV. B, each worker takes
charge of each user. They transmit respective gradients of DNNs to the global
PPO. The global PPO updates policy parameters
${{\mbox{\boldmath{$\omega$}}}^{0}}$ and DQN’s parameters
${{\mbox{\boldmath{$\upsilon$}}}^{0}}$ based on the estimated advantage
functions ${\hat{\bf A}}^{t}\left({\bf
H},{\mbox{\boldmath{$\omega$}}}\right)$. Each worker (i.e. user) utilizes the
original primal-dual PPO algorithm in Algorithm 1.
## VI Numerical Simulation and Analysis
The simulation results in this section. We simulated in an x64 workstation
with Intel i7 CPU and the Microsoft Windows 10 Operation System. The version
of Python and TensorFlow is v3.7.9 and v2.1.0, respectively. According to the
work in [36], the carrier frequency is set as 0.55 THz and the factor of
molecular absorption loss ${k_{abs}}$ is $6.7141\times{10^{-4}}$. The paper
simulated IRS-aided ISAC system where several users are randomly distributed
in need. The targets’ azimuth angles $\theta_{k}$ are assumed in
${\text{-40}}^{\circ}$, ${\text{0}}^{\circ}$, and ${\text{40}}^{\circ}$. The
$\Delta$ in the formula (17) is set as ${\text{5}}^{\circ}$ and the ideal beam
pattern $\theta_{p}$ consists of three directions, including
$\theta_{1}=-40^{\circ}$, $\theta_{2}=0^{\circ}$, and $\theta_{3}=40^{\circ}$
[37].
The PPO’s structure is divided into two DNNs, including the actor part and the
critic part. The primal-dual PPO optimization is performed with Adam optimizer
for these two DNNs parameters update. The layer of actor part is a single
dense layer with 30 units and its activation function is ReLU. The policy
${\mbox{\boldmath{$\pi$}}}_{\omega}$ is derived from the truncated Gaussian
distribution. In the truncated Gaussian distribution, the mean is the output
of dense layer with tanh activation function and the standard deviation is the
output of dense layer with softplus activation function. The action of policy
${\bf W}_{k}\in{{\mathbb{C}}^{M\times 1}}$ is fixed on the domain
$\left[0,P_{max}\right]$. The maximum time step in one episode is set as 50.
There exists a single dense layer with 20 units in the critic part. And the
activation function of the dense layer is ReLU. Besides output of value
function, the critic part also outputs the Q-value function. It has the
function of adjusting the IRS’elements and it can generate the indexes of
phases shift. The action choice of Q-value function follows
$\varepsilon\text{-greedy}$ strategy. The parameter of
$\varepsilon\text{-greedy}$ strategy is set as $\varepsilon=0.95$. Thus, the
critic part outputs two values, including the baseline function [35] and
Q-value function. For updating the primal and dual variables, the batch size
of replay experience is set to 32. And the primal dual upgrade steps are set
as
${\gamma_{\rm{1}}}{\rm{=}}{\gamma_{\rm{2}}}{\rm{=}}{\gamma_{\rm{3}}}{\rm{=}}{\gamma_{\rm{4}}}{\rm{=0}}{\rm{.001}}$,
and the gradient upgrades are set as
${\alpha_{\rm{1}}}{\rm{=}}{\alpha_{\rm{2}}}{\rm{=}}{\alpha_{\rm{3}}}{\rm{=}}{\rm{0}}{\rm{.001}}$.
### VI-A Multi-user MISO Scenario
In the simulation of multi-user MISO scenario, we deploy a ISAC BS equipped
with 5 antennas and 4 users with single antenna in IRS-aided ISAC system. In
this case, the proposed primal-dual PPO algorithm obtains each user’s channel
state ${\bf H}_{k}\in{{\mathbb{C}}^{M\times 1}}$ in turn within each episode.
Figure 3: The reward of primal-dual PPO algorithm with different learning
rates.
In Fig. 3, the reward of the proposed primal-dual PPO is simulated in
different learning rates. The appropriate learning rate can avoid unnecessary
training and achieve quicker optimization. As shown in Fig. 3, when the
learning rate of actor equals to
${\text{1}}\times{\text{1}}{{\text{0}}^{{\text{-3}}}}$ and the learning rate
of critic equals to ${\text{2}}\times{\text{1}}{{\text{0}}^{{\text{-3}}}}$,
the moving averaged episode reward can convergence after 1200 episodes.
However, when the order of magnitude is $-\text{2}$ in learning rate (the
learning rate of the actor part is
${\text{1}}\times{\text{1}}{{\text{0}}^{{\text{-2}}}}$ and the learning rate
of the critic part is ${\text{2}}\times{\text{1}}{{\text{0}}^{{\text{-2}}}}$),
it remains in around 90. Similarly, $-\text{5}$ order of magnitude (the
learning rate of the actor part is
${\text{1}}\times{\text{1}}{{\text{0}}^{{\text{-5}}}}$ and the learning rate
of the critic part is ${\text{2}}\times{\text{1}}{{\text{0}}^{{\text{-5}}}}$)
also keeps in around 70 due to too small learning rate. Although $-\text{4}$
order of magnitude (the learning rate of the actor part is
${\text{1}}\times{\text{1}}{{\text{0}}^{{\text{-4}}}}$ and the learning rate
of the critic part is ${\text{2}}\times{\text{1}}{{\text{0}}^{{\text{-4}}}}$)
has tendency to converge, it still requires more unnecessary episodes to
converge. The step size of learning rate is so big that the agent does not
learn the suitable action. Using so many learning parameters, it can be seen
that the learning rate of actor equals to
${\text{1}}\times{\text{1}}{{\text{0}}^{{\text{-3}}}}$ and the learning rate
of critic equals to ${\text{2}}\times{\text{1}}{{\text{0}}^{{\text{-3}}}}$,
the moving averaged episode reward can convergence after 1200 episodes. Too
high or too small learning rate is not suitable in the designed model.
Figure 4: Comparison of performance using primal-dual PPO algorithm and other
algorithms.
Fig. 4 shows the capacity of IRS-aided ISAC system based on different
algorithms. The zero force (ZF) transmit beamforming and maximum ratio
transmission (MRT) are introduced to compare with the proposed primal dual PPO
algorithm [38]. Initially, the performance function $f_{0}\left({\bf
x}\right)$ (i.e. total capacity) with the primal-dual PPO algorithm is lower
than the previous two methods slightly. Nevertheless, the obtained system
capacity converges after 200 episodes. Due to the shifting environment in each
episode, it can be seen that ZF and MRT can only obtain around 1 bps capacity
in the same training conditions. These traditional optimization occupy too
high computational cost so that is not suitable the practical IRS-aided ISAC
system. Referring to the primal-dual PPO, a primal-dual advantage actor-critic
(A2C) algorithm is designed. Although primal-dual A2C can obtain the similar
convergence, there still exists around 0.5 bps gap between two different
primal-dual learning algorithms. This is because PPO’s important sampling
ensures new policy can not deviate too far. The new improved policy can be
modified by the old policy constantly. Compared with the primal-dual A2C
algorithm, the proposed primal-dual PPO method is more efficient and robust.
Figure 5: Comparison of capacity versus transmit power under different users.
Fig. 5 studies the capacity comparison versus different transmit powers for
different users. It is clear that as the ISAC BS transmit power increases, the
performance function increases as well. It reflects that more transmit power
contributes to realizing higher capacity of IRS-aided ISAC system. Meanwhile,
the gap between different users does not change significantly. The simulation
result reveals the more receive users obtain higher capacity on the condition
that equal transmit power. When transmit power is big enough, the increasing
speed of capacity tends to slow. The reason is that the logarithmic sum
relationship between ${f_{0}}\left({{{\mathbf{x}}}}\right)$ and policy
${\mbox{\boldmath{$\pi$}}}\left({{\bf{H}},{\mbox{\boldmath{$\omega$}}}}\right)$.
Figure 6: Comparison of beam patterns using primal-dual PPO algorithm and ZF
algorithm.
Fig. 6 shows the beam pattern is optimized by the primal-dual PPO algorithm
and the ZF algorithm. The transmit beam pattern is calculated by the
covariance of transmit waveform $\bf R$. It shows beam pattern against
different angle spaces, where the angle ranges from ${-\rm{90}}^{\circ}$ to
${\rm{90}}^{\circ}$. It can be seen that the performance of detecting targets
is focused on the desired angles, including ${-\rm{40}}^{\circ}$,
${\rm{0}}^{\circ}$, and ${\rm{40}}^{\circ}$. Compared to the ZF beam pattern,
the beam pattern optimized by the primal-dual PPO algorithm has a better
detecting performance in the proposed multi-user MISO scenario.
Figure 7: Beam pattern error versus different thresholds $\varepsilon$.
Fig. 7 shows the beam pattern error $L_{r}\left({\bf R}\right)$ under
different thresholds $\ell$. The different thresholds are compared among 1, 2,
and 3. It can be seen that the lower threshold $\ell$ can obtain the lower
beam pattern error $L_{r}\left({\bf R}\right)$. When $L_{r}\left({\bf
R}\right)\leq 1$, the beam pattern error $L_{r}\left({\bf R}\right)$
fluctuates among 0.2. However, the lower threshold $\ell$, the learning time
requested is longer. This is because the higher sensing accuracy need longer
time to learning.
Figure 8: Radar-communication trade-off performance.
Fig. 8 indicates the radar-communication trade-off performance in the proposed
ISAC system. The different thresholds are increasing form 2 to 10. It is
obvious that the larger given threshold, the obtained system capacity will be
larger in the proposed algorithm. This is because the tolerance of radar
performance, MSE, relieve the pressure from sensing function. The higher
communication performance will be obtained when $\ell$ equals 10.
Figure 9: System capacity versus the number of reflective elements.
Fig. 9 reveals the system capacity versus the number of reflective elements.
We compared with two different methods, including with IRS relaying and
without IRS relaying. It can be seen that the system capacity increases with
the increasing of the elements of IRS, under the condition of IRS relaying.
Obviously, when the IRS is not aided in the ISAC system, the capacity does not
change. It suggests the function of IRS is significant in the field of
enhancing capacity.
### VI-B Multi-user MIMO Scenario
After discussing the simulation results in multi-user MISO scenario in IRS-
aided ISAC system. The figures about multi-user MIMO scenario are shown in
this subsection. The number of the transmit antennas of the ISAC BS is set as
5 and the received antennas of each user is set as 4. The number of DPPO’s
workers equals to the number of severed users. The learning rate of the actor
part and the critic part is $1\times 10^{-3}$ and $2\times 10^{-3}$,
respectively. The primal dual upgrade steps $\gamma_{1}$, $\gamma_{2}$,
$\gamma_{3}$, $\gamma_{4}$ and gradient upgrades $\alpha_{1}$, $\alpha_{2}$,
$\alpha_{3}$ are the same as the case of multi-user MISO scenario.
Fig. 10 shows that the moving averaged episode reward with the growing of
episode. To verify the effectiveness of the proposed primal-dual DPPO
algorithm, A3C algorithm and weighted minimum mean squared error (WMMSE)
algorithm [29] are introduced in this simulation. More detailed discussion
about these two algorithms are introduced in the Appendix B. As shown in Fig.
10, the A3C algorithm also has a tendency to increase, but the speed of
learning is too slow under the same learning rate. This is the reason that the
learning rate is not suitable in A3C method, resulting in the learning effect
is slow. Meanwhile, we also introduce the WMMSE algorithm as an
unparameterized method and calculate its reward in each episode. Obviously,
the proposed primal-dual DPPO converges after 800 episodes, and achieves a
higher reward than two other algorithms’.
Figure 10: Convergence of moving averaged episode reward using different
algorithms.
Fig. 11 compares the capacity of different distributed DRL algorithms.
Obviously, the proposed primal-dual DPPO algorithm can converge after 90
episodes and it shows more stable performance. This is because the global PPO
collects local data rather than policy gradients from different workers.
Compared with the parallel optimization in the A3C algorithm, the primal-dual
DPPO algorithm can save computation time and realize quicker optimization.
Figure 11: Comparison of performance using primal-dual DPPO algorithm and A3C
algorithm.
Fig. 12 illustrates the beam patterns of 4 users in the multi-user MIMO
scenario. The number of transmit antennas is set as 5. The targets are also
focused on the desired angles, including ${-\rm{40}}^{\circ}$,
${\rm{0}}^{\circ}$, and ${\rm{40}}^{\circ}$. It can be seen that each user
realizes the desired performance of detecting targets via the proposed primal-
dual DPPO algorithm. The amplitudes of 4 users may be different. The reason is
that the different users may face with different channel states observations.
Different workers take charges of different users, it implies that the
utilization rate of process is fully scheduled.
Figure 12: Beam patterns in different users via primal-dual DPPO algorithm.
## VII Conclusion
In this paper, we have investigated capacity maximization in the IRS-aided
ISAC system and considered the joint transmit beamforming and passive phase
optimization. The optimization problem was transformed into an ergodic form to
capture the long-term ISAC system performance. Considering the power budget
and target response, a beamforming optimization scheme combining PPO with
primal-dual was adopted in the multi-user MISO-based scenario. Furthermore, we
proposed to utilize the concept of multi-threading derived from the DPPO, to
realize the joint optimization of transmit beamforming and phase-shift design
in the multi-user MIMO scenario. Simulations results have verified the
effectiveness of the primal-dual PPO algorithm and the primal-dual DPPO
algorithm. With the aid of the actor-critic structure in DRL, the joint
optimization transmit beamforming and phase-shift design achieved higher
capacity.
## Appendix A Proof of the Trust Region Policy Optimization
The goal of intensive learning is to obtain the best strategy that maximizes
the desired reward function.
$\eta\left({{{\mbox{\boldmath{$\pi$}}}^{t+1}}}\right)={\mathbb{E}_{{{\mathbf{H}}_{0}},{p_{0}},...,{{\mbox{\boldmath{$\pi$}}}^{t}}}}\left[{\sum\limits_{t=0}^{\infty}{{\gamma^{t}}{{\bf
A}^{t}}\left({{{\mathbf{H}}^{t}},{\mbox{\boldmath{$\omega$}}}}\right)}}\right]$
(45)
The discounted visitation frequencies
${\rho_{\mbox{\boldmath{$\pi$}}}}\left({\mathbf{H}}\right)=\sum\limits_{t=0}^{\infty}{{\gamma^{t}}P\left({{{\mathbf{H}}^{t}}={\mathbf{H}}}\right)}$
are introduced to the formula (45), it can be transformed into
$\eta\left({{{\mbox{\boldmath{$\pi$}}}^{t+1}}}\right)=\sum\limits_{\mathbf{H}}{{\rho_{{{\mbox{\boldmath{$\pi$}}}^{t+1}}}}\left({\mathbf{H}}\right)}\sum\limits_{p}{{{\mbox{\boldmath{$\pi$}}}^{t+1}}\left({p\left|{\mathbf{H}}\right.}\right)}{{\bf
A}^{t}}\left({{{\mathbf{H}}^{t}},{\mbox{\boldmath{$\omega$}}}}\right)$ (46)
However, the discounted visitation frequencies
${\rho_{{\mbox{\boldmath{$\pi$}}}}}\left({\mathbf{H}}\right)$ of new policy
${\mbox{\boldmath{$\pi$}}}^{t+1}$ in the formula (46) has to be calculated,
which requires too high computation cost. The surrogate function
${L_{\mbox{\boldmath{$\pi$}}}}\left({{{\mbox{\boldmath{$\pi$}}}^{t+1}}}\right)$
is introduced to approximate the
$\eta\left({{{\mbox{\boldmath{$\pi$}}}^{t+1}}}\right)$:
${L_{\mbox{\boldmath{$\pi$}}}}\left({{{\mbox{\boldmath{$\pi$}}}^{t+1}}}\right)=\sum\limits_{\mathbf{H}}{{\rho_{\mbox{\boldmath{$\pi$}}}}\left({\mathbf{H}}\right)}\sum\limits_{p}{{{\mbox{\boldmath{$\pi$}}}^{t+1}}\left({p\left|{\mathbf{H}}\right.}\right)}{{\bf
A}^{t}}\left({{{\mathbf{H}}^{t}},{\mbox{\boldmath{$\omega$}}}}\right)$ (47)
The difference between
${L_{\mbox{\boldmath{$\pi$}}}}\left({{{\mbox{\boldmath{$\pi$}}}^{t+1}}}\right)$
and $\eta\left({{{\mbox{\boldmath{$\pi$}}}^{t+1}}}\right)$ is different
${\rho_{\mbox{\boldmath{$\pi$}}}}\left({\mathbf{H}}\right)$. The discounted
visitation frequencies
${\rho_{\mbox{\boldmath{$\pi$}}}}\left({\mathbf{H}}\right)$ of the surrogate
function
${L_{\mbox{\boldmath{$\pi$}}}}\left({{{\mbox{\boldmath{$\pi$}}}^{t+1}}}\right)$
can be obtained by the old policy ${\mbox{\boldmath{$\pi$}}}^{t}$. When
${\mbox{\boldmath{$\pi$}}}^{t}$ and ${\mbox{\boldmath{$\pi$}}}^{t+1}$ satisfy
the constraints, they can be equal.
Herein, the total variation (TV) divergence $D_{TV}$ is utilized to deign the
constraints. For two discrete probabilities $p$ and $q$, the TV divergence
${D_{TV}}\left({p\left\|q\right.}\right)=\frac{1}{2}\sum\limits_{i}{\left|{{p_{i}}-{q_{i}}}\right|}$.
It is assumed that
$D_{TV}^{\max}\left({{{\mathbf{\pi}}^{t}},{{\mathbf{\pi}}^{t+1}}}\right)=\mathop{\max}\limits_{\mathbf{H}}{D_{TV}}\left({{{{\mbox{\boldmath{$\pi$}}}}^{t}}\left({\mathbf{H}}\right)\left\|{{{{\mbox{\boldmath{$\pi$}}}}^{t+1}}\left({\mathbf{H}}\right)}\right.}\right)$,
the difference of the surrogate function
${L_{\mbox{\boldmath{$\pi$}}}}\left({{{\mbox{\boldmath{$\pi$}}}^{t+1}}}\right)$
and the cumulative reward $\eta\left({\mbox{\boldmath{$\pi$}}}\right)$ is
given by
$\eta\left({{\mbox{\boldmath{$\pi$}}}}\right)\geqslant{L_{\pi}}\left({{\mbox{\boldmath{$\pi$}}}}\right)-\frac{{4\varepsilon\gamma}}{{{{\left({1-\gamma}\right)}^{2}}}}D_{TV}^{\max}\left({{{{\mbox{\boldmath{$\pi$}}}}^{t}},{{{\mbox{\boldmath{$\pi$}}}}^{t+1}}}\right)$
(48)
The relationship between TV divergence and KL divergence satisfy
${D_{TV}}{\left({p\left\|q\right.}\right)^{2}}\leqslant{D_{KL}}\left({p\left\|q\right.}\right)$,
the constraint (48) can be reformulated by the DL divergence.
$\eta\left({{{{\mbox{\boldmath{$\pi$}}}}^{t+1}}}\right)\geqslant{L_{\pi}}\left({{{{\mbox{\boldmath{$\pi$}}}}^{t+1}}}\right)-\frac{{4\varepsilon\gamma}}{{{{\left({1-\gamma}\right)}^{2}}}}D_{KL}^{\max}\left({{{{\mbox{\boldmath{$\pi$}}}}^{t}},{{{\mbox{\boldmath{$\pi$}}}}^{t+1}}}\right)$
(49)
With the constraint of the KL divergence, the lower bound of the cumulative
reward $\eta\left({\mbox{\boldmath{$\pi$}}}\right)$ can be obtained.
Additionally, the true objective $\eta\left({\mbox{\boldmath{$\pi$}}}\right)$
is non-decreasing. The following constraint also can guarantee the
optimization of the cumulative reward
$\eta\left({\mbox{\boldmath{$\pi$}}}\right)$.
$\mathop{\max}\limits_{\theta}{L_{{{\mathbf{\pi}}_{{\theta_{0}}}}}}\left({{{\mathbf{\pi}}_{\theta}}}\right)-\frac{{4\varepsilon\gamma}}{{{{\left({1-\gamma}\right)}^{2}}}}D_{KL}^{\max}\left({{{\mathbf{\pi}}_{\theta}},{{\mathbf{\pi}}_{{\theta_{0}}}}}\right)$
(50)
In practical, the penalty coefficient
$\frac{{4\varepsilon\gamma}}{{{{\left({1-\gamma}\right)}^{2}}}}$ will cause
the step length of policy iteration small. One way to take larger steps is to
use a constraint based on the KL divergence between the new policy and the old
policy, i.e., a trust region constraint.
$\begin{array}[]{*{20}{l}}{\mathop{\max}\limits_{\mbox{\boldmath{$\omega$}}}}&{{\mathbb{E}_{{\rho_{\omega}}\left(\tau\right)}}\left[{\sum\nolimits_{t}{{\gamma^{t-1}}\frac{{{\mbox{\boldmath{$\pi$}}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}}{{{{\mbox{\boldmath{$\pi$}}}_{old}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}}{{\mathbf{A}}_{old}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}}\right]},\\\
{s.t.}&{{D_{KL}}\left[{\frac{{{\mbox{\boldmath{$\pi$}}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}}{{{{\mbox{\boldmath{$\pi$}}}_{old}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}}}\right]\leqslant\delta}.\end{array}$
(51)
## Appendix B Introduction of A3C algorithm and WMMSE algorithm
### B-A A3C Algorithm
The A3C algorithm is one of the methods for asynchronous DRL in a multi-
threading mode. The A3C algorithm essentially puts actor-critic algorithm into
multiple threads for synchronous training. On the one hand, it is based on the
policy-based method, so it can deal with continuous state and action spaces.
On the other hand, A3C algorithm employs asynchronous method to relieve data
correlation, in which data is not generated at the same time. Compared with
the DQN algorithm, the A3C algorithm does not need to use the experience pool
to store historical samples. This asynchronous method can save storage space
and accelerate the sampling speed of data. All the actor learners update the
policy
${\mbox{\boldmath{$\pi$}}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)$
and the advantage function
${\mathbf{A}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)$
according to gradient loss. Based on the advantage function, the gradient loss
function is given by
${L^{A3C}}\left({\mbox{\boldmath{$\omega$}}}\right){\text{ =
}}\mathbb{E}\left[{{\nabla_{\mbox{\boldmath{$\omega$}}}}\log{\mbox{\boldmath{$\pi$}}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right){\mathbf{A}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)}\right]$
(52)
The update formulas for $\omega$ and $\upsilon$ are
$\begin{array}[]{*{20}{l}}{{{\mbox{\boldmath{$\omega$}}}^{t+1}}={{\mbox{\boldmath{$\omega$}}}^{t}}+{\nabla_{\mbox{\boldmath{$\omega$}}}}\log{\mbox{\boldmath{$\pi$}}}\left({{\mathbf{H}},{\mbox{\boldmath{$\omega$}}}}\right)\left({r-v\left({{\mathbf{H}},{{\mbox{\boldmath{$\upsilon$}}}^{t}}}\right)}\right)}\\\
{{{\mbox{\boldmath{$\upsilon$}}}^{t+1}}={{\mbox{\boldmath{$\upsilon$}}}^{t}}+\partial{{\left({r-v\left({{\mathbf{H}},{{\mbox{\boldmath{$\upsilon$}}}^{t}}}\right)}\right)}\mathord{\left/{\vphantom{{\left({r-v\left({{\mathbf{H}},{{\mbox{\boldmath{$\upsilon$}}}^{t}}}\right)}\right)}{\partial{\mbox{\boldmath{$\upsilon$}}}}}}\right.\kern-1.2pt}{\partial{\mbox{\boldmath{$\upsilon$}}}}}}\end{array}$
(53)
### B-B WMMSE Algorithm
This algorithm transforms the weighted sum-rate maximization problem into a
higher dimensional space, using the well-known minimum mean squared error. The
sum-rate maximization is equivalent to the following weighted sum-MSE
minimization,
$\begin{array}[]{*{20}{l}}{\mathop{\min}\limits_{\varsigma,\vartheta,W}}&{\sum\limits_{k=1}^{K}{\left({{\varsigma_{k}}{e_{k}}-\log\left({{\varsigma_{k}}}\right)}\right)}}\\\
{s.t.}&{0\leqslant{W_{k}}\leqslant\sqrt{{P_{\max}}},k=1,2,...,K.}\end{array}$
(54)
where $e_{k}$ is the meansquare estimation error and $\vartheta_{k}$ equals
the ${SINR}_{k}$.
${e_{k}}={\left({{\vartheta_{k}}{{\mathbf{H}}_{k}}{{\mathbf{W}}_{k}}-1}\right)^{2}}+\sum\limits_{l\neq
k}{{{\left({{\vartheta_{k}}{{\mathbf{H}}_{l}}{{\mathbf{W}}_{l}}}\right)}^{2}}+\sigma_{c}^{2}\vartheta_{k}^{2}}$
(55)
The $\varsigma_{k}=e_{k}^{-1}$. Here by equivalent we meant that all
stationary solutions of the sum-MSE minimization is identical to maximizing
sum rate.
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| Xiangnan Liu received the B.S. degree from the School of Computer and
Communication Engineering, University of Science and Technology of Beijing,
Beijing, China, in 2019. He is currently pursuing his Ph.D. degree at
University of Science and Technology Beijing, China. His research interests
include access control, beamforming, and resource allocation in 6G wireless
communication.
---|---
| Haijun Zhang (M’13, SM’17) is currently a Full Professor and Associate Dean
at University of Science and Technology Beijing, China. He was a Postdoctoral
Research Fellow in Department of Electrical and Computer Engineering, the
University of British Columbia (UBC), Canada. He serves/served as Track Co-
Chair of WCNC 2020, Symposium Chair of Globecom’19, TPC Co-Chair of INFOCOM
2018 Workshop on Integrating Edge Computing, Caching, and Offloading in Next
Generation Networks, and General Co-Chair of GameNets’16. He serves as an
Editor of IEEE Transactions on Communications, IEEE Transactions on Network
Science and Engineering, and IEEE Transactions on Vehicular Technology. He
received the IEEE CSIM Technical Committee Best Journal Paper Award in 2018,
IEEE ComSoc Young Author Best Paper Award in 2017, and IEEE ComSoc Asia-
Pacific Best Young Researcher Award in 2019.
---|---
| Keping Long (SM’06) received the M.S. and Ph.D. degrees from the University
of Electronic Science and Technology of China, Chengdu, in 1995 and 1998,
respectively. From September 1998 to August 2000, he was a Postdoctoral
Research Fellow at the National Laboratory of Switching Technology and
Telecommunication Networks, Beijing University of Posts and Telecommunications
(BUPT), China. From September 2000 to June 2001, he was an Associate Professor
at BUPT. From July 2001 to November 2002, he was a Research Fellow with the
ARC Special Research Centre for Ultra Broadband Information Networks (CUBIN),
University of Melbourne, Australia. He is currently a professor and Dean at
the School of Computer and Communication Engineering, University of Science
and Technology Beijing. He has published more than 200 papers, 20 keynote
speeches, and invited talks at international and local conferences. His
research interests are optical Internet technology, new generation network
technology, wireless information networks, value added services, and secure
technology of networks. Dr. Long has been aTPC or ISC member of COIN
2003/04/05/06/07/08/09/10, IEEE IWCN2010, ICON2004/06, APOC2004/06/08, Co-
Chair of the organization Committee for IWCMC2006, TPC Chair of COIN 2005/08,
and TPC Co-Chair of COIN 2008/10. He was awarded by the National Science Fund
for Distinguished Young Scholars of China in 2007 and selected as the Chang
Jiang Scholars Program Professor of China in 2008. He is a member of the
Editorial Committees of Sciences in China Series F and China Communications.
---|---
| Mingyu Zhou received the Ph.D. degree from the Beijing University of Posts
and Telecommunications (BUPT) as a focus on research into key technologies of
wireless communications. After graduation in 2008, he became a Senior Engineer
in Huawei Technologies, dedicated in 3GPP standardization and patent
application. In 2014, he joined Baicells Technologies as the Research
Director. Until now, he has applied for more than 100 patents (tens of them
are PCTs), published more than 20 academic papers, and finished more than 100
standardization proposals. He was qualified in the Beijing Nova Program and
has participated in several science and technology projects, targeting to
bring more innovation to the 5G system.
---|---
| Yonghui Li (M’04, SM’09, F’19) received his PhD degree in November 2002
from Beijing University of Aeronautics and Astronautics. Since 2003, he has
been with the Centre of Excellence in Telecommunications, the University of
Sydney, Australia. He is now a Professor and Director of Wireless Engineering
Laboratory in School of Electrical and Information Engineering, University of
Sydney. He is the recipient of the Australian Queen Elizabeth II Fellowship in
2008 and the Australian Future Fellowship in 2012. He is a Fellow of IEEE. His
current research interests are in the area of wireless communications, with a
particular focus on MIMO, millimeter wave communications, machine to machine
communications, coding techniques and cooperative communications. He holds a
number of patents granted and pending in these fields. He is now an editor for
IEEE transactions on communications, IEEE transactions on vehicular
technology. He also served as the guest editor for several IEEE journals, such
as IEEE JSAC, IEEE Communications Magazine, IEEE IoT journal, IEEE Access. He
received the best paper awards from IEEE International Conference on
Communications (ICC) 2014, IEEE PIRMC 2017 and IEEE Wireless Days Conferences
(WD) 2014.
---|---
| H. Vincent Poor (S’72, M’77, SM’82, F’87) received the Ph.D. degree in EECS
from Princeton University in 1977. From 1977 until 1990, he was on the faculty
of the University of Illinois at Urbana-Champaign. Since 1990 he has been on
the faculty at Princeton, where he is currently the Michael Henry Strater
University Professor. During 2006 to 2016, he served as the dean of
Princeton’s School of Engineering and Applied Science. He has also held
visiting appointments at several other universities, including most recently
at Berkeley and Cambridge. His research interests are in the areas of
information theory, machine learning and network science, and their
applications in wireless networks, energy systems and related fields. Among
his publications in these areas is the forthcoming book Machine Learning and
Wireless Communications. (Cambridge University Press). Dr. Poor is a member of
the National Academy of Engineering and the National Academy of Sciences and
is a foreign member of the Chinese Academy of Sciences, the Royal Society, and
other national and international academies. He received the IEEE Alexander
Graham Bell Medal in 2017.
---|---
|
# What Sentiment and Fun Facts We Learnt Before FIFA World Cup Qatar 2022
Using Twitter and AI
James She2, Kamilla Swart-Arries2, Mohammad Belal2 and Simon Wong4 2College
of Science and Engineering, Hamad Bin Khalifa University, Qatar
4Department of Electronic and Computer Engineering, HKUST, Hong Kong
Email: 2{pshe, kswartarries<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Twitter is a social media platform bridging most countries and allows real-
time news discovery. Since the tweets on Twitter are usually short and express
public feelings, thus provide a source for opinion mining and sentiment
analysis for global events. This paper proposed an effective solution, in
providing a sentiment on tweets related to the FIFA World Cup. At least 350k
tweets, as the first in the community, are collected and implemented as a
dataset to evaluate the performance of the proposed machine learning solution.
These tweets are collected with the related hashtags and keywords of the Qatar
World Cup 2022. The Vader algorithm is used in this paper for sentiment
analysis. Through the machine learning method and collected Twitter tweets, we
discovered the sentiments and fun facts of several aspects important to the
period before the World Cup. The result shows people are positive to the
opening of the World Cup.
###### Index Terms:
Sentiment Analysis, Social Media, Machine Learning
## 1 Introduction
Social media is an important source for retrieving opinions from the public
and provides interactions for different users in the online world. Famous
social media platforms, like Twitter, Facebook, Youtube and Tiktok, consist of
over 70one of the popular international platforms for social media and micro-
blogging services, and the publicly posted microblogs are called tweets.
Tweets allow users to connect and update the topics and events related to
their interests. Although the tweets are usually short, the content can be
completed by different components like text content, emoji, and hashtags. All
these components provide sufficient information for opinion mining and
sentiment analysis, and the evaluation of these components can reflect the
feelings or polarity of the stand from the users. More than 400 million tweets
are posted every day on Twitter. Thus, a large dataset can be collected to
provide a convincing reflection of the public attitude towards some special
events. For example, more than half a billion tweets have been obtained in the
previous World Cup.
With the large dataset and sentiment analysis, the public reaction to the
events during the World Cup period can be analyzed, providing information on
public feelings and attitudes. This paper investigates the relationship
between the sentiment based on Twitter tweet and the World Cup 2022 and
contributes the following,
1. 1.
Provide an open-source and updated dataset about FIFA 2022 with Twitter
tweets;
2. 2.
Analyze the excitement of the users about the World Cup 2022;
3. 3.
Evaluate the popularity of football stars and football teams;
4. 4.
Suggest an emoji handling solution for the tweet sentiment analysis.
All the results are evaluated with the Vader algorithm, explanation and event
spotting with the timeline. Over 350k tweets are investigated to provide a
sufficient and convincing dataset from the public with the official hashtag of
the FIFA World Cup 2022.
## 2 Related Works
Sentiment analysis is a natural language processing (NLP) technique that
determines the opinion regarding the input text. There have been various
methods to assess the sentiment of a text. Machine Learning methods have been
implemented to classify them automatically. Both supervised and unsupervised
learning methods have been shown to work well. Yue, Le et al. [1] have
demonstrated multiple supervised and unsupervised learning techniques for
sentiment analysis. Vader [2]is one of the unsupervised rule-based sentiment
analysis models whose lexicon is specially tuned for social media. The Vader
model works well on social media data and is efficient. It has the edge over
its competition when used for Twitter data [3]. Vader algorithm also
incorporates emoticons in its lexicon, which could help in better
classification [2].
There have been previous studies in identifying the sentiment of people during
some crucial events [4] [5] [6]. A similar study [5] was done for the 2014
FIFA World cup to identify people’s sentiments during the event. They used
manual labelling to train the Bayesian Logistic Regression model. Elbagir et
al. [4] have used the Vader model on Twitter data for the 2016 US elections.
The result shows the model’s effectiveness and efficiency. The multi-class
classification helped them understand people’s sentiments during the
elections. Meier et al. [6] have shown the politicization of Mega sports. They
have demonstrated how sentiments change before and after the FIFA World Cup
2018. Most topics, which were frequent before the World Cup, diminished as
soon as the tournament started.
### 2.1 Vader Algorithm
Valence Aware Dictionary and sEntiment Reasoner (VADER) algorithm [2] [4] is a
lexicon and rule-based sentiment analysis tool that specifically attunes to
sentiments expressed in social media. Besides, it provides a dictionary
consisting of emoji sentiment, which other sentiment methods do not implement.
A compound sentiment is computed based on the word and emoji in the tweets and
normalized into the range $[-1,1]$. The sentiment, $S(x)$, for given input,
$x$, is evaluated as,
$S(x)=\sum_{i=1}^{N}\lambda_{i}\times f_{i}$ (1)
where $x:\\{f_{i}|0<i\leq N\\}$, $i$ is the feature index, $N$ is the total
number of the extracted features, $\lambda_{i}$ is the sentiment of each
lexicon, $f_{i}$ is the feature vector in each tweet.
TABLE I: The value of $\lambda_{i}$ for different lexicons Lexicon | $\lambda_{i}$
---|---
affected | -0.6
| 0.0
loyalty | 2.5
| 1.9
Table I provides some examples of the sentiments evaluated from some words or
emojis. With the given sentiment based on the features, the compound
sentiment, $S_{c}(x)$, can be evaluated as,
$S_{c}(x)=\frac{S(x)}{\sqrt{S(x)^{2}+\alpha}}$ (2)
where $\alpha$ is the maximum expected value of the score, $15$, thus, the
sentiment is bounded within $[-1,1]$.
Figure 1: Emoji frequency in the collected tweets
Except for the features of text content, emojis also play an essential role in
representing the users’ feelings. Fig. 1 shows the frequency of the emojis in
the collected tweets, and around 30% of tweets contain emojis. Therefore,
sentiment analysis of emojis is essential to be considered.
TABLE II: The influence of emoji in Vader algorithm Tweets | Sentiment
---|---
The FIFA is coming. I’m excited about it! | 0.4003
The FIFA is coming. I’m excited about it! | 0.4003
The FIFA is coming. I’m excited about it! | 0.69
The FIFA is coming. I’m excited about it! | 0.8268
Vader algorithm has special handling for the emojis in the tweets. The
selection and the amount of emojis also affect the evaluated sentiment of the
tweets which contain the same text content. Table 2 shows the addition or the
amount of emojis may influence the sentiment. The emoji ”” provides a neutral
polarity and the emoji ”” provides a positive polarity to the content. The
increased number of ”” also makes the sentiment polarity more positive.
Figure 2: The flow and step of the sentiment analysis
## 3 Proposed Sentiment Analysis with Text and Emoji
This section involved the flow, the proposed machine learning method and the
details of each component in the sentiment analysis process.
### 3.1 The Flow and Step of the Sentiment Analysis
The proposed solution involves multiple steps to retrieve the Twitter tweets
and indicate their sentiment and polarity. The procedures are shown in Fig. 2
and described as,
1. 1.
Keyword selection - select suitable keywords for the Twitter tweet collection;
2. 2.
Data collection - collect and organize tweets by Twitter API with query
parameters;
3. 3.
Data pre-processing - remove the noise from the collected tweets;
4. 4.
Sentiment classification - identify the polarity and calculate the sentiment
of tweets;
5. 5.
Sentiment analysis - evaluate the calculated polarity and sentiment of tweets.
Throughout the whole process, the visualization of sentiment analysis is
presented in Section 4.
### 3.2 Selection of Keyword
Figure 3: Frequency of keywords ($k_{a}$: #FIFAWorldCup, $k_{b}$: #Qatar2022,
$k_{c}$: football world cup, $k_{d}$: qatar world cup, $k_{e}$: soccer world
cup, $k_{f}$: worldcup2022), only $k_{a}$ and $k_{b}$ are official keywords
Keyword selection is an essential part that influences the quality of the
collected tweets for the sentiment analysis. In this paper, official keywords
and hashtags, i.e., Qatar2022 and FIFAWorldCup, are used for the crawling in
the first stage. However, the amount of collected tweets are small based on
these official hashtags, i.e., less than 2k. We have also selected some other
keywords and hashtags which frequently exist in the tweets collected from the
official keywords, e.g., football world cup, worldcup2022, soccer world cup
and Qatar world cup. Fig. 3 shows the official keywords occupy most of the
frequency of keyword usage in tweets and are followed by the suggested
keywords based on the tweet statistics. Thus, the selected keywords are the
most frequently used and related to the World Cup event.
### 3.3 Data Collection
After selecting the suitable keywords, we have used the Twitter academic API
for the data collection. It helps us to retrieve all the tweets containing
these keywords. Except the keywords, different parameters can be added to the
API calls for the tweets needed based on the task requirement. Therefore, we
considered the filter of retweets to avoid duplicated tweets or domination of
certain tweet contents. And we only collect the tweets in English everyday in
the period of 1 Aug., 2022 to 17 Sept., 2022. Around 130k tweets are collected
within the 48-day period
### 3.4 Data Pre-processing
Pre-processing text is the first step in sentiment analysis. Proper
methodologies are required to remove the noise from text without losing the
essential semantic information. This important step involves dealing with
hashtags, usernames, and emoticons. Our algorithm handles the emojis, and it
has an impact on the sentiment. The unrelated tweets need to be filtered. Here
is an example,
> Amazing giveaway i really excited
> @Ajay8307 @Tarun54552170
> #NFTGiveaways #NFTs #Qatar2022
The tweet is not related to the FIFA World Cup, but it used the official
hashtag suggested by FIFA. If this tweet is used in the sentiment analysis, it
will lead to a faulty result regarding the public feeling about World Cup
2022. The Twitter API has a parameter named exclude, which is used to filter
out unwanted tweets. Here, by using #NFTs, we can exclude the tweets with our
keyword and the #NFTs. Afterwards, the text of the tweet would be split using
spaces. Each word would be used as a feature for our model.
### 3.5 Sentiment Classification
Given the collected tweets are pre-processed, all the tweets are taken into
consideration of these classifications, (1) the sentiment; (2) the polarity;
(3) the number of emoji that exists in the tweet. The Vader algorithm
evaluates the sentiment, which handles both the text and emoji content. After
the compound sentiment, $S_{c}(x)$, is evaluated for each tweet, the polarity
is considered as positive if $S_{c}(x)>0$, and it will be considered as
negative if $S_{c}(x)<0$. Otherwise, the remaining will be considered neutral.
All data are organized in excel format and stored inside a server and Kaggle.
### 3.6 Sentiment Result Analysis
After the polarity and sentiment are stored on the server, we query each day’s
average sentiment and polarity within the collection period. Besides, the
sentiment and polarity of all the tweets based on different keywords and
selected football stars are evaluated. Based on the evaluation result, the
observations are summarized in Section 4.
## 4 Sentiment Finding and Fun Facts
This section mentions the finding. The sentiments are compared versus time,
and different football stars are investigated. The whole process is done
within the period of World Cup 2022. Fun facts about the sentiment of tweets
over time and about football stars are investigated.
### 4.1 Sentiment analysis over time
The sentiment is investigated versus the time during the FIFA World Cup
period. The polarity of public feelings keeps positive within the 48 days
before the opening of the World Cup.
Figure 4: Sentiment analysis based on time
Fig. 4 shows the average sentiment of the tweets related to the World Cup over
time. All tweets that only use official keywords or all suggested keywords
show the same trend of public expression in the World Cup.
### 4.2 Sentiment analysis over football stars
Figure 5: Sentiment analysis of tweets about football stars Figure 6: Number
of tweets about football stars
Tweets could help us to reflect the public sentiment on players’ on-field
performance during the FIFA World Cup. Different events concerning players or
team members affect how people think and talk about them. We are interested in
evaluating people’s sentiment toward the top players and how it changes during
the World Cup. The popularity of the players may have some significance on how
the players are perceived by the general public.
## 5 Future Work
We would try to understand the public sentiment to help us to identify the
golden ball winner in World Cup. Football players’ sentiment must indicate
their on-field performances, and a higher sentiment could help us to identify
the winner. Some of the tweets consist of images, which are part of the
feeling expressed by the public. Therefore, the image can provide extra
information about the public’s emotions. The sentiment correlation between
text content and image can be investigated to enhance sentiment accuracy.
## 6 Conclusion
FIFA World Cup is a global event every four years, and Twitter, one of the
most commonly used social platforms, provides tons of tweets to reflect the
public’s expression of these global events. With the data from Twitter, the
public feelings can be analyzed based on the sentiment proposed solution with
the collected tweets. And the main contributions can be summarized as follows,
1. 1.
Provide an open-source and updated dataset about FIFA 2022;
2. 2.
Provide a sentiment analysis that reflects the users are getting excited about
the World Cup 2022 and proves that Twitter could be a social media sensor for
the vibe of global users;
3. 3.
Suggest an emoji handling solution for the tweet sentiment analysis.
The sentiment accuracy can be further improved with a correlation study of the
posted image and text content inside tweets instead of the pure emoji and text
content study.
## Acknowledgments
This work was initiated by the College of Science and Engineering at HBKU and
HKUST-NIE Social Media Lab. at Hong Kong University of Science & Technology.
The visualization and dataset storage is supported by CyPhy Media Limited.
## References
* [1] L. Yue, W. Chen, X. Li, W. Zuo, and M. Yin, “A survey of sentiment analysis in social media,” _Knowl. Inf. Syst._ , vol. 60, no. 2, p. 617–663, aug 2019\. [Online]. Available: https://doi.org/10.1007/s10115-018-1236-4
* [2] C. J. Hutto and E. Gilbert, “Vader: A parsimonious rule-based model for sentiment analysis of social media text,” in _ICWSM_ , 2014.
* [3] F. N. Ribeiro, M. Araújo, P. Gonçalves, F. Benevenuto, and M. A. Gonçalves, “A benchmark comparison of state-of-the-practice sentiment analysis methods,” _CoRR_ , vol. abs/1512.01818, 2015. [Online]. Available: http://arxiv.org/abs/1512.01818
* [4] S. Elbagir and J. Yang, “Analysis using natural language toolkit and vader sentiment.”
* [5] P. Barnaghi, P. Barnaghi, P. Ghaffari, J. G. Breslin, and J. G. Breslin, “Analysis and sentiment polarity on fifa world cup 2014 tweets,” 2015.
* [6] H. E. Meier, M. Mutz, J. Glathe, M. Jetzke, and M. Hölzen, “Politicization of a contested mega event: The 2018 fifa world cup on twitter,” _Communication & Sport_, vol. 9, no. 5, pp. 785–810, 2021. [Online]. Available: https://doi.org/10.1177/2167479519892579
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Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).
CLEF 2022: Conference and Labs of the Evaluation Forum, September 5–8, 2022,
Bologna, Italy
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# An End-to-End Set Transformer for User-Level Classification of Depression
and Gambling Disorder
Ana-Maria Bucur Interdisciplinary School of Doctoral Studies, University of
Bucharest, Romania PRHLT Research Center, Universitat Politècnica de
València, Spain Adrian Cosma Politehnica University of Bucharest, Romania
Liviu P. Dinu Faculty of Mathematics and Computer Science, University of
Bucharest, Romania Human Language Technologies Research Center, University of
Bucharest, Romania Paolo Rosso
(2022)
###### Abstract
This work proposes a transformer architecture for user-level classification of
gambling addiction and depression that is trainable end-to-end. As opposed to
other methods that operate at the post level, we process a set of social media
posts from a particular individual, to make use of the interactions between
posts and eliminate label noise at the post level. We exploit the fact that,
by not injecting positional encodings, multi-head attention is permutation
invariant and we process randomly sampled sets of texts from a user after
being encoded with a modern pretrained sentence encoder (RoBERTa / MiniLM).
Moreover, our architecture is interpretable with modern feature attribution
methods and allows for automatic dataset creation by identifying
discriminating posts in a user’s text-set. We perform ablation studies on
hyper-parameters and evaluate our method for the eRisk 2022 Lab on early
detection of signs of pathological gambling and early risk detection of
depression. The method proposed by our team BLUE obtained the best ERDE5 score
of 0.015, and the second-best ERDE50 score of 0.009 for pathological gambling
detection. For the early detection of depression, we obtained the second-best
ERDE50 of 0.027.
###### keywords:
set transformer, sentence encoder, gambling disorder detection, depression
detection, social media
## 1 Introduction
How much can one know about someone from their social media interactions?
Billions of people111https://www.statista.com/statistics/272014/global-social-
networks-ranked-by-number-of-users/ use social media sites like Facebook,
Instagram, Twitter, and Reddit every day. While some sites like Facebook and
Instagram encourage users to use their real names, websites such as Reddit are
often praised for enabling users to hide between a pseudonym, offering the
illusion of privacy. Under the guise of anonymity, users tend to post more
personal information related to their lives and their everyday struggles
instead of striving to maintain an image and a persona when their identities
are open [1]. Many aspects of a user’s personal life can be uncovered in their
posting history. Of course, not one single post can be all-encompassing, but
rather the information is scattered across many unrelated comments and posts.
For instance, on the
r/relationship_advice222https://www.reddit.com/r/relationship_advice/
subreddit a user might reveal their gender and age when discussing intimate
relationship struggles, while on
r/depression333https://www.reddit.com/r/depression/ a user might provide clues
for their internal conflicts and experiences.
In the task of mental health disorders detection from social media text, many
approaches operate on the post-level [2, 3, 4], considering that, for
instance, if a user is depressed, then all their posts might contain some
information regarding this issue. However, we posit that this method of post-
level classification is unsuitable - many posts are unrelated and
uninformative to the particular task. Their interaction, however, might
contain clues to the mental well-being of a user.
As such, we propose an architecture that performs user-level classification by
processing a set of posts from a user. We exploit the fact that the multi-head
attention operation in transformers is permutation invariant and inputs
multiple texts from a single user into the network, modeling their interaction
and classifying the user. This approach has several advantages: (i) it is
trainable end-to-end, mitigating the need for hand-crafted construction of
global user features (ii) it is robust to label noise, as some posts might be
uninformative, the network learns to ignore them in the decision and (iii) it
is interpretable, using feature attribution methods [5] we can extract the
most important posts for the decision.
The Early Risk Prediction on the Internet (eRisk)444https://erisk.irlab.org/
Lab started in 2017 with one pilot task and, since then, tacked the early risk
detection of several mental illnesses: depression, self-harm, eating
disorders, and pathological gambling. This work showcases team BLUE’s proposed
approach for Tasks 1 and 2 of eRisk 2022 Lab [6], of gambling and depression
detection, respectively.
The paper makes the following contributions:
1. 1.
We propose a set-based transformer architecture for user-level classification,
which makes a decision by processing multiple texts of a particular user.
2. 2.
We show that our architecture is robust to label noise and is interpretable
with modern feature attribution methods, allowing it to be used as a dataset
filtering tool.
3. 3.
We obtained promising results on the eRisk 2022 tasks on early risk detection
of pathological gambling (best ERDE5555Early Risk Detection Error, introduced
in Section 5.1 score of 0.015 and the second-best ERDE50 score of 0.009) and
depression detection (second-best ERDE50 of 0.027).
## 2 Related Work
Pathological Gambling For the detection of gambling disorder, the eRisk Lab is
the first to use social media data for the assessment of gambling risk.
Usually, the automated methods use data from behavioral markers [7, 8] or
personality biomarkers [9]. In the first iteration of the task for gambling
addiction detection, the best-performing systems were developed by Maupomé et
al. [10] and Loyola et al. [11]. Maupomé et al. [10] used a user-level
approach based on the similarity distance between the vector of topic
probabilities of the users’ texts to be assessed for pathological gambling
risk and testimonials or items from a self-evaluation questionnaire for
compulsive gamblers. By using this method, the authors obtain the best ERDE5
of 0.048. Loyola et al. [11] attain the best ERDE50 (0.020) and latency-
weighted F1 (0.693) through a post-level rule-based early alert policy on bag-
of-words text representation classified with SVM.
Depression Depression detection from social media data is an interdisciplinary
topic, and efforts have been made by researchers from both NLP and Psychology
to detect different markers of depression found in the online discourse of
individuals. Some depression cues found in language are: greater use of the
first-person singular pronouns "I" [12], lesser use of first-person plural
"we" [13], increased use of negative or absolutist terms (e.g., "never",
"forever") [14], greater use of verbs at past tense [15].
For the task of early detection of depression, the best systems from the first
iteration of the task (eRisk 2017) used as input linguistic meta information
extracted from the texts such as LIWC [16], readability and hand-crafted
features [17] obtaining the best ERDE5 (12.70$\%$) or a combination of
linguistic information and temporal variation of terms from users’ posts [18]
achieving the best ERDE50 (9.68$\%$). The best-performing systems from eRisk
2018 were the ones from Funez et al. [19] and Trotzek et al. [20]. Funez et
al. [19] propose a user-level approach using an SVM classifier on semantic
representations that take into account the temporal variation of terms between
the users’ posts and achieve an ERDE5 of 8.78$\%$. On the other hand, the best
ERDE50 (6.44$\%$) is attained by Trotzek et al. [20] using a chunk-level666in
2018 the test data was released in chunks of posts, not one post at a time as
it is the case in this year’s tasks approach using an ensemble of logistic
regression classifiers on bag-of-words features. The dataset from the
depression detection task from the eRisk Lab was an important resource later
used in different research articles tackling the detection problem using
approaches such as a neural network architecture on topic modeling features
[21], SVM or deep learning architectures using fine-grained emotions features
[22] or deep learning methods using content, writing style and emotion
features [23].
## 3 Method
The transformer encoder, as proposed by Vaswani et al. [24], essentially
consists of multiple sequential layers of multi-head attention. Scaled dot-
product attention of a query $Q$ relative to a set of values $V$ and a set of
keys $K$ is computed using the following equation ($d_{k}$ is the
dimensionality of the query and keys):
$\textrm{Attention}(Q,K,V)=\textrm{softmax}(\frac{QK^{T}}{\sqrt{d_{k}}})V$ (1)
As such, multi-head attention consists of multiple applications of the
attention mechanism to the same input. The multi-head attention is defined as:
$\displaystyle\textrm{MultiHead}(Q,K,V)=\textrm{Concat}(\textrm{head}_{1},\textrm{head}_{2}\dots\textrm{head}_{h})W^{O}$
(2)
$\displaystyle\textrm{head}_{i}=\textrm{Attention}(QW_{i}^{Q},KW_{i}^{K},VW_{i}^{V})$
In this formulation, multi-head attention is permutation invariant, and the
current way to inject temporal information into the input sequence is by
employing positional encodings [25]. This is useful when processing sequential
data such as texts. However, by omitting positional encodings, the transformer
essentially acts as a set encoder. Lee et al. [26] introduced the Set
Transformer, in which they prove that multi-head attention is permutation
invariant and that the Set Transformer is a universal approximator of
permutation invariant functions. We make use of this fact to perform user-
level classification by processing sets of texts (in the form of social media
posts) from a particular user. The intuition behind processing a set of texts
from a user is that no single social media post is sufficiently informative
for a classifier decision, but rather their interaction and the user behavior
as a whole. Moreover, through mean pooling, the inevitable noise (in terms of
unrelated posts) is dampened, which aids classification in weakly-supervised
scenarios, such as ours, in which a user is labeled rather than all of their
posts.
We consider a user $i$ to contain multiple social media posts $U_{i}$. A set
of $K$ texts $t$ are randomly sampled from $U_{i}$, which defines our text-set
$S_{i}=\\{t^{j}\sim U_{i},j\in(1\dots K)\\}$. We sample $K$ posts from the
user’s history, instead of processing all of them due to memory limitations -
some individuals have thousands of posts while others have only in the order
of tens. Moreover, stochasticity is introduced in the training procedure,
which prevents overfitting. As such, for training, an input batch of size $n$
is defined by the concatenation of $n$ such text-sets:
$B=\\{S_{b_{1}},S_{b_{2}},\dots S_{b_{n}}\\}$. We do not consider the relative
order of the texts for a particular user, and text-sets are fed into the
transformer encoder without using positional encoding. Since some users have a
total number of texts smaller than $K$, creating a batch of text-sets is
impossible without padding and masking. However, to alleviate this problem, we
train with an effective batch size of 1 and chose to employ gradient
accumulation to simulate a larger batch size.
Figure 1: Proposed model architecture. We perform user-level classification by
operating on a sample of K texts from a user. Texts are encoded with a
pretrained sentence encoder and processed by a permutation-invariant
transformer network. Binary cross-entropy loss is applied at the user level
for a text-set.
Figure 1 showcases our proposed model architecture for user-level
classification. Each text in a text-set is embedded into a fixed-size vector
using available pretrained sentence encoder models (i.e., RoBERTa / MiniLM).
The text embeddings are fed into the transformer encoder network, and after
processing, we perform mean pooling and output the decision. We compute binary
cross-entropy at the user-level, for a text-set. The pretrained sentence
encoder is frozen and not updated during training.
Figure 2: Performance of our model across training steps, in terms of F1
score, for different sentence encoders (RoBERTa / MiniLM). We show the mean
and standard deviation of F1 score across multiple values of $K$. For both
tasks, RoBERTa yields consistent superior performance compared to MiniLM. Best
viewed in color. Figure 3: Performance of our model across training steps, in
terms of validation F1 score, for RoBERTa sentence embeddings and varying the
$K$, the number of texts per user. For Tasks 1 and 2, the best performance is
attained with $K=16$ and $K=32$, respectively. Best viewed in color.
Baytas et al. [27] proposed to use a T-LSTM to process social media posts
sequentially as a time-series. The authors modify the LSTM architecture to
include a relative time component. However, in our case it is unclear how to
incorporate such a mechanism into the transformer architecture, aside from
using a relative positional encoding [28], which ignores long-ranged
dependencies between posts. As such, we chose to ignore the temporal order of
the posts and process them directly as a set. The main reason for considering
the posts as a set is that in a user’s post history, many posts are
uninformative to the modeling task, and by processing a set of texts, label
noise is reduced naturally as a direct consequence of the attention mechanism,
which assigns more importance to informative posts. However, training with a
sufficiently large dataset might achieve the same effect, but previous
attempts at post-level classification have proven ineffective [4].
In order to assess the impact of the sentence representations, we chose two
different sentence encoders: RoBERTa [29] and MiniLM [30]. We chose RoBERTa
since it is one of the best performing English language models in downstream
tasks [29], and MiniLM, a multi-lingual model, since some users have social
media posts in languages other than English. Figure 2 showcases the
performance gap between the two sentence encoders, averaged across multiple
values of $K$. RoBERTa yields a consistently superior performance across
training steps. Similarly, to assess the impact of the text-set size $K$, we
performed an ablation study, as shown in Figure 3. We kept the sentence
encoder fixed to RoBERTa, and vary the number of texts per user
$K\in\\{4,8,16,32,64,128\\}$. The best performance was achieved with $K=16$
and $K=32$ for Tasks 1 and 2, respectively.
In our final submission, we chose RoBERTa as a sentence encoder and sampled
$K=16$ texts per user for Task 1 and $K=32$ for Task 2. We used the standard
formulation of the transformer network [24], with 4 encoder layers, 8
attention heads each and a dimensionality of 256. Both networks were trained
for 120 epochs, with AdamW optimizer [31], with a cyclical learning rate [32]
ranging from 0.00001 to 0.0001 across 6 epochs and a batch size of 128. To
account for class imbalance, we computed balanced class weights with respect
to each dataset and adjusted the loss function accordingly. Finally, we opted
for a very high threshold when predicting the final decision.
Our proposed architecture can be easily interpretable using modern
explainability methods for feature attribution [33, 34, 5], such as Integrated
Gradients [5]. It automatically identifies social media posts containing signs
of mental health disorders and filters out uninformative posts.
## 4 Interpretability
Since our model operates on sets of social media texts from a particular user,
we can employ model explainability methods to assess the importance of a piece
of text to the model decision. Through this, automatic filtering and selection
of the most indicative posts of a user can be made for use in dataset
creation. This idea is similar to Ríssola et al. [3], which employed a series
of heuristics to recognize posts portraying depression symptoms for use in
constructing a post-level training set from existing depression datasets
annotated at the user level. As such, we use Integrated Gradients [5] to
compute attribution scores for a text-set. The integrated gradients method has
been used in NLP to explore the contribution of individual words and phrases
to a decision made by a classifier. Since we are not operating on words, but
rather on whole texts, this method computes the most important text to the
classifier decision.
Figure 4: Texts from a particular user, relatively ranked in terms of
attribution scores (contribution to a positive decision by the model) computed
with the Integrated Gradients method. For each task, all texts belong to a
single text-set of a user. The model is able to identify posts with a clear
discriminating information for each task. Best viewed in color. Examples have
been paraphrased for anonymity.
Figure 4 showcases selected samples ordered by their attribution score from
the validation set of each task. All samples belong to the same user for each
task, and the attribution scores are computed within the respective text-set.
Posts with a high positive contribution to the decision contain more explicit
descriptions of symptoms, while posts with more negative contributions are
mainly unrelated to the particular mental illness. We use the integrated
gradients method in one of our runs to select the most important posts in the
user history. However, we emphasize that the best application of this approach
is for automatic dataset creation in scenarios of weak supervision, which we
aim to explore in future work.
## 5 Results
### 5.1 Evaluation
There are two kinds of evaluation used for measuring the performance of the
systems, decision-based and raking-based. The decision-based evaluation is
used for quantifying the capacity of a system to perform the binary
classification and predicting if a user is from the positive class (i.e.,
pathological gambling or depression) or the negative one. It is comprised of
standard measures for classification (Precision, Recall, F1) and measures for
this specific task of early detection that consider the delay and the speed of
the decision. The early risk detection error (ERDE) [35] measures the correct
predictions considering a late decision penalty (for predictions taken after
the 5 or 50 first submissions of a user). To overcome the limitations of this
metric [36], the latency-weighted F1 score [37] was also proposed to measure
the performance of early risk detection. Latency measures the delay in
detecting true positives based on the median number of submissions seen by the
system before taking a decision. The speed of a system that correctly predicts
true positives from the first submission is equal to 1, while a slow system
which decides after processing hundreds of texts. The latency-weighted F1
combines the F1-score with the delay in decision-taking for true positives. A
perfect system should achieve a latency-weighted F1 of 1. Besides the binary
classification decisions, the participating teams were asked to also submit a
score for estimating the risk of users for the ranking-based evaluation. These
scores are used to rank users’ risk for pathological gambling or depression.
Standard IR metrics (P@10, NDCG@10, and NDCG@100) are used to measure the
models’ ranking-based performance after processing 1, 100, 500, or 1000
submissions.
### 5.2 Task 1: Early Detection of Signs of Pathological Gambling
The first task proposes the detection of gambling addiction from social media
data. This being the second edition of this task, the organizers provided the
last year’s test data for training the systems. The dataset was collected from
Reddit, following the methodology described by Losada and Crestani [35] and
contains a chronological sequence of posts from each user. The training
dataset was comprised of 164 pathological gamblers, with a total of 54,674
submissions, and 2,184 control users with 1,073,883 submissions. The test
dataset contains 81 users with gambling addiction, summing 14,627 posts, and
1,998 control users with a total of 1,014,122 posts. For the testing phase,
the submissions of users were released sequentially, the systems proposed by
the participating teams received one submission at a time from all the users.
We submitted three runs for the early detection of pathological gambling: Run
0 is comprised of the text-set transformer model using the most recent $K=16$
posts for prediction; the system for Run 1 is the same text-set transformer
model using as input the set of $K=16$ texts that are most important in a
user’s history, selected with Integrated Gradients; Run 2 is a baseline run,
using the proposed model architecture for predicting at post-level, on one
sample at a time.
Table 1: Decision-based evaluation on Task 1: Early Detection of Signs of
Pathological Gambling. We show the performance of our systems compared to the
best-performing run from each team.
Team | Run ID | P | R | F1 | ERDE5 | ERDE50 | LatencyTP | Speed | Latency-Weighted F1
---|---|---|---|---|---|---|---|---|---
BLUE | 0 | 0.260 | 0.975 | 0.410 | 0.015 | 0.009 | 1.0 | 1.000 | 0.410
BLUE | 1 | 0.123 | 0.988 | 0.219 | 0.021 | 0.015 | 1.0 | 1.000 | 0.219
BLUE | 2 | 0.052 | 1.000 | 0.099 | 0.037 | 0.028 | 1.0 | 1.000 | 0.099
UNED-NLP | 4 | 0.809 | 0.938 | 0.869 | 0.020 | 0.008 | 3.0 | 0.992 | 0.862
SINAI | 1 | 0.575 | 0.802 | 0.670 | 0.015 | 0.009 | 1.0 | 1.000 | 0.670
BioInfo$\\_$UAVR | 4 | 0.192 | 0.988 | 0.321 | 0.033 | 0.011 | 5.0 | 0.984 | 0.316
RELAI | 2 | 0.052 | 0.963 | 0.099 | 0.036 | 0.029 | 1.0 | 1.000 | 0.099
BioNLP-UniBuc | 4 | 0.046 | 1.000 | 0.089 | 0.032 | 0.031 | 1.0 | 1.000 | 0.089
UNSL | 1 | 0.461 | 0.938 | 0.618 | 0.041 | 0.008 | 11.0 | 0.961 | 0.594
NLPGroup-IISERB | 4 | 1.000 | 0.074 | 0.138 | 0.038 | 0.037 | 41.5 | 0.843 | 0.116
stezmo3 | 4 | 0.160 | 0.901 | 0.271 | 0.043 | 0.011 | 7.0 | 0.977 | 0.265
Table 2: Ranking-based evaluation on Task 1: Early Detection of Signs of
Pathological Gambling.
| | 1 writing | 100 writings | 500 writings | 1000 writings
---|---|---|---|---|---
Team | Run ID | P@10 | NDCG@10 | NDCG@100 | P@10 | NDCG@10 | NDCG@100 | P@10 | NDCG@10 | NDCG@100 | P@10 | NDCG@10 | NDCG@100
BLUE | 0 | 1.00 | 1.00 | 0.76 | 1.00 | 1.00 | 0.81 | 1.00 | 1.00 | 0.89 | 1.00 | 1.00 | 0.89
BLUE | 1 | 1.00 | 1.00 | 0.76 | 1.00 | 1.00 | 0.89 | 1.00 | 1.00 | 0.91 | 1.00 | 1.00 | 0.91
BLUE | 2 | 1.00 | 1.00 | 0.69 | 1.00 | 1.00 | 0.40 | 0.00 | 0.00 | 0.02 | 0.00 | 0.00 | 0.01
UNED-NLP | 4 | 1.00 | 1.00 | 0.56 | 1.00 | 1.00 | 0.88 | 1.00 | 1.00 | 0.95 | 1.00 | 1.00 | 0.95
UNSL | 0 | 1.00 | 1.00 | 0.68 | 1.00 | 1.00 | 0.90 | 1.00 | 1.00 | 0.93 | 1.00 | 1.00 | 0.95
Table 1 showcases the performance of the systems measured using the decision-
based measures. Regarding ERDE, our first run (Run 0), using the transformer
architecture on the most recent texts from each user, manages to achieve the
best ERDE5 score of 0.015, and the second-best ERDE50 score of 0.009,
demonstrating that the system could detect early the true positive cases. The
perfect scores for latencyTP and speed show that our models were successful at
detecting the true positive cases after the first writing. As expected, the
baseline run using a post-level approach (Run 2) has the lowest performance.
Regarding Run 2, we expected it to achieve the best performance from our
submitted runs, as this approach is more aggressive in taking decisions by
using for classification the most informative posts from users’ history.
Furthermore, our best run from this year’s task surpasses all the runs from
our participation in the first iteration of the task in 2021 [4], showing that
a user-level approach considering a set of texts from each individual is more
suitable than a post-level approach. In Table 2 we show the results of the
ranking-based evaluation, in which each team had to submit the rankings of
users’ risk for pathological gambling. Our team has excellent results for NDCG
and P@10 in all the situations (after 1, 100, 1000, 5000 writings).
### 5.3 Task 2: Early Detection of Depression
This year marks the third iteration of the early detection of depression task,
continuing the 2017 T1 and 2018 T2 tasks. The organizers provided the data
from the previous two editions for training the models. Users from the
depression class were labeled by their mention of diagnosis on their Reddit
posts (e.g., "I was diagnosed with depression"). In contrast, users from the
control class are users who do not have any mention of diagnosis in their
posts [35]. The training dataset comprises 214 users diagnosed with depression
with 270,666 submissions and 1493 control users with a total of 2,959,080
submissions. The test set contains 98 users with depression with 35,332 posts,
and 1,302 users in the control group with a total of 687,228 posts. The texts
for making the predictions for the testing phase were released sequentially,
and the systems from the participating teams had to decide on firing a
decision for a specific user or waiting for more data. We submitted three runs
for the early detection of depression: Run 0 is the text-set transformer model
using the most recent $K=32$ posts for prediction; for Run 1 we employ the
same text-set transformer model using as input the set of $K=32$ texts that
are most important in a user’s history, selected with Integrated Gradients;
Run 2 is a baseline run, using the proposed model architecture for predicting
at post-level, on one sample at a time.
Table 3: Decision-based evaluation on Task 2: Early Detection of Depression.
We show the performance of our systems compared to the best-performing run
from each team.
Team | Run ID | P | R | F1 | ERDE5 | ERDE50 | LatencyTP | Speed | Latency-Weighted F1
---|---|---|---|---|---|---|---|---|---
BLUE | 0 | 0.395 | 0.898 | 0.548 | 0.047 | 0.027 | 5.0 | 0.984 | 0.540
BLUE | 1 | 0.213 | 0.939 | 0.347 | 0.054 | 0.033 | 4.5 | 0.986 | 0.342
BLUE | 2 | 0.106 | 1.000 | 0.192 | 0.074 | 0.048 | 4.0 | 0.988 | 0.190
CYUT | 0 | 0.165 | 0.918 | 0.280 | 0.053 | 0.032 | 3.0 | 0.992 | 0.277
LauSAn | 4 | 0.201 | 0.724 | 0.315 | 0.039 | 0.033 | 1.0 | 1.000 | 0.315
BioInfo$\\_$UAVR | 4 | 0.378 | 0.857 | 0.525 | 0.069 | 0.031 | 16.0 | 0.942 | 0.494
TUA1 | 4 | 0.159 | 0.959 | 0.272 | 0.052 | 0.036 | 3.0 | 0.992 | 0.270
NLPGroup-IISERB | 0 | 0.682 | 0.745 | 0.712 | 0.055 | 0.032 | 9.0 | 0.969 | 0.690
RELAI | 0 | 0.085 | 0.847 | 0.155 | 0.114 | 0.092 | 51.0 | 0.807 | 0.125
UNED-MED | 1 | 0.139 | 0.980 | 0.244 | 0.079 | 0.046 | 13.0 | 0.953 | 0.233
Sunday-Rocker2 | 1 | 0.355 | 0.786 | 0.489 | 0.068 | 0.041 | 27.0 | 0.899 | 0.439
SCIR2 | 3 | 0.316 | 0.847 | 0.460 | 0.079 | 0.026 | 44.0 | 0.834 | 0.383
UNSL | 2 | 0.400 | 0.755 | 0.523 | 0.045 | 0.026 | 3.0 | 0.992 | 0.519
E8-IJS | 0 | 0.684 | 0.133 | 0.222 | 0.061 | 0.061 | 1.0 | 1.000 | 0.222
NITK-NLP2 | 3 | 0.149 | 0.724 | 0.248 | 0.049 | 0.039 | 2.0 | 0.996 | 0.247
In Table 3 we present the performance of the systems using the decision-based
metrics. Our best performing run is the transformer architecture using the
most recent texts from users (Run 0), followed by the system that considers
only the most informative submissions from each user for the model’s decisions
(Run 1). The post-level system (Run 2) has the worst performance. Our three
submitted runs achieve high Recall at the expense of lower Precision scores.
The precision of our models can be improved by incorporating a mechanism for
weighting user posts according to the prevalence of signs of depression [38].
As such, a text-set containing few posts with signs of depression will not
induce a positive prediction. Regarding the early detection evaluation, our
team has the second-best score on the ERDE50 metric (0.027), while our ERDE5
score is close to the best one. Compared to the best metrics from the 2018
edition of this task, when the best ERDE5 and ERDE50 were 0.087 and 0.064,
respectively, current systems surpass these scores due to more data being
available for training the models and the advancements in the field of machine
learning in the last few years. Regarding the standard metrics for
classification, a slight improvement was made in terms of F1 score, from 0.64
in 2018 to 0.71 in 2022. The ranking-based evaluation performance from Table 4
shows that for 1 and 1000 writings, our systems attain some of the best scores
for P@10 and NDCG.
Table 4: Ranking-based evaluation on Task 2: Early Detection of Depression.
| | 1 writing | 100 writings | 500 writings | 1000 writings
---|---|---|---|---|---
Team | Run ID | P@10 | NDCG@10 | NDCG@100 | P@10 | NDCG@10 | NDCG@100 | P@10 | NDCG@10 | NDCG@100 | P@10 | NDCG@10 | NDCG@100
BLUE | 0 | 0.80 | 0.88 | 0.54 | 0.60 | 0.56 | 0.59 | 0.80 | 0.81 | 0.66 | 0.80 | 0.80 | 0.68
BLUE | 1 | 0.80 | 0.88 | 0.54 | 0.70 | 0.64 | 0.67 | 0.80 | 0.84 | 0.74 | 0.80 | 0.86 | 0.72
BLUE | 2 | 0.80 | 0.75 | 0.46 | 0.40 | 0.40 | 0.30 | 0.30 | 0.35 | 0.20 | 0.30 | 0.38 | 0.16
NLPGroup-IISERB | 0 | 0.00 | 0.00 | 0.02 | 0.90 | 0.92 | 0.30 | 0.90 | 0.92 | 0.33 | 0.00 | 0.00 | 0.00
Sunday-Rocker2 | 1 | 0.70 | 0.81 | 0.39 | 0.90 | 0.93 | 0.66 | 0.90 | 0.88 | 0.65 | 0.00 | 0.00 | 0.00
## 6 Conclusion
In this work, we proposed a transformer architecture that performs user-level
classification of gambling addiction and depression detection. For each
individual, the transformer processes a set of texts encoded by a pretrained
sentence encoder to model the interactions between posts and mitigate noise in
the dataset. Our network is interpretable and allows for automatic dataset
creation by filtering uninformative posts in a user’s history. Our method is a
promising approach, especially for social media text processing, where a user
has many texts: some informative and some unrelated to the particular modeling
task. However, their interaction is indicative of the mental state of the
user. We attained the best ERDE5 score of 0.015, and the second-best ERDE50
score of 0.009 for pathological gambling detection. For the early detection of
depression, we obtained the second-best ERDE50 (0.027).
For future work, we aim to extend our method and construct a mechanism for
encoding the relative order of a user’s posts with a modified version of
relative positional embeddings [39]. While we chose an approach that ignores
temporal ordering and processes posts as a set, preserving order is a natural
way to increase the expressive power in modeling a user’s entire social media
interactions, similar to architectures such as the time-aware LSTM [27].
###### Acknowledgements.
The work of Ana-Maria Bucur was in the framework of the research project
NPRP13S-0206-200281. The work of Paolo Rosso was in the framework of the
research project PROMETEO/2019/121 (DeepPattern) by the Generalitat
Valenciana. The authors thank the EU-FEDER Comunitat Valenciana 2014–2020
grant IDIFEDER/2018/025.
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|
# _I can’t believe there’s no images!_
Learning Visual Tasks Using Only Language Supervision
Sophia Gu Christopher Clark11footnotemark: 1 Aniruddha Kembhavi
Allen Institute for Artificial Intelligence
{sophiag, chrisc<EMAIL_ADDRESS>Equal contribution
###### Abstract
Many high-level skills that are required for computer vision tasks, such as
parsing questions, comparing and contrasting semantics, and writing
descriptions, are also required in other domains such as natural language
processing. In this paper, we ask whether it is possible to learn those skills
from text data and then transfer them to vision tasks without ever training on
visual training data. Key to our approach is exploiting the joint embedding
space of contrastively trained vision and language encoders. In practice,
there can be systematic differences between embedding spaces for different
modalities in contrastive models, and we analyze how these differences affect
our approach and study strategies to mitigate this concern. We produce models
using only text training data on four representative tasks: image captioning,
visual entailment, visual question answering and visual news captioning, and
evaluate them on standard benchmarks using images. We find these models
perform close to models trained on images, while surpassing prior work for
captioning and visual entailment in this text-only setting by over 9 points,
and outperforming all prior work on visual news by over 30 points. We also
showcase a variety of stylistic image captioning models that are trained using
no image data and no human-curated language data, but instead using readily-
available text data from books, the web, or language models.
## 1 Introduction
Although vision and natural language processing (NLP) tasks are typically
thought of as being very distinct, there is often a high degree of overlap in
the skills needed to complete them. Visual question answering and reading
comprehension question answering both require parsing and understanding
questions, visual entailment and textual entailment require comparing
different semantic meanings, and captioning and summarization require writing
text that summarizes the semantics of the input. This raises an intriguing
possibility: if a model learned to complete one of these tasks using a high-
level semantic representation of the input text, then in theory it could
immediately be able to complete the corresponding visual task as long as the
input image is encoded in the same semantic representation. We call this
challenge zero-shot cross-modal transfer because it requires applying skills
learned from one modality to a different one. Achieving this would be a step
towards building multi-modal models that can generalize skills across
modalities without needing expensive training data for each modality, and has
potential applications for tasks where visual training data is scarce but text
data is relatively easy to collect.
Accomplishing this requires encoding images and text into a shared semantic
space. We use vision and language (V&L) models trained with a contrastive loss
for this purpose [51, 25]. These models learn to embed text and images into
vectors such that the vectors for matching images and captions are close
together, and vectors for unrelated images and captions are far apart.
Although this loss was originally intended for representation learning and
zero-shot classification, here we show it also facilitates cross-modal
transfer.
To do this, we propose a method called Cross modaL transfer On Semantic
Embeddings (CLOSE). An outline of CLOSE is shown in Figure 1. During training,
the text inputs are encoded into a vector using the (frozen) text encoder from
a contrastive model, which is then used as an input to a model. During
testing, the visual input is embedded with a (frozen) image encoder and used
in place of the text embedding. Because these encoders were explicitly trained
to produce embeddings that encode semantics in similar ways, learning to read
and process the text vector should naturally translate to the ability to read
and process the image vector. Although we focus on text-to-image transfer in
this paper, our approach is applicable to other contrastive models such as
videos [75], point clouds [1], and audio [22, 11, 73], potentially allowing
transfer between many other modalities.
Figure 1: Overview of CLOSE. During training, input text is encoded into a
vector with a text encoder and adapted with an adaptation method. A model
learns to use the vector to perform a task such as VQA, captioning, or visual
entailment. During testing, an input image is encoded with an image encoder
instead to allow cross-modal transfer.
One potential difficulty with this approach is that, while contrastive
embeddings do share some structure between modalities, there can still be
significant differences between the image and text vectors in practice [39].
To mitigate this, we propose to additionally use adapters that modify the text
vectors being used during training. We find adding Gaussian noise to be very
effective in boosting performance, but consider other approaches as well in
our analyses.
Text-to-image transfer is a relatively unexplored setting, so we first conduct
extensive experiments to establish that CLOSE can handle the text-to-image
domain shift without a major performance drop. We compare models trained with
CLOSE on text alone to models trained with images and text on three standard
V&L tasks: captioning, visual questioning answers (VQA) and visual entailment,
and the more complex task of visual news captioning [40]. We find the text-
only models generally perform reasonably close to versions trained with
images, showing that CLOSE can effectively transfer many skills across
modalities. We surpass the previous best text-only method in captioning [79]
by 17 CIDEr (78.2 vs. 95.4) and visual entailment [57] by 9 points (66.6 vs.
75.9), making our method state-of-the-art for these settings by a large
margin. There are no prior results for VQA and visual news in this setting,
however we do surpass the previously best reported result in visual news even
with images [40] (50.5 vs 80.8 CIDEr).
These experiments show that efficient text-to-image transfer is possible. This
has important practical implications because text training data can be
directly constructed by annotators, mined from many existing text datasets, or
even generated by a large language model such as GPT-3 [4], and can therefore
be significantly less expensive than constructing visual training data. We
demonstrate this potential by training effective CLOSE captioning models from
text generated by large language models [4], meaning the only human annotation
required was for prompt construction. We also train several stylistic
captioning models without any labeled images (see Figure 2). We collect text
with various styles from a diverse set of sources, including internet reviews,
books, and GPT-3 generations, and demonstrate that CLOSE models trained on
this text can produce accurate and stylistically correct captions for images.
Finally, we complete two analyses: A sensitivity analysis showing that CLOSE
is robust to cases where text and image vectors differ by a constant offset,
which therefore allows CLOSE to work despite seemingly large differences
between the image/text embeddings. Additionally, a study on the effectiveness
of using an auxiliary vision and language corpus to build an improved adapter.
We find that improvements are possible but vary depending on the source of
that data and that a particularly effective approach is to use the auxiliary
data to compute a structured covariance matrix for use when adding Gaussian
noise.
Figure 2: Using CLOSE to learn stylistic captioning without image data. Text
examples of the desired style are gathered from sources such as the web,
books, or GPT-3. Models are trained on text only and then applied to images.
In summary, our contributions include: (i) introducing the CLOSE model for
zero-shot cross-modal transfer; (ii) showing that training CLOSE with text
data alone, on four V&L tasks, gives results close to models trained on both
images and text; (iii) SoTA results when using only text for three of the
tasks; (iv) demonstrating an application of CLOSE for stylistic captioning;
(v) analyzing how differences between image/text vectors in contrastive models
and how different adapters affect CLOSE’s performance. To facilitate future
work in the community, we release our code111https://github.com/allenai/close.
## 2 Method
(a) No Adapter
(b) Gaussian Noise
(c) Mean Shift
(d) Mean Shift + Noise
(e) CC3M Mean Shift
Figure 3: t-SNE [65] plots for various adapters on 350 randomly selected image
vectors (blue) and paired caption vectors (orange) from Coco captions. The
first two panels demonstrate CLOSE, and the remaining three show additional
adapters we study in our analysis (Section 4).
Model. Our approach uses the image/text encoder from a contrastive model to
encode the input, and then follows many prior works (e.g., [27, 7]) by fine-
tuning a pre-trained language model to process the input vector, along with
any additional input text, to generate output text. First, the input image or
text vector is normalized to have unit length to match what is used in the
contrastive loss. Then that vector is converted into a number of vectors, we
use 4 in our experiments, of the same dimensionality as the language model’s
embedding layer using a linear layer. Next, other input text (e.g., the
hypothesis in visual entailment or the question in VQA) is tokenized and
embedded with the language model’s embedding layer. Those embeddings are
concatenated with the embeddings built from the input vector to construct an
input sequence for the language model.
For the sake of simplicity, we train the model generatively for all tasks [20,
8]. The model generates a caption, a free-form question answer, or a class
name for the tasks of captioning, VQA, and visual entailment respectively.
During training, the language model and linear layer are fine-tuned, but the
text encoder is kept frozen to ensure the correspondence between text and
image vectors learned during pre-training is preserved.
Modality Gap. In practice, text and image vectors from contrastive models can
be far apart, a phenomenon known as the modality gap [39]. For example, on
Coco captions [6] the average cosine similarity between an image and paired
caption is only 0.26, while the average similarity between two unrelated
captions is 0.35. Figure 3(a) shows this gap causes image and text vectors to
fall into separate clusters in the vector space. The root cause is that the
cross-entropy loss used by contrastive models only requires paired image and
text vectors to be close relative to random image and text pairs, which does
not necessarily mean they are close in absolute terms, see Liang _et al_. [39]
for more discussion.
We thus adopt a simple and effective solution – adding Gaussian noise that is
drawn from a standard normal distribution and then scaled by a hyper-parameter
$w$, to the text vectors during training. Intuitively, this noise helps to
close the modality gap by spreading out the text vectors and overlapping them
with the image vectors. Figure 3(b) visually shows that even a small amount of
noise leads to much better overlapping of the image and text vector spaces.
The noise also encourages the model to be more robust to minor changes or
variations to the input vectors, and thus be better prepared for the shift
caused by switching from text to image vectors.
A second motivation for using random noise is the observation that image
vectors capture certain subtle visual details like lighting, background, or
camera position that are not reflected in the text vectors. To illustrate
this, we show a small case study in Appendix 5 where we observe that semantic
changes (e.g., changing the subject of a caption or image from “dog” to “cat”)
result in a relatively consistent directional shift for text vectors, but has
a more erratic effect on image vectors. Adding noise to the text embedding
helps to mitigate this problem by simulating the fact that, even for
semantically similar inputs, image and text vectors can still have minor
differences due to the additional information encoded in the images.
After adding the noise we re-normalize the vector to unit length to match the
image vectors that will be used during evaluation. We study the modality gap
and other approaches to handling it in more detail in Section 4.
## 3 Experiments
Model | Text-Only | Cap. (Single) | Cap. (Mult.) | VE | VQA | E-VQA | VN
---|---|---|---|---|---|---|---
Prior Work | ✓ | - | ESPER Style [79] | CLIP Cls. [57] | TAP-C [57] | - | -
78.2 | 66.6 | 38.7
CLOSE w/o Noise | ✓ | 16.4 | 68.7 | 68.2 | 60.2 | 59.8 | 32.1
CLOSE (Ours) | ✓ | 80.5 | 95.3 | 75.9 | 59.6 | 62.9 | 80.8
CLOSE w/Tuned Noise | | 95.4 | 98.4 | 75.9 | 61.9 | 64.3 | 80.8
CLOSE w/Images | | 113.2 | 113.2 | 77.7 | 65.4 | 67.9 | 105.7
Table 1: Results on V&L tasks. Models in the last two rows require images and
so are upper bounds for CLOSE. We report CIDEr [66] for captioning with single
and multiple captions, visual entailment test accuracy, VQA 2.0 test-dev
accuracy, E-VQA validation accuracy, visual news test CIDEr. See Appendix 2
for other metrics and more detailed results.
We report results on four V&L tasks: captioning, visual entailment, VQA and
visual news, and when training CLOSE using only text generated by a language
model.
### 3.1 Setup
We construct pure-text training datasets for these tasks using the text
annotations from the relevant training datasets, and, for some tasks, text
captions of the training images. Our primary point of comparison is a CLOSE
model trained with the training images, in which case the images are encoded
with the image encoder during training in the same manner as done during
testing. This model does not experience domain shift, so we view it as an
upper bound. We emphasize that in practice the text training data could come
from many other possible sources, see Sect. 5 and Sect. 3.3 for additional
experiments that demonstrate this, we use these text sources since they
closely match the data the models with images are trained on and therefore
allow us to better isolate and study what performance is lost due to the
image-text domain shift.
We use T5base [52] and CLIPViT-L/14 [51], a noise level of 0.08, and a fixed
set of hyper-parameters for all tasks to demonstrate our method is effective
even when there is no image/text validation set to tune on. See Appendix 1 for
hyper-parameter details. We additionally show results when the noise level is
tuned on validation sets, and when the noise is removed, to study the effect
of noise on CLOSE.
### 3.2 Results
Results are shown in Table 1. Due to space constraints, we only report one
metric for each task here and include more results in Appendix 2. We also show
the best method from prior work, when present, that does not use images.
Image Captioning. For captioning, we use text captions as both the input text
and the target output text. However we find that, if multiple captions about
one scene are available, it is beneficial to use different captions about the
same image as the input and target text. We call the first setting captioning
(single) and the second captioning (multiple) and evaluate both since they
facilitate different training setups. We evaluate on Coco Captioning [6] using
the Karpathy split [28]. We train our text-only models using just the captions
in the training data. We treat all captions per image as a group for the
multiple-caption setting and use each caption individually in the single-
caption setting.
CLOSE reaches 95.3 CIDEr in the multiple caption setting, showing high
captioning competency despite not using images. In the single caption setting,
performance is reduced but can be increased to 95.4 with higher noise levels.
Our approach is substantially better than recent zero-shot methods such as
MAGIC (49.3) [61] and Socratic Models (44.5) [81], and is 17 points ahead of
ESPER Style (78.2) [79] which also uses text captions.
Visual Entailment. Visual entailment requires determining whether a premise
image either entails, contradicts, or is neutral with respect to a hypothesis
sentence. During training, a text premise is used instead of an image. The
hypothesis sentence is always text and is encoded with the language model. We
train on SNLI [45] (a language-only dataset) and evaluate on SNLI-VE [74] (a
vision and language dataset). Despite not using images, CLOSE achieves similar
performance to the image model. Song _et al_. [57] also experiment with this
task, but we find adding Gaussian noise allows us to surpass their result by
over 9 points.
VQA. To train a VQA model we use data that contains a sentence describing a
scene (encoded with the text encoder), a question (encoded with the language
model), and a target answer. We consider two datasets. First, we pair Coco
captions with questions about the same image from VQA 2.0 [17]. However, in
this dataset, the questions might ask about details of the image not included
in the caption, and thus cannot be answered by the text-only model. Hence we
also train and evaluate on VQA-E [34] which contains a subset of the VQA 2.0
questions paired with Coco captions that have been verified to contain the
answer.
These training sets have significantly different question distributions due to
the filtering done in VQA-E, so we evaluate models either on the VQA 2.0 test-
dev set or the VQA-E validation set222VQA-E does not have a test set depending
on what train set was used. There is no prior work for this task in the text-
only setting, however CLOSE does outperform TAP-CViT-B/16 [57], a CLIP-based
zero-shot approach.
For VQA-E, we observe only a 3.5 point drop in accuracy relative to image
training while surpassing the baselines. The gap is more significant on VQA
2.0, which we attribute to the sometimes poor alignment between the captions
and questions, although our method is still within 5 points of the model
trained on images.
Visual News. Visual news requires captioning an image in the context of a news
article, and which therefore often requires mentioning the people, locations,
and events from the article text [40]. CLOSE is easily extended to this
setting by using the caption as both the image text and the target output,
while the article is given as additional context to the language model. For
this task, we randomly sample 15% of the training data each epoch due to the
large dataset size, and use OpenCLIP instead of CLIP since our previous
experiments found it slightly improves performance. CLOSE with images achieves
over 105 CIDEr, a significant improvement over the previous best benchmark of
50.5 CIDEr [40]. Training without images also outperforms the previous state-
of-the-art, obtaining a respectable 80.8 CIDEr. See Appendix 5 for qualitative
examples.
Discussion. Overall, performance is comparable to the model trained with
images showing CLOSE is able to transfer skills between modalities. Tuning the
noise level can benefit some tasks, therefore better heuristics for choosing
the noise level or leveraging a small image/text validation set could
additionally improve performance. On the other hand, removing the noise
reduces performance drastically across almost all tasks. This is because the
noise plays an important role in addressing the modality gap.
### 3.3 Training with Data from a Language Model
Figure 4: Prompt used to generate a synthetic caption from a language model. The language model’s continuation (highlighted text) is used as a synthetic caption. Model | B-4 | M | C | S
---|---|---|---|---
MAGIC [58] | 12.9 | 17.4 | 49.3 | 11.3
CLOSE w/Coco | 29.5 | 25.6 | 98.4 | 18.3
CLOSE w/GPT-J RNG | 19.6 | 20.9 | 63.2 | 13.8
CLOSE w/GPT-J Unigram | 23.2 | 22.2 | 78.9 | 15.6
CLOSE w/OpenAI Curie | 18.5 | 21.2 | 69.0 | 14.9
Table 2: BLEU-4, METEOR, CIDEr, and SPICE on the Coco validation set when
training on synthetic captions.
Next, we use CLOSE to train a captioning model on synthetic data generated by
a language model. We first construct a prompt that includes a natural language
instruction and some example captions following an in-context learning
approach [4], shown in Figure 4. To generate a diverse set of captions, we
prefix each caption with two keywords that occur in that caption, and end the
prompt with two new keywords to be used in the caption to be generated (“fire”
and “hydrant” in Figure 4). Then diverse captions can be constructed by
changing the ending keyword pair. To reduce the chance of caption style
affecting the quantitative evaluation, we take steps to better match the style
of the Coco captions, although in settings where the precise style is of less
importance this would not be required. We generate 100k examples from three
generation methods:
GPT-J RNG. Examples are generated using a 6 billion parameter open source
language model, GPT-J[68], with 50 in-context examples. Keywords are sampled
uniformly at random from keywords in the Coco training data.
GPT-J Unigram. Keywords are instead sampled to match the unigram distribution
of Coco captions.
Curie Unigram. Generations are from OpenAI
Curie333https://beta.openai.com/docs/models/gpt-3 with 20 examples and
unigram-matching.
Results on Coco are shown in Table 2. Our best result achieves 78.9 CIDEr.
Inspection shows that, even with our keyword sampling approach, many errors
are still caused by style issues, and that style also explains the reduced
performance of the Curie model. For example, the synthetic captions from the
Curie model are 23 times more likely than the Coco and the GPT-J captions to
use the word “opens” (e.g., “a living room that opens onto the balcony”), and
use “cellphone” while “cell phone” is much more common in Coco. More details
are in Appendix 3. This illustrates how, when using this method, the choice of
language model can have subtle effects on the style of captioning that will be
learned. Despite this issue, this is still a very strong result that surpasses
the zero-shot method MAGIC [58].
## 4 Analysis
Our approach opens up two intriguing questions: (1) Why does embedding
substitution work even when text and image vectors are generally quite far
apart? (2) Can methods that leverage additional data to better close the
modality gap improve upon this approach? We do two analyses to answer these
questions. Furthermore, we study how different choices for the contrastive
embedding model or for the language model affect our method’s performance.
### 4.1 Sensitivity Analysis
Bias | Mag. | MG | $\Delta$ | Cap. | VE | VQA
---|---|---|---|---|---|---
none | 0.0 | 0.26 | 1.00 | 94.4 | 64.3 | 75.9
mean | 0.8 | 0.62 | 0.69 | 92.8 | 64.7 | 75.4
-mean | 0.8 | -0.10 | 0.85 | 84.3 | 62.0 | 71.8
RNG | 0.2 | 0.25 | 0.98 | 93.5 | 63.9 | 75.3
RNG | 0.5 | 0.24 | 0.89 | 92.5 | 64.2 | 75.3
RNG | 0.8 | 0.20 | 0.78 | 89.3 | 63.7 | 74.8
RNG | 1.0 | 0.18 | 0.71 | 87.2 | 63.8 | 74.2
RNG | 2.0 | 0.11 | 0.45 | 73.7 | 61.4 | 71.3
Table 3: Text vector translation-sensitivity analysis. The first three columns
show the translation magnitude, the resulting modality gap on Coco, and the
cosine similarity to the original vectors. The following columns show CIDEr
captioning score, accuracy on VQA-E, and accuracy on visual entailment on
validation sets.
To help answer the first question, we perform a sensitivity analysis on the
input text vectors. To do this, the model is trained while adding a constant
vector to the normalized text vectors and then re-normalizing, and tested on
the unaltered image vectors as before. This alteration will change how the
text vectors are distributed relative to the image vectors, but will not
change how the text vectors are distributed relative to one another. We show
results when using a random vector (note the same vector is used for all of
training, it will just be selected randomly at the start of training) of
different magnitudes, the mean difference of text and image vectors to
represent a shift towards the image vectors, and the negation of that vector
to shift away from the image vectors. In all cases, we continue to add
Gaussian noise as before.
Results are shown in Table 3. For random vectors (RNG), we report the average
of three runs with 3 different vectors. Overall, we see only minor degradation
when using random vectors until very large shifts are used, showing the model
is generally insensitive to shifting the text vectors during training.
Shifting the vectors towards the images (mean) can result in a slight gain in
performance, and shifting the vectors away from them (-mean) results in a more
significant decrease, showing the model is not completely insensitive. However
it is still notable that vector substitutions work well even as the text
vector’s positions are significantly randomized.
We hypothesize that this insensitivity is due to two reasons. First, most
directions in the shifted feature space are predictive of the output in the
same manner as before because the text vectors do not change relative
positions. Second, the Gaussian noise trains the model to be insensitive to
shifts in unimportant directions in the feature space, which often include the
direction of the shift. This insensitivity provides part of the answer to
question 1. A major source of the modality gap is a constant shift between the
image and text vectors [38]. However, addressing this is not as important as
one might expect because CLOSE is not highly sensitive to the absolute
positioning of the text vectors.
Figure 5: Examples of stylistic captions produced by CLOSE trained with only
text data, and then applied 0-shot to images.
### 4.2 Learned Adapter Analysis
Method | MG | Cap. | VE | VQA | VN
---|---|---|---|---|---
CLOSE | 0.26 | 94.3 | 75.9 | 64.3 | 80.8
+Cov. (Coco) | 0.62 | 106.5 | 75.5 | 65.5 | 84.1
+Cov. (CC3M) | 0.58 | 95.1 | 75.8 | 65.0 | -
+Linear (Coco) | 0.81 | 99.5 | 76.0 | 65.7 | -
+Linear (CC3M) | 0.75 | 81.8 | 75.5 | 64.9 | -
Table 4: Results with adapters built with paired data. The modality gap on
Coco captions, captioning CIDEr, visual entailment accuracy, VQA-E accuracy
and visual news CIDEr are shown. The last task is more complex and so we only
experiment it with one promising adapter.
As suggested by Figure 3(c), mean shift might not be perfect at aligning the
text and image vectors, so we hypothesize more sophisticated adaption methods
could improve performance. More complex adapters generally require a paired
image/text corpus to train on, so we avoid using them in our main CLOSE
method. However, here we investigate them to better understand how much
performance they could potentially contribute. To study the difference between
using high-quality annotated data or web data we use both Coco captions and
Conceptual Captions 3 Million (CC3M) [54]. For Coco we use the 30k captions
from the “restval” set of the Karapathy split, which do not appear in our
train, eval or test sets, and for CC3M we use a random sample of 100k
image/text pairs. We consider two adapters:
Linear Adapter. We learn the modality shift by training a linear model to
minimize the Euclidean distance between the adapted text vector and its paired
image vector. We continue to add Gaussian noise after applying this model.
Structured Noise with Covariance Matrix. Even in principle, we do not expect
there to be a perfect one-to-one mapping between text and image vectors
because an image vector can be similar to many different texts that describe
different parts or details of the image. This motivates us to approach the
problem from the perspective of better understanding how text vectors are
distributed around its related image vectors, instead of just trying to learn
a simple mapping function. In Appendix 4, we provide insight into how the
vector differences from Coco image-caption pairs follow a particular shape. To
capture this shaped relationship between text and images, we add Gaussian
noise whose mean and covariance are learned from the differences between text-
image vectors in the auxiliary corpus, to the text during training. This noise
is expected to better simulate the text-image shift that will occur during
evaluation.
Results are shown in Table 4. We observe large improvements on captioning,
modest improvements on VQA and visual news444We only test one adapter on this
task due to the longer training times, and similar performance on visual
entailment using the adapters from Coco, with the structured noise approach
being significantly better on captioning, and slightly worse on the other
tasks. The CC3M adapter also achieves mild gains, although it is less
effective. This shows the training data used for the adapter is important, a
point that can be qualitatively observed in Figure 3(c) and Figure 3(e).
### 4.3 Performance Analysis of Different CLIP and T5 Models
CLIP Model | T5 Model | Cap. | VE | VQA
---|---|---|---|---
ViT-L/14 | small | 94.4 | 74.9 | 59.9
ViT-L/14 | base | 95.4 | 76.1 | 64.3
ViT-L/14 | large | 93.9 | 75.1 | 65.2
ViT-B/32 | base | 91.1 | 75.3 | 61.4
RN101 | base | 90.0 | 75.4 | 59.8
RN50 | base | 90.2 | 75.3 | 60.4
RN50$\times{4}$ | base | 92.0 | 75.3 | 61.5
RN50$\times{16}$ | base | 93.4 | 74.4 | 62.5
RN50$\times{64}$ | base | 96.1 | 75.8 | 64.2
OpenCLIP [24] | base | 99.2 | 76.3 | 65.1
EVA-CLIP [13] | base | 101.7 | 75.53 | 66.6
Table 5: Ablations with different contrastive and language models. The first
column indicates which CLIP model was used, with OpenCLIP indicating we use
the ViT-L/14 OpenCLIP model trained on Laion 400m [24]. The last three columns
show CIDEr on Coco captioning in the single caption setting, accuracy on
visual entailment, and overall accuracy on VQA-E on the validation sets.
Finally, we study how different choices for the contrastive embedding model or
for the language model affect the performance of our method. Results for
captioning, visual entailment, and E-VQA are shown in Table 5. For these
experiments we use the tuned noise values in order to compare best-case
performance. We find the optimal noise level for these models generally does
not change as these components are altered, so we use the same noise levels as
our main results for all these experiments.
There is a consistent decrease in performance when using CLIP versions other
than ViT-L/14, with only RN50×64 being comparable, showing that CLOSE gains
effectiveness as the contrastive model becomes more powerful. We also observe
much less dependence on the size of the T5 model, with the large model
increasing performance on VQA but not on the other tasks. The OpenCLIP model
is generally more effective and boosts the captioning results to nearly 100
CIDEr. The EVA-CLIP model [13] further boosts VQA scores, approaching our main
result with images (67.9), showing that CLOSE’s performance can be improved by
enhancing the contrastive model.
## 5 Stylistic Captioning
We demonstrate an application of our method by applying it to the task of
constructing captions with specific writing styles. Our general approach is to
gather text-only training data that exemplifies the style we want the model to
use, train on them as if they were text captions as done in Section 3.2, and
then apply the model to images. To show that a diverse range of natural
language data sources can be used to learn different styles we show four
captioning styles, each of which uses a different method of collecting
training data.
Ego-Centric. Section 3.3 shows that our model can be trained using data
generated by a language model. Now we demonstrate an application of that
approach by using the language model to generate captions in an ego-centric
style. We use the same prompt format as before (Figure 4), only now with 20
examples of manually authored captions written from a first-person
perspective. We again sample keywords randomly from those found in Coco
training captions to generate diverse prompts and obtain 20k captions using
OpenAI’s GPT-3 model. We apply this model to Coco validation images, shown in
the top row of Figure 5, and observe it learns to use a variety of first-
person language while accurately describing the image.
Uplifting. We use a publicly available dataset [14] to collect 6k examples of
uplifting captions (no images). Results are shown in the second row in Figure
5, where we observe the model adds warm and optimistic details to its
captions.
Character-Based. Next, we target character-based captions that use proper
nouns and describe images as if they were from a story. Using proper nouns
would be a significant hurdle for many existing systems due to the lack of
image/name paired data in existing datasets. However, CLOSE can leverage
CLIP’s ability of recognizing names of famous people [51] to handle that
problem. We first pick 33 Harry Potter characters. Then only a few excerpts
from the Harry Potter books or fan fictions are manually collected and used,
together with the characters, as prompts to GPT-3 to create 13k captions.
Results on relevant photos are shown in the third row of Figure 5. The model
uses the correct names and image content, while sometimes making up plausible
events that could give additional context to the image as if it was a scene in
a book or a movie.
Reviews. We train a model to write captions like a customer writing a review.
For training data, we gather publicly-available Amazon product
reviews555https://www.kaggle.com/datasets/bittlingmayer/amazonreviews and
select positive reviews that are a maximum of 40 tokens long. As shown in
Figure 5 bottom row, the captions use a variety of language to write positive
reviews of the items in the photos.
## 6 Related Work
Using Contrastive Models. Many vision and language contrastive models have
been constructed, including CLIP [51], ALIGN [25], UniCL [76] and OpenCLIP
[24], and recent multi-modal models that contain a contrastive training
component [78, 80, 32]. Typically these models are used either zero-shot,
which is effective for image classification but challenging for more complex
tasks like captioning or visual entailment [57, 61, 81], or as feature
extractors for down-stream tasks [55, 29, 18, 12, 44, 50, 82, 72]. Our work
offers a compromise between those two approaches by allowing models to be
trained with only textual data, which substantially improves upon zero-shot
performance without requiring annotated images.
Zero-Shot Vision Using Language Models. Several recent works have combined
large language models with pre-trained vision models to perform vision tasks
zero-shot. Methods include using reinforcement learning to learn how to
generate text that matches a CLIP Embedding [79], using CLIP to guide
inference in the LLM [62], or using a pre-trained model to generate text
describing an image to pass into the language model [81]. Compared to these
methods our approach of leveraging text training has several advantages. Fine-
tuning on text-only data enables our model to learn task-specific details and
subtleties that are challenging for fully zero-shot methods, such as the style
of captions to be generated. Our approach also works effectively with smaller
language models (CLOSE only uses 220M trainable parameters) which
significantly reduces the computational demand.
Cross-Modal Transfer Learning. Transfer learning has typically focused on
transferring skills from one modality to the same modality. CROMA is an
exception and uses a modality-invariant feature space to achieve transfer
similar to our work, however, it is limited to classification tasks and is
few-shot rather than zero-shot [38]. Pre-trained language models have been
shown to learn skills that can transfer to new modalities [42], however, this
will be ineffective for task-specific skills such as a desired captioning
style or learning the space of output labels. Several multi-modal/multi-task
models have learned many tasks in different modalities simultaneously [41, 70,
37, 26] and could thus potentially transfer skills between them, with HighMMT
in particular showing positive results [37]. Our work studies the more
challenging zero-shot setting (meaning no training data in the target modality
is available), and therefore requires all the needed skills to be learned from
a modality different than the one used in evaluation.
Recently, Song _et al_. [57] use a similar vector-substitution trick with CLIP
to train visual entailment models, however they do no use noise or other
methods that address the modality gap. Yu _et al_. [79] use reinforcement
learning to train a model to generate text that CLIP ranks as being close to
input images, and text data to learn captioning styles, although they do not
directly train on text versions of the vision tasks. Concurrently with our
work, Nukrai _et al_. [48] and Wei _et al_. [35] propose text-only approaches
leveraging CLIP with either Gaussian noise similar to CLOSE, or using a
projection of the text embeddings. Our work does additional analysis, covers
more tasks including experiments using data generated by a language model, and
achieves better captioning results.
Domain Invariant Representations. Using domain-invariant features to achieve
out-of-domain generalization has a long history in transfer learning. Work in
this area has shown such features can be built from multi-domain training data
[69, 19], small amounts of labelled data in the target domain [9, 64], and
unsupervised data [71, 59]. Methods include using adversarial learning to
remove domain-dependent features [16, 36, 63], using maximum mean discrepancy
to ensure features are distributed similarly across multiple domains [31, 3]
and various data augmentation approaches to prevent models from learning
domain-dependent features [85, 84, 67, 53]. The effectiveness of Gaussian
noise in making models robust to domain shifts in these features has also been
observed in image classification [33]. While we also use domain-invariant
features, the domain shift we study is more extreme than what is typically
studied due to the change in modalities, and we show large-scale contrastive
models can be an effective source of invariant features if used correctly.
Stylistic Captioning. Stylistic captioning models can be built by authoring
captions of the desired style [46, 14, 21, 56] and applying standard
captioning methods. However, since creating such annotations is expensive,
many stylistic captioning methods additionally transfer from captions with
other styles by pre-training or multi-tasking [46, 47, 77]. Other methods have
combined unstylized captioning data with text data in the desired style
through methods such as adversarial learning [5], multi-tasking with language
modelling [14], or factoring caption writing into style and context components
so that the style component can be learned from the text [14, 83]. Most
similar to our work, Tan _et al_. [60] train a model to generate text from
either images or text using a shared encoding space and learned style
embeddings. Unlike these methods, our approach does not require the use of any
paired image/caption data.
## 7 Conclusion
We have shown that the multi-modal semantic vector space learned by
contrastive models can be used for cross-modal generalization through CLOSE,
and studied its sensitivity and what improvements can be made with trained
adapters. We have also conducted experiments on multiple vision and language
tasks and demonstrated a specific application to stylistic captioning. Beyond
stylistic captioning, CLOSE is applicable to many other cases where training
data is abundant in one modality but scarce in another. Possible use-cases
include: training a captioning model for 3D scenes using image captioning
data; training a model to summarize a video using text summarization data; and
training a model to perform tasks like VQA or captioning for less-studied
modalities like tables, graphs, or sensors without having to annotate
additional data for all modalities. As more powerful contrastive models that
span more modalities are trained, we expect CLOSE to yield better results and
gain more use cases.
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## Appendix
## 1 Hyperparameters
For all tasks, we fine-tune our model with the Adam optimizer [30] with a
linear decaying learning rate starting at $3\text{e-}4$, $\beta_{1}=0.9$ and
$\beta_{1}=0.999$, batch size of 128, and train for 8 epochs. We use beam
search with a beam size of 5 for evaluations. When tuning the noise level, we
select 0.04 for VQA, 0.08 for visual entailment and visual news, 0.14 for
captioning in the single caption setting, and 0.04 for captioning in the
multiple captioning setting.
## 2 Detailed Results
To facilitate more detailed comparisons with other works, we present results
across more metrics of our evaluated datasets. In all tables, upper bounds
that use images are shown above the dashed line.
Model | Mode | B-4 | M | C | S
---|---|---|---|---|---
CLOSE w/Images | - | 34.4 | 27.8 | 113.2 | 20.4
CLOSE w/Tuned Noise | S | 28.6 | 25.2 | 95.4 | 18.1
CLOSE w/Tuned Noise | M | 29.5 | 25.6 | 98.4 | 18.3
ESPER Style [79] | - | 21.9 | 21.9 | 78.2 | -
CLOSE w/o Noise | S | 4.2 | 12.2 | 16.4 | 6.5
CLOSE w/o Noise | M | 21.9 | 20.6 | 68.7 | 13.5
CLOSE | S | 22.1 | 23.7 | 81.2 | 17.7
CLOSE | M | 29.5 | 25.7 | 97.8 | 18.3
Table 1: Results on the caption test set in single-caption setting and
multiple captioning setting, M indicates the multiple caption setting and S
indicates the single caption setting.
.
Model | Yes/No | Num. | Other | All
---|---|---|---|---
CLOSE w/Images | 83.2 | 44.8 | 54.9 | 65.4
CLOSE w/Tuned Noise | 79.4 | 43.4 | 51.1 | 61.9
TAP-CViT-B/16 [57] | 71.4 | 20.9 | 18.6 | 38.7
CLOSE | 77.1 | 42.1 | 48.6 | 59.6
CLOSE w/o Noise | 78.6 | 40.6 | 49.0 | 60.2
Table 2: Results on the VQA 2.0 test-dev set. Model | Yes/No | Num. | Other | All
---|---|---|---|---
CLOSE w/Images | 80.4 | 48.4 | 64.1 | 67.9
CLOSE w/Tuned Noise | 78.2 | 46.0 | 59.5 | 64.3
CLOSE | 74.9 | 45.2 | 59.2 | 62.9
CLOSE w/o Noise | 76.8 | 36.8 | 53.9 | 59.8
Table 3: Results on the VQA-E validation set. Model | Val | Test
---|---|---
CLOSE w/Images | 77.0 | 77.7
CLOSE w/Tuned Noise | 75.9 | 75.9
CLIP Classifier [57] | 67.2 | 66.6
CLOSE | 75.9 | 75.9
CLOSE w/o Noise | 68.7 | 68.2
Table 4: Results on the visual entailment test and validation set. Model | B-4 | M | R | C
---|---|---|---|---
VNC w/Images [40] | 5.3 | 8.2 | 17.9 | 50.5
CLOSE w/Images | 9.3 | 10.9 | 25 | 105.7
CLOSE | 5.4 | 8.2 | 19.7 | 80.8
CLOSE w/o Noise | 2.1 | 4.9 | 12.7 | 32.1
Table 5: Results on the visual news test set.
Captioning. We present results in Table 1 for BLEU-4 [49], METEOR [10], CIDEr
[66] and SPICE [2].
VQA. We present results by question-type for VQA 2.0 in Table 2 and VQA-E in
Table 3.
Visual Entailment. We present visual entailment results on the test and dev
set in Table 4.
Visual News. We present results with BLEU-4 [49], METEOR [10], ROUGE [15] and
CIDEr [66] following [40] in Table 5. To the best of our knowledge the
previous best reported results is from Liu _et al_. [40] which does not make
use of a pre-trained language model like CLOSE does. Qualitative results are
show Section 5.
## 3 Generating Synthetic Captions using Language Models
Figure 1: Examples of words that are over-produced by the captioning model trained on the OpenAI Curie synthetic captions relative to the model trained on the Coco captions. The first column shows the word and how much more common it is across captions generated for images in the Coco validation set. The remaining columns provide an example image and a caption from both models with the CIDEr score computed using human-annotated captions. Model | Individual | Any
---|---|---
OpenAI Curie | 58.8 | 85.0
GPT-J | 42.7 | 81.9
Table 6: How often generated captions contain the target keywords when
generating synthetic captions using different language models. The second
column shows the success rate for individual generations, and the third column
shows how often any caption in the 5 captions generated per a prompt contain
both keywords.
In this section, we give more details about how we generate captions using
language models and the results from Section 3.3. When generating captions, we
use nucleus sampling [23] at $p=0.95$ and a temperate of 1, which we find
generally improves results. It is not uncommon for the caption to fail to
contain both input keywords, so we sample 5 captions for each prompt and then
select a caption containing the keywords if one exists, and select one
randomly otherwise. The in-context example captions are prefixed by randomly
chosen words that exist within that caption (excluding stop words), and we use
randomly selected captions from Coco training captions as the examples. During
sampling, we randomly shuffle both the order of the in-context examples and
what keywords are used as prefixes for those examples to improve the diversity
of the outputs. If doing unigram sampling, we keep track of the distribution
of words found in the captions generated so far, and sample new keywords in
proportion to how under-represented they are, while never sampling over-
represented words.
Statistics for how often the input keywords are correctly included in the
caption are shown in Table 6. The success rate is less than 60%, although
selecting from 5 generations brings the success rate up considerably. GPT-J is
worse than OpenAI Curie, but sampling extra captions helps make up for this
deficiency. Future work could integrate a constrained beam search method to
address this difficulty [43].
We find that about 10% of GPT-J captions are not coherent or do not describe a
visual scene, while these kinds of captions almost never occur with OpenAI
Curie. Overall, for GPT-J, producing 100k captions took about 50 GPU hours
using a NVIDIA RTX A6000. For OpenAI Curie, each generation requires
approximately 500 tokens per a query, so the total cost was about 100$111At
the current rate of 0.002$ per 1k tokens on 11/16/2022. Both methods are far
cheaper than annotating data.
As discussed, we observe stylistic differences occur between models trained on
synthetic captions and models trained on Coco captions. A particular issue is
that, while unigram sampling prevents words becoming under-represented, it
still allows some words to become over-represented if the language model has a
natural tendency to generate them. Figure 1 contains some examples where the
model trained on OpenAI Curie captions uses words like “pictured”, “lays” or
“cityscape” that almost never occur in Coco captions and thus lead to low
quantitative scores even when used correctly. Interestingly, we find GPT-J is
not as affected by this issue, which likely stems from differences in what
data the language model was trained on. Nevertheless, the captions do still
correspond well to the image content, as shown by reasonably good captioning
scores despite these stylistic issues, showing it is possible to learn
captioning using only synthetic data.
## 4 The Relationship Between Image and Text Vectors
Figure 2: An example of how image/text feature vectors shift with a specific
change in species (vertically) or position (horizontally). Text adjacent to
each arrow shows any significant changes in the text (purple) or image (red)
vector that occurred because of the shift.
Figure 3: Plots analyzing the differences between image and text vectors for
image/caption pairs in Coco captions. Only the first 200 features are shown.
We perform a small case study by selecting four image/caption pairs that
represent two different semantic changes in terms of animal species and
positions (the result is shown in Figure 2) and examine how the image or text
vectors shift according to these changes. We observe that text vectors move
more consistently when either the species or positions of the animals change.
This disparity is likely due to random shifts in image semantics that
correlate with conceptual changes in the text, such as subtle alterations in
the animals’ appearance, textures, or background.
We further analyze how image and text vectors typically differ by computing
the differences between image/text pairs in an auxiliary corpus of COCO. We
center these differences and apply PCA. The first two plots in Figure 3 show
that the first few PCA dimensions explain a large portion of the variance in
these differences, showing that differences often occur in similar directions.
We also plot the Pearson correlation coefficient for the most related features
in the third plot, showing that a number of these features are highly
correlated. Indeed, image/text pairs tend to move in a structured manner that
follows a particular ”shape”. We capture this subtle relationship by studying
the covariance matrix of the differences between text-image vectors. We then
modify our Gaussian noise that is added to the text during training to better
simulate this co-movement.
## 5 Visual News Qualitative Examples
We show some qualitative examples for visual news in Figure 5. We observe that
close to $50\%$ of time, the predicted captions can be more descriptive (i.e.,
they can include more details), indicating there is room for this visual news
captioner to grow. There are also some cases in which the predicted captions
are better than the ones provided by human (the target captions). But overall,
the general sense of both the news images and articles are present in the
captions produced by CLOSE.
(a) (a) Figure 5: Examples of visual news captions produced by CLOSE trained
on text captions and news articles alone, and then applied zero-shot to news
images and articles.
|
# KeyGen2Vec: Learning Document Embedding via Multi-label Keyword Generation
in Question-Answering
Iftitahu Ni’mah♣,♠ Samaneh Khoshrou♣ Vlado Menkovski♣ Mykola Pechenizkiy♣
♣ Eindhoven University of Technology ♠ BRIN Indonesia
{i.nimah, v.menkovski<EMAIL_ADDRESS>
###### Abstract
Representing documents into high dimensional embedding space while preserving
the structural similarity between document sources has been an ultimate goal
for many works on text representation learning. Current embedding models,
however, mainly rely on the availability of label supervision to increase the
expressiveness of the resulting embeddings. In contrast, unsupervised
embeddings are cheap, but they often cannot capture implicit structure in
target corpus, particularly for samples that come from different distribution
with the pretraining source.
Our study aims to loosen up the dependency on label supervision by learning
document embeddings via Sequence-to-Sequence (Seq2Seq) text generator.
Specifically, we reformulate keyphrase generation task into multi-label
keyword generation in community-based Question Answering (cQA). Our empirical
results show that KeyGen2Vec in general is superior than multi-label keyword
classifier by up to 14.7% based on Purity, Normalized Mutual Information
(NMI), and F1-Score metrics. Interestingly, although in general the absolute
advantage of learning embeddings through label supervision is highly positive
across evaluation datasets, KeyGen2Vec is shown to be competitive with
classifier that exploits topic label supervision in Yahoo cQA with larger
number of latent topic labels. 111The empirical study was completed in 2020 at
Eindhoven University of Technology.
## 1 Introduction
(a) Doc2Vec clusters
(b) Fitting labels
(c) Supervised clusters
(d) Fitting labels
Figure 1: The expressiveness between unsupervised and supervised document
embeddings3332-D Projection is based on t-SNE. Points are colored according to
the predicted cluster ids $\Omega$ (left) and exact classes $\mathbb{C}$
(right). For the consistency plotting of cluster membership colors, exact
classes was mapped to prediction cluster ids by “Hungarian” algorithm.. Upper:
unsupervised embeddings via Doc2Vec. Lower: supervised embeddings via a multi-
class classifier.
Keywords or _tags_ have been widely used in community-driven Question-Answer
(cQA) systems and many online platforms, such as Twitter, and online
bibliographic databases and search engines, as metadata describing sub-topics
of an article. Obtaining keywords as supervisory information for training any
machine learning models is considered to be cheaper than obtaining topic
labels since there exists automated keyword extraction methods, e.g. TfIdf,
TextRank Mihalcea and Tarau (2004), SingleRank Wan and Xiao (2008), Maui
Medelyan et al. (2009), which in practice is often accompanied by human
validation to control the quality of keyword labels.
Figure 2: KeyGen2Vec. Model is trained in _autoregressive_ mode. In teacher
forcing mode, $t-1$ shifted version of keyword is given to the model as input
for decoder. Circles represent RNN states in encoder (left/source) and decoder
(right/target) network.
Despite many potential benefits of keywords or tags, such as providing
predictable patterns Golder and Huberman (2005, 2006); Nimah et al. (2019,
2021); knowledge organization and resource discovery Macgregor and McCulloch
(2006); Ames and Naaman (2007); improving retrieval performance Hotho et al.
(2006); Nimah et al. (2019), less attention has been paid to incorporate
keywords as a condition to embed documents in high dimensional embedding
space. Existing approaches that incorporate keywords, tags, or phrases as
additional semantic knowledge for document clustering can be divided into two:
(1) approaches that focus on improving the quality of document embeddings
during training Pu et al. (2015); Sato et al. (2017); and (2) approaches that
focus on improving the clustering algorithm or subsequent tasks by providing
additional post-pipelines Ramage et al. (2009); Rosa et al. (2011); Dao et al.
(2018) to intensify the expressiveness of representations in latent space,
such that semantically similar points in that space are close together
compared to dissimilar points. However, these works depend on multiple
pipelines, which consequently hinder their reproducibility and adaptation as
end-to-end system in many real world NLP applications.
stepwise/.style n args=2 edge path= [draw, edge] (!u.parent anchor) —-
+(#1,#2) —- (.child anchor)edge label; , my shading/.style= for tree=
text/.wrap pgfmath arg=black!##1!#110*level(), edge/.wrap pgfmath arg=-¿,
draw=black!##1!#110*level(), , , for tree= edge=-¿, grow=east, align=left,
child anchor=west, edge path= [draw, edge] (!u.parent anchor) —- (.child
anchor)edge label; , font= [Approaches [3. Distributed [e. Discrete predictor
(Supervised) [2. Multi-Label Classifier; [ 4, 5 ] in table 1, name=multilbl]
[1. Multi-Class Classifier; [ 3 ] in table 1, name=upperbound] ] [d.
Pretrained Sentence Encoder: S-BERT Reimers and Gurevych (2019); [ 6 ] in
table 1] [c. Paragraph Vector (Doc2Vec) Le and Mikolov (2014) [ 8,9 ] in table
1] [b. Seq2Seq [2. KeyGen2Vec
[ 1 ] in table 1, my shading=vermilion, name=framework] [1. Autoencoder
Sutskever et al. (2014); Cho et al. (2014); [ 2 ] in table 1] ] [a. Bottom-Up
[2. Covariance Matrix Nikolentzos et al. (2017); Torki (2018) [b. Neural Word
Embedding;
[ 17,18,20 ] in table 1 ] [a. PMI; [ 19 ] in table 1] ] [1. Mean Embedding [d.
Pretrained $\rightarrow$ Wiki2Vec Yamada et al. (2020)
and fastText Bojanowski et al. (2017); [ 11, 13 ] in table 1] [c. Pretrained
$\rightarrow$ GloVe
Pennington et al. (2014); [ 10, 12 ] in table 1] [b. Trainable $\rightarrow$
Word2Vec
Mikolov et al. (2013b, a); [ 14, 15 ] in table 1 ] [a. Pointwise Mutual
Information (PMI)
Levy and Goldberg (2014); [ 16 ] in table 1 ] ] ] ] [2. Probabilistic
[Dirichlet-based Topic Model (LDA)
Blei et al. (2010); Blei (2012);
[7] in table 1 ] ] [1. Non-Distributed [TfIdf Salton et al. (1975); [21] in
table 1 ] ] ] [draw=vermilion, solid, line width=1mm, inner sep=.4em,
fit=(framework), label=[align=left]0: Proposed Framework ] ;
Figure 3: Document embedding approaches in this study.
Our work mainly focuses on topical clustering of cQA archives as a subsequent
task to evaluate currently available document embedding approaches, including
the proposed KeyGen2Vec framework. As a motivating example, Figure 1(a)-1(b)
illustrate how unsupervised-based embeddings are likely random, indicating the
model’s incapability to capture semantic aspects such as latent topics
structure inferred in target data. By contrast, supervised approach is more
expressive, producing separable clusters in latent space that are coherent
with topics, as shown in Figure 1(c)-1(d). However, the latter model requires
learning document embeddings with topics as label supervision. So, it is more
costly than the unsupervised embedding approaches.
To negotiate the trade-offs between utilizing unsupervised and supervised
approaches for learning document embeddings, we utilize keywords as sub-latent
structure in corpora to train Seq2Seq networks referred to as KeyGen2Vec. Our
work holds an assumption that learning a conditioned sequence-to-sequence
mapping between documents and their corresponding keywords equals to learning
the structural similarity that hierarchically links contents in document,
keywords as explicit document abstractions, and topics as latent variables
that further group documents based on keywords co-occurrences. For a fair
comparison, we also train Multi-label and Multi-class Neural Network
classifiers as supervised approaches to learn document embeddings on cQA data.
The main difference between our proposal and classifier-based approaches is
that the classifiers view keywords and topics as discrete labels $Y\in R^{d}$,
while the proposed KeyGen2Vec sees keywords as a sequence of discrete
structure $Y\in\Sigma^{*}$.
Summarizing, our main contributions are:
* •
We introduce KeyGen2Vec, a simple Seq2Seq framework that can be utilized as a
general tool to learn document embeddings conditioned on sub-topics
information, such as keywords.
* •
We comprehensively investigate currently available approaches for learning
document embeddings
We empirically show that unsupervised approaches often produce clusters that
are incoherent with hidden semantics or latent structure inferred in target
data.
* •
We empirically show that training Seq2Seq networks on multi-label keyword
generation is analogous to indirectly incorporating label dependency
assumption.
We demonstrate that the proposed KeyGen2Vec is superior than a classifier that
is trained on multi-label classification task with document source as inputs
and keywords as target outputs for the models.
## 2 Background
### 2.1 Community-based Question Answering
Our study focuses on investigating the potential usefulness of state-of-the-
art document embeddings for clustering cQA archives with topics as latent
structural similarity. Most of previous studies on cQA archives are
centralized on the exploration of retrieval issues, such as learning latent
topics for question retrieval Cai et al. (2011), a retrieval framework with
neural network embedding P et al. (2017), hybrid approach of neural network
and latent topic clustering to rank the candidate answers given question Yoon
et al. (2018); and textual similarity problems between questions and their
candidate answers Wang et al. (2010); Tan et al. (2016); Yang et al. (2018).
Whereas, previous works on clustering cQA archives mainly focus on improving
clustering algorithm based on simple feature extractor method (e.g. TfIdf)
Momtazi and Klakow (2009); P (2016). Topical clustering itself is previously
studied by Rosa et al. (2011) to organize large unstructured twitter posts
into topically coherent clusters with hashtags as a means of guidance.
### 2.2 Document Embedding
No | Model | GLO | SUB | SEQ | PRE | TRA | DIM
---|---|---|---|---|---|---|---
1 | KeyGen2Vec | - | ✓ | ✓ | - | ✓ | 200
2 | S2S-AE | - | - | ✓ | - | ✓ | 200
3 | FC-Mult-Cls ∗) | ✓ | - | - | - | ✓ | 100
4 | Sigm-Mult-Lbl | - | ✓ | - | - | ✓ | 100
5 | Softm-Mult-Lbl | - | ✓ | - | - | ✓ | 100
6 | S-BERT | - | - | - | ✓ | - | 768
7 | LDA-Topic | - | - | - | - | ✓ | ∗
8 | D2V-DBOW100 | - | - | - | - | ✓ | 100
9 | D2V-PVDM100 | - | - | - | - | ✓ | 100
10 | Avg-GloVe100 | - | - | - | ✓ | - | 100
11 | Avg-w2v100 | - | - | - | ✓ | - | 100
12 | Avg-GloVe300 | - | - | - | ✓ | - | 300
13 | Avg-w2v300 | - | - | - | ✓ | - | 300
14 | Avg-w2v50-tr-sm | - | - | - | - | ✓ | 50
15 | Avg-w2v50-tr-lg | - | - | - | - | ✓ | 50
16 | Avg-PMI50 | - | - | - | - | ✓ | 50
17 | DC-GloVe100 | - | - | - | ✓ | - | $100^{2}/2$
18 | DC-w2v100 | - | - | - | ✓ | - | $100^{2}/2$
19 | DC-PMI50 | - | - | - | - | ✓ | $50^{2}/2$
20 | DC-w2v50-tr-lg | - | - | - | - | ✓ | $50^{2}/2$
21 | TfIdf | - | - | - | - | ✓ | ∗∗
Table 1: Properties of models compared in this study: GLO: Global semantics
(topic labels) are exposed to the model; SUB: Sub-semantic structures
(keywords) are exposed to the model; SEQ: Model with sequential assumption;
PRE: Use pretrained embedding - no finetuning; TRA: training on observed
corpus; DIM: Dimension of embeddings; ∗ depends on the chosen hyper-parameter
($N$ topics) ; ∗∗ \- depends on the vocabulary size in corpus. ∗) is upper
bound model.
Our study on currently available document embeddings is constrained on
approaches that are domain independent. Since most space is devoted to the
proposed framework and model evaluation, we refer the future readers to the
original papers. Figure 3 shows document embedding approaches that are being
observed in this study, which we broadly divided based on three categories:
(1) Non-distributed (frequency-based) approach; (2) Probabilistic approach;
and (3) Distributed (neural-based) embedding learning. The property of each
embedding model is briefly described in Table 1. For a fair comparison, we
include methods that learn embeddings based on global semantic structure
(GLO), sub-semantic structure (SUB), sequential assumption (SEQ), pretrained
embeddings (PRE), and directly trained embeddings on the target corpus (TRA).
## 3 KeyGen2Vec Framework
Figure 4: Corpus as hierarchical semantic network of documents, keywords, and
topics.
KeyGen2Vec is built based on a hierarchical semantic assumption of a corpus,
as briefly illustrated in Figure 4. The assumption is that documents and their
corresponding keyword labels form sub-structures or _sub-networks_ of latent
topic structure as global semantics. Our work adopts Seq2Seq-based keyphrase
generation introduced by Meng et al. (2017); Chen et al. (2018). While these
preliminary works are motivated by the intuition of Seq2Seq capturing document
semantics, there is currently neither analysis nor empirical evidence to
support the claim that the learnt context representation has encapsulated
latent semantic concept of document source conditioned on its keyword labels.
We hypothesize that Seq2Seq network that has been trained on a keyword
generation task is capable of capturing such latent semantic structure
inferred in data.
(a)
(b)
(c)
Figure 5: Training and Inference stages of the proposed KeyGen2Vec; (a)
Observation set; (b) Training; (c) Prediction in test set; $c$ is latent
topic.
Figure 5 illustrates the reformulation of multi-label keyword generation as
the training objective of KeyGen2Vec. The objective of the task is to
approximate the mapping function $f:X\mapsto Y$ \- where $X$ denotes a
collection of documents and $Y$ denotes the corresponding set of keywords in
observation set. These sets of observations
$\\{(x_{i},\\{y_{i}^{1},y_{i}^{2}\\})\\}$ were transformed into one-to-one
training examples $\\{x_{i},y_{i}^{1}\\},\\{x_{i},y_{i}^{k}\\}$ (fig. 5(b)).
Each training example is represented as sequences, $\mathcal{X}:\Sigma^{*}$
and $\mathcal{Y}:\Sigma^{*}$. In inference stage, to evaluate how well the
trained Seq2Seq capture the semantic structure inferred in $f$, the
parameterized encoder decoder model $g$ was further utilized as a decoder
framework, to generate keywords given unseen documents. Details of
architecture used is further explained in sec.3.1.
### 3.1 Architecture
Our framework is built based on a standard Sequence-to-Sequence (Seq2Seq)
encoder-decoder framework. An encoder first maps a sequence of words to a
vector $c$ – where $c$ serves as the resulting document embedding. Given the
encoded embedding of document source $c$, the decoder then generates target
sequences.
$\displaystyle\texttt{ENC:}x=\\{w_{1},\ldots,w_{T_{x}}\\}\mapsto
c\in\mathbb{R}^{d}$ $\displaystyle\texttt{DEC:}c\in\mathbb{R}^{d}\mapsto
y=\\{w_{1},\ldots,w_{T_{y}}\\}$
#### Encoder
The encoder network is constructed of bidirectional GRU units for mapping
sequence of embedded words $e_{t\cdots T_{x}}$ into a sequence of intermediate
state representation $h_{t\cdots T_{x}}$, which is a concatenation of forward
and backward hidden states $h_{t\cdots
T_{x}}=[\overrightarrow{h},\overleftarrow{h}]$.
$\displaystyle\overrightarrow{h_{t\ldots
T_{x}}}=\overrightarrow{GRU}(e_{t\cdots T_{x}})$
$\displaystyle\overleftarrow{h_{t\ldots T_{x}}}=\overleftarrow{GRU}(e_{t\cdots
T_{x}})$
#### Decoder
The decoder is a neural language model based on forward GRU network that
conditions on context embedding of encoder $c$. $s_{t-1}$ is decoder state at
previous time step. $y_{t-1}$ denotes prediction at $t-1$. Here, $g(.)$
denotes prediction layer (dense network) with softmax activation function.
$\displaystyle s_{t}=\overrightarrow{\rm GRU}(y_{t-1},s_{t-1},c)$
$\displaystyle p(y_{t}|y_{1,\ldots,t-1},x)=g(y_{t-1},s_{t},c)$
#### Attention
We use Bahdanau’s MLP attention scoring function Bahdanau et al. (2014) to
calculate attention score $\alpha$ corresponds to the importance weight of
words in source sequence given embedding of words in target sequence.
$\displaystyle\alpha_{t}=\frac{exp(score(s_{t}^{(j)},h_{t\cdots
T_{x}}^{(i)}))}{\sum_{t}^{T_{x}}{exp(score(s_{t}^{(j)},h_{t\cdots
T_{x}}^{(i)}))}}$
#### Context (Document) Embeddings
The final document embedding $c$ is computed based on weighted sum between a
sequence of encoder states and attention score.
$\displaystyle{c}_{t}=\sum_{t=1}^{T_{x}}{\alpha_{t}h_{t}}$
### 3.2 On Label Dependency Assumption
In our proposed KeyGen2Vec framework, keywords as target variables are
represented as sequences of words. The probability of a particular keyword
chosen in inference stage equals to the joint probability of words in sequence
$p(w_{1:T})$. Softmax activation function is used for projecting decoder
states $s_{t-1}\in R^{d}$ into probabilistic values over $\mathcal{V}$
vocabulary size, $\in R^{\mathcal{V}}$.
$\displaystyle
p(w_{t}|w_{1},\ldots,w_{t-1};\theta)=\texttt{softmax}(Ws_{t-1}+b)$
$\displaystyle p(w_{1:T})=\prod_{t}p(w_{t}|w_{1},\ldots,w_{t-1})$
where softmax function is formally given by:
$\displaystyle\texttt{softmax}(z)_{i}=\frac{e^{z_{i}}}{\sum_{j=1}^{K}e^{z_{j}}}$
By dividing each softmax unit (the probability of each word $w_{t}$ in
vocabulary $\mathcal{V}$) with the sum of all units, the total probability of
words in $\mathcal{V}$ is ensured to be $1$. An increase of one class
probability $p(y_{t}|x,\theta)$ causes the probability of other class
decreases.
We hypothesize that by transforming one-to-many training objective in multi-
label keyword generation task into one-to-one multi-class learning scheme, as
shown in Figure 5), we indirectly incorporate label dependency assumption
during training stage. The trained model treats each sample as mutually
exclusive event via softmax normalization and outputs final prediction
$\hat{y_{t}}=\operatorname*{argmax}p(y_{t}|x,\theta)$. This results in an
indirect dependent assumption between a pair of keyword labels since the
probability of particular pair of keywords given the same document source
$p(y_{1}^{1}|x_{1})$ and $p(y_{1}^{2}|x_{1})$ are dependent each other. By
contrast, standard multi-label learning commonly uses independent Bernoulli
assumption via Sigmoid function, disregarding the dependency between labels.
We further investigate this problem by comparing models with Softmax-based
multi-class classification loss and a standard Sigmoid-based Multi-label
classifier.
## 4 Experiments
### 4.1 Data
We use the following data constructed from cQA archives as gold standard for
learning and evaluation. The three data sets represent data with different
level of difficulties w.r.t. sentence length, noise-level, and number of
unique keywords and topic labels. Toy data is considered to be less noisy and
balance – each sub-class category is composed of sentences and their
paraphrases, forming natural cluster structure. Yahoo! data sets with 5 topic
categories (5-T) and 11 topics (11-T) are considered to be more noisy and
imbalanced due to many non-informative words (e.g. digits, measures, url-
links, query about address or web sources) and domain specific terms (e.g.
medical and automotive terms).
Data set | #Topics | #Keywords | #Train | #Test | Sentence
---|---|---|---|---|---
| (GLO) | (SUB) | | | Length
Wikianswer | NA | NA | 700M | NA | 9 $\pm$ 3
Toy data | 12 | 77 | 1158 | 290 | 9 $\pm$ 3
5-T Yahoo! cQA | 5 | 120 | 23824 | 5957 | 36 $\pm$ 28
11-T Yahoo! cQA | 11 | 179 | 70962 | 17741 | 35 $\pm$ 28
Table 2: Data set; Wikianswer (original corpus of Toy data) is used to train
Word2Vec (w2v50-tr-lg) and PMI method.
#### Toy Data
We created a small set of hand-labelled sentence-keywords-topic pairs (1448
sentences) from WikiAnswer
444http://knowitall.cs.washington.edu/oqa/data/wikianswers/. WikiAnswer is a
data set composed of millions of questions asked by humans, where each
sentence example is accompanied by its paraphrased versions, forming a
paraphrase cluster of one particular question. We use the original WikiAnswer
corpus to train large scale Word2Vec and PMI models incrementally, to inspect
how the scale of data affects model performance. Table. 9 shows a training
example in Toy data. The number of keywords and topic assignment per sentence
were made fixed, i.e. two keywords and one topic for each sentence.
Source: “the sporozoan plasmodium carried from host to host by mosquitoes
causes what serious infection?
---
Keywords: malaria; plasmodium parasite
Topic: virus and diseases
Table 3: Sentence examples in Toy Data set Source: “what is diabetes mellitus?
diabetes mellitus is medical disorder characterized by varying or persistent
hyperglycemia elevated blood sugar levels, $\dots$”
---
Keywords: diabetes; diseases and conditions
Topic: health
Table 4: Sentence example (concatenated cQA pair) in Yahoo! Answer cQA
#### Yahoo! Answer Comprehensive cQA
We reproduce and extend our result on real world cQA archives consisting of
question-answers pairs, accompanied by keywords (tags) and the corresponding
topic. Data was obtained from Yahoo! Answer Comprehensive cQA dataset
555https://webscope.sandbox.yahoo.com/catalog.php, originated from the query
log of Yahoo! Answer. We constructed two corpora: corpus with 5 topic
categorization – referred to as 5-T cQA and corpus with 11 topics – referred
to as 11-T cQA. The training and test examples were constructed by
concatenating each question and the corresponding answers. Table 4 shows a
training example obtained from Yahoo! Answer cQA archives. Likewise, each
document corresponds to a fixed membership: two keywords and one topic
describing the document semantic abstraction.
### 4.2 Training and Hyper-parameters
For training KeyGen2Vec, we use negative log-likelihood loss function with an
adaptive learning rate optimization (Adam Kingma and Ba (2014)),
$lr=0.001,betas=(0.9,0.98),eps=1e-9$. Curriculum learning Bengio et al. (2015)
was employed to sampling whether to use a teacher forcing method during
training stage. For the other models, we refer the reader to the provided code
documentation.
For LDA, trainable Word2Vec, Paragraph Vector, we used Gensim implementation
666https://radimrehurek.com/gensim/. BERT pretrained sentence encoder is taken
from a recent sentence similarity task Reimers and Gurevych (2019). Specific
for Toy data experiment, we trained two Word2Vec models: small scale model
Avg-w2v50-tr-sm was trained on the constructed set of Toy data; and large
scale model Avg-w2v50-tr-lg was trained incrementally on WikiAnswer (the
original large scale corpora of Toy data) – to inspect how model performance
differs based on the scale of data. Classifiers (Multi-class and Multi-label)
were constructed from fully-connected network (MLP) since we do not find a
significant performance differences between using different types of networks
(i.e. dense, convolutional, and recurrent). We trained two types of Multi-
label classifiers (Sigm-Cls and Softm-Cls) to inspect the effect of
incorporating label dependency in multi-label learning.
### 4.3 Clustering as Evaluation
We use K-Means clustering 777scikit-learn.org/../sklearn.cluster.KMeans.html
to evaluate the quality (clusterability) of document embeddings in this study
(table 1). The hyper-parameter choices of K-means is kept as minimum as
possible (init=’random’, n_clusters=$K$, n_init=$10$, max_iter=$50$). This is
to make sure that the clustering is not overly parameterized, which can
obscure the actual quality of the learnt embeddings. Given the actual global
semantic classes (topic labels) $\mathbb{C}$ in the current observed corpora
and the predicted classes $\Omega$ from K-Means method, we employ Purity,
Normalized Mutual Information (NMI), and F1-score Manning et al. (2008)
metrics to objectively measure whether the resulting clustering $\Omega$ can
recreate or approximate the exact classes $\mathbb{C}$.
### 4.4 $\chi^{2}$ Feature selection
Dining Out | Health | Travel | Cars
---|---|---|---
hamburger | medicine | trip | jeep
taco | heart | map | vehicle
sandwich | symptom | disney | auto
buffet | virus | vacation | manual
cafe | treatment | ticket | nisan
Table 5: Example of influential words per topic category – selected based on
$\chi^{2}$ feature selection method on 5-T Yahoo! cQA.
We employ feature selection based on $\chi^{2}$ method Manning et al. (2008)
to select $N-$ most influential words per topic category. Each training
example is then represented as Bag-of-Influential words with
$N\in\\{20,50,100,250\\}$ for Toy data and
$N\in\\{20,50,100,250,500,1000,2500\\}$ for Yahoo! cQA data. The larger the
size of influential words per category, the more noises preserved in the
training data. This experiment was conducted to investigate: (1) the effect of
noises on the clusterability of embeddings; (2) the effect of incorporating
label dependency via Softmax-based loss on the clusterability of embeddings.
(a) Upper-bound
(b) KeyGen2Vec
(c) Sigm-MultLbl
(d) Softm-MultLbl
(e) Avg-w2v-tr
Figure 6: Example of Clustering Visualization on 5-T Yahoo! Q&A.
## 5 Results and Discussion
We summarize our empirical findings as follows:
#### KeyGen2Vec outperforms multi-label classifiers
Based on the clustering performance on three data sets, as shown in Table 6-8,
we demonstrate that although the model does not exploit the actual topic
labels during training stage, the proposed KeyGen2Vec has a capability of
preserving topical proximity in latent space, outperforming its counterparts –
models trained on multi-label classifiers (Sigm-Mult-Lbl and Softm-Mult-Lbl).
Approach | Purity | NMI | F1-score
---|---|---|---
| All | Test | All | Test | All | Test
KeyGen2Vec | 0.734 | 0.726 | 0.769 | 0.774 | 0.631 | 0.614
S2S-AE | 0.288 | 0.317 | 0.213 | 0.269 | 0.167 | 0.164
FC-Mult-Cls∗ | 0.961 | 0.938 | 0.968 | 0.940 | 0.947 | 0.910
Sigm-Mult-Lbl | 0.590 | 0.605 | 0.629 | 0.649 | 0.435 | 0.438
Softm-Mult-Lbl | 0.659 | 0.687 | 0.673 | 0.710 | 0.532 | 0.556
BERT | 0.636 | 0.644 | 0.680 | 0.678 | 0.527 | 0.502 2
LDA-Topic | 0.474 | 0.551 | 0.472 | 0.584 | 0.368 | 0.454
D2V-DBOW100 | 0.179 | 0.207 | 0.049 | 0.115 | 0.111 | 0.100
D2V-PVDM100 | 0.171 | 0.206 | 0.035 | 0.104 | 0.102 | 0.094
Avg-GloVe100 | 0.648 | 0.649 | 0.656 | 0.672 | 0.507 | 0.485
Avg-W2V100 | 0.686 | 0.679 | 0.694 | 0.689 | 0.535 | 0.489
Avg-GloVe300 | 0.668 | 0.693 | 0.688 | 0.719 | 0.523 | 0.531
Avg-W2V300 | 0.655 | 0.654 | 0.685 | 0.688 | 0.511 | 0.475
Avg-W2v50-tr-sm | 0.208 | 0.243 | 0.076 | 0.152 | 0.116 | 0.112
Avg-W2v50-tr-lg | 0.643 | 0.651 | 0.671 | 0.673 | 0.538 | 0.506
Avg-PMI50 | 0.305 | 0.325 | 0.212 | 0.290 | 0.180 | 0.176
DC-GloVe100 | 0.374 | 0.326 | 0.351 | 0.321 | 0.222 | 0.185
DC-W2V100 | 0.407 | 0.312 | 0.447 | 0.325 | 0.225 | 0.180
DC-PMI50 | 0.282 | 0.313 | 0.180 | 0.266 | 0.162 | 0.167
DC-W2V50-tr-lg | 0.555 | 0.546 | 0.545 | 0.567 | 0.376 | 0.359
TfIdf | 0.564 | 0.611 | 0.615 | 0.649 | 0.377 | 0.401
Table 6: Clustering Evaluation on Toy Data. FC-Mult-Cls∗ is an upper bound
model. Scores were calculated based on average score in 10 iterations of
K-means clustering. Higher is better.
Approach | Purity | NMI | F1-score
---|---|---|---
| All | Test | All | Test | All | Test
KeyGen2Vec | 0.801 | 0.784 | 0.657 | 0.603 | 0.668 | 0.630
S2S-AE | 0.337 | 0.341 | 0.012 | 0.014 | 0.241 | 0.242
FC-Mult-Cls∗ | 0.970 | 0.853 | 0.903 | 0.721 | 0.957 | 0.794
Sigm-Mult-Lbl | 0.763 | 0.738 | 0.567 | 0.522 | 0.635 | 0.606
Softm-Mult-Lbl | 0.772 | 0.744 | 0.579 | 0.533 | 0.657 | 0.622
BERT | 0.329 | 0.325 | 0.005 | 0.007 | 0.278 | 0.283
LDA-Topic | 0.541 | 0.583 | 0.222 | 0.269 | 0.479 | 0.488
D2V-DBOW100 | 0.335 | 0.335 | 0.018 | 0.019 | 0.245 | 0.246
D2V-PVDM100 | 0.337 | 0.339 | 0.012 | 0.012 | 0.274 | 0.276
Avg-GloVe100 | 0.495 | 0.502 | 0.122 | 0.133 | 0.330 | 0.349
Avg-W2V100 | 0.531 | 0.533 | 0.202 | 0.207 | 0.371 | 0.379
Avg-GloVe300 | 0.483 | 0.498 | 0.117 | 0.132 | 0.334 | 0.352
Avg-W2V300 | 0.463 | 0.462 | 0.117 | 0.117 | 0.342 | 0.347
Avg-W2V50-tr | 0.609 | 0.621 | 0.318 | 0.327 | 0.477 | 0.491
Avg-PMI50 | 0.339 | 0.343 | 0.024 | 0.026 | 0.276 | 0.273
DC-GloVe100 | 0.359 | 0.361 | 0.031 | 0.034 | 0.296 | 0.311
DC-W2V100 | 0.426 | 0.413 | 0.117 | 0.105 | 0.356 | 0.353
DC-PMI50 | 0.326 | 0.329 | 0.018 | 0.019 | 0.295 | 0.292
DC-W2V50-tr | 0.329 | 0.328 | 0.012 | 0.013 | 0.343 | 0.342
TfIdf | 0.357 | 0.383 | 0.047 | 0.077 | 0.304 | 0.308
Table 7: Clustering Evaluation on 5-T Yahoo! cQA.
Approach | Purity | NMI | F1-score
---|---|---|---
| All | Test | All | Test | All | Test
KeyGen2Vec | 0.841 | 0.797 | 0.723 | 0.662 | 0.655 | 0.603
S2S-AE | 0.306 | 0.303 | 0.037 | 0.036 | 0.135 | 0.133
FC-Mult-Cls∗ | 0.862 | 0.768 | 0.774 | 0.643 | 0.717 | 0.564
Sigm-Mult-Lbl | 0.729 | 0.723 | 0.545 | 0.535 | 0.487 | 0.479
Softm-Mult-Lbl | 0.739 | 0.718 | 0.589 | 0.538 | 0.508 | 0.493
BERT | 0.274 | 0.229 | 0.016 | 0.016 | 0.211 | 0.229
LDA-Topic | 0.518 | 0.534 | 0.275 | 0.300 | 0.304 | 0.293
D2V-DBOW100 | 0.279 | 0.278 | 0.022 | 0.024 | 0.133 | 0.133
D2V-PVDM100 | 0.281 | 0.280 | 0.019 | 0.018 | 0.167 | 0.162
Avg-GloVe100 | 0.435 | 0.434 | 0.193 | 0.199 | 0.212 | 0.214
Avg-W2V100 | 0.407 | 0.409 | 0.155 | 0.159 | 0.198 | 0.201
Avg-GloVe300 | 0.427 | 0.425 | 0.188 | 0.190 | 0.213 | 0.205
Avg-W2V300 | 0.448 | 0.449 | 0.216 | 0.221 | 0.216 | 0.218
Avg-W2V50-tr | 0.551 | 0.552 | 0.309 | 0.311 | 0.294 | 0.287
Avg-PMI50 | 0.283 | 0.283 | 0.032 | 0.035 | 0.138 | 0.139
DC-GloVe100 | 0.303 | 0.307 | 0.053 | 0.059 | 0.169 | 0.182
DC-W2V100 | 0.328 | 0.321 | 0.088 | 0.089 | 0.203 | 0.209
DC-PMI50 | 0.273 | 0.271 | 0.018 | 0.019 | 0.161 | 0.160
DC-W2V50-tr | 0.303 | 0.301 | 0.043 | 0.043 | 0.203 | 0.203
TfIdf | 0.306 | 0.303 | 0.037 | 0.037 | 0.135 | 0.133
Table 8: Clustering Evaluation on 11-T Yahoo! cQA.
#### Pretrained unsupervised embeddings are more random on noisy datasets
Specific to unsupervised pretrained models (S-BERT, Wikipedia2Vec Avg-w2v100
and GloVe Avg-GloVe300), the results show that while these models perform well
in Toy data, their performance degrades in the other two data sets. This
indicates that the embeddings generalized from the source domain in which the
model is trained on are not sufficient for the target corpora (Yahoo! data).
Fine tuning the models or expanding the vocabulary, however, is beyond the
scope of this study.
This finding specifically challenges the prior belief stating an _off-the-
shell_ encoder that has been trained on large scale data or multi-tasks
learning (e.g. Skip-Thought vector, Universal Sentence Encoder, Sentence-BERT,
pretrained word embeddings) can produce highly generic embedding that performs
well in practice. We argue that a generic pretrained embedding may best fit
for tasks that are less noisy and complement to the pretrained source domain,
exemplified in our Toy data experiment.
#### Trained unsupervised embeddings rely on large-scale data
We observe that unsupervised neural embeddings that are trained on the
observed corpora (e.g. Word2Vec, Doc2Vec, Seq2Seq Autoencoder) seemingly rely
on the scale of data. See how small scale Word2Vec (Avg-w2v50-tr-sm) results
in a notably low performance on Toy data (similar to Autoencoder S2S-AE and
Doc2Vec), as compared to large scale Word2Vec (Avg-w2v50-tr-lg).
We argue that the low quality of unsupervised embeddings in the current study
is due to the models mainly depend on local information in document contents –
there is no strong assumption on differentiating salient features (words)
w.r.t. global semantic aspects, which may hinder their direct utilization on a
subsequent predictive analytics tasks. Specific to LDA topic model, we argue
that their low performance in the current task is due to no strong assumption
on distinguishing between local (keywords - or more specific document theme)
and global (more general) latent topics.
#### The effects of noises on embedding quality
We argue that the main reason why the current clustering task is challenging
for all observed models, specifically unsupervised ones is mainly due to the
_noisy_ characteristic of cQA archives. For instance, topic ”Health” and
“Dining out” may both contain queries about dietary or source of healthy food.
Topic “Cars”, “Travel”, “Local Business” may all contain queries about car
rental and service. We empirically show that in a clean scenario – where
training examples only contain $N$-most influential words w.r.t. topic
category (Toy data experiment in fig.7(a)) unsupervised methods sufficiently
perform well. The performance, however, degrades in the occurrence of noises
(larger pre-selected feature size). By contrast, KeyGen2Vec can maintain its
considerably high performance (fig.7(a)-7(c)) regardless the presence of
noises. This indicates the exposure of keywords as sub-topical information
benefits the model to obtain high quality embeddings.
#### Problem reformulation improves the expressiveness of embeddings
Redefining one-to-many multi-label learning into one-to-one multi-class
learning scheme via Softmax normalization, which we argue is analogous to
indirectly incorporating label dependency (sec. 3.2), benefits KeyGen2Vec and
Multi-label learning in the current study, resulting in a more accurate
embedding (higher $F_{1}$-score, in table 6-8 and fig.7(a)-7(c)).
(a) Toy Data
(b) 5-T Yahoo! Answer Data
(c) 11-T Yahoo! Answer Data
Figure 7: Effect of noises on F1-Score; _best viewed in color. The larger pre-
selected feature set size (x-axis), the more noisy the documents are._
## 6 Conclusion
We extensively investigate document embedding approaches for topical
clustering of cQA archives. We show current limitations of unsupervised
embeddings on dealing with noisy articles, indicating the need of
incorporating strong assumption either on learning approach or data. Our
empirical results highlight the capability of the proposed KeyGen2Vec in
preserving topical proximity in latent space via multi-label multi-class
learning.
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## Appendix A Document Embedding Methods
### A.1 Non-distributed approach
#### Frequency-based (TfIdf)
_TfIdf_ Salton et al. (1975) – also referred to as “bag-of-words” model, is
commonly used as a standard approach to transform text document into numerical
representation. A document is represented as vector of semantics, where each
dimension reflects a degree of importance (based on relative frequency-based
weight) of a particular word $w$ in the corresponding document $d_{i}$ and
across documents in corpus $d_{i}\in D$, $w\in\mathcal{V}$ – where
$\mathcal{V}$ denotes vocabulary size of the corpus.
### A.2 Probabilistic approach
#### Probabilistic Topic Model
Latent Dirichlet Allocation (LDA) Blei et al. (2010); Blei (2012) – commonly
known as Topic Model, is a probabilistic mixture model that views a document
as a collection of un-ordered words (“bag-of-words”). The document is
represented as a mixture membership of topics - where each dimension of the
document vector corresponds to a probabilistic distribution of topic-$c_{j}$
in the corresponding document, $c_{j}\in\mathbb{C}$. Each topic is represented
as a vector of most probable words defining the topic.
### A.3 Distributed approach
#### Mean Embedding
– a bottom-up approach. A document is viewed as a collection of bag-of-word
embeddings. The word-level embedding is learnt through non-linear mapping of
neural network architecture Mikolov et al. (2013b, a). The document
representation is computed based on averaging word-level representations occur
in the corresponding document. We acquire word embedding $w\in\mathcal{V}$
through both pretrained models, i.e. Word2Vec pretrained on Wikipedia corpus
Yamada et al. (2020), GloVe vector Pennington et al. (2014), and trainable
model – i.e. by training the Word2Vec model Mikolov et al. (2013b, a) and
Pointwise Mutual Information (PMI) - based embedding on the current corpus. In
table 1, the pretrained models of word embedding are represented by {Avg-
GloVe100, Avg-w2v100, Avg-GloVe300, Avg-w2v300}. The trainable models are
represented by {Avg-w2v50-tr-sm, Avg-w2v50-tr-lg, Avg-PMI50}.
#### Document Covariance Matrix
– a bottom-up approach, represented by {DC-GloVe100, DC-w2v100, DC-PMI50,
DC-w2v50-tr-lg} in table 1. A document is represented as the multivariate
gaussian embedding (covariance matrix) of its word-level representation
Nikolentzos et al. (2017); Torki (2018). To construct a document covariance
matrix, we use two types of word-level representations: (1) Pointwise Mutual
Information (PMI) Levy and Goldberg (2014) that measures the association of
words $w$ and their context $c$ by calculating the cooccurrence of words and
their neighboured words in sentence or document; (2) Word embedding Mikolov et
al. (2013b) that learn distributed representation of words-context words via a
neural network architecture - or log linear mapping function (_here_ , we use
both pretrained and trainable word embedding models).
#### Paragraph Vector
– represented by {D2V-DBOW100, D2V-PVDM100} in table 1. Document and the
corresponding words are mapped into a share vector space – where the objective
is to predicting target words, given document and context words in Distributed
Memory model (PV-DM); and to predict context words, given a document in
Distributed Bag-of-Words model (PV-DBOW) Le and Mikolov (2014).
#### Multi-class classifier
We utilize neural network with dense connections (MLP, denoted as FC-Mult-Cls
in table 1), representing an upper-bound model in this study. Discrete global
class structure in the observed data sets $\mathbb{C}$ is exposed as training
objective to learn and condition document features from this MLP-based model.
The network architecture is composed of an embedding layer, dropout networks,
a pooling layer as a flattening mechanism, and a stack of two fully-connected
(FC) networks. The objective of the study, thus is to find the best feature
extractor that closer to the quality of features based Multi-class classifier.
#### Multi-label classifier
To provide a _fair_ comparison with the proposed Seq2Seq framework, we utilize
a Multi-label classifier based on dense networks (FC-Mult-Lbl in table 1) to
model document features condition on multiple dependent labels. Compared to
Multi-Class classifier (FC-Mult-Cls) that holds independent assumption of
$X\mapsto Y$ mapping tasks, Multi-label classifier sees the tasks as mutually
inclusive, as such one document can correspond to multiple labels (e.g. tags,
keywords). Compared to Seq2Seq that learn to predict a set of sequences based
on tree-based mutually dependent structure $Y\in\Sigma^{*}$ , the Multi-label
classifier estimates the probability of multiple classes for one source
instance independently (_sigmoid_ , instead of normalized _softmax_
probability outputs).
#### Pretrained Model: BERT
BERT Devlin et al. (2019) is the most recent language representation model
surprisingly performed well in diverse language understanding benchmark -
indicating the network has a capacity to capture structural information from
natural language data Xu et al. (2019); Jawahar et al. (2019); Reimers and
Gurevych (2019). Unlike its deep architecture counterparts commonly composed
of recurrent network to hold the main assumption of sequential data (e.g.
Seq2Seq), BERT network is mainly composed of dense connections - referred to
as “self-attention” network. We utilize a pretrained BERT as static universal
sentence encoder – i.e. to transform documents into vectors unsupervisedly
without fine-tuning, assuming the global and sub semantic structure is
_unobserved_. We use the latest implementation of BERT for sentence embedding
(S-BERT) Reimers and Gurevych (2019), which has shown a better generalization
performance than vanilla BERT Devlin et al. (2019).
## Appendix B Seq2Seq Networks
### B.1 Attention Network
We use Bahdanau’s concat attention scoring function Bahdanau et al. (2014),
illustrated in fig. 8, to calculate attention weights $\alpha\in R^{T_{x}}$ of
encoder state representation conditioned by decoder output state
representation.
Figure 8: Attention network Bahdanau et al. (2014).
### B.2 Teacher forcing
A common strategy to train a recurrent-based Seq2Seq model in a generation
task is incorporating teacher forcing, i.e. by exposing the actual or expected
output $Y_{t<T_{y}}$ at the current decoding time step $t<T_{y}$, rather than
the output generated by the network ($s_{t<T_{y}}$). The drawback, however,
during evaluation stage the network only relies on its own prediction from
previous time steps, resulting a performance degradation referred as
“training-evaluation loss mismatch” in Wiseman and Rush (2016). In this study,
we employ curriculum learning Bengio et al. (2015), i.e. approach to sampling
whether to use teacher forcing or not during training stage in the current
sequence prediction problem. Here, the probability of incorporating teacher
forcing (ratio of teacher forcing) is calculated based on inverse sigmoid
function of scheduled sampling (after $xx-$ batch examples seen during
training).
## Appendix C Evaluation Metrics
#### Clustering Evaluation
A normalized mutual information (NMI) is used to measure whether the
clustering method can recreate the true or exact structure of the observed
data. We use the following metrics to evaluate the structure representation in
a clustering task, in addition to $F_{1}$-score and purity measure.
$NMI(\Omega,\mathbb{C})=\frac{I(\Omega,\mathbb{C})}{0.5(H(\Omega)+H(\mathbb{C}))}$
(1)
Where $I(\Omega,\mathbb{C})$ measures the mutual information between the
formed cluster membership and exact class. And, $H(\Omega),H(\mathbb{C})$
denotes entropy of the formed structure and exact class respectively.
$I(\Omega,\mathbb{C})=\sum_{k}\sum_{j}P(\omega_{k}\cap
c_{j})\log\frac{P(\omega_{k}\cap c_{j})}{P(\omega_{k})P(c_{j})}$ (2)
$H(\mathbb{C})=-\sum_{k}P(\omega_{k})\log P(\omega_{k})$ (3)
$P(\omega_{k}),P(c_{j}),P(\omega_{k}\cap c_{j})$ are the probability of
document feature being in cluster $\omega_{k}$, class $c_{j}$, and both
memberships.
#### Purity
Each cluster is assigned to the class which has the most frequent members in
the cluster, as such the purity of clusters is computed by
$\texttt{purity}(\Omega,\mathbb{C})=\frac{1}{N}\sum_{k}\texttt{max}_{j}|\omega_{k}\cap
c_{j}|$. $\Omega=\\{\omega_{1},\omega_{2},\ldots,\omega_{k}\\}$ denotes a set
of $k$ clusters, while $\mathbb{C}=\\{c_{1},c_{2},\ldots,c_{j}\\}$ is a set of
$j$ actual classes. A perfect clustering has a purity of $1$. Note: the purity
metric disregards the _uniqueness_ of the cluster since it is computed based
only on the majority class and number of members of the majority class.
#### Normalized Mutual Information (NMI)
NMI or a Mutual Information-based metric measures the amount of information
needed to predict the actual class of a cluster, given a knowledge about
documents in that cluster.
$\displaystyle
NMI(\Omega,\mathbb{C})=\frac{I(\Omega,\mathbb{C})}{[H(\Omega)+H(\mathbb{C})]/2}$
$\displaystyle I(\Omega,\mathbb{C})=\sum_{k}\sum_{j}P(\omega_{k}\cap
c_{j})log\frac{P(\omega_{k}\cap c_{j})}{P(\omega_{k})P(c_{j})}$
where $I$ is mutual information between the predicted clusters $\Omega$ and
the actual classes $\mathbb{C}$, which is measured based on the overlapping
document membership between clusters and actual classes. $H$ denotes an
entropy measure $H(P)=-\sum_{x\in X}P(x)log_{2}P(x)$.
#### F1-Score
We measure F1-score (harmonic mean of Precision and Recall) of document
clustering based on the notion of how a pair of documents (points in latent
space) is assigned into clusters Manning et al. (2008). True Positive (TP)
assignment implies that two _similar_ documents are assigned to the same
cluster. True Negative (TN) assignment refers to the assignment of two
_dissimilar_ documents to different clusters. False Positive (FP) assigns two
dissimilar documents into the same cluster, while False Negative (FN) assigns
two similar documents into different clusters. Precision (P) and Recall (R)
are then computed based on $P=TP/(TP+FP)$ and $R=TP/(TP+FN)$; while
$F_{1}$-score is a harmonic mean of both metrics $F_{1}=(2PR)/(P+R)$.
## Appendix D Data
Sentence | Keywords | Topic
---|---|---
| (SUB) | (GLO)
“what was so important about the battle of quebec” | battle of quebec; american revolutionary war | history
“who were the commanders that died in the battle of quebec in 1759” | battle of quebec; american revolutionary war | history
“the sporozoan plasmodium carried from host to host by mosquitoes causes what serious infection” | malaria; plasmodium parasite | virus and diseases
“plasmodium is a malaria causing sporozoan which is transmited by mosquito” | malaria; plasmodium parasite | virus and diseases
Table 9: Sentence examples in Toy Data set
“diabetes mellitus is medical disorder characterized by varying or persistent
hyperglycemia elevated blood sugar levels, especially after eating. all types
of diabetes mellitus share similar symptoms and complications at advanced
stages. hyperglycemia itself can lead to dehydration and ketoacidosis. longer
term complications include cardiovascular disease doubled risk, chronic renal
failure it is the main cause for dialysis, retinal damage which can lead to
blindness, nerve damage which can lead to erectile dysfunction impotence,
gangrene with risk of amputation of toes, feet, and even legs”
---
keywords: diabetes; diseases and conditions
topic: health
Table 10: Sentence examples in Yahoo! Answer Q&A
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 9: Corpus visualization as graph network (nodes represent documents,
keywords, and topic categorization); $(a-b)$ Toy data; $(c-h)$ 5-T Yahoo! cQA
data.
## Appendix E Corpus Visualization
Figure 9 shows a corpus visualization as graph network where nodes represent
document sources that are connected by keywords as sub-structure and topic
labels as global structure.
## Appendix F More Results
### F.1 KeyGen2Vec as Embedding Models
“Under which condition KeyGen2Vec is better than the other embedding models?”
#### On Scalability Aspect of Model
Compared to unsupervised methods and pretrained models, KeyGen2Vec has shown a
consistent good performance on capturing semantic structure in data,
outperforming the other models, regardless the number of training examples.
_Note_ : This specifically holds on scale-free network (Figure 9 and Figure
LABEL:fig:net_define), i.e. the growth of the network is independent with the
underlying structure of the network. While word-level embedding and Seq2Seq
for autoencoding suffer on small set of training examples (Toy data, where the
evaluation result is shown in Table 6), Seq2Seq for keyword generation (STS-
KG) shows an ability to learn useful features even in small data set,
indicating the model can be utilized as feature extractor for both small and
large scale data.
#### On Semantic Structure Inferred in Data
While sub-structure can promote the learning of _latent_ global structure
inferred in data, shown in our empirical results, an overlapping sub semantic
structures exemplified by the two real world data sets in the current study,
as shown in Figure 9(c)-9(h), may potentially introduce noises in the
learning. This type of loss is also shown on the extracted features from both
models that learn representation from sub semantic structures: Seq2Seq for
keyword generation STS-KG (Figure 10(a)) and multi-label classifier (Figure
10(b)). The overlapping points in different colors (Figure 10) represent
documents in three category topic labels: ’dining out’, ’travel’, ’local
business’ that shares common set of keywords (‘‘London’’, ‘‘UK’’).
(a)
(b)
Figure 10: Loss in approximating global structure is mainly due to overlapping
sub-structures; (a) STS-KG; (b) Multi-Label Classifier
(a)
(b)
(c)
(d)
Figure 11: (a) Average silhouette score on three data sets; (b) Multi-Class
Classifier on 5-T Yahoo!; (c) STS-KG on 5-T Yahoo!; (d) Multi-Label Classifier
on 5-T Yahoo!;
#### KeyGen2Vec vs. Seq2Seq Autoencoding
The clustering evaluation results presented in table 6-8 notably shows that
Seq2Seq autoencoder (STS-AE) cannot adequately capture the implicit global
semantic structure of data. Similar to the small version of word embedding-
based model in the current work (Avg-w2v-tr-sm table 1), we argue that an
autoencoder heavily relies on the “goodness” in the data, indicating the model
may be useful if meaningful sentences that promote a content-based semantic
structure learning are available abundantly as training examples. If such case
is not available, the performance of feature extractor may be improved by
inferring bias to the model architecture or training objective. For instance,
instead of an autoencoding task, the network can be trained to predict the
semantic structure of the document source, exemplified by Seq2Seq framework
for keyword generation in the current study.
#### KeyGen2Vec vs. Multi-Label Classifier
Figure 11 shows the comparison of cluster separability of the three model
based on silhouette measure: (1) Multi-class classifier as an upper bound
model; (2) Seq2Seq for keyword generator; and (3) Multi-label classifier. The
vertical dashed lines in fig 11(b)-11(d) represents the average score of all
points in the resulting clusters, while the size of the bar plot represents
the size of cluster. Silhouette score was calculated based on the average
distances for all points in the same cluster and the average distances for
points in the closest cluster. The score values range from $[-1,1]$ – where
score$=-1$ indicates sample is assigned to the wrong cluster, average score of
$0$ indicates that the inter-cluster distance is small, and average score of
$1$ indicates the inter-cluster distance is large enough to form separable
clusters. Intuitively, the silhouette score is expected to close to $1$ for a
good quality of clusters. A good quality of clusters infers a good quality of
extracted features.
Overall, without having the exact classes to evaluate the quality of document
clustering, Both features from STS-KG and Multi-Label classifier can
approximate the cluster separability of an upper bound model, indicating both
models has a capacity to extract “good” features according to the current
definition of global semantic structure in the current study. Although the
clustered features based on Seq2Seq (STS-KG) has a slightly higher score than
multi -label classifier clustered features, the actual quality of extracted
features by Seq2Seq outperformed multi-label classifier features when exact
classes are projected on the resulting clusters.
#### KeyGen2Vec vs. LDA Topic Model
While KeyGen2Vec has outperformed all models in the current study, this
performance comes at cost of providing sub semantic structure information. As
a comparison, we discuss the performance trade-offs of LDA Topic Model as an
alternative model for unsupervised approach, in addition to the aforementioned
unsupervised and pretrained models (sec. LABEL:sec:unsup).
Figure 12: Deciding hypothesis space (number of topics) Figure 13: LDA
Performance on different preprocessing steps
While LDA topic model in the current study (results in table 6-8) does not
show an impressive performance w.r.t. the semantic quality of the learnt
clusters, the model promotes a consistent quality as document feature
extractor among three data sets, similar to a trainable word embedding-based
model (Avg-w2v-tr). Nevertheless, deciding a hypothesis space of topic model
is non-trivial. Too small $K$ number of topics results in a very broaden
topics (words corresponds to the topic semantic definition is too general).
While, too large $K$ results in a repetitive topics – i.e. different topics
contain an overlapping set of words. While the hypothesis space of topic model
can be evaluated by coherence measure during training, the measure does not
necesarily correlate to the actual quality of learnt features or an
interpretability aspect of the resulting features in topic space , illustrated
in fig. 12. We refer the “interpretability” here as a degree to which the two
different clusters are adequately far or _separable_.
The performance of topic model in capturing document semantic structure also
highly depends on heavy preprocessing steps. Figure 13 shows how the quality
of extracted features varies, depending on (a) whether the stopword and noisy
words have been removed (e.g. removing question words and non meaningful
abbreviation or short character in dat_sw); (b) whether data is lemmatized
(dat_lem); (c) document length (fuldat_lem); or (d) a balance distribution of
topic categorization labels (fuldat_lem_ba). If such knowledge (e.g. POS tag
linguistic structure in lemmatization step) is not available in particular
data or language, the performance can be expected to degrade.
|
# On the noncommutative fields method in the three-dimensional Yang-Mills
theory
J. R. Nascimento, A. Yu. Petrov, E. O. Silva Departamento de Física,
Universidade Federal da Paraíba
Caixa Postal 5008, 58051-970, João Pessoa, Paraíba, Brazil
<EMAIL_ADDRESS>
###### Abstract
We apply the noncommutative fields method to the three-dimensional non-Abelian
gauge theory. We find that, first, implementing the noncommutativity between
the canonical momenta implies in generation of the non-Abelian Chern-Simons
term, second, if one introduces the noncommutativity between the field
operators, the higher derivative terms would arise.
The noncommutativity is treated now as a fundamental quantum property of the
space-time geometry. Beside of the known scheme of introducing the
noncommutativity via the Moyal product SW , an alternative one was recently
developed, that is, so-called noncommutative fields method, in which, instead
of the spacetime coordinates, fields themselves are noncommutative, thus, the
canonical commutation relations turn out to be deformed Gamb0 . This method
turned out to be a new method of generating the Lorentz-breaking correction
after it was shown that the known Lorentz-breaking term initially introduced
by Jackiw and Kostelecky JK naturally emerges within this formalism Gamb1 .
Further, the non-Abelian analog of this term was generated via the
noncommutative fields method Gamb2 , and in our paper ourgra , this method was
applied to generate the Lorentz symmetry breaking in the linearized gravity.
At the same time, the situation in three-dimensional space-time is different.
Indeed, we have shown in NPR that application of the noncommutative field
method to three-dimensional electrodynamics, instead of the Lorentz-breaking
terms generates a gauge invariant mass term, that is, the Chern-Simons term,
with the mass turns out to be proportional to the noncommutativity parameter
NPR . We would like to notice that unlike of common perturbative approach (see
f.e. Redlich ), the essence of the noncommutative fields method consists in
possibility to generate new terms without coupling to extra matter fields. The
very natural development of this study would consist in generalization of the
noncommutative fields method for the non-Abelian case, where it is natural to
expect that not only quadratic term but also the interaction term for the
gauge field will arise. Different aspects of the Chern-Simons term, both in
Abelian and non-Abelian cases, such as non-trivial topological nature of this
term DJT and quantization of the Chern-Simons coefficient quCS were studied.
In other worlds, it is natural to expect that in this case, the three-
dimensional non-Abelian Chern-Simons term
$\displaystyle L_{CS}=\frac{1}{2}m\epsilon^{\mu\nu\lambda}{\rm
tr}(A_{\mu}\partial_{\nu}A_{\lambda}+\frac{2}{3}gA_{\mu}A_{\nu}A_{\lambda})$
(1)
will be generated. From the other side, we are planning to generalize the
noncommutative field method by introducing of a more general deformation of
the canonical algebra which in principle could imply in arising of the
Lorentz-breaking terms. These problems are considered in the paper.
Let us start our study of the three-dimensional Yang-Mills theory, whose
action is
$\displaystyle S=-\frac{1}{4}\int d^{3}x{\rm tr}F_{mn}F^{mn},$ (2)
with the $F_{mn}=F_{mn}^{a}T^{a}$ is a stress tensor constructed on the base
of the Lie-algebra valued gauge field $A_{m}(x)=A_{m}^{a}(x)T^{a}$ (with ${\rm
tr}(T^{a}T^{b})=\delta^{ab}$, and $[T^{a},T^{b}]=f^{abc}T^{c}$):
$\displaystyle
F_{mn}^{a}=\partial_{m}A_{n}^{a}-\partial_{n}A_{m}^{a}+gf^{abc}A^{b}_{m}A^{c}_{n},$
(3)
so, the Lagrangian, after splitting of the indices into time (zero) and space
ones (denoted by $i,j,k$) looks like
$\displaystyle L=-\frac{1}{4}\int
d^{3}xF_{mn}^{a}F^{mn\,a}=-\frac{1}{4}F^{a}_{ij}F^{a}_{ij}+\frac{1}{2}(\dot{A}^{a}_{i}-\partial_{i}A^{a}_{0}+gf^{abc}A^{b}_{i}A^{c}_{0})^{2}.$
(4)
Let the signature be $diag(-++)$. First, we carry out the canonical
quantization of the theory. The canonical momentum of the theory is
$\displaystyle p^{a}_{m}=\frac{\partial
L}{\partial\dot{A}^{a\,m}}=F^{a}_{0m}.$ (5)
It is clear that $p^{a}_{0}=0$, so, we find the primary constraint
$\Phi^{(1)a}=p^{a}_{0}$. The velocities can be expressed as
$\displaystyle\dot{A}^{a}_{i}=p_{i}^{a}-gf^{abc}A^{b}_{0}A^{c}_{i}+\partial_{i}A_{0}^{a}.$
(6)
Thus, the Hamiltonian is
$\displaystyle
H=p^{a}_{i}\dot{A}^{a}_{i}-L=\frac{1}{2}p^{a}_{i}p^{a}_{i}+\frac{1}{4}F^{a}_{ij}F^{a}_{ij}+p_{i}^{a}(-gf^{abc}A^{b}_{0}A^{c}_{i}+\partial_{i}A_{0}^{a}).$
(7)
The secondary constraint looks like
$\displaystyle\Phi^{(2)b}\equiv\Delta^{a}=\\{p^{a}_{0},H\\}=-\frac{\partial
H}{\partial
A_{0}^{a}}=-(\partial_{i}p_{i}^{a}+gf^{abc}A^{b}_{i}p_{i}^{c})\equiv-{\cal
D}^{ab}_{i}p^{b}_{i}.$ (8)
This constraint evidently generates the gauge transformations:
$\displaystyle\delta A_{i}^{a}$ $\displaystyle=$
$\displaystyle\\{A_{i}^{a},\int
d^{2}\vec{x}\xi^{b}(\vec{x})\Delta^{b}(\vec{x})\\}=\partial_{i}\xi^{a}(\vec{x})+gf^{abc}A^{b}_{i}(\vec{x})\xi^{c}(\vec{x})\,(\equiv{\cal
D}^{ac}_{i}\xi^{c}(\vec{x}));$ $\displaystyle\delta p_{i}^{a}$
$\displaystyle=$ $\displaystyle\\{p_{i}^{a},\int
d^{2}\vec{x}\xi^{b}(\vec{x})\Delta^{b}(\vec{x})\\}=-gf^{abc}\xi^{b}(\vec{x})p^{c}_{i}(\vec{x}),$
(9)
which evidently reproduces the known gauge transformation for the connection
and stress tensor. Here the ${\cal D}^{ac}$ is a gauge covariant derivative.
It is easy to check that the primary and secondary constraints mutually
commute, $\\{\Phi^{(1)a},\Phi^{(2)b}\\}=0$. Further, one can find that
$\\{\Phi^{(2)b},H\\}=0$, thus, no new constraints arise (see also Park ; Wo
for discussion of the canonical structure of the theories with the Chern-
Simons term).
The canonical quantization of the theory can be carried out in a standard way,
that is, we define the canonical variables $A^{a}_{i}$ and $p^{a}_{i}$ to be
operators with the commutation relation
$[A^{a}_{i}(\vec{x}),p^{b}_{j}(\vec{y})]=i\delta_{ij}\delta^{ab}\delta(\vec{x}-\vec{y})$,
with all other commutators of the canonical variables be zero.
Now, let us implement the noncommutative fields method. To do it, we deform
the canonical commutation relations to be
$\displaystyle[A^{a}_{i}(\vec{x}),p^{b}_{j}(\vec{y})]=i\delta_{ij}\delta^{ab}\delta(\vec{x}-\vec{y});$
$\displaystyle[p^{a}_{i}(\vec{x}),p^{b}_{j}(\vec{y})]=i\theta_{ij}\delta^{ab}\delta(\vec{x}-\vec{y});$
$\displaystyle[A^{a}_{i}(\vec{x}),A^{b}_{j}(\vec{y})]=0.$ (10)
Our aim is to deform the secondary constraint $\Delta^{b}$ in a manner
preserving the gauge transformations (On the noncommutative fields method in
the three-dimensional Yang-Mills theory). It is easy to see that this can be
achieved if we modify the secondary constraint as
$\displaystyle\tilde{\Delta}^{b}=-(\partial_{i}p_{i}^{b}+gf^{bcd}A^{c}_{i}p_{i}^{d})+\theta_{ij}(\partial_{i}A^{b}_{j}+\frac{1}{2}gf^{bcd}A^{c}_{i}A^{d}_{j}).$
(11)
This modification of the secondary constraint implies in the modification of
the Hamiltonian which acquires the form
$\displaystyle\tilde{H}=\frac{1}{2}p^{a}_{i}p^{a}_{i}+\frac{1}{4}F^{a}_{ij}F^{a}_{ij}+A_{0}^{b}\theta_{ij}(\partial_{i}A^{b}_{j}+\frac{1}{2}gf^{bcd}A^{c}_{i}A^{d}_{j}).$
(12)
Then, we can introduce the canonical momenta
$\displaystyle\pi_{i}^{a}=p_{i}^{a}-\frac{1}{2}\theta_{ij}A_{j}^{a},$ (13)
and they satisfy the commutation relation $[\pi_{i}^{a},\pi_{j}^{b}]=0$.
The new Lagrangian is
$\displaystyle\tilde{L}=\pi_{i}^{a}\dot{A}_{i}^{a}-\tilde{H}.$ (14)
Substituting the canonical momenta (13) and the modfified Hamiltonian (12) to
this expression, we find that the new Lagrangian can be written as
$\displaystyle\tilde{L}=L+\Delta L\equiv
L-\frac{1}{2}\theta_{ij}\dot{A}_{i}^{a}A_{j}^{a}-A_{0}^{b}\theta_{ij}(\partial_{i}A^{b}_{j}+\frac{1}{2}gf^{bcd}A^{c}_{i}A^{d}_{j}).$
(15)
As a result, we find
$\displaystyle\Delta
L=\theta_{ij}(-\frac{1}{2}\dot{A}_{i}^{a}A_{j}^{a}-A_{0}^{a}\partial_{i}A_{j}^{a}+\frac{1}{2}gf^{bcd}A_{0}^{b}A_{i}^{c}A^{d}_{j}).$
(16)
After an appropriate symmetrization, introducing
$\theta_{ij}=\epsilon_{0ij}\theta$, we find
$\displaystyle\Delta
L=\frac{1}{2}\theta\epsilon^{\mu\nu\lambda}(A_{\mu}^{a}\partial_{\nu}A_{\lambda}^{a}+\frac{1}{3}gf^{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\lambda}^{c})=\frac{1}{2}\theta\epsilon^{\mu\nu\lambda}{\rm
tr}(A_{\mu}\partial_{\nu}A_{\lambda}+\frac{2}{3}gA_{\mu}A_{\nu}A_{\lambda}),$
(17)
which reproduces the structure of the well known non-Abelian Chern-Simons
term, with the mass is proportional to the noncommutativity parameter, just as
in NPR .
We can try to implement a more general deformation of the canonical algebra,
that is,
$\displaystyle[A^{a}_{i}(\vec{x}),p^{b}_{j}(\vec{y})]=i\delta_{ij}\delta^{ab}\delta(\vec{x}-\vec{y});$
$\displaystyle[p^{a}_{i}(\vec{x}),p^{b}_{j}(\vec{y})]=i\theta_{ij}\delta^{ab}\delta(\vec{x}-\vec{y});$
$\displaystyle[A^{a}_{i}(\vec{x}),A^{b}_{j}(\vec{y})]=i\tilde{\theta}_{ij}\delta^{ab}\delta(\vec{x}-\vec{y}).$
(18)
Let us impose again a requirement that the gauge transformations should have
the form (On the noncommutative fields method in the three-dimensional Yang-
Mills theory). First of all, since $\theta_{ij}$ and $\tilde{\theta}_{ij}$ are
constants, we suggest from the beginning that
$\theta_{ij}=\theta\epsilon_{ij}$,
$\tilde{\theta}_{ij}=\tilde{\theta}\epsilon_{ij}$.
To do it, let us suggest the following form of the modified secondary
constraint which is the most general expression of no higher than second order
in canonical variables:
$\displaystyle\Phi^{(2)b}$ $\displaystyle=$
$\displaystyle-\partial_{i}p_{i}^{b}+k_{1}gf^{bcd}A^{c}_{i}p_{i}^{d}+k_{2}\epsilon_{ij}\partial_{i}A^{b}_{j}+k_{3}\epsilon_{ij}\partial_{i}p^{b}_{j}+k_{4}\epsilon_{ij}gf^{bcd}A^{c}_{i}p^{d}_{j}+$
(19) $\displaystyle+$ $\displaystyle
k_{5}gf^{bcd}\epsilon_{ij}p^{c}_{i}p^{d}_{j}+k_{6}gf^{bcd}\epsilon_{ij}A^{c}_{i}A^{d}_{j}.$
Here the coefficients $k_{1}\ldots k_{6}$ depend on $\theta,\tilde{\theta}$.
The corresponding variations of the fields look like
$\displaystyle\delta A^{a}_{n}$ $\displaystyle=$
$\displaystyle\\{A^{a}_{n},\Phi^{(2)b}\\}\xi^{b}=\partial_{n}\xi^{a}-k_{1}gf^{abc}\xi^{b}(\tilde{\theta}\epsilon_{ni}p^{c}_{i}-A^{c}_{n})-k_{2}\tilde{\theta}\partial_{n}\xi^{a}+k_{3}\epsilon_{ni}\partial_{i}\xi^{a}+$
$\displaystyle+$ $\displaystyle
k_{4}gf^{abc}\xi^{b}(\tilde{\theta}p^{c}_{n}-\epsilon_{ni}A^{c}_{i})-2k_{5}g\epsilon_{ni}f^{abc}\xi^{b}p^{c}_{i}+2k_{6}gf^{abc}\tilde{\theta}\xi^{b}A^{c}_{n};$
$\displaystyle\delta p^{a}_{n}$ $\displaystyle=$
$\displaystyle\\{p^{a}_{n},\Phi^{(2)b}\\}\xi^{b}=\theta\epsilon_{ni}\partial_{i}\xi^{a}+k_{1}gf^{abc}\xi^{b}p^{c}_{n}+k_{1}\theta
gf^{abc}\epsilon_{ni}\xi^{b}A^{c}_{i}-k_{2}\epsilon_{ni}\partial_{i}\xi^{a}-k_{3}\theta\partial_{n}\xi^{a}+$
(20) $\displaystyle+$ $\displaystyle
k_{4}\epsilon_{ni}gf^{abc}\xi^{b}p^{c}_{i}+k_{4}\theta
gf^{abc}A^{c}_{n}\xi^{b}+2k_{5}gf^{abc}\theta\xi^{b}p^{c}_{n}-2k_{6}gf^{abc}\epsilon_{ni}\xi^{b}A^{c}_{i}.$
We want these transformations to reproduce (On the noncommutative fields
method in the three-dimensional Yang-Mills theory). For the variation of
$A^{a}_{n}$ this requirement yields $k_{3}=0,k_{4}=0$, so, we will not
consider these terms in the equation for $\delta p^{a}_{i}$. Also, we find
$\displaystyle
k_{2}\tilde{\theta}=0;\quad\,k_{1}+2k_{6}\tilde{\theta}=-1,\quad\,k_{1}\tilde{\theta}+2k_{5}=0.$
(21)
For the second equation, after substituting $k_{3}=k_{4}=0$, we get
$\displaystyle
k_{2}=\theta,\quad\,k_{1}+2k_{5}\theta=-1,\quad\,k_{1}\theta-2k_{6}=0.$ (22)
Comparing these equations, we find that the variations of the fields (On the
noncommutative fields method in the three-dimensional Yang-Mills theory)
reproduce the form of variations under the gauge transformations if and only
if $\theta\tilde{\theta}=0$. Hence, we must have either $\tilde{\theta}=0$,
which is exactly the case studied above, or $\theta=0$. Thus, we conclude that
we cannot impose noncommutativity both in field and momentum sectors in a
manner compatible with the gauge symmetry.
It remains only to finish the study in the case when $\theta=0$. In this case,
the modified constraint is
$\displaystyle\Phi^{(2)b}$ $\displaystyle=$
$\displaystyle-\partial_{i}p_{i}^{b}-gf^{bcd}A^{c}_{i}p_{i}^{d}+\frac{\tilde{\theta}}{2}gf^{bcd}\epsilon_{ij}p^{c}_{i}p^{d}_{j},$
(23)
and the modified Hamiltonian is
$\displaystyle\tilde{H}=\frac{1}{2}p^{a}_{i}p^{a}_{i}+\frac{1}{4}F^{a}_{ij}F^{a}_{ij}+A_{0}^{b}[-\partial_{i}p_{i}^{b}-gf^{bcd}A^{c}_{i}p_{i}^{d}+\frac{\tilde{\theta}}{2}gf^{bcd}\epsilon_{ij}p^{c}_{i}p^{d}_{j}].$
(24)
Since commutation relations between momenta are not modified in this case, the
momenta $p^{a}_{i}$ continue to be canonical ones, whereas the coordinates –
do not more. The correct ”new” canonical coordinates, whose commutators are
equal to zero, are
$\displaystyle\tilde{A}^{a}_{i}=A^{a}_{i}-\frac{1}{2}\tilde{\theta}\epsilon_{ij}p^{a}_{j},$
(25)
with the ”old” velocities are related with momenta as
$\displaystyle\dot{A}^{b}_{i}=\frac{\partial\tilde{H}}{\partial
p^{b}_{i}}=p^{b}_{i}+\partial_{i}A^{b}_{0}+gf^{abc}A^{a}_{0}A^{c}_{i}+g\tilde{\theta}f^{abc}A^{a}_{0}\epsilon_{ij}p^{c}_{j},$
(26)
which for $\tilde{\theta}=0$ evidently reduces to the common expression (6).
Unfortunately, this equation, whose equivalent form is
$\displaystyle
p^{c}_{j}(\delta^{bc}\delta_{ij}+g\tilde{\theta}f^{abc}A^{a}_{0}\epsilon_{ij})=\dot{A}^{b}_{i}-\partial_{i}A^{b}_{0}+gf^{bac}A^{a}_{0}A^{c}_{i}\quad\,(=F^{b}_{0i}),$
(27)
cannot be solved exactly, we can use only iterative approach (however, we
would like to point out that this problem does not arises in the Abelian case
where one finds $p^{b}_{i}=F^{b}_{0i}$). As a zeroth approximation (which,
however, is sufficient to find the corrections in the effective Lagrangian up
to the first order in $\tilde{\theta}$), we can use the $\tilde{\theta}=0$
expression for the canonical momentum $p^{a}_{i}$ (5), thus, the Lagrangian
$\tilde{L}=p^{a}_{i}\dot{\tilde{A}}^{a}_{i}-\tilde{H}$ acquires a correction
$\Delta L$ generated by modifications both of the Hamiltonian and
$\dot{A}^{a}_{i}$. This correction, being expressed in terms of the canonical
momenta, looks like:
$\displaystyle\Delta
L=-\frac{1}{2}\tilde{\theta}\epsilon_{ij}p^{a}_{i}\dot{p}^{a}_{j}-\frac{1}{2}gf^{bcd}A^{b}_{0}\tilde{\theta}\epsilon_{ij}p^{c}_{i}p^{d}_{j}.$
(28)
This expresion is exact, without any approximations. After elimination of
momenta, where we must employ the approximate expressions for $p^{a}_{i}$ in
terms of velocities, we find that
$\displaystyle\Delta
L=-\frac{1}{2}\tilde{\theta}\epsilon_{ij}F^{a}_{0i}\dot{F}^{a}_{0j}-\frac{1}{2}gf^{bcd}A^{b}_{0}\tilde{\theta}\epsilon_{ij}F^{c}_{0i}F^{d}_{0j}+O(\tilde{\theta}^{2}).$
(29)
Thus, one can see that, as a result, the modified Lagrangian in the case of
noncommuting field operators involves higher derivatives (since $F^{a}_{0i}$
contain first temporal derivative). The similar conclusion, that is,
generation of higher derivatives in the case of noncommuting fields (which can
be treated as UV limit of the theory, see discussion of scales in the
noncommutative fields method in Gamb0 ), was obtained in Gamb . Also, we note
that, as this correction to the Lagrangian has quite ugly form, we can
conclude that in this case, unlike of the case of noncommuting momenta, we
meet an explicit Lorentz symmetry breaking.
Let us discuss the results. We studied the generalized version of the
noncommutative field method, in which, differently from the most popular
version Gamb1 ; Gamb2 ; ourgra not only the commutation relations between
canonical momenta are deformed but also the commutation relations between
canonical field coordinates. The most important conclusions are the following
ones. First, one cannot deform these two canonical commutation relations
simultaneously in a manner compatible with the gauge symmetry. This fact can
be treated as a need to choose between study of the low-energy behaviour
(which corresponds to deformation of commutation relation between canonical
momenta) and study of the high-energy behaviour (which corresponds to
deformation of commutation relation between canonical fields) with no
possibility to consider two limits at the same time. Second, in the low-energy
limit the complete, non-linearized Chern-Simons term is generated, which is a
natural non-Abelian generalization of the result obtained in NPR where the
quadratic Chern-Simons term was generated for the electrodynamics, with no
Lorentz symmetry breaking terms arises in this case, and both the mass term
and cubic interaction term with a correct coefficient are generated. However,
the new term arisen in the high-energy limit turns out to break the Lorentz
symmetry explicitly, and, moreover, it involves higher derivatives as it was
predicted in Gamb . The natural treating of this result is that the breaking
of the Lorentz symmetry at high energies can be related to the GZK effect and
many other studies predicting Lorentz symmetry breaking namely for high energy
scales (see f.e. Mag ).
Acknowledgments. The work by A. Yu. P. has been supported by CNPq-FAPESQ DCR
program, CNPq project No. 350400/2005-9.
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|
# Information Harvesting for Far-Field Wireless Power Transfer
Mehmet C. Ilter1, Risto Wichman1, Mikko Saily2 and Jyri Hämäläinen3
1Department of Signal Processing and Acoustics, Aalto University, 02150 Espoo,
Finland 2Network and Architecture Group, Nokia Bell Labs, 02610 Espoo,
Finland 3Department of Communications and Networking, Aalto University, 02150
Espoo, Finland
###### Abstract
Considering ubiquitous connectivity and advanced information processing
capability, huge amount of low-power IoT devices are deployed nowadays and the
maintenance of those devices which includes firmware/software updates and
recharging the units has become a bottleneck for IoT systems. For addressing
limited battery constraints, wireless power transfer is a promising approach
such that it does not require any physical link between energy harvester and
power transfer. Furthermore, combining wireless power transfer with
information transmission has become more appealing. In the systems that apply
radio signals the wireless power transfer has become a popular trend to
harvest RF-radiated energy from received information signal in IoT devices.
For those systems, design frameworks mainly deal with the trade-off between
information capacity and energy harvesting efficiency. Therein various
signaling design frameworks have been proposed for different system
preferences between power and information. In addition to this, protecting the
information part from potential eavesdropping activity in a service area
introduces security considerations for those systems. In this paper, we
propose a novel concept, Information Harvesting, for wireless power transfer
systems. It introduces a novel protocol design from opposite perspective
compared to the existing studies. Particularly, Information Harvesting aims to
transmit information through existing wireless power transfer mechanism
without interfering/interrupting power transfer. To do so, only intended
information receivers are able to decode the transmitted information after
they initiate an information transfer process at wireless power transmitter
while energy harvesters continue energy harvesting procedure without noticing
this procedure. Considering the diversity of IoT networks and the availability
of wireless power transfer infrastructure, proposed Information Harvesting
principles may turn out to a pivotal methodology especially for the cases
where a large number of IoT devices require the software/firmware updates
along with periodical battery recharging needs.
###### Index Terms:
Wireless power and information transfer, information harvesting, Internet of
Things, energy harvesting.
## I Introduction
The Internet of Things (IoT) creates the potential for new generation industry
use cases and smart cities. With uniquely identifiable devices capable of
communicating through wireless environment, billions of devices are able to
sense and interact with everything and everyone. The vision of IoT is set to
be a communication platform for broadcast and point-to-point communication.
From other aspect, wireless power transfer enables the next stage in the
current consumer electronics revolution, including battery-less sensors,
passive RF identification (RFID), passive wireless sensors, and machine-to-
machine solutions. The origin of wireless power transmitted by using RF
spectrum was pioneered by Tesla and it is based on radiative energy transfer
through a medium (i.e. air) with the help of antennas [1].
There are two types of power transfer: near-field power and far-field power
transfer. The first one depends on inductive/capacitive coupling between
energy harvester and power transmitter so its range is less than a meter. The
latter one exploits radio frequency transmission and due to propagation
characteristics of RF signals, it can be used over longer ranges [2]. For
instance, prototype RF-based energy harvesting circuits operating below 1 GHz
are capable of harvesting microwatts to milliwatts of power over the range of
10 m with similar transmit power of a Wi-Fi router [3].
The main objective of radiative power transfer system is to maximize the
harvested DC power subject to a given transmit power constraint, in other
words, enhancing the end-to-end power transfer efficiency. Designing an
efficient rectenna, the unit that consists of a rectifier and an antenna, to
maximize RF-to-DC conversion has been the traditional line of research in
earlier power transfer literature [4]. Furthermore, signal design for wireless
power mechanism is getting more popular thanks to the availability of wider
range of optimization techniques. For instance, different from grid-based and
lattice-based constellations in conventional communication systems where lower
peak-to-average power ratio is typically desired, real Gaussian signals, flash
signaling and linear frequency-modulated signals are the preferred
constellations for energy harvesting mechanisms since higher PAPR values
result in higher DC voltage by a rectifier leading to better end-to-end power
efficiency [4] at the cost of higher complexity and power consumption in the
transmitter.
Rather than radiating only energy into a particular service area, combining
RF-based power transfer with simultaneous information transmission has become
appealing with the recently introduced simultaneous wireless information and
power transfer (SWIPT) concept. In SWIPT systems, the power and information
components of the transmitting signal are separable by using the energy domain
(power splitting), the time domain (time splitting) and the space domain
(antenna splitting) [5]. For those systems, the receiver architecture is
generally optimized for increasing total harvested energy while decoding
information with minimum power consumption at the same time without
considerable loss in achievable data rates. Therefore, a trade-off exists
between information transfer and energy transfer in such systems.
In practice, sensitivities of information receivers and energy harvesters are
quite different. Specifically, the minimum requirement for the received power
at an information receiver is around –60 dBm, while that of an energy
harvester is –10 dBm. For instance under Friis assumption, 2.4 GHz frequency
introduces a decay of 40 dB decay per meter. Therefore, energy harvester is
usually located closer to wireless power transceiver than information
receivers. However, this setting may create a potential eavesdropping activity
where receiver close by energy harvester can act as a malicious device who
exploits better received signal statistics resulting in a high risk of
eavesdropping and information interception. Similar phenomenon is also
observed in power domain-non-orthogonal multiple access systems where
successive interference cancellation techniques are implemented in the nearby
users in order to eliminate other users’ symbols thanks to having better
channel conditions in the process of extracting its own intended signal from
superposed transmitting symbol. Therefore, nearby users have an advantage to
decode far user information as well and there is a need to combat against
eavesdropping risk of nearby users by applying physical layer security
schemes.
Physical layer (PHY)-security techniques which mainly turn the randomness in
wireless channels into an advantage by exploiting the physical aspects of
communication channels between the nodes provide a powerful tool to achieve
secure communication. Although encryption mechanism is quite useful for
protecting ongoing data transmission in many cases, considering the diversity
in protocols and networks in 5G and beyond, those mechanism might not be
suitable for all available cases. Even with limited computational capability,
it was shown that the PHY-security is able to provide secure communications
for a massive number of IoT devices [6].
Due to huge amount of devices, IoT networks require frequent software updates
which include security updates, bug fixes, and software extensions. From this
perspective, there is considerable amount of data transmission procedure
occurring periodically. For those activities, encryption techniques can
contribute up to 30% of total energy consumption [7]. Furthermore, most IoT
devices/sensors are simple nodes having limited computational capabilities so
the complexity of encryption-decryption procedures are beyond the capabilities
of those devices. It is claimed that no encryption is required in
integrated/tiered IoT devices [8] and SigFox protocol [9]. Those trends can be
easily verified by checking the statistics about existing IoT traffic showing
that the big portion of IoT traffic is unencrypted [10]. In addition, even
encrypted data might not be enough if malicious users have enough time to
analyze the traffic patterns to obtain the information. Therefore
additional/complimentary solution is required.
To tackle these challenges this paper introduces Information Harvesting (IH)
in wireless power transfer mechanism, which proposes data communication
ability on top of existing power transfer procedure; in other words, wireless
power transfer is partnered with a information transfer protocol which allows
secure transfer. The pivotal point of the proposed protocol is that the power
receivers are not aware of information broadcasting and it is required to use
a transmitter entity for the purpose of transmitting broadcasting packets
without disturbing ongoing power transfer. From this perspective, IH can
dissolve wireless power transfer and information broadcasting in one
mechanism. This mechanism can be exploited through updating low complexity
devices without any encryption (or low level of encryption for necessary
cases) by utilizing existing wireless power transceiver unit originally
deployed for charging power receivers.
The rest of the paper is organized as follows. In Section II, the system
components of IH mechanism are introduced. Then, the protocol details of IH
are represented in Section III. Then, Section IV introduces two different
practical realizations, which are spatial modulation and analog beamforming,
respectively. Besides, some additional techniques which increase the physical
layer security of the proposed protocol are also described therein. Section V
represents the feasibility of spatial modulation based-information harvesting
with simulation results generated for an indoor environment. Section VI
concludes the paper.
## II System Components
The system components of IH are illustrated in Fig. 1. Herein, wireless power
transceiver (WPT) serves as a module supplying power transfer to a portable
power receiver device(s), energy harvester (EH) and also providing information
transfer by utilizing information seeding to a portable information receiver
device, information receiver (IR). The IR contains receive antenna chain where
single/multiple receive antenna(s) is(are) deployed and the energy harvester
consists of a receive antenna chain, a battery charging unit and a battery.
The key motivation herein is to design information seeding and information
harvesting cycles so that the EHs are not able to detect the on-going
information transfer and in case of this activity detected by EHs, they are
not able to decode the embedded information.
Figure 1: The overall diagram of Information Harvesting system components.
From this point of view, information seeding protocol lies on creating
variations in transmitter entity in WPT with respect to information stream
transmitted to IR. By doing so, broadcasting channels between WPT and IR can
be configured by exploiting existing wireless power transfer mechanism. Those
changes in the chosen transmitter entity cannot be sensed in existing EHs so
broadcasted information can be only decoded at target IRs. At this point, the
procedures handled by IR can be referred to information harvesting which
includes sending request for broadcasting to WPT via request for information
(RFI) signalling, carrying out channel estimation mechanism and detecting the
variations in the transmitter entity from the existing power transmitting
signal. Note that the EH and IR are not necessarily different devices so an EH
can act as an information receiver. The system components of IH can be
described as follows:
* •
WPT senses power request/information request from portable power
receiver/portable information receiver through identification and
configuration phase. Accordingly, there are two modes for wireless power
transceiver, which are power transfer mode and information seeding mode.
During information seeding, the process of using different transmission entity
in order to send the information to the wireless information receiver is
handled.
* •
IR contains single/multiple receive antenna(s). Through this/them, the
radiated wireless power transfer signal generated by the wireless power
transceiver is received, then; the harvesting procedure starts with estimating
corresponding transmission entity which was the source of the information.
* •
EH consists of receive antenna chain, battery charging unit and battery.
Herein, battery charging unit is configured to establish a link between the
receive antenna chain and battery, wherein the manner in which power is
transferred from wireless power transceiver is controlled in accordance with
the parameters and/or state information assigned by the power management unit.
### II-A Improving PHY security:Dynamic activation pattern and artificial
noise generation
In order to prevent passive attacks (silent eavesdropper and eavesdropping
activities), dynamic transmit entity mapping [11] can be applied in order to
prevent decoding the index of transmitting entities over the air by existing
attacker. For dynamic transmit entity index, dynamic transmit entity
activation pattern can be added in WPT. Those updates on transmit entity index
can be done periodically after certain number of information frames based on
the arrangements between WPT and IR.
Another PHY security improvement can be obtained from artificial noise
generation in full-duplex information receivers. In this scheme, information
receiver can use its own sources in order to mask information transmission
from wireless power transceiver. To do so, when Information receiver sends its
RFI signalling, it can start to generate artificial noise signal. By this way,
it will be more difficult for any existing attackers to trace the transmit
entity index. Since artificial noise is generated by information receiver, the
information receiver can first use a physical layer self-interference
cancellation method, subtracting its own generated artificial noise signal
from received signal, then information harvesting phase can start.
Now, the details of the proposed protocol in information seeding and
information harvesting phases will be described in the next section.
## III Protocol Design
As mentioned above, WPT initially serves a module supplying power transfer to
a portable EHs and can provide information transfer to portable information
receiver device(s) once information request from the IRs is received. In order
to distinguish the cycles of power transfer and information transfer at WPT,
IH protocol should have pre-signaling, identification and configuration
phases.
For wireless power transfer mode when there is only EH in a service area, the
identification and configuration phases are based on establishing wireless
power transfer link between WPT and EH illustrated in Fig. 2 with blue blocks.
After the wireless power transfer request is initiated by EH, WPT can sense
this request, and the identification of the EH and its configuration is
exchanged between them. Once it is completed, WPT unit turns into power
transfer mode. Any failure in this configuration can be detected by a control
error mechanism and the configuration starts from the beginning if any error
is detected during power transfer period.
Figure 2: The detailed procedures between WPT-EH and WPT-IR during wireless
power and information transfer phases.
In case of IR enters the area, the changes in the existing power transfer
mechanism procedure is highlighted with green blocks in Fig. 2. Basically, the
arrival of IR triggers two different cycles; information seeding and
information harvesting. The first one appears in WPT while the latter is
executed in IR. At the beginning, information harvesting starts with sensing
ongoing wireless power transfer mechanism in the service area. Once it is
sensed by IR, the channel estimation process between WPT-IR links are
initiated and the parameters will be attached to RFI signal based on the
channel estimation process. For instance, this process can determine the
required data rate in the IR so the requirement for how frequent a chosen
transmit entity is varied at WPT during information seeding can be sent
through the RFI signals. Then, extra PHY security enhancement described in
Section II.A can be added based on the IR preference.
Following those, the IR sends its information request to the power transmitter
and this request is sensed through the receiver at the WPT. After sensing this
information request, the communications controller in the transmitter becomes
active and the feature of a chosen transmission entity which was determined by
RF chain activation pattern is updated with respect to information bits. Then,
those changes imposed to the chosen transmission entity are estimated to
extract the information embedded on wireless power signals while ongoing power
transmission configuration is not affected by this procedure.
## IV Spatial modulation-based IH
Spatial modulation (SM) technique was proposed in [12] where a subset of
active transmit antennas out of multiple transmit antennas can be used in
order to increase spectral efficiency. To do so, extra information bits are
typically conveyed through the index of active transmit antenna(s), in
addition to conventional modulated symbols. We recall that SM was proposed as
a potential NR candidate for 5G networks [13] and its derivatives, index
modulation and media-based modulation, are still popular [14]. The original
version of SM only considered a single active transmit antenna over fading
channels; then, generalized SM was introduced in [15] where a group of active
transmit antennas are considered. Furthermore, there are some examples of SM
proposed for non-fading/low-rank channels beyond 5G systems, i.e. mmWave
channnels [5]. In case of exploiting spatial modulation technique, information
seeding is based on the process of using different transmit antennas at
wireless power transceiver and estimating the transmit antenna in information
receiver.
In the power transfer mode, antenna activation pattern selector is inactive
since there is no information receiver in the range. When the information
receiver appears in the range, it transmits Reguest for Information (RFI)
message. Once RFI is received in wireless power transmitting unit, antenna
activation pattern selector becomes active, so wireless charging power
transmitter turns into information seeding mode. RFI signaling can also
determine how fast active transmit antenna pattern varies. Recently, space-
time mapping has been proposed in [16] where the period of antenna activation
was extended to multiple symbol periods along with marginal increment in
detection complexity. Then, the new transmit antenna(s) can be active in the
next frame based on information bits block. During the period of information
seeding mode ON, information receiver aims to detect the index of active
transmit antenna or indexes of group of active antennas. The overall procedure
is summarized in Fig. 3.
In the case of using analog beamforming for information seeding phase, in
addition to the existing power signals transmitted through EH along the
selected beam pairs, the activated transmit beam indices, when correctly
identified at the IR, can be used to convey additional information.
Figure 3: Spatial modulation-based Information Harvesting. (upper-left) Power
transfer mode. (upper-right) IR-initiated request for information signalling.
(bottom-left) Information seeding mechanism. (bottom-right) Information
harvesting mechanism. TABLE I: The simulation-related parameters
Parameter List | Values
---|---
Total transmitted power at the WPT | 10 dB
Total active transmit antenna at the WPT | 64
The distance of information receiver | $\displaystyle\rm d_{\rm info}=20$ m
The distance of energy harvester | $\displaystyle\rm d_{\rm energy}=1$ m
Antenna spacing | $\displaystyle\lambda/2$
Operation frequency | $\displaystyle 2$ GHz
Array antenna gain | $\displaystyle 15$ dBi
Path loss model | $\displaystyle 128.1+37.6\log_{10}(d)$, d[km]
Angular offset’s standard deviation | $\displaystyle 2^{\circ}$
Log-normal shadowing’s standard deviation | $\displaystyle 8$ dB
Subcarrier bandwidth | $\displaystyle 15$ kHz
Obstacle cover ratio | $\displaystyle 0-0.9$
Obstacle radius | [0.3, 0.6]
Obstacle height | [5, 25]
## V Proof of Concept: Spatial Modulation based Information Harvesting
In this section, the feasibility of Information Harvesting will be
investigated through wireless power transceiver which utilizes spatial
modulation mechanism. To do so, the received signal differences at information
receiver and energy harvester will be investigated.
In the wireless power transceiver, $\displaystyle N_{t}$ transmit antennas
sharing a common RF chain are employed and a group of transmit antennas are
active during wireless power transmission as illustrated in Fig. 3. In order
to align with multi-antenna broadcast systems, the constant envelope power
transfer signals considered in [17] are used and a set of phase shifters for
each transmit antenna is deployed. The spatial correlation between transmit
antennas is modeled by the standard Kronecker correlation model which is
widely used in many studies.
The characteristics of the channels for WPT-EH and WPT-IR will vary based on
spatial features (near-distance and far-distance, line-of-sight and non-line-
of-sight) and the WPT-IR shows more variations due to larger distance from
WPT. In order to reflect a more realistic scenario, the obstacles are modeled
by using a homogeneous Poisson point process (PPP) in the calculation of line-
of-sight and non-line-of-sight probabilities for the locations of EH and IR by
using obstacle coverage ratio (OCR). Specifically, the obstacle density is
determined by the average obstacle area and total area covered by the
obstacles and the obstacles are modeled as cylinders that are spatially
distributed according to a homogeneous PPP for an indoor area.
The goal is to show the feasibility of sending information through the changes
in active transmit antenna index where energy harvesters do not notice any
considerable changes in received signals while information receiver can detect
those changes in information harvesting cycle. To do so, the energy harvester
is placed closer with respect to information receiver along with similar
obstacle coverage ratio. The list of the simulation parameters are given in
Table I.
Figure 4: The received signal variations and LoS probabilities in EH over
different distances and the OCR values.
The LoS probabilities with respect to different distances are plotted in Fig.
4. As seen from the curves, higher OCR values lead lower chance of getting LoS
link in any distance. Now, in order to see the effect of the variations in
active transmit antenna indices at wireless power transceiver, two different
distances are considered for energy harvester; 1 m and 20 m. Fig. 4
illustrates the mean received powers harvested in energy harvester at those
distances with different number of active transmit antennas out of
$\displaystyle 64$ antennas at the power transmitter. Those average power
values are obtained from $\displaystyle 1000$ different realizations where the
positions of WPT and EH and the number of active transmit antennas are fixed.
As seen from the figure, changing the transmit antenna creates negligible
differences in the harvested power levels and these variations result from
different power transfer signal for each realization.
The indexes of active transmit antennas during wireless power transmission
period can be detected so the extra information for information receivers can
be decoded without noticing any change in energy harvesters. The upper bound
for spectral efficiency achieved using spatial modulation-based IH is
illustrated in Fig. 5. Therein, different number of active transmit antennas
is considered out of $\displaystyle 64$ total transmit antenna at the WPT. For
higher number of active antennas, total combinations for information seeding
is increasing until half of the antennas turn active resulting in higher
spectral efficiency values at the IR. Higher OCR and longer distance result in
more variations in non-line-sight channel coefficient so it leads to higher
spectral efficiencies in those cases. Interestingly, when no obstacle exists
in the environment, all channels between the WPT and IR turns into roughly
identical LOS channels so spectral efficiency is zero. Note that Fig. 5
reflects the maximum performance of IH when using the spatial modulation-based
IH. Values can be reduced in practical implementation based on different
service area and device capabilities.
Figure 5: Upper bound for achievable spectral efficiency at IR with respect to
different numbers of active transmit antennas are used at WPT.
## VI Conclusion
We described a method and protocol for data communications where wireless
power transfer is enhanced with a protocol to allow secure information
broadcasting to devices which are in their information harvesting cycle. The
key motivation is the design of information seeding and information harvesting
mechanisms where energy harvesters are not able to detect the ongoing
information broadcasting since harvested energy is not affected by information
seeding mechanism occurring at the wireless power transceiver. From this
aspect, Information Harvesting can be utilized towards to the direction of
updating IoT devices by exploiting available wireless power transfer
infrastructure in the near future. In addition, Information Harvesting is a
potential candidate of 3GPP Rel-18+ study items where the reduced capability
devices may be energy harvesting devices with the radio access and
communication capability, thus operating on the same carrier frequencies with
information harvesting devices. These devices are expected to follow a
protocol allowing both energy harvesting and information harvesting cycles.
## VII Acknowledgment
This study has been supported by the Academy of Finland (grant number 334000).
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# On the energy barrier of hypergraph product codes
Guangqi Zhao Centre for Engineered Quantum Systems, School of Physics,
University of Sydney, Sydney, NSW 2006, Australia<EMAIL_ADDRESS>Andrew C. Doherty Centre for Engineered Quantum Systems, School of Physics,
University of Sydney, Sydney, NSW 2006, Australia Isaac H. Kim Department of
Computer Science, University of California, Davis, CA 95616, USA
###### Abstract
A macroscopic energy barrier is a necessary condition for self-correcting
quantum memory. In this paper, we prove tight bounds on the energy barrier
applicable to any quantum code obtained from the hypergraph product of two
classical codes. If the underlying classical codes are low-density parity-
check codes (LDPC), the energy barrier of the quantum code is shown to be the
minimum energy barrier of the underlying classical codes (and their
transposes) up to an additive $O(1)$ constant.
_Introduction_.\- Quantum computers are capable of solving problems that are
likely intractable for classical computers, such as factoring of large
integers Shor (1994) and simulation of quantum systems Feynman (1982); Lloyd
(1996). However, realistic quantum computers are noisy. In order to build a
useful quantum computer capable of carrying out such computational tasks,
quantum error correction is likely necessary.
Since the discovery of the first quantum error-correcting code Shor (1996),
much progress has been made towards finding better codes. While the leading
approach to scalable quantum error correction has been the surface code Kitaev
(2003) for nearly 20 years, lately there has been a surge of interest in using
quantum low-density parity check (LDPC) codes. This recent interest is in part
due to Gottesman, who showed that with quantum LDPC codes, one can achieve a
constant overhead for fault-tolerant quantum computation Gottesman (2014). A
well-known approach to construct such codes is the hypergraph product
construction Tillich and Zemor (2014). More recent studies led to the
development of other families of quantum LDPC codes with improved parameters
Freedman _et al._ (2002); Evra _et al._ ; Kaufman and Tessler (2021);
Hastings _et al._ (2021); Panteleev and Kalachev (2022a); Breuckmann and
Eberhardt (2021a); Panteleev and Kalachev (2022b); Leverrier and Zemor (2022);
Dinur _et al._ (2023). Moreover, recent studies suggest that there are viable
fault-tolerant quantum computing architectures that can host such codes
Tremblay _et al._ (2022); Bravyi _et al._ (2024); Xu _et al._ (2023).
One challenge in using these codes lies in the decoding. While there are
general methods such as the BP+OSD decoder Panteleev and Kalachev (2021), it
is a priori not obvious how well such a general-purpose decoder would work for
a given code. One attractive approach is to use codes that have an extensive
energy barrier. For such codes, a simple process that iteratively lowers the
energy (quantified in terms of the number of stabilizers violated) is a viable
decoder candidate. Alternatively, such a model can be viewed as a candidate
for self-correcting quantum memory, protecting quantum information by the
macroscopic energy barrier. (We note that the energy barrier is not a
sufficient condition for building a self-correcting quantum memory Haah
(2011); Michnicki (2014); Siva and Yoshida (2017), though it is nevertheless a
necessary condition.)
Such approaches work well for four-dimensional (4D) toric code Alicki _et
al._ (2010) and the quantum expander code Leverrier _et al._ (2015); Fawzi
_et al._ (2020), both of which have extensive energy barriers. Unfortunately,
rigorous bounds on the energy barrier are hard to come by and often derived
for specific codes (or family of codes) Bravyi and Haah (2011); Michnicki
(2014); Leverrier _et al._ (2015); Williamson and Baspin (2023); Lin _et
al._ (2023).
In this paper, we prove a tight bound on the energy barrier for the hypergraph
product codes Tillich and Zemor (2014), defined in terms of the energy barrier
of the underlying classical codes. The hypergraph product is defined in terms
of the parity check matrices (or equivalently, the Tanner graphs) of two
classical codes. While there are codes with better parameters Hastings _et
al._ (2021); Breuckmann and Eberhardt (2021a); Panteleev and Kalachev (2022a,
b), the hypergraph product remains a flexible framework for constructing
quantum LDPC codes with many advantages, such as the variety of decoders
Leverrier _et al._ (2015); Fawzi _et al._ (2020); Panteleev and Kalachev
(2021); Roffe _et al._ (2020); Grospellier _et al._ (2021), logical gates
Krishna and Poulin (2021); Cohen _et al._ (2022); Quintavalle _et al._
(2023), and distance-preserving syndrome extraction circuit Tremblay _et al._
(2022); Manes and Claes (2023).
Now we describe our main result. Without loss of generality, consider a
hypergraph product of two classical codes, defined in terms of the parity
check matrices $H_{1}$ and $H_{2}$. We denote the parity check matrix of the
resulting quantum code as $H_{(H_{1},H_{2})}$. For both the quantum and the
classical code, we denote the energy barrier as $\Delta(H)$, where $H$ can be
a parity check matrix of the quantum or the classical code. We prove that
under a modest condition,
$\Delta(H_{(H_{1},H_{2})})=\min(\Delta(H_{1}),\Delta(H_{2}),\Delta(H_{1}^{T}),\Delta(H_{2}^{T})),$
(1)
where $H^{T}$ is the transpose of $H$. Eq. (1) holds if the energy barriers of
the classical codes are larger than or equal to a certain sparsity parameter
of the code. Importantly, if the underlying codes are LDPC, then the sparsity
parameter is $O(1)$. Therefore, Eq. (1) holds if the energy barriers of
$H_{1},H_{2},H_{1}^{T},$ and $H_{2}^{T}$ grow as the code size grows. If this
condition is not satisfied, the energy barrier is bounded by a constant, a
fact that follows trivially from the definition of the hypergraph product
code.
The proof of Eq. (1) is based on two results. First, if two logical operators
of a quantum LDPC code are equivalent up to a stabilizer, their energy
barriers differ only by a constant related to the code’s sparsity parameters
$w_{c},w_{q}$ [Theorem 1]. Thanks to this result, proving the energy barrier
for a given code reduces to proving the energy barrier for any complete set of
logical operators, which can be a much smaller set than the set of all logical
operators. We then identify a set of logical operators for which the exact
energy barrier can be determined in terms of the energy barriers of the
underlying classical codes. Together, these two results imply Eq. (1).
_Energy barrier of codes_.\- We first present a formal definition of the
energy barrier for stabilizer codes Bravyi and Terhal (2009). Let
$\mathcal{C}$ be the code subspace of a stabilizer code, defined in terms of
the stabilizer group $S$. The code subspace can be viewed as the ground state
subspace of a Hamiltonian of the form $\hat{H}=\sum_{i=1}^{m}(I-s_{i})/2$,
where $\\{s_{1},\ldots,s_{m}\\}\subset S$ is a set of generators. Note that
the energy barrier depends on the choice of the generating set. For an
operator $P$ in the Pauli group $\mathcal{P}$, the energy of the state
$P|\psi\rangle$ is given by
$\bra{\psi}P^{\dagger}\hat{H}P\ket{\psi}=\epsilon(P)$. Here $\ket{\psi}$ is
any ground state with $\bra{\psi}\hat{H}\ket{\psi}=0$, and $\epsilon(P)$ is
the energy cost of $P$. This is the number of $s_{i}$s that anticommute with
$P$, i.e., $\epsilon(P)=|\\{i:s_{i}P=-Ps_{i}\\}|$. Equivalently, one may
define the energy barrier in terms of the parity check matrix $H$ of the
stabilizer code. Let $v(P)$ be the binary (bit-string) representation of a
Pauli $P$:
$\epsilon(P)=\text{wt}(Hv(P)).$ (2)
A sequence $P_{0},P_{1},\ldots,P_{t}$ from the Pauli group $\mathcal{P}$ forms
a _path_ from $P_{0}$ to $P_{t}$ if for each index $i$, the operators $P_{i}$
and $P_{i+1}$ differ at no more than one qubit. The notation $w(P_{0},P_{t})$
represents the collection of all such paths from $P_{0}$ to $P_{t}$. For $r\in
w(P_{0},P_{t})$, $\epsilon_{\max}(r)$ denotes the highest energy along path
$r$, i.e., $\epsilon_{\max}(r)=\max_{P_{i}\in r}\epsilon(P_{i})$, as the
energy barrier of $E$ along the path $r$.
The minimum energy associated with a Pauli $P$ is the smallest value of
$\epsilon_{\text{max}}$ across all possible paths from $0$ to $P$. This is the
energy barrier of $P$, denoted as $\Delta(P)$:
$\displaystyle\Delta(P)=\min_{r\in w(I,P)}\epsilon_{\max}(r).$ (3)
The energy barrier of the quantum code is the minimum energy barrier over the
set of nontrivial logical operators.
###### Definition 1.
Let $S$ be a stabilizer group and $L(S)$ be the set of nontrivial logical
operators. The energy barrier is
$\displaystyle\Delta(H):=\min_{\ell\in L(S)}\Delta(\ell).$ (4)
This is the smallest energy the environment has to overcome to enact a logical
operation on the encoded qubit. We can similarly define the energy barrier of
classical codes by only considering the path formed by Pauli-$X$s. We shall
denote this energy barrier also as $\Delta(H)$, where $H$ in this case is the
parity check matrix of the classical code.
_Quantum LDPC codes and their energy barriers_.\- A quantum LDPC code is a
stabilizer code with a sparse parity-check matrix; see Babar _et al._ (2015);
Breuckmann and Eberhardt (2021b) for recent reviews. The sparsity parameters
are $w_{c}$ and $w_{q}$, which are the maximum row and column weights of the
parity check matrix, respectively. These represent the maximum weight amongst
all the checks and the maximum number of checks associated with a single bit.
A code is LDPC if $w_{c},w_{q}=O(1)$.
Here, we prove a property that holds true for any quantum LDPC code. It states
that the energy barrier of two logical operators equivalent under stabilizers
are equal, provided that at least one of their energy barriers is larger than
or equal to $w_{c}w_{q}$. For quantum LDPC codes, $w_{c}w_{q}=O(1)$.
Therefore, if the energy barrier is $\Omega(1)$ for any given non-trivial
logical operator, it must also be the same energy barrier for any equivalent
logical operator.
We first prove the following bound.
###### Lemma 1.
Let $\mathcal{C}$ be a quantum code with sparsity parameters $(w_{c},w_{q})$.
For any stabilizer $s\in S$,
$\Delta(s)\leq w_{c}w_{q}.$ (5)
###### Proof.
Without loss of generality, any stabilizer $s$ can be expressed as a product
$s=s_{1}\cdots s_{m}$, where $s_{1},\ldots,s_{m}$ are the stabilizer
generators. Consider a path that applies $s_{i}$ in sequence, from $i=1$ to
$m$. For each $s_{i}$, we envision applying a (sub)sequence of Paulis in the
support of $s_{i}$. This subsequence has a length of at most $w_{c}$, and each
Pauli in the sequence affects at most $w_{q}$ stabilizers. Therefore, the
highest energy attained within this subsequence is at most $w_{c}w_{q}$. Once
$s_{i}$ is applied, the energy cost becomes zero. Therefore, $\Delta(s)\leq
w_{c}w_{q}$.
∎
###### Theorem 1.
Let $\mathcal{C}$ be a $(w_{c},w_{q})$ quantum LDPC code. For any logical
operator $L$ and stabilizer $s\in S$,
$\Delta(Ls)\leq\max(\Delta(L),w_{c}w_{q}).$ (6)
###### Proof.
Without loss of generality, consider a path $r=(\ell_{1},\ldots,\ell_{N})\in
w(0,L)$. We can consider a new path $r^{\prime}\in w(0,Ls)$ by appending $r$
with a sequence $\delta_{r}\in w(0,s)$, forming
$(\ell_{1},\ldots,\ell_{N},\ell_{1}^{\prime},\ldots,\ell_{M}^{\prime})$ where
$\delta_{r}=(\ell_{1}^{\prime},\ldots,\ell_{M}^{\prime})$. By assumption,
$\epsilon_{\max}(r^{\prime})=\max(\epsilon_{\max}(r),\Delta(s)).$ (7)
According to Lemma 1, $\Delta(s)\leq w_{c}w_{q}$. It implies that for any
given path $r$, there exists a path $r^{\prime}$ such that
$\epsilon_{\max}(r^{\prime})\leq\max(\epsilon_{\max}(r),w_{c}w_{q})$. Choose
$r$ such that $\Delta(L)=\epsilon_{\max}(r)$. Since
$\Delta(Ls)\leq\epsilon_{\max}(r^{\prime})$, we have
$\Delta(Ls)\leq\max(\Delta(L),w_{c}w_{q})$. ∎
We remark that using the same logic, one can deduce
$\Delta(L)=\Delta(Lss)\leq\max(\Delta(Ls),w_{c}w_{q})$. Consequently, if
$\Delta(L)\geq w_{c}w_{q}$ or $\Delta(Ls)\geq w_{c}w_{q},$
$\Delta(L)=\Delta(Ls)$. Therefore, to determine the asymptotic scaling of the
energy barrier of a quantum LDPC code, it suffices to consider the energy
barrier of any _fixed_ complete set of logical operators. Once an energy
barrier is obtained for such a set, the energy of all the other logical
operators is also essentially determined, thanks to Theorem 1.
However, it is a priori not obvious how to choose such a set. Below, we will
solve this problem for a large family of quantum codes known as the hypergraph
product code Tillich and Zemor (2014).
_Hypergraph product code and its logical operators_.\- Hypergraph product
codes are CSS codes formed from two classical linear codes. Without loss of
generality, let $H_{1}$ and $H_{2}$ be $r_{1}\times n_{1}$ and $r_{2}\times
n_{2}$ parity check matrices, respectively. The parity-check matrix of the
hypergraph product code becomes Tillich and Zemor (2014):
$\displaystyle H_{X}$ $\displaystyle=$
$\displaystyle\left(H_{1}\otimes\mathbf{I}_{n_{2}}\ \mathbf{I}_{r_{1}}\otimes
H_{2}^{T}\right),$ (8) $\displaystyle H_{Z}$ $\displaystyle=$
$\displaystyle\left(\mathbf{I}_{n_{1}}\otimes H_{2}\
H_{1}^{T}\otimes\mathbf{I}_{r_{2}}\right).$
Because $H_{X}H_{Z}^{T}=0$, these two parity check matrices define a CSS code,
with a quantum parity check matrix
$H_{(H_{1},H_{2})}=\left(\begin{array}[]{cc}H_{X}&0\\\
0&H_{Z}\end{array}\right).$ (9)
The classical codes defined by $H_{1},H_{2},H_{1}^{T},\text{ and }H_{2}^{T}$
form the parent classical codes. Without loss of generality, we will assume
that $H_{i}$ and $H_{i}^{T}$ define codes with parameters
$[n_{i},k_{i},d_{i}]$ and $[r_{i},k_{i}^{T},d_{i}^{T}]$, for $i=1,2$. Under
this assumption, the quantum code is a
$\left[[n_{1}n_{2}+r_{1}r_{2},k_{1}k_{2}+k_{1}^{T}k_{2}^{T},\min(d_{1},d_{2},d_{1}^{T},d_{2}^{T})]\right]$
code Tillich and Zemor (2014).
We now introduce a particularly useful set of logical operators, which we
refer to as the _canonical_ logical operators Quintavalle and Campbell (2022);
Manes and Claes (2023). For the $Z$-type logical operators, consider the
following operator
$\begin{pmatrix}\sum_{k,j}\lambda_{kj}\overline{x}_{k}\otimes y_{j}\\\
\sum_{\ell,m}\kappa_{\ell m}a_{\ell}\otimes\overline{b}_{m}\end{pmatrix},$
(10)
where (i) $H_{1}\overline{x}_{i}=H_{2}^{T}\overline{b}_{m}=0$ and (ii)
$y_{j}\not\in\text{Im}(H_{2}^{T})$ and $a_{\ell}\not\in\text{Im}(H_{1})$ are
unit vectors. We note that $j\in\\{1,\ldots,k_{2}\\}$ and
$\ell\in\\{1,\ldots,k_{1}^{T}\\}$ and that the set of logical operators
expressible in this form is complete Quintavalle and Campbell (2022); Manes
and Claes (2023), which means all $k_{1}k_{2}+k_{1}^{T}k_{2}^{T}$ logical
operators are provided. A canonical logical operator is _elementary_ if only
one of the coefficients, either $\lambda_{kj}$ or $\kappa_{\ell m}$, equals
one. A similar form of canonical $X$-type logical operators can also be
constructed. Because our discussion below can be applied to such operators
with little change, we omit the discussion about the $X$-type logical
operators.
Let $\mathcal{G}_{1}(V_{1},C_{1})$ and $\mathcal{G}_{2}(V_{2},C_{2})$ be the
Tanner graphs of codes defined by $H_{1}$ and $H_{2}$, respectively. Here,
$V_{1}$ and $V_{2}$ represent the set of bits, and $C_{1}$ and $C_{2}$ denote
the set of checks. We use $\\{v_{1}^{i}:i\in\\{1,2,\cdots n_{1}\\}\\}$ (resp.
$\\{v_{2}^{j}:j\in\\{1,2,\cdots n_{2}\\}\\}$) to refer to bit vertices in
$V_{1}$ (resp. $V_{2}$), and also as length $n_{1}$ (resp. $n_{2}$) unit
vectors with the $i$th (resp. $j$th) entry as $1$. The hypergraph product
$\mathcal{G}_{1}\times\mathcal{G}_{2}$ is a bipartite graph with vertex set
$V\cup C$, where $V=V_{1}\otimes V_{2}\cup C_{1}\otimes C_{2}$ is the qubit
set and $C=C_{1}\otimes V_{2}\cup V_{1}\otimes C_{2}$ is the stabilizer set.
The set of qubits can be partitioned into two subsets, $V_{1}\otimes V_{2}$
and $C_{1}\times C_{2}$. For $H_{X}$, $H_{1}\otimes\mathbf{I}_{n_{2}}$ acts on
$V_{1}\otimes V_{2}$ and $\mathbf{I}_{r_{1}}\otimes H_{2}^{T}$ acts on
$C_{1}\otimes C_{2}$. Moreover, the subset $V_{1}\otimes V_{2}$ can be further
partitioned into $n_{2}$ subsets $\\{V_{1}\otimes v_{2}^{1},V_{1}\otimes
v_{2}^{2},\cdots,V_{1}\otimes v_{2}^{n_{2}}\\}$, where $V_{1}\otimes
v_{2}^{k}:=\\{v\otimes v_{2}^{k}:v\in V_{1}\\}$ and
$V_{2}=\\{v_{2}^{1},\ldots,v_{2}^{n_{2}}\\}$ [Fig. 1].
Thus a $Z$-type Pauli operator can be expressed as a bit-string
$z=\left(z^{(1)},z^{(2)}\right)^{T}$, where $z^{(1)}$ is supported on the
qubit subset $V_{1}\otimes V_{2}$ with vector space
$\mathbb{F}_{2}^{n_{1}}\otimes\mathbb{F}_{2}^{n_{2}}$, and $z^{(2)}$ is
supported on the qubit set $C_{1}\otimes C_{2}$ with vector space
$\mathbb{F}_{2}^{r_{1}}\otimes\mathbb{F}_{2}^{r_{2}}$. Because $z^{(1)}$ and
$z^{(2)}$ act on a tensor product of two vector spaces, one can view them also
as a matrix. (For instance, $v_{i}\otimes u_{j}$ for unit vectors $v_{i}$ and
$u_{j}$ can be viewed as a matrix whose entry is $1$ on the $i$’th row and the
$j$’th column and zero elsewhere.) We call this procedure as vector reshaping,
and explain a few basic facts in the Appendix. After the reshaping, $z^{(1)}$
and $z^{(2)}$ become $Z^{(1)}$ and $Z^{(2)}$ respectively. Here, $Z^{(1)}$ is
an $n_{1}\times n_{2}$ matrix, and the entry $Z^{(1)}_{i,j}$ is supported on
qubit $v_{1}^{i}\otimes v_{2}^{j}$. Moreover, the $j$th column $Z^{(1)}_{j}$
is supported on qubits $\\{v_{1}^{1}\otimes v_{2}^{j},v_{1}^{2}\otimes
v_{2}^{j},\cdots,v_{1}^{n_{1}}\otimes v_{2}^{j}\\}$, which we refer to as
$V_{1}\otimes v_{2}^{j}$. Similarly, the subset $C_{1}\otimes C_{2}$ can be
partitioned into $r_{1}$ rows, and $i$th row is supported on qubit subset
$c_{1}^{i}\otimes C_{2}$ (defined similarly).
An elementary canonical logical-$Z$ operator, which is in the form
$\left(\bar{x}_{k}\otimes y_{j},0_{r_{1}r_{2}}\right)^{T}$ (where
$0_{r_{1}r_{2}}$ is the $r_{1}r_{2}$-dimensional zero vector), is supported on
the subset $V_{1}\otimes v_{2}^{j}$ if $y_{j}=v_{2}^{j}$. In the matrix form,
it is supported on the $j$th column of $Z^{(1)}$.
Figure 1: Graphical illustration the hypergraph product structure: (a) The
qubit subset $V_{1}\otimes V_{2}$ is partitioned into $\\{V_{1}\otimes
v_{2}^{1},V_{1}\otimes v_{2}^{2},\ldots,V_{1}\otimes v_{2}^{n_{2}}\\}$, where
$\\{v_{2}^{1},v_{2}^{2},\ldots,v^{n_{2}}_{2}\\}\in V_{2}$. (b) The qubit
subset $C_{1}\otimes C_{2}$ is divided into $\\{c^{1}_{1}\otimes
C_{2},c^{2}_{1}\otimes C_{2},\ldots,c^{r_{1}}_{1}\otimes C_{2}\\}$, with
$\\{c_{1}^{1},c^{2}_{1},\cdots,c^{r_{1}}_{1}\\}\in C_{1}$. An elementary
canonical logical operator $(\bar{x}_{k}\otimes y_{j},0_{r_{1}r_{2}})^{T}$
resides on the qubit subset $V_{1}\otimes v^{j}_{2}$ if $y_{j}=v_{2}^{j}$.
Similarly, the logical operator
$(0_{n_{1}n_{2}},a_{\ell}\otimes\bar{b}_{m})^{T}$ is localized on the subset
$c^{i}_{1}\otimes C_{2}$ if $c^{i}_{1}=a_{\ell}$.
It would be instructive to discuss an example of the canonical logical
operators. It is well-known that Kitaev’s toric code Kitaev (2003) can be
viewed as two copies of 1D repetition code Tillich and Zemor (2014). In this
context, the canonical logical operators are the string operators of minimal
lengths [Fig. 2].
Figure 2: The hypergraph product construction of the Toric code from two
repetition codes (periodic boundary conditions). (a) Delineates the two
subsets of qubits, $V_{1}\otimes V_{2}$ and $C_{1}\otimes C_{2}$. (b) Show the
logical $X$ operator (blue circles) and the logical-$Z$ operator (red circles)
within the subset $V_{1}\otimes V_{2}$. (c) Showcasing the logical $X$ (blue
circles) and $Z$ (red circles) operators within the subset $C_{1}\otimes
C_{2}$.
_Energy barrier of hypergraph product code._ \- We now study relationship
between the energy barrier of hypergraph product codes and their underlying
classical codes. We first upper bound the energy barrier of the hypergraph
product code in terms of the energy barrier of the classical codes. We then
prove a matching lower bound, establishing their equivalence.
_(1) Upper bound_.\- The upper bound on the energy barrier can be obtained by
considering a specific path from the identity to some logical operator. By
choosing this logical operator to be a canonical logical operator, we obtain
an upper bound. First, consider an elementary canonical logical operator of
the form of $(\bar{x}_{k}\otimes y_{j},0_{r_{1}r_{2}})^{T}$, where
$0_{r_{1}r_{2}}$ is the $r_{1}r_{2}$-dimensional zero vector. We will consider
a path consisting of the operators of the form of $(x_{i}\otimes y_{j})^{T}$
from $i=1$ to $i=N$, interpolating between $x_{1}=(0,\ldots,0)^{T}$ to
$x_{N}=\overline{x}_{k}$. The energy at the $i$th step is precisely
$\mathrm{wt}(H_{1}x_{i})$, which is the energy of the classical code $H_{1}$
evaluated with respect to $x_{i}$. Because $x_{N}=\overline{x}_{k}$ is a
codeword of $H_{1}$, the energy barrier along this path is at most
$\Delta(H_{1})$.
Similarly, we can consider a path of the form of
$(0_{n_{1}n_{2}},a_{\ell}\otimes b_{i})$ from $i=1$ to $i=N$, interpolating
between $b_{1}=(0,\ldots,0)^{T}$ to $b_{N}=\overline{b}_{m}$, where
$\overline{b}_{m}$ is a coreword of $H_{2}^{T}$. This yields an energy barrier
upper bound of $\Delta(H_{2}^{T})$. Together, we obtain an energy upper bound
of $\min(\Delta(H_{1}),\Delta(H_{2}^{T}))$ for the canonical logical-$Z$
operators. A similar argument can be applied to the canonical logical $X$
operators, yielding an upper bound of $\min(\Delta(H_{2}),\Delta(H_{1}^{T}))$.
_(2) Lower bound_.\- We now prove a matching lower bound for the energy
barrier of the canonical logical-$Z$ operators.
###### Proposition 1.
For any nontrivial canonical logical-$Z$ operator $L$,
$\Delta(L)\geq\min(\Delta(H_{1}),\Delta(H_{2}^{T})).$ (11)
We introduce two lemmas to prove this proposition. We use the following
convention in the proof. A path $r=\left\\{P_{0},P_{1},\ldots,P_{F}\right\\}$
is said to be supported on $U\subseteq V$ if the support of all $P_{i}$ in $r$
is included in $U$.
###### Lemma 2.
For any elementary canonical logical-$Z$ operator $L$ supported on
$V_{1}\otimes v^{\alpha}_{2}$ (resp. $c_{1}^{\beta}\otimes C_{2}$), its energy
barrier is attained by a path supported on $V_{1}\otimes v^{\alpha}_{2}$
(resp. $c^{\beta}_{2}\otimes C_{2}$).
###### Lemma 3.
For any nontrivial canonical logical-$Z$ operator $L$, the energy barrier
$\Delta(L)$ is greater than or equal to the minimum energy barrier of the
elementary canonical logical-$Z$ operators.
Consider an elementary canonical logical-$Z$ operator $L$ supported on
$V_{1}\otimes v_{2}^{\alpha}$. Suppose $\Delta(L)$ is given by a path $r$. The
main idea behind the proof of Lemma 2 is to deform a general path $r$ into a
new path $r^{\prime}$, supported on $V_{1}\otimes v_{2}^{\alpha}$.
Importantly, we show that the energy barrier of $r^{\prime}$ is no greater
than that of the original path $r$. A similar argument can be applied to prove
Lemma 3; see the Appendix for the proofs.
Let $L=(\bar{x}_{k}\otimes v_{2}^{j},0_{r_{1}r_{2}})^{T}$ be an elementary
canonical logical-$Z$ operator, supported on $V_{1}\otimes v_{2}^{j}$. Lemma 2
implies there is a path $r$ supported on $V_{1}\otimes v_{2}^{j}$ that attains
the energy barrier of $\Delta(L)$.
Using such a path, we can prove a lower bound on the energy barrier of $L$.
Without loss of generality, let $r=(P_{0},P_{1},\cdots,P_{F})$ with $P_{0}=I$
and $P_{F}=L$. Because this path is supported on $V_{1}\otimes v_{2}^{j}$, in
the binary representation, each $P_{i}$ becomes $u_{i}\otimes v_{2}^{j}$ for
some $u_{i}\in V_{1}$. In particular, at the $i$’th step, the energy is
$\text{wt}(H_{1}u_{i})$, which is exactly the energy of the classical code
with respect to $u_{i}\in V_{1}$. Because $u_{1}=(0,\ldots,0)$ and
$u_{F}=\bar{x}_{k}$ for some codeword $\bar{x}_{k}$ of $H_{1}$, the path
$(u_{0},\ldots,u_{F})$ must have an energy barrier of at least
$\Delta(H_{1})$. Similarly, for any elementary canonical logical-$Z$ operator
$L$ on subset $c_{1}^{\beta}\otimes C_{2}$, one can show that
$\Delta\left(L\right)\geq\Delta\left(H_{2}^{T}\right)$. Then Lemma 3
imeidately implies Proposition 1.
_(3) Energy barrier_.\- Since the lower bound and the upper bound match, the
energy barrier of the canonical logical-$Z$ operators is precisely
$\min(\Delta(H_{1}),\Delta(H_{2}^{T}))$. Similarly, the energy barrier of the
canonical logical-$X$ operators can be shown to be
$\min(\Delta(H_{2}),\Delta(H_{1}^{T}))$. Moreover, due to Theorem 1, the
energy barrier of any logical operator can be bounded in terms of the
canonical logical operators’ energy barrier and the code’s sparsity parameter.
Note that this conclusion applies to the hypergraph product of _any_ two
classical codes.
If the code is LDPC and the energy barrier is $\Omega(1)$, we conclude that
the energy barrier of the quantum code is exactly equal to
$\min(\Delta(H_{1}),\Delta(H_{2}),\Delta(H_{1}^{T}),\Delta(H_{2}^{T}))$.
###### Theorem 2.
Let $\Delta(H)$ be the energy barrier of the hypergraph product code obtained
from parity check matrices $H_{1}$ and $H_{2}$ of $O(1)$ row and column
weight. If
$\Delta(H_{1}),\Delta(H_{2}),\Delta(H_{1}^{T}),\Delta(H_{2}^{T})=\Omega(1)$,
$\Delta(H)=\min(\Delta(H_{1}),\Delta(H_{2}),\Delta(H_{1}^{T}),\Delta(H_{2}^{T})).$
(12)
More precisely, for any $\left(w_{c},w_{q}\right)$ quantum code derived from a
hypergraph product, if the parent classical codes’ energy barriers exceed
$w_{c}w_{q}$, the quantum code’s energy barrier matches the minimum energy
barrier of these parent codes. This can be used to prove tight bounds on the
energy barrier. For instance, a classical code whose Tanner graph is a
bipartite expander graph has an extensive macroscopic energy barrier. Thus our
result implies that the quantum expander code Leverrier _et al._ (2015);
Fawzi _et al._ (2020) also has an extensive energy barrier. While this is
already known Fawzi _et al._ (2020), note that this argument does not rely on
the expansion property of quantum expander code.
_Outlook_.\- We proved a tight bound on the energy barrier of the hypergraph
product code, determined in terms of the energy barriers of the underlying
classical codes. While it was expected that the energy barrier of the quantum
code is related to its counterpart Rakovszky and Khemani (2024), we provided a
first rigorous proof of this statement [Eq. (1)] to our knowledge. Looking
forward, it would be interesting to study the energy barrier of the codes such
as the homological product Bravyi and Hastings (2014), balanced product codes
Breuckmann and Eberhardt (2021a) and lifted product codes Panteleev and
Kalachev (2022a). The generalized bicycle code Kovalev and Pryadko (2013);
Panteleev and Kalachev (2022a), due to its simple structure, is another
natural candidate to explore. Understanding how our proof technique can be
generalized to these setups and how to design efficient decoders that can
leverage the energy barrier is left for future work.
_Acknowledgement_ \- We thank Niko Breuckmann, Sergey Bravyi, Earl Campbell,
Jeongwan Haah, Vedika Khemani, and Anthony Leverrier for helpful discussions.
GZ acknowledges the financial support from Sydney Quantum Academy. This work
was supported by the Australian Research Council Centre of Excellence for
Engineered Quantum Systems (EQUS, CE170100009). IK acknowledges support from
NSF Grant QCIS-FF 2013562.
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## I Appendix
Throughout the appendix, we are always working in the field $\mathbb{F}_{2}$.
Thus, all addition operations are modulo $2$ except for the computation of the
weight of vectors or matrices, i.e., the function $\mathrm{wt}(\cdot)$.
### I.1 Vector reshaping
Consider a basis $\mathcal{B}$ of the vector space
$\mathbb{F}_{2}^{n_{1}}\otimes\mathbb{F}_{2}^{n_{2}}$:
$\displaystyle\mathcal{B}=\left\\{a_{i}\otimes b_{j}\mid
i=1,\ldots,n_{1}\text{ and }j=1,\ldots,n_{2}\right\\}.$ (13)
Then any vector $v\in\mathbb{F}_{2}^{n_{1}}\otimes\mathbb{F}_{2}^{n_{2}}$ can
be written as
$\displaystyle v=\sum_{a_{i}\otimes
b_{j}\in\mathcal{B}}v_{ij}\left(a_{i}\otimes b_{j}\right)$ (14)
for some $v_{ij}\in\mathbb{F}_{2}$. We call the $n_{1}\times n_{2}$ matrix $V$
with entries $v_{ij}$ the _reshaping_ of the vector $v$. By this definition,
if $A,B$ are respectively $m_{1}\times n_{1}$ and $m_{2}\times n_{2}$
matrices, then
$\displaystyle(A\otimes B)v\Rightarrow AVB^{T}.$ (15)
Define $\mathrm{wt}(M)$ as the number of ones in the vector (or matrix) $M$.
We have $\mathrm{wt}((A\otimes B)v)=\mathrm{wt}(AVB^{T})$.
### I.2 Proof of Lemma 2
Recall that a $Z$-type Pauli error of the hypergraph product code can be
expressed as a bit-string $z=\left(z^{(1)},z^{(2)}\right)^{T}$. The
corresponding energy $\epsilon(z)=\text{wt}(H_{X}z)$ is
$\epsilon(z)=\text{wt}\left((H_{1}\otimes I_{n_{2}})z^{(1)}+(I_{r_{1}}\otimes
H_{2}^{T})z^{(2)}\right).$ (16)
By applying vector reshaping, the energy can be written as
$\epsilon(z)=\text{wt}\left(H_{1}Z^{(1)}+Z^{(2)}H_{2}\right),$ (17)
where $Z^{(1)}$ and $Z^{(2)}$ are the matrices reshaped from $z^{(1)}$ and
$z^{(2)}$, respectively. The $j$th column of $Z^{(1)}$ is supported on qubit
subset $V_{1}\otimes v_{2}^{j}$.
Using Eq. (17), we aim to prove a lower bound on the energy $\epsilon(z)$
[Lemma 4]. To that end, we shall use the following convention. Let $L_{c}$ be
a nontrivial codeword of $H_{2}$. The set of columns of $Z^{(1)}$ associated
with the nonzero entries of $L_{c}$ will play an important role. We define the
index set of such columns as $\mathcal{C}(L_{c})$:
$\mathcal{C}(L_{c}):=\\{j:(L_{c})_{j}=1\\}.$ (18)
Given a codeword $L_{c}$ of $H_{2}$, one can construct a vector $z^{1,s}$ from
$Z^{(1)}$ by summing all the columns in the set $\mathcal{C}(L_{c})$. More
formally,
$z^{1,s}_{i}=\sum_{j:j\in\mathcal{C}(L_{c})}Z^{(1)}_{ij},$ (19)
where the addition is modulo $2$. For example, if $L_{c}=110100$, we would sum
columns $1,2$, and $4$. Using Eq. (19), we can deform an arbitrary path to a
path consisting of Paulis only supported on subset $V_{1}\otimes v_{2}^{k}$
for some $k$.111The precise choice of $k$ does not matter; any
$k\in\mathcal{C}(L_{c})$ would suffice. In particular, we can prove an
inequality between the energy of the original Pauli and the deformed Pauli,
proved in Lemma 4.
###### Lemma 4.
$\mathrm{wt}(H_{1}z^{1,s})\leqslant\mathrm{wt}\left(H_{1}Z^{(1)}+Z^{(2)}H_{2}\right).$
(20)
###### Proof.
We prove this by contradiction. Consider the row spaces of $H_{1}z^{1,s}$ and
$H_{1}Z^{(1)}+Z^{(2)}H_{2}$. If
$\mathrm{wt}(H_{1}z^{1,s})>\mathrm{wt}\left(H_{1}Z^{(1)}+Z^{(2)}H_{2}\right)$,
then there exists a row $j$ such that
$\displaystyle\mathrm{wt}((H_{1}z^{1,s})_{\text{row
}j})>\mathrm{wt}\left((H_{1}Z^{(1)}+Z^{(2)}H_{2})_{\text{row }j}\right).$ (21)
We will prove that Eq. (21) cannot be satisfied, thereby proving the claim.
Without loss of generality, consider the $j$’th row. Since $H_{1}z^{1,s}$ is a
$r_{1}\times 1$ vector, $\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})$ must be
either 0 or 1. If $\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})=0$, Eq. (21)
cannot be satisfied. Therefore, we consider the
$\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})=1$ case.
If $\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})=1$, $(H_{1}Z^{(1)})_{\text{row
}j}$ must contain an odd number of ones on the columns in the set
$\mathcal{C}(L_{c})$. Otherwise, we would have had
$\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})=0$, which is a contradiction. On
the other hand, we remark that $(Z^{(2)}H_{2})_{\text{row }j}$ consists of an
even number of ones on the column set $\mathcal{C}(L_{c})$. To see why, note
that each row of $Z^{(2)}H_{2}$ is a linear combination of the checks in
$H_{2}$. Because $L_{c}$ is a codeword of $H_{2}$,
$(Z^{(2)}H_{2})_{\text{row},j}L_{c}=0$. Therefore, in the
$\mathrm{wt}((H_{1}z^{1,s})_{\text{row }j})=1$ case, the number of ones in
$H_{1}Z^{(1)}+Z^{(2)}H_{2}$ on the $j$’th row and the columns in
$\mathcal{C}(L_{c})$ is odd.
Thus we conclude $\mathrm{wt}\left((H_{1}Z^{(1)}+Z^{(2)}H_{2})_{\text{row
}j}\right)\geq 1$. As such, Eq. (21) cannot be satisfied. This completes the
proof. ∎
Now we are in a position to prove Lemma 2. We do so by identifying a path
$r^{\prime}=\left\\{P_{0}^{\prime},P_{1}^{\prime},\cdots,P_{F}^{\prime}\right\\}$
that is only supported on $V_{1}\otimes v_{2}^{\alpha}$ while ensuring that
$\epsilon_{\max}\left(r^{\prime}\right)\leqslant\epsilon_{\max}(r)$. Lemma 4
suggests a way to deform the path $r$ to the one supported on $V_{1}\otimes
v^{\alpha}_{2}$.
Without loss of generality, let $r=\left\\{P_{0},P_{1},\cdots,P_{F}\right\\}$
be a path that $\Delta(L)=\epsilon_{\max}(r)$, with $P_{0}=I$ and $P_{F}=L$.
We consider a Pauli $P_{i}$ in the path $r$. It will be convenient to work in
its binary representation, written as $(p^{(1)},p^{(2)})^{T}$, where $p^{(1)}$
and $p^{(2)}$ represent the Paulis supported on $V_{1}\otimes V_{2}$ and
$C_{1}\otimes C_{2}$, respectively. First, we remove all the Paulis supported
on $C_{1}\otimes C_{2}$ by setting $p^{(2)}$ as the zero vector. Next, we
apply the following transformations to $p^{(1)}$. We reshape $p^{(1)}$ and
refer to the reshaped matrix as $P^{(1)}$. Let $L_{c}$ be a codeword of
$H_{2}$ such that $(L_{c})_{\alpha}=1$. Consider a set of columns in $P^{(1)}$
corresponding to the index set $\mathcal{C}(L_{c})$. Denoting each column as
$u_{k}$, where $k\in\mathcal{C}(L_{c})$, we update the column $u_{\alpha}$ in
the following way:
$u_{\alpha}\to
u_{\alpha}+\sum_{k\in\mathcal{C}(L_{c})\setminus\\{\alpha\\}}u_{k}.$ (22)
Afterwards, the other columns of $P^{(1)}$ are set to the zero vector. This
yields the deformed Pauli operator $P_{i}^{\prime}$.
By construction, the resulting $P_{i}^{\prime}$ is supported on $V_{1}\otimes
v^{\alpha}_{2}$. Note that $r^{\prime}$ is a valid path because
$\mathrm{wt}(P_{i}^{\prime}P_{i+1}^{\prime})\leq 1$ for every $i$. Also,
because $P_{F}^{\prime}=P_{F}=L$, $r^{\prime}$ is still a path for $L$.
Moreover, because
$\epsilon\left(P_{i}^{\prime}\right)\leqslant\epsilon\left(P_{i}\right)$ for
all $i$ [Lemma 4], we have
$\epsilon_{\max}\left(r^{\prime}\right)\leqslant\epsilon_{\max}(r)$. Both
$r^{\prime}$ and $r$ are paths for $L$, by definition
$\epsilon_{\max}\left(r^{\prime}\right)\geqslant\epsilon_{\max}(r)$, we
conclude $\epsilon_{\max}\left(r^{\prime}\right)=\epsilon_{\max}(r)$. Thus, by
deforming $r$, we obtained a new path $r^{\prime}$ supported on $V_{1}\otimes
v^{\alpha}_{2}$ that yields the energy barrier $\Delta(L)$.
This argument can be applied to prove similar lower bounds for logical
operators on $c^{\beta}_{1}\otimes C_{2}$. To conclude, for any elementary
canonical logical operator $L$ supported on $V_{1}\otimes v^{\alpha}_{2}$
(resp. $c^{\beta}_{1}\otimes C_{2}$), their energy barrier can be given by a
path supported on $V_{1}\otimes v^{\alpha}_{2}$ (resp. $c^{\beta}_{1}\otimes
C_{2}$).
### I.3 Proof of Lemma 3
Any nontrivial canonical logical-$Z$ operator $L$ belongs to one of the
following categories:
* •
Case 1: $L$ is supported solely on the qubit subset $V_{1}\otimes V_{2}$.
* •
Case 2: $L$ is supported solely on the qubit subset $C_{1}\otimes C_{2}$.
* •
Case 3: $L$ is supported on both subsets.
We will focus solely on Case 1. Case 2 can be analyzed similarly by
considering subsets $C_{1}\otimes C_{2}$, while case 3 can be treated as Case
1 or 2.
Without loss of generality, let the energy barrier of $L$ be attained by a
path $r=\\{P_{0},P_{1},\cdots,P_{F}\\}$, with $P_{0}=I$ and $P_{F}=L$. Similar
to the approach taken in Lemma 2, we aim to deform the path $r$ to the one
supported on $V_{1}\otimes v_{2}^{k}$ for some $k$, such that the energy
barrier of the deformed path lower bounds that of the $r$.
The deformation works in the same way as in the proof of Lemma 2. We describe
this procedure again for the readers’ convenience. Let $L_{c}$ be a nontrivial
codeword of $H_{2}$ and $\mathcal{C}(L_{c})$ be its corresponding column index
set. We consider a binary representation of a Pauli $P_{i}$, written as
$(p^{(1)},p^{(2)})^{T}$. As in the proof of Lemma 2, we remove the Paulis
supported on $C_{1}\otimes C_{2}$ by setting $p^{(2)}$ as the zero vector.
Next, reshape $p^{(1)}$ into a matrix $P^{(1)}$ and update its columns in the
following way. Choose $\alpha\in\mathcal{C}(L_{c})$. This column is updated as
Eq. (22). The other columns of $P^{(1)}$ are converted to zero vectors.
Thanks to Lemma 4, we obtain a new path
$r^{\prime}=\\{P_{0}^{\prime},P_{1}^{\prime},\cdots,P_{F}^{\prime}\\}$
supported on $V_{1}\otimes v_{\alpha}^{2}$ with the property
$\epsilon_{\max}\left(r^{\prime}\right)\leqslant\epsilon_{\max}\left(r\right)$.
Note that $P_{F}^{\prime}$, in the binary representation, is of the form
$\bar{x}\otimes v_{\alpha}^{2}$, where $\bar{x}$ is a codeword of $H_{1}$.
Therefore, $P_{F}^{\prime}$ is either a nontrivial elementary canonical
logical operator or the identity. In the latter case, $P_{F}^{\prime}$ is the
trivial codeword (zero vector) in the binary representation. Henceforth, we
denote this as $L^{\prime}=P_{F}^{\prime}$.
If $L^{\prime}$ is nontrivial, we can use the relation between the energy
barriers of $L$ and $L^{\prime}$:
$\displaystyle\Delta(L)=\epsilon_{\max}(r)\geqslant\epsilon_{\max}(r^{\prime})\geqslant\Delta(L^{\prime}).$
(23)
Because $L^{\prime}$ is an elementary logical operator, $\Delta(L)$ is greater
or equal to the minimum energy barrier of elementary canonical logical
operators. Thus, if $L^{\prime}$ is nontrivial, the proof follows immediately.
If $L^{\prime}$ is an identity, the above argument does not work. Fortunately,
it turns out that for any $L^{\prime}$, one can choose $L_{c}$ (the codeword
of $H_{2}$ used in the current proof) such that $L^{\prime}$ is not an
identity.
Without loss of generality, consider a canonical logical-$Z$ operator $L$,
expressed as
$L=\begin{pmatrix}\sum_{k,j}\lambda_{kj}\bar{x}_{k}\otimes y_{j}\\\
0_{r_{1}r_{2}}\end{pmatrix},$ (24)
where (i) $H_{1}\bar{x}_{i}=0$ and (ii)
$y_{j}\notin\operatorname{Im}\left(H_{2}^{T}\right)$ are unit vectors. If a
given path ends with $L$, its deformation (using Eq. (22)) yields the
following operator $L^{\prime}$:
$L^{\prime}=\begin{pmatrix}\sum_{k}c_{k}\bar{x}_{k}\otimes y_{\alpha}\\\
0_{r_{1}r_{2}}\end{pmatrix},$ (25)
where $\alpha\in\mathcal{C}(L_{c})$ and $c_{k}$ is defined as
$c_{k}:=\sum_{j\in\mathcal{C}(L_{c})}\lambda_{kj}.$ (26)
Note that $L^{\prime}$ is trivial if and only if $c_{k}=0$ for all $k$.
Therefore, we aim to prove that there exists a choice of $L_{c}$ such that
$c_{k}=1$ for at least one $k$.
Let us prove the contrapositive. Suppose $c_{k}=0$ for all $k$, for any choice
of $L_{c}$. Consider the following vector:
$u_{k}:=\sum_{j}\lambda_{kj}y_{j}.$ (27)
Note that $c_{k}=u_{k}^{T}L_{c}$. By our assumption $c_{k}=0$ for any choice
of $L_{c}$ and so the inner product of $u_{k}$ with any codeword of $H_{2}$
must be zero. On the other hand, $u_{k}$, if it is nonzero, must lie outside
of $\text{Im}(H_{2}^{T})$ by the definition of the $y_{j}$’s. Thus, $u_{k}$ is
not an element of the row space of $H_{2}$. However, this is a contradiction
for the following reason. For a linear code, let $H$ and $G$ be the parity
check matrix and the generator matrix. Then $v^{T}G=0$ if and only if $v$ is a
vector in the row space of $H$. In our setup, if $c_{k}=0$ for any $L_{c}$,
then $u_{k}^{T}G_{2}=0$. This implies that $u_{k}$ must be in the row space of
$H_{2}$, which contradicts the fact that it lies outside of
$\text{Im}(H_{2}^{T})$. To conclude, there must be at least one $k$ such that
$c_{k}=1$. Thus, there is always a choice of $L_{c}$ such that $L^{\prime}$ is
not an identity, thereby proving the claim.
Case 2 can be analyzed similarly to Case 1 by considering rows in the subset
$C_{1}\otimes C_{2}$. For Case 3, one can treat it just as Case 1 or Case 2.
For example, when treating it as Case 1, the logical operator $L$ has a
nontrivial part in the subset $V_{1}\otimes V_{2}$. One can prove there exists
a codeword $L_{c}$ of $H_{2}$, such that after the deformation, the resulting
$L^{\prime}$ is a nontrivial elementary canonical logical operator.
|
# Cohomology and deformations of compatible Hom-Lie algebras
Apurba Das ††Department of Mathematics, Indian Institute of Technology,
Kharagpur 721302, West Bengal, India. ††Email<EMAIL_ADDRESS>
###### Abstract
In this paper, we consider compatible Hom-Lie algebras as a twisted version of
compatible Lie algebras. Compatible Hom-Lie algebras are characterized as
Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We
also define a cohomology theory for compatible Hom-Lie algebras generalizing
the recent work of Liu, Sheng and Bai. As applications of cohomology, we study
abelian extensions and deformations of compatible Hom-Lie algebras.
2020 Mathematics Subject Classification: 17B61, 17A30, 17B56, 16S80.
Keywords: Hom-Lie algebras, Compatible structures, Cohomology, Extensions,
Deformations.
###### Contents
1. 1 Introduction
2. 2 Hom-Lie algebras and bidifferential graded Lie algebras
3. 3 Compatible Hom-Lie algebras
4. 4 Cohomology of compatible Hom-Lie algebras
5. 5 Deformations of compatible Hom-Lie algebras
## 1 Introduction
The notion of Hom-algebras first appeared in the $q$-deformations of the Witt
and Virasoro algebra by the work of Hartwig, Larsson and Silvestrov [11]. More
precisely, they introduced Hom-Lie algebras as a twisted version of Lie
algebras, where the usual Jacobi identity is twisted by a linear homomorphism
(see Definition 2.1). Later, various others algebras (e.g. associative,
Leibniz, Poisson, …) twisted by homomorphisms are also studied [14, 3, 23].
These Hom-algebras are widely explored in the last 15 years. In [1, 15] the
authors study cohomology and deformations of Hom-associative and Hom-Lie
algebras. In particular, they generalize the classical Gerstenhaber bracket
and Nijenhuis-Richardson bracket on the cochain complex of Hom-associative and
Hom-Lie algebras. See also [20, 23, 24] and references therein for more on
Hom-algebras.
Two algebraic structures of the same kind are said to be compatible if their
sum also defines a same type of algebraic structure. They appeared in many
contexts of mathematics and mathematical physics. For instance, the notion of
compatibility of two Poisson structures on a manifold was first appeared in
the mathematical study of biHamiltonian mechanics [12, 16]. Using the
correspondence between Lie algebra structures on a vector space $\mathfrak{g}$
and linear Poisson structures on $\mathfrak{g}^{\ast}$, one lead to a notion
of compatible Lie algebras [12]. See [9, 19] for more study on compatible Lie
algebras. Some other compatible structures include compatible associative
algebras [18], compatible Lie bialgebras [22], compatible Lie algebroids and
Lie bialgebroids [5]. See also [7, 21] for the operadic study of compatible
algebraic structures.
Recently, a cohomology theory for compatible Lie algebras has been introduced
by Liu, Sheng and Bai [13]. This cohomology is based on the characterization
of a compatible Lie algebra as Maurer-Cartan element in a bidifferential
graded Lie algebra. It is also seen that this cohomology is related to
extensions and deformations of compatible Lie algebras. These results have
been extended to compatible associative algebras in [4]. The present paper
aims to define and study compatible Hom-Lie algebras. We observe that
compatible Hom-Lie algebras are related to compatible Hom-Poisson manifolds.
We first define representations and cohomology of a compatible Hom-Lie
algebra. We relate our cohomology of a compatible Hom-Lie algebra with the
cohomology of Hom-Lie algebra. As applications of our cohomology, we study
abelian extensions and linear, finite order deformations of a compatible Hom-
Lie algebra.
The paper is organized as follows. In Section 2 (preliminary section), we
recall Hom-Lie algebras and some basics on bidifferential graded Lie algebras.
In Section 3, we introduce compatible Hom-Lie algebras and their Maurer-Cartan
characterizations in a suitably constructed bidifferential graded Lie algebra.
We also define the notion of representation of a compatible Hom-Lie algebra
and construct the semidirect product. The cohomology of a compatible Hom-Lie
algebra with coefficients in a representation is given in Section 4. We also
introduce abelian extensions of a compatible Hom-Lie algebra and characterize
equivalence classes of abelian extensions by the second cohomology group. In
Section 5, we first define linear deformations of a compatible Hom-Lie algebra
and introduce Nijenhuis operators that induce trivial linear deformations. We
show that the equivalence classes of infinitesimal deformations of a
compatible Hom-Lie algebra are in one-to-one correspondence with the second
cohomology group with coefficients in itself. Finally, we consider finite
order deformations of compatible Hom-Lie algebra and study their extensions.
All vector spaces, (multi)linear maps, tensor products are over a field
$\mathbb{K}$ of characteristic zero.
## 2 Hom-Lie algebras and bidifferential graded Lie algebras
In this preliminary section, we recall some basics on Hom-Lie algebras [14, 1,
20] and bidifferential graded Lie algebras [13].
###### 2.1 Definition.
A Hom-Lie algebra is a vector space $\mathfrak{g}$ together with a bilinear
skew-symmetric bracket
$[~{},~{}]:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$ and a
linear map $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying the
followings:
$\displaystyle\alpha[x,y]=[\alpha(x),\alpha(y)]\qquad(\text{multiplicativity}),$
$\displaystyle[[x,y],\alpha(z)]+[[y,z],\alpha(x)]+[[z,x],\alpha(y)]=0\qquad(\text{Hom-
Jacobi identity}),$
for all $x,y,z\in\mathfrak{g}$.
A Hom-Lie algebra as above may be denoted by the triple
$(\mathfrak{g},[~{},~{}],\alpha)$ or simply by $\mathfrak{g}$ when no
confusion arises. The bracket $[~{},~{}]$ is said to be the Hom-Lie bracket on
$\mathfrak{g}$ when the twisting map $\alpha$ is clear from the context.
It follows from the above definition that Hom-Lie algebras are a twisted
version of Lie algebras. More specifically, a Hom-Lie algebra
$(\mathfrak{g},[~{},~{}],\alpha)$ with $\alpha=\mathrm{id}$ is nothing but a
Lie algebra.
###### 2.2 Example.
Let $(\mathfrak{g},[~{},~{}])$ be a Lie algebra and
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a Lie algebra homomorphism.
Then the triple
$(\mathfrak{g},[~{},~{}]_{\alpha}=\alpha\circ[~{},~{}],\alpha)$ is a Hom-Lie
algebra, called induced by composition.
###### 2.3 Example.
Let $(A,\mu,\alpha)$ be a Hom-associative algebra, i.e., $A$ is a vector
space, $\mu:A\otimes A\rightarrow A,~{}(a,b)\mapsto a\cdot b$ is a bilinear
map and $\alpha:A\rightarrow A$ a linear map satisfying $\alpha(a\cdot
b)=\alpha(a)\cdot\alpha(b)$ and the following Hom-associativity $(a\cdot
b)\cdot\alpha(c)=\alpha(a)\cdot(b\cdot c),$ for $a,b,c\in A.$ If
$(A,\mu,\alpha)$ is a Hom-associative algebra, then $(A,[~{},~{}],\alpha)$ is
a Hom-Lie algebra, where $[a,b]=a\cdot b-b\cdot a$, for $a,b\in A$.
###### 2.4 Example.
Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra. Then for any
$n\geq 0$, the triple
$(\mathfrak{g},[~{},~{}]^{(n)}=\alpha^{n}\circ[~{},~{}],\alpha^{n+1})$ is a
Hom-Lie algebra, called the $n$-th derived Hom-Lie algebra.
###### 2.5 Definition.
Let $(\mathfrak{g},[~{},~{}],\alpha)$ and
$(\mathfrak{g}^{\prime},[~{},~{}]^{\prime},\alpha^{\prime})$ be two Hom-Lie
algebras. A linear map $\phi:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ is
said to be a Hom-Lie algebra morphism if
$\alpha^{\prime}\circ\phi=\phi\circ\alpha$ and
$\phi[x,y]=[\phi(x),\phi(y)]^{\prime},$ for $x,y\in\mathfrak{g}.$
Next we recall the graded Lie bracket (called the Nijenhuis-Richardson
bracket) whose Maurer-Cartan elements are given by Hom-Lie algebra structures
[1]. This generalizes the classical Nijenhuis-Richardson bracket in the
context of Lie algebras [17]. Let $\mathfrak{g}$ be a vector space and
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a linear map. For each $n\geq
0$, consider the spaces $C^{n}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ by
$\displaystyle
C^{0}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})=\\{x\in\mathfrak{g}|~{}\alpha(x)=x\\}~{}~{}\text{
and
}~{}~{}C^{n}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})=\\{f:\wedge^{n}\mathfrak{g}\rightarrow\mathfrak{g}|~{}\alpha\circ
f=f\circ\alpha^{\wedge n}\\},~{}n\geq 1.$
Then the shifted graded vector space
$C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})=\oplus_{n\geq
0}C^{n+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ carries a graded Lie
bracket defined as follows. For $P\in
C^{m+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ and $Q\in
C^{n+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$, the Nijenhuis-Richardson
bracket $[P,Q]_{\mathsf{NR}}\in
C^{m+n+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ given by
$\displaystyle[P,Q]_{\mathsf{NR}}=P\diamond Q-(-1)^{mn}~{}Q\diamond
P,~{}\text{ where }$ $\displaystyle(P\diamond
Q)(x_{1},\ldots,x_{m+n+1})=\sum_{\sigma\in
Sh(n+1,m)}(-1)^{\sigma}~{}P\big{(}Q(x_{\sigma(1)},\ldots,x_{\sigma(n+1)}),\alpha^{n}(x_{\sigma(n+2)}),\ldots,\alpha^{n}(x_{\sigma(m+n+1)}\big{)}.$
With this notation, we have the following.
###### 2.6 Proposition.
Let $\mathfrak{g}$ be a vector space and
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a linear map. Then the Hom-Lie
brackets on $\mathfrak{g}$ are precisely the Maurer-Cartan elements in the
graded Lie algebra
$(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}}).$
In the following, we recall the Chevalley-Eilenberg cohomology of a Hom-Lie
algebra $(\mathfrak{g},[~{},~{}],\alpha)$ with coefficients in a
representation.
###### 2.7 Definition.
Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra. A representation
of it consists of a vector space $V$ together with a bilinear operation
(called the action) $\bullet:\mathfrak{g}\otimes V\rightarrow V$,
$(x,v)\mapsto x\bullet v$ and a linear map $\beta:V\rightarrow V$ satisfying
$\displaystyle\beta(x\bullet v)=\alpha(x)\bullet\beta(v),$
$\displaystyle[x,y]\bullet\beta(v)=\alpha(x)\bullet(y\bullet
v)-\alpha(y)\bullet(x\bullet v),~{}\text{ for }x,y\in\mathfrak{g},v\in V.$
We denote a representation as above by $(V,\bullet,\beta)$ or simply by $V$.
It follows that any Hom-Lie algebra $(\mathfrak{g},[~{},~{}],\alpha)$ is a
representation of itself with the action given by the bracket $[~{},~{}].$
This is called the adjoint representation.
Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra and
$(V,\bullet,\beta)$ be a representation. For each $n\geq 0$, we define the
$n$-th cochain group $C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)$ as
$\displaystyle C^{0}_{\mathrm{Hom}}(\mathfrak{g},V)=\\{v\in
V|~{}\beta(v)=v\\}~{}~{}\text{ and
}~{}~{}C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)=\\{f:\wedge^{n}\mathfrak{g}\rightarrow
V|~{}\beta\circ f=f\circ\alpha^{\wedge n}\\},~{}n\geq 1.$
The coboundary operator
$\delta_{\mathrm{Hom}}:C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{Hom}}(\mathfrak{g},V)$, for $n\geq 0$, given by
(1) $\displaystyle(\delta_{\mathrm{Hom}}v)(x)=~{}$ $\displaystyle x\bullet
v,~{}\text{ for }v\in C^{0}_{\mathrm{Hom}}(\mathfrak{g},V),x\in\mathfrak{g},$
(2) $\displaystyle(\delta_{\mathrm{Hom}}f)(x_{1},\ldots,x_{n+1})=$
$\displaystyle\sum_{i=1}^{n+1}(-1)^{i+1}~{}\alpha^{n-1}(x_{i})\bullet
f(x_{1},\ldots,\widehat{x_{i}},\ldots,x_{n+1})$ $\displaystyle+\sum_{1\leq
i<j\leq
n+1}(-1)^{i+j}~{}f([x_{i},x_{j}],\alpha(x_{1}),\ldots,\widehat{\alpha(x_{i})},\ldots,\widehat{\alpha(x_{j})},\ldots,\alpha(x_{n+1})),$
for $f\in C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)$ and
$x_{1},\ldots,x_{n+1}\in\mathfrak{g}$. The cohomology groups of the cochain
complex $\\{C^{\ast}_{\mathrm{Hom}}(\mathfrak{g},V),\delta_{\mathrm{Hom}}\\}$
are called the Chevalley-Eilenberg cohomology groups, denoted by
$H^{\ast}_{\mathrm{Hom}}(\mathfrak{g},V).$
It is important to note that the coboundary operator for the Chevalley-
Eilenberg cohomology of the Hom-Lie algebra $(\mathfrak{g},[~{},~{}],\alpha)$
with coefficients in itself is simply given by
$\displaystyle\delta_{\mathrm{Hom}}f=(-1)^{n-1}[\mu,f]_{\mathsf{NR}},~{}\text{for
}f\in C^{n}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),$
where $\mu\in C^{2}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ corresponds to
the Hom-Lie bracket $[~{},~{}]$.
Bidifferential graded Lie algebras. Next, we recall bidifferential graded Lie
algebras [13]. Before that, let us first give the definition of a differential
graded Lie algebra.
###### 2.8 Definition.
A differential graded Lie algebra is a triple $(L=\oplus L^{i},[~{},~{}],d)$
consisting of a graded Lie algebra together with a degree $+1$ differential
$d:L\rightarrow L$ which is a derivation for the bracket $[~{},~{}]$.
An element $\theta\in L^{1}$ is said to be a Maurer-Cartan element in the
differential graded Lie algebra $(L,[~{},~{}],d)$ if $\theta$ satisfies
$\displaystyle d\theta+\frac{1}{2}[\theta,\theta]=0.$
###### 2.9 Definition.
A bidifferential graded Lie algebra is a quadruple $(L=\oplus
L^{i},[~{},~{}],d_{1},d_{2})$ in which the triples $(L,[~{},~{}],d_{1})$ and
$(L,[~{},~{}],d_{2})$ are differential graded Lie algebras additionally
satisfying $d_{1}\circ d_{2}+d_{2}\circ d_{1}=0.$
###### 2.10 Remark.
Any graded Lie algebra can be considered as a bidifferential graded Lie
algebra with both the differentials $d_{1}$ and $d_{2}$ to be trivial.
###### 2.11 Definition.
Let $(L,[~{},~{}],d_{1},d_{2})$ be a bidifferential graded Lie algebra. A pair
of elements $(\theta_{1},\theta_{2})\in L^{1}\oplus L^{1}$ is said to be a
Maurer-Cartan element if
* (i)
$\theta_{1}$ is a Maurer-Cartan element in the differential graded Lie algebra
$(L,[~{},~{}],d_{1})$;
* (ii)
$\theta_{2}$ is a Maurer-Cartan element in the differential graded Lie algebra
$(L,[~{},~{}],d_{2})$;
* (iii)
the following compatibility condition holds
$\displaystyle d_{1}\theta_{2}+d_{2}\theta_{1}+[\theta_{1},\theta_{2}]=0.$
Like a differential graded Lie algebra can be twisted by a Maurer-Cartan
element, the same result holds for bidifferential graded Lie algebras.
###### 2.12 Proposition.
Let $(L,[~{},~{}],d_{1},d_{2})$ be a bidifferential graded Lie algebra and let
$(\theta_{1},\theta_{2})$ be a Maurer-Cartan element. Then the quadruple
$(L,[~{},~{}],d_{1}^{\theta_{1}},d_{2}^{\theta_{2}})$ is a bidifferential
graded Lie algebra, where
$\displaystyle d_{1}^{\theta_{1}}=d_{1}+[\theta_{1},-]~{}~{}~{}\text{ and
}~{}~{}~{}d_{2}^{\theta_{2}}=d_{2}+[\theta_{2},-].$
For any $\vartheta_{1},\vartheta_{2}\in L^{1}$, the pair
$(\theta_{1}+\vartheta_{1},\theta_{2}+\vartheta_{2})$ is a Maurer-Cartan
element in the bidifferential graded Lie algebra $(L,[~{},~{}],d_{1},d_{2})$
if and only if $(\vartheta_{1},\vartheta_{2})$ is a Maurer-Cartan element in
the bidifferential graded Lie algebra
$(L,[~{},~{}],d_{1}^{\theta_{1}},d_{2}^{\theta_{2}}).$
## 3 Compatible Hom-Lie algebras
In this section, we introduce compatible Hom-Lie algebras and give a Maurer-
Cartan characterization. We end this section by defining representations of
compatible Hom-Lie algebras.
Let $\mathfrak{g}$ be a vector space and
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a linear map.
###### 3.1 Definition.
Two Hom-Lie algebras $(\mathfrak{g},[~{},~{}]_{1},\alpha)$ and
$(\mathfrak{g},[~{},~{}]_{2},\alpha)$ are said to be compatible if for all
$\lambda,\eta\in\mathbb{K}$, the triple
$(\mathfrak{g},\lambda[~{},~{}]_{1}+\eta[~{},~{}]_{2},\alpha)$ is a Hom-Lie
algebra.
The condition in the above definition is equivalent to the following
(3)
$\displaystyle[[x,y]_{1},\alpha(z)]_{2}+[[y,z]_{1},\alpha(x)]_{2}+[[z,x]_{1},\alpha(y)]_{2}+[[x,y]_{2},\alpha(z)]_{1}+[[y,z]_{2},\alpha(x)]_{1}+[[z,x]_{2},\alpha(y)]_{1}=0,$
for all $x,y,z\in\mathfrak{g}.$
###### 3.2 Definition.
A compatible Hom-Lie algebra is a quadruple
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ in which
$(\mathfrak{g},[~{},~{}]_{1},\alpha)$ and
$(\mathfrak{g},[~{},~{}]_{2},\alpha)$ are both Hom-Lie algebras and are
compatible.
In this case, we say that the pair $([~{},~{}]_{1},[~{},~{}]_{2})$ is a
compatible Hom-Lie algebra structure on $\mathfrak{g}$ when the twisting map
$\alpha$ is clear from the context.
Compatible Hom-Lie algebras are twisted version of compatible Lie algebras
[12]. Recall that a compatible Lie algebra is a triple
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2})$ in which
$(\mathfrak{g},[~{},~{}]_{1})$ and $(\mathfrak{g},[~{},~{}]_{2})$ are Lie
algebras and are compatible in the sense that
$\lambda[~{},~{}]_{1}+\eta[~{},~{}]_{2}$ is a Lie bracket on $\mathfrak{g}$,
for all $\lambda,\eta\in\mathbb{K}$. Thus, a compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ with $\alpha=\mathrm{id}$
is nothing but a compatible Lie algebra.
It is known that compatible Lie algebras are closely related with compatible
Poisson structures [12]. Hence compatible Poisson structures can be thought of
as a motivation to study compatible Lie algebras. In the following, we study
compatible Hom-Poisson structures and relate them with compatible Hom-Lie
algebras. Recall that a Hom-Poisson manifold is a triple
$(M,\\{~{},~{}\\},\triangle)$ consisting of a smooth manifold $M$, a bilinear
skew-symmetric operation $\\{~{},~{}\\}:C^{\infty}(M)\times
C^{\infty}(M)\rightarrow C^{\infty}(M)$ and a linear map
$\triangle:C^{\infty}(M)\rightarrow C^{\infty}(M)$ satisfying for $f,g,h\in
C^{\infty}(M)$,
$\displaystyle\triangle\\{f,g\\}=\\{\triangle f,\triangle
g\\},\qquad\triangle(fg)=(\triangle f)(\triangle g),$
$\displaystyle\\{f,gh\\}=(\triangle g)\\{f,h\\}+\\{f,g\\}(\triangle h),$
$\displaystyle\\{\\{f,g\\},\triangle h\\}+\\{\\{g,h\\},\triangle
f\\}+\\{\\{h,f\\},\triangle g\\}=0.$
See [2] for more details. If $(M,\\{~{},~{}\\})$ is a Poisson manifold and
$\varphi:M\rightarrow M$ is a Poisson morphism then
$(M,\\{~{},~{}\\}_{\varphi}=\varphi^{*}\circ\\{~{},~{}\\},\triangle=\varphi^{*})$
is a Hom-Poisson manifold.
Let $(M,\\{~{},~{}\\}_{1},\triangle)$ and $(M,\\{~{},~{}\\}_{1},\triangle)$ be
two Hom-Poisson manifolds with same underlying manifold $M$ and same twisting
map $\triangle$. These two Hom-Poisson manifolds are said to be compatible if
for any $\lambda,\eta\in\mathbb{K}$, the triple
$(M,\lambda\\{~{},~{}\\}_{1}+\eta\\{~{},~{}\\}_{2},\triangle)$ is also a Hom-
Poisson manifold. In this case, we say that
$(M,\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{2},\triangle)$ is a compatible Hom-
Poisson manifold. A compatible Hom-Poisson manifold
$(M,\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{2},\triangle)$ is said to be ‘linear’ if
$M$ is a vector space and $\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{2},\triangle$
takes linear maps to linear maps.
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra. For any $f\in C^{\infty}(\mathfrak{g}^{*})$ and
$\xi\in\mathfrak{g}^{*}$, we consider the tangent map of $f$ at the point
$\alpha^{*}\xi$,
$\displaystyle\mathfrak{g}^{*}~{}\cong~{}T_{\alpha^{*}\xi}\mathfrak{g}^{*}\xrightarrow{T_{\alpha^{*}\xi}f}T_{f(\alpha^{*}\xi)}\mathbb{K}~{}\cong~{}\mathbb{K}.$
Since $T_{\alpha^{*}\xi}f$ is a linear map, it corresponds to an element of
$\mathfrak{g}$. Using this notation, we will now define brackets
$\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{1}:C^{\infty}(\mathfrak{g}^{*})\times
C^{\infty}(\mathfrak{g}^{*})\rightarrow C^{\infty}(\mathfrak{g}^{*})$ as
follows:
$\displaystyle\\{f,g\\}_{1}(\xi):=\langle[T_{\alpha^{*}\xi}f,T_{\alpha^{*}\xi}g]_{1},\xi\rangle~{}~{}~{}~{}\text{
and
}~{}~{}~{}~{}\\{f,g\\}_{2}(\xi):=\langle[T_{\alpha^{*}\xi}f,T_{\alpha^{*}\xi}g]_{2},\xi\rangle,~{}\text{
for }f,g\in C^{\infty}(\mathfrak{g}^{*}).$
By using the fact that $(\mathfrak{g},[~{},~{}]_{1},\alpha)$ is a Hom-Lie
algebra, it is easy to verify that the triple
$(\mathfrak{g}^{*},\\{~{},~{}\\}_{1},\triangle)$ is a Hom-Poisson manifold,
where $\triangle:C^{\infty}(\mathfrak{g}^{*})\rightarrow
C^{\infty}(\mathfrak{g}^{*})$ is the map $\triangle(f)=f\circ\alpha^{*}$.
Similarly, $(\mathfrak{g},[~{},~{}]_{2},\alpha)$ is a Hom-Lie algebra implies
that $(\mathfrak{g}^{*},\\{~{},~{}\\}_{2},\triangle)$ is a Hom-Poisson
manifold. Finally, the compatibility condition of the compatible Hom-Lie
algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ ensures that
$(\mathfrak{g}^{*},\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{2},\triangle)$ is a
compatible Hom-Poisson manifold. Finally, if $f,g$ are two linear functions on
$\mathfrak{g}^{*}$, then $\\{f,g\\}_{1},\\{f,g\\}_{2}$ and $\triangle(f)$ are
all linear functions on $\mathfrak{g}^{*}$. Hence it follows that
$(\mathfrak{g}^{*},\\{~{},~{}\\}_{1},\\{~{},~{}\\}_{2},\triangle)$ is a linear
compatible Hom-Poisson manifold.
###### 3.3 Definition.
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ and
$(\mathfrak{g}^{\prime},[~{},~{}]^{\prime}_{1},[~{},~{}]^{\prime}_{2},\alpha^{\prime})$
be two compatible Hom-Lie algebras. A morphism between them is a linear map
$\phi:\mathfrak{g}\rightarrow\mathfrak{g}^{\prime}$ which is a Hom-Lie algebra
morphism from $(\mathfrak{g},[~{},~{}]_{1},\alpha)$ to
$(\mathfrak{g}^{\prime},[~{},~{}]^{\prime}_{1},\alpha^{\prime})$, and a Hom-
Lie algebra morphism from $(\mathfrak{g},[~{},~{}]_{2},\alpha)$ to
$(\mathfrak{g}^{\prime},[~{},~{}]^{\prime}_{2},\alpha^{\prime}).$
In the following, we give some examples of compatible Hom-Lie algebras.
###### 3.4 Example.
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2})$ be a compatible Lie algebra
and $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a compatible Hom-Lie
algebra homomorphism, i.e., $\alpha$ is a Lie algebra homomorphism for both
the Lie algebras $(\mathfrak{g},[~{},~{}]_{1})$ and
$(\mathfrak{g},[~{},~{}]_{2})$. Then the quadruple
$(\mathfrak{g},\alpha\circ[~{},~{}]_{1},\alpha\circ[~{},~{}]_{2},\alpha)$ is a
compatible Hom-Lie algebra.
###### 3.5 Example.
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra. Then for each $n\geq 0$, the quadruple
$(\mathfrak{g},[~{},~{}]_{1}^{(n)}=\alpha^{n}\circ[~{},~{}]_{1},[~{},~{}]_{2}^{(n)}=\alpha^{n}\circ[~{},~{}]_{2},\alpha^{n+1})$
is a compatible Hom-Lie algebra. This is the $n$-th derived compatible Hom-Lie
algebra.
Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra. A Nijenhuis
operator on this Hom-Lie algebra is a linear map
$N:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying $\alpha\circ
N=N\circ\alpha$ and
$\displaystyle[Nx,Ny]=N([Nx,y]+[x,Ny]-N[x,y]),~{}\text{ for
}x,y\in\mathfrak{g}.$
Nijenhuis operators are useful to study linear deformations of a Hom-Lie
algebra [6]. Consider the $4$-dimensional Hom-Lie algebra
$(\mathfrak{g},[~{},~{}],\alpha)$, where $\mathfrak{g}=\langle
e_{1},e_{2},e_{3},e_{4}\rangle$ and structure maps are given by
$\displaystyle[e_{1},e_{2}]=ae_{1}+ae_{2},$
$\displaystyle\alpha(e_{1})=e_{2},~{}~{}~{}~{}\alpha(e_{2})=e_{1},~{}~{}~{}~{}\alpha(e_{3})=0~{}~{}~{}\text{
and }~{}~{}~{}\alpha(e_{4})=e_{3}.$
Then it is easy to verify that the map $N:\mathfrak{g}\rightarrow\mathfrak{g}$
defined by
$\displaystyle
N(e_{1})=e_{2},~{}~{}~{}~{}N(e_{2})=e_{1},~{}~{}~{}~{}N(e_{3})=e_{3},~{}~{}~{}~{}N(e_{4})=e_{4}$
is a Nijenhuis operator on the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}],\alpha)$.
###### 3.6 Example.
Let $N$ be a Nijenhuis operator on a Hom-Lie algebra
$(\mathfrak{g},[~{},~{}],\alpha)$. Then there is a deformed Hom-Lie bracket on
$\mathfrak{g}$ given by
$\displaystyle[x,y]_{N}:=[Nx,y]+[x,Ny]-N[x,y],~{}\text{ for
}x,y\in\mathfrak{g}.$
In other words $(\mathfrak{g},[~{},~{}]_{N},\alpha)$ is a Hom-Lie algebra. It
is easy to see that the quadruple
$(\mathfrak{g},[~{},~{}],[~{},~{}]_{N},\alpha)$ is a compatible Hom-Lie
algebra.
###### 3.7 Example.
Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra and
$(V,\bullet,\beta)$ be a representation of it. Suppose $f\in
C^{2}_{\mathrm{Hom}}(\mathfrak{g},V)$ is a $2$-cocycle in the Chevalley-
Eilenberg cohomology complex of the Hom-Lie algebra $\mathfrak{g}$ with
coefficients in $V$. Then the direct sum vector space $\mathfrak{g}\oplus V$
inherits a Hom-Lie algebra structure (called the $f$-twisted semidirect
product) whose bracket is given by
$\displaystyle[(x,u),(y,v)]_{\ltimes_{f}}:=([x,y],x\bullet v-y\bullet
u+f(x,y)),~{}\text{ for }(x,u),(y,v)\in\mathfrak{g}\oplus V$
and the linear twisting map on $\mathfrak{g}\oplus V$ is given by
$\alpha\oplus\beta$. It is easy to verify that the quadruple
$(\mathfrak{g}\oplus
V,[~{},~{}]_{\ltimes_{0}},[~{},~{}]_{\ltimes_{f}},\alpha\oplus\beta)$ is a
compatible Hom-Lie algebra.
Another example of a compatible Hom-Lie algebra arises from compatible Rota-
Baxter operators on a Hom-Lie algebra. Rota-Baxter operators are an algebraic
abstraction of the integral operator and operator analogue of Poisson
structures [10].
###### 3.8 Definition.
Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra. A linear map
$R:\mathfrak{g}\rightarrow\mathfrak{g}$ is said to be a Rota-Baxter operator
of weight $\lambda\in\mathbb{K}$ on the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}],\alpha)$ if $R$ satisfies $\alpha\circ
R=R\circ\alpha$ and
$\displaystyle[Rx,Ry]=R([Rx,y]+[x,Ry]+\lambda[x,y]),~{}\text{ for
}x,y\in\mathfrak{g}.$
A Rota-Baxter operator $R$ induces a new Hom-Lie algebra structure on
$\mathfrak{g}$ with the Hom-Lie bracket
$\displaystyle[x,y]_{R}:=[Rx,y]+[x,Ry]+\lambda[x,y],~{}\text{ for
}x,y\in\mathfrak{g}.$
Compatible Poisson structures first appeared in the context of bihamiltonian
mechanics [12, 16]. The operator version of compatible Poisson structures in
the Hom-Lie algebra context is given by the following.
###### 3.9 Definition.
Two Rota-Baxter operators $R$ and $S$ of same weight $\lambda\in\mathbb{K}$ on
a Hom-Lie algebra $(\mathfrak{g},[~{},~{}],\alpha)$ are said to be compatible
if
$\displaystyle[Rx,Sy]+[Sx,Ry]=R([Sx,y]+[x,Sy])+S([Rx,y]+[x,Ry]),~{}\text{ for
}x,y\in\mathfrak{g}.$
###### 3.10 Example.
Let $(\mathfrak{g},[~{},~{}],\alpha)$ be the Hom-Lie algebra given by
$\mathfrak{g}=\langle e_{1},e_{2}\rangle$ and the structure maps are given by
$\displaystyle[e_{1},e_{2}]=ae_{1}+ae_{2},~{}~{}~{}\alpha(e_{1})=e_{2},~{}~{}~{}\alpha(e_{2})=e_{1}.$
Then the map $R=\alpha$ is a Rota-Baxter operator of weight $\lambda=-1$.
Let $(\mathfrak{g},[~{},~{}],\alpha)$ be a Hom-Lie algebra and let
$R:\mathfrak{g}\rightarrow\mathfrak{g}$ be a Rota-Baxter operator of weight
$\lambda$. Then $-\lambda\mathrm{id}-R:\mathfrak{g}\rightarrow\mathfrak{g}$ is
also a Rota-Baxter operator of the same weight. Moreover, $R$ and
$-\lambda\mathrm{id}-R$ are compatible.
The proof of the following proposition is straightforward.
###### 3.11 Proposition.
Let $R$ and $S$ be two compatible Rota-Baxter operators of weight
$\lambda\in\mathbb{K}$ on a Hom-Lie algebra $(\mathfrak{g},[~{},~{}],\alpha).$
Then $(\mathfrak{g},[~{},~{}]_{R},[~{},~{}]_{S},\alpha)$ is a compatible Hom-
Lie algebra.
Let $\mathfrak{g}$ be a vector space and
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a linear map. Consider the
graded Lie algebra
$(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}})$
given in Section 2. Hence by Remark 2.10, the quadruple
$\displaystyle(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}},d_{1}=0,d_{2}=0)$
is a bidifferential graded Lie algebra. Then we have the following Maurer-
Cartan characterization of compatible Hom-Lie algebras.
###### 3.12 Theorem.
There is a one-to-one correspondence between compatible Hom-Lie algebra
structures on $\mathfrak{g}$ and Maurer-Cartan elements in the bidifferential
graded Lie algebra
$(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}},d_{1}=0,d_{2}=0)$.
###### Proof.
Let $[~{},~{}]_{1}$ and $[~{},~{}]_{2}$ be two multiplicative skew-symmetric
bilinear brackets on $\mathfrak{g}$. Then the brackets $[~{},~{}]_{1}$ and
$[~{},~{}]_{2}$ correspond to elements (say, $\mu_{1}$ and $\mu_{2}$,
respectively) in $C^{2}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$. Then
$\displaystyle[~{},~{}]_{1}\text{ is a Hom-Lie bracket }\Leftrightarrow~{}$
$\displaystyle~{}~{}[\mu_{1},\mu_{1}]_{\mathsf{NR}}=0;$
$\displaystyle[~{},~{}]_{2}\text{ is a Hom-Lie bracket }\Leftrightarrow~{}$
$\displaystyle~{}~{}[\mu_{2},\mu_{2}]_{\mathsf{NR}}=0;$ $\displaystyle\text{
compatibility condition }(\ref{comp-cond-e})~{}\Leftrightarrow~{}$
$\displaystyle~{}~{}[\mu_{1},\mu_{2}]_{\mathsf{NR}}=0.$
Hence $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is a compatible Hom-
Lie algebra if and only if $(\mu_{1},\mu_{2})$ is a Maurer-Cartan element in
the bidifferential graded Lie algebra
$(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}},d_{1}=0,d_{2}=0)$.
∎
Hence from Proposition 2.12, we get the following.
###### 3.13 Proposition.
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra. Then for any multiplicative skew-symmetric bilinear operations
$[~{},~{}]_{1}^{\prime}$ and $[~{},~{}]_{2}^{\prime}$ on $\mathfrak{g}$, the
quadruple
$\displaystyle(\mathfrak{g},[~{},~{}]_{1}+[~{},~{}]_{1}^{\prime},[~{},~{}]_{2}+[~{},~{}]_{2}^{\prime},\alpha)$
is a compatible Hom-Lie algebra if and only if
$(\mu_{1}^{\prime},\mu_{2}^{\prime})$ is a Maurer-Cartan element in the
bidifferential graded Lie algebra
$(C^{\ast+1}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g}),[~{},~{}]_{\mathsf{NR}},d_{1}=[\mu_{1},-],d_{2}=[\mu_{2},-])$.
Here $\mu_{1}^{\prime},\mu_{2}^{\prime}\in
C^{2}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ denote the elements
corresponding to the brackets $[~{},~{}]_{1}^{\prime}$ and
$[~{},~{}]_{2}^{\prime}$, respectively.
In the following, we define representations of a compatible Hom-Lie algebra.
###### 3.14 Definition.
A representation of the compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ consists of a quadruple
$(V,\bullet_{1},\bullet_{2},\beta)$ such that
* (i)
$(V,\bullet_{1},\beta)$ is a representation of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},\alpha)$;
* (ii)
$(V,\bullet_{2},\beta)$ is a representation of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{2},\alpha)$;
* (iii)
the following compatibility condition holds
$\displaystyle[x,y]_{1}\bullet_{2}\beta(v)+[x,y]_{2}\bullet_{1}\beta(v)=\alpha(x)\bullet_{1}(y\bullet_{2}v)-\alpha(y)\bullet_{2}(x\bullet_{1}v)+\alpha(x)\bullet_{2}(y\bullet_{1}v)-\alpha(y)\bullet_{1}(x\bullet_{2}v),$
for all $x,y\in\mathfrak{g}$ and $v\in V$.
It follows that any compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is a representation of
itself, where $\bullet_{1}=[~{},~{}]_{1}$ and $\bullet_{2}=[~{},~{}]_{2}$.
This is called the adjoint representation.
###### 3.15 Remark.
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra and $(V,\bullet_{1},\bullet_{2},\beta)$ be a representation of it.
Then for any $\lambda,\eta\in\mathbb{K}$, the triple
$(\mathfrak{g},\lambda[~{},~{}]_{1}+\eta[~{},~{}]_{2},\alpha)$ is a Hom-Lie
algebra and $(V,\lambda~{}\bullet_{1}+\eta~{}\bullet_{2},\beta)$ is a
representation of it.
The proof of the following proposition is similar to the standard case.
###### 3.16 Proposition.
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra and $(V,\bullet_{1},\bullet_{2},\beta)$ be a representation of it.
Then the direct sum $\mathfrak{g}\oplus V$ carries a compatible Hom-Lie
algebra structure with the linear homomorphism $\alpha\oplus\beta$, and Hom-
Lie brackets
$\displaystyle[(x,u),(y,v)]_{i}^{\ltimes}:=([x,y]_{i},x\bullet_{i}v-y\bullet_{i}u),~{}\text{
for }i=1,2\text{ and }(x,u),(y,v)\in\mathfrak{g}\oplus V.$
This is called the semidirect product.
## 4 Cohomology of compatible Hom-Lie algebras
In this section, we introduce the cohomology of a compatible Hom-Lie algebra
with coefficients in a representation. We also define abelian extensions of a
compatible Hom-Lie algebra and prove that isomorphism classes of abelian
extensions are classified by the second cohomology group.
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra and $(V,\bullet_{1},\bullet_{2},\beta)$ be a representation of it.
Let
${}^{1}\delta_{\mathrm{Hom}}:C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{Hom}}(\mathfrak{g},V)$ (resp.
${}^{2}\delta_{\mathrm{Hom}}:C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{Hom}}(\mathfrak{g},V)$ ), for $n\geq 0$, be the coboundary
operator for the Chevalley-Eilenberg cohomology of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},\alpha)$ with coefficients in the representation
$(V,\bullet_{1},\beta)$ (resp. of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{2},\alpha)$ with coefficients in the representation
$(V,\bullet_{2},\beta)$ ). Then we have
$\displaystyle({}^{1}\delta_{\mathrm{Hom}})^{2}=0~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}({}^{2}\delta_{\mathrm{Hom}})^{2}=0.$
Moreover, we have the following.
###### 4.1 Proposition.
The coboundary operators ${}^{1}\delta_{\mathrm{Hom}}$ and
${}^{2}\delta_{\mathrm{Hom}}$ satisfy the following compatibility
$\displaystyle{}^{1}\delta_{\mathrm{Hom}}\circ{}^{2}\delta_{\mathrm{Hom}}+{}^{2}\delta_{\mathrm{Hom}}\circ{}^{1}\delta_{\mathrm{Hom}}=0.$
Before we prove the above proposition, we first observe the followings. For a
compatible Hom-Lie algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$
and a representation $(V,\bullet_{1},\bullet_{2},\beta)$, we consider the
semidirect product compatible Hom-Lie algebra structure on $\mathfrak{g}\oplus
V$ given in Proposition 3.16. We denote by $\pi_{1},\pi_{2}\in
C^{2}_{\mathrm{Hom}}(\mathfrak{g}\oplus V,\mathfrak{g}\oplus V)$ the elements
corresponding to the Hom-Lie brackets $[~{},~{}]_{1}^{\ltimes}$ and
$[~{},~{}]_{2}^{\ltimes}$ on $\mathfrak{g}\oplus V$, respectively. Let
$\displaystyle\delta^{1}_{\mathrm{Hom}}:C^{n}_{\mathrm{Hom}}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V)\rightarrow C^{n+1}_{\mathrm{Hom}}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V),~{}~{}\text{ for }n\geq 0,$
$\displaystyle\delta^{2}_{\mathrm{Hom}}:C^{n}_{\mathrm{Hom}}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V)\rightarrow C^{n+1}_{\mathrm{Hom}}(\mathfrak{g}\oplus
V,\mathfrak{g}\oplus V),~{}~{}\text{ for }n\geq 0,$
denote respectively the coboundary operator for the Chevalley-Eilenberg
cohomology of the Hom-Lie algebra $(\mathfrak{g}\oplus
V,[~{},~{}]_{1}^{\ltimes},\alpha\oplus\beta)$ (resp. of the Hom-Lie algebra
$(\mathfrak{g}\oplus V,[~{},~{}]_{2}^{\ltimes},\alpha\oplus\beta)$ ) with
coefficients in itself. Note that any map $f\in
C^{n}_{\mathrm{Hom}}(\mathfrak{g}\oplus V)$ can be lifted to a map
$\widetilde{f}\in C^{n}_{\mathrm{Hom}}(\mathfrak{g}\oplus V,\mathfrak{g}\oplus
V)$ by
$\displaystyle\widetilde{f}\big{(}(x_{1},v_{1}),\ldots,(x_{n},v_{n})\big{)}=\big{(}0,f(x_{1},\ldots,x_{n})\big{)}.$
Then $f=0$ if and only if $\widetilde{f}=0$.
With these notations, for any $f\in C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)$, we
have
$\displaystyle\widetilde{({}^{1}\delta_{\mathrm{Hom}}f)}=\delta^{1}_{\mathrm{Hom}}(\widetilde{f})=(-1)^{n-1}[\pi_{1},\widetilde{f}]_{\mathsf{NR}}~{}~{}~{}~{}\text{
and
}~{}~{}~{}~{}\widetilde{({}^{2}\delta_{\mathrm{Hom}}f)}=\delta^{2}_{\mathrm{Hom}}(\widetilde{f})=(-1)^{n-1}[\pi_{2},\widetilde{f}]_{\mathsf{NR}}.$
###### Proof.
(of Proposition 4.1) For any $f\in C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)$, we
have
$\displaystyle\widetilde{({}^{1}\delta_{\mathrm{Hom}}\circ{}^{2}\delta_{\mathrm{Hom}}+{}^{2}\delta_{\mathrm{Hom}}\circ{}^{1}\delta_{\mathrm{Hom}})(f)}$
$\displaystyle=\widetilde{{}^{1}\delta_{\mathrm{Hom}}({}^{2}\delta_{\mathrm{Hom}}{f})}~{}+~{}\widetilde{{}^{2}\delta_{\mathrm{Hom}}({}^{1}\delta_{\mathrm{Hom}}{f})}$
$\displaystyle=(-1)^{n}~{}[\pi_{1},\widetilde{{}^{2}\delta_{\mathrm{Hom}}{f}}]_{\mathsf{NR}}~{}+~{}(-1)^{n}~{}[\pi_{2},\widetilde{{}^{1}\delta_{\mathrm{Hom}}{f}}]_{\mathsf{NR}}$
$\displaystyle=-[\pi_{1},[\pi_{2},\widetilde{f}]_{\mathsf{NR}}]_{\mathsf{NR}}-[\pi_{2},[\pi_{1},\widetilde{f}]_{\mathsf{NR}}]_{\mathsf{NR}}$
$\displaystyle=-[[\pi_{1},\pi_{2}]_{\mathsf{NR}},\widetilde{f}]_{\mathsf{NR}}~{}+~{}[\pi_{2},[\pi_{1},\widetilde{f}]_{\mathsf{NR}}]_{\mathsf{NR}}-[\pi_{2},[\pi_{1},\widetilde{f}]_{\mathsf{NR}}]_{\mathsf{NR}}$
$\displaystyle=0~{}~{}~{}~{}\qquad(\because~{}[\pi_{1},\pi_{2}]_{\mathsf{NR}}=0).$
Therefore, it follows that
$({}^{1}\delta_{\mathrm{Hom}}\circ{}^{2}\delta_{\mathrm{Hom}}+{}^{2}\delta_{\mathrm{Hom}}\circ{}^{1}\delta_{\mathrm{Hom}})(f)=0$.
Hence the result follows. ∎
We are now in a position to define the cohomology of a compatible Hom-Lie
algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ with coefficients
in a representation $(V,\bullet_{1},\bullet_{2},\beta)$. For each $n\geq 0$,
we define an abelian group $C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)$ as follows:
$\displaystyle C^{0}_{\mathrm{cHom}}(\mathfrak{g},V):=~{}$ $\displaystyle
C^{0}_{\mathrm{Hom}}(\mathfrak{g},V)\cap\\{v\in
V|~{}x\bullet_{1}v=x\bullet_{2}v,\forall x\in\mathfrak{g}\\}$
$\displaystyle=~{}$ $\displaystyle\\{v\in V|~{}\beta(v)=v\text{ and
}x\bullet_{1}v=x\bullet_{2}v,\forall x\in\mathfrak{g}\\},$ $\displaystyle
C^{n}_{\mathrm{cHom}}(\mathfrak{g},V):=\underbrace{C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)\oplus\cdots\oplus
C^{n}_{\mathrm{Hom}}(\mathfrak{g},V)}_{n\text{ copies }},~{}\text{ for }n\geq
1.$
Define a map
$\delta_{\mathrm{cHom}}:C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{cHom}}(\mathfrak{g},V)$, for $n\geq 0$ by
$\displaystyle\delta_{\mathrm{cHom}}(v)(x):=x\bullet_{1}v=x\bullet_{2}v,~{}\text{
for }v\in C^{0}_{\mathrm{cHom}}(\mathfrak{g},V)~{}\text{ and
}x\in\mathfrak{g},$
$\displaystyle\delta_{\mathrm{cHom}}(f_{1},\ldots,f_{n}):=\big{(}{}^{1}\delta_{\mathrm{Hom}}f_{1},\ldots,\underbrace{{}^{1}\delta_{\mathrm{Hom}}f_{i}+{}^{2}\delta_{\mathrm{Hom}}f_{i-1}}_{i\text{-th
position}},\ldots,{}^{2}\delta_{\mathrm{Hom}}f_{n}\big{)},$
for $(f_{1},\ldots,f_{n})\in C^{n}_{\mathrm{cHom}}(\mathfrak{g},V).$
Then we have the following.
###### 4.2 Proposition.
The map $\delta_{\mathrm{cHom}}$ is a coboundary map, i.e.,
$(\delta_{\mathrm{cHom}})^{2}=0$.
###### Proof.
For any $v\in C^{0}_{\mathrm{cHom}}(\mathfrak{g},V)$, we have
$\displaystyle(\delta_{\mathrm{cHom}})^{2}(v)=\delta_{\mathrm{cHom}}(\delta_{\mathrm{cHom}}v)=~{}$
$\displaystyle({}^{1}\delta_{\mathrm{Hom}}\delta_{\mathrm{cHom}}v~{},{}^{2}\delta_{\mathrm{Hom}}\delta_{\mathrm{cHom}}v)$
$\displaystyle=~{}$
$\displaystyle({}^{1}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}v~{},{}^{2}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}v)=0.$
Moreover, for any $(f_{1},\ldots,f_{n})\in
C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)$, $n\geq 1$, we have
$\displaystyle(\delta_{\mathrm{cHom}})^{2}(f_{1},\ldots,f_{n})$
$\displaystyle=\delta_{\mathrm{cHom}}\big{(}{}^{1}\delta_{\mathrm{Hom}}f_{1},\ldots,{}^{1}\delta_{\mathrm{Hom}}f_{i}+{}^{2}\delta_{\mathrm{Hom}}f_{i-1},\ldots,{}^{2}\delta_{\mathrm{Hom}}f_{n}\big{)}$
$\displaystyle=\big{(}{}^{1}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{1}~{},{}^{2}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{1}+{}^{1}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{1}+{}^{1}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{2}~{},\ldots,$
$\displaystyle\qquad\underbrace{{}^{2}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{i-2}+{}^{2}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{i-1}+{}^{1}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{i-1}+{}^{1}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{i}}_{3\leq
i\leq n-1},\ldots,$
$\displaystyle\qquad{}^{2}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{n-1}+{}^{2}\delta_{\mathrm{Hom}}{}^{1}\delta_{\mathrm{Hom}}f_{n}+{}^{1}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{n}~{},{}^{2}\delta_{\mathrm{Hom}}{}^{2}\delta_{\mathrm{Hom}}f_{n}\big{)}$
$\displaystyle=0~{}~{}~{}\quad(\text{from Proposition }\ref{d-comp}).$
This proves that $(\delta_{\mathrm{cHom}})^{2}=0$. ∎
It follows from the above proposition that
$\\{C^{\ast}_{\mathrm{cHom}}(\mathfrak{g},V),\delta_{\mathrm{cHom}}\\}$ is a
cochain complex. The corresponding cohomology groups
$\displaystyle
H^{n}_{\mathrm{cHom}}(\mathfrak{g},V):=\frac{Z^{n}_{\mathrm{cHom}}(\mathfrak{g},V)}{B^{n}_{\mathrm{cHom}}(\mathfrak{g},V)}=\frac{\mathrm{Ker~{}}\delta_{\mathrm{cHom}}:C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow
C^{n+1}_{\mathrm{cHom}}(\mathfrak{g},V)}{\mathrm{Im~{}}\delta_{\mathrm{cHom}}:C^{n-1}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow
C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)},\text{ for }n\geq 0$
are called the cohomology of the compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ with coefficients in the
representation $(V,\bullet_{1},\bullet_{2},\beta)$.
It follows from the above definition that
$\displaystyle H^{0}_{\mathrm{cHom}}(\mathfrak{g},V)=\\{v\in
V|~{}\beta(v)=v\text{ and }x\bullet_{1}v=x\bullet_{2}v=0,~{}\forall
x\in\mathfrak{g}\\}.$
A linear map $D:\mathfrak{g}\rightarrow V$ is said to be a derivation if
$\beta\circ D=D\circ\alpha$ and
$\displaystyle D[x,y]_{1}=x\bullet_{1}Dy-y\bullet_{1}Dx,\qquad
D[x,y]_{2}=x\bullet_{2}Dy-y\bullet_{2}Dx,\text{ for }x,y\in\mathfrak{g}.$
We denote the space of derivations by $\mathrm{Der}(\mathfrak{g},V)$. A
derivation $D$ is said to be inner if it is of the form
$D=-\bullet_{1}v=-\bullet_{2}v$, for some $v\in
C^{0}_{\mathrm{cHom}}(\mathfrak{g},V)$. The space of inner derivations are
denoted by $\mathrm{InnDer}(\mathfrak{g},V)$. Then we have
$\displaystyle
H^{1}_{\mathrm{cHom}}(\mathfrak{g},V)=\frac{\mathrm{Der}(\mathfrak{g},V)}{\mathrm{InnDer}(\mathfrak{g},V)}=\mathrm{OutDer}(\mathfrak{g},V),\text{
the space of outer derivations.}$
Let $\mathfrak{g}=(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a
compatible Hom-Lie algebra and $V=(V,\bullet_{1},\bullet_{2},\beta)$ be a
representation of it. Then we know from Remark 3.15 that
$\mathfrak{g}_{+}=(\mathfrak{g},[~{},~{}]_{1}+[~{},~{}]_{2},\alpha)$ is a Hom-
Lie algebra and $V_{+}=(V,\bullet_{1}+\bullet_{2},\beta)$ is a representation
of it. Consider the cochain complex
$\\{C^{\ast}_{\mathrm{cHom}}(\mathfrak{g},V),\delta_{\mathrm{cHom}}\\}$ of the
compatible Hom-Lie algebra $\mathfrak{g}$ with coefficients in the
representation $V$, and the cochain complex
$\\{C^{\ast}_{\mathrm{Hom}}(\mathfrak{g}_{+},V_{+}),\delta_{\mathrm{Hom}}\\}$
of the Hom-Lie algebra $\mathfrak{g}_{+}$ with coefficients in $V_{+}$.
For each $n\geq 0$, we define a map
$\displaystyle\triangle_{n}:C^{n}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow
C^{n}_{\mathrm{Hom}}(\mathfrak{g}_{+},V_{+})~{}~{}~{}\text{ by
}~{}~{}~{}\begin{cases}\triangle_{0}(v)=\frac{1}{2}v,\\\
\triangle_{n}((f_{1},\ldots,f_{n}))=f_{1}+\cdots+f_{n},\end{cases}$
for $v\in C^{0}_{\mathrm{cHom}}(\mathfrak{g},V)$ and $(f_{1},\ldots,f_{n})\in
C^{n\geq 1}_{\mathrm{cHom}}(\mathfrak{g},V)$. If $v\in
C^{0}_{\mathrm{cHom}}(\mathfrak{g},V)$, then
$\displaystyle(\delta_{\mathrm{Hom}}\circ\triangle_{0}(v))(x)=\frac{1}{2}(\delta_{\mathrm{Hom}}(v))(x)=\frac{1}{2}(x\bullet_{1}v+x\bullet_{2}v)=~{}$
$\displaystyle x\bullet_{1}v=x\bullet_{2}v$ $\displaystyle=~{}$
$\displaystyle\delta_{\mathrm{cHom}}(v)(x)=(\triangle_{1}\circ\delta_{\mathrm{cHom}}(v))(x).$
Moreover, if $(f_{1},\ldots,f_{n})\in C^{n\geq
1}_{\mathrm{cHom}}(\mathfrak{g},V)$, then
$\displaystyle(\delta_{\mathrm{Hom}}\circ\triangle_{n})((f_{1},\ldots,f_{n}))=~{}$
$\displaystyle\delta_{\mathrm{Hom}}(f_{1}+\cdots+f_{n})$ $\displaystyle=~{}$
$\displaystyle{}^{1}\delta_{\mathrm{Hom}}(f_{1},\ldots,f_{n})+{}^{2}\delta_{\mathrm{Hom}}(f_{1},\ldots,f_{n})$
$\displaystyle=~{}$
$\displaystyle\triangle_{n+1}\big{(}{}^{1}\delta_{\mathrm{Hom}}f_{1},\ldots,{}^{1}\delta_{\mathrm{Hom}}f_{i}+{}^{2}\delta_{\mathrm{Hom}}f_{i-1},\ldots,{}^{2}\delta_{\mathrm{Hom}}f_{n}\big{)}$
$\displaystyle=~{}$
$\displaystyle(\triangle_{n+1}\circ\delta_{\mathrm{cHom}})((f_{1},\ldots,f_{n})).$
Therefore, we get the following.
###### 4.3 Theorem.
The collection $\\{\triangle_{n}\\}_{n\geq 0}$ defines a morphism of cochain
complexes from
$\\{C^{\ast}_{\mathrm{cHom}}(\mathfrak{g},V),\delta_{\mathrm{cHom}}\\}$ to
$\\{C^{\ast}_{\mathrm{Hom}}(\mathfrak{g}_{+},V_{+}),\delta_{\mathrm{Hom}}\\}$.
Hence, it induces a morphism
$H^{\ast}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow
H^{\ast}_{\mathrm{Hom}}(\mathfrak{g}_{+},V_{+})$ between corresponding
cohomologies.
Abelian extensions of compatible Hom-Lie algebras. Here, we study abelian
extensions of compatible Hom-Lie algebras and give a classification of
equivalence classes of abelian extensions.
Let $\mathfrak{g}=(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a
compatible Hom-Lie algebra and $(V,\beta)$ be a pair of a vector space with a
linear map. Note that $(V,\beta)$ can be considered as a compatible Hom-Lie
algebra with both the Hom-Lie brackets on $V$ are trivial.
###### 4.4 Definition.
An abelian extension of $\mathfrak{g}$ by $V$ is an exact sequence of
compatible Hom-Lie algebras
(6)
together with a $\mathbb{K}$-splitting (given by $s$) satisfying
(7) $\displaystyle\alpha^{\mathfrak{h}}\circ s=s\circ\alpha.$
We denote an abelian extension as above simply by $\mathfrak{h}$ when all the
structures of the exact sequence (6) are understood. Note that an abelian
extension induces a representation of the compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ on $(V,\beta)$ with the
actions
$\displaystyle x\bullet_{1}v=[s(x),i(v)]_{1}^{\mathfrak{h}}~{}~{}~{}\text{ and
}~{}~{}~{}x\bullet_{2}v=[s(x),i(v)]_{2}^{\mathfrak{h}},\text{ for
}x\in\mathfrak{g},v\in V.$
It is easy to see that the above action is independent of the choice of $s$.
###### 4.5 Remark.
Let $(\mathfrak{h},\alpha^{\mathfrak{h}})$ and $(\mathfrak{g},\alpha)$ be two
pairs of vector spaces endowed with linear maps. Suppose
$j:\mathfrak{h}\rightarrow\mathfrak{g}$ is a linear map satisfying
$\alpha\circ j=j\circ\alpha^{\mathfrak{h}}$. Then there might not be a linear
map $s:\mathfrak{g}\rightarrow\mathfrak{h}$ that satisfies
$\alpha^{\mathfrak{h}}\circ s=s\circ\alpha$. Take $\mathfrak{h}=\langle
e_{1},e_{2}\rangle$ and $\alpha^{\mathfrak{h}}(e_{1})=e_{2}$,
$\alpha^{\mathfrak{h}}(e_{2})=0$. Also take $\mathfrak{g}=\langle f\rangle$
and $\alpha(f)=0$. Let $j:\mathfrak{h}\rightarrow\mathfrak{g}$ be the map
defined by $j(e_{1})=f$ and $j(e_{2})=0$. Let
$s:\mathfrak{g}\rightarrow\mathfrak{h}$ be a map satisfying
$\alpha^{\mathfrak{h}}\circ s=s\circ\alpha$. For $s(f)=\lambda e_{1}+\eta
e_{2}$, we have $f=(j\circ s)(f)=\lambda f$ which implies that $\lambda=1$.
Thus,
$\displaystyle 0=(s\circ\alpha)(f)=(\alpha^{\mathfrak{h}}\circ
s)(f)=\alpha^{\mathfrak{h}}(e_{1}+\eta e_{2})=e_{2}$
leads to a contradiction.
###### 4.6 Definition.
Two abelian extensions $\mathfrak{h}$ and $\mathfrak{h}^{\prime}$ are said to
be equivalent if there is a compatible Hom-Lie algebra homomorphism
$\phi:\mathfrak{h}\rightarrow\mathfrak{h}^{\prime}$ making the following
diagram commutative
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(V,0,0,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{(\mathfrak{h},[~{},~{}]_{1}^{\mathfrak{h}},[~{},~{}]_{2}^{\mathfrak{h}},\alpha^{\mathfrak{h}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{j}$$\textstyle{(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(V,0,0,\beta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{(\mathfrak{h}^{\prime},[~{},~{}]_{1}^{\mathfrak{h}^{\prime}},[~{},~{}]_{2}^{\mathfrak{h}^{\prime}},\alpha^{\mathfrak{h}^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\prime}}$$\textstyle{(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s^{\prime}}$$\textstyle{0.}$
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra and $(V,\bullet_{1},\bullet_{2},\beta)$ be a representation of it.
We denote $\mathrm{Ext}(\mathfrak{g},V)$ by the set of equivalence classes of
abelian extensions of $\mathfrak{g}$ by $V$ for which the induced
representation on $V$ is the given one.
With the above notation, we have the following.
###### 4.7 Theorem.
There is a bijection
$H^{2}_{\mathrm{cHom}}(\mathfrak{g},V)\cong\mathrm{Ext}(\mathfrak{g},V).$
###### Proof.
Let $(f_{1},f_{2})\in Z^{2}_{\mathrm{cHom}}(\mathfrak{g},V)$ be any
$2$-cocycle. Then it induces a compatible Hom-Lie algebra structure on
$\mathfrak{h}=\mathfrak{g}\oplus V$ with structure maps
$\displaystyle[(x,u),(y,v)]_{i}^{\ltimes}:=~{}$
$\displaystyle([x,y],x\bullet_{i}v-y\bullet_{i}u+f_{i}(x,y)),~{}\text{ for
}i=1,2,$ $\displaystyle\alpha^{\mathfrak{h}}(x,u):=~{}$
$\displaystyle(\alpha(x),\beta(u)),~{}\text{ for }(x,u),(y,v)\in\mathfrak{h}.$
Moreover, this compatible Hom-Lie algebra makes
into an abelian extension with the obvious splitting $s$. It is easy to see
that any other cohomologous $2$-cocycle gives rise to an equivalent abelian
extension. Hence the map
$H^{2}_{\mathrm{cHom}}(\mathfrak{g},V)\rightarrow\mathrm{Ext}(\mathfrak{g},V)$
is well defined.
Conversely, let (6) be an abelian extension with splitting $s$. Then we may
consider $\mathfrak{h}=\mathfrak{g}\oplus V$ and $s$ is the map $s(x)=(x,0)$,
for $x\in\mathfrak{g}.$ The map $i$ and $j$ are the obvious ones. Moreover, it
follows from (7) that $\alpha^{\mathfrak{h}}=(\alpha,\beta)$. Finally, since
$j$ is a compatible Hom-Lie algebra map, we have
$\displaystyle j[(x,0),(y,0)]_{i}^{\mathfrak{h}}=[x,y]_{i},~{}\text{ for
}x,y\in\mathfrak{g}.$
This implies that $[(x,0),(y,0)]_{i}^{\mathfrak{h}}=([x,y]_{i},f_{i}(x,y)),$
for $i=1,2$ and some $f_{1},f_{2}\in C^{2}_{\mathrm{Hom}}(\mathfrak{g},V)$. As
$([~{},~{}]_{1}^{\mathfrak{h}},[~{},~{}]_{2}^{\mathfrak{h}})$ is a compatible
Hom-Lie algebra structure on $\mathfrak{h}$, we get that the pair
$(f_{1},f_{2})\in Z^{2}_{\mathrm{cHom}}(\mathfrak{g},V)$ is a $2$-cocycle. It
is left to the reader to check that equivalent abelian extensions give rise to
cohomologous $2$-cocycles. Hence the map
$\mathrm{Ext}(\mathfrak{g},V)\rightarrow
H^{2}_{\mathrm{cHom}}(\mathfrak{g},V)$ is also well defined. Finally, these
two maps are inverses to each other. Hence the proof. ∎
## 5 Deformations of compatible Hom-Lie algebras
In this section, we study linear deformations and finite order deformations of
a compatible Hom-Lie algebra generalizing the classical deformation theory of
Gerstenhaber [8]. We introduce Nijenhuis operators that generate trivial
linear deformations. We also define infinitesimal deformations of a compatible
Hom-Lie algebra and characterize equivalence classes of infinitesimal
deformations in terms of cohomology. Finally, we consider extensibility of
finite order deformations.
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra and $(\omega_{1},\omega_{2})\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ be a $2$-cochain. Define two
bilinear operations on $\mathfrak{g}$ depending on the parameter $t$ as
follows:
$\displaystyle[x,y]_{1}^{t}:=[x,y]_{1}+t\omega_{1}(x,y)~{}~{}\text{ and
}~{}~{}[x,y]_{2}^{t}:=[x,y]_{2}+t\omega_{2}(x,y),\text{ for
}x,y\in\mathfrak{g}.$
###### 5.1 Definition.
We say that $(\omega_{1},\omega_{2})$ generates a linear deformation of the
compatible Hom-Lie algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$
if for all $t$, the quadruple
$(\mathfrak{g},[~{},~{}]_{1}^{t},[~{},~{}]_{2}^{t},\alpha)$ is a compatible
Hom-Lie algebra.
If $\mu_{1},\mu_{2}\in C^{2}_{\mathrm{Hom}}(\mathfrak{g},\mathfrak{g})$ denote
the elements corresponding to the Hom-Lie brackets $[~{},~{}]_{1}$ and
$[~{},~{}]_{2}$, respectively, then the above definition is equivalent to
saying that
$\displaystyle[\mu_{1}+t\omega_{1},\mu_{1}+t\omega_{1}]_{\mathsf{NR}}=0,\qquad[\mu_{2}+t\omega_{2},\mu_{2}+t\omega_{2}]_{\mathsf{NR}}=0~{}~{}\text{
and }~{}~{}[\mu_{1}+t\omega_{1},\mu_{2}+t\omega_{2}]_{\mathsf{NR}}=0.$
In other words, the followings are hold
$\displaystyle[\mu_{1},\omega_{1}]_{\mathsf{NR}}=0,\qquad[\mu_{2},\omega_{2}]_{\mathsf{NR}}=0,\qquad[\mu_{1},\omega_{2}]_{\mathsf{NR}}+[\mu_{2},\omega_{1}]_{\mathsf{NR}}=0,$
$\displaystyle[\omega_{1},\omega_{1}]_{\mathsf{NR}}=0,\qquad[\omega_{2},\omega_{2}]_{\mathsf{NR}}=0,\qquad[\omega_{1},\omega_{2}]_{\mathsf{NR}}=0.$
The first three condition implies that $(\omega_{1},\omega_{2})\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ is a $2$-cocycle in the
cohomology of the compatible Hom-Lie algebra $\mathfrak{g}$ with coefficients
in the adjoint representation. Moreover, the last three conditions implies
that $(\mathfrak{g},\omega_{1},\omega_{2},\alpha)$ is a compatible Hom-Lie
algebra.
###### 5.2 Definition.
Let $(\omega_{1},\omega_{2})$ and $(\omega_{1}^{\prime},\omega_{2}^{\prime})$
generate linear deformations
$(\mathfrak{g},[~{},~{}]_{1}^{t},[~{},~{}]_{2}^{t},\alpha)$ and
$(\mathfrak{g},[~{},~{}]_{1}^{{}^{\prime}t},[~{},~{}]_{2}^{{}^{\prime}t},\alpha)$
of a compatible Hom-Lie algebra $\mathfrak{g}.$ They are said to be equivalent
if there exists a linear map $N:\mathfrak{g}\rightarrow\mathfrak{g}$
satisfying $\alpha\circ N=N\circ\alpha$ and such that
$\displaystyle\mathrm{id}+tN:(\mathfrak{g},[~{},~{}]_{1}^{t},[~{},~{}]_{2}^{t},\alpha)\rightarrow(\mathfrak{g},[~{},~{}]_{1}^{{}^{\prime}t},[~{},~{}]_{2}^{{}^{\prime}t},\alpha)$
is a morphism of compatible Hom-Lie algebras.
One can equivalently write the explicit identities as follows:
$\displaystyle\omega_{i}(x,y)-\omega_{i}^{\prime}(x,y)=~{}$
$\displaystyle[x,Ny]_{i}+[Nx,y]_{i}-N[x,y]_{i},$ $\displaystyle
N\omega_{i}(x,y)=~{}$
$\displaystyle\omega_{i}^{\prime}(x,Ny)+\omega_{i}^{\prime}(Nx,y)+[Nx,Ny]_{i},$
$\displaystyle\omega_{i}^{\prime}(Nx,Ny)=~{}$ $\displaystyle 0,$
for $x,y\in\mathfrak{g}$ and $i=1,2$. Note that from the first identity, we
get
$\displaystyle(\omega_{1},\omega_{2})-(\omega_{1}^{\prime},\omega_{2}^{\prime})=\delta_{\mathrm{cHom}}N,$
where $N$ is considered as an element in
$C^{1}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$. Hence, summarizing the
above discussions, we get the following.
###### 5.3 Theorem.
Let $\mathfrak{g}$ be a compatible Hom-Lie algebra. Then there is a map from
the set of equivalence classes of linear deformations of $\mathfrak{g}$ to the
second cohomology group $H^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$.
Next, we introduce trivial linear deformations of a compatible Hom-Lie algebra
and introduce Nijenhuis operators that generate trivial linear deformations.
###### 5.4 Definition.
A linear deformation $([~{},~{}]_{1}+t\omega_{1},[~{},~{}]_{2}+t\omega_{2})$
of compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is said to be trivial if
the deformation is equivalent to the undeformed one
$([~{},~{}]_{1},[~{},~{}]_{2}).$
Thus, a linear deformation
$([~{},~{}]_{1}+t\omega_{1},[~{},~{}]_{2}+t\omega_{2})$ is trivial if and only
if there exists a linear map $N:\mathfrak{g}\rightarrow\mathfrak{g}$
satisfying $\alpha\circ N=N\circ\alpha$ and
$\displaystyle\omega_{i}(x,y)=~{}$
$\displaystyle[x,Ny]_{i}+[Nx,y]_{i}-N[x,y]_{i},$ $\displaystyle
N\omega_{i}(x,y)=~{}$ $\displaystyle[Nx,Ny]_{i},~{}\text{ for }i=1,2\text{ and
}x,y\in\mathfrak{g}.$
This motivates us to introduce Nijenhuis operators on a compatible Hom-Lie
algebra.
###### 5.5 Definition.
A Nijenhuis operator on a compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is a linear map
$N:\mathfrak{g}\rightarrow\mathfrak{g}$ which is a Nijenhuis operator for both
the Hom-Lie algebras $(\mathfrak{g},[~{},~{}]_{1},\alpha)$ and
$(\mathfrak{g},[~{},~{}]_{2},\alpha)$, i.e., $N$ satisfies $\alpha\circ
N=N\circ\alpha$ and
$\displaystyle[Nx,Ny]_{i}=N([Nx,y]_{i}+[x,Ny]_{i}-N[x,y]_{i}),~{}\text{ for
}i=1,2\text{ and }x,y\in\mathfrak{g}.$
It follows that any trivial linear deformation of a compatible Hom-Lie algebra
induces a Nijenhuis operator. The converse is given by the next result whose
proof is straightforward.
###### 5.6 Proposition.
Let $N:\mathfrak{g}\rightarrow\mathfrak{g}$ be a Nijenhuis operator on a
compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$. Then
$(\omega_{1},\omega_{2})$ generates a trivial linear deformation of the
compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$, where
$\displaystyle\omega_{i}(x,y)=[Nx,y]_{i}+[x,Ny]_{i}-N[x,y]_{i},~{}\text{ for
}i=1,2\text{ and }x,y\in\mathfrak{g}.$
In the following, we introduce infinitesimal deformations of a compatible Hom-
Lie algebra as a generalization of linear deformations.
###### 5.7 Definition.
An infinitesimal deformation of a compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is a linear deformation
over $\mathbb{K}[[t]]/(t^{2})$.
One can similarly define equivalences between two infinitesimal deformations.
It is easy to see that any $2$-cocycle $(\omega_{1},\omega_{2})\in
Z^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ induces an infinitesimal
deformation and cohomologous $2$-cocycles give rise to equivalent
infinitesimal deformations. Summarizing this fact with Theorem 5.3, we get the
following.
###### 5.8 Theorem.
Let $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-
Lie algebra. Then the equivalence classes of infinitesimal deformations are in
one-to-one correspondence with
$H^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g}).$
Finite order deformations. Next, we consider finite order deformations of a
compatible Hom-Lie algebra and study their extensions. Let
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ be a compatible Hom-Lie
algebra. For any natural number $p\in\mathbb{N}$, consider the ring
$\mathbb{K}[[t]]/(t^{p+1})$. Then $\mathfrak{g}[[t]]/(t^{p+1})$ is a module
over $\mathbb{K}[[t]]/(t^{p+1})$. Note that the linear map
$\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ extends to a
$\mathbb{K}[[t]]/(t^{p+1})$-linear map (denoted by the same notation) on the
space $\mathfrak{g}[[t]]/(t^{p+1})$.
###### 5.9 Definition.
An order $p$ deformation of the compatible Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is a pair
$(\mu_{1,t},\mu_{2,t})$, where $\mu_{1,t}=\sum_{i=0}^{p}t^{i}\mu_{1,i}$ and
$\mu_{2,t}=\sum_{i=0}^{p}t^{i}\mu_{2,i}$ with $\mu_{1,i},\mu_{2,i}\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$, and $\mu_{1,0}=\mu_{1}$,
$\mu_{2,0}=\mu_{2}$ which makes the quadruple
$(\mathfrak{g}[[t]]/(t^{p+1}),\mu_{1,t},\mu_{2,t},\alpha)$ into a compatible
Hom-Lie algebra over $\mathbb{K}[[t]]/(t^{p+1})$.
Therefore, in an order $p$ deformation, we have
$\displaystyle[\mu_{1,t},\mu_{1,t}]_{\mathsf{NR}}=0,\quad[\mu_{2,t},\mu_{2,t}]_{\mathsf{NR}}=0~{}~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}~{}[\mu_{1,t},\mu_{2,t}]_{\mathsf{NR}}=0.$
These are equivalent to the following system of identities
$\displaystyle{}^{1}\delta_{\mathrm{Hom}}(\mu_{1,n})=~{}$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}},$
$\displaystyle{}^{2}\delta_{\mathrm{Hom}}(\mu_{2,n})=~{}$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geq
1\end{subarray}}[\mu_{2,i},\mu_{2,j}]_{\mathsf{NR}},$
$\displaystyle{}^{1}\delta_{\mathrm{Hom}}(\mu_{2,n})+{}^{2}\delta_{\mathrm{Hom}}(\mu_{1,n})=~{}$
$\displaystyle\sum_{\begin{subarray}{c}i+j=n\\\ i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{2,j}]_{\mathsf{NR}},$
for $n=0,1,\ldots,p$.
###### 5.10 Definition.
An order $p$ deformation $(\mu_{1,t},\mu_{2,t})$ of the compatible Hom-Lie
algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is said to be
extensible if there exists an element $(\mu_{1,p+1},\mu_{2,p+1})\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ which makes the pair
$\displaystyle\big{(}\overline{\mu_{1,t}}=\mu_{1,t}+t^{p+1}\mu_{1,p+1},~{}\overline{\mu_{2,t}}=\mu_{2,t}+t^{p+1}\mu_{2,p+1}\big{)}$
into a deformation of order $p+1$.
Let $(\mu_{1,t},\mu_{2,t})$ be an order $p$ deformation of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$. Define an element
$Ob_{(\mu_{1,t},\mu_{2,t})}\in
C^{3}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ by
$\displaystyle
Ob_{(\mu_{1,t},\mu_{2,t})}=\big{(}\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}},~{}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{2,j}]_{\mathsf{NR}},~{}\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq 1\end{subarray}}[\mu_{2,i},\mu_{2,j}]_{\mathsf{NR}}\big{)}.$
Since $\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\ i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}}$ is the obstruction
cochain to extend the order $p$ deformation of the Hom-Lie algebra
$(\mathfrak{g},[~{},~{}]_{1},\alpha)$, we have
${}^{1}\delta_{\mathrm{Hom}}(\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq 1\end{subarray}}[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}})=0$. Similarly,
${}^{2}\delta_{\mathrm{Hom}}(\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq 1\end{subarray}}[\mu_{2,i},\mu_{2,j}]_{\mathsf{NR}})=0$. Moreover, we
have
$\displaystyle{}^{1}\delta_{\mathrm{Hom}}\big{(}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{1,i},\mu_{2,j}]_{\mathsf{NR}}\big{)}~{}+~{}{}^{2}\delta_{\mathrm{Hom}}\big{(}\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq 1\end{subarray}}[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}}\big{)}$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=p+1\\\ i,j\geq
1\end{subarray}}[\mu_{1},[\mu_{1,i},\mu_{2,j}]_{\mathsf{NR}}]_{\mathsf{NR}}~{}+~{}\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{2},[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}}]_{\mathsf{NR}}$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=p+1\\\ i,j\geq
1\end{subarray}}\big{(}[[\mu_{1},\mu_{1,i}]_{\mathsf{NR}},\mu_{2,j}]_{\mathsf{NR}}-[\mu_{1,i},[\mu_{1},\mu_{2,j}]_{\mathsf{NR}}]_{\mathsf{NR}}\big{)}$
$\displaystyle\quad+\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\ i,j\geq
1\end{subarray}}\big{(}[[\mu_{2},\mu_{1,i}]_{\mathsf{NR}},\mu_{1,j}]_{\mathsf{NR}}-[\mu_{1,i},[\mu_{2},\mu_{1,j}]_{\mathsf{NR}}]_{\mathsf{NR}}\big{)}$
$\displaystyle=\sum_{\begin{subarray}{c}i+j=p+1\\\ i,j\geq
1\end{subarray}}[[\mu_{1},\mu_{1,i}]_{\mathsf{NR}},\mu_{2,j}]_{\mathsf{NR}}-\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{1,i},[\mu_{1},\mu_{2,j}]_{\mathsf{NR}}+[\mu_{2},\mu_{1,j}]_{\mathsf{NR}}]_{\mathsf{NR}}$
$\displaystyle=-\sum_{\begin{subarray}{c}i_{1}+i_{2}+j=p+1\\\
i_{1},i_{2},j\geq
1\end{subarray}}\frac{1}{2}[[\mu_{1,i_{1}},\mu_{1,i_{2}}]_{\mathsf{NR}},\mu_{2,j}]_{\mathsf{NR}}+\sum_{\begin{subarray}{c}i+j_{1}+j_{2}=p+1\\\
i,j_{1},j_{2}\geq
1\end{subarray}}[\mu_{1,i},[\mu_{1,j_{1}},\mu_{2,j_{2}}]_{\mathsf{NR}}]_{\mathsf{NR}}$
$\displaystyle=-\sum_{\begin{subarray}{c}i+j+k=p+1\\\ i,j,k\geq
1\end{subarray}}\frac{1}{2}[[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}},\mu_{2,k}]_{\mathsf{NR}}+\sum_{\begin{subarray}{c}i+j+k=p+1\\\
i,j,k\geq
1\end{subarray}}\frac{1}{2}[[\mu_{1,i},\mu_{1,j}]_{\mathsf{NR}},\mu_{2,k}]_{\mathsf{NR}}=0.$
Similarly, we can show that
$\displaystyle{}^{1}\delta_{\mathrm{Hom}}\big{(}\frac{1}{2}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq
1\end{subarray}}[\mu_{2,i},\mu_{2,j}]_{\mathsf{NR}}\big{)}~{}+~{}{}^{2}\delta_{\mathrm{Hom}}\big{(}\sum_{\begin{subarray}{c}i+j=p+1\\\
i,j\geq 1\end{subarray}}[\mu_{1,i},\mu_{2,j}]_{\mathsf{NR}}\big{)}=0.$
Therefore, we have $\delta_{\mathrm{cHom}}(Ob_{(\mu_{1,t},\mu_{2,t})})=0$. The
corresponding cohomology class $[Ob_{(\mu_{1,t},\mu_{2,t})}]\in
H^{3}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ is called the obstruction
class to extend the deformation $(\mu_{1,t},\mu_{2,t}).$
###### 5.11 Theorem.
An order $p$ deformation $(\mu_{1,p},\mu_{2,p})$ of the compatible Hom-Lie
algebra $(\mathfrak{g},[~{},~{}]_{1},[~{},~{}]_{2},\alpha)$ is extensible if
and only if the corresponding obstruction class
$[Ob_{(\mu_{1,t},\mu_{2,t})}]\in
H^{3}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ is trivial.
###### Proof.
Suppose $(\mu_{1,t},\mu_{2,t})$ is extensible. Let
$(\mu_{1,p+1},\mu_{2,p+1})\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ be an element which makes
the pair
$\big{(}\overline{\mu_{1,t}}=\mu_{1,t}+t^{p+1}\mu_{1,p+1},~{}\overline{\mu_{2,t}}=\mu_{2,t}+t^{p+1}\mu_{2,p+1}\big{)}$
into a deformation of order $p+1$. Then it follows from the expression of
$Ob_{(\mu_{1,t},\mu_{2,t})}$ that
$\displaystyle
Ob_{(\mu_{1,t},\mu_{2,t})}=\delta_{\mathrm{cHom}}\big{(}(\mu_{1,p+1},\mu_{2,p+1})\big{)}.$
This shows that the cohomology class $[Ob_{(\mu_{1,t},\mu_{2,t})}]$ is
trivial.
Conversely, suppose $(\mu_{1,t},\mu_{2,t})$ is an order $p$ deformation for
which $[Ob_{(\mu_{1,t},\mu_{2,t})}]\in
H^{3}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$ is trivial. Then we have
$Ob_{(\mu_{1,t},\mu_{2,t})}=\delta_{\mathrm{cHom}}\big{(}(\mu_{1,p+1},\mu_{2,p+1})\big{)}$,
for some $(\mu_{1,p+1},\mu_{2,p+1})\in
C^{2}_{\mathrm{cHom}}(\mathfrak{g},\mathfrak{g})$. Then it is easy to see that
$\big{(}\overline{\mu_{1,t}}=\mu_{1,t}+t^{p+1}\mu_{1,p+1},~{}\overline{\mu_{2,t}}=\mu_{2,t}+t^{p+1}\mu_{2,p+1}\big{)}$
is a deformation of order $p+1$. In other words, $(\mu_{1,t},\mu_{2,t})$ is
extensible. ∎
Acknowledgements. The author would like to thank Indian Institute of
Technology (IIT) Kharagpur for providing the beautiful academic atmosphere
where the research has been carried out.
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* [18] A. Odesskii and V. Sokolov, Pairs of compatible associative algebras, classical Yang-Baxter equation and quiver representations, Comm. Math. Phys. 278 (2008), 83-99.
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|
††thanks: corresponding author<EMAIL_ADDRESS>
# Ultrahigh-performance superlattice mid-infrared nBn photodetectors at high
operating temperatures
Rohit Kumar1 Bhaskaran Muralidharan1 1Department of Electrical Engineering,
Indian Institute of Technology Bombay, Powai, Mumbai-400076, India
(August 28, 2024)
###### Abstract
While advancing a physics-based comprehensive photodetector-simulation model,
we propose a novel device design of the mid-wavelength infrared nBn
photodetectors by exploiting the inherit flexibility of the InAs1-xSbx ternary
alloy material system. To further explicate the physics of such
photodetectors, we calculate several crucial transport and optoelectronic
parameters, including the dark current density, absorption coefficient,
responsivity, and the quantum efficiency of nBn photodetectors. A remarkable
maximum efficiency of 57.39% is achieved at room temperature at a bias of
-0.25 V, coupled with a radiation power density of 50 mW/$cm^{2}$. The
proposed structure features a maximum quantum efficiency of 44.18% and 37.87%
at 60% and 70% of the $\lambda_{c}$, respectively. Furthermore, a maximum
responsivity of 0.9257 A/W is shown within the mid-wavelength infrared
spectrum. Through our comprehensive analysis, we also demonstrate that our
proposed device design effectively reduces the dark current density by
confining the electric field inside the barrier while preserving a superior
level of quantum efficiency, and the current in such detectors is diffusion-
limited. Insights uncovered here could be of broad interest to critically
evaluate the potential of the nBn structures for mid-wavelength infrared
photodetectors.
Mid-wavelength infrared (MWIR) photodetectors Soibel et al. (2014); Kumar et
al. (2023); Soibel et al. (2019); Gautam et al. (2012); Wu et al. (2020) are
in high demand for a wide range of civilian, military, and space applications,
including environmental monitoring, chemical sensing, medical diagnostics, and
infrared (IR) imaging. This is because they meet high operating temperature
requirements, exhibit better performance, and have numerous advantages over
other photodetector candidates Martyniuk and Rogalski (2013); Martyniuk et al.
(2014); Klipstein (2008); Rogalski et al. (2020). The design of such a high
performance MWIR photodetector relies on achieving a balance between low dark
current and high quantum efficiency. A majority of photodiodes in the market
today are p-n junction photodiodes made from conventional materials like
HgCdTe (MCT) Itsuno et al. (2012), InGaAs Martinelli et al. (1988) and InSb
Shimatani et al. (2020), etc., which are plagued by space-charge generation-
recombination (G-R) dark currents that significantly restrict their efficacy
for applications demanding high sensitivity at lower temperatures Reine et al.
(2014). In order to prevent excessive dark currents, these devices typically
need to be cooled down to cryogenic temperatures.
The nBn photodetector Rodriguez et al. (2007); Chen et al. (2021); Maimon and
Wicks (2006); Kim et al. (2012); Haddadi et al. (2017); Baril et al. (2016);
Craig et al. (2013), in which the barrier is sandwiched between two n-type
regions, features a distinct design that is less susceptible to crystalline
defects and effectively reduces the dark current and noise brought on by the
Shockley-Read-Hall (SRH) generation, surface states, and various other
processes. Due to the high cost of fabrication and the complexity of such
structures, an accurate theoretical modeling is essential for developing the
physics of such IR photodetectors Kumar et al. (2023). In this work, we
develop an accurate physics-based theoretical reliable simulation model and
propose a novel device design for the nBn photodetector Reine et al. (2014);
Maimon and Wicks (2006) that offers relatively high performance and excellent
quantum efficiency within the MWIR spectrum D’souza et al. (2012). We then use
our in-depth computational analysis to elucidate intriguing physics and
predict the various performance limiting factors for an InAs1-xSbx (IAS) Krier
et al. (2007); Lackner et al. (2009); Rogalski et al. (2020); Martyniuk and
Rogalski (2013); Rogalski (1989); Shaveisi and Aliparast (2023) based nBn MWIR
photodetector, where the barrier is designed with a large band gap
Al0.7In0.3As0.3Sb0.7 (AIAS) Bank et al. (2017); Maddox et al. (2016); Ren et
al. (2016) material.
The device structure, depicted in Fig. 1, consists of three principal layers:
a thick absorber layer (AL) of n-type narrow gap IAS material with a thickness
of 2.7 $\mu$m, a barrier layer (BL) of n-type lattice-matched AIAS material
with a thickness of 0.25 $\mu$m, and a contact layer (CL) of n-type IAS with a
thickness of 0.27 $\mu$m. The thickness of the BL (tBL) is considered to be
sufficiently large to inhibit electron tunneling between the CL and the AL
layers. As a result, the majority current is impeded by the barrier material
when a proper bias is applied. The absence of a significant electric field in
the narrow gap material prevents the SRH generation and the Band-to-band (BTB)
tunneling thereby, the nBn photodetectors operating in the MWIR region exhibit
lower levels of dark currents and noise Martyniuk and Rogalski (2013). This
characteristic enables a reduction in the cooling demands associated with
these devices.
FIG. 1: Preliminaries for the proposed MWIR nBn photodetector (a) schematic
illustration of the carrier transport in the nBn photodetector (b) layout of
the considered nBn photodetector with IAS as the AL and the CL, and AIAS as
the BL that hinders the movement of electrons.
FIG. 2: Energy band profiles for the nBn photodetector under (a) positive
bias, (b) equilibrium, i.e., zero bias, and (c) negative applied bias voltage.
The high potential barrier in the VB that appears within the BL at zero bias
prevents holes from moving to the CL on the left. Under equilibrium, Efh is
superimposed to Efe due to their identical spatial alignment. At negative
bias, the holes will be able to traverse the potential barrier and reach the
CL. The presence of a strong electric field within the barrier region
facilitates the movement of carriers, thus contributing to the overall drift
current. The electrons within the CL are unable to traverse to the AL due to
the large CBO.
FIG. 3: Electrostatic potential and the absorption coefficient of the MWIR nBn
photodetector under consideration (a) variation in the electrostatic potential
across the device structure as a function of temperature with an applied bias
of -0.5 V (b) behavior of the absorption coefficient, $\alpha$, in the
absorber region as a function of the incoming radiation wavelength at various
temperatures.
Our methodology comprises the finite difference method in conjunction with the
linear interpolation technique to solve the Poisson and the continuity
equations for the carriers while taking into account the temperature, doping,
and structural parameters of the ternary alloy material system. We obtain the
electrostatic potential of the heterojunction, hole quasi-Fermi-level outside
the thermal equilibrium to build the band structure of the considered device
design. The BTB tunneling, the trap-assisted tunneling (TAT), the SRH G-R, and
the Auger G-R processes are some of the primary sources of the dark current,
as shown in Fig. 1 (a).
Incorporating a BL into the design of IR detectors has the potential to reduce
the unfavorable extrinsic SRH G-R contribution substantially. In this study,
we thereby focus on the radiative recombination, the Auger G-R, and other
thermally generated processes that exhibit dominance in nBn photodetectors.
The device layout using GaSb as the substrate is shown in Fig. 1 (b). The n+
CL (i.e., bottom contact) made up of IAS material is the collector layer for
the photogenerated electrons, which also serves as the buffer layer to reduce
epitaxial strain between the GaSb and the AL for the device. The nBn structure
is shown on top of this buffer layer. We use an iterative approach to solve
the specific equations and relations used in the device’s modeling. The entire
sequence of the numerical simulation is outlined in Fig. A1. The performance
of the proposed device is analyzed based on the various calculated transport
and optoelectronic parameters such as the carrier density, the electric field,
the electrostatic potential, the absorption coefficient for the absorber
region, the dark current density, the responsivity, and the quantum
efficiency, etc., as a function of the applied bias voltage, operating
temperatures, and the structural parameters. The specifications for the CL,
BL, and AL to design the considered nBn MWIR photodetectors are given in Tab.
1.
FIG. 4: Bias dependence of the diffusion-limited dark current density for the
nBn photodetectors at various operating temperatures for (a)
$t_{{}_{AL}}=2.7\leavevmode\nobreak\ \mu m$ and
$N_{{}_{D}}\leavevmode\nobreak\ (BL)=1\times 10^{16}\leavevmode\nobreak\
cm^{-3}$, (b) $t_{{}_{AL}}=3.0\leavevmode\nobreak\ \mu m$ and
$N_{{}_{D}}\leavevmode\nobreak\ (BL)=1\times 10^{16}\leavevmode\nobreak\
cm^{-3}$, and (c) $t_{{}_{AL}}=2.7\leavevmode\nobreak\ \mu m$ and
$N_{{}_{D}}\leavevmode\nobreak\ (BL)=2\times 10^{16}\leavevmode\nobreak\
cm^{-3}$. The dark current density, as shown in (a) and (b), exhibits an
exponential relationship at low bias until it reaches the saturation voltage.
Beyond this voltage, the dark current plateaus at a constant value. In
contrast, the saturation level shifts to higher voltages when the doping of
the BL is increased, as shown in (c).
Figures 2 (a), (b), and (c) depict the potential energy profile of the nBn
structure at three distinct applied voltages: positive bias, equilibrium, and
negative bias. It is assumed that the electron quasi-Fermi levels in the BL
and AL are in equilibrium with each other. As shown in Fig. 2 (c), by applying
a voltage of V = - 0.23 V to the structure, the energy barrier for holes is
lowered by nearly two orders of magnitude (from 163 to 80 meV) compared to an
equilibrium condition. It is evident that the applied voltage primarily
affects $\Delta E_{v}$, and when the nBn detector is reverse-biased, it
indicates that a positive voltage is applied to the absorber contact.
The nBn band structure, as illustrated in Fig. 2 (c), hinders the flow of the
majority carriers (electrons) through a large conduction band offset (CBO) but
enables the flow of the minority carriers (holes) through a near-zero valence
band (VB) offset. Therefore, when a relatively low operating voltage is
applied, it falls almost entirely across the barrier, thereby separating the
photogenerated carriers. For $\Delta E_{v}$ $>$ 3$k_{B}$T, the minority
carriers in the nBn architecture are effectively blocked therefore, the
condition for unhindered minority carrier transport to the CL ($\Delta E_{v}$
$<$ 3$k_{B}$T) is met when applied bias exceeds –0.23 V. The band bending
phenomena is also observed at the boundaries of the BL, which signifies the
accumulation of the majority charge carriers in the vicinity of the space
charge region, which spans the depth of the BL. The mobile charges have
migrated to the neighboring IAS-based AL and CL, depleting the entire BL.
TABLE 1: Parameters used to design the device structures. $x_{{}_{Sb}}$ | 0.09 |
---|---|---
T | 300 K |
Cut-off wavelength ($\lambda_{c}$) | 4.33 $\mu$m |
Layers | Thickness / Depth | Doping
CL | 0.27 $\mu$m | $5\times 10^{15}$ $cm^{-3}$
BL | 0.25 $\mu$m | $8\times 10^{15}$ $cm^{-3}$
AL | 2.7 $\mu$m | $1\times 10^{16}$ $cm^{-3}$
In Appendix A, an in-depth investigation of the electric field and the carrier
density is provided. There, we demonstrate how the electric field and the
carrier density in the nBn photodetector are affected by the temperature and
applied bias.
In Fig. 3 (a), we show the distribution of the electrostatic potential across
the nBn detector as a function of the temperature and the layer thicknesses,
which has a direct impact on the carrier concentration. An inference can be
made that as the temperature increases, the electrostatic potential becomes
more concentrated within the width of the BL. The potential drop in the AL
varies hardly with temperature therefore, the complete depth of the AL is not
shown, whereas a noticeable change can be seen in the CL.
FIG. 5: Room temperature responsivity and the quantum efficiency for the MWIR
nBn photodetector (a) responsivity as a function of the incoming radiation
wavelength (b) wavelength dependence of the quantum efficiency at various AL
thicknesses (c) bias dependence of the quantum efficiency at various
percentages of cut-off wavelength.
In Fig. 3 (b), we show the absorption coefficient of the IAS absorber in
relation to the incoming radiation wavelength and temperature. The energy
threshold of absorption is expected to shift as the band gap of AL material
varies with temperature. The absorption coefficient exhibits maximum values
within the wavelength range of 3.1 to 4.3 $\mu$m for all simulated
temperatures. The corresponding $\alpha$ values fall within the range of
1557-1644 $cm^{-1}$. The absorption coefficient exhibits a decreasing trend in
its maximum value as temperature increases. This behavior appears to be
affected by the band gap of the absorber material.
In Figs. 4 (a), (b), and (c), we demonstrate the dark current density of the
nBn MWIR photodetector. It is important to note that the nBn structure
operates in a minority carrier manner, hence the hole transport from AL to CL
is the primary cause of the dark current. The “turn-on voltage” is assumed to
be V $\approx$ –(0.23 - 0.25) V. This is the voltage after which the dark
current saturates and $\Delta E_{v}$ falls below 3$k_{B}$T. The dark current
density is shown in Fig. 4 (b) when the $t_{AL}$ is slightly increased and
held constant at 3 $\mu$m. Figure 4 (c) demonstrates the dark current density
when the BL doping is increased from $1\times 10^{16}\leavevmode\nobreak\
cm^{-3}$ to $2\times 10^{16}\leavevmode\nobreak\ cm^{-3}$. Changing the
thickness of the AL has a low impact, while changing the doping of the BL has
a noticeable effect, as shown in Figs. 4 (b) and (c), respectively. The dark
current density is shown to be highly voltage-dependent between 0 to –0.23 V.
Alternatively, it is less sensitive to voltage changes between -0.23 V and
-0.7 V, as shown in Figs. 4 (a) and (b). As previously mentioned, the $\Delta
E_{v}$ value for V = -0.23 V is comparable with 3$k_{B}$T, indicating that the
holes are almost freely transported to the CL and hence contribute to the net
dark current. It is observed that for reverse voltages V $<$ –0.23 V, the dark
current density rises sharply due to a rapid rise in the hole concentration,
whereas above V $>$ –0.23 V, the dark current approaches saturation. But when
BL doping is increased, the saturation level shifts to the higher voltages, as
shown in Fig. 4 (c). Moreover, the diffusion-limited barrier structure
prevents tunneling in the simulated voltage range.
The spectral response of the considered nBn photodetector as a function of the
wavelength at a temperature of 300 K for a constant bias of -0.25 V with a
radiation power density of 50 mW/$cm^{2}$ is presented in Fig. 5(a). The
maximum responsivity obtained at lower and higher wavelengths within the MWIR
spectrum is recorded as 0.9257 A/W. The same responsivity value is also
recorded at $\lambda_{c}$ = 4.33 $\mu$m. The nBn device here consist $t_{AL}$
of 2.7 $\mu$m. The VB barrier reduces carrier collection, resulting in
slightly lower responsivity values close to the cut-off wavelength. The
effects of photogenerated carriers altering the nBn barrier height at higher
temperatures could be responsible for the slight increase in the responsivity
observed at shorter wavelengths. Due to its large responsivity and low dark
current density, the proposed nBn design has a very high external quantum
efficiency.
The quantum efficiency, which in turn affects the flux of the photogenerated
carriers transported to the CL, is directly affected by the $\Delta E_{v}$.
The dependence of the quantum efficiency on the wavelength and applied bias
voltage is depicted in Figs. 5 (b) and (c). In Fig. 5 (b), we present the
dependence of the quantum efficiency on the wavelength at a temperature of 300
K for two different AL thicknesses. The photodetector exhibits a maximum
efficiency of 57.39% and 63.74% at a bias voltage of -0.25 V for 2.7 $\mu$m
and 3.0 $\mu$m AL thicknesses, respectively, with a radiation power density of
50 mW/$cm^{2}$. In Fig. 5 (c), we show the quantum efficiency at two different
wavelengths, 60%, and 70% of the $\lambda_{c}$ and it rises sharply to a
maximum value of 44.18% and 37.87%, respectively, at voltage nearly -0.25 V as
the reverse bias increases. Beyond this threshold of applied reverse bias, the
quantum efficiency remains unaffected by the VB offset. Since there is no
tunneling contribution and the dark current density slightly saturates,
showing the photoconductive effect, while quantum efficiency reaches its
maximum value, it can be stressed that such nBn structure may be operated
above -0.25 V.
In conclusion, a physics-based theoretical simulation model has been developed
via an iterative approach for the Poisson and the continuity solver to develop
a framework for the nBn MWIR photodetectors. A remarkable maximum efficiency
of 57.39% was achieved at room temperature when applying a bias of -0.25 V,
coupled with a radiation power density of 50 mW/$cm^{2}$. We have recorded the
maximum quantum efficiency of 44.18% and 37.87% at 60% and 70% of the
$\lambda_{c}$, respectively. Furthermore, a maximum responsivity of 0.9257 A/W
has been recorded within the MWIR spectrum. Through a comprehensive analysis,
we have demonstrated that our proposed device design effectively reduces the
dark current density by confining the electric field inside the barrier while
preserving a superior level of quantum efficiency, and the current in such
detectors is diffusion-limited. Hence, the G-R and tunneling currents do not
limit the high performance of the nBn architecture. Insights uncovered here
could be of broad interest to critically evaluate the potential of the nBn
structures for MWIR IR photodetectors.
## Acknowledgments
The authors acknowledge funding from ISRO, under the ISRO-IIT Bombay Space
Technology Cell. B.M. also acknowledges funding from the Science and
Engineering Research Board (SERB), Government of India, under Grant No.
CRG/2021/003102.
## Author declarations
### Conflict of Interest
The authors have no conflicts to disclose.
### Author Contributions
Rohit Kumar: Investigation (equal); Conceptualization (equal); Visualization
(lead); Data curation (lead); Methodology (lead); Formal analysis (lead);
Validation (lead); Writing-original draft (lead); Writing-review & editing
(lead). Bhaskaran Muralidharan: Investigation (equal); Conceptualization
(equal); Funding acquisition (lead); Project administration (lead); Resources
(lead); Supervision (lead); Writing- review & editing (equal).
## Data Availability
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
## Appendix A Methodology
We solve the Poisson and the continuity equations for the carriers using the
finite difference method and obtain the electrostatic potential of the
heterojunction, hole quasi Fermi-level outside thermal equilibrium to build
the band structure of the considered device design. We assume that no G-R
processes are taking place within the BL, and the absorber region is uniformly
illuminated. The entire sequence of the numerical simulation is outlined in
Fig. A1. The subsequent equations presented herein are relevant for the
calculation of various transport and optoelectronics parameters associated
with the MWIR nBn photodetectors.
FIG. A1: Flowchart to calculate the transport and optoelectronics parameters
using the Poisson and continuity solver.
By using the linear interpolation technique, the lattice constant or the
mobility of IAS can be expressed as Adachi (2017)
$(a/\mu)_{{}_{IAS}}=x_{{}_{Sb}}\times(a/\mu)_{{}_{InSb}}+(1-x_{{}_{Sb}})\times(a/\mu)_{{}_{InAs}}\>,$
(1)
where $a$ denotes the lattice constant, $\mu$ represents the mobility of the
carriers and $x_{{}_{Sb}}$ is the molar fraction. In order to determine the
mobility at any given temperature, the mathematical expression is employed as
Schuster et al. (2014)
$\mu|_{{}_{T}}=\mu|_{{}_{(300\leavevmode\nobreak\
K)}}\times\Big{[}\dfrac{T}{300}\Big{]}^{-\zeta}\>,$ (2)
where $\mu|_{{}_{(300\leavevmode\nobreak\ K)}}$ is the mobility of the
carriers at room temperature. The value of $\zeta$ can be extracted from TABLE
S1. The diffusion current in the nBn photodetector is limited by the intrinsic
carrier concentration $n_{i}$ of the IAS material, which can be calculated
asRogalski and Jóźwikowski (1989); Rogalski (1989)
$\displaystyle(n_{i})_{{}_{IAS}}=(8.50\leavevmode\nobreak\
x_{{}_{Sb}}-6.73\leavevmode\nobreak\ x_{{}_{Sb}}^{2}-1.53\times
10^{-3}x_{{}_{Sb}}\leavevmode\nobreak\ T$ (3) $\displaystyle+4.22\times
10^{-3}\leavevmode\nobreak\ T+1.35)E_{g}^{0.75}\leavevmode\nobreak\
T^{1.5}\exp\Big(\dfrac{-E_{g}}{2k_{B}T}\Big{missing})\times 10^{14}\>,$
where $E_{g}$ is the bandgap of the material. The band gap of the IAS-based CL
and the AL as a function of molar composition and temperature has been
calculated using the following relation Wieder and Clawson (1973); Shaveisi
and Aliparast (2023); Schuster et al. (2014); Rogalski (1989); D’souza et al.
(2012)
TABLE A1: Parameters for the binary materials at T = 300 K Schuster et al. (2014); Rogalski (1989); Adachi (2017); InS ; InA ; Vurgaftman et al. (2001); Sai-Halasz et al. (1977); Rogalski et al. (2020); Martyniuk et al. (2014). Material | $a$ (Å) | $\mu_{e}\leavevmode\nobreak\ (cm^{2}\leavevmode\nobreak\ V^{-1}\leavevmode\nobreak\ s^{-1}$) | $\mu_{h}\leavevmode\nobreak\ (cm^{2}\leavevmode\nobreak\ V^{-1}\leavevmode\nobreak\ s^{-1}$) | $\zeta_{e}$ | $\zeta_{h}$
---|---|---|---|---|---
InSb | 6.4794 | $8\times 10^{4}$ | 800 | 1.8572 | 1.8572
InAs | 6.0583 | $3\times 10^{4}$ | 500 | 0.7212 | 0.5097
TABLE A2: Parameters used for the device simulations Lackner et al. (2009); IAS ; Ren et al. (2016); Maddox et al. (2016); Krier et al. (2007); Bank et al. (2017). Parameters | Unit | CL | BL | AL
---|---|---|---|---
$E_{g}|\leavevmode\nobreak\ _{(300\leavevmode\nobreak\ K)}$ | meV | 286 | 1160 | 286
$\epsilon_{r}$ | - | 15.298 | 15.5 | 15.298
$m^{*}_{e}$ | $m_{0}$ | 0.0197 | 0.071 | 0.0197
$m^{*}_{h}$ | $m_{0}$ | 0.416 | 0.35 | 0.416
$\mu_{h}|\leavevmode\nobreak\ _{(300\leavevmode\nobreak\ K)}$ | $cm^{2}\leavevmode\nobreak\ V^{-1}\leavevmode\nobreak\ s^{-1}$ | $5.27\times 10^{2}$ | $3\times 10^{3}$ | $5.27\times 10^{2}$
$D_{h}|\leavevmode\nobreak\ _{(300\leavevmode\nobreak\ K)}$ | $cm^{2}\leavevmode\nobreak\ s^{-1}$ | 13.636 | 77.625 | 13.636
$P_{d}$ | $mW/cm^{2}$ | 50 | |
$|F_{1}F_{2}|$ | - | 0.2 | |
$m_{0}$ | Kg | $9.109\times 10^{-31}$ | |
$\begin{split}E_{g}(x_{{}_{Sb}},T)=0.411-\bigg{[}\frac{(3.4\times
10^{-4})\leavevmode\nobreak\ T^{2}}{(T+210)}\bigg{]}-0.876\>\\\
x_{{}_{Sb}}+\>0.70\>x^{2}_{{}_{Sb}}+\>(3.4\times
10^{-4})\>x_{{}_{Sb}}\leavevmode\nobreak\ T\leavevmode\nobreak\
(1-x_{{}_{Sb}})\>,\end{split}$ (4)
where $x_{{}_{Sb}}$ is the molar fraction, $E_{g}$ is the bandgap, and T is
the operating temperature. The absorption coefficient $\alpha$ for the IAS-
based AL depends on the incoming radiation wavelength (or photon energy) and
temperature and can be calculated using the following equations D’souza et al.
(2012); Shaveisi and Aliparast (2023); Schuster et al. (2014). When ($E_{g}$
$\geq$ $h\nu$), the absorption coefficient can be expressed using the Urbach
tail model as
$\alpha=948.23\leavevmode\nobreak\ \times\leavevmode\nobreak\
\exp\>\>[170\leavevmode\nobreak\ (h\nu-E_{g}-0.001)]\>,$ (5)
where $h\nu$ is the photon energy. Similarly, when ($E_{g}$ ¡ $h\nu$), the
absorption coefficient can be expressed as D’souza et al. (2012); Shaveisi and
Aliparast (2023); Schuster et al. (2014)
$\begin{split}\alpha=800\leavevmode\nobreak\ +\leavevmode\nobreak\
&\frac{K(h\nu-E_{g}-\Xi)\sqrt{(h\nu-E_{g}-\Xi)^{2}-\Xi^{2}}}{h\nu}\>,\\\
&\Xi=\frac{E_{g}}{2}+0.1\>,\\\ &K=10^{4}\leavevmode\nobreak\
[1+2\>E_{g}]\>.\end{split}$ (6)
We demonstrate the temperature and bias dependence of the carrier density of
the nBn photodetector in Fig. A2. The exponential relationship between the
carrier concentration and applied bias voltage results in a significant change
of the carrier density when there is a relatively small change in the
electrostatic potential distribution, as depicted in Fig. A2. As a result of
the reverse bias applied over the heterojunction, changes are observed in the
BL and the accumulation regions. The potential drop is anticipated to occur in
the BL because it is the constituent layer with more dopant atoms. An increase
in the reverse bias voltage results in a notable increase in the charge
density within the accumulation region at the contact-barrier junction. In
contrast, a substantial drop in the charge density within the accumulation
region at the barrier-absorber junction is observed, resulting in a decrease
in the carrier density at this junction. A positive carrier density indicates
the presence of ionized donors or holes in that region, whereas a negative
carrier density indicates that the majority charge carriers occupy that
region. Up to V $\approx$ -0.5 V, the carrier density in the BL is constant,
implying that only ionized donors are present. This behavior persists up to a
voltage of -0.5 V when the flat band condition is met, indicating that there
is no charge density at the barrier absorber edge. Figure A2 (b) provides a
clear illustration of the carrier density both before and after it meets the
flat band condition.
We demonstrate the dependence of the electric field on temperature and bias in
Fig. A3. The analysis of the nBn photodetector involves considering the
electric field, which provides valuable insights into the electrostatic
properties. In addition, it is an important consideration in the carrier
collection, as a high electric field in the BL may improve the photodetector’s
response time. The occurrence of undesired phenomena, such as BTB tunneling,
can be attributed to the presence of a high electric field. However, this
limitation in the nBn photodetector is effectively reduced through the
incorporation of a substantial CBO. The magnitude of the electric field in the
AL decreases as the applied bias becomes more negative, as illustrated in Fig.
A3. At approximately -0.5 V, the flat band condition takes place, and the
electric field is close to zero. As the AL is depleted in response to an
increasing negative applied bias, a negative electric field is produced.
However, the magnitude of the electric field at the contact barrier junction
grows as the applied bias is made more negative.
FIG. A2: Variations in the carrier density across the heterostructure in
relation to the applied bias voltages at (a) T = 192 K, (b) T = 250 K, and (c)
T = 300 K. The full extent of the AL region is not shown. Carrier density with
a negative value signifies the allocation of electrons in that region, whereas
a positive value indicates the presence of holes and positively ionized
donors.
FIG. A3: Bias and position dependence of the electric field across the
heterostructure at (a) T = 192 K, (b) T = 250 K, and (c) T = 300 K. The
complete width of the AL region is not depicted in the illustration as the
electric field remains constant beyond the depletion region.
## Appendix B Recombination rate calculations
The radiative recombination mechanism is characterized by the recombination of
an electron in the conduction band with a hole in the valence band, resulting
in the emission of a photon due to an excess of energy. Therefore, the BTB
recombination coefficient, B included in our model, can be written as Bellotti
and D’Orsogna (2006); Pierret and Neudeck (1987); Rogalski (1989); Hall (1959)
$\begin{split}B=\Bigg{[}(5.8&\times
10^{-13})\>\epsilon^{0.5}_{\infty}\Big{(}\dfrac{m_{0}}{m^{*}_{e}+m^{*}_{h}}\Big{)}^{1.5}\Big{(}1+\dfrac{m_{0}}{m^{*}_{e}}+\dfrac{m_{0}}{m^{*}_{h}}\Big{)}\\\
\times&\Big{(}\dfrac{300}{T}\Big{)}^{1.5}(E^{2}_{g}+3k_{B}TE_{g}+3.75\leavevmode\nobreak\
k^{2}_{B}T^{2})\Bigg{]}\>,\\\ &R_{rad}=B\>(np-n^{2}_{i})\>,\end{split}$ (7)
where $R_{rad}$ is the BTB recombination rate, $k_{B}$ is the Boltzmann
constant, $\epsilon_{\infty}$ is the high-frequency dielectric constant, and
$m^{*}_{e}$ and $m^{*}_{h}$ are the effective masses of the electrons and
holes, respectively. Depending on the shape of the bands involved, there are
various types of Auger recombination processes. In the context of n-type
material, such as the IAS layers studied in the present work, the predominant
recombination mechanism is Auger 1. The Auger 1 process, characterized by its
non-radiative nature, exhibits dominance at higher temperatures. The Auger
carrier coefficients can be defined as Schuster et al. (2014)
$\begin{split}C_{n}=&\dfrac{\Big{(}\dfrac{m^{*}_{e}}{m_{0}}\Big{)}\leavevmode\nobreak\
|F_{1}F_{2}|^{2}}{2(n_{i}\leavevmode\nobreak\
\epsilon_{\infty})^{2}\leavevmode\nobreak\ (3.8\times
10^{-18})\Big{(}1+\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}^{0.5}\Big{(}1+2\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}}\times\\\
&\Big{(}\dfrac{E_{g}}{k_{B}T}\Big{)}^{-1.5}\exp[-\dfrac{\Big{(}1+2\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}E_{g}}{\Big{(}1+\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}k_{B}T}\Bigg{]}\>,\end{split}$ (8)
$C_{p}=C_{n}\left[\dfrac{1-\dfrac{3E_{g}}{k_{B}T}}{6\Bigg{(}1-\dfrac{5E_{g}}{4k_{B}T}\Bigg{)}}\right]\>,\\\
$ (9)
$R_{A1}=\Big{[}p\leavevmode\nobreak\ C_{p}+n\leavevmode\nobreak\
C_{n}\Big{]}\>(np-n^{2}_{i})\>,\\\ $ (10)
where $R_{A1}$ is the Auger 1 recombination rate. The overlap integral
$|F_{1}F_{2}|$ values range from 0.1 to 0.3. The key factor constraining the
carrier lifetime in the IAS layers is the Auger 1 recombination. Given that
holes are the sole carrier type capable of moving around the heterostructure,
the intrinsic Auger 1 carrier lifetime $\tau^{A1}_{i}$ can be expressed as
Bellotti and D’Orsogna (2006); Rogalski and Orman (1985)
$\begin{split}\tau^{A1}_{i}=&3.8\times 10^{-18}\leavevmode\nobreak\
\dfrac{\epsilon^{2}_{\infty}\Big{(}1+\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}^{0.5}\Big{(}1+2\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}}{\Big{(}\dfrac{m^{*}_{e}}{m_{0}}\Big{)}\leavevmode\nobreak\
|F_{1}F_{2}|^{2}}\times\\\
&\Big{(}\dfrac{E_{g}}{k_{B}T}\Big{)}^{1.5}\exp[\dfrac{\Big{(}1+2\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}E_{g}}{\Big{(}1+\leavevmode\nobreak\
\dfrac{m^{*}_{e}}{m^{*}_{h}}\Big{)}k_{B}T}\Bigg{]}\>,\\\ \end{split}$ (11)
$\tau^{A1}_{h}=\dfrac{2\tau^{A1}_{i}}{1+\Big{(}\dfrac{n_{0}}{n_{i}}\Big{)}^{2}}\>,$
(12)
where $\tau^{A1}_{h}$ is the hole carrier lifetime due to the Auger 1
recombination process, and $n_{0}$ is the equilibrium electron concentration.
## Appendix C Analysis of the dark current, optical responsivity, and the
quantum efficiency
The nBn architecture effectively mitigates the dark current contribution
resulting from the SRH recombination. Consequently, the dark current in this
architecture is mainly diffusion limited and can be calculated as Martyniuk et
al. (2013); Klipstein (2008); Savich et al. (2015); Kwan et al. (2021);
Martyniuk et al. (2014); Alchaar et al. (2019)
$J_{D}=q\dfrac{n^{2}_{i}}{N_{D}}\dfrac{L_{D}}{\tau}\dfrac{\tanh\Big(\dfrac{t_{{}_{AL}}}{L_{D}}\Big{missing})+\beta}{1+\beta\leavevmode\nobreak\
\tanh\Big(\dfrac{t_{{}_{AL}}}{L_{D}}\Big{missing})}\>,$ (13)
where $L_{D}$ is the diffusion length, $N_{D}$ is the donor density in the
absorber region, $\tau$ is the minority carrier lifetime, and $t_{{}_{AL}}$ is
the thickness of the AL. The surface recombination velocity, denoted as
$\beta$, can be neglected due to the boundary conditions that enforce the
absence of hole current at the interface between the AL and the CL. The
valence band potential barrier resulting from the electrostatics of the
junction has been taken into account. Only the thermally generated holes in
the AL that possess sufficient kinetic energy to overcome the barrier will
contribute to the dark current. Therefore, the dark current equation can be
modified as
$J_{D}=q\dfrac{n^{2}_{i}}{N_{D}}\dfrac{L_{D}}{\tau}\tanh\Big(\dfrac{t_{{}_{AL}}}{L_{D}}\Big{missing})\exp[\dfrac{E_{a}-3k_{B}T}{k_{B}T}\Big{]}\>.$
(14)
The photocurrent density $J_{photo}$ caused by the incident power density
$P_{d}$ was used to calculate the optical responsivity R as Itsuno et al.
(2011); Martyniuk et al. (2013); Shaveisi and Aliparast (2023); Schuster et
al. (2014); Kwan et al. (2021)
$R=\dfrac{J_{photo}}{P_{d}}\>.$ (15)
The quantum efficiency is a crucial parameter for assessing the optical
response of the nBn detector and can be calculated as Shaveisi and Aliparast
(2023); Martyniuk et al. (2013); Itsuno et al. (2011); Schuster et al. (2014);
Kwan et al. (2021); Akhavan et al. (2022); Martyniuk and Rogalski (2013)
$\eta=R\times\dfrac{h\nu}{q}\>.$ (16)
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FROTE: Feedback Rule-Driven Oversampling for Editing Models
Anonymous Authors1
###### Abstract
Machine learning (ML) models may involve decision boundaries that change over
time due to updates to rules and regulations, such as in loan approvals or
claims management. However, in such scenarios, it may take time for sufficient
training data to accumulate in order to retrain the model to reflect the new
decision boundaries. While work has been done to reinforce existing decision
boundaries, very little has been done to cover these scenarios where decision
boundaries of the ML models should change in order to reflect new rules. In
this paper, we focus on user-provided feedback _rules_ as a way to expedite
the ML models’ update process, and we formally introduce the problem of pre-
processing training data to edit an ML model in response to feedback rules
such that once the model is retrained on the pre-processed data, its decision
boundaries align more closely with the rules. To solve this problem, we
propose a novel data augmentation method, the Feedback Rule-Based Oversampling
Technique (FROTE). Extensive experiments using different ML models and real
world datasets demonstrate the effectiveness of the method, in particular the
benefit of augmentation and the ability to handle many feedback rules.
††footnotetext: 1Anonymous Institution, Anonymous City, Anonymous Region,
Anonymous Country. Correspondence to: Anonymous Author
<EMAIL_ADDRESS>
Preliminary work. Under review by the Machine Learning and Systems (MLSys)
Conference. Do not distribute.
## 1 Introduction
Machine learning (ML) classifiers are increasingly employed in critical
decision-making processes such as loan approvals, credit score assignment
Khandani et al. (2010), and claims management Singh & Urolagin (2020). Much
focus in the research community has been on improving accuracy of such ML
models, evaluated on test data with a similar distribution as the training
data. However, to deploy such ML models in the real world, one must address
problems that arise from the model being inherently governed and limited by
the training data. In many applications, domain expert knowledge could be used
to improve performance either where data coverage is sparse, or where decision
boundaries may have changed over time. Loan approval policies are an example
where training data may reflect historical policies but not new policies with
shifted decision boundaries.
Naive options for incorporating expert feedback include manually relabelling
historical data and labelling new data. Both are costly in terms of human
intervention, and doing the latter alone compromises the accuracy of the
deployed model until enough new data is collected. While active learning can
reduce the amount of new data needed, the burden may still be too high Cakmak
et al. (2010); Guillory & Bilmes (2011), and moreover during deployment, it
may not be possible to select which instances to label. Recent work Daly et
al. (2021) has proposed a more efficient feedback mechanism using rules. This
approach uses algorithms for learning decision rules Lakkaraju et al. (2016);
Ribeiro et al. (2018); Dash et al. (2018) to provide explanations for
arbitrary ML classifiers. The expert’s task is then limited to reviewing and
modifying a set of classifier predictions and rule-based explanations,
resulting in a _feedback rule set_ (FRS). Daly et al. (2021) propose a post-
processing layer to account for the feedback rules; however, the feedback is
not incorporated into the underlying model.
(a)
(b)
(c)
Figure 1: Left: Original classification boundary. Middle: FROTE generates
synthetic instances to move decision boundary (after relabelling and removing
if permitted). Right: Generating synthetic instances where existing data is
limited.
In this paper, we propose an algorithm called FROTE (Feedback Rule-Based
Oversampling Technique) to edit an ML model for tabular data in response to
user feedback rules. FROTE thus complements the input transformation method of
Daly et al. (2021). Given an input dataset, the algorithm first modifies the
training data if allowed, and then augments it so that re-training the model
on the augmented data results in better alignment with the feedback rules.
FROTE can thus be used with any classification algorithm that takes training
data as input and produces a classifier as output; the algorithm (which could
be proprietary) is treated as a black box. Unlike Daly et al. (2021), the user
feedback is directly encoded in the model.
We use Figure 1 to be suggestive of a loan approval scenario and to illustrate
our solution. Suppose there is a new policy to lower the ages of applicants
for whom loans are approved. Rather than crafting rules from scratch, the user
relies on the existing ML model and accompanying rule-based explanations to
capture relevant dependencies among a potentially large number of features,
and only modifies rules that involve age. Given the resulting feedback rule
set, the user may wish to relabel and remove existing instances as shown in
Figure 1(b). FROTE then generates synthetic instances that reflect both the
feedback rules as well as the existing data. Synthetic data generation can
address the challenge of insufficient training data in the region to be
adjusted, as seen in Figure 1(c). For data generation, we build upon the SMOTE
method Chawla et al. (2002) in several ways; other methods could also be
adapted.
Our contributions can be summarized as follows: 1) We formulate the problem of
editing an ML model by pre-processing a dataset based on user feedback rules.
2) A novel data augmentation-based solution, FROTE, is presented. 3) FROTE is
extensively evaluated using different ML models, real-world datasets, and
feedback rule set parameters to demonstrate its effectiveness, in particular
the benefit of augmentation, improved performance over the state-of-the-art,
and the ability to handle many feedback rules.
## 2 Related Work
To the best of our knowledge, the problem studied in this paper is novel in
that it differs in at least one of the following aspects from the existing
literature: 1) general editing of ML models 2) based on user-specified
feedback rules 3) via model-agnostic data augmentation/pre-processing, where
the rules can enforce existing boundaries, or introduce new boundaries through
changing the dataset.
Data augmentation/pre-processing has been explored in different problem
settings. The class imbalance problem, which deals with the unequal
distribution of classes in training data, was tackled in the seminal work of
Chawla et al. (2002). Their Synthetic Minority Oversampling Technique (SMOTE)
randomly selects minority data points as base instances and generates new data
points that are convex combinations of the base instances and their $k$
nearest neighbours. Han et al. (2005) extend SMOTE by synthesizing data points
that reinforce existing decision boundaries. Due to its simplicity in the
design of the procedure, as well as its robustness, SMOTE has been applied to
different type of problems and has proven successful in a variety of
applications from several different domains Fernández et al. (2018). While we
build on SMOTE for data generation, our model editing use case differs in
going beyond reinforcing existing boundaries to adjusting and introducing new
ones. While our contributions build upon these prior works in terms of
generating synthetic data instances, our use case is not only to reinforce
existing decision boundaries, but also to enable a user both to adjust those
decision boundaries and introduce new ones.
More recently, a more specific use case has gained attention, where data is
processed in order to understand and mitigate underlying biases through
focusing on fairness. Within the fairness and bias mitigation literature, pre-
processing methods such as relabelling and reweighing Calders et al. (2009),
data synthesis Sharma et al. (2020), and data transformation Calmon et al.
(2017) have been proposed.
We argue that the problem we tackle is a more general form of user feedback
that can support user concerns through feedback rules, rather than the ones
based on only the specified protected features.
Within the transfer learning literature, Dai et al. (2007); Eaton & desJardins
(2011) address a similar problem where test data does not follow the same
distribution as training data. They propose an iterative mechanism that re-
weights the old data to minimize error observed on the new data. In Eaton &
desJardins (2011), desJardins and Eaton pursue a similar strategy. Neither
approach however generates synthetic instances.
In the generative models domain, synthetic data generation is used for several
tasks. For example, generative adversarial networks (GANs) aim to improve the
realism of generated samples until the adversary cannot distinguish real from
synthetic data Goodfellow et al. (2014). In Tanaka & Aranha (2019); Douzas &
Bacao (2018); Xu et al. (2019), GANs and conditional GANs with different
network architectures are used to generate synthetic data to overcome class
imbalance as well as privacy issues. In Douzas & Bacao (2018), a conditional
version of GAN (cGAN) is used to generate data for the minority class of
various imbalanced datasets. Overall, when comparing the performance of the
classifiers on imbalanced data sets that were augmented by the GAN and SMOTE,
the former provides better results but with the cost of an higher complexity
correlated to the training of the networks. Xu et al. Xu et al. (2019)
generate tabular synthetic data using conditional tabular GANs. Again these do
not support model editing based on rules.
Incorporating prior knowledge into support vector machines (SVM) was reviewed
by Lauer & Bloch (2008). Two forms of prior knowledge were considered: 1.
invariances to transformations, to permutations and in domains of input space,
2. knowledge on the unlabelled data, the imbalance of the training set or the
quality of the data. Maclin et al. (2006) make use of knowledge bases of rules
and virtual support vectors to add constraints to the optimization. Different
from our solution, these works target only SVM models. Another work from
Kapoor et al. Kapoor et al. (2010) support user influence over ML algorithms
by manipulating confusion matrices, where the user is allowed to manipulate
the initial confusion matrix over the different classes.
Leveraging expert rules has been explored in the assisted labelling
literature. Snorkel Ratner et al. (2017) takes a weak supervision approach to
labelling training data by bringing together label predictions from different
sources, including labelling functions that can be expert-provided patterns.
The labelling sources include labelling functions which can be expert provided
patterns and heuristics to predict labels. A generative model is built to
estimate the accuracy and correlations of the different labelling sources and
produces probabilistic training data where each data point has a probabilities
distribution over all the labels and then can be used to train a model.
Awasthi et al. (2020) consider hybrid supervision from labelled instances as
well as rules that generalize them. The assisted labelling problem is
different from ours in that they seek to label unlabelled data whereas we
already have a model trained on a labelled dataset and wish to edit the model,
without negatively impacting accuracy for data unaffected by the rules. In
addition, in assisted labelling, experts have to devise rules from scratch
whereas in model editing, they may only have to modify rules that capture what
the model has already learned. They provide a solution where labels are
unavailable or noisy and seek to label unlabelled data. Our goal is somewhat
different, where we assume the presence of a dataset and a model that may be
considered trusted and validated but the user wants to make some adjustments
or edits without negatively impacting the model accuracy for unaffected data
which should remain unchanged.
The most closely related work by Daly et al. (2021) addresses user feedback
rules, but not by editing the ML model. Instead, transformations that map
between the original and feedback rules are obtained to yield a post-
processing layer called Overlay. When a new data point arrives for prediction,
Overlay checks to see if a feedback rule corresponds to the data point and if
so, applies the transformation, returning the prediction of the transformed
data point. While Overlay enables immediate changes to an ML system by
applying the above transformations to the input, without retraining the model,
Daly et al. (2021) note that it is a “patch”. As more feedback rules and their
corresponding patches are produced, the overall system consisting of the ML
model and these patches may become overly complex and difficult to maintain.
It is not difficult to imagine that even a single expert could generate a
large number of feedback rules. Additionally, experiments by Daly et al.
(2021) suggest and our experiments confirm (Table 2) that one limitation of
Overlay occurs when a feedback rule differs too significantly from the
underlying model, a limitation that FROTE overcomes. Moreover, in applications
such as finance or spam detection, Overlay’s transformations may incur
additional undesirable latency. For the reasons above, once short-term patches
have been applied, it may be preferable to directly incorporate user feedback
into the model, which is the problem that FROTE solves.
## 3 Preliminaries
As discussed in the Introduction, the premise of this work is that 1) the
distribution of future data (i.e. test data) is different from that of
training data, due for example to a policy change or to the training data not
being representative, and 2) a domain expert understands the nature of the
change and communicates that through a set $\mathcal{F}$ of _feedback rules_ ,
i.e. a _feedback rule set_ (FRS). To establish notation, let
$\mathbf{x}\in\mathcal{X}\subset\mathbb{R}^{d}$ denote a set of attributes for
decision-making, and $y\in\mathcal{Y}=\\{c_{1},c_{2},\dots,c_{l}\\}$ denote a
class label. The existing training data is a set of $n$ instances
$(\mathbf{x}_{i},y_{i})$, $i=1,\dots,n$, assumed to be drawn i.i.d. from a
joint distribution $p_{X,Y}$.
### 3.1 Feedback Rules
We consider a generalization of decision rules beyond recent works Lakkaraju
et al. (2016); Molnar (2019) to allow feedback rules that are _probabilistic_.
A feedback rule $R=(s,\pi)$ is thus a statement of the form IF the clause $s$
is true THEN the class label $Y$ is distributed according to $\pi$. These are
discussed in turn below.
#### Clauses and coverage.
A clause is a conjunction of one or more predicates (also referred to as
conditions) of the form (attribute, operator, value). In our solution, the
operators allowed for categorical attributes are {=, $\neq$}, and for numeric
attributes are {=, $>$, $\geq$, $<$, $\leq$}. An example of a clause with
three predicates is age $<$ 29 AND marital-status = ‘single’ AND income $>$
150K. We say that $\mathbf{x}\in\mathcal{X}$ satisfies a clause $s$, and
reciprocally, a rule $(s,\pi)$ _covers_ $\mathbf{x}$, if all the predicates in
$s$ are true when evaluated on $\mathbf{x}$. Given a dataset $D$, _coverage_
of a rule $(s,\pi)$ and an FRS $\mathcal{F}=\\{(s_{r},\pi_{r})\\}_{r=1}^{m}$
of $m$ feedback rules are defined as follows:
$\displaystyle\operatorname{cov}(s,{D})$
$\displaystyle=\\{(\mathbf{x},y)\in{D}:\mathbf{x}\text{ satisfies }s\\},$ (1)
$\displaystyle\operatorname{cov}(\mathcal{F},{D})$
$\displaystyle=\bigcup_{r=1}^{m}\operatorname{cov}(s_{r},{D}).$ (2)
Note that coverage involves only clauses $s$ and attributes $\mathbf{x}$. If
$D$ is omitted as in $\operatorname{cov}(s)$, then it is understood to be the
entire domain $\mathcal{X}$.
The reason for using logical clauses as above is that they semantically
resemble natural language and the way humans think Zhang & Deng (2015); Letham
et al. (2015); Molnar (2019). Therefore it can be more natural for users to
provide feedback in the form of a rule, either of their own creation or by
modifying an algorithm-provided rule-based explanation. This does require the
rule’s conditions to be built from intelligible features and favours smaller
numbers of conditions and rules Lakkaraju et al. (2016).
#### Label distribution.
Given a feedback rule $(s,\pi)$ and $\mathbf{x}\in\operatorname{cov}(s)$, we
assume that the class label is distributed as $Y\sim\pi$. We will mostly work
with the _deterministic_ case where $\pi$ is the Kronecker delta distribution
for a class $c$, i.e., $Y=c$ with probability $1$. This is the easiest case
for a human expert, who only has to specify the class $c$. However, allowing
probabilistic rules is useful for at least two reasons: 1) accommodating
conflicts between rules (discussed next), and 2) allowing uncertainty in rules
and providing robustness against over-confident rules.
#### Rule conflicts.
When feedback from multiple experts is to be considered, the possibility of
conflicts should be taken into account due to contradictory opinions. Two
rules $(s_{1},\pi_{1})$, $(s_{2},\pi_{2})$ are conflicting if their coverages
intersect,
$\operatorname{cov}(s_{1})\cap\operatorname{cov}(s_{2})\neq\emptyset$, and
$\pi_{1}\neq\pi_{2}$. We assume that all such conflicts are resolved, for
example through one of the following options:
1. 1.
Removal of the intersection, i.e., clause $s_{1}$ is changed to $s_{1}$ AND
NOT $s_{2}$, and $s_{2}$ to $s_{2}$ AND NOT $s_{1}$.
2. 2.
Creation of a new rule for the intersection with a mixture of the
distributions, e.g. $(\pi_{1}+\pi_{2})/2$ or a more general weighting. The
intersection is then excluded from the two original rules as in option 1.
3. 3.
If the two rules are provided by different experts, asking them to come to a
consensus.
We assume that the final FRS is _conflict-free_ through repeated application
of the above operations for conflict resolution.
### 3.2 Problem Formalization
We are given 1) a conflict-free feedback rule set $\mathcal{F}$, 2) an initial
training dataset $D$, and 3) a classification _algorithm_ $A$ that, given a
dataset $D$, trains a classification model $M_{D}$. The task is to create a
dataset ${\hat{D}}$ by augmenting $D$ such that when the model is retrained on
${\hat{D}}$ using $A$ to yield ${M}_{\hat{D}}$, the objective function in (3)
is minimized. To define the objective function, let
$L_{1},L_{2}:\mathcal{Y}\times\mathcal{Y}\mapsto\mathbb{R}$ be two loss
functions that compare two labels. We also assume for ease of exposition that
the rule coverage sets are disjoint, which can be achieved by 1) resolving
conflicts as described above, and 2) merging rules that overlap but do not
conflict. Then the objective function can be written as;
$\displaystyle
J\bigl{(}M_{\hat{D}},\mathcal{F}\bigr{)}=\sum_{(s_{r},\pi_{r})\in\mathcal{F}}\Pr(X\in\operatorname{cov}(s_{r}))$
$\displaystyle{}\times\mathbb{E}_{X\sim
p_{X},Y\sim\pi_{r}}\left[L_{1}\bigl{(}M_{\hat{D}}(X),Y\bigr{)}\>|\>X\in\operatorname{cov}(s_{r})\right]$
$\displaystyle{}+\Pr(X\notin\operatorname{cov}(\mathcal{F}))\mathbb{E}_{X,Y\sim
p_{X,Y}}\left[L_{2}\bigl{(}M_{\hat{D}}(X),Y\bigr{)}\>|\>X\notin\operatorname{cov}(\mathcal{F})\right].$
(3)
The summation in (3) applies to instances in the coverage of the FRS and
evaluates the retrained model’s predictions $M_{\hat{D}}(X)$ against labels
$Y$ distributed according to each feedback rule’s $\pi_{r}$. We refer to the
complement of this term (i.e. $1$ minus it) as _model-rule agreement_ (MRA).
The motivation for the name MRA comes from the case where $L_{1}$ is the
$0$-$1$ loss. Then the expectation of $1-L_{1}(M_{\hat{D}}(X),Y)$ is the
probability of agreement between $M_{\hat{D}}(X)$ and $Y$.
The last term in (3) applies to instances outside
$\operatorname{cov}(\mathcal{F})$ and evaluates the predictions against labels
following the original distribution $p_{X,Y}$. We refer to this term as
outside-coverage performance.
## 4 Proposed Solution
Given an input dataset $D$, the goal of our proposed solution FROTE is to
produce an augmented dataset $\hat{D}$ so that retraining the model on
$\hat{D}$ minimizes the loss function defined in equation (3). The initial
dataset $D$ could be the one used to train the original model, or it could be
a modified version of this dataset. We show in the Experiments section and
supplement that FROTE works with different types of initial datasets. The
steps of FROTE are given in Algorithm 1.
Base instance selection. The adaptation of SMOTE used by FROTE requires a set
of _base instances_ chosen from the original dataset. These provide the basis
for augmentation to ensure that generated instances are similar to original
instances. Base instance selection occurs in two steps: pre-selection of a
_base population_ (BP), denoted $\mathcal{P}$, before the main augmentation
loop (line 4), and selection of subsets of the BP, denoted $\mathcal{B}$,
within the loop (line 7). These are described in the Base Instance Selection
subsection.
Augmentation loop. In each iteration of FROTE, base instances are selected
from the BP (line 7) and corresponding synthetic instances are generated (line
8) as described in the Synthetic Instance Generation subsection. A temporary
dataset $D^{\prime}$ is created (line 9) by combining these synthetic
instances with $\hat{D}$, the current active dataset. The model is retrained
on $D^{\prime}$ (line 10) and the loss function $J$ is evaluated (line 11). If
the loss decreases (lines 12-15), $D^{\prime}$ becomes the current active
dataset $\hat{D}$. Otherwise, the generated instances are discarded and
$\hat{D}$ is unchanged. This augmentation loop proceeds until one of the
termination criteria is met: 1\. the oversampling quota (controlled by
oversampling fraction $q$) is used up, or 2. the iteration limit $\tau$ is
exceeded.
User Constraints. We regard $\tau$ and $q$ as constraints determined by user
preferences: $\tau$ is the number of times the user is willing to run training
algorithm $A$, and $q$ is the allowed amount of augmentation relative to the
initial dataset. Given $\tau$ and $q$, the number of generated instances per
iteration is set to $q\lvert D\rvert/\tau$ (line 1) to uniformly distribute
the quota.
Input: input dataset $D$, ML algorithm $A$, feedback rule set $\mathcal{F}$
User Constraints: iteration limit $\tau$, oversampling fraction $\mathit{q}$
Output: output dataset $\hat{D}$
$\eta\leftarrow q\lvert D\rvert/\tau$, $\hat{D}\leftarrow D$
$M_{\hat{D}}\leftarrow$ apply training algorithm $A$ to $\hat{D}$
$\hat{j}\leftarrow\hat{J}_{\hat{D}}(M_{\hat{D}},\mathcal{F})$
$\mathcal{P}\leftarrow\mathrm{PreSelectBP}(\hat{D},\mathcal{F})$
$\mathit{i},\mathit{N}\leftarrow 0$
while _$i <\tau$ and $\mathit{N}\leq\mathit{q}\times|D|$_ do
2 $\mathcal{B}\leftarrow\mathrm{SelectBaseInstances}(\mathcal{P},\eta)$
$\mathcal{S}\leftarrow\mathrm{Generate}(\mathcal{B})$
${D}^{\prime}\leftarrow\hat{D}\cup\mathcal{S}$ $M_{D^{\prime}}\leftarrow$
apply training algorithm $A$ to $D^{\prime}$ $j^{\prime}$
$\leftarrow\hat{J}_{\hat{D}}(M_{D^{\prime}},\mathcal{F})$ if _$j^{\prime}$
$<\hat{j}$_ then
3 $\hat{D}\leftarrow D^{\prime}$, $\mathit{N}\leftarrow\mathit{N}+|S|$
$\hat{j}\leftarrow j^{\prime}$
$\mathcal{P}\leftarrow\mathrm{PreSelectBP}(\hat{D},\mathcal{F})$,
4 end if
5 $\mathit{i}\leftarrow\mathit{i}+1$
6 end while
Algorithm 1 FROTE
### 4.1 Base Instance Selection
Whereas SMOTE randomly selects data points from the minority class as the base
population, our problem is more challenging as it is driven by the loss $J$ in
(3) and the ideal selection of base instances would maximally decrease this
loss. Referring to Algorithm 1, we denote by $\mathcal{B}$ the set of selected
base instances, $\mathcal{S}=\mathrm{Generate}(\mathcal{B})$ the synthetic
instances generated from $\mathcal{B}$, and $A(D^{\prime})$ the model obtained
from the temporary dataset $D^{\prime}=\hat{D}\cup\mathcal{S}$. Then the goal
is to choose $\mathcal{B}$ to minimize
$J(M_{D^{\prime}},\mathcal{F})=J\bigl{(}A\bigl{(}\hat{D}\cup\mathrm{Generate}(\mathcal{B})\bigr{)},\mathcal{F}\bigr{)}.$
(4)
There are multiple challenges in minimizing (4): 1) Choosing $\mathcal{B}$ is
a combinatorial subset selection problem. The size of the subset
$\lvert\mathcal{B}\rvert=\eta$ may be large (e.g. $100$), and the size of the
BP $\mathcal{P}$ is larger still. 2) The training algorithm $A$ is a black
box. Furthermore, it may be expensive to run to evaluate (4). 3) The
expectations in $J$ must be approximated with empirical averages. We address
this by using the current active dataset $\hat{D}$, replacing $J$ with the
empirical approximation $\hat{J}_{\hat{D}}$ over $\hat{D}$ (lines 3, 11). As a
consequence however, even evaluating (4) for all singleton $\mathcal{B}$, e.g.
all instances in $\mathcal{P}$, would incur complexity of at least
$O(\lvert\mathcal{P}\rvert\lvert\hat{D}\rvert)$. This implies that even a
greedy selection algorithm, which would evaluate
$O(\eta\lvert\mathcal{P}\rvert)$ subsets, would have cubic complexity
$O(q\lvert D\rvert^{3}/\tau)$ assuming $\eta=q\lvert D\rvert/\tau$ and
$\lvert\mathcal{P}\rvert\propto\lvert D\rvert$.
Herein we take a simple approach to base instance selection, consisting of 1)
pre-selecting a BP to focus only on the coverage set
$\operatorname{cov}(\mathcal{F},D)$, 2) selecting subsets _randomly_ , and 3)
exploring more informed strategies that maintain low computational complexity.
Base population pre-selection (line 4). Motivated by the MRA term in equation
(3), we restrict the BP to the coverage $\operatorname{cov}(\mathcal{F},D)$.
In our implementation, we maintain _per-rule_ BPs, i.e., $\mathcal{P}[r]$ for
$R_{r}\in\mathcal{F}$, and accordingly initialize
$\mathcal{P}[r]=\operatorname{cov}(s_{r},D)$. However, rules may have little
or no coverage in the original dataset $D$, and the method described in the
Synthetic Instance Generation subsection requires coverage of at least $k+1$.
To handle this scenario, FROTE uses rule relaxation to obtain a _maximal_
partial rule set, denoted as $\tilde{\mathcal{F}}$. During augmentation, an
instance is selected to be part of the base population if it is _strongly
covered_ , i.e. the instance matches a rule within $\mathcal{F}$ exactly, or
if it is _weakly covered_ , i.e. the instance only matches a rule partially.
The latter case is designed to handle a relaxed case when a rule in
$\mathcal{F}$ has zero support. In this case, we determine the maximal partial
rule, a version of the rule with the minimal condition deletion that gives the
maximum support. In other words, we tried to find out the minimum change we
can make to the rule to give us the largest non-zero support. Since the number
of conditions within each rule set is low, such a maximal partial rule can be
determined by a breath-first search exhaustively by first removing one
condition and then two and so on.
#### Base population pre-selection.
Input: input dataset $D$,
feedback rule set $\mathcal{F}$,
number of nearest neighbours $\mathit{k}$
Output: initial base population $BP$
$L\leftarrow k+1$
$BP\leftarrow\emptyset$
for _each rule $R$ in $\mathcal{F}$_ do
2 if _$cov(R,D)$ $<L$_ then
3 max_sup $\leftarrow 0$ max_cond_R $\leftarrow nil$ while _max_sup $<L$_ do
4 for _each condition $c$ in $R^{s}$_ do
5 $R^{\prime}\leftarrow R$ remove condition $c$ from $R^{\prime s}$ if
_$R^{\prime s}$ is empty_ then
6 max_sup $\leftarrow|D|$ max_cond_R $\leftarrow R^{\prime}$
7 end if
8 else
9 if _$cov(R^{\prime},D)$ $>$ max_sup_ then
10 max_sup $\leftarrow cov(R^{\prime},D)$ max_cond_R $\leftarrow R^{\prime}$
11 end if
12
13 end if
14 $R\leftarrow$ max_cond_R
15 end for
16
17 end while
18
19 end if
20 $BP\leftarrow BP\cup cov(R,D)$
21 end for
Algorithm 2 PreSelectBP
Base population pre-selection procedure PreSelectBP is outlined in Algorithm
2. For each rule in the feedback rule set $\mathcal{F}$, FROTE requires
coverage of at least $k+1$ to generate synthetic instances, where $k$
represents the number of nearest neighbors. Therefore, conditions of a
feedback rule $R$ are relaxed if the coverage of $R$ is less than $k+1$ (lines
7-18). During rule relaxation, the goal is to remove minimum number of
conditions from $R^{s}$ that will result in a maximum rule coverage. To
achieve this, PreSelectBP performs a breadth first search on a tree of
|$R^{s}$| levels, where at each level the nodes are the remaining conditions
in $R^{s}$. At each level, PreSelectBP chooses a condition whose removal
results in maximum coverage in comparison with other conditions that exist at
that level (lines 8-18). The procedure returns the union of the instances
within the coverage of the relaxed feedback rules.
Random subset selection (line 7). The simplest choice for selecting base
instances is to randomly select $\eta$ instances from the BP, motivated in
part by Chawla et al. (2002). We refer to this strategy as random in the
paper. More specifically, base instances are selected on a per-rule basis as
detailed in the supplement. Despite its simplicity, we find during the
experiments that random appears to work well empirically.
Subset selection via integer programming (line 7). We also consider an integer
programming (IP) approach, referred to as IP. Unlike random, IP takes into
account the current ML model $M_{\hat{D}}$ in seeking to generate synthetic
instances that have a greater effect on the objective $J$. The model is
accounted for using borderline instances, which are data points that lie close
to the decision boundaries of the model and thus have more impact Han et al.
(2005).
To quantify the value of different base instances, we associate a weight
$w_{i}$ with each base instance $i$ in the BP $\mathcal{P}$. Weights are pre-
computed using a similar strategy followed in Han et al. (2005), where
instances are classified as noisy, safe, or borderline based on the number of
nearest neighbours with the same and different class labels, and the highest
weight is assigned to borderline instances (see supplement for details).
Let $z_{i}$ be a binary variable such that $z_{i}=1$ if the $i$-th instance in
the BP $\mathcal{P}$ is selected, and $z_{i}=0$ otherwise. Given
$\mathcal{P}$, we define a matrix $\mathbf{A}$ with entries $a_{ji}$ and
dimensions $m\times p$, where $m$ represents the number of rules and
$p=\lvert\mathcal{P}\rvert$, such that $a_{ji}=1$ if instance $i$ is covered
by feedback rule $j$ and $a_{ji}=0$ otherwise. Then the problem of selecting
base instances can be stated as the following IP:
$\max_{z\in\\{0,1\\}^{p}}\sum_{i\in\mathcal{P}}w_{i}z_{i},\textrm{s.t.},k+1\leq\sum_{i\in\mathcal{P}}a_{ji}z_{i}\leq\frac{\eta}{m},j=1,\dots,m.$
(5)
The objective is to maximize the weighted selection of base instances subject
to lower and upper bounds on the number of instances selected for each rule.
Since the data augmentation step described in the next section seeks $k$
neighbours, the lower bound is set to $k+1$. This also preserves the per-rule
diversity in the BP. The upper bound is the number of instances to generate
divided by the number of rules. Non-uniform allocations of instances to rules
are also possible.
Despite (5) being an IP, in practice it can be solved quickly as linear
relaxations directly provide integral optimal solutions in most cases.
Furthermore, IP avoids any evaluation of the objective function (4) in
selecting base instances. In the supplement, we also discuss an approach that
simplifies the evaluation of (4) by using online learning in place of the more
expensive black-box algorithm $A$.
### 4.2 Synthetic Instance Generation
Motivated from SMOTE and its extension to categorical attributes, SMOTE-NC
Chawla et al. (2002), we design a methodology to generate synthetic instances
(line 8 of Algorithm 1) for each selected base instance in line 7. SMOTE
generates synthetic instances that lie between a base instance and one of its
$k$ nearest neighbours, selected at random. For numerical attributes, the
generated value is distributed uniformly on the line segment between the base
instance and the neighbour. For categorical attributes, the value is the
majority value among the neighbours. Following the recommendation of Chawla et
al. (2002); Han et al. (2005), we set the number of neighbours $k=5$.
FROTE’s generation method differs from SMOTE in the following ways: First,
nearest neighbours are found without the constraint that they have the same
class label as the base instance, but with the constraint that they satisfy
the same feedback rule (possibly relaxed). Second, we require that the
generated instance satisfies the conditions of the original, _unrelaxed_ rule.
This happens automatically if the rule was not relaxed, but if it was, then
special logic is needed as described in the supplement. Third, the class label
for the generated instance is sampled from the distribution $\pi$ of the rule
(or simply assigned if the rule is deterministic) rather than being equal to
the label of the base instance.
Figure 2: Experiments with models trained on initial training dataset
(initial), after relabelling (relabel), and after FROTE completes augmentation
(final). random selection strategy is used. Standard box plot shows
interquartile range (IQR) and whiskers show $1.5\times IQR$ based on 30 draws
for each of $\lvert\mathcal{F}\rvert\in\\{1,3,5\\}$. Results with other
datasets included in Section B.
## 5 Experimental Evaluation
### 5.1 Experimental Setup
Datasets, ML Models, Feedback Rules
Table 1: Properties of the datasets used during the experiments. #Ins, #Labels, and #Feat. stands for the number of instances, number of class labels and number of features (numeric/nominal) of the datasets, respectively. Dataset | #Ins. | #Feat. | #Labels
---|---|---|---
Adult | 45222 | 12(4/8) | 2
Breast Cancer | 569 | 32(32/-) | 2
Nursery | 12958 | 8(-/8) | 4
Wine Quality (white) | 4898 | 11(11/-) | 7
Mushroom | 8124 | 21(-/21) | 2
Contraceptive | 1473 | 9(2/7) | 3
Car | 1728 | 6(-/6) | 4
Splice | 3190 | 60(-/60) | 3
To evaluate the effectiveness of FROTE, we experimented with eight real-world
benchmark datasets from UCI111https://archive.ics.uci.edu/ml/datasets.php,
properties of which are provided in Table 1. To generate realistic feedback
rules, we follow the process mentioned in the Introduction by leveraging
Boolean Rules via Column Generation (BRCG) algorithm Dash et al. (2018) to
obtain a rule set explanation for an initial ML model, and then artificially
perturbing these rules to simulate users providing feedback that deviates from
the model’s predictions. For each rule extracted from Dash et al. (2018), we
performed the following three perturbations until we generate $100$ rules for
each dataset with coverage satisfying
$0.05\leq\lvert\operatorname{cov}(s,D)\rvert/\lvert D\rvert<0.25$: For each
rule extracted, 1. A predicate is randomly selected from the rule’s clause and
the operator is reversed. For instance, if the operator is $\neq$, it is
changed to $=$, and similarly if the operator is $\leq$, it is changed to
$\geq$, respectively. 2. Value of the selected predicate is updated based on
its values in the training dataset. For instance, for categorical attributes,
any randomly selected value other than the value of the current predicate is
picked and assigned. Similarly for the numerical attributes, a value within
the range of the minimum and the maximum values of that attribute observed in
the training dataset is assigned. 3. An existing condition from any other rule
is randomly picked and added to the rule’s conditions. We generated $100$
feedback rules in this manner for each dataset, where each generated rule has
coverage satisfying $0.05\leq\lvert\operatorname{cov}(s,D)\rvert/\lvert
D\rvert<0.25$. Rules are deterministic except for the probabilistic rules
experiment in Section B.
Classification models. We used three classification algorithms: scikit-learn’s
Random Forest (RF) and Logistic Regression (LR), and LightGBM (LGBM) Ke et al.
(2017). Default parameter settings are used except for max_iter $=500$ for LR
and max_depth $=3$ for RF. For finding nearest neighbours in FROTE, scikit-
learn’s Nearest Neighbors Pedregosa et al. (2011) algorithm with
algorithm=ball_tree is utilized.
FRS selection and train-test splitting. We experimented with FRS sizes
$\lvert\mathcal{F}\rvert\in\\{1,3,5,8,10,15,20\\}$, and for each run, we
randomly draw this many rules from the pools of $100$ generated as described
above. We used the following mechanism to vary the level of support of the FRS
in the initial training data. For each dataset $D$ and FRS $\mathcal{F}$, $D$
is partitioned into coverage ($\operatorname{cov}(\mathcal{F},D)$) and
outside-coverage ($D-\operatorname{cov}(\mathcal{F},D)$) sets.
$D-\operatorname{cov}(\mathcal{F},D)$ is randomly partitioned on a
$(80\%-20\%)$ basis into training and test. For the coverage set
$\operatorname{cov}(\mathcal{F},D)$, we vary the training coverage fraction
($tcf$), i.e. the fraction of the coverage set included in the training set.
That is, $tcf\times\lvert\operatorname{cov}(\mathcal{F},D)\rvert$ randomly
selected instances are added to the training partition of
$D-\operatorname{cov}(\mathcal{F},D)$, and the remainder to the test partition
of $D-\operatorname{cov}(\mathcal{F},D)$. We experimented with
$tcf\in\\{0,0.05,0.1,0.15,0.2,0.3,0.4\\}$. $tcf=0$ tests the scenario where
the FRS has no coverage in the initial training set, for example when a new
rule emerges.
We perform $30$ to $50$ runs as described in the previous paragraph for each
experimental setting, depending on the size of the dataset. This method of
randomly drawing a new rule set and train-test split for each run increases
the variability of rule sets tested (and their impact on the results) compared
to fixing a rule set and performing cross-validation with it. All algorithm
variations are compared using the same rule sets and splits.
Metrics. FROTE uses only the training dataset for augmentation and all
evaluation results are reported on the held-out test set. We report values of
the complement of $J$, $\overline{J}=1-J$, where $\overline{J}$ is a weighted
average as in (3), weighted by rule coverage probabilities
$\Pr(\operatorname{cov}(s_{r},D))$ in the test set, the first term is the MRA
discussed previously (with $L_{1}$ as $0$-$1$ loss), and the last term is
$F_{1}$ score to evaluate model performance on the outside-coverage
population. In running FROTE however, we simply use a $0.5$-$0.5$ weighting
between MRA and $F_{1}$ score in evaluating $\hat{J}_{\hat{D}}$. This is
because the test set coverage probabilities are not known to FROTE and may not
be equal to the training set probabilities.
Input dataset choices. We experiment with three choices of input dataset $D$
to FROTE. In addition to 1) taking the training dataset as it is (denoted none
for no modification), instances in $\operatorname{cov}(\mathcal{F},D)$ that do
not have the same class label as the feedback rules covering those instances
may be 2) relabelled to agree with the covering rules (relabel) or 3) dropped
(drop). relabel is used in all experiments except for the one that evaluates
input dataset choices. It is important to note that relabel and drop may not
be possible if the user is reluctant to make changes to the existing dataset
for various data integrity reasons.
Configuration. The number of instances generated per iteration ($\eta$) is set
to $200$ for Adult dataset, $50$ for Nursery, Mushroom, Splice, and Wine
datasets, and $20$ for Car, Contraceptive and Breast Cancer datasets.
$\mathit{\tau=200}$ is used as the iteration limit for all the experiments. We
used $k=5$ and $q=0.5$ for all the experiments except the ones we evaluated
the effect of these two parameters. All experiments were limited to $24$ hours
and runs that exceed this time limit were terminated. We ran all experiments
on a 2.6GHz CPU with 20GB of RAM and they were run deterministically with
consistent random number generator seed (42).
### 5.2 Results and Discussion
Benefit of augmentation. In Figure 2, we compare the test set $\overline{J}$
values obtained from models trained on 1) the initial training dataset, 2)
after relabelling based on the FRS (relabel), and 3) after FROTE completes
augmentation. The comparison is shown for the three ML models, a range of
training coverages, and three of the datasets with the remainder in Section B.
Even after relabelling, FROTE’s augmentation improves $\overline{J}$ for all
models and datasets compared to relabelling alone (final vs. relabel). This
finding is further supported by similar plots in Section B of _differences_ in
$\overline{J}$ between final and relabel, the vast majority of which are
positive. Not surprisingly, the same conclusion holds more strongly for the
drop and none options (see Section B).
Two trends are evident from Figure 2. First, the improvement over relabel is
larger for smaller $tcf$, and notably for the difficult case of $tcf=0$ in
which the initial training dataset has no coverage of the FRS. This shows that
relabel is not sufficient and there is a greater need for augmentation when
$tcf$ is low. Second, the improvement is larger for LR, which indicates that
linear models may require more data to push decision boundaries.
Comparison with the existing work. To the best of our knowledge, the closest
work to ours is Overlay Daly et al. (2021), which includes two approaches,
Soft Constraints and Hard Constraints. The former treats the user feedback as
a soft constraint and uses the prediction on the transformed instance, and the
latter considers the feedback as a hard constraint and uses the feedback
rules’ prediction for all applicable instances. A similar setting as in the
previous experiments is used for this comparison. For each run with a dataset,
3 rules are randomly selected and provided as the Full Knowledge Rule Set
(FKRS) Daly et al. (2021) for Overlay, and as the FRS for FROTE. For each rule
set, $50\%$ of the coverage population is included in the training data and
rest in the test data. Similarly, for the outside-coverage population, a
$50\%-50\%$ split is performed. The model is trained on the training dataset,
and FROTE, Soft Constraints and Hard Constraints are evaluated on the held-out
test set. Overlay is presented for binary classification problem and the
experiments reported in Daly et al. (2021) are performed using binary
datasets. Therefore we experimented with only the 3 binary datasets (out of
8), and results are displayed in Table 2 (Results with the adult dataset
together with separate MRA and F-Scores are in Section B.) We observe that
FROTE performs significantly better than both approaches of Overlay for all
datasets. The performance of Soft Constraints and Hard Constraints differs
greatly, which suggests the user feedback rules are too divergent from the
decision boundaries of the initial ML model for Overlay to perform well, in
line with the findings of Daly et al. (2021). This demonstrates that our
solution for integrating user feedback into models through pre-processing
achieves a better performance in comparison with a state-of-art post-
processing approach.
Table 2: Comparison with Overlay-Soft (soft constraints) and Overlay-Hard (hard constraints) of Daly et al. (2021) on BreastCancer and Mushroom datasets. Means and standard deviations computed from 50 runs. Dataset | Model | $\Delta\overline{J}$
---|---|---
| | Overlay-Soft | Overlay-Hard | FROTE
B.Cancer | LR | $\mathllap{\shortminus}0.008\pm 0.045$ | $\mathllap{\shortminus}0.237\pm 0.212$ | $0.030\pm 0.008$
| RF | $0.001\pm 0.003$ | $\mathllap{\shortminus}0.215\pm 0.204$ | $0.041\pm 0.018$
| LGBM | $0.006\pm 0.011$ | $\mathllap{\shortminus}0.207\pm 0.180$ | $0.033\pm 0.015$
Mushr. | LR | $0.001\pm 0.004$ | $\mathllap{\shortminus}0.158\pm 0.213$ | $0.014\pm 0.015$
| RF | $0.001\pm 0.004$ | $\mathllap{\shortminus}0.153\pm 0.208$ | $0.009\pm 0.008$
| LGBM | $\mathllap{\shortminus}0.017\pm 0.091$ | $\mathllap{\shortminus}0.150\pm 0.206$ | $0.009\pm 0.009$
Figure 3: Effect of feedback rule set size for the Breast Cancer dataset and
random selection strategy. The same comparison as in Figure 2 is shown between
initial, after relabel, and final (after FROTE). Each box and whiskers is
computed from $30$ runs with $tcf=0.2$.
Number of feedback rules. One advantage of FROTE is its capability to work
with rule sets containing any number of rules. Figure 3 displays
$\overline{J}$ values in the same manner as Figure 2 for feedback rule sets
having $8,10,15$ and $20$ rules. The improvement in $\overline{J}$ is
maintained up to $20$ rules. Results with other datasets are provided in
Section B. Overall, they demonstrate the efficacy of FROTE with larger rule
sets.
Table 3: Comparison of random and IP base instance selection strategies. Means and standard deviations computed from all runs for a given dataset and model. Dataset | Model | $\Delta\overline{J}$
---|---|---
| | random | IP
B. Cancer | RF | $0.000\pm 0.003$ | $0.001\pm 0.006$
| LR | $0.006\pm 0.022$ | $0.006\pm 0.026$
| LGBM | $0.001\pm 0.008$ | $0.002\pm 0.010$
Car | RF | $0.005\pm 0.020$ | $0.006\pm 0.020$
| LR | $0.022\pm 0.034$ | $0.020\pm 0.029$
| LGBM | $0.008\pm 0.033$ | $0.008\pm 0.027$
Mushroom | RF | $0.001\pm 0.017$ | $0.004\pm 0.034$
| LR | $0.005\pm 0.023$ | $0.011\pm 0.049$
| LGBM | $0.004\pm 0.037$ | $0.006\pm 0.041$
Adult | RF | $0.003\pm 0.014$ | $0.003\pm 0.011$
| LR | $0.008\pm 0.023$ | $0.004\pm 0.012$
| LGBM | $0.004\pm 0.015$ | $0.003\pm 0.011$
Wine | RF | $0.001\pm 0.007$ | $0.001\pm 0.007$
| LR | $0.056\pm 0.096$ | $0.055\pm 0.094$
| LGBM | $0.003\pm 0.015$ | $0.003\pm 0.010$
Contracep. | RF | $0.032\pm 0.081$ | $0.038\pm 0.085$
| LR | $0.041\pm 0.099$ | $0.051\pm 0.102$
| LGBM | $0.027\pm 0.066$ | $0.026\pm 0.057$
Nursery | RF | $0.031\pm 0.099$ | $0.023\pm 0.076$
| LR | $0.043\pm 0.088$ | $0.029\pm 0.069$
| LGBM | $0.035\pm 0.108$ | $0.030\pm 0.096$
Splice | RF | $0.003\pm 0.017$ | $0.002\pm 0.012$
| LR | $0.011\pm 0.031$ | $0.007\pm 0.018$
| LGBM | $0.014\pm 0.049$ | $0.009\pm 0.037$
Base instance selection strategy. We now compare the performance of the two
base instance selection strategies, random and IP. Table 3 shows the
$\overline{J}$ improvements for models trained on the final augmented dataset
relative to the initial dataset. The amount of augmentation required (as a
fraction of the input dataset size) for these improvements for both strategies
is included in Section B. There is not a clear winner between random and IP in
terms of $\overline{J}$ (the “win-loss-tie” record based on 3 decimal places
is 11-8-5), although IP generally adds fewer instances to the dataset. One
possible reason behind relatively good performance of random is although IP
appears more informed, random may avoid “overfitting”, in the sense of
selecting base instances that improve the objective function evaluated on the
augmented training dataset but not on the held-out test set. Looking at the
MRA and F-Score separately (provided in Section B) for the results in Table 3,
we see an improvement in MRA without significant decrease (in some cases an
increase) in F-Score for both techniques, for all results. However, the degree
of improvement is dependent on the dataset and model.
## 6 Broader Impact and Discussion
One important point to note is that there is generally an inflection point in
terms of the number of data points added where the cost to overall model
performance starts to outweigh the improvement in MRA. This inflection point
also depends on the model used and the dataset. It can be explained by the
data difficulty factors described in Stefanowski (2016), namely an effect of
too strong overlap between classes, and a presence of too many examples of one
class inside the other class’s region.
One limitation of the work is that it may be restricted to tabular data,
however, we believe similar mechanisms could be used when considering images
where Boolean rules could show relevant images segments or features.
Our work supports model editing where the final ML model will encode the
decision processes of not just the underlying data but also external
knowledge. This ability can be leveraged to correct incorrect assumptions in
the original data or encoded updated policies. On the other hand this
introduces the ability for the model builder to influence the model outcomes
which could intentionally or unintentionally introduce bias. The user feedback
however is interpretable and transparent and user influence is in the form of
a Boolean feedback rule. This supports easy integrating into a governance
framework such as proposed in Arnold et al. (2019) where clear auditing of the
original data, the feedback rules and the newly created dataset can be stored
to transparently log the updates to the model and capture the lineage of the
data. Post processing analysis to compare the original and the resulting model
could also be leveraged to ensure unintended biases have not been introduces
Bellamy et al. (2019) along with generating an interpretable model comparison
of the two models as proposed by Nair et. al. Nair et al. (2021).
Additionally, FROTE achieves this while trying to minimise the model accuracy
for other segments of the dataset. This is in contrast to human labelling or
relabelling tasks where the downstream impact of the newly labeled data points
may be unclear. Additionally, the source of the newly labeled data, their
level or expertise, familiarity with the data are all opaque. One could argue
peer reviewing a feedback rule set to obtain consensus among stake holders is
relatively easy compared to ensuring a consistent view is being used among
data labellers.
## 7 Conclusion
We presented the problem of pre-processing training dataset to edit an ML
model based on feedback rules. We proposed FROTE, a novel technique based on
data augmentation, to solve this problem. Empirical studies on real datasets
with different ML models demonstrate its effectiveness. Our work supports
model editing where the final model encodes decision processes of not just the
underlying data but also external knowledge. This ability can be leveraged to
correct deficiencies in the original data or adapt to updated policies. User
feedback is interpretable and transparent as it is in the form of Boolean
rules, supporting clear auditing and governance. A promising future direction
is to experiment on different base population selection strategies and
optimization techniques to select the base instances and their neighbors
together, in order to improve the performance.
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## Appendix A Solution Details
#### Subset selection via integer programming.
We elaborate on the integer programming formulation for the subset selection
problem presented in the main paper. For a given $\mathcal{F}$ we would like
to determine a subset of the training data within $\mathcal{F}$ that has the
greatest influence on the model decision boundaries.
The weight $w_{i}$ reflects the value of a data point for the final selection.
Instances near the decision boundary are more valuable, as it has a greater
potentially to influence the model. This weight is pre-computed as follows:
For each $j,\,j\in D$ compute, $p$ as the number of $k$ neighbours who have
the same label, and $q$ as the number of $k$ neighbours who have a different
label. Here, the label refers to the predicted label from a model we seek to
edit. If $q>>p$, the observation can be considered _noisy_ , $p>>q$, then the
observation can be considered as _safe_ , and if $p\approx q$ the observation
can be considered as _borderline_ Han et al. (2005). Correspondingly, the
weights $w_{i}$ can be assigned based on these three cases such that, the
borderline points is assigned the largest weight. In our experiments, we set
$w_{i}=3$ for borderline and $w_{i}=1$ for noisy and safe data points computed
within $k=10$ nearest neighbors.
#### Subset selection with online learning.
As mentioned in the main text, we also considered the use of online learning
to simplify the evaluation of objective function (3), and specifically to
avoid running training algorithm $A$. We instead take a proxy approach in
which 1) the current model $M_{\hat{D}}=A(\hat{D})$ is approximated by a
(parametric) model $\hat{M}$ to which online learning can be applied, and 2)
the retrained model $M_{D^{\prime}}$ is approximated by the result of online
learning, starting from $\hat{M}$ and updating based on the generated
instances $\mathrm{Generate}(\mathcal{B})$. Recalling that $J$ is also
replaced by its empirical approximation $\hat{J}_{\hat{D}}$ over $\hat{D}$,
the online learning approximation can thus be written as
$\displaystyle
J\left(A\left(\hat{D}\cup\mathrm{Generate}(\mathcal{B})\right),\mathcal{F}\right)\approx$
(6)
$\displaystyle\hat{J}_{\hat{D}}\left(\mathrm{OL}\left(\hat{M},\mathrm{Generate}(\mathcal{B})\right),\mathcal{F}\right).$
We investigated the use of LABEL:eqn:OLproxy to approximate objective function
(3) for singleton sets $\mathcal{B}=\\{i\\}$, $i\in\mathcal{P}$. Such
evaluations on singletons might be summed to provide a crude approximation to
(3) for non-singleton $\mathcal{B}$; the IP objective function (4) is also a
sum approximation in this sense. They could also constitute the first
iteration in a greedy algorithm for selecting $\mathcal{B}$.
Our experience thus far however is that even the evaluation of
(LABEL:eqn:OLproxy) is still too computationally intensive to be practical (at
least in terms of facilitating experimentation). To be more specific, we used
the Vowpal Wabbit library222https://vowpalwabbit.org for online learning with
a plain logistic regression model $\hat{M}$. Step 1) of approximating
$M_{\hat{D}}$ with $\hat{M}$ is done by training $\hat{M}$ on dataset
$\hat{D}$ and the outputs of $M_{\hat{D}}$ on $\hat{D}$. This has
computational complexity $O(\lvert\hat{D}\rvert)$. Likewise, step 2), i.e.
approximating $M_{D^{\prime}}$ by updating $\hat{M}$ for each generated
instance $\mathrm{Generate}(\\{i\\})$, $i\in\mathcal{P}$, also has complexity
$O(\lvert\mathcal{P}\rvert)=O(\lvert\hat{D}\rvert)$. However, evaluating
$\hat{J}_{\hat{D}}$ for each of these updated models results in complexity
$O(\lvert\hat{D}\rvert^{2})$, and we have found this to be the slow step in
our limited experiments. Future work could consider further approximations to
the objective function that avoid higher than first-order complexity in
$\lvert\hat{D}\rvert$.
#### Synthetic instance generation.
Synthetic instance generation is used by the Generate() procedure within
FROTE, as outlined in Algorithm 1 of main paper (line 9). It is called for
each base instance and a randomly selected neighbor of it in order to generate
synthetic instances. Synthetic instance generation uses two subroutines for
populating categorical and numerical attributes.
For populating categorical attributes, algorithm iterates through each
categorical attribute to assign a value. For each categorical attribute,
initially, all possible values for that attribute are calculated and stored.
These attribute values are sorted in the decreasing order of the number of
times they occur in the neighbors. Therefore, the first element in the list is
the value that occurs in the majority of the $k$ nearest neighbor instances of
the base instance. If the corresponding attribute is part of one of the
conditions of the rule, then a special check is needed to make sure that the
assigned value satisfies the corresponding condition(s). For instance, the
algorithm ensures that for a condition with "$\neq$" operator, the value
assigned to the corresponding attribute is different than the value of that
corresponding condition.
The procedure iterates over each of the numerical attributes, and for each
attribute, if it is not part of any of the conditions of the rule, the value
to the corresponding numerical attribute is assigned using a similar approach
to SMOTE Chawla et al. (2002). If the attribute exists in a condition where
the operator is ’$=$’, then the value of the corresponding condition is
assigned. However, if the attribute exists in a condition where the operator
is one of {’$>$’,’$\geq$’,’$<$’,’$\leq$’}, extra checks are performed to
ensure that the generated value satisfies the corresponding conditions.
Specifically, a window is defined with a minimum and maximum value (lines
21-29) based on the specific operators. These bounds keep track of the minimum
and/or the maximum values that can be assigned to the corresponding feature of
the new instance. They are further adjusted based on the base and neighbor
instance values to make sure that the new value that will be assigned will
stay within the value limits defined by the comparison operators. Finally a
diff value is assigned based on a tightest window determined by these minimum
and maximum values together with the base and the neighbor instances’
corresponding attribute values, and diff is then used to generate a value for
the corresponding attribute.
## Appendix B Experimental Evaluation
### B.1 Further Experimental Results
Benefit of augmentation. We compare the test set $\overline{J}$ values
obtained from the models that are trained on 1) the initial dataset before
FROTE, 2) after applying the modification strategy, and 3) after FROTE
completes augmentation. In Figure 4, additional plots for Figure 3 of the main
paper are given, where the results with Splice, Nursery, Breast Cancer,
Mushroom and Car datasets are included. In Figure 4, the improvements of the
$\overline{J}$ values observed after 1. modification strategy is applied, and
2. between the augmentation process and mod strategy, is displayed. Both
Figure 3 of the main paper and Figure 4 show the results with the relabel
strategy. Figure 5 and Figure 6 show the results with the none strategy, and
Figure 7 and Figure 8 show the results with the drop strategy. As can be
observed from the figures, the variance appears to be higher for both none and
drop strategies, since for the former, existing contradictory instances are
remained in the dataset, and for the latter, the base instances are selected
through rule relaxation which increases the variety in the base instances.
However, for all mod strategies, we can conclude that augmentation can improve
MRA without much compromise-in some cases increase- in F1-Score.
Figure 4: Additional plots for Figure 2 in the main paper. Experiments with
models trained on the initial dataset before FROTE (initial), after applying
the relabel mod strategy, and after FROTE completes augmentation (final). The
comparison is shown as a function of the training coverage fraction of the
feedback rule sets and for different ML models and the Splice, Nursery and
Breast Cancer, Mushroom and Car datasets. The random selection strategy is
used. Standard box plot showing interquartile range (IQR) and whiskers showing
$1.5$ times IQR based on 30 random draws for each of
$\lvert\mathcal{F}\rvert\in\\{1,3,5\\}$. Figure 5: Additional plots for Figure
2 in the main paper. Experiments with models trained on the initial dataset
before FROTE (initial), after applying the none strategy, and after FROTE
completes augmentation (final). $mod-imp$ and $final-imp$ represent the
differences in $\overline{J}$ between mod and initial and final and mod,
respectively. The comparison is shown as a function of the training coverage
fraction of the feedback rule sets and for different ML models and all the
datasets. The random selection strategy is used. Standard box plot showing
interquartile range (IQR) and whiskers showing $1.5$ times IQR based on 30
random draws for each of $\lvert\mathcal{F}\rvert\in\\{1,3,5\\}$.
Figure 6: Additional plots for Figure 2 in the main paper. Experiments with
models trained on the initial dataset before FROTE (initial), after applying
the none strategy, and after FROTE completes augmentation (final). $mod-imp$
and $final-imp$ represent the differences in $\overline{J}$ between mod and
initial and final and mod, respectively. The comparison is shown as a function
of the training coverage fraction of the feedback rule sets and for different
ML models and all the datasets. The random selection strategy is used.
Standard box plot showing interquartile range (IQR) and whiskers showing $1.5$
times IQR based on 30 random draws for each of
$\lvert\mathcal{F}\rvert\in\\{1,3,5\\}$. Figure 7: Similar setting with Figure
1 except results are presented for drop modification strategy.
Figure 8: Similar setting with Figure 1 except results are presented for drop
modification strategy. Figure 9: Augmentation progress evaluated on the held-
out test set for different models and $tcf$ values on the Adult dataset. The
objective function $\overline{J}$ (median and 5-95 percentiles) is shown as a
function of the number of instances added to the dataset during augmentation.
Results are averaged over 90 runs, and for all runs, $|\mathcal{F}|=3$, the
mod-strategy is relabel, and random selection is used.
Comparison with the existing work. Additional results for the comparison
experiments with Daly et al. (2021) are included in Tables 7 and 8. We observe
from the tables that our solution performs better than a state-of-art post-
processing approach, which confirms with the findings presented in the main
paper. When we examine the results in Table 8, we see that even Hard
Constraints has a significantly higher MRA than the Soft Constraints for all
datasets, it performs very poorly on the outside coverage population, as can
be seen from the $\Delta$F-Score values. This demonstrates that a pure post-
processing approach can suffer if the rules are deviated from the underlying
model. Similar findings are observed for the Soft Constraints, however Soft
Constraints suffers less from the deviation in the rules, since it considers
models decisions after applying changes to the data instance based on the
rules learnt so far.
Augmentation progress. In Figure 9, we evaluate $\overline{J}$ on the held-out
test set for intermediate models trained on $D^{\prime}$ (i.e. augmented
training dataset at the end of each iteration) as a function of the number of
synthetic instances added, to illustrate how these change for different models
and $tcf$ values. For all models, $\overline{J}$ improves more quickly for
lower training coverage. RF needs fewer instances to reach $\overline{J}=1$ in
comparison with LR and LGBM. This again suggests that non-linear models like
RF may require less data to edit than linear models.
Number of feedback rules. Additional plots for displaying the effect of number
of rules on the performance of the solution are given in Figure 10. For all
datasets, we experimented with |$\mathcal{F}$|={8,10,15,20}, however for some
datasets, for |$\mathcal{F}$|=15 and |$\mathcal{F}$|=20, no such conflict-free
$\mathcal{F}$ can be found out of $500$ rules. Therefore, we included the
results for the experiments for which a conflict-free rule set with the
experimented size can be formed.
As it is observed from the results, FROTE improves $J$ both after the relabel
modification strategy and after the data augmentation. Overall, results
demonstrate the efficacy of the approach even with larger rule sets.
Figure 10: Additional plots for Figure 3 in the main paper. Effect of feedback
rule set size for the Car, Contraceptive, Nursery and Splice datasets are
given using the random selection strategy. The same comparison as in Figure 1
is shown between initial (before FROTE), after relabel, and final (after
augmentation). Each box and whiskers is computed from 20 runs with $tcf=0.2$,
$\alpha=0.8$, $k=5$.
Base instance selection strategy. Performance of the two base instance
selection strategies, IP and random were compared in the main paper using the
improvement in $\overline{J}$. In Table 4, we have included the number of
instances added to achieve those improvements. In Table 5, improvements in MRA
and F-Score are reported separately. We observe that the improvement in
$\overline{J}$ is highly dominated by the improvement in MRA.
Table 4: Experiments with IP and random selection strategies for all datasets and models. Number of instances are included as an additional column to Table 2 of the main paper. $\Delta\\#Ins/|D|$ is the number of instances added (as a fraction of the dataset size) that leads to the reported improvements. Means and standard deviations are computed from all runs performed with a given dataset and model. Dataset | Model | $\Delta\overline{J}$ (random) | $\Delta\overline{J}$ (IP) | $\Delta$#Ins$/|D|$ (random) | $\Delta$#Ins$/|D|$ (IP)
---|---|---|---|---|---
B.Cancer | RF | $0.000\pm 0.003$ | $0.001\pm 0.006$ | $0.011\pm 0.016$ | $0.015\pm 0.042$
| LR | $0.006\pm 0.022$ | $0.006\pm 0.026$ | $0.298\pm 0.326$ | $0.199\pm 0.266$
| LGBM | $0.001\pm 0.008$ | $0.002\pm 0.010$ | $0.011\pm 0.016$ | $0.014\pm 0.036$
Car | RF | $0.005\pm 0.020$ | $0.006\pm 0.020$ | $0.001\pm 0.003$ | $0.001\pm 0.004$
| LR | $0.022\pm 0.034$ | $0.020\pm 0.029$ | $0.227\pm 0.225$ | $0.113\pm 0.183$
| LGBM | $0.008\pm 0.033$ | $0.008\pm 0.027$ | $0.001\pm 0.003$ | $0.001\pm 0.004$
Mushroom | RF | $0.001\pm 0.017$ | $0.004\pm 0.034$ | $0.001\pm 0.002$ | $0.001\pm 0.002$
| LR | $0.005\pm 0.023$ | $0.011\pm 0.049$ | $0.036\pm 0.136$ | $0.016\pm 0.064$
| LGBM | $0.004\pm 0.037$ | $0.006\pm 0.041$ | $0.001\pm 0.002$ | $0.001\pm 0.002$
Adult | RF | $0.003\pm 0.014$ | $0.003\pm 0.011$ | $0.005\pm 0.047$ | $0.004\pm 0.009$
| LR | $0.008\pm 0.023$ | $0.004\pm 0.012$ | $0.356\pm 0.507$ | $0.185\pm 0.352$
| LGBM | $0.004\pm 0.015$ | $0.003\pm 0.011$ | $0.059\pm 0.096$ | $0.046\pm 0.083$
Wine | RF | $0.001\pm 0.007$ | $0.001\pm 0.007$ | $0.004\pm 0.033$ | $0.003\pm 0.016$
| LR | $0.056\pm 0.096$ | $0.055\pm 0.094$ | $0.136\pm 0.135$ | $0.096\pm 0.098$
| LGBM | $0.003\pm 0.015$ | $0.003\pm 0.01$ | $0.002\pm 0.009$ | $0.003\pm 0.009$
Contracep. | RF | $0.032\pm 0.081$ | $0.038\pm 0.085$ | $0.000\pm 0.001$ | $0.001\pm 0.001$
| LR | $0.041\pm 0.099$ | $0.051\pm 0.102$ | $0.011\pm 0.019$ | $0.008\pm 0.013$
| LGBM | $0.027\pm 0.066$ | $0.026\pm 0.057$ | $0.001\pm 0.003$ | $0.001\pm 0.003$
Nursery | RF | $0.031\pm 0.099$ | $0.023\pm 0.076$ | $0.001\pm 0.003$ | $0.001\pm 0.002$
| LR | $0.043\pm 0.088$ | $0.029\pm 0.069$ | $0.144\pm 0.162$ | $0.031\pm 0.044$
| LGBM | $0.035\pm 0.108$ | $0.030\pm 0.096$ | $0.001\pm 0.003$ | $0.001\pm 0.002$
Splice | RF | $0.003\pm 0.017$ | $0.002\pm 0.012$ | $0.009\pm 0.047$ | $0.008\pm 0.044$
| LR | $0.011\pm 0.031$ | $0.007\pm 0.018$ | $0.091\pm 0.116$ | $0.046\pm 0.079$
| LGBM | $0.014\pm 0.049$ | $0.009\pm 0.037$ | $0.009\pm 0.047$ | $0.007\pm 0.040$
Table 5: MRA and F-Score reported separately for the results in Table 1 of the main paper. Same with Table 1, results are reported for IP and random selection strategies.$\Delta$MRA and $\Delta$F-Score represent the improvement in the corresponding metrics ($mean\pm std$). Means and standard deviations are computed from all runs performed with a given dataset and model. Dataset | Model | $\Delta$ MRA (IP) | $\Delta$MRA (random) | $\Delta$F-Score (IP) | $\Delta$F-Score (random)
---|---|---|---|---|---
Breastcancer | RF | $0.003\pm 0.042$ | $0.002\pm 0.038$ | $0.000\pm 0.005$ | $0.000\pm 0.003$
| LR | $0.047\pm 0.116$ | $0.039\pm 0.102$ | $\mathllap{-}0.006\pm 0.014$ | $\mathllap{-}0.006\pm 0.015$
| LGBM | $0.013\pm 0.092$ | $0.014\pm 0.098$ | $0.000\pm 0.006$ | $0.000\pm 0.005$
Car | RF | $0.018\pm 0.063$ | $0.015\pm 0.069$ | $0.000\pm 0.003$ | $0.000\pm 0.003$
| LR | $0.096\pm 0.112$ | $0.109\pm 0.135$ | $\mathllap{-}0.020\pm 0.028$ | $\mathllap{-}0.026\pm 0.031$
| LGBM | $0.024\pm 0.083$ | $0.024\pm 0.099$ | $0.000\pm 0.002$ | $0.000\pm 0.002$
Mushroom | RF | $0.009\pm 0.081$ | $0.002\pm 0.027$ | $\mathllap{-}0.000\pm 0.000$ | $\mathllap{-}0.000\pm 0.000$
| LR | $0.045\pm 0.158$ | $0.024\pm 0.111$ | $\mathllap{-}0.000\pm 0.001$ | $\mathllap{-}0.000\pm 0.001$
| LGBM | $0.024\pm 0.141$ | $0.018\pm 0.128$ | $\mathllap{-}0.000\pm 0.000$ | $\mathllap{-}0.000\pm 0.000$
Adult | RF | $0.011\pm 0.053$ | $0.012\pm 0.073$ | $\mathllap{-}0.000\pm 0.001$ | $-0.0\pm 0.001$
| LR | $0.072\pm 0.170$ | $0.075\pm 0.192$ | $\mathllap{-}0.003\pm 0.005$ | $\mathllap{-}0.003\pm 0.007$
| LGBM | $0.026\pm 0.108$ | $0.026\pm 0.117$ | $0.000\pm 0.001$ | $0.000\pm 0.001$
Wine | RF | $0.018\pm 0.096$ | $0.020\pm 0.107$ | $\mathllap{-}0.001\pm 0.005$ | $\mathllap{-}0.000\pm 0.005$
| LR | $0.360\pm 0.306$ | $0.354\pm 0.309$ | $\mathllap{-}0.020\pm 0.023$ | $\mathllap{-}0.023\pm 0.026$
| LGBM | $0.043\pm 0.169$ | $0.037\pm 0.155$ | $0.001\pm 0.005$ | $0.001\pm 0.008$
Contraceptive | RF | $0.070\pm 0.151$ | $0.059\pm 0.144$ | $\mathllap{-}0.000\pm 0.009$ | $\mathllap{-}0.000\pm 0.007$
| LR | $0.115\pm 0.214$ | $0.095\pm 0.203$ | $\mathllap{-}0.007\pm 0.019$ | $\mathllap{-}0.010\pm 0.025$
| LGBM | $0.048\pm 0.104$ | $0.049\pm 0.119$ | $\mathllap{-}0.001\pm 0.010$ | $\mathllap{-}0.000\pm 0.009$
Nursery | RF | $0.059\pm 0.192$ | $0.074\pm 0.226$ | $\mathllap{-}0.000\pm 0.001$ | $\mathllap{-}0.000\pm 0.001$
| LR | $0.097\pm 0.217$ | $0.131\pm 0.240$ | $\mathllap{-}0.002\pm 0.004$ | $\mathllap{-}0.008\pm 0.013$
| LGBM | $0.073\pm 0.227$ | $0.082\pm 0.242$ | $\mathllap{-}0.000\pm 0.000$ | $\mathllap{-}0.000\pm 0.000$
Splice | RF | $0.006\pm 0.026$ | $0.009\pm 0.036$ | $\mathllap{-}0.001\pm 0.004$ | $\mathllap{-}0.001\pm 0.004$
| LR | $0.025\pm 0.046$ | $0.035\pm 0.071$ | $\mathllap{-}0.002\pm 0.006$ | $\mathllap{-}0.004\pm 0.010$
| LGBM | $0.022\pm 0.098$ | $0.032\pm 0.116$ | $0.000\pm 0.002$ | $0.000\pm 0.002$
#### Probabilistic rules.
In this experiment, we consider probabilistic rules, where the label
distribution $\pi$ is not just a Kronecker delta for one of the classes. The
experiment provides a brief demonstration of the ability of probabilistic
rules to represent uncertainty and mitigate the effect of an over-confident
expert rule. We consider an extreme case of this where the expert provides a
single feedback rule, but the test distribution remains the same as the
training distribution, i.e., the expert is wrong and the rule does not take
effect. (We use only a single feedback rule to try to isolate the effect of
having a probabilistic rule and avoid interactions among rules.) We also set
$tcf=0$ (so relabel and drop initializations are not applicable).
We run FROTE with the following label distribution $\pi$ for instances
generated under the rule: With probability $p$, the label is equal to the
class $c$ specified by the feedback rule. With probability $1-p$, it is equal
to the label of the corresponding base instance, except when that label is
$c$, in which case the label of the generated instance is chosen uniformly at
random from classes other than $c$. Thus overall, the labels of generated
instances are equal to $c$ with probability $p$, and otherwise they
approximately follow the distribution of the training data (as represented by
the base instances) restricted to classes other than $c$. The case $p=1$ is
the deterministic case used in the other experiments. With $p<1$, the user of
FROTE can express less than full confidence in the expert rule and rely more
on the existing training data.
Table 6 shows the MRA and $\overline{J}$ improvements for different
probabilities $p$. In this case, since the feedback rule is not in effect for
test data, MRA just measures agreement with respect to labels following the
original distribution $p_{X,Y}$, within the coverage of the rule. The MRA
column shows that setting $p=1.0$, i.e., completely following the expert rule,
does not give as good a performance as setting $p$ to a lower, less confident
value. This pattern however is not as clear looking at the $\overline{J}$
column. In reality, the best value of $p$ is not known a priori as it depends
on the exact extent to which the test data (in this case, the distribution
$p_{X,Y}$) conforms to the expert rule. Nevertheless, Table 6 suggests that
there is a benefit to using a probabilistic rule with $p<1$ if there is reason
to be less confident in the validity of the feedback rules.
Table 6: Experiments with probabilistic rules. Means and standard deviations computed from $50$ runs for LR model and for the given datasets. For each run, $|FRS|=1$, and $tcf=0$. random selection strategy is utilized during the experiments. Dataset | Probability | $\Delta{mra}$ | $\Delta\overline{J}$
---|---|---|---
Mushroom | $p=0.4$ | $0.206\pm 0344$ | $0.007\pm 0.012$
| $p=0.6$ | $0.242\pm 0.386$ | $0.009\pm 0.014$
| $p=0.8$ | $0.249\pm 0.390$ | $0.009\pm 0.014$
| $p=1.0$ | $0.173\pm 0.296$ | $0.006\pm 0.011$
Wine | $p=0.4$ | $0.416\pm 0.305$ | $\mathllap{\shortminus}0.011\pm 0.024$
| $p=0.6$ | $0.448\pm 0.317$ | $\mathllap{\shortminus}0.010\pm 0.021$
| $p=0.8$ | $0.423\pm 0.348$ | $\mathllap{\shortminus}0.011\pm 0.020$
| $p=1.0$ | $0.338\pm 0.327$ | $\mathllap{\shortminus}0.008\pm 0.016$
B. Cancer | $p=0.4$ | $0.005\pm 0.015$ | $0.003\pm 0.007$
| $p=0.6$ | $0.005\pm 0.015$ | $0.002\pm 0.006$
| $p=0.8$ | $0.007\pm 0.015$ | $0.002\pm 0.007$
| $p=1.0$ | $0.005\pm 0.015$ | $0.003\pm 0.006$
Table 7: Comparison with Overlay-Soft (soft constraints) and Overlay-Hard (hard constraints) of Daly et al. (2021) on Adult dataset. Means and standard deviations computed from 50 runs. Dataset | Model | $\Delta\overline{J}$
---|---|---
| | Overlay-Soft | Overlay-Hard | FROTE
Adult | LR | $\mathllap{\shortminus}0.015\pm 0.034$ | $\mathllap{\shortminus}0.107\pm 0.111$ | $0.025\pm 0.039$
| RF | $0.114\pm 0.013$ | $\mathllap{\shortminus}0.121\pm 0.019$ | $0.036\pm 0.039$
| LGBM | $0.102\pm 0.021$ | $\mathllap{\shortminus}0.018\pm 0.180$ | $0.240\pm 0.043$
Table 8: Experiments with Overlay Daly et al. (2021). Overlay-Hard and Overlay-Soft refers to the Hard Constraints and Soft Constraints approaches of Overlay. The comparison is shown for different ML models and the Breast Cancer, Mushroom and Adult datasets. random selection strategy is used for FROTE. Means and standard deviations are computed from 50 runs, where for each run a different set of $3$ rules are used. Model | $\Delta$MRA | $\Delta$F-Score
---|---|---
B. Cancer
| Overlay-Soft | Overlay-Hard | FROTE | Overlay-Soft | Overlay-Hard | FROTE
LR | $0.008\pm 0.021$ | $0.021\pm 0.232$ | $0.080\pm 0.168$ | $\mathllap{-}0.012\pm 0.058$ | $\mathllap{-}0.313\pm 0.248$ | $\mathllap{-}0.014\pm 0.022$
RF | $0.005\pm 0.015$ | $0.071\pm 0.224$ | $0.097\pm 0.194$ | $0.000\pm 0.000$ | $\mathllap{-}0.299\pm 0.238$ | $\mathllap{-}0.002\pm 0.007$
LGBM | $0.016\pm 0.032$ | $0.072\pm 0.218$ | $0.880\pm 0.238$ | $\mathllap{-}0.000\pm 0.001$ | $\mathllap{-}0.272\pm 0.198$ | $\mathllap{-}0.001\pm 0.009$
Mushroom
| Overlay-Soft | Overlay-Hard | FROTE | Overlay-Soft | Overlay-Hard | FROTE
LR | $0.046\pm 0.091$ | $0.202\pm 0.34$ | $0.049\pm 0.033$ | $\mathllap{-}0.001\pm 0.004$ | $\mathllap{-}0.168\pm 0.223$ | $\mathllap{-}0.000\pm 0.001$
RF | $0.021\pm 0.114$ | $0.205\pm 0.34$ | $0.040\pm 0.032$ | $0.000\pm 0.000$ | $\mathllap{-}0.166\pm 0.220$ | $0.000\pm 0.000$
LGBM | $0.155\pm 0.302$ | $0.208\pm 0.34$ | $0.049\pm 0.033$ | $\mathllap{-}0.023\pm 0.093$ | $\mathllap{-}0.163\pm 0.218$ | $0.000\pm 0.000$
|
0
11institutetext: Indian Institute of Technology Bombay, Mumbai, India
11email<EMAIL_ADDRESS>22institutetext: TCS Research, Pune,
India
22email<EMAIL_ADDRESS>
# Diffy: Inductive Reasoning of Array Programs using Difference Invariants
Supratik Chakraborty 11 Ashutosh Gupta 11 Divyesh Unadkat 1122
###### Abstract
We present a novel verification technique to prove interesting properties of a
class of array programs with a symbolic parameter $N$ denoting the size of
arrays. The technique relies on constructing two slightly different versions
of the same program. It infers difference relations between the corresponding
variables at key control points of the joint control-flow graph of the two
program versions. The desired post-condition is then proved by inducting on
the program parameter $N$, wherein the difference invariants are crucially
used in the inductive step. This contrasts with classical techniques that rely
on finding potentially complex loop invaraints for each loop in the program.
Our synergistic combination of inductive reasoning and finding simple
difference invariants helps prove properties of programs that cannot be proved
even by the winner of Arrays sub-category from SV-COMP 2021. We have
implemented a prototype tool called Diffy to demonstrate these ideas. We
present results comparing the performance of Diffy with that of state-of-the-
art tools.
## 1 Introduction
Software used in a wide range of applications use arrays to store and update
data, often using loops to read and write arrays. Verifying correctness
properties of such array programs is important, yet challenging. A variety of
techniques have been proposed in the literature to address this problem,
including inference of quantified loop invariants [20]. However, it is often
difficult to automatically infer such invariants, especially when programs
have loops that are sequentially composed and/or nested within each other, and
have complex control flows. This has spurred recent interest in mathematical
induction-based techniques for verifying parametric properties of array
manipulating programs [12, 42, 11, 44]. While induction-based techniques are
efficient and quite powerful, their Achilles heel is the automation of the
inductive argument. Indeed, this often becomes the limiting step in
applications of induction-based techniques. Automating the induction step and
expanding the class of array manipulating programs to which induction-based
techniques can be applied forms the primary motivation for our work. Rather
than being a stand-alone technique, we envisage our work being used as part of
a portfolio of techniques in a modern program verification tool.
We propose a novel and practically efficient induction-based technique that
advances the state-of-the-art in automating the inductive step when reasoning
about array manipulating programs. This allows us to automatically verify
interesting properties of a large class of array manipulating programs that
are beyond the reach of state-of-the-art induction-based techniques, viz. [12,
42]. The work that comes closest to us is Vajra [12], which is part of the
portfolio of techniques in VeriAbs [1] – the winner of SV-COMP 2021 in the
Arrays Reach sub-category. Our work addresses several key limitations of the
technique implemented in Vajra, thereby making it possible to analyze a much
larger class of array manipulating programs than can be done by VeriAbs.
Significantly, this includes programs with nested loops that have hitherto
been beyond the reach of automated techniques that use mathematical induction
[12, 42, 44].
A key innovation in our approach is the construction of two slightly different
versions of a given program that have identical control flow structures but
slightly different data operations. We automatically identify simple
relations, called _difference invariants_ , between corresponding variables in
the two versions of a program at key control flow points. Interestingly, these
relations often turn out to be significantly simpler than inductive invariants
required to prove the property directly. This is not entirely surprising,
since the difference invariants depend less on what individual statements in
the programs are doing, and more on the difference between what they are doing
in the two versions of the program. We show how the two versions of a given
program can be automatically constructed, and how differences in individual
statements can be analyzed to infer simple difference invariants. Finally, we
show how these difference invariants can be used to simplify the reasoning in
the inductive step of our technique.
We consider programs with (possibly nested) loops manipulating arrays, where
the size of each array is a symbolic integer parameter $N~{}(>0)$111For a more
general class of programs supported by our technique, please see [13].. We
verify (a sub-class of) quantified and quantifier-free properties that may
depend on the symbolic parameter $N$. Like in [12], we view the verification
problem as one of proving the validity of a parameterized Hoare triple
$\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$ for all values of
$N~{}(>0)$, where arrays are of size $N$ in the program $\mathsf{P}_{N}$, and
$N$ is a free variable in $\varphi(\cdot)$ and $\psi(\cdot)$.
To illustrate the kind of programs that are amenable to our technique,
consider the program shown in Fig. 1(a), adapted from an SV-COMP benchmark.
This program has a couple of sequentially composed loops that update arrays
and scalars. The scalars S and F are initialized to $0$ and $1$ respectively
before the first loop starts iterating. Subsequently, the first loop computes
a recurrence in variable S and initializes elements of the array B to $1$ if
the corresponding elements of array A have non-negative values, and to $0$
otherwise. The outermost branch condition in the body of the second loop
evaluates to true only if the program parameter $N$ and the variable S have
same values. The value of F is reset based on some conditions depending on
corresponding entries of arrays A and B. The pre-condition of this program is
true; the post-condition asserts that F is never reset in the second loop.
State-of-the-art techniques find it difficult to prove the assertion in this
program. Specifically, Vajra [12] is unable to prove the property, since it
cannot reason about the branch condition (in the second loop) whose value
depends on the program parameter $N$. VeriAbs [1], which employs a sequence of
techniques such as loop shrinking, loop pruning, and inductive reasoning using
[12] is also unable to verify the assertion shown in this program. Indeed, the
loops in this program cannot be merged as the final value of S computed by the
first loop is required in the second loop; hence loop shrinking does not help.
Also, loop pruning does not work due to the complex dependencies in the
program and the fact that the exact value of the recurrence variable S is
required to verify the program. Subsequent abstractions and techniques applied
by VeriAbs from its portfolio are also unable to verify the given post-
condition. VIAP [42] translates the program to a quantified first-order logic
formula in the theory of equality and uninterpreted functions [32]. It applies
a sequence of tactics to simplify and prove the generated formula. These
tactics include computing closed forms of recurrences, induction over array
indices and the like to prove the property. However, its sequence of tactics
is unable to verify this example within our time limit of $1$ minute.
// assume(true)
1. S = 0; F = 1;
2. for(i = 0; i< N; i++) {
3. S = S + 1;
4. if ( A[i] >= 0 ) B[i] = 1;
5. else B[i] = 0;
6. }
7. for(j = 0; j< N; j++) {
8. if(S == N) {
9. if ( A[j] >= 0 && !B[j] ) F = 0;
10. if ( A[j] < 0 && B[j] ) F = 0;
11. }
12.}
// assert(F == 1)
(a) |
// assume(true)
1. S = 0;
2. for(i=0; i<N; i++) A[i] = 0;
3. for(j=0; j<N; j++) S = S + 1;
4. for(k=0; k<N; k++) {
5. for(l=0; l<N; l++) A[l] = A[l] + 1;
6. A[k] = A[k] + S;
7. }
// assert(forall x in [0,N), A[x]==2*N)
(b)
---|---
Figure 1: Motivating Examples
Benchmarks with nested loops are a long standing challenge for most verifiers.
Consider the program shown in Fig. 1(b) with a nested loop in addition to
sequentially composed loops. The first loop initializes entries in array A to
$0$. The second loop aggregates a constant value in the scalar S. The third
loop is a nested loop that updates array A based on the value of S. The
entries of A are updated in the inner as well as outer loop. The property
asserts that on termination, each array element equals twice the value of the
parameter $N$.
While the inductive reasoning of Vajra and the tactics in VIAP do not support
nested loops, the sequence of techniques used by VeriAbs is also unable to
prove the given post-condition in this program. In sharp contrast, our
prototype tool Diffy is able to verify the assertions in both these programs
automatically within a few seconds. This illustrates the power of the
inductive technique proposed in this paper.
The technical contributions of the paper can be summarized as follows:
* •
We present a novel technique based on mathematical induction to prove
interesting properties of a class of programs that manipulate arrays. The
crucial inductive step in our technique uses difference invariants from two
slightly different versions of the same program, and differs significantly
from other induction-based techniques proposed in the literature [12, 42, 11,
44].
* •
We describe algorithms to transform the input program for use in our inductive
verification technique. We also present techniques to infer simple difference
invariants from the two slightly different program versions, and to complete
the inductive step using these difference invariants.
* •
We describe a prototype tool Diffy that implements our algorithms.
* •
We compare Diffy vis-a-vis state-of-the-art tools for verification of C
programs that manipulate arrays on a large set of benchmarks. We demonstrate
that Diffy significantly outperforms the winners of SV-COMP 2019, 2020 and
2021 in the Array Reach sub-category.
## 2 Overview and Relation to Earlier Work
In this section, we provide an overview of the main ideas underlying our
technique. We also highlight how our technique differs from [12], which comes
closest to our work. To keep the exposition simple, we consider the program
$\mathsf{P}_{N}$, shown in the first column of Fig. 2, where $N$ is a symbolic
parameter denoting the sizes of arrays a and b. We assume that we are given a
parameterized pre-condition $\varphi(N)$, and our goal is to establish the
parameterized post-condition $\psi(N)$, for all $N>0$. In [12, 44], techniques
based on mathematical induction (on $N$) were proposed to solve this class of
problems. As with any induction-based technique, these approaches consist of
three steps. First, they check if the _base case_ holds, i.e. if the Hoare
triple $\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$ holds for small
values of $N$, say $1\leq N\leq M$, for some $M>0$. Next, they assume that the
_inductive hypothesis_ $\\{\varphi(N-1)\\}\;\mathsf{P}_{N-1}\;\\{\psi(N-1)\\}$
holds for some $N\geq M+1$. Finally, in the _inductive step_ , they show that
if the inductive hypothesis holds, so does
$\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$. It is not hard to see that
the inductive step is the most crucial step in this style of reasoning. It is
also often the limiting step, since not all programs and properties allow for
efficient inferencing of $\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$
from $\\{\varphi(N-1)\\}\;\mathsf{P}_{N-1}\;\\{\psi(N-1)\\}$.
for(i=0; i$<$N; i++)for(j=0; j$<$N; j++)x = x + N*N;a[i] = a[i] +
N;$\mathsf{P}_{N}$b[j] = x + j;for(i=0; i$<$N-1; i++)for(j=0; j$<$N-1; j++)x =
x + N*N;a[i] = a[i] + N ;b[j] = x + j;x = x + N*N;a[N-1] = a[N-1]+N;b[N-1] = x
+ N-1;for(i=0; i$<$N-1; i++)for(j=0; j$<$N-1; j++)x = x + N*N;a[i] = a[i] + N
;b[j] = x+N*N+ j;x = x + N*N ;a[N-1] = a[N-1]+N;b[N-1] = x +
N-1;$\mathsf{Q}_{N-1}$$\mathsf{peel}(\mathsf{P}_{N})$for(i=0; i$<$N-1;
i++)for(j=0; j$<$N-1; j++)x=x+(N-1)*(N-1);a[i] = a[i] + N-1;x = x + N*N;a[N-1]
= a[N-1]+N;$\mathsf{P}_{N-1}$$\partial\mathsf{P}_{N}$for(k=0; k$<$N-1;
k++)b[k] = b[k] + (N-1)*(2*N-1)+N*N;x = 0;x = 0;x = 0;x = 0;b[j] = x +
j;for(i=0; i$<$N-1; i++)x = x + 2*N-1;a[i] = a[i] + 1;b[N-1] = x + N-1;//
$\mathtt{\varphi(N)=true}$//$\mathtt{\psi(N)=}$$\mathtt{(\forall
j.~{}b[j]=j+N^{3})}$
Figure 2: Pictorial Depiction of our Program Transformations
Like in [12, 44], our technique uses induction on $N$ to prove the Hoare
triple $\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$ for all $N>0$. Hence,
our base case and inductive hypothesis are the same as those in [12, 44].
However, our reasoning in the crucial inductive step is significantly
different from that in [12, 44], and this is where our primary contribution
lies. As we show later, not only does this allow a much larger class of
programs to be efficiently verified compared to [12, 44], it also permits
reasoning about classes of programs with nested loops, that are beyond the
reach of [12, 44]. Since the work of [12] significantly generalizes that of
[44], henceforth, we only refer to [12] when talking of earlier work that uses
induction on $N$.
In order to better understand our contribution and its difference vis-a-vis
the work of [12], a quick recap of the inductive step used in [12] is
essential. The inductive step in [12] crucially relies on finding a
“difference program” $\partial\mathsf{P}_{N}$ and a “difference pre-condition”
$\partial\varphi(N)$ such that: (i) $\mathsf{P}_{N}$ is semantically
equivalent to $\mathsf{P}_{N-1};\partial\mathsf{P}_{N}$, where ’;’ denotes
sequential composition of programs222Although the authors of [12] mention that
it suffices to find a $\partial\mathsf{P}_{N}$ that satisfies
$\\{\varphi(N)\\}\;\mathsf{P}_{N-1};\partial\mathsf{P}_{N}\;\\{\psi(N)\\}$,
they do not discuss any technique that takes $\varphi(N)$ or $\psi(N)$ into
account when generating $\partial\mathsf{P}_{N}$., (ii)
$\varphi(N)\Rightarrow\varphi(N-1)\wedge\partial\varphi(N)$, and (iii) no
variable/array element in $\partial\varphi(N)$ is modified by
$\mathsf{P}_{N-1}$. As shown in [12], once $\partial\mathsf{P}_{N}$ and
$\partial\varphi(N)$ satisfying these conditions are obtained, the problem of
proving $\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$ can be reduced to
that of proving
$\\{\psi(N-1)\wedge\partial\varphi(N)\\}\;\partial\mathsf{P}_{N}\;\\{\psi(N)\\}$.
This approach can be very effective if (i) $\partial\mathsf{P}_{N}$ is
“simpler” (e.g. has fewer loops or strictly less deeply nested loops) than
$\mathsf{P}_{N}$ and can be computed efficiently, and (ii) a formula
$\partial\varphi(N)$ satisfying the conditions mentioned above exists and can
be computed efficiently.
The requirement of $\mathsf{P}_{N}$ being semantically equivalent to
$\mathsf{P}_{N-1};\partial\mathsf{P}_{N}$ is a very stringent one, and finding
such a program $\partial\mathsf{P}_{N}$ is non-trivial in general. In fact,
the authors of [12] simply provide a set of syntax-guided conditionally sound
heuristics for computing $\partial\mathsf{P}_{N}$. Unfortunately, when these
conditions are violated (we have found many simple programs where they are
violated), there are no known algorithmic techniques to generate
$\partial\mathsf{P}_{N}$ in a sound manner. Even if a program
$\partial\mathsf{P}_{N}$ were to be found in an ad-hoc manner, it may be as
“complex” as $\mathsf{P}_{N}$ itself. This makes the approach of [12]
ineffective for analyzing such programs. As an example, the fourth column of
Fig. 2 shows $\mathsf{P}_{N-1}$ followed by one possible
$\partial\mathsf{P}_{N}$ that ensures $\mathsf{P}_{N}$ (shown in the first
column of the same figure) is semantically equivalent to
$\mathsf{P}_{N-1};\partial\mathsf{P}_{N}$. Notice that
$\partial\mathsf{P}_{N}$ in this example has two sequentially composed loops,
just like $\mathsf{P}_{N}$ had. In addition, the assignment statement in the
body of the second loop uses a more complex expression than that present in
the corresponding loop of $\mathsf{P}_{N}$. Proving
$\\{\psi(N-1)\wedge\partial\varphi(N)\\}\;\partial\mathsf{P}_{N}\;\\{\psi(N)\\}$
may therefore not be any simpler (perhaps even more difficult) than proving
$\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$.
In addition to the difficulty of computing $\partial\mathsf{P}_{N}$, it may be
impossible to find a formula $\partial\varphi(N)$ such that
$\varphi(N)\Rightarrow\varphi(N-1)\wedge\partial\varphi(N)$, as required by
[12]. This can happen even for fairly routine pre-conditions, such as
$\varphi(N)\equiv\big{(}\bigwedge_{i=0}^{N-1}A[i]=N\big{)}$. Notice that there
is no $\partial\varphi(N)$ that satisfies
$\varphi(N)\Rightarrow\varphi(N-1)\wedge\partial\varphi(N)$ in this case. In
such cases, the technique of [12] cannot be used at all, even if
$\mathsf{P}_{N}$, $\varphi(N)$ and $\psi(N)$ are such that there exists a
trivial proof of $\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$.
The inductive step proposed in this paper largely mitigates the above
problems, thereby making it possible to efficiently reason about a much larger
class of programs than that possible using the technique of [12]. Our
inductive step proceeds as follows. Given $\mathsf{P}_{N}$, we first
algorithmically construct two programs $\mathsf{Q}_{N-1}$ and
$\mathsf{peel}(\mathsf{P}_{N})$, such that $\mathsf{P}_{N}$ is semantically
equivalent to $\mathsf{Q}_{N-1};\mathsf{peel}(\mathsf{P}_{N})$. Intuitively,
$\mathsf{Q}_{N-1}$ is the same as $\mathsf{P}_{N}$, but with all loop bounds
that depend on $N$ now modified to depend on $N-1$ instead. Note that this is
different from $\mathsf{P}_{N-1}$, which is obtained by replacing _all uses_
(not just in loop bounds) of $N$ in $\mathsf{P}_{N}$ by $N-1$. As we will see,
this simple difference makes the generation of $\mathsf{peel}(\mathsf{P}_{N})$
significantly simpler than generation of $\partial\mathsf{P}_{N}$, as in [12].
While generating $\mathsf{Q}_{N-1}$ and $\mathsf{peel}(P_{N})$ may sound
similar to generating $\mathsf{P}_{N-1}$ and $\partial\mathsf{P}_{N}$ [12],
there are fundamental differences between the two approaches. First, as noted
above, $\mathsf{P}_{N-1}$ is semantically different from $\mathsf{Q}_{N-1}$.
Similarly, $\mathsf{peel}(\mathsf{P}_{N})$ is also semantically different from
$\partial\mathsf{P}_{N}$. Second, we provide an algorithm for generating
$\mathsf{Q}_{N-1}$ and $\mathsf{peel}(\mathsf{P}_{N})$ that works for a
significantly larger class of programs than that for which the technique of
[12] works. Specifically, our algorithm works for all programs amenable to the
technique of [12], and also for programs that violate the restrictions imposed
by the grammar and conditional heuristics in [12]. For example, we can
algorithmically generate $\mathsf{Q}_{N-1}$ and
$\mathsf{peel}(\mathsf{P}_{N})$ even for a class of programs with arbitrarily
nested loops – a program feature explicitly disallowed by the grammar in [12].
Third, we guarantee that $\mathsf{peel}(\mathsf{P}_{N})$ is “simpler” than
$\mathsf{P}_{N}$ in the sense that the maximum nesting depth of loops in
$\mathsf{peel}(\mathsf{P}_{N})$ is _strictly less_ than that in
$\mathsf{P}_{N}$. Thus, if $\mathsf{P}_{N}$ has no nested loops (all programs
amenable to analysis by [12] belong to this class),
$\mathsf{peel}(\mathsf{P}_{N})$ is guaranteed to be loop-free. As demonstrated
by the fourth column of Fig. 2, no such guarantees can be given for
$\partial\mathsf{P}_{N}$ generated by the technique of [12]. This is a
significant difference, since it greatly simplifies the analysis of
$\mathsf{peel}(\mathsf{P}_{N})$ vis-a-vis that of $\partial\mathsf{P}_{N}$.
We had mentioned earlier that some pre-conditions $\varphi(N)$ do not admit
any $\partial\varphi(N)$ such that
$\varphi(N)\Rightarrow\varphi(N-1)\wedge\partial\varphi(N)$. It is, however,
often easy to compute formulas $\varphi^{\prime}(N-1)$ and
$\Delta\varphi^{\prime}(N)$ in such cases such that
$\varphi(N)\Rightarrow\varphi^{\prime}(N-1)\wedge\Delta\varphi^{\prime}(N)$,
and the variables/array elements in $\Delta\varphi^{\prime}(N)$ are not
modified by either $\mathsf{P}_{N-1}$ or $\mathsf{Q}_{N-1}$. For example, if
we were to consider a (new) pre-condition
$\varphi(N)\equiv\big{(}\bigwedge_{i=0}^{N-1}A[i]=N\big{)}$ for the program
$\mathsf{P}_{N}$ shown in the first column of Fig. 2, then we have
$\varphi^{\prime}(N-1)\equiv\big{(}\bigwedge_{i=0}^{N-2}A[i]=N\big{)}$ and
$\Delta\varphi^{\prime}(N)\equiv\big{(}A[N-1]=N\big{)}$. We assume the
availability of such a $\varphi^{\prime}(N-1)$ and $\Delta\varphi^{\prime}(N)$
for the given $\varphi(N)$. This significantly relaxes the requirement on pre-
conditions and allows a much larger class of Hoare triples to be proved using
our technique vis-a-vis that of [12].
The third column of Fig. 2 shows $\mathsf{Q}_{N-1}$ and
$\mathsf{peel}(\mathsf{P}_{N})$ generated by our algorithm for the program
$\mathsf{P}_{N}$ in the first column of the figure. It is illustrative to
compare these with $\mathsf{P}_{N-1}$ and $\partial\mathsf{P}_{N}$ shown in
the fourth column of Fig. 2. Notice that $\mathsf{Q}_{N-1}$ has the same
control flow structure as $\mathsf{P}_{N-1}$, but is not semantically
equivalent to $\mathsf{P}_{N-1}$. In fact, $\mathsf{Q}_{N-1}$ and
$\mathsf{P}_{N-1}$ may be viewed as closely related versions of the same
program. Let $V_{\mathsf{Q}}$ and $V_{\mathsf{P}}$ denote the set of variables
of $\mathsf{Q}_{N-1}$ and $\mathsf{P}_{N-1}$ respectively. We assume
$V_{\mathsf{Q}}$ is disjoint from $V_{\mathsf{P}}$, and analyze the joint
execution of $\mathsf{Q}_{N-1}$ starting from a state satisfying the pre-
condition $\varphi^{\prime}(N-1)$, and $\mathsf{P}_{N-1}$ starting from a
state satisfying $\varphi(N-1)$. The purpose of this analysis is to compute a
difference predicate $D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)$ that relates
corresponding variables in $\mathsf{Q}_{N-1}$ and $\mathsf{P}_{N-1}$ at the
end of their joint execution. The above problem is reminiscent of (yet,
different from) translation validation [40, 49, 48, 4, 46, 17, 24], and
indeed, our calculation of $D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)$ is motivated
by techniques from the translation validation literature. An important finding
of our study is that corresponding variables in $\mathsf{Q}_{N-1}$ and
$\mathsf{P}_{N-1}$ are often related by simple expressions on $N$, regardless
of the complexity of $\mathsf{P}_{N}$, $\varphi(N)$ or $\psi(N)$. Indeed, in
all our experiments, we didn’t need to go beyond quadratic expressions on $N$
to compute $D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)$.
Once the steps described above are completed, we have
$\Delta\varphi^{\prime}(N)$, $\mathsf{peel}(\mathsf{P}_{N})$ and
$D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)$. It can now be shown that if the
inductive hypothesis, i.e.
$\\{\varphi(N-1)\\}\;\mathsf{P}_{N-1}\;\\{\psi(N-1)\\}$ holds, then proving
$\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$ reduces to proving
$\\{\Delta\varphi^{\prime}(N)~{}\wedge~{}\psi^{\prime}(N-1)\\}\;\mathsf{peel}(\mathsf{P}_{N})\;\\{\psi(N)\\}$,
where $\psi^{\prime}(N-1)\equiv\exists V_{\mathsf{P}}\big{(}\psi(N-1)\wedge
D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)\big{)}$. A few points are worth
emphasizing here. First, if $D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)$ is obtained
as a set of equalities, the existential quantifier in the formula
$\psi^{\prime}(N-1)$ can often be eliminated simply by substitution. We can
also use quantifier elimination capabilities of modern SMT solvers, viz. Z3
[39], to eliminate the quantifier, if needed. Second, recall that unlike
$\partial\mathsf{P}_{N}$ generated by the technique of [12],
$\mathsf{peel}(\mathsf{P}_{N})$ is guaranteed to be “simpler” than
$\mathsf{P}_{N}$, and is indeed loop-free if $\mathsf{P}_{N}$ has no nested
loops. Therefore, proving
$\\{\Delta\varphi^{\prime}(N)~{}\wedge~{}\psi^{\prime}(N-1)\\}\;\mathsf{peel}(\mathsf{P}_{N})\;\\{\psi(N)\\}$
is typically significantly simpler than proving
$\\{\psi(N-1)\wedge\partial\varphi(N)\\}\;\partial\mathsf{P}_{N}\;\\{\psi(N)\\}$.
Finally, it may happen that the pre-condition in
$\\{\Delta\varphi^{\prime}(N)~{}\wedge~{}\psi^{\prime}(N-1)\\}\;\mathsf{peel}(\mathsf{P}_{N})\;\\{\psi(N)\\}$
is not strong enough to yield a proof of the Hoare triple. In such cases, we
need to strengthen the existing pre-condition by a formula, say
$\xi^{\prime}(N-1)$, such that the strengthened pre-condition implies the
weakest pre-condition of $\psi(N)$ under $\mathsf{peel}(\mathsf{P}_{N})$.
Having a simple structure for $\mathsf{peel}(\mathsf{P}_{N})$ (e.g., loop-free
for the entire class of programs for which [12] works) makes it significantly
easier to compute the weakest pre-condition. Note that $\xi^{\prime}(N-1)$ is
defined over the variables in $V_{\mathsf{Q}}$. In order to ensure that the
inductive proof goes through, we need to strengthen the post-condition of the
original program by $\xi(N)$ such that $\xi(N-1)\wedge
D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)\Rightarrow\xi^{\prime}(N-1)$. Computing
$\xi(N-1)$ requires a special form of logical abduction that ensures that
$\xi(N-1)$ refers only to variables in $V_{P}$. However, if
$D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)$ is given as a set of equalities (as is
often the case), $\xi(N-1)$ can be computed from $\xi^{\prime}(N-1)$ simply by
substitution. This process of strengthening the pre-condition and post-
condition may need to iterate a few times until a fixed point is reached,
similar to what happens in the inductive step of [12]. Note that the fixed
point iterations may not always converge (verification is undecidable in
general). However, in our experiments, convergence always happened within a
few iterations. If $\xi^{\prime}(N-1)$ denotes the formula obtained on
reaching the fixed point, the final Hoare triple to be proved is
$\\{\xi^{\prime}(N-1)\wedge\Delta\varphi^{\prime}(N)~{}\wedge~{}\psi^{\prime}(N-1)\\}\;\mathsf{peel}(\mathsf{P}_{N})\;\\{\xi(N)\wedge\psi(N)\\}$,
where $\psi^{\prime}(N-1)\equiv\exists V_{\mathsf{P}}\big{(}\psi(N-1)\wedge
D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)\big{)}$. Having a simple (often loop-
free) $\mathsf{peel}(\mathsf{P}_{N})$ significantly simplifies the above
process.
We conclude this section by giving an overview of how $\mathsf{Q}_{N-1}$ and
$\mathsf{peel}(\mathsf{P}_{N})$ are computed for the program $\mathsf{P}_{N}$
shown in the first column of Fig. 2. The second column of this figure shows
the program obtained from $\mathsf{P}_{N}$ by peeling the last iteration of
each loop of the program. Clearly, the programs in the first and second
columns are semantically equivalent. Since there are no nested loops in
$\mathsf{P}_{N}$, the peels (shown in solid boxes) in the second column are
loop-free program fragments. For each such peel, we identify variables/array
elements modified in the peel and used in subsequent non-peeled parts of the
program. For example, the variable x is modified in the peel of the first loop
and used in the body of the second loop, as shown by the arrow in the second
column of Fig. 2. We replace all such uses (if needed, transitively) by
expressions on the right-hand side of assignments in the peel until no
variable/array element modified in the peel is used in any subsequent non-
peeled part of the program. Thus, the use of x in the body of the second loop
is replaced by the expression x + N*N in the third column of Fig. 2. The
peeled iteration of the first loop can now be moved to the end of the program,
since the variables modified in this peel are no longer used in any subsequent
non-peeled part of the program. Repeating the above steps for the peeled
iteration of the second loop, we get the program shown in the third column of
Fig. 2. This effectively gives a transformed program that can be divided into
two parts: (i) a program $\mathsf{Q}_{N-1}$ that differs from $\mathsf{P}_{N}$
only in that all loops are truncated to iterate $N-1$ (instead of $N$) times,
and (ii) a program $\mathsf{peel}(\mathsf{P}_{N})$ that is obtained by
concatenating the peels of loops in $\mathsf{P}_{N}$ in the same order in
which the loops appeared in $\mathsf{P}_{N}$. It is not hard to see that
$\mathsf{P}_{N}$, shown in the first column of Fig. 2, is semantically
equivalent to $\mathsf{Q}_{N-1};\mathsf{peel}(\mathsf{P}_{N})$. Notice that
the construction of $\mathsf{Q}_{N-1}$ and $\mathsf{peel}(\mathsf{P}_{N})$ was
fairly straightforward, and did not require any complex reasoning. In sharp
contrast, construction of $\partial\mathsf{P}_{N}$, as shown in the bottom
half of fourth column of Fig. 2, requires non-trivial reasoning, and produces
a program with two sequentially composed loops.
## 3 Preliminaries and Notation
We consider programs generated by the grammar shown below:
$\mathsf{PB}$ | ::= | $\mathsf{St}$
---|---|---
$\mathsf{St}$ | ::= | $\mathsf{St}$ ; $\mathsf{St}$ $\mid$ $v$ := $\mathsf{E}$ $\mid$ $A$[$\mathsf{E}$] := $\mathsf{E}$ $\mid$ $\mathbf{if}$($\mathsf{BoolE}$) $\mathbf{then}$ $\mathsf{St}$ $\mathbf{else}$ $\mathsf{St}$ $\mid$
| | $\mathbf{for}$ ($\ell$ := 0; $\ell$ $<$ $\mathsf{UB}$; $\ell$ := $\ell$+1) {$\mathsf{St}$}
$\mathsf{E}$ | ::= | $\mathsf{E}$ $\mathsf{op}$ $\mathsf{E}$ $\mid$ $A$[$\mathsf{E}$] $\mid$ $v$ $\mid$ $\ell$ $\mid$ $\mathsf{c}$ $\mid$ $N$
$\mathsf{op}$ | ::= | \+ $\mid$ \- $\mid$ * $\mid$ /
$\mathsf{UB}$ | ::= | $\mathsf{UB}$ $\mathsf{op}$ $\mathsf{UB}$ $\mid$ $\ell$ $\mid$ $\mathsf{c}$ $\mid$ $N$
$\mathsf{BoolE}$ | ::= | $\mathsf{E}$ $\mathsf{relop}$ $\mathsf{E}$ $\mid$ $\mathsf{BoolE}$ $\mathsf{AND}$ $\mathsf{BoolE}$ $\mid$ $\mathsf{NOT}$ $\mathsf{BoolE}$ $\mid$ $\mathsf{BoolE}$ $\mathsf{OR}$ $\mathsf{BoolE}$
Formally, we consider a program $\mathsf{P}_{N}$ to be a tuple
$(\mathcal{V},\mathcal{L},\mathcal{A},{\mathsf{PB}},N)$, where $\mathcal{V}$
is a set of scalar variables, $\mathcal{L}\subseteq\mathcal{V}$ is a set of
scalar loop counter variables, $\mathcal{A}$ is a set of array variables,
${\mathsf{PB}}$ is the program body, and $N$ is a special symbol denoting a
positive integer parameter of the program. In the grammar shown above, we
assume that ${A}\in\mathcal{A}$, ${v}\in\mathcal{V}\setminus\mathcal{L}$,
${\ell}\in\mathcal{L}$ and ${\mathsf{c}}\in\mathbb{Z}$. We also assume that
each loop ${\mathsf{L}}$ has a unique loop counter variable $\ell$ that is
initialized at the beginning of ${\mathsf{L}}$ and is incremented by $1$ at
the end of each iteration. We assume that the assignments in the body of
${\mathsf{L}}$ do not update $\ell$. For each loop $\mathsf{L}$ with
termination condition $\ell<\mathsf{UB}$, we require that $\mathsf{UB}$ is an
expression in terms of $N$, variables in $\mathcal{L}$ representing loop
counters of loops that nest $\mathsf{L}$, and constants as shown in the
grammar. Our grammar allows a large class of programs (with nested loops) to
be analyzed using our technique, and that are beyond the reach of state-of-
the-art tools like [1, 12, 42].
We verify Hoare triples of the form
$\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$, where the formulas
$\varphi(N)$ and $\psi(N)$ are either universally quantified formulas of the
form $\forall
I\,\left(\alpha(I,N)\Rightarrow\beta(\mathcal{A},\mathcal{V},I,N)\right)$ or
quantifier-free formulas of the form $\eta(\mathcal{A},\mathcal{V},N)$. In
these formulas, $I$ is a sequence of array index variables, $\alpha$ is a
quantifier-free formula in the theory of arithmetic over integers, and $\beta$
and $\eta$ are quantifier-free formulas in the combined theory of arrays and
arithmetic over integers. Our technique can also verify a restricted set of
existentially quantified post-conditions. We give a few illustrative examples
in the Appendix.
For technical reasons, we rename all scalar and array variables in the program
in a pre-processing step as follows. We rename each scalar variable using the
well-known Static Single Assignment (SSA) [43] technique, such that the
variable is written at (at most) one location in the program. We also rename
arrays in the program such that each loop updates its own version of an array
and multiple writes to an array element within the same loop are performed on
different versions of that array. We use techniques for array SSA [30]
renaming studied earlier in the context of compilers, for this purpose. In the
subsequent exposition, we assume that scalar and array variables in the
program are already SSA renamed, and that all array and scalar variables
referred to in the pre- and post-conditions are also expressed in terms of SSA
renamed arrays and scalars.
## 4 Verification using Difference Invariants
The key steps in the application of our technique, as discussed in Section 2,
are
* A1:
Generation of $\mathsf{Q}_{N-1}$ and $\mathsf{peel}(\mathsf{P}_{N})$ from a
given $\mathsf{P}_{N}$.
* A2:
Generation of $\varphi^{\prime}(N-1)$ and $\Delta\varphi^{\prime}(N)$ from a
given $\varphi(N)$.
* A3:
Generation of the difference invariant $D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)$,
given $\varphi(N-1)$, $\varphi^{\prime}(N-1)$, $\mathsf{Q}_{N-1}$ and
$\mathsf{P}_{N-1}$.
* A4:
Proving $\\{\Delta\varphi^{\prime}(N)~{}\wedge~{}\exists
V_{\mathsf{P}}\big{(}\psi(N-1)\wedge
D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)\big{)}\\}\;\mathsf{peel}(\mathsf{P}_{N})\;\\{\psi(N)\\}$,
possibly by generation of $\xi^{\prime}(N-1)$ and $\xi(N)$ to strengthen the
pre- and post-conditions, respectively.
We now discuss techniques for solving each of these sub-problems.
### 4.1 Generating $\mathsf{Q}_{N-1}$ and $\mathsf{peel}(\mathsf{P}_{N})$
The procedure illustrated in Fig. 2 (going from the first column to the third
column) is fairly straightforward if none of the loops have any nested loops
within them. It is easy to extend this to arbitrary sequential compositions of
non-nested loops. Having all variables and arrays in SSA-renamed forms makes
it particularly easy to carry out the substitution exemplified by the arrow
shown in the second column of Fig. 2. Hence, we don’t discuss any further the
generation of $\mathsf{Q}_{N-1}$ and $\mathsf{peel}(\mathsf{P}_{N})$ when all
loops are non-nested.
for($\displaystyle\ell_{1}$=0;
$\displaystyle\ell_{1}$$\displaystyle<$$\displaystyle N$;
$\displaystyle\ell_{1}$++)for($\displaystyle\ell_{2}$=0;
$\displaystyle\ell_{2}$$\displaystyle<$$\displaystyle N$;
$\displaystyle\ell_{2}$++)$\displaystyle\mathsf{L}_{2}$$\displaystyle\mathsf{L}_{1}$B1B2B3
Figure 3: A Generic Nested Loop
The case of nested loops is, however, challenging and requires additional
discussion. Before we present an algorithm for handling this case, we discuss
the intuition using an abstract example. Consider a pair of nested loops,
$\mathsf{L}_{1}$ and $\mathsf{L}_{2}$, as shown in Fig. 3. Suppose that B1 and
B3 are loop-free code fragments in the body of $\mathsf{L}_{1}$ that precede
and succeed the nested loop $\mathsf{L}_{2}$. Suppose further that the loop
body, B2, of $\mathsf{L}_{2}$ is loop-free. To focus on the key aspects of
computing peels of nested loops, we make two simplifying assumptions: (i) no
scalar variable or array element modified in B2 is used subsequently
(including transitively) in either B3 or B1, and (ii) every scalar variable or
array element that is modified in B1 and used subsequently in B2, is not
modified again in either B1, B2 or B3. Note that these assumptions are made
primarily to simplify the exposition. For a detailed discussion on how our
technique can be used even with some relaxations of these assumptions, the
reader is referred to [13]. The peel of the abstract loops $\mathsf{L}_{1}$
and $\mathsf{L}_{2}$ is as shown in Fig. 4. The first loop in the peel
includes the last iteration of $\mathsf{L}_{2}$ in each of the $N-1$
iterations of $\mathsf{L}_{1}$, that was missed in $\mathsf{Q}_{N-1}$. The
subsequent code includes the last iteration of $\mathsf{L}_{1}$ that was
missed in $\mathsf{Q}_{N-1}$.
for($\displaystyle\ell_{1}$=0;
$\displaystyle\ell_{1}$$\displaystyle<$$\displaystyle N-1$;
$\displaystyle\ell_{1}$++)for($\displaystyle\ell_{2}$=0;
$\displaystyle\ell_{2}$$\displaystyle<$$\displaystyle N$;
$\displaystyle\ell_{2}$++)B2B2B1B3
Figure 4: Peel of the Nested Loop
Formally, we use the notation $\mathsf{L}_{1}$(N) to denote a loop
$\mathsf{L}_{1}$ that has no nested loops within it, and its loop counter, say
$\ell_{1}$, increases from $0$ to an upper bound that is given by an
expression in $N$. Similarly, we use $\mathsf{L}_{1}$(N, $\mathsf{L}_{2}$(N))
to denote a loop $\mathsf{L}_{1}$ that has another loop $\mathsf{L}_{2}$
nested within it. The loop counter $\ell_{1}$ of $\mathsf{L}_{1}$ increases
from $0$ to an upper bound expression in $N$, while the loop counter
$\ell_{2}$ of $\mathsf{L}_{2}$ increases from $0$ to an upper bound expression
in $\ell_{1}$ and $N$. Using this notation, $\mathsf{L}_{1}$(N,
$\mathsf{L}_{2}$(N, $\mathsf{L}_{3}$(N))) represents three nested loops, and
so on. Notice that the upper bound expression for a nested loop can depend not
only on $N$ but also on the loop counters of other loops nesting it. For
notational clarity, we also use LPeel($\mathsf{L}_{i}$, a, b) to denote the
peel of loop $\mathsf{L}_{i}$ consisting of all iterations of $\mathsf{L}_{i}$
where the value of $\ell_{i}$ ranges from a to b-1, both inclusive. Note that
if b-a is a constant, this corresponds to the concatenation of (b-a) peels of
$\mathsf{L}_{i}$.
for($\ell_{1}$=0; $\ell_{1}$<$U_{\mathsf{L}_{1}}$(N-1); $\ell_{1}$++)
LPeel($\mathsf{L}_{2}$, $U_{\mathsf{L}_{2}}$($\ell_{1}$,N-1), $U_{\mathsf{L}_{2}}$($\ell_{1}$,N))
LPeel($\mathsf{L}_{1}$, $U_{\mathsf{L}_{1}}$(N-1), $U_{\mathsf{L}_{1}}$(N))
Figure 5: Peel of $\mathsf{L}_{1}$(N, $\mathsf{L}_{2}$(N))
We will now try to see how we can implement the transformation from the first
column to the second column of Fig. 2 for a nested loop $\mathsf{L}_{1}$(N,
$\mathsf{L}_{2}$(N)). The first step is to truncate all loops to use $N-1$
instead of $N$ in the upper bound expressions. Using the notation introduced
above, this gives the loop $\mathsf{L}_{1}$(N-1, $\mathsf{L}_{2}$(N-1)). Note
that all uses of $N$ other than in loop upper bound expressions stay unchanged
as we go from $\mathsf{L}_{1}$(N, $\mathsf{L}_{2}$(N)) to
$\mathsf{L}_{1}$(N-1, $\mathsf{L}_{2}$(N-1)). We now ask: _Which are the loop
iterations of $\mathsf{L}_{1}$(N, $\mathsf{L}_{2}$(N)) that have been missed
(or skipped) in going to $\mathsf{L}_{1}$(N-1, $\mathsf{L}_{2}$(N-1))?_ Let
the upper bound expression of $\mathsf{L}_{1}$ in $\mathsf{L}_{1}$(N,
$\mathsf{L}_{2}$(N)) be $U_{\mathsf{L}_{1}}(N)$, and that of $\mathsf{L}_{2}$
be $U_{\mathsf{L}_{2}}(\ell_{1},N)$. It is not hard to see that in every
iteration $\ell_{1}$ of $\mathsf{L}_{1}$, where
$0\leq\ell_{1}<U_{\mathsf{L}_{1}}(N-1)$, the iterations corresponding to
$\ell_{2}\in\\{U_{\mathsf{L}_{2}}(\ell_{1},N-1),\ldots,U_{\mathsf{L}_{2}}(\ell_{1},N)-1\\}$
have been missed. In addition, all iterations of $\mathsf{L}_{1}$
corresponding to
$\ell_{1}\in\\{U_{\mathsf{L}_{1}}(N-1),\ldots,U_{\mathsf{L}_{1}}(N)-1\\}$ have
also been missed. This implies that the “peel” of $\mathsf{L}_{1}$(N,
$\mathsf{L}_{2}$(N)) must include all the above missed iterations. This peel
therefore is the program fragment shown in Fig. 5.
for($\ell_{1}$=0; $\ell_{1}$<$U_{\mathsf{L}_{1}}$(N-1); $\ell_{1}$++) {
for($\ell_{2}$=0; $\ell_{2}$<$U_{\mathsf{L}_{2}}$($\ell_{1}$,N-1); $\ell_{2}$++)
LPeel($\mathsf{L}_{3}$, $U_{\mathsf{L}_{3}}$($\ell_{1}$,$\ell_{2}$,N-1), $U_{\mathsf{L}_{3}}$($\ell_{1}$,$\ell_{2}$,N))
LPeel($\mathsf{L}_{2}$, $U_{\mathsf{L}_{2}}$($\ell_{1}$,N-1), $U_{\mathsf{L}_{2}}$($\ell_{1}$,N))
}
LPeel($\mathsf{L}_{1}$, $U_{\mathsf{L}_{1}}$(N-1), $U_{\mathsf{L}_{1}}$(N))
Figure 6: Peel of $\mathsf{L}_{1}$(N, $\mathsf{L}_{2}$(N,
$\mathsf{L}_{3}$(N)))
Notice that if $U_{\mathsf{L}_{2}}$($\ell_{1}$,N) -
$U_{\mathsf{L}_{2}}$($\ell_{1}$,N-1) is a constant (as is the case if
$U_{\mathsf{L}_{2}}$($\ell_{1}$,N) is any linear function of $\ell_{1}$ and
$N$), then the peel does not have any loop with nesting depth 2. Hence, the
maximum nesting depth of loops in the peel is strictly less than that in
$\mathsf{L}_{1}$(N, $\mathsf{L}_{2}$(N)), yielding a peel that is “simpler”
than the original program. This argument can be easily generalized to loops
with arbitrarily large nesting depths. The peel of $\mathsf{L}_{1}$(N,
$\mathsf{L}_{2}$(N, $\mathsf{L}_{3}$(N))) is as shown in Fig. 6.
for(i=0; i<N; i++)
for(j=0; j<N; j++)
A[i][j] = N;
(a) | |
for(i=0; i<N-1; i++)
A[i][N-1] = N;
for(j=0; j<N; j++)
A[N-1][j] = N;
(b)
---|---|---
Figure 7: (a) Nested Loop & (b) Peel
As an illustrative example, let us consider the program in Fig. 7(a), and
suppose we wish to compute the peel of this program containing nested loops.
In this case, the upper bounds of the loops are
$U_{\mathsf{L}_{1}}(N)=U_{\mathsf{L}_{2}}(N)=N$. The peel is shown in Fig.
7(b) and consists of two sequentially composed non-nested loops. The first
loop takes into account the missed iterations of the inner loop (a single
iteration in this example) that are executed in $\mathsf{P}_{N}$ but are
missed in $\mathsf{Q}_{N-1}$. The second loop takes into account the missed
iterations of the outer loop in $\mathsf{Q}_{N-1}$ compared to
$\mathsf{P}_{N}$.
Algorithm 1 GenQandPeel($\mathsf{P}_{N}$: program)
1:Let sequentially composed loops in $\mathsf{P}_{N}$ be in the order
$\mathsf{L}_{1}$, $\mathsf{L}_{2}$, $\ldots$, $\mathsf{L}_{m}$;
2:for each loop $\mathsf{L}_{i}\in\textsc{TopLevelLoops}(\mathsf{P}_{N})$ do
3:
$\langle\mathsf{Q}_{\mathsf{L}_{i}},\mathsf{R}_{\mathsf{L}_{i}}\rangle\leftarrow$
GenQandPeelForLoop($\mathsf{L}_{i}$);
4:while $\exists v.{\it use(v)}\in\mathsf{Q}_{\mathsf{L}_{i}}$ $\wedge$ def(v)
$\in\mathsf{R}_{\mathsf{L}_{j}}$, for some $1\leq j<i\leq N$ do
$\triangleright$ $v$ is var/array element
5: Substitute rhs expression for $v$ from $\mathsf{R}_{\mathsf{L}_{j}}$ in
$\mathsf{Q}_{\mathsf{L}_{i}}$; $\triangleright$ If
$\mathsf{R}_{\mathsf{L}_{j}}$ is a loop, abort
6:$\mathsf{Q}_{N-1}\leftarrow\mathsf{Q}_{\mathsf{L}_{1}};\mathsf{Q}_{\mathsf{L}_{2}};\ldots;\mathsf{Q}_{\mathsf{L}_{m}}$;
7:$\mathsf{peel}(\mathsf{P}_{N})\leftarrow\mathsf{R}_{\mathsf{L}_{1}};\mathsf{R}_{\mathsf{L}_{2}};\ldots;\mathsf{R}_{\mathsf{L}_{m}}$;
8:return $\langle\mathsf{Q}_{N-1},\mathsf{peel}(\mathsf{P}_{N})\rangle$;
9:procedure GenQandPeelForLoop($\mathsf{L}$: loop)
10: Let $U_{\mathsf{L}}(N)$ be the $\mathsf{UB}$ expression of loop
$\mathsf{L}$;
11: $\mathsf{Q}_{\mathsf{L}}\leftarrow\mathsf{L}$ with $N-1$ substituted for
$N$ in all $\mathsf{UB}$ expressions (including for nested loops);
12: if $\mathsf{L}$ has subloops then
13: $t\leftarrow$ nesting depth of inner-most nested loop in $\mathsf{L}$;
14: $\mathsf{R}_{t+1}\leftarrow$ empty program with no statements;
15: for $k=t;k\geq 2;k$– do
16: for each subloop $SL_{j}$ in $\mathsf{L}_{i}$ at nesting depth $k$ do
$\triangleright$ Ordered $SL_{1}$, $SL_{2}$, $\ldots$, $SL_{j}$
17: $\mathsf{R}_{SL_{j}}\leftarrow$ LPeel($SL_{j}$,
$U_{SL_{j}}$($\ell_{1},\ldots,\ell_{k-1},N-1$),
$U_{SL_{j}}$($\ell_{1},\ldots,\ell_{k-1},N$));
18: $\mathsf{R}_{k}\leftarrow$ for (i=0; i$<$$U_{\mathsf{L}_{k-1}}$($N-1$);
i++) {
$\mathsf{R}_{k+1}$;$\mathsf{R}_{SL_{1}}$;$\mathsf{R}_{SL_{2}}$;…;$\mathsf{R}_{SL_{j}}$
};
19: $\mathsf{R}_{\mathsf{L}}\leftarrow$ $\mathsf{R}_{2}$ ; LPeel($\mathsf{L}$,
$U_{\mathsf{L}}$($N-1$), $U_{\mathsf{L}}$($N$));
20: else
21: $\mathsf{R}_{\mathsf{L}}\leftarrow$ LPeel($\mathsf{L}$,
$U_{\mathsf{L}}$($N-1$), $U_{\mathsf{L}}$($N$));
22: return $\langle\mathsf{Q}_{\mathsf{L}},\mathsf{R}_{\mathsf{L}}\rangle$;
Generalizing the above intuition, Algorithm 1 presents function GenQandPeel
for computing $\mathsf{Q}_{N-1}$ and $\mathsf{peel}(\mathsf{P}_{N})$ for a
given $\mathsf{P}_{N}$ that has sequentially composed loops with potentially
nested loops. Due to the grammar of our programs, our loops are well nested.
The method works by traversing over the structure of loops in the program. In
this algorithm $\mathsf{Q}_{\mathsf{L}_{i}}$ and $\mathsf{R}_{\mathsf{L}_{i}}$
represent the counterparts of $\mathsf{Q}_{N-1}$ and
$\mathsf{peel}(\mathsf{P}_{N})$ for loop $\mathsf{L}_{i}$. We create the
program $\mathsf{Q}_{N-1}$ by peeling each loop in the program and then
propagating these peels across subsequent loops. We identify the missed
iterations of each loop in the program $\mathsf{P}_{N}$ from the upper bound
expression $\mathsf{UB}$. Recall that the upper bound of each loop
${\mathsf{L}_{k}}$ at nesting depth $k$, denoted by $U_{\mathsf{L}_{k}}$ is in
terms of the loop counters $\ell$ of outer loops and the program parameter
$N$. We need to peel
$U_{\mathsf{L}_{k}}(\ell_{1},\ell_{2},\ldots,\ell_{k-1},N)-U_{\mathsf{L}_{k}}(\ell_{1},\ell_{2},\ldots,\ell_{k-1},N-1)$
number of iterations from each loop, where
$\ell_{1}\leq\ell_{2}\leq\ldots\leq\ell_{k-1}$ are counters of the outer
nesting loops. As discussed above, whenever this difference is a constant
value, we are guaranteed that the loop nesting depth reduces by one. It may so
happen that there are multiple sequentially composed loops $SL_{j}$ at nesting
depth $k$ and not just a single loop $\mathsf{L}_{k}$. At line 2, we iterate
over top level loops and call function GenQandPeelForLoop($\mathsf{L}_{i}$)
for each sequentially composed loop $\mathsf{L}_{i}$ in $\mathsf{P}_{N}$. At
line 11, we construct $\mathsf{Q}_{\mathsf{L}}$ for loop $\mathsf{L}$. If the
loop $\mathsf{L}$ has no nested loops, then the peel is the last iterations
computed using the upper bound in line 21. For nested loops, the loop at line
15 builds the peel for all loops inside $\mathsf{L}$ following the above
intuition. The peels of all sub-loops are collected and inserted in the peel
of $\mathsf{L}$ at line 19. Since all the peeled iterations are moved after
$Q_{\mathsf{L}}$ of each loop, we need to repair expressions appearing in
$Q_{\mathsf{L}}$. The repairs are applied by the loop at line 4. In the repair
step, we identify the right hand side expressions for all the variables and
array elements assigned in the peeled iterations. Subsequently, the uses of
the variables and arrays in $\mathsf{Q}_{\mathsf{L}_{i}}$ that are assigned in
$\mathsf{R}_{\mathsf{L}_{j}}$ are replaced with the assigned expressions
whenever $j<i$. If $\mathsf{R}_{\mathsf{L}_{j}}$ is a loop, this step is more
involved and hence currently not considered. Finally at line 8, the peels and
$Q$s of all top level loops are stitched and returned.
Note that lines 4 and 5 of Algorithm 1 implement the substitution represented
by the arrow in the second column of Fig. 2. This is necessary in order to
move the peel of a loop to the end of the program. If either of the loops
$\mathsf{L}_{i}$ or $\mathsf{L}_{j}$ use array elements as index to other
arrays then it can be difficult to identify what expression to use in
$\mathsf{Q}_{\mathsf{L}_{i}}$ for the substitution. However, such scenarios
are observed less often, and hence, they hardly impact the effectiveness of
the technique on programs seen in practice. The peel
$\mathsf{R}_{\mathsf{L}_{j}}$, from which the expression to be substituted in
$\mathsf{Q}_{\mathsf{L}_{i}}$ has to be taken, itself may have a loop. In such
cases, it can be significantly more challenging to identify what expression to
use in $\mathsf{Q}_{\mathsf{L}_{i}}$. We use several optimizations to
transform the peeled loop before trying to identify such an expression. If the
modified values in the peel can be summarized as closed form expressions, then
we can replace the loop in the peel with its summary. For example consider the
peeled loop, for ($\ell_{1}$=0; $\ell_{1}$<N; $\ell_{1}$++) { S = S + 1; }.
This loop is summarized as S = S + N; before it can be moved across subsequent
code. If the variables modified in the peel of a nested loop are not used
later, then the peel can be trivially moved. In many cases, the loop in the
peel can also be substituted with its conservative over-approximation. We have
implemented some of these optimizations in our tool and are able to verify
several benchmarks with sequentially composed nested loops. It may not always
be possible to move the peel of a nested loop across subsequent loops but we
have observed that these optimizations suffice for many programs seen in
practice.
###### Theorem 4.1
Let $\mathsf{Q}_{N-1}$ and $\mathsf{peel}(\mathsf{P}_{N})$ be generated by
application of function GenQandPeel from Algorithm 1 on program
$\mathsf{P}_{N}$. Then $\mathsf{P}_{N}$ is semantically equivalent to
$\mathsf{Q}_{N-1};\mathsf{peel}(\mathsf{P}_{N})$.
###### Lemma 1
Suppose the following conditions hold;
* •
Program $\mathsf{P}_{N}$ satisfies our syntactic restrictions (see Section 3).
* •
The upper bound expressions of all loops are linear expressions in $N$ and in
the loop counters of outer nesting loops.
Then, the max nesting depth of loops in $\mathsf{peel}(\mathsf{P}_{N})$ is
strictly less than that in $\mathsf{P}_{N}$.
###### Proof
Let $U_{\mathsf{L}_{k}}(\ell_{1},\ldots,\ell_{k-1},N)$ be the upper bound
expression of a loop $\mathsf{L}_{k}$ at nesting depth $k$. Suppose
$U_{\mathsf{L}_{k}}=c_{1}.\ell_{1}+\cdots c_{k-1}.\ell_{k-1}+C.N+D$, where
$c_{1},\ldots c_{k-1},C$ and $D$ are constants. Then
$U_{\mathsf{L}_{k}}(\ell_{1},\ldots,\ell_{k-1},N)$ $-$
$U_{\mathsf{L}_{k}}(\ell_{1},\ldots\ell_{k-1},N-1)=C$, i.e. a constant. Now,
recalling the discussion in Section 4.1, we see that LPeel($\mathsf{L}_{k}$,
$U_{k}$($\ell_{1},\ldots,\ell_{k-1},N-1$),
$U_{k}$($\ell_{1},\ldots,\ell_{k-1},N$)) simply results in concatenating a
constant number of peels of the loop $\mathsf{L}_{k}$. Hence, the maximum
nesting depth of loops in LPeel($\mathsf{L}_{k}$,
$U_{k}$($\ell_{1},\ldots,\ell_{k-1},N-1$),
$U_{k}$($\ell_{1},\ldots,\ell_{k-1},N$)) is strictly less than the maximum
nesting depth of loops in $\mathsf{L}_{k}$.
Suppose loop $\mathsf{L}$ with nested loops (having maximum nesting depth $t$)
is passed as the argument of function GenQandPeelForLoop (see Algorithm 1). In
line 15 of function GenQandPeelForLoop, we iterate over all loops at nesting
depth $2$ and above within $\mathsf{L}$. Let $\mathsf{L}_{k}$ be a loop at
nesting depth $k$, where $2\leq k\leq t$. Clearly, $\mathsf{L}_{k}$ can have
at most $t-k$ nested levels of loops within it. Therefore, when LPeel is
invoked on such a loop, the maximum nesting depth of loops in the peel
generated for $\mathsf{L}_{k}$ can be at most $t-k-1$. From lines 18 and 19 of
function GenQandPeelForLoop, we also know that this LPeel can itself appear at
nesting depth $k$ of the overall peel $\mathsf{R}_{\mathsf{L}}$. Hence, the
maximum nesting depth of loops in $\mathsf{R}_{\mathsf{L}}$ can be $t-k-1+k$,
i.e. $t-1$. This is strictly less than the maximum nesting depth of loops in
$\mathsf{L}$.∎
###### Corollary 1
If $\mathsf{P}_{N}$ has no nested loops, then $\mathsf{peel}(\mathsf{P}_{N})$
is loop-free.
### 4.2 Generating $\varphi^{\prime}(N-1)$ and $\Delta\varphi^{\prime}(N)$
Given $\varphi(N)$, we check if it is of the form
$\bigwedge_{i=0}^{N-1}\rho_{i}$ (or $\bigvee_{i=0}^{N-1}\rho_{i}$), where
$\rho_{i}$ is a formula on the $i^{th}$ elements of one or more arrays, and
scalars used in $\mathsf{P}_{N}$. If so, we infer $\varphi^{\prime}(N-1)$ to
be $\bigwedge_{i=0}^{N-2}\rho_{i}$ (or $\bigvee_{i=0}^{N-2}\rho_{i}$) and
$\Delta\varphi^{\prime}(N)$ to be $\rho_{N-1}$ (assuming variables/array
elements in $\rho_{N-1}$ are not modified by $\mathsf{Q}_{N-1}$). Note that
all uses of $N$ in $\rho_{i}$ are retained as is (i.e. not changed to $N-1$)
in $\varphi^{\prime}(N-1)$. In general, when deriving $\varphi^{\prime}(N-1)$,
we do not replace any use of $N$ in $\varphi(N)$ by $N-1$ unless it is the
limit of an iterated conjunct/disjunct as discussed above. Specifically, if
$\varphi(N)$ doesn’t contain an iterated conjunct/disjunct as above, then we
consider $\varphi^{\prime}(N-1)$ to be the same as $\varphi(N)$ and
$\Delta\varphi^{\prime}(N)$ to be True. Thus, our generation of
$\varphi^{\prime}(N-1)$ and $\Delta\varphi^{\prime}(N)$ differs from that of
[12]. As discussed earlier, this makes it possible to reason about a much
larger class of pre-conditions than that admissible by the technique of [12].
### 4.3 Inferring Inductive Difference Invariants
Once we have $\mathsf{P}_{N-1}$, $\mathsf{Q}_{N-1}$, $\varphi(N-1)$ and
$\varphi^{\prime}(N-1)$, we infer _difference invariants_. We construct the
standard cross-product of programs $\mathsf{Q}_{N-1}$ and $\mathsf{P}_{N-1}$,
denoted as $\mathsf{Q}_{N-1}\times\mathsf{P}_{N-1}$, and infer difference
invariants at key control points. Note that $\mathsf{P}_{N-1}$ and
$\mathsf{Q}_{N-1}$ are guaranteed to have synchronized iterations of
corresponding loops (both are obtained by restricting the upper bounds of all
loops to use $N-1$ instead of $N$). However, the conditional statements within
the loop body may not be synchronized. Thus, whenever we can infer that the
corresponding conditions are equivalent, we synchronize the branches of the
conditional statement. Otherwise, we consider all four possibilities of the
branch conditions. It can be seen that the net effect of the cross-product is
executing the programs $\mathsf{P}_{N-1}$ and $\mathsf{Q}_{N-1}$ one after the
other.
We run a dataflow analysis pass over the constructed product graph to infer
difference invariants at loop head, loop exit and at each branch condition.
The only dataflow values of interest are differences between corresponding
variables in $\mathsf{Q}_{N-1}$ and $\mathsf{P}_{N-1}$. Indeed, since
structure and variables of $\mathsf{Q}_{N-1}$ and $\mathsf{P}_{N-1}$ are
similar, we can create the correspondence map between the variables. We start
the difference invariant generation by considering relations between
corresponding variables/array elements appearing in pre-conditions of the two
programs. We apply static analysis that can track equality expressions
(including disjunctions over equality expressions) over variables as we
traverse the program. These equality expressions are our difference
invariants.
We observed in our experiments the most of the inferred equality expressions
are simple expressions of $N$ (atmost quadratic in $N$). This not totally
surprising and similar observations have also been independently made in [24,
15, 4]. Note that the difference invariants may not always be equalities. We
can easily extend our analysis to learn inequalities using interval domains in
static analysis. We can also use a library of expressions to infer difference
invariants using a guess-and-check framework. Moreover, guessing difference
invariants can be easy as in many cases the difference expressions may be
independent of the program constructs, for example, the equality expression
$\mathtt{v=v^{\prime}}$ where $\mathtt{v}\in\mathsf{P}_{N-1}$ and
$\mathtt{v^{\prime}}\in\mathsf{Q}_{N-1}$ does not depend on any other variable
from the two programs.
For the example in Fig. 2, the difference invariant at the head of the first
loop of $Q_{N-1}\times P_{N-1}$ is
$D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)\equiv\mathtt{(x^{\prime}-x=i\times(2\times
N-1)}$ $\wedge$ $\mathtt{\forall i\in[0,N-1),}$
$\mathtt{a^{\prime}[i]-a[i]=1)}$, where $\mathtt{x,a}\in V_{\mathsf{P}}$ and
$\mathtt{x^{\prime},a^{\prime}}\in V_{\mathsf{Q}}$. Given this, we easily get
$\mathtt{x^{\prime}-x=(N-1)\times(2\times N-1)}$ when the first loop
terminates. For the second loop,
$D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)\equiv(\mathtt{\forall
j\in[0,N-1),\;b^{\prime}[j]-b[j]=(x^{\prime}-x)+N^{2}=(N-1)\times}$
$\mathtt{(2\times N-1)+N^{2}})$.
Note that the difference invariants and its computation are agnostic of the
given post-condition. Hence, our technique does not need to re-run this
analysis for proving a different post-condition for the same program.
### 4.4 Verification using Inductive Difference Invariants
We present our method Diffy for verification of programs using inductive
difference invariants in Algorithm 2. It takes a Hoare triple
$\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$ as input, where $\varphi(N)$
and $\psi(N)$ are pre- and post-condition formulas. We check the base in line
1 to verify the Hoare triple for $N=1$. If this check fails, we report a
counterexample. Subsequently, we compute $\mathsf{Q}_{N-1}$ and
$\mathsf{peel}(\mathsf{P}_{N})$ as described in section 4.1 using the function
GenQandPeel from Algorithm 1. At line 5, we compute the formulas
$\varphi^{\prime}(N-1)$ and $\Delta\varphi^{\prime}(N)$ as described in
section 4.2. For automation, we analyze the quantifiers appearing in
$\varphi(N)$ and modify the quantifier ranges such that the conditions in
section 4.2 hold. We infer difference invariants
$D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)$ on line 6 using the method described in
section 4.3, wherein $V_{\mathsf{Q}}$ and $V_{\mathsf{P}}$ are sets of
variables from $\mathsf{Q}_{N-1}$ and $\mathsf{P}_{N-1}$ respectively. At line
7, we compute $\psi^{\prime}(N-1)$ by eliminating variables $V_{\mathsf{P}}$
from $\mathsf{P}_{N-1}$ from $\psi(N-1)\land
D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)$. At line 8, we check the inductive step
of our analysis. If the inductive step succeeds, then we conclude that the
assertion holds. If that is not the case then, we try to iteratively
strengthen both the pre- and post-condition of $\mathsf{peel}(\mathsf{P}_{N})$
simultaneously by invoking Strengthen.
Algorithm 2 Diffy( {$\varphi(N)$} $\mathsf{P}_{N}$ {$\psi(N)$} )
1:if {$\varphi(1)$} $\mathsf{P}_{1}$ {$\psi(1)$} fails then $\triangleright$
Base case for N=1
2: print “Counterexample found!”;
3: return $\mathsf{False}$;
4:$\langle\mathsf{Q}_{N-1},\mathsf{peel}(\mathsf{P}_{N})\rangle\leftarrow$
GenQandPeel($\mathsf{P}_{N}$);
5:$\langle\varphi^{\prime}(N-1),\Delta\varphi^{\prime}(N)\rangle\leftarrow$
FormulaDiff($\varphi(N)$); $\triangleright$
$\varphi(N)\Rightarrow\varphi^{\prime}(N-1)\wedge\Delta\varphi^{\prime}(N)$
6:$D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)\leftarrow$
InferDiffInvs$(\mathsf{Q}_{N-1},\mathsf{P}_{N-1},\varphi^{\prime}(N-1),\varphi(N-1))$;
7:$\psi^{\prime}(N-1)\leftarrow$ QE$(V_{\mathsf{P}},\psi(N-1)\wedge
D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1))$;
8:if {$\psi^{\prime}(N-1)\wedge\Delta\varphi^{\prime}(N)$}
$\mathsf{peel}(\mathsf{P}_{N})$ {$\psi(N)$} then
9: return $\mathsf{True}$; $\triangleright$ Verification Successful
10:else
11: return
Strengthen$(\mathsf{P}_{N},\mathsf{peel}(\mathsf{P}_{N}),\varphi(N),\psi(N),\psi^{\prime}(N-1),\Delta\varphi^{\prime}(N),D(V_{\mathsf{Q}},V_{\mathsf{P}},N))$;
12:procedure Strengthen($\mathsf{P}_{N}$, $\mathsf{peel}(\mathsf{P}_{N})$,
$\varphi(N)$, $\psi(N)$, $\psi^{\prime}(N-1)$, $\Delta\varphi^{\prime}(N)$,
$D(V_{\mathsf{Q}},V_{\mathsf{P}},N)$)
13: $\chi(N)\leftarrow\psi(N)$;
14: $\xi(N)\leftarrow\mathsf{True}$;
15: $\xi^{\prime}(N-1)\leftarrow\mathsf{True}$;
16: repeat
17:
$\chi^{\prime}(N-1)\leftarrow\textsc{WP}(\chi(N),\mathsf{peel}(\mathsf{P}_{N}))$;
$\triangleright$ Dijkstra’s $\mathsf{WP}$ for loop free code
18: if $\chi^{\prime}(N-1)=\emptyset$ then
19: if $\mathsf{peel}(\mathsf{P}_{N})$ has a loop then
20: return
Diffy({$\xi^{\prime}(N-1)\wedge\Delta\varphi^{\prime}(N)\wedge\psi^{\prime}(N-1)$}
$\mathsf{peel}(\mathsf{P}_{N})$ {$\xi(N)\wedge\psi(N)$});
21: else
22: return $\mathsf{False}$; $\triangleright$ Unable to prove
23: $\chi(N)\leftarrow$ QE$(V_{\mathsf{Q}},\chi^{\prime}(N)\wedge
D(V_{\mathsf{Q}},V_{\mathsf{P}},N))$;
24: $\xi(N)\leftarrow\xi(N)\wedge\chi(N)$;
25: $\xi^{\prime}(N-1)\leftarrow\xi^{\prime}(N-1)\wedge\chi^{\prime}(N-1)$;
26: if {$\varphi(1)$} $\mathsf{P}_{1}$ {$\xi(1)$} fails then
27: return $\mathsf{False}$; $\triangleright$ Unable to prove
28: if
{$\xi^{\prime}(N-1)\wedge\Delta\varphi^{\prime}(N)\wedge\psi^{\prime}(N-1)$}
$\mathsf{peel}(\mathsf{P}_{N})$ {$\xi(N)\wedge\psi(N)$} holds then
29: return $\mathsf{True}$; $\triangleright$ Verification Successful
30: until timeout;
31: return $\mathsf{False}$;
The function Strengthen first initializes the formula $\chi(N)$ with $\psi(N)$
and the formulas $\xi(N)$ and $\xi^{\prime}(N-1)$ to $\mathsf{True}$. To
strengthen the pre-condition of $\mathsf{peel}(\mathsf{P}_{N})$, we infer a
formula $\chi^{\prime}(N-1)$ using Dijkstra’s weakest pre-condition
computation of $\chi(N)$ over the $\mathsf{peel}(\mathsf{P}_{N})$ in line 17.
It may happen that we are unable to infer such a formula. In such a case, if
the program $\mathsf{peel}(\mathsf{P}_{N})$ has loops then we recursively
invoke Diffy at line 20 to further simplify the program. Otherwise, we abandon
the verification effort (line 22). We use quantifier elimination to infer
$\chi(N-1)$ from $\chi^{\prime}(N-1)$ and
$D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1))$ at line 7.
The inferred pre-conditions $\chi(N)$ and $\chi^{\prime}(N-1)$ are accumulated
in $\xi(N)$ and $\xi^{\prime}(N-1)$, which strengthen the post-conditions of
$\mathsf{P}_{N}$ and $\mathsf{Q}_{N-1}$ respectively in lines 24 \- 25. We
again check the base case for the inferred formulas in $\xi(N)$ at line 26. If
the check fails we abandon the verification attempt at line 27. If the base
case succeeds, we then proceed to the inductive step. When the inductive step
succeeds, we conclude that the assertion is verified. Otherwise, we continue
in the loop and try to infer more pre-conditions untill we run out of time.
The pre-condition in Fig. 2 is $\phi(N)\equiv\mathsf{True}$ and the post-
condition is $\psi(N)\equiv\mathtt{\forall j\in[0,N),\;b[j]=j+N^{3})}$. At
line 5, $\phi^{\prime}(N-1)$ and $\Delta\phi^{\prime}(N-1)$ are computed to be
$\mathsf{True}$. $D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)$ is the formula
computed in Section 4.3. At line 7, $\psi^{\prime}(N-1)\equiv(\mathtt{\forall
j\in[0,N-1),\;b^{\prime}[j]=j+(N-1)^{3}+(N-1)\times(2\times N-1)+N^{2}=}$
$\mathtt{j+N^{3}})$. The algortihm then invokes Strengthen at line 11 which
infers the formulas $\chi^{\prime}(N-1)\equiv\mathtt{(x^{\prime}=(N-1)^{3})}$
at line 17 and $\chi(N)\equiv\mathtt{(x=N^{3})}$ at line 23. These are
accumulated in $\xi^{\prime}(N-1)$ and $\xi(N)$, simultaneosuly strengthening
the pre- and post-condition. Verification succeeds after this strengthening
iteration.
The following theorem guarantees the soundness of our technique.
###### Theorem 4.2
Suppose there exist formulas $\xi^{\prime}(N)$ and $\xi(N)$ and an integer
$M>0$ such that the following hold
* •
$\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\wedge\xi(N)\\}$ holds for $1\leq
N\leq M$, for some $M>0$.
* •
$\xi(N)\wedge D(V_{\mathsf{Q}},V_{\mathsf{P}},N)\Rightarrow\xi^{\prime}(N)$
for all $N>0$.
* •
$\\{\xi^{\prime}(N-1)\wedge\Delta\varphi^{\prime}(N)~{}\wedge~{}\psi^{\prime}(N-1)\\}\;\mathsf{peel}(\mathsf{P}_{N})\;\\{\xi(N)\wedge\psi(N)\\}$
holds for all $N\geq M$, where $\psi^{\prime}(N-1)\equiv\exists
V_{\mathsf{P}}\big{(}\psi(N-1)\wedge
D(V_{\mathsf{Q}},V_{\mathsf{P}},N-1)\big{)}$.
Then $\\{\varphi(N)\\}\;\mathsf{P}_{N}\;\\{\psi(N)\\}$ holds for all $N>0$.
## 5 Experimental Evaluation
We have instantiated our technique in a prototype tool called Diffy. It is
written in C++ and is built using the LLVM(v$6.0.0$) [31] compiler. We use the
SMT solver Z3(v$4.8.7$) [39] for proving Hoare triples of loop-free programs.
Diffy and the supporting data to replicate the experiments are openly
available at [14].
Program | Diffy | Vajra | VeriAbs | VIAP
---|---|---|---|---
Category | S | U | TO | S | U | S | TO | S | U | TO
Safe C1 | 110 | 110 | 0 | 0 | 110 | 0 | 96 | 14 | 16 | 1 | 93
Safe C2 | 24 | 21 | 0 | 3 | 0 | 24 | 5 | 19 | 4 | 0 | 20
Safe C3 | 23 | 20 | 3 | 0 | 0 | 23 | 9 | 14 | 0 | 23 | 0
Total | 157 | 151 | 3 | 3 | 110 | 47 | 110 | 47 | 20 | 24 | 113
Unsafe C1 | 99 | 98 | 1 | 0 | 98 | 1 | 84 | 15 | 98 | 0 | 1
Unsafe C2 | 24 | 24 | 0 | 0 | 17 | 7 | 19 | 5 | 22 | 0 | 2
Unsafe C3 | 23 | 20 | 3 | 0 | 0 | 23 | 22 | 1 | 0 | 23 | 0
Total | 146 | 142 | 4 | 0 | 115 | 31 | 125 | 21 | 120 | 23 | 3
Table 1: Summary of the experimental results. S is successful result. U is
inconclusive result. TO is timeout.
Setup. All experiments were performed on a machine with Intel i7-6500U CPU,
16GB RAM, running at 2.5 GHz, and Ubuntu 18.04.5 LTS operating system. We have
compared the results obtained from Diffy with Vajra(v1.0) [12], VIAP(v1.1)
[42] and VeriAbs(v1.4.1-12) [1]. We choose Vajra which also employs inductive
reasoning for proving array programs and verify the benchmarks in its test-
suite. We compared with VeriAbs as it is the winner of the arrays sub-category
in SV-COMP 2020 [6] and 2021 [7]. VeriAbs applies a sequence of techniques
from its portfolio to verify array programs. We compared with VIAP which was
the winner in arrays sub-category in SV-COMP 2019 [5]. VIAP also employs a
sequence of tactics, implemented for proving a variety of array programs.
Diffy does not use multiple techniques, however we choose to compare it with
these portfolio verifiers to show that it performs well on a class of programs
and can be a part of their portfolio. All tools take C programs in the SV-COMP
format as input. Timeout of 60 seconds was set for each tool. A summary of the
results is presented in Table 1.
Benchmarks. We have evaluated Diffy on a set of $303$ array benchmarks,
comprising of the entire test-suite of [12], enhanced with challenging
benchmarks to test the efficacy of our approach. These benchmarks take a
symbolic parameter $N$ which specifies the size of each array. Assertions are
(in-)equalities over array elements, scalars and (non-)linear polynomial terms
over $N$. We have divided both the safe and unsafe benchmarks in three
categories. Benchmarks in C1 category have standard array operations such as
min, max, init, copy, compare as well as benchmarks that compute polynomials.
In these benchmarks, branch conditions are not affected by the value of $N$,
operations such as modulo and nested loops are not present. There are $110$
safe and $99$ unsafe programs in the C1 category in our test-suite. In C2
category, the branch conditions are affected by change in the program
parameter $N$ and operations such as modulo are used in these benchmarks.
These benchmarks do not have nested loops in them. There are $24$ safe and
unsafe benchmarks in the C2 category. Benchmarks in category C3 are programs
with atleast one nested loop in them. There are $23$ safe and unsafe programs
in category C3 in our test-suite. The test-suite has a total of $157$ safe and
$146$ unsafe programs.
|
---|---
(a) | (b)
Figure 8: Cactus Plots (a) All Safe Benchmarks (b) All Unsafe Benchmarks
Analysis. Diffy verified $151$ safe benchmarks, compared to $110$ verified by
Vajra as well as VeriAbs and $20$ verified by VIAP. Diffy was unable to verify
$6$ safe benchmarks. In $3$ cases, the smt solver timed out while trying to
prove the induction step since the formulated query had a modulus operation
and in $3$ cases it was unable to compute the predicates needed to prove the
assertions. Vajra was unable to verify $47$ programs from categories C2 and
C3. These are programs with nested loops, branch conditions affected by $N$,
and cases where it could not compute the difference program. The sequence of
techniques employed by VeriAbs, ran out of time on $47$ programs while trying
to prove the given assertion. VeriAbs proved $2$ benchmarks in category C2 and
$3$ benchmarks in category C3 where Diffy was inconclusive or timed out.
VeriAbs spends considerable amount of time on different techniques in its
portfolio before it resorts to Vajra and hence it could not verify $14$
programs that Vajra was able to prove efficiently. VIAP was inconclusive on
$24$ programs which had nested loops or constructs that could not be handled
by the tool. It ran out of time on $113$ benchmarks as the initial tactics in
its sequence took up the allotted time but could not verify the benchmarks.
Diffy was able to verify all programs that VIAP and Vajra were able to verify
within the specified time limit.
|
---|---
(a) | (b)
Figure 9: Cactus Plots (a) Safe C1 Benchmarks (b) Unsafe C1 Benchmarks
The cactus plot in Figure 8(a) shows the performance of each tool on all safe
benchmarks. Diffy was able to prove most of the programs within three seconds.
The cactus plot in Figure 9(a) shows the performance of each tool on safe
benchmarks in C1 category. Vajra and Diffy perform equally well in the C1
category. This is due to the fact that both tools perform efficient inductive
reasoning. Diffy outperforms VeriAbs and VIAP in this category. The cactus
plot in Figure 10(a) shows the performance of each tool on safe benchmarks in
the combined categories C2 and C3, that are difficult for Vajra as most of
these programs are not within its scope. Diffy out performs all other tools in
categories C2 and C3. VeriAbs was an order of magnitude slower on programs it
was able to verify, as compared to Diffy. VeriAbs spends significant amount of
time in trying techniques from its portfolio, including Vajra, before one of
them succeeds in verifying the assertion or takes up the entire time allotted
to it. VIAP took $70$ seconds more on an average as compared to Diffy to
verify the given benchmark. VIAP also spends a large portion of time in trying
different tactics implemented in the tool and solving the recurrence relations
in programs.
|
---|---
(a) | (b)
Figure 10: Cactus Plots (a) Safe C2 & C3 Benchmarks (b) Unsafe C2 & C3
Benchmarks
Our technique reports property violations when the base case of the analysis
fails for small fixed values of $N$. While the focus of our work is on proving
assertions, we report results on unsafe versions of the safe benchmarks from
our test-suite. Diffy was able to detect a property violation in $142$ unsafe
programs and was inconclusive on $4$ benchmarks. Vajra detected violations in
$115$ programs and was inconclusive on $31$ programs. VeriAbs reported $125$
programs as unsafe and ran out of time on $21$ programs. VIAP reported
property violation in $120$ programs, was inconclusive on $23$ programs and
timed out on $3$ programs.
The cactus plot in Figure 8(b) shows the performance of each tool on all
unsafe benchmarks. Diffy was able to detect a violation faster than all other
tools and on more benchmarks from the test-suite. Figure 9(b) and Figure 10(b)
give a finer glimpse of the performance of these tools on the categories that
we have defined. In the C1 category, Diffy and Vajra have comparable
performance and Diffy disproves the same number of benchmarks as Vajra and
VIAP. In C2 and C3 categories, we are able to detect property violations in
more benchmarks than other tools in less time.
To observe any changes in the performance of these, we also ran them with an
increased time out of $100$ seconds. Performance remains unchanged for Diffy,
Vajra and VeriAbs on both safe and unsafe benchmarks, and of VIAP on unsafe
benchmarks. VIAP was able to additionally verify $89$ safe programs in
categories C1 and C2 with the increased time limit.
|
---|---
(a) | (b)
Figure 11: Cactus Plots. TO=100s. (a) Safe Benchmarks (b) Unsafe Benchmarks
## 6 Related Work
Techniques based on Induction. Our work is related to several efforts that
apply inductive reasoning to verify properties of array programs. Our work
subsumes the full-program induction technique in [12] that works by inducting
on the entire program via a program parameter $N$. We propose a principled
method for computation and use of difference invariants, instead of computing
difference programs which is more challenging. An approach to construct safety
proofs by automatically synthesizing squeezing functions that shrink program
traces is proposed in [27]. Such functions are not easy to synthesize, whereas
difference invariants are relatively easy to infer. In [11], the post-
condition is inductively established by identifying a tiling relation between
the loop counter and array indices used in the program. Our technique can
verify programs from [11], when supplied with the _tiling_ relation. [44]
identifies recurrent program fragments for induction using the loop counter.
They require restrictive data dependencies, called _commutativity of
statements_ , to move peeled iterations across subsequent loops.
Unfortunately, these restrictions are not satisfied by a large class of
programs in practice, where our technique succeeds.
Difference Computation. Computing differences of program expressions has been
studied for incremental computation of expensive expressions [41, 35],
optimizing programs with arrays [34], and checking data-structure invariants
[45]. These differences are not always well suited for verifying properties,
in contrast with the difference invariants which enable inductive reasoning in
our case.
Logic based reasoning. In [21], trace logic that implicitly captures inductive
loop invariants is described. They use theorem provers to introduce and prove
lemmas at arbitrary time points in the program, whereas we infer and prove
lemmas at key control points during the inductive step using SMT solvers. VIAP
[42] translates the program to an quantified first-order logic formula using
the scheme proposed in [32]. It uses a portfolio of tactics to simplify and
prove the generated formulas. Dedicated solvers for recurrences are used
whereas our technique adapts induction for handling recurrences.
Invariant Generation. Several techniques generate invariants for array
programs. QUIC3 [25], FreqHorn [19] and [9] infer universally quantified
invariants over arrays for Constrained Horn Clauses (CHCs). Template-based
techniques [23, 47, 8] search for inductive quantified invariants by
instantiating parameters of a fixed set of templates. We generate relational
invariants, which are often easier to infer compared to inductive quantified
invariants for each loop.
Abstraction-based Techniques. Counterexample-guided abstraction refinement
using prophecy variables for programs with arrays is proposed in [36]. VeriAbs
[1] uses a portfolio of techniques, specifically to identify loops that can be
soundly abstracted by a bounded number of iterations. Vaphor [38] transforms
array programs to array-free Horn formulas to track bounded number of array
cells. Booster [3] combines lazy abstraction based interpolation[2] and
acceleration [10, 28] for array programs. Abstractions in [16, 18, 22, 26, 29,
33, 37] implicitly or explicitly partition the range array indices to infer
and prove facts on array segments. In contrast, our method does not rely on
abstractions.
## 7 Conclusion
We presented a novel verification technique that combines generation of
difference invariants and inductive reasoning. These invariants relate
corresponding variables and arrays from two versions of a program and are easy
to infer and prove. These invariants facilitate inductive reasoning by
assisting in the inductive step. We have instantiated these techniques in our
prototype Diffy. Experiments shows that Diffy out-performs the tools that won
the Arrays sub-category in SV-COMP 2019, 2020 and 2021. Investigations in
using synthesis techniques for automatic generation of difference invariants
to verify properties of array manipulating programs is a part of future work.
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## Appendix 0.A Programs with $\exists$ in the Post-Condition
In this section, we present examples with existentially quantified formulas as
post-condtions that can be verified using our technique.
// assume(true)
1. void Max(int A[], int N) {
2. int Max=A[0];
3. for(i=0; i<N; i++) {
4. if(Max < A[i])
5. Max = A[i];
6. }
7. }
// assert($\exists$i $\in$ [0,N), Max = A[i])
(a) |
// assume($\forall$i $\in$ [0,N), A[i] > 0)
1. void Average(int A[], int N)
2. float B[N], sum = 0;
3. for (int i=0; i<N; i=i+1)
4. sum = sum + A[i];
5. for (int j=0; j<N; j=j+1)
6. B[j] = A[j]/sum;
7.
// assert( sum > 0 and
// ($\exists$i $\in$ [0,N), B[i] >= 1/N) and
// ($\exists$j $\in$ [0,N), B[j] <= 1/N) )
(b)
---|---
// assume(N>1)
1. void ExtGZN(int A[], int N) {
2. for(i=0; i<N; i++) {
3. A[i] = i;
4. }
5. }
// assert($\exists$i $\in$ [0,N), A[i] > 0)
// assert($\exists$j $\in$ [0,N), A[j] >= N-1)
(c) |
// assume(N>1)
1. void EvenOdd(int A[], int N) {
2. for(i = 0; i<N; i++) {
3. if(i%2 == 0) A[i] = 0;
4. else A[i] = 1;
5. }
6. }
// assert($\exists$i $\in$ [0,N), A[i] = 1)
// assert($\exists$j $\in$ [0,N), A[j] = 0)
(d)
// assume($\forall$i $\in$ [0,N), A[i] = 1)
1. void Sum(int A[], int N) {
2. int B[N], sum = 0;
3. for (int i=0; i<N; i=i+1)
4. sum = sum + A[i];
5. for(int j=0; j<N; j++)
6. B[j] = sum;
7. }
// assert($\exists$i $\in$ [0,N), B[i] = N)
(e) |
// assume($\forall$i $\in$ [0,N), A[i] = N)
1. void Sum2(int A[], int N) {
2. int B[N], sum = 0;
3. for (int i=0; i<N; i=i+1)
4. sum = sum + A[i];
5. for(int j=0; j<N; j++)
6. B[j] = sum + j;
7. }
// assert($\exists$i $\in$ [0,N), B[i] = i + N*N)
(f)
Figure 12: Examples with Existentially Quantified Post-conditions
|
∎
11institutetext: Nguyen Hieu Thao22institutetext: Delft Center for Systems and
Control, Delft University of Technology, 2628CD Delft, The Netherlands.
Department of Mathematics, School of Education, Can Tho University, Can Tho,
Vietnam. 22email<EMAIL_ADDRESS><EMAIL_ADDRESS>
Oleg Soloviev33institutetext: Delft Center for Systems and Control, Delft
University of Technology, 2628CD Delft, The Netherlands. Flexible Optical
B.V., Polakweg 10-11, 2288 GG Rijswijk, The Netherlands. 33email:
<EMAIL_ADDRESS>
Jacques Noom44institutetext: Delft Center for Systems and Control, Delft
University of Technology, 2628CD Delft, The Netherlands. 44email:
<EMAIL_ADDRESS>
Michel Verhaegen55institutetext: Delft Center for Systems and Control, Delft
University of Technology, 2628CD Delft, The Netherlands. 55email:
<EMAIL_ADDRESS>
# Subpixel image reconstruction using nonuniform defocused images ††thanks:
This project has received funding from the ECSEL Joint Undertaking (JU) under
grant agreement No. 826589. The JU receives support from the European Union’s
Horizon 2020 research and innovation programme and Netherlands, Belgium,
Germany, France, Italy, Austria, Hungary, Romania, Sweden and Israel.
Nguyen Hieu Thao Oleg Soloviev Jacques Noom and Michel Verhaegen
###### Abstract
This paper considers the problem of reconstructing an object with high-
resolution using several low-resolution images, which are degraded due to
nonuniform defocus effects caused by angular misalignment of the subpixel
motions. The new algorithm, indicated by the Superresolution And Nonuniform
Defocus Removal (SANDR) algorithm, simultaneously performs the nonuniform
defocus removal as well as the superresolution reconstruction. The SANDR
algorithm combines non-sequentially the nonuniform defocus removal method
recently developed by Thao _et al._ and the least squares approach for
subpixel image reconstruction. Hence, it inherits global convergence from its
two component techniques and avoids the typical error amplification of multi-
step optimization contributing to its robustness. Further, existing
acceleration techniques for optimization have been proposed that assure fast
convergence of the SANDR algorithm going from rate $\mathcal{O}(1/k)$ to
$\mathcal{O}(1/k^{2})$ compared to most existing superresolution (SR)
techniques using the gradient descent method. An extensive simulation study
evaluating the new SANDR algorithm has been conducted. As no algorithms are
available to address the combined problem, in this simulation study we
restrict the comparison of SANDR with other SR algorithms neglecting the
defocus aberrations. Even for this case the advantages of the SANDR algorithm
have been demonstrated.
###### Keywords:
Superresolution, Image reconstruction, Computational imaging, Deconvolution,
Inverse problems
## 1 Introduction
Image-based quality control is one of the important tools used during the
manufacturing and end quality checks in semiconductor HUANG20151 , automotive
Zhou2019 , and many other industries. For Industry 4.0, requiring a fully
automatized quality checks, the spatial resolution (size of the smallest
feature that can be inspected) is the key factor that affects the overall
efficiency and throughput of the control tool. For high-quality imaging
systems used in these tools, the spatial resolution is defined as the quotient
between the pixel size and the magnification, and thus for a higher
resolution, either a smaller pixel size or a larger magnification is required.
Larger magnification corresponds to smaller field of view (FOV in Fig. 1),
often it is desirable to have a smaller pixel size. However, there are
technological and design limits to the magnification and the smallest pixels
that can be manufactured and/or used in these tools and thus a computational
approach to increasing spatial resolution provides an interesting alternative.
The reconstruction of an object with high-resolution from several low-
resolution (LR) images capturing the object at subpixel-offset positions,
called the SuperResolution (SR) problem, has been studied for many decades
PelKerSch87 ; UrGro92 . A number of solution approaches have been proposed for
the SR problem, including direct methods KimBosVal90 ; KimSu93 and iterative
algorithms SauAll87 ; NguMilGol01 ; FarRobElaMil04 ; SroCriFlu07 ; TakKan08 ;
LiHuGaoTaoNin10 ; ZhaYuaSheLi11 ; ZhaLiShiLin11 ; LagGhaHakRag16 ;
WanLinDenAn17 ; HuaSunYanFanLin17 .
Superresolution reconstruction is possible if the LR images are registered for
different subpixel-offset positions of the object. In practice, shifting the
object at subpixel scale can be a major challenge to the SR problem and gives
rise to a number of important questions that need to be addressed. Camera
shake and motion blur induced by the shifts have been analyzed in BasBlaZis96
; KanMil13 . Inaccuracy of subpixel registration has been considered in
LeeKan03 ; TakMilProEla09 . Ideally, the shifting process should not cause any
variations in the object orientation with respect to the camera. However, this
is not always the case in practice and such deviations cause undesirable
deterioration of the data images and thus the reconstruction. Imprecise
displacements with respect to the optical axis would introduce defocus blurs
in the acquired images. More challenging, the shifting process can induce
rotational movements of the object causing nonuniform defocus effects in the
data. To the best of our knowledge, the latter challenge has not been
considered in the literature of SR by subpixel motions.
In this paper, we consider the problem of reconstructing a Superresolution
Image using Nonuniform Defocused images, called the SIND problem. Our
consideration was primarily motivated by the inspection of wafers in
semiconductor industry and the basic hypotheses are mainly inspired by its
practical context, but the resulting solution is also scalable for similar
applications of computer vision. As an alternative to shifting the object, LR
images can be registered using multiple cameras whose optical axes are
typically at different directions towards the object. This also results in
_nonuniform defocus blurs_ in the acquired data, and the SIND problem covers
this challenge as a special case with known and fixed blurs.
Solution approaches to the SIND problem should address three main tasks,
including estimation of nonuniform defocus models, removal of nonuniform
defocus effects, and reconstruction of an SR image. Assuming that the
nonuniform defocus models have been estimated, this paper is devoted to the
last two tasks. More specifically, we propose a new algorithm to
simultaneously perform both Superresolution reconstruction And Nonuniform
Defocus Removal (SANDR). The SANDR algorithm combines the nonuniform defocus
removal method recently developed in ThaOleJacMic21 and the least squares
approach VerVer07 for subpixel image reconstruction but not in a sequential
manner. Hence, it inherits global convergence from its two component
techniques and avoids the typical error amplification of multi-step
optimization contributing to its robustness. Further, existing acceleration
techniques for optimization BecTeb09 have been proposed that assure fast
convergence of the SANDR algorithm going from rate $\mathcal{O}(1/k)$ to
$\mathcal{O}(1/k^{2})$ compared to most existing SR techniques using the
gradient descent method, where $k$ is the number of iterations.
As, to our knowledge, no algorithms are available to address the SIND problem,
we demonstrate the advantages of the SANDR algorithm over other SR algorithms
neglecting the defocus aberrations, see Sect. 4.1. It is important to mention
that the Projected Gradient (PG) and the so-called Sequential Minimization
(SM) algorithms reported along with SANDR in the numerical section are also
considered for the SIND problem for the first time. Hence, comparing the SANDR
algorithm with them is not a goal of this paper.
## 2 Problem formulation
### 2.1 Superresolution by subpixel motions
a)
b)
Figure 1: a) Imaging scheme used in the problem formulation. An imaging lens
with magnification $Q$ creates image $o(x,y)$ of some planar object, which is
registered by a detector with pixel size $s$ to obtain $M$ sampled images
$o_{m}[i,j]$ (green, blue), with introduced subpixel offsets $v_{m}$ in each
of them. The superresolution problem is to restore $O[i,j]$ representing
$o(x,y)$ sampled with a finer grid (orange). b) Side view of the imaging
scheme with an example of misalignment of the object and detector planes
creating position-dependent defocus blur.
Let $o_{m}$ be the LR images, created by an imaging system, that sample image
$o$ of some planar object, see Fig. 1. Let each $o_{m}$ be registered with
some subpixel offset $v_{m}$, with coordinates expressed in pixels of $o_{m}$.
We have the following sampling:
$o_{m}\sim T_{v_{m}}(o),\;(m=1,2,\ldots,M),$ (1)
where $T_{v}$ denotes the translation by a vector $v$.
Let $O$ be the SR image to be reconstructed. The ratio between the sizes of
$O$ and $o_{m}$ is called the superresolution factor and denoted by $\tau$. In
this paper, the images are assumed to be square and the superresolution factor
is the same in both row and column directions for the sake of brevity. As the
shifts are measured in pixels of $o_{m}$, their coordinates with respect to
$O$ should be scaled up by $\tau$. Then we also have the following sampling:
$T_{\tau v_{m}}(O)\sim T_{v_{m}}(o),\;(m=1,2,\ldots,M).$ (2)
The combination of (1) and (2) leads to the following superresolution model:
$o_{m}\simeq D_{\tau}\circ T_{\tau v_{m}}(O),\;(m=1,2,\ldots,M),$ (3)
where $D_{\tau}$ is the downsampling operator with rate $\tau$, see Sect. 2.5
for the definition.
###### Remark 1 (external blurs)
The imaging model (3) can be extended as follows IraPel91 ; FarRobElaMil03 ;
FarRobElaMil03.2 ; ParParKan03 ; SroCriFlu07 ; TakKan08 ; LiHuGaoTaoNin10 ;
ZhaYuaSheLi11 ; ZhaLiShiLin11 ; LokSolSavVdo11 ; LagGhaHakRag16 ;
WanLinDenAn17 ; HuaSunYanFanLin17 :
$o_{m}\simeq D_{\tau}\circ H_{m}\circ T_{\tau v_{m}}(O),\;(m=1,2,\ldots,M),$
where $H_{m}$ denote the external blurs often modelled as isoplanatic
convolutions and assumed to be known. For our target application in wafer
inspection, external blurs are not so relevant and thus left for brevity
though they do not add major challenge to the problem under consideration.
Instead, we handle the more challenging anisoplanatic blurs induced by angular
misalignment of the shifts as detailed in the next section.
### 2.2 Nonuniform defocus effects
In practice, the subpixel shifts can be accomplished by moving either the
sample or the detector chip. In both cases, some angular misalignment can be
introduced, which can be difficult or costly (e.g., in terms of time/overall
throughput) to eliminate completely. Figure 1b shows an example of a
misaligned sample; a similar picture could be drawn for a misaligned detector,
where geometrical distortions might also appear.111In this paper, we consider
the geometrical distortions to be negligible compared to the position-
dependent blur. For simplicity, we do not discriminate between the object and
detector misalignment. Depending on the particular realisation and on the
optical magnification of the system, this might create presence of position-
dependent defocus blur in the image, which, as we show later, prevents the
direct application of existing superresolution algorithms.
We consider the challenge that the displacement process induces undesirable
rotational movements of the object and the acquired images are degraded by
nonuniform defocus blurs. In this case, the theoretical LR images $o_{m}$ and
the measured ones $i_{m}$ are related by
$i_{m}\simeq B_{m}(o_{m}),\;(m=1,2,\ldots,M),$ (4)
where the blur operators $B_{m}$ will be detailed shortly.
Let $o_{m}$ situate in $N$ defocus zones denoted by $D_{n}$
$(n=1,2,\ldots,N)$, for each of which the Point Spread Function (PSF) is
modelled using the Fourier transform Goo05 :
$p_{n}=\left|\mathcal{F}\left(A\cdot{\rm e}^{{\rm
j}d_{n}Z_{2}^{0}}\right)\right|^{2}\quad(n=1,2,\ldots,N),$ (5)
where the amplitude, product and square operations are elementwise,
$\mathcal{F}$ is the two-dimensional Fourier transform, $A$ is the binary mask
representing the camera aperture,222We assume that the diffraction-limited PSF
is not resolved by the camera pixels. ${\rm j}=\sqrt{-1}$ is the imaginary
unit, $d_{n}$ is the (directional) distance from $D_{n}$ to the focal plane,
and $Z_{2}^{0}$ is the Zernike polynomial of order two and azimuthal frequency
zero.
We make use of the following model of nonuniform defocus blurs, whose physical
relevance has been demonstrated, e.g., in ThaOleJacMic21 :
$B_{m}(o_{m})=\sum_{n=1}^{N}\left(\mu_{mn}\cdot
o_{m}\right)*p_{n},\;(m=1,2,\ldots,M),$ (6)
where $*$ is the two-dimensional convolution, $\mu_{mn}$ are the mask
functions of $o_{m}$ defined by: for $n=1,2,\ldots,N$,
$\mu_{mn}[i,j]=\begin{cases}1&\text{if }\;o_{m}[i,j]\in D_{n},\\\
0&\text{otherwise}.\end{cases}$
This paper considers planar objects and $d_{n}$ take the following form
(ThaOleJacMic21, , Sect. IIC,):
$d_{n}=d(n_{0}-n),\quad(n=1,2,\ldots,N),$ (7)
where $d$ is the Depth of Focus (DoF) and $n_{0}$ is the _focal position_. The
number of defocus zones $N$ defines the degree of defocus in an image and can
be different for each $o_{m}$. For brevity, it is taken the same in this
paper.
The combination of (5), (6) and (7) yields the following blur model:
($m=1,2,\ldots,M$)
$B_{m}(o_{m})=\sum_{n=1}^{N}\left(\mu_{mn}\cdot
o_{m}\right)*\left|\mathcal{F}\left(A\cdot{\rm e}^{{\rm
j}d(n_{0}-n)Z_{2}^{0}}\right)\right|^{2}.$ (8)
### 2.3 The SIND problem
We consider the problem of reconstructing the Superresolution Image $O$ from
the Nonuniform Defocused images $i_{m}$ according to the relations (3) and
(4), where $B_{m}$ are given by (8). It is referred to as the SIND problem.
### 2.4 Optimization formulations
Combining (3) and (4) yields the imaging model:
$i_{m}=B_{m}\circ D_{\tau}\circ T_{\tau
v_{m}}(O)+w_{m},\quad(m=1,2,\ldots,M),$ (9)
where $B_{m}$ are given by (8), and $w_{m}$ represent the discrepancies
between the theoretical and the measured data, e.g., due to noise and model
deviations.
In this paper, $w_{m}$ are assumed to be independent zero-mean random
variables with jointly Gaussian-distributed entries.333This assumption is
ubiquitous and it does not rule out the case of Poisson noise as the latter
can be well approximated by a Gaussian distribution in view of the central
limit theorem provided that the image is registered with a sufficiently large
number of photon counts. For each $m=1,2,\ldots,M$, let $\mathcal{W}_{m}$ be
the covariance matrix of $\mathtt{Vec}\left(w_{m}\right)$, where
$\mathtt{Vec}$ denotes the vectorization operator. Then the maximum-likelihood
approach, e.g., (VerVer07, , Sect. 4.5.5,), applied to (9) leads to the
following minimization problem:
$\min_{O}\;\quad f(O)+\mathcal{G}(O),$ (10)
where $\mathcal{G}$ is the regularization capturing the physical attributes of
$O$ (see Sect. 3.1), and $f$ represents the data fidelity given by
$\displaystyle f(O)$
$\displaystyle=\sum_{m=1}^{M}R_{m}(O)^{T}\mathcal{W}_{m}^{-1}\,R_{m}(O),$ (11)
where $R_{m}$ ($m=1,2,\ldots,M$) are the fitting residual errors for the
(blurred) LR images:
$\displaystyle R_{m}(O)$ $\displaystyle=\mathtt{Vec}\left(B_{m}\circ
D_{\tau}\circ T_{\tau v_{m}}(O)-i_{m}\right).$
###### Remark 2 (sequential optimization)
The residual error in (3) is mainly due to the inaccuracy of the subpixel
shifts while the one in (4) is more related to measurement noise and model
deviations of (8). When the latter is less severe than the former,444This is
relevant to wafer inspection, where the camera is high-quality while
inexactness of the subpixel shifts poses the major challenge. one can also
address (4) and (3) sequentially via the following two-step optimization:
$\min_{O}\;\sum_{m=1}^{M}S_{m}(O)^{T}{\Sigma}_{m}^{-1}\,S_{m}(O)+\mathcal{G}(O),$
(12)
where for $m=1,2,\ldots,M$,
$\displaystyle S_{m}(O)=\mathtt{Vec}\left(D_{\tau}\circ T_{\tau
v_{m}}(O)-\widehat{o}_{m}\right),$
$\displaystyle\widehat{o}_{m}\in\arg\min_{o_{m}}Q_{m}(o_{m})^{T}\mathcal{E}_{m}^{-1}\,Q_{m}(o_{m})+\mathcal{H}\left(o_{m}\right),$
(13) $\displaystyle Q_{m}(o_{m})=\mathtt{Vec}\left(B_{m}(o_{m})-i_{m}\right).$
In the above, $\Sigma_{m}$ and $\mathcal{E}_{m}$ are respectively the
covariance matrices representing the noise in (4) and (3), and $\mathcal{H}$
is the regularization capturing the physical attributes of $o_{m}$.
Sequentially minimizing (13) and (12) gives rise to the so-called SM algorithm
(see Sect. 3.2), which suffers the typical error amplification of multi-step
optimization compared to the proposed solution method for solving (10), see
Sect. 4.3&4.4.
### 2.5 Downsampling operators
The downsampling operator with integer rate $\tau$ is given by
$D_{\tau}([u])=\frac{1}{\tau^{2}}\,\mathtt{conv}_{\tau}\left(u,\mathbf{1}_{\tau}\right),\quad(\forall\,u),$
(14)
where $\mathtt{conv}_{\tau}$ denotes the bivariate convolution operation with
striking sizes $\tau\times\tau$,555The terminology is standard in the field of
convolutional neural networks. and $\mathbf{1}_{\tau}$ is the all-ones matrix
of size $\tau\times\tau$. The striking sizes define the size reduction in row
and column directions. $D_{\tau}$ produces only the average intensity value of
every $\tau\times\tau$-block and hence it is not invertible without additional
information of $u$.
## 3 Solution approaches
Solution approaches to the SIND problem should address three main tasks,
including estimation of nonuniform defocus models, removal of nonuniform
defocus effects, and reconstruction of an SR image. Assuming that the
nonuniform defocus models have been estimated, this paper is devoted to the
last two tasks. We first discuss regularization schemes for the SIND problem.
### 3.1 Regularization functions
SR methods often minimize a cost function consisting of data fidelity and
regularization IraPel91 ; NguMilGol01 ; FarRobElaMil03 ; FarRobElaMil03.2 ;
ParParKan03 ; FarRobElaMil04 ; SroCriFlu07 ; TakKan08 ; LiHuGaoTaoNin10 ;
ZhaYuaSheLi11 ; ZhaLiShiLin11 ; LokSolSavVdo11 ; LagGhaHakRag16 ;
WanLinDenAn17 ; HuaSunYanFanLin17 . Data fidelity is typically a norm of the
residual between the theoretical and the measured data while regularization is
driven by the _a priori_ known physical attributes of the solution. The latter
pertains to each particular application and is the main difference between
existing SR techniques. Total variation and Tikhonov regularization were
considered in, e.g., NguMilGol01 ; FarRobElaMil03 . The Bilateral Total
Variation (BTV) was introduced in FarRobElaMil03 and later adapted in
FarRobElaMil04 ; LiHuGaoTaoNin10 ; ZhaLiShiLin11 ; LagGhaHakRag16 ;
WanLinDenAn17 . In LagGhaHakRag16 BTV was used in combination with the
Laplace operator while in LiHuGaoTaoNin10 it was used in combination with
another regularization to enhance the consistence of the gradient variation.
In this paper, the images are assumed to have intensities in $[0,1]$, and the
set of matrices satisfying this constraint is denoted by $\Omega$. This
constraint is easy to handle, but essential for the success of our proposed
algorithms, where acceleration optimization mechanisms are exploited. Its
effectiveness has been widely known in the literature of deconvolution, see,
e.g., WilSolPozVdoVer17 ; ThaOleJacMic21 . There are several approaches to
this constraint, e.g., the penalty approaches using the associated distance
function or its square. In this paper, we make use of the _indicator function_
VA :
$\iota_{\Omega}(x)=\begin{cases}0&\text{ if }x\in\Omega,\\\ \infty&\text{
otherwise}.\end{cases}$ (15)
In our simulation results, this constraint is a precise regularization and
hence its advantages over the other schemes are clearly observed, see Sect.
4.1.
### 3.2 The proposed algorithms
In view of Remark 2, the SIND problem can be addressed by solving (13) and
(12) sequentially. For each $m=1,2,\ldots,M$, (13) is the single-frame
nonuniform defocus removal problem recently studied in ThaOleJacMic21 . Hence,
it can be solved by the algorithm proposed in that paper, where its challenges
including the typical ill-posedness were also discussed and global convergence
of the proposed algorithm was also established. The main challenge of (12) is
that the downsampling operator $D_{\tau}$ is not invertible, in particular,
closed-form solutions for it are not available. We propose to apply the
regularization (15) and make use of the fast proximal gradient method
introduced in BecTeb09 , often known as FISTA, for solving (12).
The algorithm resulted from this sequential approach will be referred to as
the Sequential Minimization (SM) algorithm for the SIND problem. However, we
chose to skip its details for the sake of brevity. The main advantages of the
SM algorithm include its simplicity and the parallelism of (13) while its
major disadvantage is the typical error amplification of multi-step
optimization.
To overcome the drawback of SM, we next propose a new algorithm to
simultaneously handle both Superresolution reconstruction And Nonuniform
Defocus Removal (SANDR). The SANDR algorithm combines the nonuniform defocus
removal method developed in ThaOleJacMic21 and the least squares approach
VerVer07 for subpixel image reconstruction but not in a sequential manner.
Hence, it inherits global convergence from its two component techniques and
avoids the typical error amplification of multi-step optimization contributing
to its robustness, see Sects. 4.3&4.4. Making use of the acceleration
techniques for optimization of FISTA assures fast convergence of the SANDR
algorithm going from rate $\mathcal{O}(1/k)$ to $\mathcal{O}(1/k^{2})$
compared to most existing SR techniques using the gradient descent method.
For simplicity, the noise covariance matrices $\mathcal{W}_{m}$ in (11) are
taken to be the identity matrix in the sequel. The repetitive term
$(m=1,2,\ldots,M)$ following the subscript $m$ will be omitted for brevity.
In the sequel, $U_{\tau}$ will denote a right inverse of the downsampling
operator $D_{\tau}$ defined in (14), i.e., $D_{\tau}\circ
U_{\tau}=\operatorname{Id}$, the identity mapping.666$U_{\tau}$ is not unique
and in general $D_{\tau}\circ U_{\tau}\neq U_{\tau}\circ D_{\tau}$. $U_{\tau}$
can be understood as a numerical upsampling operator, and in our numerical
results, it is taken to be the interpolation with block constant values.
Recall that the translation by a vector with integer coordinates
$(v_{x},v_{y})$, is given by
$T_{(v_{x},v_{y})}(u)(r,c)=u(r-v_{x},c-v_{y}),\quad(\forall\,u),$ (16)
where $(r,c)$ are the row-column coordinates of the pixels.
###### Algorithm 1 (the SANDR algorithm)
_Input:_ $i_{m}$ – LR images, $B_{m}$ – blur operators, $\lambda$ – stepsize,
$t^{(0)}$ – initial acceleration stepsize, $K$ – number of iterations, and
$\varepsilon>0$.
_Initialization:_ $X^{(0)}=O^{(0)}=\frac{1}{M}\sum_{m=1}^{M}T_{(-\tau
v_{m})}(U_{\tau}(i_{m}))$.
_Iteration process_ : given $X^{(k)}$, $O^{(k)}$, $t^{(k)}$
$\displaystyle G_{m}^{(k)}=T_{(-\tau v_{m})}\circ U_{\tau}\left(\nabla
f_{m}\left(D_{\tau}\circ T_{\tau v_{m}}\left(O^{(k)}\right)\right)\right),$
$\displaystyle X_{m}^{(k+1)}=P_{\Omega}\left(O^{(k)}-\lambda
G_{m}^{(k)}\right),$ $\displaystyle
X^{(k+1)}=\frac{1}{M}\sum_{m=1}^{M}X_{m}^{(k+1)},$ $\displaystyle
t^{(k+1)}=\frac{1+\sqrt{1+4{t^{(k)}}^{2}}}{2},$ $\displaystyle
O^{(k+1)}=X^{(k+1)}+\frac{t^{(k)}-1}{t^{(k+1)}}\left(X^{(k+1)}-X^{(k)}\right).$
_Stopping criteria_ : $k>K$ or
$\sum_{m=1}^{M}\left\|G_{m}^{(k)}\right\|>\sum_{m=1}^{M}\left\|G_{m}^{(k-1)}\right\|+\varepsilon.$
(17)
_Output:_ $\widehat{O}=P_{\Omega}\left(O^{(\mathtt{end})}\right)$.
In Algorithm 1, $P_{\Omega}$ is the projection operator associated with
$\Omega$ and the functions $f_{m}$ are given by
$f_{m}(x)=\frac{1}{2}\left\|B_{m}(x)-i_{m}\right\|^{2},\;(m=1,2,\ldots,M).$
## 4 Numerical simulations
As explained in Sect. 2.2, a higher degree of defocus in an (LR) image
corresponds to a larger number of defocus zones and smaller supports (nonzero
entries) of the mask functions and vice versa. To simplify simulation of
random defocus levels in LR images, we chose to fix these parameters, but
consider the DoF $d$ in (7) as the single parameter quantifying the defocus in
each image, called the blur coefficient of the image in the sequel. It is
important to mention that our choice for convenience does not contradict the
fact that DoF is a fixed physical parameter of the camera because
underestimation of DoF does not introduce model deviations.777It only costs
computational time as the number of defocus zones increases accordingly. The
larger the blur coefficient is, the more the defocus blur in the image.
Simulation data is generated according to the forward imaging model (9).
Except for the analysis regarding the number of input images in Sect. 4.5,
each data set consists of four images corresponding to the shift vectors
$v_{1}=(0,0)$, $v_{2}=(1/2,0)$, $v_{3}=(0,1/2)$ and $v_{4}=(1/2,1/2)$. Half of
the images contain defocus blur varying in the vertical direction and half in
the horizontal direction. Unless otherwise specified, the common parameters
are as in Table 1.
Table 1: Parameters used in Sect. 4. $M$ is the number of LR images, $\mu$ – size of the supports of mask functions in pixel rows/columns, $\rho$ – PSF size (square), $\tau$ – SR factor, $\lambda$ – stepsize, and $t^{(0)}$ – initial acceleration stepsize. Parameter | $M$ | $\mu$ | $\rho$ | $\tau$ | $\lambda$ | $t^{(0)}$
---|---|---|---|---|---|---
Value | 4 | 3 | 11 | 2 | 1 | 1
Except for the noise analysis in Sect. 4.4, the data is corrupted with Poisson
noise using the MATLAB function $\mathtt{imnoise}$. The quality of SR
reconstruction is measured by the Root Mean Square (RMS) error of the restored
SR image relative to the ideal one:
${\|\widehat{O}-O\|}\big{/}{\left\|O\right\|}$. The stopping criterion (17) is
not implemented as it is not so relevant for simulations.
As no algorithms are available to address the SIND problem, we can only
demonstrate its advantages over other SR methods neglecting the defocus
effects. It is important to mention that the Projected Gradient (PG) and the
Sequential Minimization (SM) algorithms are also first considered for the SIND
problem, and hence comparing the SANDR algorithm with them is not a goal of
this section. Instead, their own advantages and disadvantages in various
problem settings will be of our primary interest.
### 4.1 Comparison to known SR methods
Figure 2: LR image (left) and the ideal SR (right). The ROIs are shown in Fig.
3 for visual comparison of different SR methods.
Figure 3: SR images obtained by L1-BTV, L1-BTV-L, L2-BTV and SANDR are shown
together with an LR image and the ideal SR. Only the ROIs are shown for
clarity. The relative RMS error of each ROI is also reported. The SANDR
algorithm outperforms the other SR methods.
Most existing SR methods minimize a cost function consisting of data fidelity
and regularization using the gradient descent method FarRobElaMil04 ;
SroCriFlu07 ; TakKan08 ; LiHuGaoTaoNin10 ; ZhaYuaSheLi11 ; ZhaLiShiLin11 ;
LagGhaHakRag16 ; WanLinDenAn17 ; HuaSunYanFanLin17 . Data fidelity is
typically the (weighted) $L_{p}$-norm ($1\leq p\leq 2$) of the residual
between the theoretical and the measured data while regularization is driven
by the _a priori_ known physical attributes of the solution, see Sect. 3.1. In
this section, we compare the SANDR algorithm with three existing SR methods
minimizing (1) the $L_{1}$-norm with bilateral total variation (L1-BTV)
FarRobElaMil04 , (2) the $L_{1}$-norm with BTV and Laplace operator (L1-BTV-L)
LagGhaHakRag16 , and (3) the $L_{2}$-norm with BTV (L2-BTV) WanLinDenAn17 .
Each iteration of the algorithms is additionally followed by a projection on
the constraint $\Omega$ to improve their performance, especially in terms of
stability. Note that without defocus effects, the SM and the SANDR algorithms
coincide.
Figure 4: Relative RMS errors of the SR images obtained by L1-BTV, L1-BTV-L,
L2-BTV and SANDR are shown in iterations. The SANDR algorithm is superior to
the other methods in both convergence speed and accuracy.
Figure 2 shows an LR image (left) and the ideal SR (right). The SR images
obtained by L1-BTV, L1-BTV-L, L2-BTV and SANDR are shown in Fig. 3 together
with an LR image and the ideal SR. Only the ROIs are shown for clarity. The
relative RMS error of each ROI is also reported. The SANDR algorithm clearly
outperforms the other methods both visually and in terms of RMS errors. This
is further explained in Fig. 4, where the RMS errors are shown in iterations.
The SANDR algorithm is far superior to the others in both convergence speed
and accuracy. Faster convergence is due to the acceleration feature of SANDR
while higher accuracy can be explained by the fact that $\Omega$ is a precise
regularization in this simulation problem. Ripple behaviours of L1-BTV and
L1-BTV-L in Fig. 4 can be explained by the step-size being larger than the
distance from the iteration to a local minimum. This phenomenon is more likely
to happen to $L_{1}$-norm cost functions since their gradient includes the
sign function, which does not depend on the residual gap of the current
iteration.888An advantage of $L_{1}$-norm cost functions is their potential to
suppress outliers. Gradually decreasing the stepsize is a possible remedy for
this issue, however, we chose not to distract the reader further in that
direction because there does not exist a unified recipe for such tasks while
the methods are not applicable to the SIND problem.
### 4.2 Convergence properties
Figure 5: LR image with defocus varying in the vertical direction (left) and
the unblurred one (right). Figure 6: RMS errors of the SR images obtained by
PG, SM and SANDR are shown in iterations. The algorithms exhibit convergence
properties and without acceleration, PG (red) converges slower than SM (black)
and SANDR (blue). The RMS error with 150 PG iterations is 0.79%, corresponding
to about 30 iterations of SANDR.
In this section, we demonstrate convergence properties of the SANDR algorithm
along with the PG and SM methods. We consider LR images of size $330\times
330$ pixels with $110$ defocus zones and blur coefficients randomly taken in
the interval $[0.001,0.06]$. The other parameters are as in Table 1. One of
the LR images with defocus effects varying in the vertical direction and its
unblurred version are shown in Fig. 5.
In Fig. 6 the RMS errors of the SR images obtained by PG, SM and SANDR are
shown in iterations. The algorithms exhibit convergence properties and without
acceleration, PG (red) converges slower than SM (black) and SANDR (blue). The
RMS error with 150 iterations of PG is 0.79%, corresponding to about 30
iterations of SANDR. In Fig. 7 the ROIs of the SR images obtained by 5, 15, 50
and 150 iterations of PG, SM and SANDR are shown in comparison with the ones
of an LR image and the ideal SR. The relative RMS error of each ROI is also
shown. Note that the RMS error of each unblurred LR image is around 13.83%.
a)
b)
c)
d)
e)
Figure 7: ROIs of the SR images obtained by a) 5, b) 15, c) 50, d) 150
iterations of PG, SM and SANDR are shown in comparison with e) the ones of an
LR image and the ideal SR. The RMS error of each ROI is also shown. The RMS
error of each unblurred LR image is around 13.83%.
### 4.3 Solvability analysis
As explained at the beginning of Sect. 4, blur coefficients quantify the
defocus in the LR images. The larger they are, the more challenging the SIND
problem is. In this section, we analyse the solvability of PG, SM and SANDR
with respect to this parameter. For each experiment, four blur coefficients of
the LR images are randomly taken in the interval $[0.001,d_{\mathtt{max}}]$
with $d_{\mathtt{max}}$ ranging from 0.06 to 0.3. The other parameters are as
in Table 2.999It is a trade-off between the computational complexity (number
of iterations) and the restoration accuracy, in view of Figs. 4 and 6 we chose
to run $50$ iterations for each experiment.
Table 2: Parameters used in Sect. 4.3. $d_{\mathtt{max}}$ is the upper bound of (random) blur coefficients, $N$ – # defocus zones, $n_{0}$ – focal position, $(l,w)$ – size of LR images, and $K$ – # iterations. Parameter | $d_{\mathtt{max}}$ | $N$ | $n_{0}$ | $l=w$ | $K$
---|---|---|---|---|---
Value | 0.06 – 0.3 | 55 | 28 | 165 | 50
Figure 8: Solvability analysis of PG, SM and SANDR with respect to the
distortion level of data images quantified by the blur coefficients. One
hundred experiments are reported for each value of $d_{\mathtt{max}}$ ranging
from 0.06 to 0.3. The restoration errors increase for larger
$d_{\mathtt{max}}$. SM works best for $d_{\mathtt{max}}$ up to 0.06, but it
quickly becomes problematic for $d_{\mathtt{max}}$ from 0.12 due to its
sequential optimization. SANDR is effective for blur coefficients up to 0.3
and outperforms PG and SM (for $d_{\mathtt{max}}\geq 0.12$) in both accuracy
and consistency. The superiority becomes more significant for larger values of
$d_{\mathtt{max}}$.
For each value of $d_{\mathtt{max}}$, one hundred experiments with PG, SM and
SANDR are reported in Fig. 8, where the RMS errors of the obtained SR images
with respect to the ideal one are presented. The restoration errors increase
for larger values of $d_{\mathtt{max}}$. SM works best for $d_{\mathtt{max}}$
up to 0.06, but it quickly becomes problematic for $d_{\mathtt{max}}$ from
0.12 due to its sequential optimization. SANDR is effective for blur
coefficients up to 0.3. It outperforms PG and SM (for $d_{\mathtt{max}}\geq
0.12$) in both accuracy and consistency, and the superiority becomes more
significant for larger $d_{\mathtt{max}}$. Higher accuracy is reflected by its
smaller average restoration errors while more consistency is indicated by its
smaller variances of the errors. To visualize the _blur coefficient_
parameter, the PSFs for the 30th and 50th defocus zones (counted from the
focal position) with blur coefficient 0.06 are shown in Fig. 9. Recall that
the distortion level of an image is proportional to the product of the blur
coefficient and the zone position in view of (7).
Figure 9: PSFs for the 30th (left) and 50th (right) defocus zones with blur
coefficient 0.06.
### 4.4 Noise analysis
We analyse the influence of noise on the performance of PG, SM and SANDR. Five
levels of Gaussian noise ranging from 45 to 65 dB (decibels) are considered.
Recall that the signal-to-noise ratio (SNR) in decibels is defined by:
$\text{SNR}=10\ln\left({P}/{P_{0}}\right)$, where $P$ and $P_{0}$ are the
powers of the signal and noise, respectively. To visualize the noise, an LR
image with SNR 45 dB is shown in Fig. 12 together with its residual relative
to the noiseless one.
Figure 10: Noise analysis of PG, SM and SANDR. One hundred experiments are
reported for each SNR from 45 to 65 dB. The reconstruction is more accurate
for higher SNR. SM works best for SNR from 55 dB, but it quickly becomes
problematic for SNR decreasing from 50 dB. Since SM and SANDR are acceleration
variants of PG, they are less robust than PG. Figure 11: Experiments similar
to those reported in Fig. 10 but with $d_{\mathtt{max}}=0.12$ instead of
$0.09$ show that SM deteriorates much faster than PG and SANDR for larger blur
coefficients.
For each SNR, one hundred experiments with $d_{\mathtt{max}}=0.09$ and the
other parameters as in Table 2 are reported in Fig. 10, where the RMS errors
of the SR images obtained by PG, SM and SANDR are presented. The
reconstruction is more accurate for higher SNR. SM works best for SNR from 55
dB, but it quickly becomes problematic for SNR decreasing from 50 dB. Its less
robustness against noise compared to SANDR is due to its two-step
optimization, see also Sect. 4.3. It is not a surprise that SM and SANDR are
less robust than PG because the former are acceleration variants of the latter
and there is a typical trade-off between robustness and convergence speed. In
view of Fig. 10, it is worth thinking about PG for the SIND problem with SNR
below 45 dB, but for higher SNR it is outperformed by the others. It is
important to recall that the conclusions drawn for the SM algorithm from Fig.
10 are valid only for $d_{\mathtt{max}}$ up to 0.09, which seems to be a limit
for it, see also Fig. 8. To demonstrate this point, we do similar experiments
but with slightly larger blur coefficients, $d_{\mathtt{max}}=0.12$ in place
of $0.09$. The results are summarized in Fig. 11, where SM deteriorates much
more than PG and SANDR in comparison with Fig. 10.
Figure 12: A noisy LR image (left) with SNR 45 dB and its residual (right)
with respect to the noiseless one.
### 4.5 Number of input images
The major practical challenge of SR by subpixel motions is to perform the
shifts accurately. Let us suppose that we are able to perform shifts at scale
$1/\tau$ pixel, where $1<\tau\in\mathbb{Z}$.101010In this study, inaccuracy of
subpixel registration is subsumed in noise. Then there are at most $\tau^{2}$
LR images and one cannot expect to gain a SR factor greater than $\tau$. In
this section, we briefly study the influence of the number of input images on
the quality of SR. We consider $\tau=4$ and construct the SR image using 2, 4,
8 and 16 LR images, respectively. In this experiment, $d_{\mathtt{max}}=0.06$
and the other parameters are as in Table 2.
The numerical results are summarized in Fig. 13, where only the ROIs and their
RMS errors are shown for brevity. It is clear that more input images result in
higher quality of the SR and the observation is consistent for PG, SM and
SANDR. SR images obtained with two LR images (the first row) already shows
improvement even in comparison with the unblurred LR images (the second in the
last row).
Figure 13: Influence of the number of input images on the SR images obtained
by PG, SM and SANDR. ROIs of the SR images obtained with 2, 4, 8 and 16 LR
images are shown together with their RMS errors. More input images result in
higher quality of SR.
### 4.6 Image cropping
Cropping the data images would introduce deviations to the imaging model (9).
This issue does not arise in the previous sections since the simulation object
there has almost constant intensity near the boundary. In this section, we
study the influence of image cropping on the performance of the PG, SM and
SANDR algorithms.
Table 3: Parameters used in Sect. 4.6. $d_{\mathtt{max}}$ is the upper bound of blur coefficients, $N$ – # defocus zones, $n_{0}$ – focal position, $(l,w)$ – size of LR images, and $K$ – # iterations. Parameter | $d_{\mathtt{max}}$ | $N$ | $n_{0}$ | $l=w$ | $K$
---|---|---|---|---|---
Value | 0.09 | 85 | 43 | 255 | 50
Figure 14: SR images obtained by PG, SM and SANDR are shown together with an
LR image (bottom left) and the ideal SR (bottom right). The restoration error
is smaller in the central region and becomes larger towards the boundary. The
SANDR algorithm is the most effective for this problem while the SM algorithm
is problematic due to high level of defocus effect. The SANDR and SM
algorithms suffer more boundary effects than PG. Figure 15: The ROIs marked in
Fig. 14 are shown for a better inspection of finer details. Their RMS errors
with respect to the ideal SR are also reported.
Four images are generated according to (9) with the parameters as in Table 3.
They are then windowed using Butterworth function to yield the LR images, one
of which is shown at the bottom left of Fig. 14. The SR images obtained by PG,
SM and SANDR are shown in Fig. 14 in comparison with an LR image and the ideal
SR. The SANDR algorithm is the most effective for this problem while the SM
algorithm is problematic due to high level of defocus effect as discussed in
Sect. 4.3. The SM and SANDR algorithms suffer more boundary effects than PG
since the former are more sensitive to noise than the latter as analyzed in
Sect. 4.4. To reduce the restoration errors near the boundary, the Butterworth
filter also need to be applied to every iteration update of the algorithms.
The RMS errors are computed for the central regions with 90% in radius of the
images. The reconstruction error is smaller in the central region and
increases towards the boundary. The ROIs are zoomed out in Fig. 15 for a
better inspection of finer details. The RMS error of each ROI is also
reported.
## 5 Concluding Remarks
We have investigated the problem of constructing an object with high-
resolution using several nonuniform defocused images, called the SIND problem.
Nonuniform defocus effects can arise in both standard techniques of data
registration, including the use of multiple cameras and moving the object.
However, the SIND problem has not been studied before. We have proposed the
efficient algorithm for SIND, called the SANDR algorithm, that can process
both subpixel image reconstruction and nonuniform defocus removal
simultaneously. Important theoretical and practical aspects of the SANDR
algorithm have been analyzed, including its global convergence, solvability,
noise robustness, dependence on the number of LR images, and sensitivity to
model deviations due to image croping. We have demonstrated advantages of the
SANDR algorithm over a number of existing superresolution methods without
considering defocus effects because the latter cannot handle this additional
challenge. Our consideration was primarily motivated by the inspection of
wafers in semiconductor industry, but the SANDR algorithm can be scalable for
similar applications of computer vision in Industry 4.0.
Funding. This project has received funding from the ECSEL Joint Undertaking
(JU) under grant agreement No. 826589. The JU receives support from the
European Union’s Horizon 2020 research and innovation programme and
Netherlands, Belgium, Germany, France, Italy, Austria, Hungary, Romania,
Sweden and Israel.
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# Dynamics of Long-lived Axion Domain Walls and Its Cosmological Implications
Chia-Feng Chang<EMAIL_ADDRESS>Yanou Cui<EMAIL_ADDRESS>Department of Physics and Astronomy, University of California, Riverside, CA
92521, USA
###### Abstract
We perform an updated analysis on a long-lived axion domain wall (DW) network.
By simulating the axion field on a 3D lattice and fitting an analytical model
for the DW evolution, we identify the leading energy loss mechanisms of the
DWs and compute the spectrum of axions emitted from the network. The
contribution from the DWs to axion dark matter (DM) density is derived, with
viable parameter space given. The application to both QCD axions and general
axion-like particles (ALPs) are considered. Due to the new approaches taken,
while our results bear consistency with earlier literature, notable
discrepancies are also revealed, such as the prediction for DM abundance,
which may have a profound impact on axion phenomenology at large.
## I Introduction
Axions are ultra-light particles that are originally proposed as a compelling
solution to the Strong CP problem in quantum chromodynamics (QCD) [1, 2, 3].
Recent years have seen a significantly increased interest in QCD axions and
more general axion-like particles (ALPs), as dark matter (DM) candidates
alternative to WIMPs [4, 5, 6, 7]. While most existing studies on axion
phenomenology and detection focused on the axion particle per se, the impact
of the accompanying axion topological defects, i.e. axion strings and domain
walls (DWs), can be substantial, yet still not well understood. Such axion
topological defects are indispensable companions of axion particles for post-
inflationary PQ symmetry breaking, with potentially significant contribution
to axion relic abundance [8, 9, 10, 11, 12, 13], and may provide complementary
search avenues for axion models [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26]. A growing effort has been made in the past few years along this
direction. However, there are still debates to be resolved and clarifications
to be made, in part due to the technical challenges with simulating axion
topological defects [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].
Axion cosmic strings form as the PQ breaking phase transition (PT) occurs at a
high energy scale $f_{a}$, and prevail till the pseudo-goldstone boson (axion)
later acquires a nonzero mass $m_{a}$ and DWs form. The structure of the DWs
depends on the model specifics of the axion potential and is characterized by
the axion mass and the DW number $N_{\rm DW}$. The case with $N_{\rm DW}=1$ is
most studied in recent years, where the DWs are short-lived and strings
dominate the dynamics of the axion topological defects [36, 32, 34]. On the
other hand, more generally for the $N_{\rm DW}>1$ models e.g. Dine-Fischler-
Srednicki-Zhitnitsky model [40, 41], the DWs are stable and problematic as
they would over-close the Universe. Nevertheless, the $N_{\rm DW}>1$ cases can
be innocuous with the presence of a small symmetry-breaking bias term in the
axion potential, which yields the DWs that are long-lived but collapse before
the BBN [42, 43]. Upon collapsing, long-lived DWs can leave observable
imprints in the form of axion dark matter relic density, gravitational waves
(GWs), as well as the impact on cosmic structure formation [12, 44]. A clear
understanding of the evolution and dynamics of the DW network is crucial for
predicting and probing such potentially rich phenomenology. However, the
literature on the dynamics of metastable DWs (axion-associated or more
general) is still relatively scarce [45, 46, 47, 48, 49, 12, 44], and further
investigation is required to advance and clarify our understanding.
In this work, we conduct an updated analysis for the long-lived axion DWs and
predict axion relic abundance produced from the axion DWs (with $N_{\rm DW}$=2
as a benchmark). We perform a 3D field theory lattice simulation for the axion
field with grid size $N^{3}=1536^{3}$ in a radiation-dominated background,
including a bias term in the axion potential, and solve the axion field
equation of motion exactly. This differs from earlier simulation work, with
the promise of potential improvement: e.g. the analysis of metastable DWs in
[12] and [38] is based on a 2D simulation, while the 3D simulation in [47, 48]
employs Higgs DWs with Press-Ryden-Spergel (PRS) [50] approximation. In order
to elucidate the physics of the dynamics of DW evolution, we investigated the
DW radiation mechanisms by capturing and zooming in the snapshots of
animations from our simulation and by analyzing the axion spectrum and zoom-
in. In addition to obtaining results based on numerical simulation, through
analytical fitting, we also present the velocity-dependent one-scale (VOS)
model applicable to the metastable DW evolution. This is a notable extension
of the framework of the VOS model which previously has been widely used to
describe the evolution of other types of topological defects such as cosmic
strings [51, 52] and, only recently a few attempts on stable DWs [47, 53, 48,
54, 55, 56]. By combining numerical and analytical approaches, our analysis
leads to an updated prediction for the spectrum and relic abundance of axions
radiated from DWs, as well as new insights into the evolution of DW
substructures. This study may shed new light on the cosmological implication
of axion topological defects and their role in axion physics at large.
In the following, we will first introduce the axion model and simulation setup
that we adopted. Then we will present the essential results on the dynamics of
axion DWs derived from the simulation, and demonstrate how these can be used
to calibrate the analytical VOS model. Cosmological implications related to
axion DM will be discussed before we conclude.
## II Axion model
We first introduce the benchmark axion model that we consider and the
essentials in our simulation. As a pseudo-Nambu-Goldstone boson, axion is
associated with the angular mode of a complex scalar field whose VEV
spontaneously breaks a global U(1) symmetry. The U(1) symmetry breaking occurs
at a relatively high scale $T\sim f_{a}$ when the radial mode acquires a mass
$m_{R}\sim f_{a}$. The original shift symmetry possessed by the axion is
broken at a much later time $T\sim\Lambda\simeq\sqrt{m_{a}f_{a}}$ (e.g.
$\Lambda_{\rm QCD}$ for QCD axion), when the axion acquires a mass $m_{a}$ and
DW forms. At an even later time when $H\ll f_{a}$, the effective Lagrangian
for axion field $a=a(\textbf{x},t)$ with the radial mode integrated out reads
$\displaystyle\mathcal{L}=|\partial_{\mu}a|^{2}-V(a).$ (1)
We consider a biased potential
$\displaystyle V(a)=\frac{m_{a}^{2}f_{a}^{2}}{N_{\rm
DW}^{2}}\Bigg{[}1-\cos\left(N_{\rm
DW}\frac{a}{f_{a}}\right)+\epsilon\left(1+\cos\frac{a}{f_{a}}\right)\Bigg{]},$
(2)
where $\epsilon\ll 1$ is the bias parameter that causes the DW to collapse. We
consider $N_{\rm DW}=2$, which implies one true vacuum and one false vacuum in
the model 111It is worth mentioning that the bias term in Eq.(2) doesn’t shift
the true vacuum in the axion potential, which is for avoiding the axion
quality problem, see a review in [31].. This is a representative choice that
involves a simple DW structure which eases the simulation analysis and also
allows us to extrapolate our results to the string-wall scenario, which we
will discuss in more detail in the Appendix A.
We estimate the DW surface tension based on the axion potential in Eq.(2) as:
$\displaystyle\sigma_{\rm DW}\simeq\eta_{\rm DW}\,\frac{m_{a}f_{a}^{2}}{N_{\rm
DW}^{2}},$ (3)
where $\eta_{\rm DW}=8$ for the potential in Eq.(2), which we will use in this
study. $\eta_{\rm DW}=8.97(5)$ for QCD axion with pion contribution included
[58]. The DWs become dynamical at cosmic time $t\sim 1/m_{a}$ when the horizon
becomes comparable to the DW thickness $\delta\sim 1/m_{a}$.
## III Simulation
### III.1 Setup
The equation of motion (EoM) of the axion field in a flat homogeneous and
isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) universe is
$\displaystyle\frac{\partial^{2}a}{\partial\tau^{2}}+2\left(\frac{d\hbox{ln}R}{d\hbox{ln}\tau}\right)\frac{1}{\tau}\frac{\partial
a}{\partial\tau}-\frac{\partial^{2}a}{\partial x_{i}^{2}}=-R^{2}\frac{\partial
V}{\partial a},$ (4)
where $R(t)$ is the scale factor, $x_{i}$ is comoving spatial coordinate,
$\tau$ is comoving time, and $\nabla$ is the Laplacian in physical
coordinates. We start our simulations at a time that is slightly earlier than
the DW formation time.
For the initial condition (IC) of the field of our simulation, a random and
uniform distribution of the axion field is consistent with the consequence of
stochastic inflation under the assumption that the axion potential scale
$\sqrt{m_{a}f_{a}}$ is far below the inflation scale $H_{I}$ (see [59] and a
review of the stochastic method [60]). We thus consider a simpler scenario in
that we randomly assign field value $a=0$ or $\pi$ (the two vacuums in the
potential) to realize an unbiased IC where half of the points on the lattice
are in a true vacuum and assume zero initial field velocity $\dot{a}(t_{i})\to
0$. As we will see once the DW network enters the attractive solution, the so-
called scaling regime, the DW network evolution would no longer be sensitive
to IC. This phenomena has been observed in earlier simulations [61, 28, 17,
54, 44], and see [48] for a discussion of the effect of a biased IC on the PRS
DW evolution (and earlier references [49, 62, 63]).
Other simulation setups are as follows. We normalized all parameters according
to $f_{a}\to 1$. The lattice size is $N^{3}=1536^{3}$, and the simulation
period starts from $1/H(t_{i})=R(t_{i})\Delta x_{i}$, and ends at
$1/H(t_{f})=(N/2)\Delta x_{f}$, where $\Delta x_{i}=1$ is initial lattice
spacing, $R(t_{i})=1$ is initial scaling factor, $\Delta x_{f}=R(t_{f})\Delta
x_{i}$ is comoving spacing at the end of simulation, and a radiation
background is assumed with $R(t)\propto t^{1/2}$. We fix the time interval
$\Delta\tau=0.1$ and test convergence by re-running with smaller time
intervals, where $\tau$ is the comoving time. We further fix the physical DW
thickness as
$\displaystyle\delta\sim\frac{1}{m_{a}}=\frac{1}{(N/2)R(t_{i})\Delta x_{i}}.$
(5)
These choices imply that the simulation starts at the time when the horizon
size equals lattice spacing $\Delta x_{i}$, and ends when the horizon expands
to half of the full lattice size. On the other hand, the DW thickness $\delta$
occupies $N/2$ lattice grids at $t_{i}$, then as the coordinate expands, the
simulation ends when $\delta$ occupies two grids. We chose such simulation
setups for the following reasons:
(1) $\delta$ cannot be smaller than the size of two grids for sufficient
resolution of the DW. Lower resolution leads to incorrect and insensible
simulation results such as a frozen DW in the lattice because the gradient
$\nabla^{2}a$ in the equation of motion Eq.(4) would be incorrectly calculated
in the simulation. In addition, a lower resolution would incorrectly induce a
wrong tail in the axion kinetic spectrum around axion momentum of $k\sim
2\pi/\Delta x_{f}$.
(2) We simulated with two types of boundary conditions (b.c.’s), periodic and
symmetric, and investigated the results’ robustness against the choice of b.c.
As the simulation results are expected to be inevitably subject to b.c.
(albeit not significantly as we found), in order to mitigate the effect we
conservatively collect simulation data from the central $1/8$ of the
simulation box and discard the rest. This data collection range equals the
Hubble box size at the end of the simulation.
In order to present a free axion spectrum by filtering out the DW
contribution, we employ a mask function on the axion field as in previous
studies [37, 12] (originally applied in CMB analysis [64]). The method is to
mask $\dot{a}(x)$ by a window function
$\displaystyle\dot{a}(x)\to\theta(x-d)\dot{a}(x),$ (6)
where $x$ is the coordinate that origin at the DW center where
$V(a(x=0))=V_{\rm max}$, $d$ is a mask function parameter, and $\theta(x)$ is
the Heaviside step function. We fix $d=\delta/2$ in our simulation to exclude
the DWs contribution to the power spectrum. But due to the influence on the
DWs exerted by the background axion field, $\delta$ would not be perfectly a
constant. Thus we cannot fully erase the DW contribution to the free axion
spectrum, yet our approach should provide a good estimation. A more effective
algorithm to erase such a contribution may be developed with dedicated future
work. The kinetic power spectrum is found to be insensitive to the choice of
$d$ that is not too far from $\delta$, i.e. $\delta/4\lesssim d\lesssim
2\delta$. We found that applying the mask function on the axion field itself
$a(x)\to\theta(x-d)a(x)$ causes an insensible result on the gradient energy
and potential, i.e. a variation on the blue tail of spectrum ($k\sim 1/m_{a}$)
sensitive to the variation of $d$. This may be caused by the oscillation
behavior of the axion field around the vacuum such as the contribution from
sub-horizon compact DW or oscillons (see the red points at the end of the
simulation, i.e. the far right panel in Fig. 1) that cannot be fully removed
by the mask function. Thus to estimate the total energy of the radiated free
axions we only apply the mask function for the axion kinetic energy and assume
that the free axions are all in harmonic mode i.e. its kinetic energy takes
half of its total energy .
Our DW simulation was run with various simulation conditions and ALP model
benchmarks as follows. We conducted 5 simulations for each benchmark with
$\epsilon\gtrsim 10^{-3}$ (to ensure that all the DWs decayed away by the end
of simulation) while keeping the aforementioned parameters constant as
described in last three paragraphs. Subsequently, based on the simulation
data, we will construct a model for the DW dynamics and then extrapolate it to
lower $\epsilon$ values and a wider range of $m_{a}$ via analyzing the axion
spectrum as well as monitoring the evolution of the DWs and the free axion
background field informed by the snapshots of simulation and the spectrum
analysis in Sec. IV.
Besides the main simulation runs, we also conducted test runs under various
conditions and ALP model benchmarks to ensure that our analysis result would
not be affected by the specific simulation parameters that we have set. In
particular, the test runs are set as the following. We assessed the impact of
varying simulation parameters (with 5 testing runs for each benchmark as well)
such as axion mass $m_{a}$, spanning a range from 0.5 to 2, initial scaling
rate $R(t_{i})$ with values of 0.5, 1, and 2, and $x_{i}$ with values of 0.1,
1, and 10. Additionally, we considered different lattice sizes $N$ (512, 1024,
and 1536) and the mask function parameter $d$ as previously mentioned. As
expected for free axion spectrum as shown in Sec. IV, and consequently, our
conclusions remained unaffected.
### III.2 Application to Other Models
Although we simulated a network for a simple DW model, our results can be
applied to a variety of more complex models if they satisfy the following
conditions:
(1) The DW network has enough time to enter the scaling regime before its
decay. For instance, in our model a large $\epsilon\gtrsim 5\times 10^{-3}$
(see Sec. IV.2) would cause the false vacuum to collapse too early for the DW
area to have time to converge to a constant, i.e. enter the scaling regime
(see Sec. IV.2 and Eq.(10) therein).
(2) Essential properties of the DW should be (approximately) the same as in
our simulation. For instance, the DW thickness $\delta$ should be kept as a
constant during the scaling regime and before the DW starts to decay.
Meanwhile, DW number should be $N_{\rm DW}=2$ as considered in this study.
The first condition eliminates the dependence on the DW initial distribution
effect when applied to different models. The second ensures that the DW
dynamics are congruent with our findings. As an example, in the following, we
explain how our simulation can apply to certain QCD axion models. Firstly, a
simple condition for a QCD model to be mimicked by our DW-only simulation is
for the DW structure to be absent from the model, which can be satisfied in
the scenario of a pre-inflationary PQ symmetry breaking or if the vacuum
manifold after the PQ phase transition is simply connected (see later
discussion in this section and Appendix. A for a more complex case: a possible
application to QCD models with cosmic strings). Secondly, the QCD axion model
needs to have the same $N_{\rm DW}=2$ and the presence of a nonzero $\epsilon$
term in order to avoid the DW over-closure problem. Furthermore, the DW
thickness in the QCD model needs to be effectively constant during the
simulation time window. Consider that unlike in the model we considered in
Sec. VII where $m_{a}$ and thus DW thickness is a constant, in QCD the DW
thickness generally takes a time-dependent form as
$\displaystyle\frac{1}{\delta_{\rm QCD}}\simeq
m_{a}(T)\simeq\left\\{\begin{aligned} m_{a}\left(\frac{\Lambda_{\rm
QCD}}{T}\right)^{4}\;\;\hbox{for}\;\;T>\Lambda_{\rm QCD},\\\
m_{a}\;\;\;\;\;\;\;\;\hbox{for}\;\;T\leq\Lambda_{\rm
QCD},\end{aligned}\right.$ (7)
where the QCD scale $\Lambda_{\rm QCD}=400\,$MeV, $T$ is the cosmic
temperature, and the expression is derived from a diluted instanton gas
approximation [65, 66, 67, 68] (also see the results from lattice simulation
[69, 58]). The QCD axion DW thickness $\delta_{\rm QCD}$ approaches a constant
at the transition time $t_{a}$ when $T\simeq\Lambda_{\rm QCD}$, and
afterwards, the QCD axion DW would evolve as in our simulation.
We did not simulate a time-dependent thickness $\delta_{\rm QCD}$ due to the
computational limitations imposed by the lattice. The DW thickness, which
rapidly shrinks as $\delta_{\rm QCD}\propto R(t)^{-4}$ in Eq.(7), imposes a
significant demand on the evolution time range in our simulation, because the
thickness should be at least larger than the lattice spacing for accurate
resolution. Due to this limitation, we choose to focus on simulating the cases
where $\delta$ can be treated as a constant. In order for our model to
approximate a $\delta_{\rm QCD}$ during the simulation time window, we should
consider a small $\epsilon\lesssim 10^{-4}$ such that the DW can live long
enough to enter the scaling regime after $t_{a}$. We will discuss the concrete
application of this condition on the parameter space in Fig. 15 in Appendix.
B.
In addition to the issue of constant vs. time-dependent DW thickness as
discussed above, another key potential difference between our simple model and
the QCD case is that some QCD axion models may also involve cosmic strings in
the axion topological defect structure, such as in the scenario of post-
inflationary $U(1)$ symmetry breaking, where QCD axion strings persist until
DW formation. In such a case with $N_{\rm DW}=2$, two DWs attach to a single
cosmic string, forming a string-wall network that differs significantly from
what the DW-only structure that we considered in our study. Nevertheless, we
find that the influence of cosmic strings is negligible when the DW tension
dominates the network [70], specifically when the condition
$\displaystyle\sigma_{\rm DW}t/\mu>1$ (8)
is satisfied, where $\mu\simeq 2\pi^{2}f_{a}^{2}\hbox{ln}(tf_{a})$ is the
cosmic string tension. Under this condition, the string-wall structure is well
approximated by our simulation. However, for higher values of $N_{\rm DW}>2$,
where multiple DWs attach to a single string, a more complex scenario arises
with the attachment of multi-DWs. We have chosen to leave the investigation of
such complex scenarios with $N_{\rm DW}>2$ for future work. We will present
the viable parameter space satisfying the condition given in Eq.(8), and
discuss the application to the QCD axion model with cosmic strings in the
Appendix.A.
Furthermore, our decision to focus on the simplified case without string
contribution is also influenced by technical considerations. Due to
limitations in our simulation resources, the lattice size imposes constraints
on extending the simulation period sufficiently to observe DW decay if cosmic
strings are included. The scale hierarchy between the width of the string
($\sim 1/f_{a}$) and the Hubble scale at the time of DW decay prevents us from
adequately observing the network in our simulation with the current lattice
size.
Finally, note that our simulation results not only can apply to the
aforementioned QCD axion models, but also to other axion-like particle models
that satisfy the two conditions that we identified above.
## IV Domain wall dynamics
Figure 1: Visualization of lattice simulation with bias parameter
$\epsilon=0.0013$: snapshots in a time series (left to right:
$m_{a}t=21,43,97,385$). The yellow (blue) region indicates a false (true)
vacuum, and the red region represents DWs. The Hubble volume is shown as a
black cube in the bottom-left corner of each snapshot (see animation for
$\epsilon=0.0012$). The small red dots are defined as sub-horizon compact DW
or oscillon, which are the axion field that oscillates around the false
vacuum, surrounded by DWs (For a zoomed-in simulation for the dissipation of
the small red dots, see: animation link) Figure 2: DW area parameter (defined
in Eq.(10)) as a function of the cosmic time in our simulation, with varying
bias parameter $\epsilon$ (defined in Eq.(2)). Figure 3: Visualization of the
lattice simulation with the bias parameter $\epsilon=0.0012$. The leftmost
figure displays a snapshot of the entire simulation scale, where the domain
wall (DW) is highlighted in red color. The upper row shows a zoomed-in region
of our simulation with a $160^{3}$ lattice, accompanied by a further zoomed-in
time series depicted at the bottom. The lower row comprises smaller lattice
sizes. Both sets of sub-figures encompass a range of features discovered in
this study, and a detailed discussion of these features is provided in the
main text of Section IV.1. Figure 4: A cartoon illustration for the DW
collapse process. Figure 5: A cartoon for DW self-chopping process. Figure 6:
A cartoon for two DWs chopping. This process is rarely observed in the
simulation. Figure 7: A cartoon for DW contraction process with DW velocity
$v$. The Hubble box enlarges over time, while the compact DW is contracting.
The zoomed-in subfigure illustrates the DW tension and pressure that are also
shown in Fig. 8. Figure 8: Force diagram for domain wall tension and vacuum
pressures. This is used to illustrate the origin of the DW flattening motion.
### IV.1 Features Observed in the Simulation
In this subsection, we will discuss the features identified from the snapshots
of our simulations, and will further discuss their corresponding energy
contributions and dynamic behaviors later in Sec. V and Sec. VI. We find 6
distinguishable objects in simulation, and they are connected through 3
different dynamic motions, including their creation, annihilation, and motion.
As illustrated in Fig. 3, the objects observed in the simulation can be
categorized as follows:
(1) Super-horizon sized DWs: represented as the red wall-like structures in
Fig. 1 and Fig. 3, with different shapes (either planar or compact). These
super-horizon sized DWs are formed due to the initial field distribution of
the simulation.
(2) Horizon-sized compact DW: also shown as red wall-like structures in Fig. 1
and Fig. 3, but with a compact geometry. These horizon-sized compact DWs are
formed by the contraction or self-chopping process (which will be discussed)
of super-horizon sized DWs. Such DWs release energy through flattening motion,
self-chopping into smaller compact DWs, and then collapse (to be defined
later).
(3) Sub-horizon compact DWs or oscillons: DWs with typical sizes of $\sim
1/m_{a}$ (in our simulation it is found that larger, sub-horizon sized compact
DW rapidly contract down to the size of $\sim 1/m_{a}$), much smaller than the
horizon scale. These structures are mainly formed through self-chopping due to
the fluctuations on the DW surface, and the collapse of the horizon-sized
compact DWs, see Fig. 3 and the red dots in Fig. 1. Distinguishing between
sub-horizon compact DWs and oscillons is challenging due to limited lattice
resolution, as both structures occupy only a few lattice spacings. Therefore,
sub-horizon compact DW and oscillon are two interchangeable terms in this
study. At the end of our simulation, sub-horizon compact DWs/oscillons are
found to contribute to the residual energy density . However, their
contribution is subdominant when compared to that from free axion fields, such
as axion clouds and mechanical waves (will be introduced next).
(4) Axion clouds: background axion field distributed around the vacua, on
average with relatively large momentum of $k\gtrsim m_{a}$. They are shown as
blue regions in the true vacuum and yellow regions in the false vacuum in Fig.
3. The formation of axion clouds can be induced by heating the background
axion field, i.e. increasing the oscillation amplitude (and thus the energy
density) of the background axion field around the vacua through DW movements,
specifically, processes like flattening and compact DW collapsing, which will
be elaborated on shortly.
(5) Axion Mechanical Wave: the ripple-like structure in Fig. 3, originated
from the axion waves propagating outward from the collapsing DWs (see Fig. 4).
Compared to axion clouds, they have relatively lower momentum $k\lesssim
m_{a}$.
(6) Resonance: the phenomenon where a region of the axion clouds are divided
into small wave-packets as particle-like structures, with a scale of $k\sim
3.68/m_{a}$ (this characteristic scale will be demonstrated by spectral
analysis later in Sec. V). This characteristic $k$ value is obtained by first
visualizing it in spatial dimension and then converting it to momentum space
by Fourier transformation. We have shown the resonance in the lowermost-right
sub-figure in Fig. 3.
These objects are connected to each other via transformative processes (e.g.
creation, annihilation) which can be categorized as the following dynamic
motions as identified from our simulation:
(1) Flattening motion: This DW motion is analogous to laying a piece of paper
flat (for example, as illustrated in Fig. 16), and therefore we refer to this
motion as “flattening”, which originates from the DW tension and vacuum
pressures, see Fig. 8. As a result of such a flattening process, the DW
curvature and surface fluctuation are reduced, resulting in the heating of the
background axion field. Additionally, the flattening process induces the
contraction of compact domain walls, causing larger domain walls to transform
into smaller ones. For instance, a super-horizon-sized compact domain wall
contracts into a horizon-sized domain wall, as illustrated in Fig. 7.
(2) Self-chopping: refers to the phenomenon where a segment of the DW shrinks
and eventually breaks off from the ‘parent’ DW, leading to the splitting of
the DW into two parts. This mechanism plays a crucial role in the DW network
evolution, affecting any size of DW. The upper-row subfigures in Fig.3
illustrate the sub-horizon sized DW self-chopping (the first three subfigures)
and horizon-sized DW self-chopping (the third subfigure) processes, while a
cartoon illustration can be found in Fig. 5. This process is in analogy to
self-intersection in cosmic string dynamics. Note that self-chopping is an
intermediate process of transforming the DW energy from larger DWs to smaller
DWs, which would further decay to the final outcome (mostly free axions)
through the collapse process as defined below. Therefore we do not count self-
chopping as an effective mechanism of DW energy release, unlike the flattening
and collapse processes.
(3) Collapse of horizon-sized compact DWs: the process during the final stage
of DW evolution when a horizon-sized compact DW rapidly contracts and
subsequently collapses while radiating the axion field in the form of
mechanical wave and heating the background axion field. Such a process is
illustrated in the upper-row subfigures in Fig. 3, and also as a cartoon in
Fig. 4.
The complete evolution process of DWs can then be summarized as follows: At
the beginning of simulation, super-horizon-sized DWs transform into horizon-
sized DWs via either contracting (by flattening) or dividing (by self-
chopping). Following this, the horizon-sized DWs undergo a collapse, resulting
in the emergence of axion mechanical waves and axion clouds, while also
releasing a smaller amount of energy in form of subhorizon compact DWs.
Throughout this entire process, the sub-horizon-sized DWs and oscillons
undergo continuous self-chopping, while the background axion field continues
to heat up, resulting in the formation of an axion cloud.
The energy released during the evolution of DWs can be categorized based on
two key mechanisms: flattening and collapsing. In Section VI, we will delve
into a detailed discussion and analysis of the energy release, with a specific
emphasis on these two aspects i.e. collapsing, leading to $\rho_{2}$ in
Eq.(21) and flattening, leading to $\rho_{3}$ in Eq.(27), respectively. Note
that it may not be feasible to precisely separate the energy contributions
arising from these two mechanisms, as both of them lead to the heating of the
background axion field. There are additional contributions from processes such
as the self-chopping and the subsequent decay of sub-horizon compact DWs. But
these influences are comparatively insignificant when compared to the
essential processes mentioned above.
It is worth noting that in the analogous VOS model of cosmic strings, the
majority of energy is released through the formation of loops primarily
generated by the interaction of two long strings [71]. In contrast to the
chopping process of cosmic strings, the probability of chopping due to the
intersection of two DWs (cartoon illustration in Fig. 6) is negligibly low,
and the majority of energy loss is due to the two mechanisms-flattening and
collapse outlined above. The energy contribution from the self-chopping of
sub-horizon compact DWs is negligible when compared to that of horizon-sized
compact DWs, as found in the simulation. Furthermore, it is observed that
horizon-sized and sub-horizon sized compact DWs typically do not originate
from the chopping of two horizon-sized DW, as shown in Fig. 6, rather, from
contraction or self-chopping of larger, super-horizon or horizon-sized DWs.
### IV.2 Scaling Regime
In our simulation, we track the evolution of DWs and the pattern of energy
loss from the DW network. A snapshot of the evolution is shown in Fig. 1, and
for comparison, its counterpart with non-biased potential is shown in Fig. 16
in the Appendix.B. The left-most snapshot is taken as the network enters the
scaling regime when the DWs flatten while expanding. Shortly after its
formation ($\Delta t\lesssim 10/m_{a}$), the network approaches an attractive
solution called the scaling regime while releasing energy through the two
mechanisms which were introduced in the last section: (1) the collapsing of
horizon-sized compact DWs; (2) The flattening motion. Meanwhile, the super-
horizon DWs enter into the horizon continuously, which consequently
compensates for the energy loss due to both mechanisms, so that the DW area
per horizon volume $A_{v}$ remains constant. This constant solution is the
feature of the scaling regime. Such a feature has been identified in
literature [54, 53, 17, 12, 44], and also agrees with our findings as shown in
Fig. 2. At a later time, the DWs start to decay around $t_{\rm decay}$, and
the scaling solution breaks down. In the scaling regime, the DW energy density
takes the following form:
$\displaystyle\rho_{\rm DW}=\gamma^{2}\frac{\sigma_{\rm DW}A_{v}}{t},$ (9)
where $\gamma\simeq 1$ is the Lorentz factor that represents the contribution
of the kinetic energy of the DW, and the DW area parameter is given by
(originally introduced in [50])
$\displaystyle
A_{v}\equiv\frac{A_{w}t}{R(t)V}=0.67^{+0.04}_{-0.04},\;\;\;\;\;\;\;\hbox{for}\;\;\epsilon=0,$
(10)
where $A_{w}$ is the DW comoving area, and $V$ is the comoving volume. The
result largely agrees with the previous simulation studies [12, 44, 38, 44],
but it is about $30\%$ less than the prediction by the simulation assuming PRS
approximation for the DW network [47]. On the other hand, in the metastable DW
scenario, we find
$\displaystyle
A_{v}=c_{1}+c_{2}\,\hbox{Exp}\left[-c_{3}\,\left(\epsilon\sqrt{m_{a}t}\right)^{c_{4}}\right],\;\;\;\;\hbox{for}\;\;\epsilon>0,$
(11)
with
$\displaystyle
c_{1}=0.0088^{+0.0009}_{-0.0009},\;\;\;c_{2}=0.62^{+0.06}_{-0.05},$
$\displaystyle c_{3}=3.98^{+0.40}_{-0.40}\times
10^{6},\;\;\;c_{4}=3.57^{+0.08}_{-0.11},$
where the parameter $c_{1}$ term represents the residual compact DWs and
oscillons at the end of the simulation. As mentioned we cannot distinguish
whether these are sub-horizon (i.e. much smaller than the horizon scale)
compact DWs or oscillons due to the limitation of the simulation period and
resolution. The fitting model Eq.(11) is inspired by field theory analysis
[45] that employs mean-field approximation method and Gaussian ansatz on the
field probability distribution in the limit of a small bias term $\epsilon\ll
1$. Moreover, the parameter $c_{4}\sim 3$ is approximately the spatial
dimension as predicted in [45]. The fitting model in Eq.(11) also fits the
data from other DW simulation studies [63, 49, 48]. As the axion kinetic
energy reduces due to redshifting, the true vacuum pressure force gradually
overcomes the DW tension, which causes energy loss of the DW network. We
define the characteristic decay time of the DW, $t_{\rm decay}$, as when the
DW area $A_{v}$ becomes $\sim 10\%$ of the pre-collapsing value i.e.
$0.1A_{v}(t\to 0)=A_{v}(t_{\rm decay})$. $t_{\rm decay}$ can be estimated by
Eq.(11) as
$\displaystyle t_{\rm decay}$
$\displaystyle\simeq\frac{\epsilon^{-2}}{m_{a}}\left(\frac{c_{\mu}}{c_{3}}\right)^{2/c_{4}}$
(12) $\displaystyle\simeq\frac{\epsilon^{-2}}{m_{a}}(3.22\pm 0.94)\times
10^{-4},$
where the factor
$\displaystyle c_{\mu}=2.32_{-0.60}^{+0.61}.$ (13)
Note that other semi-analytical estimation studies [12, 43] compare the
pressure gap between vacua and use a power-law model to fit their data, and
predict $t_{\rm decay}\propto 1/\epsilon$. This causes a notable difference
from our results in the prediction for the axion relic abundance as shown in
Sec. VII.
### IV.3 Domain Wall Velocity
Figure 9: The average $\gamma v$ versus $m_{a}t$ with varying bias parameter
$\epsilon$. The uncertainty bands are shown as shaded areas. The network
enters the scaling regime at about $m_{a}t\sim 15$, thus the earlier peak at
$m_{a}t\sim 10$ is due to the initial condition. Figure 10: Bias parameter
$\epsilon$ versus axion mass times decaying time $m_{a}t_{\rm decay}$. The red
bars are the decay time calculated by Eq.(12) using the fitting result of this
study with Eq.(11). The black bars are estimated from the peaks in Fig. 9,
which is the time that DW velocity starts deceleration.
In DW dynamics, its velocity plays an important role in its equation of
motion. We measure the velocity by tracking the movement of the maximum of the
axion potential $V(a(x,t))=V_{\rm max}$ in the simulation. The observed DW
velocity is shown in Fig. 9 for varying $\epsilon$. The DW network during the
scaling regime at first decelerates (relative to the initial velocity set by
initial condition) due to the Hubble friction and the DW flattening motion,
then experiences acceleration due to the pressure difference between the true
and false vacua, during the decaying period $t\sim t_{\rm decay}$, then
decelerates again when the network decays away during the later stage of
$t>t_{\rm decay}$. The peak of each curve is thus located at about $t_{\rm
decay}$, see Fig. 10, where we show that the comparison of the decay time
$t_{\rm decay}$ as defined in Eq.(12) and the peak of the observed velocity.
To fit the DW velocity function, we consider the following model:
$\displaystyle\gamma v=\frac{0.923\pm 0.136}{(m_{a}t)^{0.614\pm
0.031}}+\alpha_{v}e^{-(t-t_{\rm decay})^{2}/(2\sigma_{v}^{2})},$ (14)
with
$\displaystyle\alpha_{v}=(0.241\pm
0.039),\;\;\;\hbox{and}\;\;\;\sigma_{v}=(52\pm 20)\frac{1}{m_{a}}.$
The second term in Eq.(14) indicates the effect of the pressure difference
between the true and false vacua in the decay phase, $\alpha_{v}$ represents
the magnitude of the acceleration, $\sigma_{v}$ is the uncertainty in our
observation and the exponential indicates that the acceleration stops at about
$t\simeq t_{\rm decay}$.
This section analyzed the the domain wall’s velocity, which along with earlier
discussions, paves the way for the next section, where we will investigate the
free axion spectrum, resulting from of the decay of the DWs.
## V Free Axion Spectral Analysis
Figure 11: Free axion energy density spectrum $\partial\rho_{a}/\partial k$
as a function of physical momentum $k$, assuming the bias parameter
$\epsilon=0.0011$. The early to later spectrum is shown as blue to red. The
spectrum can be split into three Gaussian distributions as shown as dashed
gray curves corresponding to the 3 contributing terms in Eq.(17). From low $k$
to higher $k$, these three Gaussian distributions present the energy density
from misalignment ($k/m_{a}\lesssim 0.2$), free axions radiated by compact DW
self-chopping, and collapsing ($k/m_{a}\lesssim 1$), and the small structure
axion field such as the axion clouds with the resonance at
($k\sim\mathcal{O}(m_{a})$ ), respectively. The smaller $k<0.01m_{a}$ region
is lacking data because of the simulation lattice size, and higher $k$ has
been cut at Nyquist frequency as discussed in Sec. V.
We discuss the details of the spectral analysis for free axion energy density
in this section, which would be the key input for estimating axion dark matter
relic density in Sec. VII. As discussed in Sec. III, we estimate the total
free axion energy as twice the masked axion kinetic energy. We then compute
the free axion spectrum according to [35, 38] as
$\displaystyle\rho_{a}=\int dk\partial\rho_{a}/\partial
k,\;\;\;\;\hbox{with}\;\;\;\;\rho_{a}=\langle\dot{a}^{2}\rangle,$ (15)
where the axion spectrum $\partial\rho_{a}/\partial k$ is given by
$\displaystyle\frac{\partial\rho_{a}}{\partial k}=\frac{k^{2}}{(2\pi
L)^{3}}\int d\Omega_{k}|\tilde{\dot{a}}(k)|^{2},$ (16)
where $\tilde{\dot{a}}(k)$ is the Fourier transform of $\dot{a}(x)$,
$L=(N/2)R(t)\Delta x_{i}$ is the collected data range, and the momentum
$\vec{k}\equiv\frac{2\pi\vec{n}}{L}$. In addition, we cut off the momentum
that is higher than the Nyquist frequency $k_{\rm Ny}=\pi/(R(t)\Delta x_{i})$
to prevent corrupted data caused by insufficient resolution.
In Fig. 11 we show the free axion energy spectrum with snapshots for the
cosmic time evolution, using different colors. The dark blue curve
($m_{a}t=15$) represents the spectrum when the network just enters the scaling
regime, and the red curve ($m_{a}t=360$) presents the spectrum near the end of
the simulation. We find that the spectrum can be fitted as a sum of three
Gaussian distributions corresponding to distinct physics origins (to be
explained later):
$\displaystyle\frac{\partial\rho_{a}}{\partial k}$
$\displaystyle=\sum_{i=1}^{3}\frac{\partial\rho_{i}(A_{i},k_{i},\sigma_{i})}{\partial
k},$ (17)
where the labels $i=\\{1,2,3\\}$ denote the 3 gray-dashed curves from low k to
higher in Fig. 11, associated with the first, second, and third peak, as
indicated respectively. These curves are parameterized by
$\displaystyle\frac{\partial\rho_{i}(A_{i},k_{i},\sigma_{i})}{\partial
k}\equiv A_{i}\exp^{-(k-k_{i})^{2}/\sigma_{i}^{2}},$ (18)
where we set $k_{1}\simeq 0$ due to the lack of data within the large scale
range of $k\leq 0.02$ as limited by the simulation size of $N=1536$, and
$k_{2}\simeq 0$ since the first peak dominates over the lower $k$ range
associated with the second peak, making it challenging to discern the
contribution of the second peak to the measurement, and
$\displaystyle k_{3}=(3.68\pm 0.03)m_{a},$ (19)
which decreases as $1/R(t)$ due to redshift after DW decay. We fit the
parameters in Eq.(17) with data from each cosmic time snapshot from every
simulation run (we show data from a single run in Fig. 11 as an example), then
analyze their time dependence in the next section. The fitting results for the
parameters and energy densities in Eq.(17) are given in Appendix. B. We have
verified the robustness of the peak at $k_{3}$ by conducting additional test
runs involving variations in the value of $m_{a}$ and lattice spacing, as
outlined in Sec. III.1, but the magnitude of the peak may be subject to the
inherent resolution of the simulation during the later stage of the simulation
(roughly when $t\gtrsim 300/m_{a}$), as $k_{3}$ in Eq.(19) closely approaches
the Nyquist frequency during this later stage.
We observe that $\rho_{1}$ is in reasonable agreement with the energy density
of axions produced through the misalignment mechanism, specifically,
$\rho_{1}\sim m_{a}^{2}f_{a}^{2}/2N_{\rm DW}^{2}$ 222Note that the initial
condition that sets the axion fields on vacuums seems to exclude the axion
energy from the misalignment mechanism. However, it just stores the energy on
the gradient energy budget at the onset of simulation., at the early stage of
simulation, then redshifts like matter. As a result of this redshift, the
spectral line associated with this contribution progressively shifts towards
the lower frequencies over time.
The free axion energy density $\rho_{2}$ in Eq.(15) carries the energy
contribution with the scale $k\lesssim m_{a}$. We attribute this energy
component to axion mechanical wave originated from collapsing of horizon-sized
compact DWs. There are two reasons for this explanation: (1) The energy
spectrum of $\rho_{2}$ is consistent with the scale range of the axion
mechanical wave, i.e. $k\lesssim m_{a}$. (2) $\rho_{2}$ aligns well with the
production process of the compact DW according to the data fitting (see
Eq.(22, and the details will be provided in the next section), as predicted by
the DW VOS model in the context of DW chopping [54]. It is important to note
that while we observe the self-chopping phenomenon (as discussed in Sec.
IV.1), it differs from the definition of two DWs chopping in the VOS model.
Nonetheless, they share a similar energy loss form in the equation of motion,
as we will see in Sec. VI.
The energy density $\rho_{3}$ can be interpreted as the contribution from
axion clouds with a resonance at $k_{3}$. This energy arises from various
processes, as discussed in Sec. IV.1. We anticipate that the primary
contribution to this energy comes from the annihilation of fluctuations on the
DW surface through the flattening motion, because the estimation of the energy
released from these fluctuations aligns well with the energy density
$\rho_{3}$ as demonstrated in Sec. VI.
The energy release mechanisms discussed in Sec. IV.1 occur in both the scaling
regime and decaying period, and the compact DW collapse is more likely to
occur in the decaying period. In other words, the biased potential
significantly accelerates the DW flattening, contraction, and self-chopping.
During the decaying phase, we find that the production of axion clouds
($\rho_{3}$) increases by about $\sim 70\%$, and the radiation for larger
wavelength axion mechanical waves ($\rho_{2}$) is enhanced by about $\sim
30\%$, compared to the scaling regime. The percentage is estimated at the time
$t_{\rm decay}(\epsilon\to 0.0012)$, and by comparing the outcome from the
$\epsilon=0.0012$ and $\epsilon=0$ scenarios.
## VI Model for Domain Wall Evolution
In this section, we present the coupled evolution equations for the energy
densities of the DW network and of the free axions emitted from the DWs. The
two components of axion energy densities sourced by different DW dynamics,
$\rho_{2}$ and $\rho_{3}$, as identified via spectral analysis and monitoring
simulation evolution in Sec. IV.1 and Sec. V, are key inputs in this section.
Here we will quantitatively model these contributions, $\rho_{2}$ and
$\rho_{3}$, respectively, by numerically fitting simulation data. We extract
time-dependent data from simulations in Sec. V, and we further fit them into
the DW evolution equations in this section.
We first generalize the DW evolution equation in the VOS model for a stable DW
network [54, 53] as follows:
$\displaystyle\frac{d\rho_{\rm DW}}{dt}=-(1+3v^{2})H\rho_{\rm
DW}-\frac{d\rho_{\rm DW}}{dt}\Bigg{|}_{\rm to2}-\frac{d\rho_{\rm
DW}}{dt}\Bigg{|}_{\rm to3},$ (20)
where the right-hand side of the equation represents, in order, the redshift
effect, the DW energy loss to $\rho_{2}$ and $\rho_{3}$, respectively. Here we
have reasonably assumed that the final form of DW energy release is free
axions, as gravitational wave radiation albeit inevitable, is expected to be
subleading.
By energy conservation, the latter two terms in Eq. 20 also enter the
evolution equations of the free axions, which is essential for solving the
axion relic abundance. As revealed via the spectral analysis based on
simulation results, free axion production from DWs can be roughly divided into
two kinetic regions associated with distinct DW dynamics, corresponding to
$\rho_{2}$, $\rho_{3}$. It is thus reasonable to consider the evolution of
$\rho_{2}$ and $\rho_{3}$ components separately, then sum up their solution
for the total axion abundance. We first write down the evolution equation for
$\rho_{2}$, which originates from the collapse of compact DWs:
$\displaystyle\frac{d\rho_{2}}{dt}=-3H\rho_{2}+\frac{d\rho_{\rm
DW}}{dt}\Bigg{|}_{\rm to2},$ (21)
where $3H$ reflects the finding that this spectral component of axions
generally has a longer wavelength and behaves like cold matter, and
corresponds to the axion mechanical wave as introduced earlier. The second
term on the right-hand side reflects energy conservation and the
aforementioned reasonable assumption that the DW energy release $100\%$ goes
to axions. As the second term descends from the formation of compact DWs
through DW self-chopping, we can explicitly model its evolution as follows:
$\displaystyle\frac{d\rho_{\rm DW}}{dt}\Bigg{|}_{\rm
to2}=\tilde{c}_{v}v\frac{\rho_{\rm DW}}{L_{\rm DW}},$ (22)
where the self-chopping efficiency parameter, $\tilde{c}_{v}$, can be modeled
as
$\displaystyle\tilde{c}_{v}\equiv
c_{v}\gamma^{c_{\gamma}}\mathcal{A}_{F}^{-c_{\mathcal{A}}},$ (23)
with
$\displaystyle
c_{v}=0.36^{+0.07}_{-0.03},\;\;\;\;c_{\gamma}=3.36^{+0.93}_{-0.58},\;\;\;\;c_{\mathcal{A}}=1.55^{+0.04}_{-0.06}$
(24)
where $L_{\rm DW}=\gamma^{2}\sigma_{\rm DW}/\rho_{\rm DW}$ is the DW
correlation length. The value of the parameters $c_{v}$, $c_{\gamma}$, and
$c_{\mathcal{A}}$ are calibrated by the simulation data. A single run data is
shown in Fig. 11 and Fig. 12. $\mathcal{A}_{F}$ is the area fraction
parameter:
$\displaystyle\mathcal{A}_{F}\equiv\frac{A_{v}(\epsilon)}{A_{v}(\epsilon\to
0)}.$ (25)
where $A_{v}$ is defined in Eq.(11). In the limit of non-relativistic and
stable DW, i.e., $\gamma\to 1$ and $\epsilon\to 0$, Eq.(22) approaches the
expression $c_{v}v\frac{\rho_{\rm DW}}{L_{\rm DW}}$ which was used to describe
the energy loss resulting from the intersection of DWs, leading to the
creation of compact DWs that eventually collapse. This term was originally
introduced by Kibble in the context of the cosmic string network [71], and
later applied to the stable DW VOS model [54] for two DWs chopping. We
slightly modify its physical interpretation to self-chopping and utilize it to
explain our data (see Fig. 12). The factor $\mathcal{A}_{F}^{-3/2}$ captures
the simulation finding that compact DW production is more efficient during the
decay phase, $v\rho_{\rm DW}/L_{\rm DW}$ represents the likelihood of DW self-
chopping, and $\gamma^{c_{\gamma}}$ indicates that an accelerated DW velocity
increases the rate of self-chopping.
We further estimate the solution of $\rho_{2}$ by numerically solving the
axion radiation equation Eq.(21) with Eq.(22), and can be fitted as:
$\displaystyle\rho_{2}\left(\frac{R(t)}{R(t_{\rm decay})}\right)^{3}\simeq
2\tilde{c}_{v}v\rho_{\rm DW}\bigg{|}_{\epsilon\to 0,\;t\to t_{\rm decay}}.$
(26)
The dominant DW contribution to the $\rho_{2}$ component of the axions is from
the era around $t_{\rm decay}$, and the radiated axions redshift like matter
afterward. This solution can be understood as resulting from energy
conservation.
Next, we consider the evolution equation for the component of $\rho_{3}$,
mostly due to the axion clouds production from the DWs flattening motion as
discussed in Sec. IV.1. By analogy of Eq. 21 for $\rho_{2}$, we have:
$\displaystyle\frac{d\rho_{3}}{dt}=-\lambda_{3}H\rho_{3}+\frac{d\rho_{\rm
DW}}{dt}\Bigg{|}_{\rm to3},$ (27)
where $\lambda_{3}$ represents the time-dependent redshift of this component
of axion energy density. As shown in the spectral analysis, at production
these axions are on average (semi-)relativistic with a shorter wavelength,
thus radiation-like and $\lambda_{3}\simeq 4$; then the axions cool down and
become matter-like with $\lambda_{3}=3$ 333The emitted axions can be thought
of as hot axions at first. Our simulations have confirmed that when the
initial conditions of the axion field are such that the time derivative
$\dot{\theta}\gg m_{a}$ and the spatial gradient $\nabla_{x}\theta\gg m_{a}$
(with $\theta=a(x)/f_{a}$). This means that during the early stage the kinetic
and gradient components dominate over the potential energy $V(a)\sim
m_{a}^{2}f_{a}^{2}$. In this scenario, the axion energy at first oscillates
harmonically between the kinetic and gradient components, when the total
energy density dilutes like radiation. As the kinetic energy later becomes
comparable to the potential energy, the axion rolls down to the potential
minimum and starts exhibiting characteristics as a matter-like component.. For
simplicity, we use the following function for $\lambda_{3}$ to fit the
spectrum,
$\displaystyle\lambda_{3}=\left\\{\begin{aligned}
\,4\;\;\;\;\hbox{for}\;\;\;\;t<t_{\rm decay},\\\
\,3\;\;\;\;\hbox{for}\;\;\;\;t\geq t_{\rm decay}.\end{aligned}\right.$ (28)
The evolution of DW energy loss that leads to this component of axion
production can be modeled as (to be explained later):
$\displaystyle\frac{d\rho_{\rm DW}}{dt}\Bigg{|}_{\rm to3}=$
$\displaystyle\;\frac{1}{2}\frac{d}{dt}\left[\rho_{\rm
DW}(1-v^{2})^{c_{f2}}\left(\frac{m_{a}}{H}\right)^{c_{f1}(1-\mathcal{A}_{F})}\right],$
$\displaystyle\equiv$
$\displaystyle\;\frac{1}{2}\frac{d}{dt}\mathcal{F}_{A}(t),$ (29)
where the parameters are calibrated by simulation data as:
$\displaystyle
c_{f1}=0.44^{+0.20}_{-0.20},\;\;\;\;c_{f2}=3.61^{+0.90}_{-0.98}.$ (30)
We also show a fitting result for $\epsilon=0.0012$ in Fig. 13 as an example.
Similar to the case of $\rho_{2}$, the numerical solution of Eq. (27) can be
fitted as
$\displaystyle\rho_{3}\left(\frac{R(t)}{R(t_{\rm
decay})}\right)^{3}\simeq\mathcal{F}_{A}\bigg{|}_{\epsilon\to 0,\;t\to t_{\rm
decay}}.$ (31)
We have chosen the model fitting form given by Eq. (VI) for the following
reasons. Firstly, the energy of the perturbation per unit area of the DW
increases with the scalar (axion) mass $m_{a}$ as estimated in [74].
Additionally, the total area of the horizon-sized DWs within a horizon
decreases as $H$ increases, and it is expected that the energy loss of DWs is
greater for higher overall DW energy density $\rho_{\rm DW}$. These
considerations are captured by the variables $1/H$, and $\rho_{\rm DW}$,
respectively, along with their functional form in Eq. (VI). In addition, the
power of $c_{f1}(1-\mathcal{A}_{F})$ renders the dimensionless parameter
$m_{a}/H$ negligible in the scaling regime, which captures the fact that the
DW fluctuations release energy becomes more significant in the scenario of
metastable DWs (i.e. $\epsilon\neq 0$). We also introduced a simple velocity
dependence to Eq. (VI) as preferred by numerical fitting, which implies that a
significant contribution to $\rho_{3}$ occurs around the peaks shown in Fig.
9, i.e. when $t\sim t_{\rm decay}$.
It is important to note that the DW fluctuation (scalar perturbation)
radiation term as described in [47] represents the axion radiation resulting
from the annihilation of surface fluctuations, which corresponds to $\rho_{3}$
component in this study. They find that the chopping effect 444As mentioned in
Sec. IV.1, the word ’chopping’ is used in [47] for two DW chopping, but we use
’self-chopping’ which was defined in the text and found to dominate over two
DW chopping from our simulation. in the VOS model that results in $\rho_{2}$
in this study is negligible in their simulation. Their conclusion does not
align well with the axion spectrum depicted in Fig. 11 as found from our
simulation. This discrepancy may be attributed to the utilization of the PRS
algorithm [50] in [47], which can inaccurately model the DW dynamics at small-
scale structures, as pointed out in [44].
There are also caveats identified from our detailed analysis that are worth
reiterating. Firstly, $\rho_{3}$ encompasses not only the radiation from the
flattening the surface fluctuations of the DWs, but also the (sub-dominant)
contributions from, for instance, the collapse of horizon-sized compact DW
that also leads to the heating of background axion field, as discussed in Sec.
IV.1. Secondly, in the later stages of the simulation, the characteristic
energy scale of $\rho_{3}$ become close to the Nyquist frequency, which may
result in considerable observational uncertainties, as discussed in Sec. V.
In this section we introduced the coupled evolution equations for DW network
and free axions from DWs, using $\rho_{2}$ and $\rho_{3}$ from spectral
analysis, and provided an estimate for axion production. The DW evolution
equation, considering the redshift effects and energy loss to $\rho_{2}$ and
$\rho_{3}$, demonstrates the relation between DW energy loss and axion
production. Separate equations for $\rho_{2}$ and $\rho_{3}$ capture the
horizon compact DW creation and collapse, and axion cloud production and axion
field resonance, respectively. In the next section, we will apply the results
obtained here for the prediction of $\Omega_{a}$.
Figure 12: The energy density of the second Gaussian fitting function
(related to $\rho_{2}$) as given in Fig. 11 and Eq.(17) where we fix
$\epsilon=0.0012$. The black curve presents the prediction of axion production
model Eq.(21) which implies the energy loss in the DW network through the
horizon compact DW collapse as discussed in Sec. IV.1. Figure 13: The energy
density for the third Gaussian fitting function (related to $\rho_{3}$) as
given in Fig. 11 and Eq.(17), with $\epsilon=0.0012$. The blue curve presents
the prediction of axion production model Eq.(27) which implies the energy loss
in the DW network through the DW flattening motionas discussed in Sec. IV.1.
The vertical line $A_{v}\to 10\%$ corresponds to the time $t_{\rm decay}$ when
$A_{v}$ becomes $10\%$ of the value at scaling regime (Eq. 10).
## VII Cosmological implication
Figure 14: Viable parameter region of axion model considering the DW
contribution to axion relic density as estimated by this work (assuming
$\Lambda=\Lambda_{\rm QCD}$). The white region indicates that the axion relic
abundance is sufficient to account for the observed dark matter as measured by
the Planck Observatory ($\Omega_{\rm DM}=(0.12\pm 0.0012)h^{-2}$) [76], taking
into account both the misalignment mechanism and the DW contribution. The
width of the white region presents the uncertainty associated with
extrapolation, which expands as $\epsilon$ decreases. Above the black-dashed
horizontal line, the DW has not entered the scaling regime before its decay.
The yellow area indicates that the produced axion partially contributes to
dark matter, while the orange area indicates an overproduction of dark matter.
The blue-dashed line represents the prediction of $\Omega_{\rm
DM}=\Omega_{a}^{\rm DW}\propto\epsilon^{-1/2}$ from a previous simulation
study [12]: the area to the lower left of the line indicates overproduction.
The result from [12] is shown as a thin line as the error bar given there is
tiny. The grey/dark grey areas are excluded by BBN constraint and CMB
observation, respectively, as DWs must decay prior to the BBN and CMB eras
($t_{\rm decay}<0.01$s) [77, 78].
In this section, we will estimate the contribution of DWs to the relic density
of axions based on the results obtained in earlier sections and present the
viable parameter space of our model. We will apply our result to the $N_{\rm
DW}=2$ ALP model (see Eq.(2)) with pre-inflationary PQ symmetry breaking (so
that cosmic strings are simply absent) as a concrete example. We then present
an illustrative analysis that includes the potential contribution of cosmic
strings to the axion relic abundance in the Appendix. A.
The contribution of the standard misalignment mechanism to the axion relic
density is found to be negligible compared to the DW contribution in the
parameter space of our interest ($\epsilon\lesssim 10^{-3}$): $\rho_{\rm
mis}/\rho_{\rm DW}<1\%$, where $\rho_{\rm mis}\simeq
m_{a}^{2}f_{a}^{2}/2N_{\rm DW}^{2}$ is the axion energy density from the
misalignment mechanism, and $\rho_{\rm DW}$ is the contribution from DW decay.
We thus neglect its contribution in the subsequent discussions.
The DW contribution to the relic axions is given by the solutions to the
evolution equation of motion Eq.(21) and Eq.(27) along with their numerical
results in Eq.(26) and Eq.(31), respectively. The total axion energy density
is $\rho_{a}=\rho_{2}+\rho_{3}$. In order to estimate the axion relic density
$\Omega_{a}$ from DWs, we numerically fit $\rho_{2}+\rho_{3}$ based on data
points in Fig. 9, and extrapolate the result to lower $\epsilon$’s and a wider
range of $m_{a}$. Our fitting result for the DW contribution to $\Omega_{a}$
is
$\displaystyle\Omega_{\rm a}^{\rm DW}h^{2}\simeq$
$\displaystyle\,0.116\left(\frac{m_{a}}{2\times
10^{-4}\,\hbox{eV}}\right)^{-1.50^{+0.02}_{-0.02}}$
$\displaystyle\times\left(\frac{\Lambda}{400\,\hbox{MeV}}\right)^{4}\left(\frac{\epsilon}{10^{-4}}\right)^{-1.87^{-0.35}_{+0.44}},$
(32)
where the uncertainties are fitted within the $m_{a}$ and $\epsilon$ ranges as
given in Fig. 14. The benchmark example with $\Lambda=\Lambda_{\rm QCD}$ is
shown in Fig. 14, where the parameter region that predicts the observed axion
DM relic density $\Omega_{\rm a}=(0.12\pm 0.0012)h^{-2}$ lies in the white
area. We also considered the BBN constraint $t_{\rm decay}<0.01$s [77, 78],
and the CMB constraint that DWs should decay before photon decoupling. In
addition, the region above the black horizontal dashed line corresponding to
$\epsilon=5\times 10^{-3}$ (also see Fig. 25 in Appendix. B indicates that the
DW network does not have sufficient time to transition into the scaling regime
before its decay. Furthermore, we have fixed $\Lambda=\Lambda_{\rm QCD}$ in
Fig. 14 as a QCD example, but Eq. VII can apply to general ALPs by varying
$\Lambda$, and the constraints shown in Fig. 14 related to axion relic
abundance would be relieved for smaller $\Lambda$’s.
Fig. 14 also includes a comparison between the results from our study and
those from the previous 2D simulation for the metastable DW [12, 38]. We use
the dashed blue curve to represent the prediction 555The authors in [12] used
a different notation compared to our Eq.(2). Here we used a conversion:
$\Xi\simeq\epsilon\frac{m_{a}^{2}}{2f_{a}^{2}N_{\rm DW}^{2}}$, where $\Xi$ is
the bias parameter used in Eq.(3.1) in [12]. of the DW contribution to the
axion relic abundance as presented in [12]. Both studies have technical
limitations that restrict their simulations to relatively large values of
$\epsilon\gtrsim\mathcal{O}(10^{-3})$, and extrapolations are made to smaller
$\epsilon$ and different $m_{a}$ values. Our estimate of the axion relic
abundance for $\epsilon\sim 10^{-4}$ to $10^{-3}$ roughly agrees with that of
[12], but a discrepancy becomes increasingly significant as $\epsilon$
decreases. For example, the DW network produces more axion energy density in
our finding compared to [12] in the range of smaller bias region of
$\epsilon\lesssim 10^{-4}$, while results in less axion energy density for
$\epsilon\gtrsim 10^{-4}$. In [12] the fitting for axion relic density from
DWs is $\Omega\propto\epsilon^{-1/2}m_{a}^{-3/2}$. The discrepancy between
their and our result may arise from the differences in the fitting models
chosen for DW dynamics, especially the DW decay behavior $A_{v}$. This $A_{v}$
controls the energy density of DW and explains its decaying process, and thus
consequently influences the axion production. We adopt the fitting model
described by Eq.(11), whereas [12] employs a power-law form $A_{v}\propto
t^{1-p}$ with a pressure calibration parameter $p$. This power-law model was
investigated in [63, 80]. They analyze the pressure gap between different
vacuums, then conclude that the collapse of DWs occurs when the pressure in
the true vacuum overcomes the one in the false vacuum, which takes place at
$t_{\rm decay}\sim\sigma_{\rm DW}/\Delta V\propto\frac{\epsilon}{m_{a}}$,
where $\Delta V$ represents the difference in potential between the vacua.
However, the fitting model described by Eq.(11) and Eq.(12) in our work
provides a much better fit to our simulation results. These fitting formulae
that we used are inspired by the mean-field approximation method analysis in
[45] as discussed in Sec. II.
## VIII Conclusion
This work presents an updated study on the dynamics and evolution of long-
lived, metastable axion DWs, with a DW number of $N_{\rm DW}=2$ as a
benchmark. The study incorporates 3D lattice simulations and a semi-analytical
approach based on the VOS model. Our analysis includes analyzing the DW
evolution dynamics by monitoring the simulation snapshots, and a detailed
examination of the axion kinetic energy spectrum. We infer the mechanisms of
axion production sourced by the DWs and the corresponding energy loss
mechanisms of the DWs. The contribution to the relic abundance of axions from
the DW is then derived by numerical fitting and extrapolation, and is found to
be significantly greater than that from the misalignment mechanism for a small
bias parameter $\epsilon\lesssim 5\times 10^{-3}$.
Based on the features in the axion energy spectrum obtained from our
simulation (see Sec. IV.1), we identified two distinct components or kinetic
energy regimes of the axions: the shorter wave-length axion clouds with
resonance around $k\sim 3.68\,m_{a}$, with larger impact on the small-scale
region in the axion spectrum; and the longer wave-length axion mechanical
waves with $k\lesssim m_{a}$. These two features are sourced by different DW
dynamics. The axion clouds primarily arise from the flattening motion of the
horizon-scale DWs, which smooths the fluctuations on the DW surface while
heating (i.e. enlarging the oscillation amplitude of) the background axion
fields. On the other hand, the axion mechanical wave is mostly generated by
the collapse of the horizon-sized compact DWs which are formed by self-
chopping or contraction processes of the super-horizon sized DWs.
Based on these identified features and the corresponding sources, we derive
equations governing the evolution of the DWs, built upon the existing VOS
model (for stable DWs) while extending it to incorporate the decay phase of
the DWs. By energy conservation, the evolution equation of the DWs is coupled
to that of the free axions. By solving these equations numerically, we
determine the present-day relic abundance of axions. Our findings align with
some earlier literature in terms of the scaling solution, the DW area $A_{v}$
in Eq.(10) and the self-chopping effect in the VOS model. Meanwhile, notable
differences are identified and thoroughly discussed. Particularly, our
prediction for $\Omega_{a}(m_{a},~{}\epsilon,~{}\Lambda)$ takes a different
form compared to the results found in [12, 38], as shown in Eq.(VII) and Fig.
14. This discrepancy, which is likely caused by the mathematical fitting model
for DW area evolution $A_{v}$, has potentially significant implications for
axion dark matter physics and related experimental probes. Consequently, we
predict a larger $\Omega_{a}$ from the DW decay process in the range of
$\epsilon\lesssim 10^{-4}$ compared to the earlier simulation study [12], and
a smaller $\Omega_{a}$ for larger $\epsilon$.
While we directly simulated a simple axion model using the potential described
in Eq.(2), we have demonstrated that the results can be applied to certain ALP
models and the QCD axion models, with a bias parameter $\epsilon\lesssim
10^{-3}-10^{-4}$ that ensure that the DW thickness can be treated as a
constant before DWs decay away. See discussion in Sec. III.2 for the
conditions of general applicability, and Sec. VII and Appendix. A for
numerical examples of the application to stringless ALP/QCD models and QCD
axion string-wall networks, respectively. In particular, we considered the
benchmark of axion mass in the range of $10^{-6}\leq m_{a}\leq 1~{}$eV with a
fixed DW phase transition scale $\Lambda=\Lambda_{\rm QCD}$ as a benchmark.
Notably, our study improves upon existing literature by including the biased
potential in the 3D field simulation without relying on approximations such as
the PRS algorithm. To ensure efficient simulation with this more accurate
treatment, we focus on the benchmark case of $N_{\rm DW}=2$ and decouple the
radial mode, which is a reasonable assumption for the relevant time range of
DW formation. It is worth exploring further by considering $N_{\rm DW}>2$ and
simulating the full complex scalar field. The dynamics of DWs identified in
this study can provide new insights into the physics of axion-like DWs and
other types of DWs, such as those arising from GUT models. The updated results
on axion DW dynamics presented here are also expected to implications for
astrophysical observables related to axion physics, including gravitational
wave signals from axion DWs and the formation of axion minihalos as relic
overdense energy regions originated from DWs decay.
## Acknowledgement
The authors are supported in part by the US Department of Energy under award
number DE-SC0008541. The simulation in this work was performed with the UCR
HPCC.
## Appendix A The application to QCD axion case with string-wall network
In order to estimate the axion energy density generated by cosmic strings, we
employ a conservative estimation outlined in [32]. They simulated the QCD
axion cosmic string evolution in the scenario with a post-inflationary PQ
symmetry breaking and a short-lived DW ($N_{\rm DW}=1$) that formed at the QCD
phase transition. We considered $N_{\rm DW}=2$ in this study, however, we can
still apply the cosmic string contribution to the axion field as the result
given in [32, 36, 35] to our study. There are two main reasons for this.
First, the contribution from cosmic strings that decayed prior to the QCD
phase transition should match the simulations presented in references [32, 36,
35], as this is sourced before DW formation and thus independent of DW
details. Second, shortly after the QCD phase transition, the DW tension
becomes dominant within the string-DW network, as long as the condition
introduced in Eq. 8 is met. Therefore in this regime, the string contribution
to the axion abundance would be subleading relative to that from the DW, and
the possible variance compared to the $N_{\rm DW}=1$ case would be
insignificant.
As discussed in Sec. III.2, our simulation result can be applied to the DW
domination period in the QCD axion string-wall network if the following two
conditions are met:
(1) The domain wall (DW) becomes dominant within the string-wall network
($t>\mu/\sigma_{\rm DW}$, as shown in Eq.(8)), and subsequently, it has
sufficient time ($\Delta t\sim 10/m_{a}$) to transition into the scaling
regime before its eventual decay. This condition ensures that the influence of
the cosmic string on the network becomes negligible and eliminates sensitivity
to the initial field distribution, thanks to the attractive (scaling) solution
offered by the DW. Furthermore, we specifically consider a scenario with a
domain wall number of $N_{\rm DW}=2$, where two DWs are attached to a single
cosmic string. In this case, the DW tension prevails on $t>\mu/\sigma_{\rm
DW}$, rendering the impact of the cosmic string negligible. Consequently, the
network behaves no differently from a scenario with a single DW ($N_{\rm
DW}=2$ without string) in our simulation. This alignment with the network’s
evolution after the DW tension dominates is consistent with the findings of
our study.
(2) The QCD axion domain wall thickness is time-dependent as shown in Eq.(7)
until cosmic temperature $T=\Lambda_{QCD}$, while our simulation considered a
constant thickness. Therefore in order to be self-consistent, the second
condition is that the DW network should be long-lived enough to enter the
scaling region after $T=\Lambda_{QCD}$.
As will be discussed in the following paragraphs, Fig. 15 shows those two
conditions with a red solid line, and a green dashed line, respectively.
The calculation of the axion energy density produced by cosmic strings in the
references [32, 36, 35] considers two distinct contributions from these cosmic
structures:
(A) Axion radiation during the evolutionary phase, which starts from the
cosmic string formation and ends around the QCD phase transition: this
contribution arises from the emission of axion radiation by cosmic strings as
they evolve. This emission takes place during the earlier phases of cosmic
string evolution. This component is particularly significant in determining
the axion relic abundance. As per the QCD axion model (referenced as Eq. (7)),
the mass of a single axion particle $m_{a}(T)$ is inversely proportional to
the energy levels at earlier times, following the relationship
$m_{a}(T)\propto T^{-4}$. Consequently, axions with lighter masses are
produced during this phase.
(B) Decay of the remaining cosmic strings at QCD phase transition: The second
contribution stems from the complete decay of the cosmic strings that remain
after their evolutionary phase. This decay occurs at the QCD phase transition.
The dominant role in determining the axion relic abundance is played by the
first contribution (A), where lighter axion particles are produced. This is
because the required energy thresholds for axion production are lower during
the earlier stages of the universe. The second contribution (B), involving the
decay of cosmic strings at the QCD phase transition, accounts for about half
of the overall contribution as found in [32].
It is important to note that the specific case being discussed involves a
network of strings attached to walls (referred to as a string-walls network).
Not all of these strings decay immediately during the QCD phase transition.
Some of these strings persist until later stages, and their contribution to
the axion abundance is ignored due to the dominance of domain walls (DWs) at
that later time, see Eq.(8). The remaining strings that have not yet decayed
at the QCD phase transition will mostly eventually decay along with the
decaying domain walls. Some of them will decay before the DW dominates, but
the decay rate should be gradually suppressed by DW tension. The mass of axion
particles in this scenario is higher than the mass during the earlier phases,
thus fewer axion particles are produced. As a result, the estimation presented
in [32], which considered the contributions from (A) and from the immediate
string decay through (B), could potentially predict a higher axion abundance
compared to the string-walls scenario.
As shown in Fig. 15, the prediction for the observed axion DM relic density
$\Omega_{\rm a}=(0.12\pm 0.0012)h^{-2}$ lies in the white area. The BBN and
CMB constraints, scaling region, and a comparison to the early simulation work
[61] are discussed in Fig. 14 and Sec. VII. Furthermore, we present condition
(1) as the red line, and condition (2) has been shown as the green dashed line
in Fig. 15. The prediction of DW-produced axion relic abundance is given in
Eq.(VII).
The estimated contribution from cosmic strings is found to be considerably
higher than the energy contribution of axions resulting from the misalignment
mechanism. Additionally, when domain walls (DWs) have a sufficiently long
lifetime, i.e., $\epsilon\lesssim 10^{-3}$, their contribution can surpass
that of cosmic strings.
Figure 15: Viable parameter region of axion model as estimated by this work,
considering both the DW and cosmic strings’ contribution to axion relic
density ($N_{\rm DW}=2$). The white region indicates that the axion relic
abundance is sufficient to account for the observed dark matter as measured by
the Planck Observatory ($\Omega_{\rm DW}=(0.12\pm 0.0012)h^{-2}$) [76], taking
into account both the misalignment mechanism, cosmic string [32], and the DW.
Above the black-dashed horizontal line, the DW has not entered the scaling
regime before its decay. Above the red-solid curve, if a cosmic string exists,
string tension dominates the network until the DW decay. Below the green-
dashed curve, the thickness of the QCD axion DWs approaches a constant before
its collapse, as given by Eq.(7). The yellow area indicates that the produced
axion partially contributes to dark matter, while the orange area indicates an
overproduction of dark matter. The blue-dashed line represents the prediction
$\Omega_{a}^{\rm DW}\propto\epsilon^{-1/2}$ from a previous simulation study
[12]. The grey/dark grey areas are excluded by BBN constraint and CMB
observation, respectively, as DWs must decay prior to the BBN and CMB eras
($t_{\rm decay}<0.01$s) [77, 78].
## Appendix B Supplemental Data
Figure 16: Virtualization of lattice simulation with no biased ($\epsilon=0$)
potential from early cosmic time to later (left to right). It is more clear to
see the flattening motion of the DW on the right-most and second-right
snapshots, in which the DW flats its surface curvature. Figure 17: Axion
energy density spectrum $\partial\rho_{a}/\partial k$ versus physical momentum
$k$ with no biased potential. Figure 18: Axion energy density spectrum
$\partial\rho_{a}/\partial k$ versus physical momentum $k$ with bias parameter
$\epsilon=0.0012$. Figure 19: The energy density with $\epsilon=0$ for the
third Gaussian fitting function, see Eq.(17). The blue curve presents the
prediction of energy loss model Eq.(27). We excluded the data from the early
time $m_{a}t<100$ because its amplitude is too small, and the fitting result
has big uncertainty. The later time data $m_{a}t>200$ has also been excluded
because the peak of $\rho_{3}$ is out of $k_{\rm N_{y}}$, and we are thus not
able to fit the model nicely. Figure 20: The energy density with
$\epsilon=0.0011$ for the third Gaussian fitting function as given in Eq.(17).
The blue curve presents the prediction of energy loss model Eq.(27). Figure
21: The energy density with $\epsilon=0.0013$ for the third Gaussian fitting
function as given in Eq.(17). The blue curve presents the prediction of energy
loss model Eq.(27). Figure 22: The energy density with $\epsilon=0.0014$ for
the third Gaussian fitting function as given in Eq.(17). The blue curve
presents the prediction of energy loss model Eq.(27). Figure 23: The energy
density of the second Gaussian fitting function as given in Eq.(17) where we
provide a variation of $\epsilon$ as marked in the figure. The black curve
presents the prediction of energy loss model Eq.(21). Figure 24: Averaged DW
velocity with a relativistic factor $<\gamma v>$ versus $m_{a}t$ with
benchmark $\epsilon=0$. The black error bars are observation data in the
simulation. The red area presents a constant fit. The dashed purple curve fits
with whole time ranges. The dashed green curve fits with $m_{a}t\geq 15$ which
corresponds to the scaling regime. And the dashed blue curve fits with
$m_{a}t\geq 20$ that excludes the high-velocity data point at $m_{a}t\sim 15$
where the network just right entered the scaling regime (see Fig. 2, the
$A_{v}$ becomes a constant). Figure 25: Domain wall area parameter to
simulation cosmic time with varying bias parameter $\epsilon$. All the
benchmarks converge to $A_{v}=0.009^{+0.0012}_{-0.0012}$. As the yellow curve,
$\epsilon=0.005$, the DW enters the scaling regime with a short period
$11\lesssim m_{a}t\lesssim 17$, then decays shortly after. We thus conclude
that the DW has enough time to enter the scaling regime for $\epsilon\lesssim
0.005$.
In this appendix, we provide supplementary data for the following purposes:
$\bullet$ We present a simulation animation for a no biased potential
$\epsilon=0$ in Fig. 16. The right-most and the second-right snapshots clearly
present the flattening motion of the DW.
$\bullet$ Axion kinetic energy density spectrum with benchmarks $\epsilon=0$
and $\epsilon=0.0012$ are given in Fig. 17 and Fig. 18, respectively.
$\bullet$ The model fits for $\rho_{3}$ with benchmarks $\epsilon=0$,
$\epsilon=0.0011$, $\epsilon=0.0013$, and $\epsilon=0.0014$ are shown in Fig.
19, Fig. 20, Fig. 21, and Fig. 22, respectively.
$\bullet$ The model fits for $\rho_{2}$ with different benchmarks are shown in
Fig. 23.
$\bullet$ Fig.24 displays the various potential model fit options for the DW
velocity $\langle\gamma v\rangle$ when $\epsilon=0$, which corresponds to
fitting the first term in Eq.(14). The interpolation results for later times
$m_{a}t\gg 1$ are significantly influenced by different assumptions made about
the data, such as when the network enters the scaling regime. In this
particular study, we assumed that the network enters the scaling regime when
$A_{v}$ becomes constant, i.e.$m_{a}t$, as shown in Fig. 2.
$\bullet$ We increase the bias parameter $\epsilon$ from $0.002$ to $0.005$ to
verify a limitation of $\epsilon$ that whether the DW network enters into the
scaling region before its decay in our simulation. Fig. 25.
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|
# Sharp total variation rates of convergence
for fluctuations of linear statistics of $\beta$-ensembles
Jürgen ANGST IRMAR, Université de Rennes 1<EMAIL_ADDRESS>,
Ronan HERRY IRMAR, Université de Rennes 1<EMAIL_ADDRESS>https://orcid.org/0000-0001-6313-1372 , Dominique MALICET LAMA, Université
Gustave Eiffel<EMAIL_ADDRESS>https://orcid.org/0000-0003-2768-0125 and Guillaume POLY IRMAR, Université
de Rennes 1<EMAIL_ADDRESS>
###### Abstract.
In this article, we revisit the question of fluctuations of linear statistics
of beta ensembles in the single cut and non-critical regime for general
potentials $V$ under mild regularity and growth assumptions. Our main
objective is to establish sharp quantitative Central Limit Theorems (CLT) for
strong distances, such as the total variation distance, which to the best of
our knowledge, is new for general potentials, even qualitatively. Namely,
setting $\mu_{V}$ the equilibrium measure, for a test function
$\xi\in\mathscr{C}^{14}$, we establish the convergence in total variation of
$X_{n}=\sum_{i=1}^{n}\xi(\lambda_{i})-n\langle\xi,\mu_{V}\rangle$ to an
explicit Gaussian variable at the sharp speed $1/n$. Under the same
assumptions, we also establish multivariate CLTs for vectors of linear
statistics in $p-$Wasserstein distances for any $p\geq 1$, with the optimal
rate $1/n$, a result which already in dimension one sharpens the speed of
convergence established in the recent contribution [26] as well as the
required regularity on the test functions. A second objective of this paper,
in a more qualitative direction, is to establish the so-called super-
convergence of linear statistics, that is to say the convergence of all
derivatives of the densities of $X_{n}$ uniformly on $\mathbb{R}$, provided
that $\xi\in\mathscr{C}^{\infty}(\mathbb{R})$ and is not too degenerated in
some sense.
As far as quantitative results are concerned, our approach shares similar
tools with the latter reference as we also rely on Stein’s method and
integration by part formalism provided by the Dyson generator $\mathsf{L}$.
Nevertheless, we adopt a different and innovative strategy which consists in
proving that, up to centering, the above linear statistics $X_{n}$ can be
written in the form $\mathsf{L}F_{n}+Z_{n}$ where $Z_{n}$ is some small
remainder and $\Gamma[X_{n},F_{n}]$ converges to some constant as direct
consequences of the law of large numbers. We emphasize that this approach
bypasses the costly requirement of the invertibility of the Markov operator
which is central in Malliavin-Stein’s method and seems robust enough to be
implemented for various models of Gibbs measures. On the qualitative side, the
proof of the super-convergence of densities, for its part, is a consequence of
the existence of negative moments for the carré du champ of linear statistics,
associated with integration by parts techniques.
###### Contents
1. 1 Introduction and statement of the results
2. 2 Stein’s method for Markov diffusive operators
3. 3 The master equation for linear statistics
4. 4 Proofs of the main results
## 1\. Introduction and statement of the results
### 1.1. Overview of our contributions
Given a parameter $\beta>0$ interpreted as an inverse temperature, the
_$\beta$ -ensemble_, with $n$ particles associated with a continuous potential
$V\colon\mathbb{R}\to\mathbb{R}$ such that
$\liminf_{x\to\pm\infty}V(x)-\log\lvert x\rvert>0$, is the Gibbs probability
distribution on $\mathbb{R}^{n}$ associated with the _energy_
$H_{n}(\lambda):=\sum_{i<j}\log\frac{1}{\lvert\lambda_{i}-\lambda_{j}\rvert}+n\sum_{i}V(\lambda_{i}).$
Namely, the $\beta$-ensemble is the probability measure
(1.1)
$\operatorname{\mathbf{P}}(\mathtt{d}\lambda)=\operatorname{\mathbf{P}}_{n,\beta}(\mathtt{d}\lambda):=\frac{1}{Z_{n,\beta}}\mathrm{e}^{-\beta
H_{n}(\lambda)}\mathtt{d}\lambda,$
where $Z_{n,\beta}:=\int\mathrm{e}^{-\beta H_{n}}\mathtt{d}\lambda$ is the
_partition function_ , and $\mathtt{d}\lambda$ stands for the Lebesgue measure
on $\mathbb{R}^{n}$. In this paper, we establish both qualitative and
quantitative Central Limit Theorems (CLT) for _linear statistics_ of
$\beta$-ensemble, which are random variables of the form
$X_{n}:=\sum_{i=1}^{n}\xi(\lambda_{i})-n\int\xi\mathtt{d}\mu_{V},\qquad
n\in\mathbb{N},$
for some non-constant test function $\xi\colon\mathbb{R}\to\mathbb{R}$
satisfying some mild regularity assumptions, where $\mu_{V}$ is the so-called
_equilibrium measure_ , and where the random vector
$(\lambda_{1},\dots,\lambda_{n})$ is drawn from the Gibbs measure
$\operatorname{\mathbf{P}}$.
Our first main results quantifies precisely the fluctuations of such linear
statistics under mild assumptions on the potential $V$ and the test function
$\xi$. Whenever $V\in\mathscr{C}^{7}(\mathbb{R})$ is semi-convex, regular, and
that we are in the single-cut regime (see Assumption 1 below), we indeed
obtain a CLT for linear statistics, in both total variation and
$p-$Wasserstein distances ($1\leq p<\infty$), with the following rates:
* •
(Theorem 1.2) If $\xi\in\mathscr{C}^{{14}}(\mathbb{R})$, we obtain the optimal
rate $\frac{1}{n}$.
* •
(Theorem 1.1) If $\xi\in\mathscr{C}^{6}(\mathbb{R})$, we obtain the almost
optimal rate $\frac{1}{n^{1-\alpha}}$ for all $\alpha>0$.
* •
(Theorem 1.3) If $\xi\in\mathscr{C}^{1,\gamma}(\mathbb{R})$ with
$\gamma\in(0,1)$, we obtain an explicit rate of convergence depending on the
Hölder exponent $\gamma$.
Our bounds in $p-$Wasserstein distance are furthermore obtained in the
multivariate case. In fact, we obtain precise rates of convergence, with
explicit upper bounds involving $n$, $\beta$ and norms of derivatives of
$\xi$. This could allow for limit theorems with varying temperature and/or
with a test functions $\xi$ also depending on $n$.
Our next main result Theorem 1.4 is of qualitative nature and show that the
convergence in law of a linear statistic to a Gaussian automatically upgrades
to a very strong form of convergence. Namely, if the potential $V$ and the
test function $\xi$ are of class $\mathscr{C}^{\infty}$ and if the latter is
not degenerate is some sense, then $(X_{n})$ _super-converges_ to a Gaussian
(see Section 1.3.3 below for definition of the super-convergence). Informally
it means that the density of $(X_{n})$ converges in the
$\mathscr{C}^{\infty}$-topology to that of a Gaussian. In particular, it
entails that the convergence of linear statistics holds in relative entropy.
### 1.2. Motivations
The $\beta$-ensembles, also known as _$1d$ -log gas_, are related to random
matrix theory. We refer to the monographs [33, 20] for details on the subject,
as well as [36] for a broader introduction to Coulomb gases. Indeed, in the
case $\beta\in\\{1,2,4\\}$ and $V(x)=x^{2}$, the corresponding
$\beta$-ensemble describes the joint law of the spectrum of an $n\times n$
random matrix whose density is proportional to $\exp(-\frac{\beta
n}{4}\operatorname{Tr}V(M))\mathtt{d}M$, where $\mathrm{d}M$ is the Haar
measure on the sets of symmetric, hermitian, or symplectic matrices
respectively. The observation that certain $\beta$-ensembles relate to
eigenvalues of Gaussian ensembles goes back at least to [17].
The spectral macroscopic properties of large Gaussian matrices first observed
by [41], actually extend to more general $\beta$-ensembles, and it is by now
well-understood, see for instance [33, Thm. 11.1.2], that
$\frac{1}{n}\sum_{i=1}^{n}\xi(\lambda_{i})\xrightarrow[n\to\infty]{a.s}\int\xi\mathtt{d}\mu_{V},$
where the non-random _equilibrium measure_ $\mu_{V}$ is the unique minimizer
of the _mean-field energy_
$\mathcal{I}_{V}(\mu):=\int V\mathtt{d}\mu-\frac{1}{2}\iint\log\lvert
x-y\rvert\mu(\mathtt{d}x)\mu(\mathtt{d}y).$
In this article, our goal is to study the fluctuations associated with the
above “law of large numbers”, with different motivations that we detail below.
#### 1.2.1. Sharp quantitative Central Limit Theorems
The seminal work of [24] shows that, in the matrix case and under suitable
assumptions on the potential $V$ and the test function $\xi$, the fluctuations
of $\sum_{i=1}^{n}\xi(\lambda_{i})$ around the value
$n\int\xi\mathtt{d}\mu_{V}$ are Gaussian. The quantitative aspect of this
convergence has attracted a lot of attention in the last decade: $i)$ in [13],
total variation bounds are provided in the matrix case via second order
Poincaré inequalities; $ii)$ in [3], still in the matrix case, near optimal
rates of convergence are derived for Kolmogorov distance; $iii)$ for general
$\beta$-ensembles, polynomial rates of convergence are obtained for
$2-$Wasserstein distance in [26] using a variant of Stein’s method due to
Meckes. However, in the last reference, the rates obtained are at most
$O(n^{-2/3+\varepsilon})$ for general potential and of near optimal order
$O(n^{-1+\varepsilon})$ only in the quadratic case. Our primary motivation in
this work is to derive total variation estimates, which is a much stronger
distance than the Wasserstein ones, but also to get optimal rate of
convergence of order $O(n^{-1})$. We also provide sharp rates of convergence
in Wasserstein metric in both univariate and multivariate settings, and our
findings require less regularity on the test functions than previous results
on this question, see [26, 5].
#### 1.2.2. Implementing a robust Stein’s approach for $\beta$-ensembles
Historically, Stein’s method [38] has proved its remarkable efficientcy to
obtain quantitative CLTs in strong probabilistic metrics such as total
variation or relative entropy. Despite numerous applications to various
probabilistic models such as: (a) Erdös–Rényi random graphs [35]; (b)
infinite-dimensional Gaussian fields [32]; (c) Poisson point processes [34];
(d) free probability [19]; to the best of our knowledge, Stein’s method to
study $\beta$-ensemble has only been implemented in [26, 21], through a
variant due to Meckes [30], which in this context consists in approaching
linear statistics by infinite sums of approximate eigenvectors of the Dyson
generator. This demanding task requires strong regularity assumptions on the
test function $\xi$ and prevents one to both get optimal rates and to handle
total variation metric. A second motivation for this works is to develop a
Stein’s method for $\beta$-ensembles, sufficiently robust and general to be
possibly applied to other models in statistical physics involving Gibbs
measures.
#### 1.2.3. Super-convergence phenomenon
For sums of independent and identically distributed random variables, due to
the convolution structure and provided the common law has some small initial
regularity, the regularity in fact improves along the classical CLT. In
particular, it is possible to reinforce the classical convergence in law to
the get the $\mathscr{C}^{\infty}$ convergence of densities, see [28]. This
particular behavior also appears in the free CLT [8], from which the
terminology of _super-convergence_ proceeds. Recently, three of the authors
have revisited this regularization phenomenon on Wiener chaoses [23] and for
quadratic forms [22]. In the setting of $\beta$-ensembles, despite the linear
nature of the statistics in consideration, no convolution structure arises due
to the interaction between the particles. Establishing better-than-expected
limit theorems in this dependent setting serves as the third motivation of
this article.
### 1.3. Statement of the main results
Let us now precise our assumptions, fix our notations and state formally our
main results.
#### 1.3.1. Assumption and notations
In the rest of the paper, we will always assume that the potential $V$
satisfies the following conditions. Note that the latter are classical, in
particular they coincide with the assumptions in [26].
###### Assumption 1.
The potential $V$ and associated equilibrium measure $\mu_{V}$ are such that
* •
(Smoothness) $V\in\mathscr{C}^{7}(\mathbb{R})$.
* •
(Single-cut) The equilibrium measure $\mu_{V}$ is supported on the interval
$\Sigma_{V}:=[-1,1]$.
* •
(Semi-convexity) $\inf_{\mathbb{R}}V^{\prime\prime}>-\infty$.
* •
(Regularity) The measure $\mu_{V}$ has a positive density with respect to the
semi-circle law.
Recall that the measure $\mu_{V}$ is here the unique minimizer of the mean-
field energy $\mathcal{I}_{V}$. In particular, it satisfies the Euler–Lagrange
equation, for some constant $C_{V}\in\mathbb{R}$:
$V(x)-\int\log(|x-y|)\mu_{V}(\mathtt{d}y)=C_{V},\qquad\forall x\in\Sigma_{V}.$
Under the above assumptions, the equilibrium measure $\mu_{V}$ admits a
density with respect to the semi-circular law
$\mu_{sc}(\mathtt{d}x):=\frac{2}{\pi}\sqrt{1-x^{2}}\mathds{1}_{[-1,1]}(x)\mathtt{d}x$,
namely setting $\rho(\mathtt{d}x):=\frac{\mathtt{d}x}{\pi\sqrt{1-x^{2}}}$
$\mu_{V}(\mathtt{d}x)=S(x)\mu_{sc}(\mathtt{d}x),\quad\text{where}\quad
S(x):=\frac{1}{2}\int_{-1}^{1}\frac{V^{\prime}(x)-V^{\prime}(y)}{x-y}\rho(\mathtt{d}y).$
As already mentioned, our main objective is to provide (multivariate)
qualitative and quantitative CLTs for linear statistics of $\beta$-ensembles.
We consider test functions
$\xi_{1},\dots,\xi_{d}\colon\mathbb{R}\to\mathbb{R}$ and we define the random
vector $X=(X_{1},\dots,X_{d})$ by
$X_{k}:=\sum_{i=1}^{n}\xi_{k}(\lambda_{i})-n\int\xi_{k}\mathtt{d}\mu_{V},\quad
1\leq k\leq d.$
For such a vector $X$, as $n$ goes to infinity, following Equations $(1.10)$
and $(1.11)$ in [26], the candidate limiting mean $m=(m_{i})_{1\leq i\leq d}$
and covariance matrix $C=(c_{i,j})_{1\leq i,j,d}$ are given by
$m_{i}:=\left(\frac{1}{2}-\frac{1}{\beta}\right)\left[\frac{\xi_{i}(-1)+\xi_{i}(1)}{2}-\int_{\Sigma_{V}}\xi_{i}(x)\rho(\mathtt{d}x)-\frac{1}{2}\int_{\Sigma_{V}^{2}}\frac{S^{\prime}(x)}{S(x)}\frac{\xi_{i}(x)-\xi_{i}(y)}{x-y}\rho(\mathtt{d}y)\mu_{sc}(\mathtt{d}x)\right],$
$c_{ij}:=\frac{1}{2\beta}\int\frac{\xi_{i}(x)-\xi_{i}(y)}{x-y}\frac{\xi_{j}(x)-\xi_{j}(y)}{x-y}(1-xy)\rho(\mathtt{d}x)\rho(\mathtt{d}y).$
The above matrix $C$ is well-defined as soon as the $\xi_{i}$’s are
$1/2$-Hölder continuous. We shall see in Section 4.1.1 below that $C$ is
indeed a covariance matrix as soon as
(1.2) $1,\xi_{1},\dots,\xi_{d}\ \text{are linearly independent}.$
For an open set $U\subset\mathbb{R}$, we write $\mathscr{C}^{r}(U)$ for the
space of $r$ times continuously differentiable functions on $U$ with bounded
derivatives. We endow it with the Banach norm
$\lVert\xi\rVert_{\mathscr{C}^{r}(U)}:=\max_{r^{\prime}\leq r}\sup_{x\in
U}\lvert\xi^{(r^{\prime})}(x)\rvert.$
When $\xi=(\xi_{1},\dots,\xi_{d})$, we also write
$\lVert\xi\rVert_{\mathscr{C}^{r}(U)}=\sum_{k=1}^{d}\lVert\xi_{k}\rVert_{\mathscr{C}^{r}(U)}.$
We write indifferently $\lvert\cdot\rvert$ for the absolute value of a real
number, the Euclidean norm of a vector, or the Euclidean norm of a square
matrix, also known as its Hilbert–Schmidt norm. For a square symmetric matrix
$A$, we write $\lVert A\rVert_{op}$ for its _operator norm_ , that is its
largest singular value. Given two random variables $X$ and $Y$, the _$p-$
Wasserstein distance_ is
$\mathbf{W}_{p}(X,Y):=\inf\left\\{\operatorname{\mathbf{E}}\left[\lvert
X^{\prime}-Y^{\prime}\rvert^{p}\right]:X^{\prime}\overset{\operatorname{\mathbf{law}}}{=}X,Y^{\prime}\overset{\operatorname{\mathbf{law}}}{=}Y\right\\}.$
The Wasserstein distances only depends on $X$ and $Y$ through their respective
laws but we favor a probabilistic notation more suited for the approximation
theorems we establish. The $p-$Wasserstein distance induces a topology
corresponding to convergence in law together with convergence of the $p$-th
moment [39, Thm. 6.9]. We also work with the total variation distance of real-
valued random variables
$\begin{split}\mathbf{TV}(X,Y)&:=\sup\left\\{\operatorname{\mathbf{P}}[X\in
B]-\operatorname{\mathbf{P}}[Y\in B]:B\ \text{Borel}\right\\}.\end{split}$
Let us finally define
$\displaystyle A_{\beta}:=\left\lparen\frac{1}{\beta}\lVert
C^{1/2}\rVert_{op}\lvert C^{-1}\rvert\lVert
N\rVert_{L^{p}}+\left\lvert\frac{1}{2}-\frac{1}{\beta}\right\rvert\lVert
C^{1/2}\rVert_{op}\right\rparen;$ $\displaystyle
a_{\beta}:=\left\lparen\frac{1}{\beta}\frac{1}{\sigma^{2}}+\left\lvert\frac{1}{2}-\frac{1}{\beta}\right\rvert\frac{\sqrt{\pi}}{2}\right\rparen.$
#### 1.3.2. Quantitative CLT in total variation and Wasserstein distances
We can now state our main results concerning the quantitative behavior of the
fluctuations of linear statistics of $\beta$-ensembles. The following bounds
are optimal or nearly optimal depending on the regularity of the test
functions.
###### Theorem 1.1.
Let us fix $1\leq p<\infty$ and $\alpha>0$. There exists $K_{p,\alpha}>0$ such
that for all test functions
$\xi_{1},\dots,\xi_{d}\in\mathscr{C}^{{6}}(\mathbb{R})$ satisfying 1.2, we
have, with $N$ a standard Gaussian in $\mathbb{R}^{d}$
$\mathbf{W}_{p}(X,C^{1/2}N+m)\leq
K_{p,\alpha}A_{\beta}\frac{\lVert\xi\rVert_{\mathscr{C}^{6}(\mathbb{R})}+\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}^{2}}{n^{1-\alpha}}.$
In the univariate case $d=1$, with $\sigma:=\sqrt{c_{11}}$ and $m=m_{1}$, we
also have
$\mathbf{TV}(X,\sigma N+m)\leq
K_{1,\alpha}a_{\beta}\frac{\lVert\xi\rVert_{\mathscr{C}^{6}(\mathbb{R})}+\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}^{2}}{n^{1-\alpha}}.$
Assuming more regularity on the test functions, we obtain an optimal rate of
convergence.
###### Theorem 1.2.
For $1\leq p<\infty$ and $\alpha>0$, there exists $K_{p,\alpha}>0$ such that
for all test functions
$\xi_{1},\dots,\xi_{d}\in\mathscr{C}^{{14}}(\mathbb{R})$ satisfying 1.2, we
have, with $N$ a standard Gaussian in $\mathbb{R}^{d}$
$\mathbf{W}_{p}(X,C^{1/2}N+m)\leq
K_{p}A_{\beta}\frac{\lVert\xi\rVert_{\mathscr{C}^{14}(\mathbb{R})}+\lVert\xi\rVert_{\mathscr{C}^{7}(\mathbb{R})}^{2}}{n}.$
In the univariate case $d=1$, with $\sigma:=\sqrt{c_{11}}$ and $m=m_{1}$, we
also have
$\mathbf{TV}(X,\sigma N+m)\leq
K_{1}a_{\beta}\frac{\lVert\xi\rVert_{\mathscr{C}^{14}(\mathbb{R})}+\lVert\xi\rVert_{\mathscr{C}^{7}(\mathbb{R})}^{2}}{n}.$
We also prove a theorem for functions of lower regularity. For
$\gamma\in(0,1)$, we write $\xi\in\mathscr{C}^{1,\gamma}(\mathbb{R})$,
provided $\xi\in\mathscr{C}^{1}(\mathbb{R})$ and it satisfies
$\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})}:=\lVert\xi\rVert_{\mathscr{C}^{1}(\mathbb{R})}+\sup_{x\neq
y}\frac{\lvert\xi^{\prime}(x)-\xi^{\prime}(y)\rvert}{\lvert
x-y\rvert^{\gamma}}<\infty.$
###### Theorem 1.3.
Let $\xi\in\mathscr{C}^{1,\gamma}(\mathbb{R})$ for some $\gamma\in(0,1)$.
Setting $\sigma:=\sqrt{c_{11}}$ and $m=m_{1}$, and for any
$a<\frac{\gamma}{6+\gamma}$, we have
$\mathbf{TV}(X,\sigma N+m)\leq
K_{a}a_{\beta}\frac{\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})}}{n^{a}}.$
#### 1.3.3. Super-convergence
We now turn to our qualitative results and the reinforcement of the mode of
convergence is the smooth case. We say that a sequence $(X_{n})$ _super-
converges_ to a non-degenerate Gaussian $N\sim\mathcal{N}(m,\sigma^{2})$ with
density $\varphi_{m,\sigma^{2}}$ provided that, for all $r\in\mathbb{N}$,
there exists $n_{0}\in\mathbb{N}$ such that for $n\geq n_{0}$, the law of
$X_{n}$ admits a density $\varphi_{n}\in\mathscr{C}^{r}(\mathbb{R})$, and
$\limsup_{n\to\infty}\lVert\varphi_{n}-\varphi_{m,\sigma^{2}}\rVert_{\mathscr{C}^{r}(\mathbb{R})}=0.$
Regarding linear statistics, we obtain that convergence in law can easily be
upgraded to super-convergence. For $\alpha>0$, we say that
$\xi\in\mathscr{C}^{1}(\mathbb{R})$ is _$\alpha$ -regular_ provided that
$\mathsf{Leb}(x\in\mathbb{R}^{d}:|\xi^{\prime}(x)|\leq\epsilon)\lesssim\epsilon^{\alpha}.$
It is in particular true for any polynomial function $\xi$.
###### Theorem 1.4.
Let $\xi\in\mathscr{C}^{\infty}(\mathbb{R})$ be $\alpha$-regular such that
$(X_{n})$ converges in law to a Gaussian variable
$N\sim\mathcal{N}(m,\sigma^{2})$. Then, $(X_{n})$ super-converges to $N$.
### 1.4. Comparison with existing results
#### 1.4.1. Gaussian fluctuations for linear statistics
The mathematical study of fluctuations of linear statistics around their
equilibrium starts with the seminal work [24]. His method consists in writing
the Laplace transform of a linear statistics as the ratio of a partition
function associated with a perturbed potential by that associated with the
initial potential. Interestingly, the perturbation involves the so-called
master operator, noted $\Theta_{V}$ in the sequel, see section 3.1 for precise
definition. This operators plays also a central role in our approach. The
method developed in [24] has been continued in [25, 37, 9, 10] to obtain
qualitative CLT with increasing levels of generality, allowing, for instance,
results in the multi-cut regime. All the above results require the potential
$V$ to be analytic. Using the same approach on the Laplace transform, [5]
proves a CLT when $V\in\mathscr{C}^{5}(\mathbb{R})$ and
$\xi\in\mathscr{C}^{3}_{c}(\mathbb{R})$. To the best of our knowledge, these
are the best results available regarding the regularity of $V$ and $\xi$. In
contrast, we obtain a CLT for $V\in\mathscr{C}^{7}(\mathbb{R})$ but allowing
to lower the regularity to $\xi\in\mathscr{C}^{1,\gamma}(\mathbb{R})$. Our
results are also limited to the single-cut case. We stress that in the multi-
cut regime, non-Gaussian fluctuations are known, and additional compatibility
conditions on $\xi$ are required to ensure a CLT. We also mention [6] that
derives a CLT for linear statistics at all meso-scales, that is for linear
statistics where the particles are rescaled by a factor $n^{-\alpha}$ for
$\alpha\in(0,1)$. Even if our main result allows to recover CLT at some meso-
scale, we cannot reach their full range with our method.
#### 1.4.2. Stein’s method and $\beta$-ensemble
To the best of our knowledge, the reference [26] is the first to implement
Stein’s method in the setting of general $\beta$-ensembles and to provide
quantitative bounds in $2-$Wasserstein distance which are near optimal for
$V(x)=x^{2}$ but sub-optimal for more general potentials. Their method works
for linear statistics that are close to be eigenvalues of the Dyson generator,
and technically completing their requires, among others, the tour de force of
diagonalizing the so-called master operator. Relying on the concept of
exchangeable pairs which plays an important role in Stein’s method (see e.g.
[30]), quantitative CLTs for linear statistics of Haar distributed random
variables on compact classical groups are established in [16, 15] and later
on, in the context of circular $\beta$-ensembles, in [40] where exchangeable
pairs are built through the $n$-dimensional circular Dyson Brownian motion.
Finally, the contribution [21] which builds upon the techniques given in [26]
to provide quantitative (near optimal) CLTs in the high temperature regime.
Let us also mention the seminal article [13]. There, the author introduced a
variation around Stein’s method, based on _second-order Poincaré inequality_
in order to study fluctuations of eigenvalues of matrices with random
coefficients, possibly not identically distributed, whose distributions admit
suitable densities. It is somewhat complicated to compare the results there
with ours, simply because for general potentials $V$ we cannot a priori
interpret $\beta$-ensembles as spectrum of random matrices with independent
coefficients. Besides, [13, Thm. 4.2] obtains convergence to a Gaussian of
random variables of the form $\operatorname{Tr}A_{n}^{p_{n}}$ where $(A_{n})$
are some random matrices as above, and $p_{n}=o(\log n)$. Note that, in
comparison, in our context, it should be possible to get CLTs for
$\xi_{n}(x)=x^{p_{n}}$ when $p_{n}$ grows polynomially although the models are
pretty different and only coincide for $V(x)=x^{2}$.
We would like to point out that despite the aforementioned articles and ours
being inspired by the philosophy of Stein’s method, they strongly differ in
the way Stein’s method is implemented. Indeed, [26] relies on ideas coined in
[30] using exchangeable pairs while our builds on a novel refinement of the
well established Malliavin Stein’s approach, which allows to handle random
variables which are not in the image of the underlying Markov generator.
Indeed, whenever $X$ belongs to the image of $\mathsf{L}$ one can directly
provide a Stein’s kernel by setting
$\tau(x)=\mathbb{E}\left[\Gamma\left[X,-L^{-1}X\right]\right]$ which fulfils
the equation $\mathbb{E}\left[\phi^{\prime}(X)\tau(X)-X\phi(X)\right]=0$. The
great advantage of this point of view is to control the total variation
distance (among others) by $\mathbb{E}\left[|1-\tau(X)|\right]$ but it
requires the costly assumption that $\mathsf{L}$ is invertible which is not
true in general. To overcome this obstacle, we notice that it is enough to
show that $X$ is near $\text{Im}(Z)$ in the following sense: $X=LF+Z$ for $Z$
small in probability. This simple remark, which is new to the best of our
knowledge, increases considerably the applicability of the above Malliavin-
Stein’s approach and still enables to provide total variation bounds but is a
priori not sufficient to build a Stein’s kernel. While this being noticed, our
strategy then consists in proving that any linear statistics is close to the
range of $\mathsf{L}$, a step which proceeds from the invertibility of the
master operator. As a result, our findings thus refine those of [26] by
providing optimal rates of convergence for stronger metrics such as the total
variation distance and we believe that our approach could also be successfully
implemented in the framework of circular $\beta$-ensembles and Haar
distributed matrices on compact groups.
#### 1.4.3. Super-convergence
To the best of our knowledge, in the context of random matrices or
$\beta$-ensembles, the question of establishing convergence in metrics
stronger than total variation or Wasserstein distances has not been considered
yet. We deploy here ideas that are classical in the framework of Malliavin
calculus and stochastic analysis and which merely consists in establishing
negative moments for the carré-du-champ operator applied to the considered
linear statistics. Then, relying on integration by parts techniques, we are
able to prove strong forms of convergence for the densities of the underlying
sequences of random variables.
### 1.5. Outline of the proofs an plan of the paper
Let us give here more details on our strategy of proof and on the plan of the
paper.
#### 1.5.1. The generator of the Dyson Brownian motion
The overall strategy behind our quantitative and qualitative estimates
leverage the characterization of the $\beta$-ensemble as the unique invariant
distribution of the Dyson Brownian motion. Namely, consider the _generator of
the Dyson Brownian motion_
(1.3) $\mathsf{L}=\mathsf{L}_{n,\beta,V}=:=\Delta-\beta\nabla
H\cdot\nabla=\sum_{i=1}^{n}\partial_{\lambda_{i}}^{2}-\beta
n\sum_{i=1}^{n}V^{\prime}(\lambda_{i})\partial_{\lambda_{i}}+\frac{\beta}{2n}\sum_{i\neq
j}\frac{\partial_{\lambda_{i}}-\partial_{\lambda_{j}}}{\lambda_{i}-\lambda_{j}}.$
The operator $\mathsf{L}$ is the diffusive Markov generator canonically
associated with the $\beta$-ensemble. We have indeed the following
characterization of the $\beta$-ensemble Gibbs measure
$\operatorname{\mathbf{P}}$:
$\tilde{\operatorname{\mathbf{P}}}=\operatorname{\mathbf{P}}\Leftrightarrow\left\lparen\tilde{\operatorname{\mathbf{E}}}\mathsf{L}F=0,\
\forall
F\in\mathscr{C}^{\infty}_{c}(\mathbb{R}^{n})\right\rparen\Leftrightarrow\left\lparen\tilde{\operatorname{\mathbf{E}}}\left[F\mathsf{L}G\right]=\tilde{\operatorname{\mathbf{E}}}[G\mathsf{L}F],\
\forall F,G\in\mathscr{C}^{\infty}_{c}(\mathbb{R}^{n})\right\rparen.$
The differential structure induced by $\mathsf{L}$ on $\mathbb{R}^{n}$,
technically called a _Dirichlet structure_ , comes with a _carré du champ_
operator
$\Gamma[F,G]:=\nabla F\cdot\nabla G,\qquad
F,\,G\in\mathscr{C}^{1}(\mathbb{R}^{n}).$
We have the following integration by parts formula
(1.4)
$\operatorname{\mathbf{E}}\left[\Gamma[F,G]\right]=-\operatorname{\mathbf{E}}\left[F\mathsf{L}G\right],\qquad
F,\,G\in\mathscr{C}^{\infty}_{c}(\mathbb{R}^{n}).$
#### 1.5.2. $\Gamma$-Stein’s method for $\beta$-ensemble
At an informal level, the data of a diffusive Markov generator generally
combines well with Stein’s method to provide quantitative bounds for normal
approximation. Provided $\ker\mathsf{L}$ is limited to constant functions,
$\mathsf{L}$ is invertible on mean-zero functions, and the celebrated
_$\Gamma$ -Stein_ approach [32, 34, 1, 27], or _Malliavin–Stein_ approach in
the setting of Gaussian fields or Poisson point processes, yields that the
variance of $\Gamma[X,-\mathsf{L}^{-1}X]$ controls the Gaussian fluctuations
of $X$, in total variation or Wasserstein distance. However, in the case
$\beta$-ensembles, the operator $\mathsf{L}$ is in general _not invertible_.
To overcome this difficulty, we amend the classical $\Gamma$-Stein approach.
Intuitively, whenever $X=\mathsf{L}F$ for some $F$, $\mathsf{L}^{-1}X$ makes
sense despite the non-invertibility of $\mathsf{L}$. In this case, it is
natural to expect that $\Gamma[X,-\mathsf{L}^{-1}X]=\Gamma[X,-F]$ controls the
Gaussian fluctuations of $X$. In the next Section 2 and more precisely in
Theorem 2.1 below, we formalize this intuition by proving quantitative bounds
for the normal approximation, in Wasserstein distance and total variation, of
random variables of the form $X=\mathsf{L}F+Z$. Provided, $Z$ is small and
that $\Gamma[X,-F]$ is close to a constant $\sigma^{2}$, then $X$ is close to
a Gaussian with variance $\sigma^{2}$. This theorem should not come as a
surprise to Stein’s method aficionados and its proof is rather
straightforward. However, this observation has, to the best of our knowledge,
never been remarked and this “almost invertibility” decomposition is precisely
what allows us to provide an efficient proof. We stress that our bounds hold
for generic abstract diffusive Markov generators and we believe it could prove
useful in other statistical physics models where the Gibbs measure has an
explicit density.
#### 1.5.3. The master equation
The next Section 3 of the paper consists in checking that the Dyson generator
is indeed in line with the global strategy developed in Section 2. In order to
apply our abstract bound, we need to show that our linear statistics $X$ is of
the form $\mathsf{L}F+Z$. With the notations of Section 3, using mostly
algebraic manipulations together with the minimality of $\mu_{V}$ with respect
to the mean-field energy $\mathcal{I}_{V}$, we show, in Theorem 3.8, that
whenever $F:=\sum_{i=1}^{n}f(\lambda_{i})$ is a linear statistic, setting
$m_{f}:=\left\lparen\frac{1}{2}-\frac{1}{\beta}\right\rparen\langle
f^{\prime\prime}|\mu_{V}\rangle$ which is such that the term in parenthesis
below is asymptotically centered, we have
$\frac{1}{n\beta}\mathsf{L}F=\left(\sum_{i=1}^{n}(\Theta_{V}f^{\prime})(\lambda_{i})-n\int(\Theta_{V}f^{\prime})\mathtt{d}\mu_{V}-m_{f}\right)+\frac{Z}{\beta},$
where $Z$ is a quadratic (which should be thought as a remainder) term and
$\Theta_{V}$ is the so-called _master operator_. In particular, under
Assumption 1, $\Theta_{V}$ is invertible in a neighborhood of $[-1,1]$.
Namely, there exists an open neighborhood $U$ of $[-1,1]$, such that given
$\xi\in\mathscr{C}^{6}(\mathbb{R})$, we can find
$\psi\in\mathscr{C}^{5}_{c}(\mathbb{R})$ and $c_{\xi}\in\mathbb{R}$ satisfying
$(\Theta_{V}\psi)(x)=\xi(x)+c_{\xi},\qquad x\in U.$
Under Assumption 1, by [11] the $(\lambda_{i})$ enjoy a strong rigidity
property, precisely recalled in Theorem 3.3. Informally, the probability that
the $\lambda_{i}$’s deviate from the grid given by the $i/n$-th quantiles of
$\mu_{V}$ is overwhelmingly small. Note that this rigidity phenomenon is also
used in [26]. Thus up to paying a loss factor, that is negligible compared to
our bound, we can assume that all the $\lambda_{i}$’s are localized in the
neighborhood $U$. By the master equation, choosing $f$ any primitive of
$\psi$, we are thus left with establishing the quantitative CLT for the random
variable
$X=\frac{1}{n\beta}\mathsf{L}F-\frac{Z}{\beta},$
which falls precisely under the scope of our Stein’s bound.
#### 1.5.4. Controlling the carré du champ and the remainder
In Section 4, we concretely implement our strategy and complete the proofs of
our main results. We first establish the quantitative statements, namely
Theorem 1.1, Theorem 1.2 and Theorem 1.3. In view of our strategy, we show
that $(n\beta)^{-1}\Gamma[X,-F]$ is close to a constant and that
$(\beta)^{-1}Z$ is small. We control the term involving the carré du champ by
relying on a simple yet remarkable property of the carré du champ: it
preserves linear statistics. Precisely, we have that
$\Gamma\left[\sum\varphi(\lambda_{i}),\sum\chi(\lambda_{i})\right]=\sum\varphi^{\prime}(\lambda_{i})\chi^{\prime}(\lambda_{i}).$
It follows that
$\frac{1}{n}\Gamma[X,F]=\frac{1}{n}\sum\xi^{\prime}(\lambda_{i})f^{\prime}(\lambda_{i}).$
By the convergence to equilibrium, the above quantity converges to
$\int\xi^{\prime}f^{\prime}\mathtt{d}\mu_{V}$. Using the minimality property
of $\mu_{V}$, it can be shown that this last integral coincides with limiting
covariance given above. To derive quantitative bounds from there, we rely on a
quantitative convergence to equilibrium which also follows from the rigidity
estimate from [11].
Handling the quadratic remainder $Z$ is more tedious. We use Fourier inversion
to decompose $Z$ into a product of linear statistics. By the aforementioned
quantitative law, one of term goes to $0$ at a given rate while the other
stays bounded. This allows us to conclude for the proof Theorem 1.1. From
there, Theorem 1.2 follows by a bootstrap argument: we essentially redo the
same proof but instead of using the sub-optimal quantitative law of large
numbers provided by [11], we use the CLT we just established that gives us a
law of large numbers at speed $1/n$. To conclude for Theorem 1.3, we
approximate $\xi\in\mathscr{C}^{1,\gamma}(\mathbb{R})$ by a sequence of smooth
functional $\xi_{\varepsilon}$ while choosing $\varepsilon\simeq n^{a}$, for a
well-chosen $a$.
#### 1.5.5. Super-convergence
The proof of our qualitative result Theorem 1.4 is finally given in Section
4.4. The derivation of the super-convergence relies on a lemma, classical in
Dirichlet forms / Malliavin calculus theory, stating that negative moments of
$\Gamma[X,X]$ control the Sobolev norms of the density of $X$. We rely again
on the fact that $\Gamma$ preserves linear statistics. In this precise case,
$\Gamma{[X,X]}=\sum(\xi^{\prime}(\lambda_{i}))^{2}.$
We obtain negative moments through a direct control on
$\operatorname{\mathbf{P}}\left[\sum(\xi^{\prime}(\lambda_{i}))^{2}\leq\varepsilon\right]$.
## 2\. Stein’s method for Markov diffusive operators
In order to carry out our program, we need to establish quantitative bounds in
total variation for random variables which are close in some sense to the
range of a given Markov diffusive operator $\mathsf{L}$. To the best of our
knowledge, the following estimates seem to be new in the well-studied area of
Malliavin–Stein’s method and are of independent interest. We stress that all
the results of this section are valid for any diffusive Markov operator
$\mathsf{L}$ associated with $\Gamma$ the so-called carré du champ and
$\operatorname{\mathbf{P}}$ the invariant measure for $\mathsf{L}$. We refer
to [2] for definitions in this abstract setting. Indeed, the only properties
used in our proofs are the chain rule and the integration by parts which hold
in full generality. The cornerstone of our method relies on the following
Theorem which will proved in Section 2.3 based on the content of Sections 2.1
and 2.2.
In case of vector-valued random variables $F$ and $G$, we extend our
definition to the _matrix-valued carré du champ_
$\Gamma[F,G]_{ij}:=\Gamma[F_{i},G_{j}],\qquad i,j=1,\dots,d.$
###### Theorem 2.1.
Take $F_{1},\dots,F_{d}\in\operatorname{\mathbb{D}om}\mathsf{L}$,
$Z_{1},\dots,Z_{d}\in L^{p}$, and
$X:=(\mathsf{L}F_{1}+Z_{1},\dots,\mathsf{L}F_{d}+Z_{d})$. Let $\Sigma$ be a
positive symmetric matrix, $C=\Sigma^{2}$ and $N$ be standard Gaussian in
$\mathbb{R}^{d}$. Then, we have
(2.1) $\begin{split}\mathbf{W}_{p}(X,\Sigma
N)&\leq\lVert\Sigma\rVert_{op}\lVert
N\rVert_{L^{p}}\lVert\mathrm{id}-\Sigma^{-1}\Gamma[X,-F]\Sigma^{-1}\rVert_{L^{p}}+\lVert\Sigma\rVert_{op}\lVert
Z\rVert_{L^{p}}\\\ &\leq\lVert\Sigma\rVert_{op}\lVert C^{-1}\rVert\lVert
N\rVert_{L^{p}}\lVert
C-\Gamma[X,-F]\rVert_{L^{p}}+\lVert\Sigma\rVert_{op}\lVert
Z\rVert_{L^{p}}.\end{split}$
Moreover, in the univariate case $d=1$, setting $\sigma=\Sigma_{11}$, we
obtain
(2.2) $\mathbf{TV}(X,\sigma
N)=\mathbf{TV}(\sigma^{-1}X,N)\leq\frac{2}{\sigma^{2}}\lVert\sigma-\Gamma[X,-F]\rVert_{L^{1}}+\frac{\sqrt{\pi}}{2}\lVert
Z\rVert_{L^{1}}.$
### 2.1. Reminders on Stein kernels
Let us first recall important results regarding Stein kernels. Here we work on
a general probability space $(\Omega,\mathfrak{W},\operatorname{\mathbf{P}})$.
Given a multivariate random variable $X=(X_{1},\dots,X_{d})$, we say that a
matrix-valued random variable $\tau$, measurable with respect to $X$, is a
_Stein kernel_ for $X$ provided
$\operatorname{\mathbf{E}}\left[X\cdot\nabla\varphi(X)\right]=\operatorname{\mathbf{E}}\left[\tau\cdot\nabla^{2}\varphi(X)\right],\qquad\varphi\in\mathscr{C}^{\infty}(\mathbb{R}^{d}).$
Heuristically, Stein kernels are relevant for normal approximation, since $X$
is a standard multivariate normal variable if and only if it admits
$\tau=\mathrm{id}$ as a Stein kernel. A key observation at the heart of
Stein’s method are the following quantitative normal approximation
inequalities.
###### Lemma 2.2 (Stein kernel bounds [27, 18, 31] ).
Let $X=(X_{1},\dots,X_{d})$, let $N$ be a standard normal variable on
$\mathbb{R}^{d}$, and $\tau$ be Stein kernel for $X$.
1. (i)
Let $p\in[1,\infty)$. If $X\in L^{p}(\Omega)$, then
$\mathbf{W}_{p}(X,N)^{p}\leq\operatorname{\mathbf{E}}\left[\lvert
N\rvert^{p}\right]\operatorname{\mathbf{E}}\left[\lVert\tau-\mathrm{id}\rVert^{p}\right].$
2. (ii)
In the univariate setting $d=1$, we have
$\mathbf{TV}(X,N)\leq
2\operatorname{\mathbf{E}}\left[\lvert\tau-1\rvert\right].$
### 2.2. Computations of Stein kernels on $\operatorname{Im}\mathsf{L}$
The celebrated Malliavin–Stein method [32] puts forward a strikingly efficient
way to compute a Stein kernel for sufficiently smooth functionals of an
infinite-dimensional Gaussian field.
###### Lemma 2.3.
Take $F_{1},\dots,F_{d}\in\operatorname{\mathbb{D}om}\mathsf{L}$. Let
$X:=(\mathsf{L}F_{1},\dots,\mathsf{L}F_{d})$, and define the matrix-valued
random variable $\tau=(\tau_{ij})$, where
$\tau_{ij}:=\operatorname{\mathbf{E}}\left[\Gamma[X_{i},-F_{j}]\nonscript\>|\allowbreak\nonscript\>\mathopen{}X\right],\qquad
i,j=1,\dots,d.$
Then, $\tau$ is a Stein kernel for $X$.
###### Proof.
Take $\varphi\in\mathscr{C}^{\infty}(\mathbb{R}^{d})$. Then, using the
integration by parts, and then the chain rule, we get
$\operatorname{\mathbf{E}}\left[X\cdot\nabla\varphi(X)\right]=\sum_{i=1}^{l}\operatorname{\mathbf{E}}\left[\mathsf{L}F_{i}\partial_{i}\varphi(X)\right]=-\sum_{i,j}\operatorname{\mathbf{E}}\left[\partial_{ij}\varphi(X)\Gamma[F_{i},X_{j}]\right].$
This concludes the proof in view of the definition of a Stein kernel. ∎
Combining Lemmas 2.2 and 2.3, we obtain the following $\Gamma$-Stein bound.
###### Theorem 2.4.
Take $F_{1},\dots,F_{d}\in\operatorname{\mathbb{D}om}\mathsf{L}$, and
$X:=(\mathsf{L}F_{1},\dots,\mathsf{L}F_{d})$. Then
$\mathbf{W}_{p}(X,N)^{p}\leq\operatorname{\mathbf{E}}\left[\lvert
N\rvert^{p}\right]\operatorname{\mathbf{E}}\left[\lVert\operatorname{id}-\Gamma[X,-F]\rVert^{p}\right].$
Moreover, in the univariate case $d=1$, we find
$\mathbf{TV}(X,N)\leq 2\operatorname{\mathbf{E}}\left[\lvert
1-\Gamma[X,-F]\rvert\right].$
###### Remark 2.5.
The classical approach on the Wiener space relies on the invertibility of the
generator of the Ornstein–Uhlenbeck process. In this case, it is known that
$\tau_{ij}:=\operatorname{\mathbf{E}}\left[\Gamma[X_{i},-\mathsf{L}^{-1}X_{j}]\nonscript\>|\allowbreak\nonscript\>\mathopen{}X\right],\qquad
i,j=1,\dots,d,$
is a Stein kernel for $X\in L^{2}(\Omega)$ with
$\operatorname{\mathbf{E}}X=0$. This strategy has also been applied in the
setting of other diffusive Markov generators with discrete spectrum, see for
instance [1, 27]. This actually could be further generalized to any diffusive
Markov generators that is invertible away from constants. Note that, the
generator of the Dyson Brownian motion $\mathsf{L}$ is in general not
invertible, in particular it is a priori not true that a given linear
statistics belongs to the range of $\mathsf{L}$. Therefore, we need to extend
the above techniques for random variables not belonging to the range of
$\mathsf{L}$ which is precisely the object of the next section.
### 2.3. Normal approximation away from $\operatorname{Im}\mathsf{L}$
Given $X$, finding $F$ such that $\mathsf{L}F=X$ is a demanding task. Indeed
as already mentioned, $\mathsf{L}$ may be not invertible. Nevertheless, as we
now demonstrate, the bounds obtained by the Stein’s method can be amended in
order to cover the case where $X$ is close to the range of $\mathsf{L}$ that
is to say of the form $X=\mathsf{L}F+Z$ with small perturbation $Z$.
###### Proof of Theorem 2.1.
We first prove the claim in the case $\Sigma=\mathrm{id}$, and then explain
how to handle the case of general non-degenerate covariance. For the upper
bound 2.1, by the triangle inequality, we have that
$\mathbf{W}_{p}(X,N)\leq\mathbf{W}_{p}(\mathsf{L}F,N)+\mathbf{W}_{p}(\mathsf{L}F,\mathsf{L}F+Z).$
We apply Lemma 2.3 on the first term on the right-hand side. For the second
term by choosing the coupling $(\mathsf{L}F,\mathsf{L}F+Z)$ in the
minimization problem defining $\mathbf{W}_{p}$, we see that
$\mathbf{W}_{p}(\mathsf{L}F,\mathsf{L}F+Z)^{p}\leq\operatorname{\mathbf{E}}\left[\lVert\mathsf{L}F-\mathsf{L}F+Z\rVert^{p}\right]=\operatorname{\mathbf{E}}\left[\lVert
Z\rVert^{p}\right].$
This concludes the proof of 2.1.
We now prove 2.2. Since, by the chain rule an integration by parts, we have
$\operatorname{\mathbf{E}}\left[\varphi^{\prime}(X)\Gamma[X,-F]\right]=\operatorname{\mathbf{E}}\left[\varphi(X)\mathsf{L}F\right]=\operatorname{\mathbf{E}}\left[\varphi(X)X\right]-\operatorname{\mathbf{E}}\left[\varphi(X)Z\right],$
we find that
$\operatorname{\mathbf{E}}\left[\varphi^{\prime}(X)-X\varphi(X)\right]=\operatorname{\mathbf{E}}\left[\varphi^{\prime}(X)(1-\Gamma[X,-F])\right]-\operatorname{\mathbf{E}}\left[\varphi(X)Z\right].$
Recalling that, by [32, Thm. 3.3.1],
$\mathbf{TV}(X,N)\leq\sup\left\\{\operatorname{\mathbf{E}}\left[\varphi^{\prime}(X)\right]-\operatorname{\mathbf{E}}\left[X\varphi(X)\right]:\lVert\varphi\rVert_{\infty}\leq\sqrt{\frac{\pi}{2}},\,\lVert\varphi^{\prime}\rVert_{\infty}\leq
2\right\\},$
this concludes the proof in the case where $\Sigma=\mathrm{id}$. Now we take
an general positive, hence invertible, covariance matrix $\Sigma$. On the one
hand, by the stability of Wasserstein distances under the Lipschitz map
$\mathbb{R}^{n}\ni v\mapsto\Sigma v$, we find that
$\mathbf{W}_{p}(X,\Sigma
N)\leq\lVert\Sigma\rVert_{op}\mathbf{W}_{p}(\Sigma^{-1}X,N).$
Recall that $\lVert\cdot\rVert_{op}$ is the operator norm with respect to the
Euclidean norm on $\mathbb{R}^{n}$. On the other hand, a routine computation
yields
$\Gamma[\Sigma^{-1}X,-\Sigma^{-1}F]=\Sigma^{-1}\Gamma[X,-F]\Sigma^{-1}.$
Thus by 2.1, we get
(2.3) $\begin{split}\mathbf{W}_{p}(X,\Sigma
N)&\leq\lVert\Sigma\rVert_{op}\lVert
N\rVert_{L^{p}}\lVert\mathrm{id}-\Sigma^{-1}\Gamma[X,-F]\Sigma^{-1}\rVert_{L^{p}}+\lVert\Sigma\rVert_{op}\lVert
Z\rVert_{L^{p}}\\\ &\leq\lVert\Sigma\rVert_{op}\lVert C^{-1}\rVert\lVert
N\rVert_{L^{p}}\lVert
C-\Gamma[X,-F]\rVert_{L^{p}}+\lVert\Sigma\rVert_{op}\lVert
Z\rVert_{L^{p}}.\end{split}$
Similarly, in the univariate case $d=1$, we obtain the bound 2.2. ∎
## 3\. The master equation for linear statistics
We now apply the content of the previous section to the specific operator
$\mathsf{L}$ which is the generator of the Dyson Brownian motion which was
given in equation (1.3).
### 3.1. Notations and introduction of relevant operators
We show here that linear statistics associated with $\beta$-ensembles can
indeed be expressed in the form $\mathsf{L}F+Z$ thus making them good
candidates to apply our Theorem 2.1. For a more concise exposition of our
results and proofs, we adopt the following notations related to the empirical
measure:
$\displaystyle\nu_{n}:=\sum_{i=1}^{n}\delta_{\lambda_{i}},\ \text{and}\quad\
\bar{\nu}_{n}:=\nu_{n}-n\mu_{V};$ $\displaystyle\mu_{n}:=\frac{\nu_{n}}{n},\
\text{and}\quad\ \bar{\mu}_{n}:=\mu_{n}-\mu_{V}.$
We will write $\langle f|\mu\rangle$ for the integration of a function $f$
against a measure $\mu$. We consider the following operators, acting on
functions $f\in\mathscr{C}^{2}(\mathbb{R})$:
$\displaystyle
T_{V}(f)(x):=\int\frac{f(x)-f(y)}{x-y}\mathtt{d}\mu_{V}(y)=\int\int_{0}^{1}f^{\prime}((1-u)x+uy)\mathtt{d}u\mu_{V}(\mathtt{d}y);$
$\displaystyle\Theta_{V}(f):=-V^{\prime}f+T_{V}(f);$ $\displaystyle
T_{n}(f)(x):=\int\int_{0}^{1}f^{\prime\prime}((1-u)x+uy)\mathtt{d}u\mu_{n}(\mathtt{d}y)=\frac{1}{n}\sum_{i=1}^{n}\frac{f^{\prime}(x)-f^{\prime}(\lambda_{i})}{x-\lambda_{i}}1_{x\neq\lambda_{i}}+f^{\prime\prime}(\lambda_{i})1_{x=\lambda_{i}};$
$\displaystyle\Theta_{n}(f):=-V^{\prime}f^{\prime}+T_{n}(f).$
To some extend, $T_{n}$ and $\Theta_{n}$ are the empirical counterparts of
$T_{V}$ and $\Theta_{V}$. Since $\mu_{V}$ is compactly supported and since $f$
is chosen $\mathscr{C}^{2}$, the integrals above are all convergent. This
shows in particular that the operators $T_{V}$ and $\Theta_{V}$ are well-
defined. We recall an _invertibility result_ for $\Theta_{V}$ taken from [5].
###### Lemma 3.1 ([5, Lem. 3.3]).
There exists $\delta>0$, such that if $U:=(-1-\delta,1+\delta)$, for all
$\xi\in\mathscr{C}^{6}(\mathbb{R})$, there exist a constant $c_{\xi}$ and
$\psi\in\mathscr{C}^{5}_{c}(\mathbb{R})$ such that
(3.1) $\displaystyle(\Theta_{V}\psi)(x)=\xi(x)+c_{\xi},\qquad x\in U,$ (3.2)
$\displaystyle\lVert\psi\rVert_{\mathscr{C}^{5}(\mathbb{R})}\leq
C\lVert\xi\rVert_{\mathscr{C}^{6}(\mathbb{R})}.$
For the rest of the paper, we fix the margin $\delta>0$ and the associated
neighborhood $U$ given by Lemma 3.1 above.
###### Remark 3.2.
1. (a)
A slight variant of this invertibility result can be found in [4, Lem. 3.2].
2. (b)
We do not assume [5, (H3)], however as noticed in this last reference, this
assumption is not used in the proof of [5, Lem. 3.3].
3. (c)
Since we are in the single-cut case, with no singular density, with the
notations of [5], we have here $\mathsf{k}=0$ and $\mathsf{m}=0$.
### 3.2. The rigidity estimates and its consequences
Many ingredients in our proof proceed from a rigidity result from [11] which
is also used in [26].
###### Theorem 3.3 ([11, Thm 2.4]).
Let $\rho_{0}:=-1$, and $\rho_{j}$ be the $j/n$-quantile of $\mu_{V}$ for
$j\in\\{1,\cdots,n\\}$. Set $\hat{\jmath}:=\min(j,n-j+1)$ and write
$(\lambda_{(j)})$ for the increasingly-ordered vector of $\lambda_{j}$’s. For
any $\varepsilon>0$, there exist $c_{\varepsilon}>0$ and $N_{\varepsilon}>0$
such that for all $n\geq N_{\varepsilon}$
(3.3) $\operatorname{\mathbf{P}}\left[\exists
j\in[1,n]:\lvert\lambda_{(j)}-\rho_{j}\rvert>\hat{\jmath}^{-\frac{1}{3}}n^{-\frac{2}{3}+\varepsilon}\right]\leq\exp(-n^{c_{\varepsilon}}).$
As anticipated, we repeatedly use in our proofs, the following corollary on
the control of the outliers. Philosophically, it allows us to work on
$U=(-1-\delta,1+\delta)$ and to discard the rest up to accepting a negligible
loss. The next result follows directly from the Theorem 3.3 and fact that the
locations $(\rho_{j})_{1\leq j\leq n}$ belong to $[-1,1]$.
###### Lemma 3.4.
There exists $C>0$ and $c>0$ such that for $n$ large enough
(3.4) $\operatorname{\mathbf{P}}\left[\max_{1\leq i\leq n}|\lambda_{i}|\geq
1+\delta\right]\leq C\exp\left\lparen-n^{c}\right\rparen.$
###### Remark 3.5.
Similar results are numerous in the literature: even with weaker assumptions
on the potential $V$ than the ones in [11] for which the rigidity is not yet
known, one can derive better rate of convergence, such as exponential decay.
Such exponential decay follows, for instance, from the large deviations
principle for the extreme positions, as in [9, Prop. 2.1]. Under strong
assumptions on the potential $V$, the reference [29, Lem. 4] offers an
exponential decay with an explicit dependence on $\delta$, which, of course we
do not need since our margin $\delta$ is fixed. See also [12, Thm. 1.12] for
similar a similar result under weaker assumptions in dimension greater than
two. As the rigidity estimate given by Theorem 3.3 is in fact used to quantify
the speed of convergence to equilibrium in the next Lemma 3.6 and since the
associated bound 3.4 is sufficient for our purpose, we have decided to use the
rigidity also to control the outliers.
As proved in [26], Theorem 3.3 provides a polynomial speed of convergence to
equilibrium. Namely, they obtain the following lemma that we reproduce below
for completeness.
###### Lemma 3.6 ([26, Lemma 5.3]).
Let $\alpha>0$ and $p\geq 1$. Then, there exists $K_{p,\alpha}>0$ such that
for any bounded and Lipschitz function $f$ we have
(3.5) $\lVert\langle
f|\mu_{n}-\mu_{V}\rangle\rVert_{L^{p}}\leq\frac{1}{n^{1-\alpha}}K_{p,\alpha}(\lVert
f\rVert_{\infty}+\lVert f^{\prime}\rVert_{\infty}).$
###### Remark 3.7.
As we shall see below, our findings allow to upgrade the rate of convergence
given in Lemma 3.6 to get an optimal rate $O(n^{-1})$ for regular enough test
functions $f$.
### 3.3. The master equation
In relation to the Dyson generator $\mathsf{L}$, the master operator
$\Theta_{V}$ allows us to derive the following _master equation_ , which is at
the heart of our argument.
###### Theorem 3.8.
Consider a test function $f\in\mathscr{C}^{2}(\mathbb{R})$ and define
$F:=\langle f|\bar{\nu}_{n}\rangle$. Then, we have the following decomposition
(3.6) $\frac{\mathsf{L}F}{n}=\frac{2-\beta}{2}\langle
f^{\prime\prime}|\mu_{n}\rangle+\beta\langle\Theta_{V}f^{\prime}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f^{\prime}-T_{V}f^{\prime}|\bar{\nu}_{n}\rangle.$
###### Proof.
Recalling the definitions of $\mathsf{L}$ and $T_{n}$, we get that
(3.7) $\begin{split}\frac{\mathsf{L}F}{n}&=\langle
f^{\prime\prime}|\mu_{n}\rangle-\beta\langle
V^{\prime}f^{\prime}|\nu_{n}\rangle+\frac{\beta}{2n}\sum_{i\neq
j}^{n}\frac{f^{\prime}(\lambda_{i})-f^{\prime}(\lambda_{j})}{\lambda_{i}-\lambda_{j}}\\\
&=\frac{2-\beta}{2}\langle f^{\prime\prime}|\mu_{n}\rangle-\beta\langle
V^{\prime}f^{\prime}|\nu_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f^{\prime}|\nu_{n}\rangle\\\ &=\frac{2-\beta}{2}\langle
f^{\prime\prime}|\mu_{n}\rangle-\beta\langle
V^{\prime}f^{\prime}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f^{\prime}|\nu_{n}\rangle-n\beta\langle
V^{\prime}f^{\prime}|\mu_{V}\rangle.\end{split}$
Recall that the Euler–Lagrange equation for the minimality of $\mu_{V}$ with
respect to the energy reads, for some $C_{V}\in\mathbb{R}$:
$V(x)-\int\log(|x-y|)\mu_{V}(\mathtt{d}y)=C_{V},\qquad x\in\Sigma_{V}.$
Differentiating with respect to $x$ leads to
$V^{\prime}(x)-\int\frac{1}{x-y}\mu_{V}(\mathtt{d}y)=0,\qquad x\in\Sigma_{V}.$
From there, multiplying by $f^{\prime}(x)$, integrating with respect to
$\mu_{V}(\mathtt{d}x)$, and symmetrizing yields
$\langle
V^{\prime}f^{\prime}|\mu_{V}\rangle=\frac{1}{2}\int\frac{f^{\prime}(x)-f^{\prime}(y)}{x-y}\mu_{V}(\mathtt{d}x)\mu_{V}(\mathtt{d}y)=\frac{1}{2}\langle
T_{V}f^{\prime}|\mu_{V}\rangle.$
Substituting the latter in 3.7, one gets
(3.8) $\begin{split}\frac{\mathsf{L}F}{n}&=\frac{2-\beta}{2}\langle
f^{\prime\prime}|\mu_{n}\rangle-\beta\langle
V^{\prime}f^{\prime}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f^{\prime}|\nu_{n}\rangle-\frac{\beta}{2}\langle
T_{V}f^{\prime}|n\mu_{V}\rangle\\\ &=\frac{2-\beta}{2}\langle
f^{\prime\prime}|\mu_{n}\rangle-\beta\langle
V^{\prime}f^{\prime}|\bar{\nu}_{n}\rangle-\frac{\beta}{2}\langle
T_{V}f^{\prime}|n\mu_{V}\rangle\\\ &+\frac{\beta}{2}\langle
T_{n}f^{\prime}-T_{V}f^{\prime}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f^{\prime}|n\mu_{V}\rangle+\frac{\beta}{2}\langle
T_{V}f^{\prime}|\bar{\nu}_{n}\rangle.\end{split}$
By Fubini Theorem, we have that $\langle
T_{n}f^{\prime}|n\mu_{V}\rangle=\langle T_{V}f^{\prime}|\nu_{n}\rangle$. This
gives
$\langle T_{n}f^{\prime}|n\mu_{V}\rangle-\langle
T_{V}f^{\prime}|n\mu_{V}\rangle=\langle T_{V}f^{\prime}|\bar{\nu}_{n}\rangle.$
Plugging this equality in 3.8 leads to the announced result. ∎
## 4\. Proofs of the main results
We now complete the proofs of our main results. Regarding quantitative normal
approximation for linear statistics of $\beta$-ensembles, we follow the
strategy described in Sections 2 and 3, applying the general quantitative
bounds given by Theorem 2.1.
### 4.1. Proof of Theorem 1.1
We first give the proof of Theorem 1.1, which gives a near optimal rate of
convergence $O(n^{-1+\alpha})$ for all $\alpha>0$.
#### 4.1.1. Handling the covariance
Let us recall the form of the limit covariance matrix $C$ of the linear
statistics and let us first prove that it is invertible. Following [26, §4.1,
in particular Eq. (4.14)], we have
$c_{ij}=\frac{1}{2\beta}\int\frac{\xi_{i}(x)-\xi_{i}(y)}{x-y}\frac{\xi_{j}(x)-\xi_{j}(y)}{x-y}(1-xy)\rho(\mathtt{d}x)\rho(\mathtt{d}y)=:\langle\xi_{i},\xi_{j}\rangle_{\mathscr{H}^{1/2}}.$
The bilinear symmetric form
$\langle\xi_{i},\xi_{j}\rangle_{\mathscr{H}^{1/2}}$ is not a scalar product,
since $\langle g,g\rangle_{\mathscr{H}^{1/2}}=0$ if and only if $g$ is
constant. Nevertheless, the matrix $C:=(c_{ij})$ is a Gram matrix with respect
to the semi-scalar product $\langle\cdot,\cdot\rangle_{\mathscr{H}^{1/2}}$.
Thus under the freeness condition 1.2, the matrix $C$ is symmetric definite
positive, and we write $\Sigma$ for its unique positive symmetric square root.
#### 4.1.2. Preparatory computations
We now use the master equation (3.6) to decompose the linear statistics in a
suitable form to apply Theorem 2.1.
Reduction to the case $\mathsf{L}F+Z$. Let us consider
$\xi\in\mathscr{C}^{6}(\mathbb{R})$, $\psi\in\mathscr{C}^{5}_{c}(\mathbb{R})$
associated with $\xi$ through Lemma 3.1 and let $f$ be any primitive of
$\psi$. We then have
$\Theta_{V}f^{\prime}=\xi+c_{\xi}+\left\lparen\Theta_{V}f^{\prime}-\xi-
c_{\xi}\right\rparen 1_{\mathbb{R}\setminus U}.$
Since $\langle 1|\bar{\nu}_{n}\rangle=0$, and that
$\operatorname{supp}\mu_{V}\subset U$, we find that
$\langle\xi|\bar{\nu}_{n}\rangle=\langle\Theta_{V}f^{\prime}|\bar{\nu}_{n}\rangle+\sum_{i=1}^{n}1_{\lvert\lambda_{i}\rvert>1+\delta}(\Theta_{V}f^{\prime}-\xi-
c_{\xi}).$
By Lemma 3.4, we have that
$\left\lVert\sum_{i=1}^{n}1_{\lvert\lambda_{i}\rvert>1+\delta}(\Theta_{V}f^{\prime}-\xi-
c_{\xi})\right\rVert_{L^{p}}=n\mathrm{e}^{-n^{c}}O\left(\lVert\xi\rVert_{\infty}+\lVert\Theta_{V}f^{\prime}\rVert_{\infty}+1\right).$
From the explicit expression of $\Theta_{V}$ and 3.2, we find that
$\lVert\Theta_{V}f^{\prime}\rVert_{\infty}\leq c(\lVert
f^{\prime}\rVert_{\infty}+\lVert f^{\prime\prime}\rVert_{\infty})\leq
c\lVert\psi\rVert_{\mathscr{C}^{5}(\mathbb{R})}\leq
c\lVert\xi\rVert_{\mathscr{C}^{6}(\mathbb{R})}.$
Combining those two estimates, and since $\langle 1|\bar{\nu}_{n}\rangle=0$,
it is sufficient to establish our bound for linear statistics of the form
$\langle\Theta_{V}f^{\prime}|\bar{\nu}_{n}\rangle$, where
$f^{\prime}=\psi\in\mathscr{C}_{c}^{5}(\mathbb{R})$ is associated with
$\xi\in\mathscr{C}^{6}(\mathbb{R})$ by Lemma 3.1. Thus, we now study
$X:=\left\lparen\langle\Theta_{V}f_{1}^{\prime}|\bar{\nu}_{n}\rangle,\dots,\langle\Theta_{V}f_{d}^{\prime}|\bar{\nu}_{n}\rangle\right\rparen.$
By Theorem 3.8, we can then decompose $X$ under the form
$X=m+\frac{1}{n}\mathsf{L}F+Z,$
where $m=(m_{i})_{1\leq i\leq d}$ with $m_{i}:=(1/2-1/\beta)\langle
f_{i}^{\prime\prime}|\mu_{V}\rangle$ where $F_{i}:=\frac{1}{\beta}\langle
f_{i}|\bar{\nu}_{n}\rangle$, and
(4.1) $Z_{i}:=\left\lparen\frac{1}{2}-\frac{1}{\beta}\right\rparen\langle
f_{i}^{\prime\prime}|\mu_{n}\rangle-m_{i}-\frac{1}{2}\langle
T_{n}f_{i}^{\prime}-T_{V}f_{i}^{\prime}|\bar{\nu}_{n}\rangle.$
In the next section, we apply Theorem 2.1 to provide quantitative bounds.
Computation of $\Gamma[X,-F]$. We repeatedly use the simple yet remarkable
observation that linear statistics are stable under the action of the carré du
champ, namely
(4.2)
$\Gamma\big{[}\langle\varphi|\nu_{n}\rangle,\langle\psi|\nu_{n}\rangle\big{]}=\langle\varphi^{\prime}\psi^{\prime}|\nu_{n}\rangle,\qquad\varphi,\psi\in\mathscr{C}^{1}(\mathbb{R}).$
Thus, we find that for $1\leq i,j\leq d$
(4.3)
$\Gamma\left[X_{i},-\frac{F_{j}}{n}\right]=\frac{1}{\beta}\langle(\Theta_{V}f_{i}^{\prime})^{\prime}f_{j}^{\prime}|\mu_{n}\rangle.$
Heuristics. As a consequence of the convergence to equilibrium,
$\mu_{n}\to\mu_{V}$, we see that the sum of the first two terms on the right-
hand side in 4.1 converges to zero. Similarly, by Equation 4.3, the term
$\Gamma[X_{i},-F_{j}/n]$ converges to
$\frac{1}{\beta}\langle\xi_{i}^{\prime}f_{j}^{\prime}|\mu_{V}\rangle$ which
will be the limit covariance of
$(\langle\xi_{i},\bar{\nu_{n}}\rangle,\langle\xi_{j},\bar{\nu_{n}}\rangle)$.
Relying on the expression of the limit variance given in [26] and by
uniqueness one must therefore have
$c_{ij}=\lim_{n}\text{Cov}(\langle\xi_{i},\bar{\nu_{n}}\rangle,\langle\xi_{j},\bar{\nu_{n}}\rangle)=\frac{1}{\beta}\langle\xi_{i}^{\prime}f_{j}^{\prime}|\mu_{V}\rangle.$
However, the determination of the asymptotics of all the terms involving the
quadratic term $\langle(T_{n}-T_{v})f_{i}^{\prime}|\bar{\nu}_{n}\rangle$ is
more delicate. Indeed, the same heuristic on the convergence to equilibrium
shows that this term is of order $o(n)$ whereas one would expect $o(1)$. We
handle this remainder in two steps:
1. (i)
We split the quadratic term in a product of linear terms, using Fourier
inversion. Indeed since $f_{i}^{\prime}\in\mathscr{C}_{c}^{5}(\mathbb{R})$ is
is in the image of the Fourier transform.
2. (ii)
We control each of the linear term as before using convergence to equilibrium.
This is where we need the finer quantitative estimates recalled in Lemma 3.6.
#### 4.1.3. Quantitative control and completion of the proof
Le us now quantify the remainders mentioned in the last Section.
Splitting the remainder by Fourier inversion. For simplicity, we omit the
indices here and write $f$ for $f_{i}$, $1\leq i\leq d$. Since
$f^{\prime}\in\mathscr{C}^{5}_{c}(\mathbb{R})$, we have
(4.4) $\lvert\widehat{f^{\prime\prime}}(t)\rvert\leq 2\frac{\lVert
f^{(6)}\rVert_{\infty}}{(1+|t|)^{4}},\qquad t\in\mathbb{R}.$
In particular, $\widehat{f^{\prime\prime}}\in L^{1}(\mathbb{R})$. By the
Fourier inversion Theorem, one finds
(4.5)
$\begin{split}\langle(T_{n}-T_{V})f^{\prime}|\bar{\nu}_{n}\rangle&=n\iint\int_{0}^{1}f^{\prime\prime}((1-u)x+uy)(\mu_{n}-\mu_{V})(\mathtt{d}x)(\mu_{n}-\mu_{V})(\mathtt{d}y)\\\
&=n\int_{\mathbb{R}}\int_{0}^{1}\widehat{f^{\prime\prime}}(t)\langle\mathrm{e}^{\mathrm{i}tu\bullet}|\mu_{n}-\mu_{V}\rangle\langle\mathrm{e}^{\mathrm{i}t(1-u)\bullet}|\mu_{n}-\mu_{V}\rangle\mathtt{d}u\mathtt{d}t.\end{split}$
By Lemma 3.6 applied to the exponential functions and by Hölder’s inequality,
we have
$n\lVert\langle\mathrm{e}^{\mathrm{i}tu\bullet}|\mu_{n}-\mu_{V}\rangle\langle\mathrm{e}^{\mathrm{i}t(1-u)\bullet}|\mu_{n}-\mu_{V}\rangle\rVert_{L^{p}}\leq\frac{1}{n^{1-2\alpha}}K_{p,\alpha}^{2}(1+\lvert
t\rvert+t^{2}).$
Thus, reporting in 4.5, and up to changing the constants from line to line, we
find that
(4.6)
$\lVert\langle(T_{n}-T_{V})f^{\prime}|\bar{\nu}_{n}\rangle\rVert_{L^{p}}\leq\frac{1}{n^{1-2\alpha}}K_{p,\alpha}\int_{\mathbb{R}}\lvert\widehat{f^{\prime\prime}}(t)\rvert\left\lparen
1+\lvert
t\rvert+t^{2}\right\rparen\mathtt{d}t\leq\frac{1}{n^{1-2\alpha}}K_{p,\alpha}\lVert
f^{(6)}\rVert_{\infty}.$
Completion of the proof. We can finally complete the proof Theorem 1.1. Again,
to simplify the expressions, we omit the indices $1\leq i\leq d.$ here. Recall
the definition of the term $Z$ given by Equation 4.1. By Lemma 3.6 and the
above Equation (4.6), we find that
$\lVert
Z\rVert_{L^{p}}\leq\left\lparen\frac{1}{2}-\frac{1}{\beta}\right\rparen\lVert\langle
f^{\prime\prime}|\mu_{n}-\mu_{V}\rangle\rVert_{L^{p}}+\lVert\langle(T_{n}-T_{V})f^{\prime}|\bar{\nu}_{n}\rangle\rVert_{L^{p}}\leq\frac{1}{n^{1-\alpha}}K_{p,\alpha}\lVert
f^{\prime}\rVert_{\mathscr{C}^{5}(\mathbb{R})}.$
Introducing the random matrix
$C_{n}(i,j):=\langle\xi^{\prime}_{i}f^{\prime}_{i}|\mu_{n}\rangle,$
we have by Equation 4.3
$\left\|\frac{1}{\beta}C-\Gamma[X,-F/n]\right\|_{L^{p}}=\frac{1}{\beta}\lVert
C-C_{n}\rVert.$
Invoking Lemma 3.6 to control $C-C_{n}$ yields, again with constants which may
change from line to line,
(4.7) $\begin{split}\lVert C-C_{n}\rVert_{L^{p}}&\leq
K_{p,\alpha}\frac{1}{n^{1-\alpha}}\left\lparen\lVert\xi^{\prime}\rVert_{\infty}\lVert
f^{\prime}\rVert_{\infty}+\lVert\xi^{\prime}\rVert_{\infty}\lVert
f^{\prime\prime}\rVert_{\infty}+\lVert\xi^{\prime\prime}\rVert_{\infty}\lVert
f^{\prime}\rVert_{\infty}\right\rparen\\\
&\leq\frac{K_{p,\alpha}}{n^{1-\alpha}}\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}\lVert
f\rVert_{\mathscr{C}^{2}(\mathbb{R})}.\end{split}$
Since by Lemma 3.1, we have $\lVert f\rVert_{\mathscr{C}^{2}(\mathbb{R})}\leq
K\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}$, all the bounds above can thus
be rewritten only in term of the norm
$\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}$. Injecting these estimates in
the general upper bound given by Theorem 2.1, we arrive at the announced
result.
### 4.2. Proof of Theorem 1.2
We now give the proof of Theorem 1.2, which gives an optimal rate of
convergence $O(n^{-1})$ in the case where the test function $\xi$ is
sufficiently regular. We work here in dimension $d=1$ for simplicity, but the
proof will extend verbatim to higher dimensions.
The strategy used in order to obtain the optimal rate of convergence consists
in bootstrapping the argument used in the previous proof of Theorem 1.1.
Except that instead of invoking Lemma 3.6, we use here a better law of large
numbers provided by Theorem 1.1 itself. Namely, for
$\xi\in\mathscr{C}^{6}(\mathbb{R})$, taking $\alpha:=1/2$ and using the
$p$-Wasserstein bound for any $p\geq 1$ given by Theorem 1.1, we have that
$\lVert\langle\xi|\bar{\nu}_{n}\rangle\rVert_{L^{p}}\leq
K_{p}\left(\frac{\lVert\xi\rVert_{\mathscr{C}^{6}(\mathbb{R})})+\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}^{2}}{n^{1/2}}\right)+\lVert\sigma
N+m\rVert_{L^{p}},$
where $N$ is a standard Gaussian and both the mean $m=m_{1}$ and the standard
deviation $\sigma=\sqrt{c_{11}}$ are controlled by
$\|\xi\|_{\mathscr{C}^{1}(\mathbb{R})}$, see Section 1.3.1 where their
explicit expressions are given. This shows that provided we can control the
$6$-th derivative, we can remove the polynomial loss in the quantitative law
of large numbers given by Lemma 3.6. At the level of the upper bound 4.6, this
implies that we have a $t^{12}$ appearing in the integral, which leads to an
upper bound controlled by $\lVert\xi\rVert_{\mathscr{C}^{14}(\mathbb{R})}$.
Similarly, in the upper bound 4.7, the right hand side now yields a control of
the form $\lVert\xi^{\prime}\rVert_{\mathscr{C}^{6}(\mathbb{R})}\lVert
f^{\prime}\rVert_{\mathscr{C}^{6}(\mathbb{R})}$ and concludes the proof.
As a result and as already mentioned in Remark 3.7, this argument implies that
any polynomial speed of convergence on the law of large numbers can be
upgraded to the optimal speed $\frac{1}{n}$, the price to pay is to impose
stronger regularity assumption on the test functions.
### 4.3. Proof of Theorem 1.3
We now show how Theorem 1.3 dealing with functions with Hölder derivative
easily follows from Theorem 1.1 together with Lemma 3.6.
Take $\eta$ a smooth probability density supported on $[-1,1]$ with finite
first moment. Defining
$\eta_{\varepsilon}:=\frac{1}{\varepsilon}\eta(\frac{\bullet}{\varepsilon})$,
and $\xi_{\varepsilon}:=\xi\ast\eta_{\varepsilon}$, we see by direct
computations that
$\displaystyle\lVert\xi-\xi_{\varepsilon}\rVert_{\infty}\leq\lVert\xi^{\prime}\rVert_{\infty}\varepsilon\int\lvert
y\rvert\eta(y),$ (4.8)
$\displaystyle\lVert\xi^{\prime}-\xi_{\varepsilon}^{\prime}\rVert_{\infty}\leq\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})}\,\varepsilon^{\gamma}\int\lvert
y\rvert^{\gamma}\eta(y).$
Thus by Lemma 3.6, for any $\alpha>0$, we have
$\lVert\langle\xi-\xi_{\varepsilon}|\bar{\nu}_{n}\rangle\rVert_{1}\leq
K_{\alpha}\varepsilon^{\gamma}n^{\alpha}\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})}.$
Now we use the master equation (3.6) for the linear statistics
$\xi_{\varepsilon}$ and we write:
$\frac{\mathsf{L}F_{\varepsilon}}{n}=\frac{2-\beta}{2}\langle
f_{\varepsilon}^{\prime\prime}|\mu_{n}\rangle+\beta\langle\Theta_{v}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f_{\varepsilon}^{\prime}-T_{V}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle,$
where $\Theta_{v}f_{\varepsilon}^{\prime}=\xi_{\varepsilon}$ on the
neighbourhood $U$. Mimicking the arguments developed in Section 4.1.2, we thus
have:
$\frac{\mathsf{L}F_{\varepsilon}}{n}=\frac{2-\beta}{2}\langle
f_{\varepsilon}^{\prime\prime}|\mu_{n}\rangle+\beta\langle\xi_{\varepsilon}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f_{\varepsilon}^{\prime}-T_{V}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle+O\left(ne^{-n^{c}}\|\xi_{\varepsilon}\|_{\mathscr{C}^{6}}\right).$
As a result, we get
$\frac{\mathsf{L}F_{\varepsilon}}{n}=\frac{2-\beta}{2}\langle
f_{\varepsilon}^{\prime\prime}|\mu_{n}\rangle+\beta\langle\xi|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f_{\varepsilon}^{\prime}-T_{V}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle+O\left(ne^{-n^{c}}\|\xi_{\varepsilon}\|_{\mathscr{C}^{6}}\right)+\langle\xi-\xi_{\varepsilon}|\bar{\nu}_{n}\rangle.$
Hence, we may decompose the linear statistics in the form
$\langle\xi|\bar{\nu}_{n}\rangle=m_{\varepsilon}+\frac{\mathsf{L}F_{\varepsilon}}{n}+Z_{\varepsilon}$
as in the proof of Theorem 1.1 with this time
$\begin{array}[]{ll}\displaystyle{m_{\varepsilon}:=\left(\frac{1}{2}-\frac{1}{\beta}\right)\langle
f_{\varepsilon}^{\prime\prime}|\mu_{V}\rangle},\\\ \\\
\displaystyle{Z_{\varepsilon}:=\frac{2-\beta}{2}\langle
f_{\varepsilon}^{\prime\prime}|\mu_{n}-\mu_{V}\rangle+\frac{\beta}{2}\langle
T_{n}f_{\varepsilon}^{\prime}-T_{V}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle+O\left(ne^{-n^{c}}\|\xi_{\varepsilon}\|_{\mathscr{C}^{6}}\right)+\langle\xi-\xi_{\varepsilon}|\bar{\nu}_{n}\rangle.}\end{array}$
Gathering the bounds given in the proof of Theorem 1.1 and the above Equation
(4.3), we then obtain the following controls
$\displaystyle\left\|\frac{2-\beta}{2}\langle
f_{\varepsilon}^{\prime\prime}|\mu_{n}-\mu_{V}\rangle+\frac{\beta}{2}\langle
T_{n}f_{\varepsilon}^{\prime}-T_{V}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle\right\|_{1}\leq\frac{K_{\alpha}}{n^{1-2\alpha}}\left(\|\xi_{\varepsilon}\|_{\mathscr{C}^{6}}+\|\xi_{\varepsilon}\|_{\mathscr{C}^{2}}^{2}\right),$
$\displaystyle\|\langle\xi-\xi_{\varepsilon}|\bar{\nu}_{n}\rangle\|_{1}\leq
K_{\alpha}\varepsilon^{\gamma}n^{\alpha}\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})},$
$\displaystyle\lVert
C_{\varepsilon}-C_{n,\varepsilon}\rVert_{L^{p}}\leq\frac{K}{n^{1-\alpha}}\|\xi_{\varepsilon}\|_{\mathscr{C}^{2}}^{2},$
where $C_{\varepsilon}$ denote the limit variance associated with the linear
statistic $\xi_{\varepsilon}$ and
$C_{n,\varepsilon}:=\langle\xi^{\prime}_{\varepsilon}f^{\prime}_{\varepsilon}|\mu_{n}\rangle$
is the empirical analogue. By Equation (2.2) in Theorem 2.1, combined with the
three above bounds we may write for some constant $K_{\alpha}$
(4.9)
$\mathbf{TV}\left(\langle\xi|\bar{\nu}_{n}\rangle,m_{\varepsilon}+\sigma_{\varepsilon}N\right)\leq
K_{\alpha}\left(\frac{1}{\varepsilon^{6}n^{1-2\alpha}}+\varepsilon^{\gamma}n^{\alpha}\right)\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})}.$
In order to complete the proof, one needs to show that $m_{\varepsilon}$ and
$\sigma_{\varepsilon}$ converge towards $m$ and $\sigma$ and to estimate
$mathbf{TV}\left(m_{\varepsilon}+\sigma_{\varepsilon}N,m+\sigma N\right)$ as
$\varepsilon$ goes to zero. First of all, relying on the explicit expressions
of the limit means and variances, we have indeed as $\varepsilon$ goes to zero
$|m_{\varepsilon}-m|=O\left(\|\xi_{\varepsilon}-\xi\|_{\mathscr{C}^{1}}\right)=O\left(\varepsilon^{\gamma}\|\xi\|_{\mathscr{C}^{1,\gamma}}\right),\quad\text{as
well
as}\quad|\sigma_{\varepsilon}-\sigma|=O\left(\varepsilon^{\gamma}\|\xi\|_{\mathscr{C}^{1,\gamma}}\right).$
Finally, using for instance Theorem 1.3 in [14], we derive that
$\mathbf{TV}\left(m_{\varepsilon}+\sigma_{\varepsilon}N,m+\sigma N\right)\leq
K\varepsilon^{\gamma}\|\xi\|_{\mathscr{C}^{1,\gamma}}.$
Optimizing in the parameters in Equation (4.9), since $\alpha$ can be chosen
arbitrarily small, we get the announced bound of $O(n^{-a})$ for any
$a<\frac{\gamma}{\gamma+6}$.
### 4.4. Proof of Theorem 1.4
We finally give the proof of Theorem 1.4 stating the super-convergence of
linear statistics of $\beta-$ensembles. If $X$ denotes our linear statistics,
the proof is based on the fact that $\Gamma[X,X]$ admit negative moments. This
fact, combined with classical integration by parts techniques indeed ensure
the convergence of densities.
#### 4.4.1. Control of the negative moments of $\Gamma$
Let us first establish the following lemma.
###### Lemma 4.1.
Take $\xi\in\mathscr{C}^{1}(\mathbb{R})$ such that there exist $\alpha>0$ such
that
$\mathsf{Leb}\left\lparen
x\in\mathbb{R}:\lvert\xi^{\prime}(x)\rvert\leq\varepsilon\right\rparen\lesssim\varepsilon^{\alpha},\qquad\varepsilon>0.$
Then, there exist $\gamma>0$, such that for $n$ large enough
(4.10)
$\operatorname{\mathbf{P}}\left[\frac{1}{n}\sum_{i=1}^{n}\xi^{\prime}(\lambda_{i})^{2}\leq\varepsilon\right]\lesssim\varepsilon^{\gamma
n},\qquad\varepsilon>0.$
###### Proof of Lemma 4.1.
To simplify the expressions in this proof, we set $g:=(\xi^{\prime})^{2}$ and
$m:=\langle g|\mu_{V}\rangle$. We distinguish here two regimes depending on
the magnitude of $\varepsilon$.
The large deviation regime. Take $\varepsilon<\frac{1}{2}m$, then by the large
deviations principle for $\beta$-ensembles [36, Thm. 2.3] (originally proved
in [7]), there exist $c=c_{g}>0$ such that, for $n$ large enough
$\operatorname{\mathbf{P}}\left[\langle
g|\mu_{n}\rangle\leq\varepsilon\right]\leq\operatorname{\mathbf{P}}\left[\lvert\langle
g|\bar{\mu}_{n}\rangle\rvert\geq\frac{m}{2}\right]\leq\exp\left\lparen-
cn^{2}\right\rparen.$
In this case, in the regime $\varepsilon\geq\mathrm{e}^{-kn}$ where $k$ is
some constant to be fixed later, we have
$\mathrm{e}^{-cn^{2}}=(\mathrm{e}^{-kn})^{n\frac{c}{k}}\leq\varepsilon^{n\frac{c}{k}}.$
The free energy regime. Let us work in the regime
$\varepsilon\leq\mathrm{e}^{-kn}$, where we recall that we have the freedom to
choose $k$. By the explicit form of the density for the $\beta$-ensemble, we
find that
$\begin{split}\operatorname{\mathbf{P}}\left[\langle
g|\mu_{n}\rangle\leq\varepsilon\right]&\leq\operatorname{\mathbf{P}}\left[g(\lambda_{i})\leq\varepsilon
n,\,\forall i\in[1,n]\right]\\\
&=\frac{1}{Z_{n,\beta}}\int_{\mathbb{R}^{n}}\prod_{i=1}^{n}\mathbf{1}_{\left\\{g(\lambda_{i})\leq
n\varepsilon\right\\}}\mathrm{e}^{-\beta
H_{n}(\lambda_{1},\dots,\lambda_{n})}\mathtt{d}\lambda_{1}\dots\mathtt{d}\lambda_{n}.\end{split}$
Thus applying Cauchy–Schwarz inequality and using the $\alpha-$regular
condition on $\xi$ hence $g$, we obtain
$\operatorname{\mathbf{P}}\left[\langle
g|\mu_{n}\rangle\leq\varepsilon\right]\leq\frac{Z_{n,2\beta}^{1/2}}{Z_{n,\beta}}(n\varepsilon)^{n\alpha/2}.$
By the large deviations principle, see for example [36, Thm. 2.3], we know
that
$\frac{1}{n^{2}}\log
Z_{n,\beta}\xrightarrow[n\to\infty]{}-\frac{\beta}{2}\mathcal{I}_{V}(\mu_{V}).$
In particular, there exists a constant $a(\beta)$ such that
$\frac{Z_{n,2\beta}^{1/2}}{Z_{n,\beta}}\leq\mathrm{e}^{n^{2}a(\beta)}.$
As a result, we have
$\operatorname{\mathbf{P}}\left[\langle
g|\mu_{n}\rangle\leq\varepsilon\right]\leq(n\varepsilon)^{n\alpha/2}\mathrm{e}^{a(\beta)n^{2}}.$
We can assume that $a(\beta)\geq 0$ otherwise the claim is trivial. In this
regime, we have
$\mathrm{e}^{a(\beta)n^{2}}\leq\varepsilon^{-a(\beta)n/k},\qquad
n^{n\alpha/2}\leq\varepsilon^{-\alpha/(2k)\log n}.$
It follows that
$\operatorname{\mathbf{P}}\left[\langle
g|\mu_{n}\rangle\leq\varepsilon\right]\leq\varepsilon^{n(\alpha/2-a(\beta)/k)-\alpha/(2k)\log
n}.$
Choosing $k$ large enough so that $\alpha/2-a(\beta)/kr>0$ then yields and
upper bound for the probability of order $\varepsilon^{cn}$, for some $c>0$. ∎
Recall that if $X=\sum_{i=1}^{n}\xi(\lambda_{i})$ is a linear statistics, then
we have $\Gamma[X,X]=\sum_{i=1}^{n}\xi^{\prime}(\lambda_{i})^{2}$. As a
result, given an exponent $\alpha>0$, we can write
$\operatorname{\mathbf{E}}\left[\frac{1}{\Gamma[X,X]^{\alpha}}\right]=\int_{\mathbb{R}^{+}}\operatorname{\mathbf{P}}\left[\frac{1}{\Gamma[X,X]^{\alpha}}>t\right]dt=\int_{\mathbb{R}^{+}}\operatorname{\mathbf{P}}\left[\Gamma[X,X]<\frac{1}{t^{1/\alpha}}\right]dt.$
As a result, the last Lemma 4.1 indeed ensures that $\Gamma[X,X]$ admits
negative moments.
#### 4.4.2. Regularity and super convergence
The proof of Theorem 1.4 now follows from a well-known integration by parts
procedure see [22, §2.1], or [23, §3.2.2], for details. We only recall the
first part of the proof to highlight that, in the setting of $\beta$-ensemble,
the correct quantities to control are the negative moments of $\Gamma[X,X]/n$,
and not those of $\Gamma[X,X]$, as it is the case in [22, 23]. Take
$\varphi\in\mathscr{C}^{1}_{c}(\mathbb{R})$ and write
(4.11)
$\operatorname{\mathbf{E}}\left[\varphi^{\prime}(X)\right]=\operatorname{\mathbf{E}}\left[\varphi(X)\left\lparen\frac{\Gamma[X,\Gamma[X,X]]}{\Gamma[X,X]^{2}}-\frac{\mathsf{L}X}{\Gamma[X,X]}\right\rparen\right].$
By Equation 4.2, we find that
$\Gamma[X,X]=\langle(\xi^{\prime})^{2}|\nu_{n}\rangle,\qquad\Gamma[X,\Gamma[X,X]]=2\langle(\xi^{\prime})^{2}\xi^{\prime\prime}|\nu_{n}\rangle.$
As a result, as $n$ goes to infinity, we have
$\frac{\Gamma[X,\Gamma[X,X]]}{\Gamma[X,X]^{2}}\to 0.$
Now in view of the master equation 3.6, we have the decomposition
(4.12)
$\begin{split}\Gamma\left[\frac{\mathsf{L}X}{n},\frac{X}{n}\right]&=\frac{2-\beta}{2}\frac{1}{n}\langle\xi^{(3)}\xi^{\prime}|\mu_{n}\rangle+\frac{1}{\beta}\langle(\Theta_{V}\xi^{\prime})^{\prime}\xi^{\prime}|\mu_{n}\rangle\\\
&-\frac{1}{2\beta}\Gamma\left[\langle(T_{V}-T_{n})\xi^{\prime}|\nu_{n}\rangle,\frac{X}{n}\right].\end{split}$
Using again the Fourier splitting, by the chain rule, dominated convergence,
and 4.2, we find that
$\begin{split}R_{n}&:=\Gamma\left[\langle(T_{n}-T_{V})\xi^{\prime}|\bar{\nu}_{n}\rangle,\langle\psi|\mu_{n}\rangle\right]\\\
&=\mathrm{i}\int_{\mathbb{R}}\widehat{\xi^{\prime\prime}}(t)t\int_{0}^{1}u\langle\mathrm{e}^{\mathrm{i}u\bullet}\psi^{\prime}|\mu_{n}-\mu_{V}\rangle\langle\mathrm{e}^{\mathrm{i}t(1-u)\bullet}|\mu_{n}-\mu_{V}\rangle\mathtt{d}u\mathrm{d}t\\\
&+\mathrm{i}\int_{\mathbb{R}}\widehat{\xi^{\prime\prime}}(t)t\int_{0}^{1}(1-u)\langle\mathrm{e}^{\mathrm{i}t(1-u)\bullet}\psi^{\prime}|\mu_{n}-\mu_{V}\rangle\langle\mathrm{e}^{\mathrm{i}tu\bullet}|\mu_{n}-\mu_{V}\rangle\mathtt{d}u\mathrm{d}t.\end{split}$
By Lemma 3.6, we then obtain that $R_{n}\to 0$ in $L^{2}$. By Lemma 2.3, this
shows that $\mathsf{L}F/n$ converges to a Gaussian with respect to
$\mathbf{W}_{2}$. In particular, its variance is of order $1$. Thus re-
writing, the last term in 4.11 as the product
$\frac{\mathsf{L}X}{n}\frac{n}{\Gamma[X,X]},$
the need to control the negative moments of $\Gamma[X,X]/n$ is more obvious.
It implies in particular that for some constant $C>0$ and any
$\phi\in\mathscr{C}^{1}$ we have
$\left|\mathbb{E}\left[\phi^{\prime}(X)\right]\right|\leq C\|\phi\|_{\infty}$.
This argument can be iterated and for any $p>1$ one would get similarly for
$n$ large enough that $\left|\mathbb{E}\left[\phi^{(p)}(X)\right]\right|\leq
C\|\phi\|_{\infty}$. The latter combined with the convergence in law proved in
Theorem 1.1 entails the announced super-convergence, see [23] for more
details.
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|
# Brownian loops on non-smooth surfaces
and the Polyakov-Alvarez formula
Minjae Park University of Chicago Joshua Pfeffer Columbia University Scott
Sheffield MIT
###### Abstract
Let $\rho$ be compactly supported on $D\subset\mathbb{R}^{2}$. Endow
$\mathbb{R}^{2}$ with the metric $e^{\rho}(dx_{1}^{2}+dx_{2}^{2})$. As
$\delta\to 0$ the set of Brownian loops centered in $D$ with length at least
$\delta$ has measure
$\frac{\mathrm{Vol}(D)}{2\pi\delta}+\frac{1}{48\pi}(\rho,\rho)_{\nabla}+o(1).$
When $\rho$ is smooth, this follows from the classical Polyakov-Alvarez
formula. We show that the above also holds if $\rho$ is not smooth, e.g. if
$\rho$ is only Lipschitz. This fact can alternatively be expressed in terms of
heat kernel traces, eigenvalue asymptotics, or zeta regularized determinants.
Variants of this statement apply to more general non-smooth manifolds on which
one considers all loops (not only those centered in a domain $D$).
We also show that the $o(1)$ error is uniform for any family of $\rho$
satisfying certain conditions. This implies that if we weight a measure $\nu$
on this family by the ($\delta$-truncated) Brownian loop soup partition
function, and take the vague $\delta\to 0$ limit, we obtain a measure whose
Radon-Nikodym derivative with respect to $\nu$ is
$\exp\bigl{(}\frac{1}{48\pi}(\rho,\rho)_{\nabla}\bigr{)}$. When the measure is
a certain regularized Liouville quantum gravity measure, a companion work
[APPS20] shows that this weighting has the effect of changing the so-called
central charge of the surface.
Acknowledgments. We thank Morris Ang, Ewain Gwynne, Camillo De Lellis, Sung-
jin Oh, and Peter Sarnak for helpful comments. The authors were partially
supported by NSF grants DMS 1712862 and DMS 2153742. J.P. was partially
supported by a NSF Postdoctoral Research Fellowship under grant 2002159.
## 1 Introduction
Let us first recall a few standard definitions and observations. On a compact
surface with boundary, the heat kernel trace can be written
$Z=Z(t)=\textrm{sp}\,e^{t\Delta}=\sum e^{t\lambda_{n}}$ where $\lambda_{n}$
are the eigenvalues of the Laplace-Beltrami operator $\Delta$. If $I(s)$ is
the number of $-\lambda_{n}$ less than $s$ then
$\int_{0}^{\infty}e^{-ts}I(s)ds=\int_{0}^{\infty}e^{-ts}\sum_{n=0}^{\infty}1_{s>-\lambda_{n}}ds=\sum_{n=0}^{\infty}\int_{-\lambda_{n}}^{\infty}e^{-ts}ds=\frac{1}{t}\sum
e^{t\lambda_{n}}=Z(t)/t.$
In other words, $Z/t$ is the Laplace transform of $I$. The asymptotics of $Z$
(as $t\to 0$) are therefore closely related to the asymptotics of
$\lambda_{n}$ (as $n\to\infty$). Weyl addressed the latter for bounded planar
domains $D$ in 1911 [Wey11] (see discussion in [MS+67]) by showing
$-\lambda_{n}\sim\frac{2\pi n}{\mathrm{Vol}(D)}$ as $n\to\infty$ which is
equivalent to
$Z\sim\frac{\mathrm{Vol}(D)}{4\pi t}$ (1.1)
as $t\to 0$. In 1966 Kac gave higher order correction terms for $Z$ on domains
with piecewise linear boundaries (accounting for boundary length and corners)
in his famously titled “Can you hear the shape of a drum?” which asks what
features of the geometry of $D$ can be deduced from $I$ (or equivalently from
$Z$) [Kac66]. McKean and Singer extended these asymptotics from planar domains
to smooth manifolds with non-zero curvature [MS+67] where the constant order
correction term is a certain curvature integral. For two dimensional surfaces,
with metric given by $e^{\rho}$ times a flat metric, the integral
$\int_{\delta}^{\infty}\frac{Z(t)}{t}dt$ turns out to be a natural quantity
whose small $\delta$ asymptotics involve a constant order term that
corresponds to the Dirichlet energy of $\rho$ (the so-called Polyakov-Alvarez
formula, also known as the Polyakov-Ray-Singer or Weyl anomaly formula) [RS73,
Pol81, Alv83, Sar87, OPS88]. This constant order Dirichlet energy term (which
can also be formulated in terms of Brownian loop soups, see below) is the main
concern of this paper.
Much of the literature assumes that $\rho$ is smooth and makes regular use of
objects like curvature that are not well defined if $\rho$ is not $C^{2}$. But
it is known [Hör68] that if $\rho$ is only $C^{2}$ then Weyl’s law still
holds, i.e. $-\lambda_{n}\sim\frac{2\pi n}{\mathrm{Vol}_{\rho}(D)}$ [AHT18,
Example 4.9] and Weyl’s law has been established in certain less smooth
settings as well.111We remark that there is a general theory of metric measure
spaces with the so-called “Riemannian curvature-dimension” (RCD) condition,
not necessarily confined to conformal changes of flat metrics. They include
Ricci limit spaces [Stu06, LV09], weighted Riemannian manifolds [Gri06],
Alexandrov spaces [Pet10], and many others. A lower bound on Ricci curvature
is a key ingredient of many useful estimates in geometric analysis, so Sturm,
Lott and Villani [Stu06, LV09] initiated the study of a class of metric
measure spaces with a generalized lower-Ricci-bound condition. This has been
an active research topic for the last decade; see [Gig18] for an overview. The
classical Weyl’s law and the short time asymptotics for heat kernels on these
non-smooth metric measure spaces still hold [ZZ17, AHT18]. Many aspects of the
theory are stable under the pointed measured Gromov-Hausdorff topology; for
example eigenvalues, heat kernels, and Green’s function converge uniformly
[Din02, ZZ17], Brownian motions converge weakly [Suz19], etc. Therefore,
Weyl’s law holds for any RCD space with a measure that can be reasonably
approximated. On the other hand, the short time expansion used to define the
functional determinant does not exist in this non-smooth setting, so it is not
clear if the zeta regularization procedure is also stable. The problem is
somewhat different when the regularity is below $C^{2}$, since curvature is no
longer well-defined everywhere and the relevant estimates no longer hold
pointwise and instead hold in an average sense. The primary purpose of this
note is to extend some of the basic results in this subject about the
conformal anomaly (the Dirichlet energy of $\rho$) to $\rho$ that are less
regular—e.g., only Lipschitz—and to show that the rates of convergence can be
made to hold uniformly across certain families of $\rho$ values.
This paper is motivated in part by another work by the authors [APPS20] in
which similar results are formulated in terms of the so-called Brownian loop
measures which were introduced in [LW04] and are related to heat kernel traces
on planar domains in e.g. [Dub09, Wan18] as well as [APPS20]. The results here
are useful in the context of [APPS20] because they strengthen the sense in
which one can say that “decorating” regularized Liouville quantum gravity
surfaces by Brownian loop soups has the effect of changing their central
charge. We will formulate the results in this paper solely in terms of
Brownian loop measures and their generalizations. (The relationship to heat
kernels is explained in [APPS20].)
In addition to the weaker regularity assumptions and the use of generalized
loop measures, there are several smaller differences between our presentation
and the classical approach in e.g. [MS+67]: we work in the conformal gauge
throughout and do all our calculations in terms of $\rho$, we index loops by
their Euclidean center rather than by a typical point on the loop (which would
be more similar to the heat kernel approach), and we establish the Polyakov-
Alvarez formula in terms of Dirichlet energy directly rather than first
establishing an equivalent curvature integral.
Although we encounter some complexity due to the non-smoothness of $\rho$, we
also take advantage of the extra simplicity of the two-dimensional setting,
where the manifold is completely determined by a conformal factor.
Finally, we note that there is a great deal of additional work in this area,
and we cannot begin to survey it all. For example, reference texts such as
[BGV03, Gil18] explore heat kernel traces in greater generality: dimensions
other than $2$, operators other than the Laplacian, etc. Other works extend
the behavior known for compact smooth manifolds to specific non-smooth
manifolds such as those with conical singularities or boundary corners (which
both correspond to logarithmic singularities in $\rho$) [Moo99, Kok13, She13,
She15, AR18, Gre21, Kal21] or to non-compact surfaces [AAR13]. There are also
many open problems in this subject, which spans probability, geometry, number
theory, mathematical physics, and analysis. We present a few of these
questions in Section 6. We hope that the techniques and perspectives presented
here will facilitate progress on these problems and perhaps also find
applications in other contexts where Weyl’s formula and the Polyakov-Alvarez
term appear.
### 1.1 Main result
Let $\mathcal{L}$ denote the set of zero-centered unit-length loops in
$\mathbb{R}^{2}$. We define the Brownian loop measure in the plane by encoding
each loop in the plane as an element of
$\mathbb{R}^{2}\times(0,\infty)\times\mathcal{L}$ and formulating the Brownian
loop measure as a measure on this product space.
###### Definition 1.1.
We express every loop $L$ in $\mathbb{R}^{2}$ by the triple $(x,t,\ell)$,
where
* •
$t=\mathrm{len}(L)$ is the _length_ of $L$, where we define the length of a
path as the length of its parametrizing interval.
* •
$x=\mathrm{cen}(L)$ is the _center_ of $L$, Euclidean center of mass of $L$,
which is equal to $t^{-1}\int_{0}^{t}L(s)ds$.
* •
$\ell$ is the zero-centered unit-length loop $s\mapsto t^{-1/2}(L(ts)-x)$
obtained from $L$ by translating the center to zero and rescaling time by
$t^{-1}$ and space by $t^{-1/2}$.
###### Definition 1.2.
We define the _Brownian loop measure_ on $\mathbb{R}^{2}$ as the measure on
loops $(x,t,\ell)$ in $\mathbb{R}^{2}$ given by
$\frac{1}{2\pi t^{2}}dx\,dt\,d\ell,$
where $dx$ denotes Lebesgue measure on $\mathbb{R}^{2}$, $dt$ is Lebesgue
measure on $(0,\infty)$ and $d\ell$ is the probabilistic law of the random
loop in $\mathcal{L}$ obtained by first sampling a two-dimensional Brownian
bridge on $[0,1]$ and then subtracting its mean.222Equivalently $d\ell$ on the
complex plane is the law of the complex-valued GFF indexed by the unit-length
circle—with additive constant chosen to make the mean zero. In particular,
$d\ell$ is invariant under rotations of that circle.
The mass of the set of Brownian loops centered in $D$ with size greater than
$\delta$ is given by
$\int_{D}\int_{\delta}^{\infty}\frac{1}{2\pi
t^{2}}\,dt\,dx=\frac{\mathrm{Vol}(D)}{2\pi\delta}.$ (1.2)
In particular, (1.2) implies that no matter how small $\delta$ is, most loops
with length $t\in(\delta,\infty)$ have length of order $\delta$: half of them
have $t<2\delta$, ninety-five percent have $t<20\delta$, and so forth. Also,
the fact that (1.2) tends to $0$ as $\delta\to\infty$ informally means that
there are very few large loops centered in $D$.
Our main result describes how this mass of Brownian loops changes when we
measure the length of loops with respect to a different metric
$e^{\rho}|dz|^{2}$ on the plane, for $\rho$ a Lipschitz function supported in
$D$. We begin by defining the length of a Brownian loop in the metric
$e^{\rho}|dz|^{2}$, which we call its _$\rho$ -length_. If the loop were a
smooth curve, we would compute its $\rho$-length by integrating $e^{\rho/2}$
along the curve. Since Brownian loops have Hausdorff dimension $2$, we instead
define its $\rho$-length by integrating $e^{\rho}$ along the loop, so that it
has the same scale factor as area.
###### Definition 1.3.
Let $(M,g)$ be a smooth two-dimensional Riemannian manifold, and let $\rho$ be
a function on $M$. We define the $\rho$-length $\mathrm{len}_{\rho}(L)$ of a
loop $L$ as $\int_{0}^{\mathrm{len}(L)}e^{\rho(L(s))}ds$. We define the
$\rho$-volume form $\mathrm{Vol}_{\rho}$ as the volume form associated to
$(M,e^{\rho}g)$, and we write $\mathrm{Vol}:=\mathrm{Vol}_{0}$.
Except in Theorem 1.9 and Section 4, we always take $(M,g)$ in Definition 1.3
to be the Euclidean plane. We first describe the space of functions in the
scope of this section.
###### Theorem 1.4.
Let $D$ be a bounded open subset of $\mathbb{R}^{2}$, and
$\operatorname{Lip}(D)$ be the space of real-valued Lipschitz functions that
vanish outside of $D$. Suppose that $\mathcal{B}\subset\operatorname{Lip}(D)$
is a collection of functions that (1) has uniformly bounded Lipschitz
constants, and (2) is precompact in $W^{1,1}(D)$.
Then as $\delta\to 0$, the $\mu^{\text{loop}}$-mass of loops centered in $D$
with $\rho$-length at least $\delta$, with respect to the Brownian loop
measure, is given by
$\frac{\mathrm{Vol}_{\rho}(D)}{2\pi\delta}+\frac{1}{48\pi}(\rho,\rho)_{\nabla}+o(1)$
(1.3)
with the convergence uniform over $\rho\in\mathcal{B}$.
###### Remark 1.5 (Uniform boundedness).
The conditions (1) and (2) imply that the functions in $\mathcal{B}$ are
uniformly bounded.
###### Remark 1.6 (General $p$).
Since we require for now that the Lipschitz constant is uniformly bounded and
the domain is bounded, precompactness in $W^{1,1}$ is equivalent to
precompactness in $W^{1,p}$ for any fixed $p\in(1,\infty)$. In particular,
Theorem 1.4 could have been formulated using precompactness in $W^{1,2}$
instead of $W^{1,1}$. Let us also remark that the space $W^{1,2}(D)$ is
equivalent to the space $H^{1}(D)$ of $\rho$ for which $(\rho,\rho)_{\nabla}$
is finite.
###### Remark 1.7 (Precompactness and uniform equicontinuity).
Recall the Fréchet-Kolmogorov theorem (e.g., see [BB11]): let
$D\subset\mathbb{R}^{n}$ be a bounded domain, and $1\leq p<\infty$. A subset
$\mathcal{A}\subset L^{p}(D)$ is precompact if and only if $\mathcal{A}$ is
bounded in $L^{p}(D)$ and
$\sup_{u\in\mathcal{A}}\,\int_{D}\left|u(x+h)-u(x)\right|^{p}\,dx\to
0\quad\text{as }h\to 0,$ (1.4)
where $u$ is extended to the function on $\mathbb{R}^{n}$ whose value outside
$D$ is zero.
We will use this equivalent characterization of precompactness in some of our
proofs, usually referred as the _uniform equicontinuity_ condition in $L^{p}$.
In particular, we will apply this, in the case $p=1$, to the set $\mathcal{A}$
of the gradients of the functions in the set $\mathcal{B}$ from the statement
of Theorem 1.4.
###### Remark 1.8 (The uniform equicontinuity condition is necessary).
As mentioned above, the precompactness hypothesis is equivalent to a type of
uniform equicontinuity hypothesis. This hypothesis—or some similar condition
on the functions $\rho\in\mathcal{B}$—is necessary for the conclusion of
Theorem 1.4 (or Theorem 1.12) to hold. Simply requiring all surfaces in
$\mathcal{B}$ to be $C^{1}$ with a universal bound on $|\nabla\rho|$ would not
suffice. For example, in the Theorem 1.4 setting, $\mathcal{B}$ could contain
a sequence $\rho_{1},\rho_{2},\ldots$ of $C^{1}$ functions that converge
uniformly to zero with $(\rho_{j},\rho_{j})_{\nabla}=1$ for each $j$. We can
construct such a sequence of functions $\rho_{j}$ by arranging for
$\nabla\rho_{j}$ to oscillate between fixed opposite values, with the
oscillation rate becoming faster as $j\to\infty$. (A simple example of such a
family of functions on the torus $[0,2\pi)^{2}$ is given by a constant
multiple of $\rho_{j}\Bigl{(}(a,b)\Bigr{)})=j^{-1}\sin(ja)$; we can define
$\rho_{j}$ similarly on the planar domain $D$ by tapering the sine functions
to zero near the boundary of $D$.) We can also perturb the functions to
arrange that $\mathrm{Vol}_{\rho_{j}}(D)=\mathrm{Vol}_{\rho}(D)$ for all $j$.
This set of functions $\mathcal{B}$ does not satisfy (1.3): for any fixed
$\delta$, one can easily show that
$\lim_{j\to\infty}\mu\\{L:\mathrm{cen}(L)\in
D,\,\mathrm{len}_{\rho_{j}}(L)\geq\delta\\}-\frac{\mathrm{Vol}(D)}{\delta}=0,$
(1.5)
even though for each fixed $j$ we have
$\lim_{\delta\to 0}\bigl{(}\mu\\{L:\mathrm{cen}(L)\in
D,\,\mathrm{len}_{\rho_{j}}(L)\geq\delta\\}-\frac{\mathrm{Vol}(D)}{\delta}\bigr{)}=b/2.$
(1.6)
If the limit in (1.6) were uniform in $j$, we could choose a $\delta$ with
$\mu\\{L:\mathrm{cen}(L)\in D,\,\mathrm{len}_{\rho_{j}}(L)\geq\delta\\}>b/4$
for all $j$, and (1.5) would not hold for that $\delta$.333One might wonder
whether it is enough have $\rho$ in the Hilbert space defined by the inner
product $(\rho,\rho)_{\nabla}$, i.e., the Sobolev space $W^{1,2}(D)=H^{1}(D)$,
with (say) zero boundary conditions. Such a $\rho$ can be nowhere
differentiable [Ser61], so one would also need to modify the condition on
$\mathcal{B}$. See Question 6.1.
We extend this result to general surfaces. The statement of the theorem
involves the notion of the zeta-regularized determinant of the Laplacian, as
defined, e.g., in [Alv83, Sar87]. (However, it is not necessary to understand
the definition of the zeta-regularized determinant to follow the proof of
Theorem 1.9.)
###### Theorem 1.9.
Let $(M,g)$ be a fixed compact smooth two-dimensional Riemannian manifold, and
we let $\mu^{\text{loop}}$ denote the Brownian loop measure on $(M,g)$. Let
$K$ be the Gaussian curvature on $M$, let $\Delta$ be the Laplacian associated
to $(M,g)$, and let $\det_{\zeta}^{\prime}\Delta$ denote its zeta-regularized
determinant. Let $\mathcal{B}$ be a family of Lipschitz functions that (1) has
uniformly bounded Lipshitz constants, and (2) is precompact in $W^{1,1}(M)$.
Then the $\mu^{\text{loop}}$-mass of loops with $\rho$-length between $\delta$
and $C$ is given by
$\displaystyle\frac{\mathrm{Vol}_{\rho}(M)}{2\pi\delta}-{\frac{\chi(M)}{6}}(\log\frac{\delta}{2}+\upgamma)+\log{C}+\upgamma+\frac{1}{48\pi}\int_{M}(\|\nabla\rho\|^{2}+2K\rho)\mathrm{Vol}(dz)$
$\displaystyle\qquad+\log\mathrm{Vol}(M)-\log\mathrm{Vol}_{\rho}(M)-\log\det\nolimits_{\zeta}^{\prime}\Delta+o_{\delta}(1)+o_{C}(C^{(-1+\varepsilon)/2}),$
(1.7)
with the convergence as $\delta\to 0$ and $C\to\infty$ uniform over
$\rho\in\mathcal{B}$, where $\upgamma\approx 0.5772$ is the Euler-Mascheroni
constant.
For simplicity, we have addressed just the compact manifold case, but one
could prove a similar result for manifolds with boundary, see Question 6.3
where we give a heuristic justification in the preceding paragraph; the
resulting expression would include a boundary term which is of order
$\delta^{-1/2}$.
Observe that, for smooth $\rho$, the expression in the second line (1.7) of
the above expression is equal to $-\log\det_{\zeta}^{\prime}\Delta_{\rho}$,
where $\Delta_{\rho}$ is the Laplacian associated to $(M,e^{\rho}g)$. The
expression (1.7) for $-\log\det_{\zeta}^{\prime}\Delta_{\rho}$ is known as the
Polyakov-Alvarez formula; see, e.g., [APPS20, Proposition 6.9]. Thus, for
smooth $\rho$, Theorem 1.9 reduces to a relation between the Brownian loop
measure and the zeta-regularized Laplacian determinant, which was shown in
[APPS20, Theorem 1.3].
In fact, we prove a slight generalization of Theorem 1.4 in which we consider
a more general class of loop measures.
###### Definition 1.10.
The _expected occupation measure_ of a random variable
$Z:[0,T]\rightarrow\mathbb{R}^{2}$ is the function
$\theta:\mathbb{R}^{2}\rightarrow(0,\infty)$ such that, for each measurable
set $A\subset\mathbb{R}^{2}$, the set $\\{t\in[0,T]:Z(t)\in T\\}$ has expected
Lebesgue measure $\int_{A}\theta(x)dx$.
###### Definition 1.11.
We define a _generalized loop measure_ $\mu$ as a measure on loops
$(x,t,\ell)$ in $\mathbb{R}^{2}$ given by
$\frac{1}{t^{2}}dx\,dt\,d\ell,$
where $dx$ denotes Lebesgue measure on $\mathbb{R}^{2}$, $dt$ is Lebesgue
measure on $(0,\infty)$ and $d\ell$ is an arbitrary rotationally invariant
measure on loops in $\mathcal{L}$ whose expected occupation measure is a
Schwartz distribution (but not necessarily Gaussian as for the Brownian loop
measure). We denote by $b$ the second central moment of the first—or
equivalently, second—coordinate of a random variable whose density is this
expected occupation measure.
Definition 1.11 is the same as Definition 1.2, except that we no longer insist
that $d\ell$ be the Brownian bridge (and we have removed the $2\pi$ factor as
it is less natural for general $d\ell$). The measure $d\ell$ can be supported
on the space of circular loops, square-shaped loops, or outer boundaries of
Brownian loops, etc. The $\mu$ from Definition 1.11 need not have the same
conformal symmetries as the Brownian loop measure. Even if $d\ell$ is
supported on smooth loops (rather than Brownian loops) we parameterize the
space of loops as in Definition 1.1, so that $(0,t,\ell)$ represents the loop
that traces $\sqrt{t}\ell$ over time duration $t$. In the special case of the
Brownian loop measure, $b=1/12$. (See Proposition 3.9.) The Schwartz
distribution assumption in Definition 1.11 does not seem necessary for Theorem
1.12 to hold, but we have included it to simplify the calculations in the
proof of Proposition 3.1 below.
###### Theorem 1.12.
Let $D$ and $\mathcal{B}$ be as in Theorem 1.4, and let $\mu$ be a generalized
loop measure in the sense of Definition 1.11. Then as $\delta\rightarrow 0$,
the $\mu$-mass of loops centered in $D$ with $\rho$-length at least $\delta$
is given by
$\frac{\mathrm{Vol}_{\rho}(D)}{\delta}+\frac{b}{2}(\rho,\rho)_{\nabla}+o(1),$
(1.8)
with the convergence uniform over $\rho\in\mathcal{B}$.
### 1.2 Proof outline
In this section, we let $\mathcal{B}$ be a fixed collection of functions
$\rho$ satisfying the hypotheses of Theorem 1.12. To prove Theorem 1.12, we
compare $\mathrm{len}_{\rho}$ to a simpler notion of the length of a loop with
respect to $e^{\rho}|dz|^{2}$, in which we approximate $e^{\rho}$ along the
loop by its value at the center of the loop.
###### Definition 1.13.
We define the _center $\rho$-length_ $\mathrm{clen}_{\rho}(L)$ of a loop $L$
in $\mathbb{R}^{2}$ centered at a point $x$ as
$\int_{0}^{t}e^{\rho(x)}ds=e^{\rho(x)}t$.
We observe that the cutoff $\mathrm{clen}_{\rho}(L)=\delta$ corresponds to a
unique value of $\mathrm{len}(L)$:
###### Proposition 1.14.
Let $\delta>0$, $x\in D$ and $\ell\in\mathcal{L}$, and set
$\beta:=e^{-\rho(x)}\delta$. The loop $L=(x,t,\ell)$ satisfies
$\mathrm{clen}_{\rho}(L)=\delta$ iff $t=\beta$, and
$\mathrm{clen}_{\rho}(L)\geq\delta$ iff $t\geq\beta$.
###### Proof.
The result follows trivially from the definition of center $\rho$-length. ∎
Proposition 1.14 immediately implies the following trivial analogue of Theorem
1.12 for center $\rho$-length.
###### Proposition 1.15.
The mass of loops $L$ centered in $D$ with $\mathrm{clen}_{\rho}\geq\delta$
with respect to the Brownian loop measure is given by
$\int_{D}\int_{\mathcal{L}}\int_{\beta}^{\infty}\frac{1}{t^{2}}\,dt\,d\ell\,dx=\int_{D}\beta^{-1}dx=\int_{D}\frac{e^{\rho(x)}}{\delta}dx=\frac{\mathrm{Vol}_{\rho}(D)}{\delta}.$
(1.9)
###### Proof.
The result is an immediate consequence of Proposition 1.14. ∎
We can therefore restate Theorem 1.12 as the assertion that if we change our
notion of loop length from $\mathrm{clen}_{\rho}$ to $\mathrm{len}_{\rho}$,
the $\mu$-mass of loops with length $\geq\delta$ increases by
$\frac{b}{2}(\rho,\rho)_{\nabla}$, up to an error that is $o(1)$ as $\delta\to
0$ uniformly in $\rho\in\mathcal{B}$.
We divide the proof of Theorem 1.12 into two stages. First, in Section 2 we
show that, up to a uniform $o(1)$ error, replacing $\mathrm{clen}_{\rho}$ with
$\mathrm{len}_{\rho}$ has the effect of subtracting $\delta^{-1}$ times the
average discrepancy between the value of $e^{\rho}$ along a Brownian loop and
the value of $e^{\rho}$ at its center.
###### Lemma 1.16.
Consider a loop sampled from $\mu$ conditioned to have its center in $D$ and
length $\beta$. Let $X$ denote its center, and let $Z$ denote a point on the
loop sampled uniformly with respect to length. Then the $\mu$-mass of loops
$L$ with center in $D$ and $\mathrm{len}_{\rho}(L)\geq\delta$ is equal to the
$\mu$-mass of loops $L$ with center in $D$ and
$\mathrm{clen}_{\rho}(L)\geq\delta$, minus
$\frac{1}{\delta}\mathbb{E}[e^{\rho(Z)}-e^{\rho(X)}]+o(1),$ (1.10)
with the $o(1)$ error tending to 0 as $\delta\to 0$ at a rate that is uniform
in $\rho\in\mathcal{B}$. (In (1.10) the expectation is w.r.t. the overall law
of $X$ and $Z$ as described above.)
We then complete the proof of Theorem 1.12 in Section 3 by showing that the
quantity (1.10) equals $\frac{b}{2}(\rho,\rho)_{\nabla}$ up to a uniform
$o(1)$ error.
###### Lemma 1.17.
The quantity (1.10) equals $\frac{b}{2}(\rho,\rho)_{\nabla}$ plus an error
term that converges to $0$ as $\delta\rightarrow 0$ uniformly in
$\rho\in\mathcal{B}$.
## 2 Loop mass difference vs. expected length discrepancy
In this section, we prove Lemma 1.16 in three steps.
Step 1: Establishing a length threshold $\alpha>0$ corresponding to
$\rho$-length $\delta$. We saw in Proposition 1.14 that, with
$\beta=e^{-\rho(x)}\delta$, we have $\mathrm{clen}_{\rho}(L)=\delta$ if and
only if $t=\beta$, and $\mathrm{clen}(L)\geq\delta$ if and only if
$t\geq\beta$. To prove Lemma 1.16, we establish a similar result for
$\rho$-length. We will show in Proposition 2.4 below that, for $x\in D$ and
$\ell\in\mathcal{L}$ with the diameter of $(x,\delta,\ell)$ sufficiently
small, there exists a threshold $\alpha>0$ such that
$\mathrm{len}_{\rho}((x,t,\ell))=\delta$ if and only if $t=\alpha$, and
$\mathrm{len}((x,t,\ell))\geq\delta$ if and only if $t\geq\alpha$. We note
that, unlike $\beta$, the threshold $\alpha$ may depend on $\ell$ as well as
$x$ and $\delta$.
Step 2: Relating the difference in the masses of loops to the quantity
$\alpha^{-1}$. We saw in Proposition 1.15 that the $\mu$-mass of loops with
center $\rho$-length $\geq\delta$ can be expressed as an integral of
$\beta^{-1}$. In Proposition 2.5 below, we similarly express the $\mu$-mass of
loops with $\rho$-length $\geq\delta$ as an integral of $\alpha^{-1}$, plus a
uniform $o(1)$ error. This reduces the task of proving Lemma 1.16 to analyzing
the difference of integrands $\alpha^{-1}-\beta^{-1}$.
Step 3: Expressing the difference $\alpha^{-1}-\beta^{-1}$ in terms of a
difference in lengths. We first express the difference $\alpha-\beta$ in terms
of a difference between the $\rho$-length and center $\rho$-length of a loop
(Proposition 2.7). We then apply this result in Proposition 2.8 to derive a
similar expression for $\alpha^{-1}-\beta^{-1}$, which immediately yields
Lemma 1.16.
Having described the main steps of the proof of Lemma 1.16, we now proceed
with Step 1—proving the existence of the threshold $\alpha$. As we observed in
Proposition 1.14, the existence of the analogous threshold $\beta$ for center
$\rho$-length is trivial, since for fixed $\rho,x,$ and $\ell$, the function
$t\mapsto\mathrm{clen}_{\rho}(L)$ is linear with slope $e^{\rho(x)}$. This is
not the case for $\rho$-length, so we proceed by showing its derivative as a
function of $t$ is positive on a sufficiently large interval. We first observe
that, since the functions $\rho\in\mathcal{B}$ are uniformly bounded above and
below, we can crudely bound the function $t\mapsto\mathrm{len}_{\rho}(L)$
between two linear functions uniformly in $\rho,x,$ and $\ell$.
###### Proposition 2.1.
There exists a constant $\Lambda>0$ such that, for each $\rho\in\mathcal{B}$,
$x\in D$ and $\ell\in\mathcal{L}$, the loop $L=(x,t,\ell)$ satisfies
$\Lambda^{-1}\leq\frac{\mathrm{len}_{\rho}(L)}{\mathrm{len}(L)}\leq\Lambda.$
(2.1)
In other words, $\mathrm{len}(L)$ and $\mathrm{len}_{\rho}(L)$ length agree up
to a universal constant factor.
###### Proof.
The lemma follows immediately from the fact that $\rho$ is bounded from above
and below by a constant uniform in $\rho\in\mathcal{B}$, as noted in Remark
1.5. ∎
In addition, the collection of functions $\\{e^{\rho}\\}_{\rho\in\mathcal{B}}$
satisfies the same conditions of $\mathcal{B}$, possibly with different
bounds.
###### Proposition 2.2.
The functions $e^{\rho}$ for $\rho\in\mathcal{B}$ also have uniformly bounded
Lipschitz constants and are precompact in $W^{1,1}(D)$.
###### Proof.
Since the functions $\rho\in\mathcal{B}$ are uniformly bounded, their images
are contained in some finite closed interval. The exponential function is
Lipschitz on any finite closed interval, so the composition $\exp\circ\rho$ is
also Lipschitz. Other conditions are straightforward to check. ∎
We now apply the uniform boundedness of the Lipschitz constants of $e^{\rho}$
for $\rho\in\mathcal{B}$ to show that, when
$\mathrm{diam}(L)=\mathrm{diam}(\ell)\sqrt{t}$ is not too large, the
derivative of the function $t\mapsto\mathrm{len}_{\rho}(L)$ is uniformly close
to that of the linear function $t\mapsto\mathrm{clen}_{\rho}(L)$.
###### Proposition 2.3.
For any $\varepsilon>0$, there exists $d=d(\varepsilon)>0$ and a family of
sets $\\{D^{\rho,\varepsilon}\\}_{\rho\in\mathcal{B}}$ with
$\mathrm{Vol}(D^{\rho,\varepsilon})\leq\varepsilon$ such that the following is
true. For each $\rho\in\mathcal{B}$, $x\in D\setminus D^{\rho,\varepsilon}$,
and $\ell\in\mathcal{L}$, the derivative
$\frac{\partial}{\partial t}\mathrm{len}_{\rho}((x,t,\ell))$
exists and differs from
$\frac{\partial}{\partial t}\mathrm{clen}_{\rho}((x,t,\ell))=e^{\rho(x)}$
by at most $\varepsilon\,\mathrm{diam}(\ell)\sqrt{t}$ for almost every $t>0$
with $\mathrm{diam}(\ell)\sqrt{t}<d$. Furthermore, there exists a constant
$\widetilde{\Lambda}>0$ such that the previous statement it true for arbitrary
$d$ and $D^{\rho,\varepsilon}=\emptyset$ when we choose
$\varepsilon=\widetilde{\Lambda}$.
###### Proof.
By Rademacher’s theorem, any Lipschitz function is differentiable almost
everywhere. In particular, by Proposition 2.2, there exists some constant
$\widetilde{\Lambda}>0$ that does not depend on $\rho$ such that
$\left|\nabla(e^{\rho})\right|<\frac{2}{3}\widetilde{\Lambda}$ for almost
every $x\in D$ for all $\rho\in\mathcal{B}$, where $\nabla(e^{\rho})$ is a
weak gradient of $e^{\rho}$. In addition, as $\mathcal{B}$ is precompact in
$W^{1,1}(D)$, it follow from (1.4) that there exists some $d>0$ such that the
set
$D^{\rho,\varepsilon}:=\\{x\in D:\operatorname*{ess\,sup}_{|h|\leq
d}\left|(\nabla e^{\rho})(x+h)-\nabla
e^{\rho}(x)\right|>\frac{2}{3}\varepsilon\\}$ (2.2)
satisfies $\mathrm{Vol}(D^{\rho,\varepsilon})\leq\varepsilon$ for each
$\rho\in\mathcal{B}$.
For fixed $\rho\in\mathcal{B}$, $x\in D$, and $\ell\in\mathcal{L}$, we can
write $\mathrm{len}_{\rho}((x,t,\ell))$ as $A(\sqrt{t})t$, where
$A(r)=A_{\rho,x,\ell}(r):=\int_{0}^{1}e^{\rho}(r\ell(s)+x)ds$. We express a
weak $t$-derivative of $\mathrm{len}_{\rho}((x,t,\ell))$ in terms of $A$ as
$\frac{\partial}{\partial
t}\mathrm{len}_{\rho}((x,t,\ell))=\frac{\partial}{\partial
t}(A(\sqrt{t})t)=\frac{1}{2}t^{-1/2}A^{\prime}(\sqrt{t})t+A(\sqrt{t})=\frac{1}{2}A^{\prime}(\sqrt{t})\sqrt{t}+A(\sqrt{t}).$
(2.3)
Since
$\displaystyle A^{\prime}(r)$ $\displaystyle=\int_{0}^{1}\ell(s)\cdot(\nabla
e^{\rho})(r\ell(s)+x)ds$ (2.4) $\displaystyle=\int_{0}^{1}\ell(s)\cdot((\nabla
e^{\rho})(r\ell(s)+x)-(\nabla e^{\rho})(x))ds,$ (2.5)
we can bound $|A^{\prime}(\sqrt{t})|$ from above by
$\mathrm{diam}(\ell)\sup_{h\in\mathbb{R}^{2}}\left|(\nabla
e^{\rho})(x+h)\right|\leq\frac{2}{3}\widetilde{\Lambda}\,\mathrm{diam}(\ell)$
for almost every $t>0$, using (2.4). On the other hand, if $x\in D\setminus
D^{\rho,\varepsilon}$, we use (2.2) and (2.5) to bound
$|A^{\prime}(\sqrt{t})|$ from above by
$\mathrm{diam}(\ell)\operatorname*{ess\,sup}_{|h|\leq\mathrm{diam}(\ell)\sqrt{t}}\left|(\nabla
e^{\rho})(x+h)-\nabla
e^{\rho}(x)\right|\leq\frac{2}{3}\varepsilon\,\mathrm{diam}(\ell)$
for almost every $t>0$ with $\mathrm{diam}(\ell)\sqrt{t}<d$.
Note that (2.3) gives
$\left|\frac{\partial}{\partial
t}\mathrm{len}_{\rho}((x,t,\ell))-A(\sqrt{t})\right|=\frac{1}{2}\left|A^{\prime}(\sqrt{t})\right|\sqrt{t},$
and we also have
$|A(\sqrt{t})-e^{\rho(x)}|=|A(\sqrt{t})-A(0)|\leq\int_{0}^{\sqrt{t}}|A^{\prime}(s)|ds\leq\sup_{0\leq
s\leq\sqrt{t}}\left|A^{\prime}(s)\right|\sqrt{t}.$ (2.6)
Combining these two inequalities with the estimates for
$\left|A^{\prime}(\sqrt{t})\right|$ in two different cases gives the desired
result. ∎
We use Proposition 2.3 to show that, just as for center $\rho$-length, there
is a _unique_ value $\alpha$ of $\mathrm{len}(L)$ corresponding to
$\mathrm{len}_{\rho}(L)=\delta$.
###### Proposition 2.4.
We can choose $d_{*}>0$ such that the following holds. For each
$\rho\in\mathcal{B}$ and $x\in D$, if $L=(x,\delta,\ell)$ is a loop with
$\mathrm{diam}(\ell)\sqrt{\delta}$ less than $d_{*}$, then there is a unique
positive $\alpha=\alpha(x,\delta,\ell,\rho)$ such that
$\mathrm{len}_{\rho}((x,t,\ell))=\delta$ iff $t=\alpha$ and
$\mathrm{len}_{\rho}((x,t,\ell))>\delta$ iff $t>\alpha$. (If
$\mathrm{diam}(\ell)\sqrt{\delta}$ is $\geq d_{*}$, then we arbitrarily set
$\alpha=\delta$, so that $\alpha$ is defined for every loop
$L=(x,\delta,\ell)$.)
###### Proof.
Let $\Lambda$ be the constant in Proposition 2.1. If $t\geq\Lambda\delta$,
then $\mathrm{len}_{\rho}((x,t,\ell))\geq\Lambda^{-1}t\geq\delta$ by
Proposition 2.1. Thus, it suffices to show that, for
$\mathrm{diam}(\ell)\sqrt{\delta}$ sufficiently small, the function
$\mathrm{len}_{\rho}((x,t,\ell))$ is strictly increasing in $t$ when
$t\in(0,\Lambda\delta)$. We prove this fact by analyzing
$\frac{\partial}{\partial t}\mathrm{len}_{\rho}((x,t,\ell))$ using Proposition
2.3.
Let $\widetilde{\Lambda}$ be the constant in Proposition 2.3 so that for all
$\rho\in\mathcal{B}$, $x\in D,\ell\in\mathcal{L}$, and almost every $t>0$,
$\left|\frac{\partial}{\partial
t}\mathrm{len}_{\rho}((x,t,\ell))-e^{\rho(x)}\right|\leq\widetilde{\Lambda}\,\mathrm{diam}(\ell)\sqrt{t}.$
(2.7)
We choose $\mathrm{diam}(\ell)\sqrt{\delta}$ sufficiently small less than
$d_{*}>0$ such that, for almost every $t\leq\Lambda\delta$, the bound
$\widetilde{\Lambda}\,\mathrm{diam}(\ell)\sqrt{t}$ in (2.7) is less than
$\frac{1}{2}e^{\rho(x)}$. This means $\frac{\partial}{\partial
t}\mathrm{len}_{\rho}((x,t,\ell))$ is strictly positive for almost every
$t\leq\Lambda\delta$. Since $\mathrm{len}_{\rho}((x,t,\ell))$ is a continous
function in $t$, we conclude that $\mathrm{len}_{\rho}((x,t,\ell))$ is
strictly increasing on $(0,\Lambda\delta)$. ∎
When $x$ and $\ell$ are fixed, $\alpha$ gives the Euclidean $t$ value of
$\mathrm{len}$ that corresponds to a $\delta$ value of $\mathrm{len}_{\rho}$.
We obtain the following analogue of Proposition 1.15 with
$\mathrm{len}_{\rho}$ in place of $\mathrm{clen}_{\rho}$.
###### Proposition 2.5.
The $\mu$-mass of the set of loops $L$ with center in $D$ and
$\mathrm{len}_{\rho}(L)\geq\delta$ is equal to
$\int_{D}\int_{\mathcal{L}}\int_{\alpha}^{\infty}\frac{1}{t^{2}}\,dt\,d\ell\,dx=\int_{D}\int_{\mathcal{L}}\alpha^{-1}\,d\ell\,dx$
(2.8)
plus an error term that tends to $0$ as $\delta\rightarrow 0$ at a rate that
is uniform in $\rho\in\mathcal{B}$.
The reason the mass of loops does not exactly equal (2.8) is that $\alpha$ is
improperly defined when $\mathrm{diam}(\ell)\sqrt{\delta}\geq d_{*}$, with
$d_{*}$ as in Proposition 2.4. Proposition 2.5 asserts that the resulting
error is negligible in the $\delta\to 0$ limit.
To prove Proposition 2.5, we use the following bound on the $d\ell$-measure of
loops with large diameter.
###### Proposition 2.6.
The $d\ell$-measure of loops $\ell$ with diameter greater than $K$ tends to
$0$ as $K\rightarrow\infty$ faster than any negative power of $K$.
###### Proof.
If $\ell$ has diameter $>K$, then Proposition 2.1 implies that
$\int_{0}^{\mathrm{len}(\ell)}\mathbf{1}_{|\ell(s)|>K/3}dt>cK$ for some
constant $c>0$. It follows from Markov’s inequality that the probability of
this event is bounded from above by
$(cK)^{-1}\mathbb{E}(\int_{0}^{\mathrm{len}(\ell)}\mathbf{1}_{|\ell(s)|>K/3}dt)$.
By the Schwartz condition in Definition 1.11, the latter decays as
$K\rightarrow\infty$ faster than any negative power of $K$. ∎
###### Proof of Proposition 2.5.
By definition of $\alpha$, the integral in (2.8) gives the $\mu$-mass of loops
$L$ with $\mathrm{cen}(L)\in D$ and either
* •
$\mathrm{diam}(\ell)\sqrt{\delta}<d_{*}$ and
$\mathrm{len}_{\rho}(L)\geq\delta$, or
* •
$\mathrm{diam}(\ell)\sqrt{\delta}\geq d_{*}$ and $t\geq\delta$.
Thus, the error term—i.e., the difference between (2.8) and the mass of loops
we consider in the proposition statement—is the $\mu$-mass of loops $L$ with
$\mathrm{diam}(\ell)\sqrt{\delta}\geq d_{*}$ and for which either
* •
$t\geq\delta$ and $\mathrm{len}_{\rho}(L)<\delta$, or
* •
$t<\delta$ and $\mathrm{len}_{\rho}(L)\geq\delta$.
To bound this error, we recall from Proposition 2.1 that
$\mathrm{len}_{\rho}(L)$ is $>\delta$ when $t>\Lambda\delta$ and $<\delta$
when $t<\Lambda^{-1}\delta$. Thus, the error is at most the $\mu$-mass of
loops $L$ with center in $D$, $\mathrm{diam}(\ell)\sqrt{\delta}\geq d_{*}$ and
$\Lambda^{-1}\delta\leq t\leq\Lambda\delta$. This mass is equal to the
Lebesgue measure of $D$ times $(\Lambda-\Lambda^{-1})/\delta$ times the
$d\ell$-measure of the set of loops $\ell$ with
$\mathrm{diam}(\ell)\sqrt{\delta}>d_{*}$. From Proposition 2.6, we deduce that
the error tends to $0$ as $\delta\rightarrow 0$ at a rate that is uniform in
$\rho\in\mathcal{B}$. ∎
Proposition 2.5 reduces the problem of proving Lemma 1.16 to the problem of
analyzing the difference of integrands $\alpha^{-1}$ and $\beta^{-1}$ in (2.8)
and (1.9). We first derive an expression for $\alpha-\beta$ in terms of the
difference in the $\rho$-length and center $\rho$-length of the loop
$(x,\beta,\ell)$.
###### Proposition 2.7.
Let $\varepsilon,K>0$ be fixed. Then, for all $\ell\in\mathcal{L}$ with
diameter at most $K$, as $\delta\to 0$,
$\alpha-\beta=e^{-\rho(x)}\Bigl{(}\mathrm{len}_{\rho}(x,\beta,\ell)-\mathrm{clen}_{\rho}(x,\beta,\ell)\Bigr{)}+o(\delta^{2}),$
(2.9)
with the error converging uniformly in the choice of $\rho\in\mathcal{B}$,
$\ell$ with diameter at most $K$, and $x\in D\setminus D^{\rho,\varepsilon}$
where $\\{D^{\rho,\varepsilon}\\}_{\rho\in D}$ is defined as in Proposition
2.3. If we remove the restriction on $\mathrm{diam}(\ell)$, then the error is
$O(\delta^{2}\mathrm{diam}(\ell)^{2})$ uniformly in $\rho\in\mathcal{B}$,
$x\in D\setminus D^{\rho,\varepsilon}$, and $\ell\in\mathcal{L}$..
Figure 1: An illustration of the quantities we consider in Proposition 2.7.
With $\rho\in\mathcal{B}$, $x\in D\setminus D^{\rho,\varepsilon}$, and
$\ell\in\mathcal{L}$ fixed, the blue curve is the graph of
$t\mapsto\mathrm{clen}_{\rho}(L)$, and the red curve is the graph of
$t\mapsto\mathrm{len}_{\rho}(L)$. The blue curve is a line with slope
$e^{\rho(x)}$, and the red curve is differentiable with the same derivative
$e^{\rho(x)}$ at the origin. Here, we consider $\delta>0$ for which
$\mathrm{diam}(\ell)\sqrt{\delta}<d_{*}$, with $d_{*}$ as in Proposition 2.4.
By Proposition 2.4, the red curve intersects the horizontal (dotted) line of
height $\delta$ at the single red point $(\alpha,\delta)$. The blue curve
intersects the line of height $\delta$ at the blue point $(\beta,\delta)$. The
green point is the point on the red curve directly above the blue point, and
the purple line segment is the segment parallel to the blue line from the
green point to the dotted line (purple point). Proposition 2.7 asserts that
the distance between the blue and red points can be approximated by the
distance between the blue and purple points. The latter distance is simply the
length of the green segment divided by the slope $e^{\rho(x)}$ of the blue
line.
###### Proof.
Let $d_{*}>0$ be as in Proposition 2.4. Observe that if we are restricting to
$\ell$ with diameter at most some constant $K$, we automatically have
$\mathrm{diam}(\ell)\sqrt{\delta}<d_{*}$ for uniformly small $\delta$. If we
do not impose the restriction $\mathrm{diam}(\ell)\leq K$, then we could have
$\mathrm{diam}(\ell)\sqrt{\delta}>d_{*}$ for arbitrarily small $\delta$.
However, the proposition statement easily holds for this range of $\delta$ and
$\ell$: by Proposition 2.1, the error in the proposition statement must be
bounded by a uniform constant times $\delta$, which is
$O(\delta^{2}\mathrm{diam}(\ell)^{2})$ when
$\mathrm{diam}(\ell)\sqrt{\delta}>d_{*}$ because
$\delta^{2}\mathrm{diam}(\ell)^{2}>d_{*}^{2}\delta$. Thus, we may assume for
the rest of the proof that $\mathrm{diam}(\ell)\sqrt{\delta}<d_{*}$.
Throughout the proof, we refer to the graphs of the functions
$t\mapsto\mathrm{clen}_{\rho}(L)$ and $t\mapsto\mathrm{len}_{\rho}(L)$ for
fixed $\rho\in\mathcal{B}$, $x\in D\setminus D^{\rho,\varepsilon}$, and
$\ell\in\mathcal{L}$ in Figure 1. See the caption of the figure for the
definitions of the red, blue and green points. By Proposition 2.4,
$\alpha-\beta$ is the distance between the red and blue points, and the
quantity
$e^{-\rho(x)}\left(\mathrm{len}_{\rho}(x,\beta,\ell)-\mathrm{clen}_{\rho}(x,\beta,\ell)\right)$
on the right-hand side of (2.9) is the distance between the blue and purple
points. We can express these two distances in terms of the slopes of the two
curves:
* •
The distance between the blue and purple points is equal to the length of the
green segment divided by the slope of the blue line (i.e., $e^{\rho(x)}$).
* •
The distance between the blue and red points is equal to the length of the
green segment divided by the average derivative of the red curve between the
red and green points.
The error that we need to bound is the difference between these two
distances—i.e., the distance between the red and purple points. By Proposition
2.3, the derivative of the red curve between the red and green points differs
from the derivative of the blue line by at most
$c\,\mathrm{diam}(\ell)\sqrt{\delta}$, where $c$ is a constant bounded
uniformly in $\rho\in\mathcal{B}$ and $x\in D\setminus D^{\rho,\varepsilon}$
that can be made arbitrarily small for $\mathrm{diam}(\ell)\sqrt{\delta}$
sufficiently small. By Proposition 2.1, this implies that the _inverses_ of
these two derivatives differ by a uniform constant multiplied by
$c\,\mathrm{diam}(\ell)\sqrt{\delta}$. Moreover, the length of the green
segment is $\mathrm{len}_{\rho}(x,\beta,\ell)-\delta$, and by (2.6),
$\mathrm{len}_{\rho}(x,\beta,\ell)-\delta=\beta(A(\sqrt{\beta})-e^{\rho(x)})\leq
c\,\mathrm{diam}(\ell)\beta^{3/2},$ (2.10)
with $c$ as above. Thus, the error term—i.e., the distance between the red and
purple points—is bounded by a uniform constant times
$c\,\mathrm{diam}(\ell)\sqrt{\delta}\cdot
c\,\mathrm{diam}(\ell)\delta^{3/2}=c^{2}\delta^{2}\mathrm{diam}(\ell)^{2}$. If
we do not restrict to $\mathrm{diam}(\ell)\leq K$, the error is
$O(\delta^{2}\mathrm{diam}(\ell)^{2})$ uniformly in $\rho\in\mathcal{B}$,
$x\in D\setminus D^{\rho,\varepsilon}$, and $\ell\in\mathcal{L}$. If we
restrict to $\mathrm{diam}(\ell)\leq K$, then $c\rightarrow 0$ as
$\delta\rightarrow 0$ at a uniform rate, so the error is $o(\delta^{2})$
uniformly in $\rho\in\mathcal{B}$, $x\in D\setminus D^{\rho,\varepsilon}$, and
$\ell$ with $\mathrm{diam}(\ell)\leq K$. ∎
Proposition 2.7 immediately yields the following expression for
$\alpha^{-1}-\beta^{-1}$.
###### Proposition 2.8.
Let $\varepsilon,K>0$ be fixed. Then, for all $\ell$ with diameter at most
$K$, as $\delta\to 0$,
$\alpha^{-1}-\beta^{-1}=-\delta^{-1}\beta^{-1}\Bigl{(}\mathrm{len}_{\rho}(x,\beta,\ell)-\mathrm{clen}_{\rho}(x,\beta,\ell)\Bigr{)}+o(1),$
(2.11)
with the error converging uniformly in the choice of $\rho\in\mathcal{B}$,
$\ell$ with diameter at most $K$, and $x\in D\setminus D^{\rho,\varepsilon}$
where $\\{D^{\rho,\varepsilon}\\}_{\rho\in D}$ is defined as in Proposition
2.3. If we remove the restriction on $\mathrm{diam}(\ell)$, then the error is
$O(\mathrm{diam}(\ell)^{2})$ \+ $o(\delta^{2})$ uniformly in
$\rho\in\mathcal{B}$, $x\in D\setminus D^{\rho,\varepsilon}$, and
$\ell\in\mathcal{L}$.
###### Proof.
Throughout the following proof, each $O(\cdot)$ and $o(\cdot)$ error converges
uniformly as $\delta\rightarrow 0$ in the choice of $\rho\in\mathcal{B}$,
$x\in D\setminus D^{\rho,\varepsilon}$, and $\ell$.
By the Taylor expansion of $f(r)=1/r$ at $\beta$, we have
$\alpha^{-1}-\beta^{-1}=-\beta^{-2}(\alpha-\beta)+2\gamma^{-3}(\alpha-\beta)^{2}$
for some $\gamma$ between $\alpha$ and $\beta$. Proposition 2.7 gives
$-\beta^{-2}(\alpha-\beta)=-\delta^{-1}\beta^{-1}\Bigl{(}\mathrm{len}_{\rho}(x,\beta,\ell)-\mathrm{clen}_{\rho}(x,\beta,\ell)\Bigr{)}+o(1).$
We now handle the $2\gamma^{-3}(\alpha-\beta)^{2}$ term. From (2.1), we have
$\Lambda^{-1}\leq\alpha/\delta\leq\Lambda$; therefore,
$\gamma^{-3}=O(\delta^{-3})$. Next, by (2.10) and Proposition 2.7,
$(\alpha-\beta)^{2}=O(c^{2}\mathrm{diam}(\ell)^{2}\delta^{3})+o(\delta^{4})$,
where $c$ is a constant bounded uniformly in $\rho\in\mathcal{B}$ that can be
made arbitrarily small for $\mathrm{diam}(\ell)\sqrt{\delta}$ sufficiently
small. Hence,
$\gamma^{-3}(\alpha-\beta)^{2}=O(c^{2}\mathrm{diam}(\ell)^{2})+o(\delta^{2})$.
The latter is $o(1)$ with the restriction on $\mathrm{diam}(\ell)$, and
$O(\mathrm{diam}(\ell)^{2})+o(\delta^{2})$ otherwise. ∎
We also give a similar estimate for bookkeeping which might be useful for
Question 6.5.
###### Proposition 2.9.
With the same assumption of Proposition 2.8 without restriction on
$\mathrm{diam}(\ell)$, we have
$\alpha^{-2}-\beta^{-2}=O(c\,\mathrm{diam}(\ell)\delta^{-1/2})$ uniformly in
$\rho\in\mathcal{B}$, $x\in D\setminus D^{\rho,\varepsilon}$, and
$\ell\in\mathcal{L}$, where $c$ is a constant bounded uniformly in
$\rho\in\mathcal{B}$ that can be made arbitrarily small for
$\mathrm{diam}(\ell)\sqrt{\delta}$ sufficiently small.
###### Proof.
By the Taylor expansion of $f(r)=1/r$ at $\beta$, we have
$\alpha^{-2}-\beta^{-2}=-2\gamma^{-3}(\alpha-\beta)$ for some $\gamma$ between
$\alpha$ and $\beta$. From (2.1), we have
$\Lambda^{-1}\leq\alpha/\delta\leq\Lambda$; therefore,
$\gamma^{-3}=O(\delta^{-3})$. Next, by (2.10) and Proposition 2.7,
$\alpha-\beta=O(c\,\mathrm{diam}(\ell)\delta^{3/2})$, where $c$ is a constant
bounded uniformly in $\rho\in\mathcal{B}$ that can be made arbitrarily small
for $\mathrm{diam}(\ell)\sqrt{\delta}$ sufficiently small. Hence,
$\gamma^{-3}(\alpha-\beta)=O(c\,\mathrm{diam}(\ell)\delta^{-1/2})$. ∎
###### Proof of Lemma 1.16.
Let $\varepsilon>0$ and $\\{D^{\rho,\varepsilon}\\}_{\rho\in D}$ be defined as
in Proposition 2.3 so that
$\mathrm{Vol}(D^{\rho,\varepsilon})\leq\varepsilon$. By Propositions 1.15 and
2.5, integrating $\alpha^{-1}-\beta^{-1}$ over $x\in D\setminus
D^{\rho,\varepsilon}$ and $\ell\in\mathcal{L}$ yields the $\mu$-mass of loops
$L$ centered in $D$ with $\mathrm{len}_{\rho}(L)\geq\delta$ minus the
$\mu$-mass of loops $L$ centered in $D$ with
$\mathrm{clen}_{\rho}(L)\geq\delta$, up to a uniform $o(1)$ error. By
Proposition 2.8, for each fixed $x\in D\setminus D^{\rho,\varepsilon}$ and
$\ell\in\mathcal{L}$, the integrand $\alpha^{-1}-\beta^{-1}$ is equal to
$-\delta^{-1}\Bigl{(}\beta^{-1}\mathrm{len}_{\rho}(x,e^{-\rho(x)}\delta,\ell)-e^{\rho(x)}\Bigr{)}$
(2.12)
plus an error term that has the following limiting behavior as
$\delta\rightarrow 0$:
1. (a)
The error is $O(\mathrm{diam}(\ell)^{2})+o(\delta^{2})$ uniformly in
$\rho\in\mathcal{B}$ and $(x,\ell)\in(D\setminus
D^{\rho,\varepsilon})\times\mathcal{L}$.
2. (b)
For each fixed $K>0$, the error is $o(1)$ uniformly in $\rho\in\mathcal{B}$
and $(x,\ell)\in(D\setminus D^{\rho,\varepsilon})\times\mathcal{L}$ with
$\mathrm{diam}(\ell)\leq K$.
We now integrate $\alpha^{-1}-\beta^{-1}$ over $x\in D\setminus
D^{\rho,\varepsilon}$ and $\ell\in\mathcal{L}$ and take the $\delta\rightarrow
0$ limit. The integral of (2.12) is exactly equal to the term
$\frac{1}{\delta}\mathbb{E}[e^{\rho(Z)}-e^{\rho(X)}]$ in (1.10), so we just
need to show that the integral of the error term in the expression for
$\alpha^{-1}-\beta^{-1}$ tends to $0$ as $\delta\rightarrow 0$ uniformly in
$\rho\in\mathcal{B}$.
To analyze the integral of this error term, we partition the domain of
integration. For any function $K(\delta)$ of $\delta>0$, we can partition
$(D\setminus D^{\rho,\varepsilon})\times\mathcal{L}$ into two subdomains: the
set of pairs $(x,\ell)$ with $\mathrm{diam}(\ell)\leq K(\delta)$, and the set
of $(x,\ell)$ with $\mathrm{diam}(\ell)>K(\delta)$. The bound (a) implies that
the integral of the error over $(x,\ell)$ with $\mathrm{diam}(\ell)>k$ equals
a uniform constant times
$\int_{\\{\mathrm{diam}(\ell)>k\\}}(\mathrm{diam}(\ell)^{2}+\delta^{2})d\ell$,
which tends to zero as $k\rightarrow\infty$ at a rate uniform in (small)
$\delta>0$ and $\rho\in\mathcal{B}$. Moreover, (b) implies that if $k>0$ is
fixed, the integral of the error over $(x,\ell)$ with $\mathrm{diam}(\ell)\leq
k$ tends to $0$ as $\delta\rightarrow 0$ uniformly in $\rho\in\mathcal{B}$.
Hence, if we choose a function $K(\delta)$ that tends to infinity sufficiently
slowly as $\delta\rightarrow 0$, the integrals of the error term over both
subdomains of $(D\setminus D^{\rho,\varepsilon})\times\mathcal{L}$ tend to $0$
as $\delta\rightarrow 0$ uniformly in $\rho\in\mathcal{B}$.
Finally, the integral of $\alpha^{-1}-\beta^{-1}$ over $x\in
D^{\rho,\varepsilon}$ and $\ell\in\mathcal{L}$ is bounded by a uniform
constant times $\varepsilon\delta$ because of (2.1), which finishes the proof.
∎
## 3 Expected length discrepancy vs. Dirichlet energy
We now complete the proof of Theorem 1.12 by proving Lemma 1.17. The main step
in proving this lemma is proving the following proposition.
###### Proposition 3.1.
With $Z$ and $X$ defined as in Lemma 1.16, the quantity
$\beta^{-1}\mathbb{E}[\rho(Z)-\rho(X)]$ converges to $0$ as $\delta\rightarrow
0$ uniformly in $\rho\in\mathcal{B}$. The expectation here is w.r.t. to the
overall law of $X$ and $Z$ as defined in Lemma 1.16.
We prove Proposition 3.1 by a Fourier analysis argument. Throughout this
section, we define the Fourier transform of a function $\phi$ as
$\widehat{\phi}(\zeta):=\int_{\mathbb{R}^{2}}e^{i\zeta\cdot z}\phi(z)\,dz$
(using the convention of characteristic functions). To prove Proposition 3.1,
we express the conditional density of $Z$ given $X=x$ in terms of the expected
occupation measure of a loop sampled from $\mu$ with given length and center.
In the following proposition, we introduce some notation for this expected
occupation measure and record some of its elementary properties.
###### Proposition 3.2.
For $z\in\mathbb{R}^{2}$ and $s>0$, let $\theta(z,s)$ be the expected
occupation measure of a loop sampled from $\mu$ and conditioned to have length
$s$ and center zero. For fixed $s$, the measure $\theta(z,s)$ is radially
symmetric Schwartz, and a probability measure. Each coordinate has second
central moment $bs$. Moreover, $\theta$ satisfies the scaling relation
$\theta(z/\sqrt{t},s)=t\theta(z,ts)$ for any $t\geq 0$. Finally, we have
$\lim_{s\to 0}\widehat{\theta}(z,s)=1$.
###### Proof.
Since $\theta(z,s)$ is the law of With $\ell$ sampled from $d\ell$, the loop
$t\mapsto\sqrt{s}\ell(t/s)$ has law $\theta(z,s)$. It follows from Definition
1.11 that $\theta(z,s)$ is radially symmetric, Schwartz, and a probability
measure with the desired second central moments. The scaling relation also
follows immediately. Finally, the second central moments imply that the
measures $\theta(z,s)$ converge weakly as $s\rightarrow 0$ to a point mass at
$0$, so their characteristic functions converge to $1$. ∎
We first reduce the task of proving Proposition 3.1 to analyzing the Dirichlet
energy of the inverse Laplacian of an appropriately chosen measure.
###### Proposition 3.3.
Suppose that $\phi:\mathbb{R}^{2}\to\mathbb{R}$ is a function satisfying
$\Delta\phi(y)=\int_{D}\beta^{-1}(\theta(y-x,\beta)-\bm{\delta}(y-x))\,dx\qquad\text{and}\qquad\lim_{\left|y\right|\to\infty}\phi(y)=0,$
(3.1)
where $\theta$ is defined in Proposition 3.2 and $\bm{\delta}$ is the point
mass at $0$. Then
$\beta^{-1}\mathbb{E}[\rho(Z)-\rho(X)]\leq(\rho,\rho)_{\nabla}^{2}(\phi,\phi)_{\nabla}^{2}$
###### Proof.
The conditional density of $Z$ given $X=x$ is $\theta(\cdot-x,\beta)$.
Therefore,
$\beta^{-1}\mathbb{E}[\rho(Z)-\rho(X)]=\int_{D}\Bigl{(}\rho,\beta^{-1}\bigl{(}\theta(\cdot-x,\beta)-\bm{\delta}(\cdot-x)\bigr{)}\Bigr{)}\,dx=(\rho,\Delta\phi)$
by the definition of $\phi$. Integrating by parts (or applying Green’s
identity) gives $(\rho,\Delta\phi)=-(\rho,\phi)_{\nabla}$, and
$(\rho,\phi)_{\nabla}^{2}\leq(\rho,\rho)_{\nabla}(\phi,\phi)_{\nabla}.$
by applying the Cauchy-Schwarz inequality. ∎
Therefore, it is enough to show $(\phi,\phi)_{\nabla}\to 0$ uniformly for the
proof. To express a function $\phi$ satisfying (3.1) using Fourier
integral444Alternatively, such $\phi$ can be written in terms of the
convolution with Green’s function., we first define a family of auxiliary
functions that we label $\Theta_{t}$ indexed by $t>0$.
###### Proposition 3.4.
With the notation
$\widehat{\theta_{s}}(\zeta,s_{0}):=\bigl{.}\frac{\partial\widehat{\theta}}{\partial
s}(\zeta,s)\bigr{|}_{s=s_{0}}$, the function $\Theta_{t}$ for $t>0$ defined as
the inverse Fourier transform of
$\left|\zeta\right|^{-2}\widehat{\theta_{s}}(\zeta,t)$ is a radially symmetric
Schwartz function. In addition, $\Theta_{t}(y)=\Theta_{1}(y/\sqrt{t})/t$ and
$\widehat{\Theta_{t}}(\zeta)$ is uniformly bounded for all $t>0$ and
$\zeta\in\mathbb{R}^{2}$.
###### Proof.
Fix $t>0$. Recall that $\widehat{\theta}(r,t)$ is Schwartz in $r\geq 0$, so we
use the notation
$\widehat{\theta_{r}}(r,t):=\bigl{.}\frac{\partial\widehat{\theta}}{\partial\left|\zeta\right|}(\left|\zeta\right|,t)\Bigr{|}_{\left|\zeta\right|=r}$
in this proof. The scaling relation and the radial symmetry of $\theta$
(Proposition 3.2) implies that
$\widehat{\Theta_{t}}(\zeta)=\left|\zeta\right|^{-2}\widehat{\theta_{s}}(\zeta,t)=\widehat{\theta_{s}}(1,t\left|\zeta\right|^{2})$.
Hence, it is enough to show that
$\widehat{\Theta_{t}}(r)=\widehat{\theta_{s}}(1,tr^{2})$, as a function of
$r\geq 0$, is a Schwartz function. This will also imply that
$\widehat{\Theta_{t}}(\zeta)=\widehat{\Theta_{1}}(\sqrt{t}\zeta)$ is uniformly
bounded by $\sup_{t\geq 0}\left|\widehat{\Theta_{1}}(t)\right|$, and
$\Theta_{t}(y)=\Theta_{1}(y/\sqrt{t})/t$ from the Fourier inversion.
First we repeatedly differentiate both sides of the relation
$\widehat{\theta}(1,tr^{2})=\widehat{\theta}(r,t)$ using the chain rule, and
observe $\partial_{s}^{k}\widehat{\theta}(1,tr^{2})\mid_{r=0}$ exists as a
constant multiple of $\partial_{r}^{2k}\widehat{\theta}(r,t)\mid_{r=0}$. Also,
the first differentiation implies
$\widehat{\Theta_{t}}(r)=\widehat{\theta_{s}}(1,tr^{2})=\frac{\widehat{\theta_{r}}(r,t)}{2tr}.$
Let $P(r)$ be any polynomial in $r$, and $k$ be any nonnegative integer. On
one hand, for a fixed $\varepsilon>0$, we have
$\sup_{r>\varepsilon}\left|P(r)\frac{d^{k}}{dr^{k}}\frac{\widehat{\theta_{r}}(r,t)}{2tr}\right|<\infty.$
because $\widehat{\theta_{r}}(r,t)$ and its derivatives with respect to $r$
are rapidly decreasing, On the other hand, note that
$\left|P(r)\frac{d^{k}}{dr^{k}}\widehat{\theta_{s}}(1,tr^{2})\right|$ is
continuous on $(0,\varepsilon]$ and bounded at 0. This proves that
$\Theta_{t}$ is Schwartz. ∎
###### Proposition 3.5.
With $\Theta_{t}$ defined in Proposition 3.4, for $x\in\mathbb{R}^{2}$, let
$\phi^{x}:\mathbb{R}^{2}\to\mathbb{R}$ be defined as
$\phi^{x}(y):=-\frac{1}{\delta}\int_{0}^{\delta}\Theta_{te^{-\rho(x)}}(y-x)\,dt.$
and $\phi:\mathbb{R}^{2}\to\mathbb{R}$ defined as
$\phi(y):=\int_{D}\phi^{x}(y)\,dx$ (3.2)
for each $y\in\mathbb{R}^{2}$. Then $\phi$ satisfies (3.1).
###### Proof.
By the Fourier transform, we have
$\displaystyle\widehat{\phi^{x}}(\zeta)$
$\displaystyle=-\frac{1}{\delta}\int_{0}^{\delta}e^{i\zeta\cdot
x}\widehat{\Theta_{te^{-\rho(x)}}}(\zeta)\,dt$
$\displaystyle=-\frac{1}{\delta}\int_{0}^{\delta}e^{i\zeta\cdot
x}\left|\zeta\right|^{-2}\widehat{\theta_{s}}\bigl{(}\zeta,te^{-\rho(x)}\bigr{)}\,dt$
$\displaystyle=-e^{i\zeta\cdot
x}\left|\zeta\right|^{-2}\beta^{-1}\bigl{(}\widehat{\theta}(\zeta,\delta
e^{-\rho(x)})-1\bigr{)},$
as we have defined $\beta=\delta e^{-\rho(x)}$. Therefore, the Fourier
inversion of
$\widehat{\Delta\phi^{x}}=-\left|\zeta\right|^{2}\widehat{\phi^{x}}$ gives the
desired result. Furthermore, the first identity shows that
$\widehat{\phi^{x}}\in L^{1}$ from Proposition 3.4. By the Riemann-Lebesgue
lemma, we conclude that $\lim_{\left|y\right|\to\infty}\phi^{x}(y)=0$, and
thus it follows from the dominated convergence theorem that
$\lim_{\left|y\right|\to\infty}\phi(y)=0$. ∎
###### Proposition 3.6.
With $\phi$ defined as (3.2) and $\Theta_{t}$ defined in Proposition 3.4,
define
$G_{\rho,r}^{y}(z):=e^{3\rho(y-rz)/2}\nabla\Theta_{1}(ze^{\rho(y-rz)/2})$
(3.3)
for each $\rho\in\mathcal{B}$, $y\in\mathbb{R}^{2}$, $r\in\mathbb{R}$, and
$z\in\mathbb{R}^{2}$. Then
$\nabla_{y}\phi(y)=\frac{1}{\delta}\int_{0}^{\delta}\frac{1}{\sqrt{t}}\int_{\frac{y-D}{\sqrt{t}}}G_{\rho,\sqrt{t}}^{y}(z)\,dz\,dt.$
(3.4)
###### Proof.
From Proposition 3.4,
$\phi(y)=-\frac{1}{\delta}\int_{0}^{\delta}\Theta_{te^{-\rho(x)}}(y-x)\,dx\,dt=-\frac{1}{\delta}\int_{0}^{\delta}\int_{D}\frac{\Theta_{1}\bigl{(}(y-x)(te^{-\rho(x)})^{-\frac{1}{2}}\bigr{)}}{te^{-\rho(x)}}\,dx\,dt,$
taking gradient and substituting $y-x=\sqrt{t}z$ gives
$\nabla_{y}\phi(y)=\frac{1}{\delta}\int_{0}^{\delta}\frac{1}{\sqrt{t}}\int_{\frac{y-D}{\sqrt{t}}}e^{3\rho(y-\sqrt{t}z)/2}\nabla\Theta_{1}(ze^{\rho(y-\sqrt{t}z)/2})\,dz\,dt,$
so the result follows. ∎
###### Proposition 3.7.
For each $\rho\in\mathcal{B}$, $r\in\mathbb{R}$, and $y\in\mathbb{R}^{2}$, let
$F_{\rho,r}(y):=\int_{\mathbb{R}^{2}}G_{\rho,r}^{y}(z)\,dz$ (3.5)
where $G_{\rho,r}^{y}$ is defined as (3.3). Then $F_{\rho,r}$ and
$\partial_{r}F_{\rho,r}$ both converges to 0 in $L^{2}(\mathbb{R}^{2})$ as
$r\to 0$ uniformly over $\rho\in\mathcal{B}$.
###### Proof.
Since $\nabla\Theta_{1}$ is rapidly decreasing, given any $\varepsilon>0$,
there exists $K=K(\epsilon)$ such that
$\int_{\left|z\right|\geq
K}\left|G_{\rho,r}^{y}(z)\right|^{2}\,dz<\varepsilon$ (3.6)
for all $r\in\mathbb{R}$, $y\in\mathbb{R}^{2}$, and $\rho\in\mathcal{B}$ as
$\rho$ is uniformly bounded. By radial symmetry, note that $F_{\rho,0}(y)=0$
for all $\rho$ and $y$ because $\nabla\Theta_{1}$ is an odd function. Now we
divide the domain of integral (3.5) into two regions depending on the sign of
$z_{1}z_{2}$ when we write $z=(z_{1},z_{2})\in\mathbb{R}^{2}$, so that the
integral on each domain becomes
$\int_{\pm
z_{1}z_{2}>0,\left|z\right|<K}G_{\rho,r}^{y}(z)\,dz=\int_{z_{1}>0,\pm
z_{2}>0,\left|z\right|<K}\bigl{(}G_{\rho,r}^{y}(z)-G_{\rho,-r}^{y}(z)\bigr{)}\,dz.$
Since $\nabla\Theta_{1}$ is Lipschitz as being Schwartz, the uniform
equicontinuity of $\rho$ in $L^{2}(D)$ (in the sense of Remark 1.7) implies
that $G_{\rho,r}$, as a function of $rz$, is also uniformly equicontinuous in
$L^{2}(\mathbb{R}^{2})$. Therefore, denoting
$A_{\pm}=\\{z\in\mathbb{R}^{2}:z_{1}>0,\pm z_{2}>0,\left|z\right|<K\\}$, there
exists $d=d(\varepsilon)>0$ such that
$\displaystyle\int_{D}\left|\int_{\left|z\right|<K}G_{\rho,r}^{y}(z)\right|^{2}dy$
$\displaystyle\leq\sum_{i\in\\{\pm\\}}\int_{A_{i}}\int_{D}\left|G_{\rho,r}^{y}(z)-G_{\rho,-r}^{y}(z)\right|^{2}\,dy\,dz\leq
2\varepsilon$
for all $\rho\in\mathcal{B}$ whenever $\left|r\right|<d$. Combined with (3.6),
we conclude that $F_{\rho,r}$ converges to 0 in $L^{2}(D)$ uniformly over
$\rho\in\mathcal{B}$ as $r\to 0$.
A similar argument applies to $\partial_{r}F_{\rho,r}$. In particular, as a
function of $\rho$ and $\nabla\rho$, note that $\nabla G_{\rho}$ is Lipschitz.
Hence the uniform equicontinuity of $\rho$ in $W^{1,2}(D)$ (in the sense of
Remark 1.7) implies the uniform equicontinuity of $\nabla G_{\rho,r}^{y}(z)$
in $L^{2}(\mathbb{R}^{2})$ over $\rho\in\mathcal{B}$ as a function of $rz$.
Since $\Theta_{1}$ is rapidly decreasing, from integration by parts,
$\left.\frac{\partial}{\partial
r}F_{\rho,r}(y)\right|_{r=0}=-\int_{\mathbb{R}^{2}}\nabla_{z}G_{\rho,0}^{y}(z)\cdot
z\,dz=\int_{\mathbb{R}^{2}}2G_{\rho,0}^{y}(z)\,dz=0$
again by the oddity of $G_{\rho,0}^{y}$. Repeating the previous argument, we
conclude that $\partial_{r}F_{\rho,r}$ converges to 0 in
$L^{2}(\mathbb{R}^{2})$ uniformly over $\rho\in\mathcal{B}$ as $r\to 0$. ∎
###### Proposition 3.8.
For each $y\in\mathbb{R}^{2}$, define
$\Phi(y):=\frac{1}{\delta}\int_{0}^{\delta}\frac{F_{\rho,\sqrt{t}}(y)}{\sqrt{t}}\,dt.$
(3.7)
Then $\nabla\phi-\Phi$ converges to 0 in $L^{2}(\mathbb{R}^{2})$ uniformly
over $\rho\in\mathcal{B}$ as $\delta\to 0$.
###### Proof.
Given $\varepsilon>0$, let $K=K(\varepsilon)$ be defined such that (3.6)
holds. Choose $d(\varepsilon)$ small enough so that as long as $0<r<d$, we
have $\left|G_{\rho,r}^{y}(z)-G_{\rho,0}^{y}(z))\right|<\varepsilon/\pi K^{2}$
and the diameter of $(y-D)/r$ is greater than $K$. Using the triangle
inequality, we obtain the desired result from (3.4) and (3.7). ∎
###### Proof of Proposition 3.1.
By the mean value theorem, with $\Phi$ is defined in Proposition 3.8, we have
$\int_{D}\left|\Phi(y)\right|^{2}\,dy\leq\frac{1}{\delta}\int_{0}^{\delta}\int_{D}\left|\frac{F_{\rho,\sqrt{t}}(y)}{\sqrt{t}}\right|^{2}\,dy\,dt=\int_{D}\left|\partial_{r}{F_{\rho,r}(y)}\right|^{2}\,dy$
for some $r\in(0,\sqrt{\delta})$, that is $\Phi$ converges to 0 in
$L^{2}(\mathbb{R}^{2})$ as $\delta\to 0$ uniformly over $\rho\in\mathcal{B}$
by Proposition 3.7. With Proposition 3.8, we conclude $\nabla\phi$ also
converges to 0 in $L^{2}(\mathbb{R}^{2})$ as $\delta\to 0$ uniformly over
$\rho\in\mathcal{B}$. This completes the proof. ∎
###### Proof of Lemma 1.17.
Let $Y=\rho(Z)-\rho(X)$, so that
$\displaystyle\delta^{-1}\mathbb{E}[e^{\rho(Z)}-e^{\rho(X)}]$
$\displaystyle=\delta^{-1}\mathbb{E}\left[e^{\rho(X)}\left(Y+\frac{Y^{2}}{2!}+\frac{Y^{3}}{3!}+\ldots\right)\right]$
$\displaystyle=\delta^{-1}\mathbb{E}[e^{\rho(X)}Y]+\delta^{-1}\mathbb{E}\left[e^{\rho(X)}\left(\frac{Y^{2}}{2!}+O(Y^{3})\right)\right]\Bigr{)}$
(3.8)
By Proposition 3.1, the term $\delta^{-1}\mathbb{E}[e^{\rho(X)}Y]$ is $o(1)$
with the convergence uniform in $\rho\in\mathcal{B}$. We now analyze the
second term in (3.8). Since the gradients $\nabla\rho$ for
$\rho\in\mathcal{B}$ are uniformly equicontinuous in $L^{1}$ (in the sense of
Remark 1.7), we can express $Y$ as $\nabla\rho(X)\cdot(Z-X)+\varepsilon$
almost surely, where $\varepsilon$ is an error term uniformly $o(|Z-X|)$ in
expectation, as $|Z-X|\to 0$. Conditional on $X$, each coordinate of $Z-X$ has
mean $X$ and variance $\beta b$. Therefore, with
$U\sim\text{Uniform}[0,2\pi]$,
$\mathbb{E}((\nabla\rho(X)\cdot(Z-X))^{2}|X)=|\nabla\rho(X)|^{2}\mathbb{E}\left[|Z-X|^{2}|X\right]\mathbb{E}(\cos^{2}(U))=\beta
b|\nabla\rho(X)|^{2}.$
Therefore,
$\displaystyle\mathbb{E}[Y^{2}|X]$
$\displaystyle=\mathbb{E}\left[(\nabla\rho(X)\cdot(Z-X)+\varepsilon)^{2}|X\right]$
$\displaystyle=\beta
b|\nabla\rho(X)|^{2}+\mathbb{E}[2\varepsilon\nabla\rho(X)\cdot(Z-X)+\varepsilon^{2}|X]$
$\displaystyle=\beta b|\nabla\rho(X)|^{2}+o(\delta),$
with the $o(\delta)$ error uniform in $\rho$ and $X$. We can similarly show
that $\mathbb{E}[Y^{3}|X]$ is $o(\delta)$ uniformly in $\rho$ and $X$.
Multiplying both sides by $\delta^{-1}e^{\rho(X)}$ and taking the expectation,
we deduce that the second term in (3.8) is equal to
$\frac{b}{2}(\rho,\rho)_{\nabla}$ plus an $o(1)$ error that converges
uniformly in $\rho\in\mathcal{B}$. ∎
###### Proof of Theorem 1.12.
The result immediately follows from combining Proposition 1.15, Lemma 1.16 and
Lemma 1.17. ∎
Finally, to deduce Theorem 1.4 from Theorem 1.12, we explicitly characterize
the expected occupation measure.
###### Proposition 3.9.
For the Brownian loop measure, the expected occupation measure of a loop
sampled from $(\mathcal{L},d\ell)$ is the density of a complex Gaussian with
variance $1/12$.
###### Proof.
The law of a loop sampled from $(\mathcal{L},d\ell)$ is that of a Brownian
bridge indexed by the circle minus its mean. The value of a Brownian bridge
indexed by the circle at any given time minus the mean value is a complex
mean-zero Gaussian random variable of variance $1/12$; this calculation
appears, for example, in [She07].555One way to see it is to consider a
Gaussian free field $h$ indexed by the circle and observe that the Dirichlet
energy on the circle parameterized by $[-1/2,1/2]$ of the function
$f(x)=x^{2}/2$ is given by $1/12$, so $(f,f)_{\nabla}=1/12$. But for a
function $g$ on the circle we have from integration by parts that
$(f(x),g(x))_{\nabla}=g(0)-\int_{-1/2}^{1/2}g(x)dx$. Then using the above and
the definition of the GFF we have
$\mathrm{Var}\bigl{(}(h,f)_{\nabla}\bigr{)}=\mathrm{Var}\bigl{(}h(1/2)-\int_{0}^{1}h(x)dx\bigr{)}=(f,f)_{\nabla}=1/12$.
By rotational symmetry, this holds if $0$ is replaced by any other number in
$[-1/2,1/2]$. The number $1/12$ is also derived in [She07] by Fourier series,
and we remark that comparing these two derivations is one way to prove
$\sum_{n=1}^{\infty}n^{-2}=\pi^{2}/6$. ∎
###### Proof of Theorem 1.12.
The result follows from Theorem 1.12 and Proposition 3.9. ∎
## 4 Brownian loops on surfaces
In this section, we prove Theorem 1.9. Throughout the section, we let $(M,g)$
be a fixed compact smooth two-dimensional Riemannian manifold, and we let
$\mu^{\text{loop}}$ denote the Brownian loop measure on $(M,g)$. We also fix
$\mathcal{B}$ as a precompact set of Lipschitz functions in $W^{1,1}(M)$ with
uniformly bounded Lipschitz constants.
To prove Theorem 1.9, we analyze the mass of “large” loops and the mass of
“small” loops with respect to $\rho$-length separately. We begin by analyzing
the mass of large loops by proving a central limit theorem for $\rho$-length
along large loops:
###### Proposition 4.1.
We can choose a constant $c>0$ such that, for every $\varepsilon,t>0$ and
$\rho\in\mathcal{B}$ and every loop $L$ sampled from $\mu(z,z;t)$,
$\left|\int_{0}^{t}e^{\rho(L(s))}ds-t\mathrm{Vol}_{\rho}(M)/\mathrm{Vol}(M)\right|\leq
ct^{(1+\varepsilon)/2}$ (4.1)
with probability $1-O(e^{-t^{\varepsilon}})$ as $t\to\infty$, with the rate
uniform in $\rho\in\mathcal{B}$.
To prove Proposition 4.1, we apply the Markov central limit theorem to
Brownian motion on $(M,g)$, and then compare Brownian motion to a loop sampled
from $\mu(z,z;t)$ by using the following Radon-Nikodym estimate.
###### Proposition 4.2.
Let $z\in M$ be fixed, and let $L$ be a loop sampled from $\mu(z,z;t)$. The
Radon-Nikodym derivative of the law of $L|_{[0,s]}$ with respect to Brownian
motion restricted to $[0,s]$ is given by $1+O(e^{-\alpha(t-s)})$ for some
$\alpha=\alpha(M,g)>0$.
###### Proof.
By Proposition 4.3,
$\|\mu(\cdot,z;t-s)/\mu(z,z;t)\|_{\infty}=1+O(e^{-\alpha(t-s)})$ for some
$\alpha=\alpha(M,g)>0$, with the error uniform in $z\in M$. This implies the
derivative bound. ∎
To apply the Markov central limit theorem to Brownian motion on $(M,g)$, we
need the following convergence result for the Brownian transition kernel
$\mu(\cdot,\cdot;t)$ as $t\to\infty$.
###### Proposition 4.3.
We have
$\|\mu(\cdot,\cdot;t)-(\mathrm{Vol}(M))^{-1}\|_{L^{\infty}(M\times
M)}=O(e^{-\alpha t}).$
for some constant $\alpha>0$.
###### Proof.
Let $0=\lambda_{0}\leq\lambda_{1}\leq\lambda_{2}\leq\cdots$ denote the
eigenvalues of $\Delta$, and let $\\{\phi_{n}\\}$ be a corresponding Hilbert
basis of eigenfunctions. It follows from the heat equation that $p$ can be
written as
$p(x,y;t)=\sum_{n=0}^{\infty}e^{-t\lambda_{n}}\phi_{n}(x)\phi_{n}(y).$
It is known that $\|\phi_{n}\|_{\infty}\leq c\lambda_{n}^{1/4}$ for some
constant $c$ depending only on $(M,g)$ [Don01]. Hence,
$\|\mu(\cdot,\cdot;t)-(\mathrm{Vol}(M))^{-1}\|_{L^{\infty}(M\times M)}$ is
bounded from above by a function of $t$ that decays exponentially as
$t\to\infty$. ∎
###### Proof of Proposition 4.1.
(In the proof that follows, the rate of convergence of the $O(\cdot)$ errors
are uniform in the choice of $\rho\in\mathcal{B}$.) Let $B_{t}$ be a Brownian
motion on $(M,g)$ started at a point sampled from the volume measure
associated to $(M,g)$. Let $Y_{n}=B|_{[n,n+1]}$, and let
$f(Y_{n})=\int_{n}^{n+1}e^{\rho(B_{t})}dt$. For $y\in M$, let $f_{*}(y)$ be
the expected value of $\int_{n}^{n+1}e^{\rho(B^{y}_{t})}dt$, where $B_{t}^{y}$
is a Brownian motion on $(M,g)$ started at $y$. By Proposition 4.3, the
conditional expectation of $f(Y_{n})$ given $Y_{0}$ is
$\int_{M}p(B_{1},y;n-1)f_{*}(y)\mathrm{Vol}(dy)=\mathbb{E}(f(Y_{0}))+O(e^{-\alpha
n})$ for some $\alpha=\alpha(M,g)$. Hence,
$\mathrm{Cov}(f(Y_{0}),f(Y_{n}))=O(e^{-\alpha n})$ for some
$\alpha=\alpha(M,g)$. Thus, we may apply the Markov chain central limit
theorem to deduce that
$\sqrt{n}\left(\frac{1}{n}\sum_{j=1}^{n}f(Y_{j})-\mathbb{E}(f(Y_{0}))\right)$
converges in the $n\to\infty$ limit to a centered Gaussian distribution whose
variance is bounded uniformly in the choice of $\rho\in\mathcal{B}$. Note that
we have
$\mathbb{E}(f(Y_{0}))=\frac{1}{\mathrm{Vol}(M)}\int_{M}e^{\rho(x)})\mathrm{Vol}(dx)=\mathrm{Vol}_{\rho}(M)/\mathrm{Vol}(M)$.
Therefore, for each fixed $\varepsilon,c>0$,
$\left|\int_{0}^{n}e^{\rho(B_{t})}dt-n\mathrm{Vol}_{\rho}(M)/\mathrm{Vol}(M)\right|\leq
cn^{(1+\varepsilon)/2}$
with probability $1-O(e^{-n^{\varepsilon}})$.
Combining this with Proposition 4.2, we deduce that, if $n$ is an integer with
$t-2\sqrt{t}\leq n\leq t-\sqrt{t}$, then
$\left|\int_{0}^{n}e^{\rho(L(s))}ds-n\mathrm{Vol}_{\rho}(M)/\mathrm{Vol}(M)\right|\leq
c\sqrt{t}^{(1+\varepsilon)/2}$
with probability $1-O(e^{-t^{\varepsilon}})$ as $t\to\infty$. Since
$\int_{n}^{t}e^{\rho(L(s))}ds$ is bounded from above by $\sqrt{t}$ times a
constant uniform in $\rho\in\mathcal{B}$, this implies the proposition. ∎
We now apply Proposition 4.1 to analyze the mass of large loops.
###### Proposition 4.4.
For each $\varepsilon>0$, the symmetric difference between
* •
the set of loops with length $\geq\delta$ and length $\leq C$ , and
* •
the set of loops with length $\geq\delta$ and $\rho$-length $\leq
C\mathrm{Vol}_{\rho}(M)/\mathrm{Vol}(M)$,
has $\mu^{\text{loop}}$-mass at most $O(C^{(-1+\varepsilon)/2})$, with the
rate of convergence uniform in $\rho\in\mathcal{B}$.
###### Proof.
(In the proof that follows, the rate of convergence of the $O(\cdot)$ errors
are uniform in the choice of $\rho\in\mathcal{B}$.) Let $\mathcal{S}$ denote
the symmetric difference between the two sets. By Proposition 2.1, each loop
in $\mathcal{S}$ has length between $\Lambda^{-1}C$ and $\Lambda C$, with
$\Lambda$ independent of the choice of $\rho\in\mathcal{B}$. By Proposition
4.1, the subset $\mathcal{S}^{*}\subset\mathcal{S}$ of loops not satisfying
(4.1) has $\mu^{\text{loop}}$-mass at most $\int_{\Lambda^{-1}C}^{\Lambda
C}O(e^{-t^{\varepsilon}}/t)dt=O(\exp(-\Lambda^{-\varepsilon}C^{\varepsilon})/C)$.
Moreover, with $a_{\pm}=C\mathrm{Vol}_{\rho}(M)/\mathrm{Vol}(M)\pm
c(C^{(1+\varepsilon)/2}$, we can bound the $\mu^{\text{loop}}$-mass of the set
of loops in $\mathcal{S}\backslash\mathcal{S}^{*}$ by
$\int_{a_{-}}^{a_{+}}\int_{M}t^{-1}\mu_{\rho}(z,z;t)\mathrm{Vol}_{\rho}(dz)dt\leq\mathrm{Vol}_{\rho}(M)\int_{a_{-}}^{a_{+}}t^{-1}dt=O(C^{(-1+\varepsilon)/2}).\qed$
Next, we analyze the mass of small loops.
###### Proposition 4.5.
For each $\rho_{1},\rho_{2}\in\mathcal{B}$, the difference in the masses of
the sets
$\\{L:\mathrm{len}_{\rho_{j}}L\geq\delta,\mathrm{len}L\leq C\\}\qquad j=1,2$
under the Brownian loop measure in $(M,g)$ is equal to the difference in the
expressions
$\frac{\mathrm{Vol}_{\rho_{j}}(M)}{2\pi\delta}+\frac{1}{48\pi}\int_{M}(\|\nabla\rho_{j}\|^{2}+2K\rho_{j})\mathrm{Vol}(dz),\quad
j=1,2.$ (4.2)
plus a term that tends to zero as $\delta\to 0$ at a rate that is uniform in
the choice of $\rho_{1},\rho_{2}\in\mathcal{B}$.
To prove Proposition 4.5, we will apply the following pair of propositions.
###### Proposition 4.6.
Let $(M,g)$ be a smooth two-dimensional Riemannian manifold. Let $U\subset M$,
and let $\mathcal{S}$ be a collection of loops in $(M,g)$ that intersects a
closed set disjoint from $\overline{U}$. Then the following holds for all
$\delta>0$ sufficiently small. Let $\rho_{1},\rho_{2}\in\mathcal{B}$ be
uniformly bounded functions that agree outside $U$. Then, for each
$L\in\mathcal{S}$, we have $\mathrm{len}_{\rho_{1}}L\geq\delta$ iff
$\mathrm{len}_{\rho_{2}}L\geq\delta$.
###### Proof.
It is straightforward as $\mathcal{B}$ is uniformly bounded and the length of
$L$ outside $U$ is bounded below. ∎
###### Proposition 4.7.
Let $f$ be a compactly supported function on a region $D\subset\mathbb{R}^{2}$
with finite Dirichlet energy, and let $h$ be a smooth compactly supported
function on $D$, and let $\nabla_{h},\nu_{h},K_{h}$ denote the gradient,
volume form and Gaussian curvature associated to $(D,e^{h}|dz|^{2})$. Then
$\int_{D}\|\nabla(f+h)\|^{2}dz-\int_{D}\|\nabla
h\|^{2}dz=\int_{D}(\|\nabla_{h}f\|^{2}+2K_{h}f)\nu_{h}(dz).$
###### Proof.
It follows from
$\displaystyle\int_{D}(\|\nabla(f+h)\|^{2}-\|\nabla
h\|^{2})dz=-\int_{D}(f\Delta f+2f\Delta h)dz$ $\displaystyle=$
$\displaystyle-\int_{D}(fe^{-h}\Delta f+2fe^{-h}\Delta
h)\nu_{h}(dz)=-\int_{D}(f\Delta_{h}f-2K_{h}f)\nu_{h}(dz)$ $\displaystyle=$
$\displaystyle-\int_{D}(fe^{-h}\Delta
f-2K_{h}f)\nu_{h}(dz)=\int_{D}(\|\nabla_{h}f\|^{2}+2K_{h}f)\nu_{h}(dz).\qed$
###### Proof of Proposition 4.5.
In the proof that follows, we consider $\rho$-lengths and $\rho$-volume forms
(as defined in Definition 1.3) for functions $\rho$ on both $(M,g)$ and a
region of the Euclidean plane. To avoid confusion between the two settings, we
use the notation $\mathrm{len}_{\rho}$ and $\mathrm{Vol}_{\rho}$ in the
$(M,g)$ setting, and $\widetilde{\mathrm{len}}_{\rho}$ and
$\widetilde{\mathrm{Vol}}_{\rho}$ in the Euclidean setting.
Let $U,V,W$ be open sets in $M$ with $\overline{U}\subset V$ and
$\overline{V}\subset W$, such that we can find a homeomorphism
$\varphi:W\rightarrow\widetilde{W}\subset\mathbb{R}^{2}$. By a partition of
unity argument, it suffices to prove the proposition under the assumption that
$\rho_{1},\rho_{2}$ agree on $U$. So, we assume that this is the case. By
Proposition 4.6, the difference in the masses of the sets
$\\{L:\mathrm{len}_{\rho_{j}}L\geq\delta,\mathrm{len}L\leq C\\}\qquad j=1,2$
(4.3)
under the Brownian loop measure in $(M,g)$ is equal to the difference in
masses with (4.3) replaced by
$\\{L:\mathrm{len}_{\rho_{j}}L\geq\delta,\mathrm{len}L\leq C,L\subset
V\\}\qquad j=1,2.$ (4.4)
Since, by Proposition 2.1, $\mathrm{len}L>C$ automatically implies
$\mathrm{len}_{\rho_{j}}L\geq\delta$ for sufficiently small $\delta$ (in a
manner that does not depend on the choice of
$\rho_{1},\rho_{2}\in\mathcal{B}$), we can replace (4.4) by the sets
$\\{L:\mathrm{len}_{\rho_{j}}L\geq\delta,L\subset V\\},\qquad j=1,2,$ (4.5)
with the condition $\mathrm{len}L\leq C$ in (4.4) removed. Let
$\widetilde{\rho}_{j}$ be the pushforward of $\rho_{j}$ via $\varphi$, and let
$e^{\sigma}|dz|^{2}$ be the pushforward of the metric $g$ via $\varphi$. Also,
set $\widetilde{V}=\varphi(V)$. By conformal invariance of the Brownian loop
measure [APPS20, Lemma 3.3], the masses of the sets (4.5) under the Brownian
loop measure in $(M,g)$ equals the masses of the sets
$\\{L:\widetilde{\mathrm{len}}_{\widetilde{\rho}_{j}+\sigma}L\geq\delta,L\subset\widetilde{V}\\}\qquad
j=1,2$ (4.6)
under the Brownian loop measure in the Euclidean plane. Now, let $\theta$ be a
smooth function on $\mathbb{R}^{2}$ that equals $1$ on
$\overline{\widetilde{V}}$ and zero outside a compact subset of
$\widetilde{W}$. Since $\theta\equiv 1$ on $\widetilde{V}$, (4.6) is equal to
the set
$\\{L:\widetilde{\mathrm{len}}_{\theta(\widetilde{\rho}_{j}+\sigma)}L\geq\delta,L\subset\widetilde{V}\\}\qquad
j=1,2$ (4.7)
By Proposition 4.6, the difference in these loop masses is unchanged if we
replace the sets (4.7) by the sets
$\\{L:\widetilde{\mathrm{len}}_{\theta(\widetilde{\rho}_{j}+\sigma)}L\geq\delta,\mathrm{cen}(L)\in\widetilde{W}\\},\qquad
j=1,2.$
By Theorem 1.4, this difference in loop masses is given by the difference in
the quantities
$\frac{\widetilde{\mathrm{Vol}}_{\theta(\widetilde{\rho}_{j}+\sigma)}(\widetilde{W})}{2\pi\delta}+\frac{1}{48\pi}\frac{1}{48\pi}\int_{\widetilde{W}}\|\nabla(\theta(\rho_{j}+\sigma))\|^{2}dz,\qquad
j=1,2.$ (4.8)
By Proposition 4.7 with $f=\theta\widetilde{\rho}_{j}$ and $h=\sigma$,
together with the fact that $\widetilde{\rho}_{1}\equiv\widetilde{\rho}_{2}$
outside $U$ and $\theta\equiv 1$ in $V$, we can write the difference in the
quantities (4.8) as the difference in the quantities
$\frac{\widetilde{\mathrm{Vol}}_{\widetilde{\rho}_{j}+\sigma}(\widetilde{W})}{2\pi\delta}+\frac{1}{48\pi}\int_{\widetilde{W}}(\|\nabla_{\sigma}\widetilde{\rho}_{j}\|^{2}+2K_{\sigma}\widetilde{\rho}_{j})\widetilde{\mathrm{Vol}}_{\sigma}(dz),\quad
j=1,2,$ (4.9)
where $\nabla_{\sigma},K_{\sigma}$ are the gradient and Gauss curvature
associated to $(\widetilde{W},e^{\sigma}|dz|^{2})$. Pulling back via
$\varphi$, we can rewrite the expressions (4.9) as
$\frac{\mathrm{Vol}_{\rho_{j}}(W)}{2\pi\delta}+\frac{1}{48\pi}\int_{W}(\|\nabla\rho_{j}\|^{2}+2K\rho_{j})\mathrm{Vol}(dz),\quad
j=1,2.$
We complete the proof by noting that, since $\rho_{1}\equiv\rho_{2}$ outside
$W$, we can replace $W$ by $M$ in (4.2). ∎
###### Proof of Theorem 1.9.
By conformal invariance of the Brownian loop measure [APPS20, Lemma 3.3], the
statement of the theorem is equivalent to the assertion that (1.7) holds up to
scaling. The result holds for $\rho\equiv 0$ by [APPS20, Theorem 1.3].666We
remark that the $\delta$ in the statement of [APPS20, Theorem 1.3] represents
the quadratic variation of Brownian loops, which is two times the time
interval length we use in this paper. Thus, for our application, we need to
substitute $\delta$ there into $\delta/2$. We deduce the result for general
$\rho\in\mathcal{B}$ by applying Propositions 4.4 and 4.5. ∎
## 5 Constructing square subdivision regularizations
Now what happens if $\rho$ is defined from a finite square subdivision as in
[APPS20, Section 6], so that it has constant Laplacians on each square in a
grid? These functions can be shown to be $C^{1}$, but along the edges of the
squares they fail to be $C^{2}$. In this section, we explain enough about
these function to make it clear that they fit into the framework of this
paper, at least if one restricts attentions to those for which the square
averages are restricted to a compact set.
One can construct functions with piecewise-constant Laplacian on squares
somewhat explicitly. Consider the function on
$f(z)=\mathrm{Re}[z^{2}\log(z^{2})]/\pi+|z|^{2}/2$ restricted to the quadrant
$Q=\\{z:\arg(z)\in(-\pi/4,\pi/4)\\}$. Note that on the boundary of the
quadrant, $z^{2}$ is purely imaginary, equal to $|z^{2}|i$ on the upper
boundary ray and $-|z^{2}|i$ on the lower, while
$\mathrm{Im}\log(z^{2})=\arg(z^{2})$ is $\pi/2$ on the upper boundary and
$-\pi/2$ on the lower. so that $\mathrm{Re}[z^{2}\log(z^{2})]/\pi=-|z^{2}|/2$
on both rays. Thus $f$ has constant Laplacian and equals zero on the boundary
$Q$. We can extend the definition of $f$ to the other three quadrants ($iQ$,
$-iQ$, and $-Q$) by imposing the relation $f(iz)=-f(z)$. In a small
neighborhood of the origin, the $f$ defined this way is negative on $\pm Q$
and positive on $\pm iQ$. The complex derivatives of
$g(z)=z^{2}\log(z^{2})=2z^{2}\log(z)$ are first $g^{\prime}(z)=4z\log(z)+2z$,
second $g^{\prime\prime}(z)=4\log(z)+6$, third
$g^{\prime\prime\prime}(z)=4/z$, etc. One can deduce from this that $f$ is
differentiable as a real-valued function (with derivative $0$ at $0$) but that
its second derivatives blow up slowly (logarithmically) near zero and also
have discontinuities along the quadrant boundaries.
The function $f(\omega z)$ (where $\omega$ is a fixed eighth root of unity) is
thus a $C^{1}$ function that has piecewise constant Laplacian on the four
standard quadrants, while being equal to zero on the boundaries of these
quadrants.
By taking linear combinations of $f(\omega z)$ and the four functions
$\bigl{[}\max\bigl{(}\mathrm{Re}(az),0\bigr{)}\bigr{]}^{2}$ (with $a\in\\{\pm
1,\pm i\\}$) we can get a differentiable function whose Laplacian matches any
function that is constant on each of the four standard quadrants — in
particular a function whose Laplacian is $1$ on one quadrant and $0$ on the
other three. By taking differences of translates of this function, we can
obtain a function whose Laplacian is $1$ on a semi-strip or a rectangle (and
$0$ elsewhere). Linear combinations of these allow us to describe a function
$\phi$ whose Laplacian is any given function that is constant on the squares
of the grid. Any other function with the same Laplacian has to then differ
from $\phi$ by a harmonic function. For example, by subtracting the harmonic
extension of the values of $\phi$ on $\partial D$ one obtains a function with
the desired Laplacian whose boundary values on $\partial D$ are zero.
## 6 Open questions
In this section, we discuss some open questions. First, as already mentioned
in Footnote 3, we expect that some of our main results can be extended to
$\rho\in W^{1,2}(D)$. In addition to the fact that the Dirichlet energy is
well-defined for such functions, here is another reason that the arguments
from the Lipschitz case might carry over to this setting. Suppose that
$-M\leq\rho\leq M$. Then the function
$f(x):=\begin{cases}e^{x}-1&\text{ for }x\in[-M,M]\\\ 0&\text{
otherwise.}\end{cases}$
is a Lipschitz function on $[-M,M]$ with $f(0)=0$. Therefore, the celebrated
Stampacchia’s theorem [EG18, §4.2.2] asserts that $\rho\in W^{1,2}(D)$ implies
$e^{\rho}-1$ is $W^{1,2}(D)$ and the chain rule of weak derivatives implies
that
$\nabla(e^{\rho(x)})=e^{\rho(x)}\nabla\rho(x)$
holds for almost every $x$. This already implies that many parts of our proofs
carry over to the general case. As an example, Lemma 1.17 follows almost
immediately with the condition of precompactness in $W^{1,2}(D)$, or
equivalently the uniform equicontinuity in $W^{1,2}(D)$, in the sense of
Remark 1.7. However, there are a few technical issues where we need to improve
our estimates. For example, our proof of Lemma 1.16 is based on a pointwise
estimate of $\nabla\rho$, which would have to be replaced by an $L^{2}$
estimate, and such an estimate does not immediately follow from our current
arguments.
###### Question 6.1.
Prove that Theorem 1.4 holds for a larger class of functions. For example, one
might consider functions in $W^{1,2}(D)$ and take $\mathcal{B}$ to be any
precompact subset of $W^{1,2}(D)$, possibly with some additional conditions.
Next, we note that Lemma 1.17 is proven for Schwartz functions $\theta$, which
is not expected to be optimal in any sense other than making the proofs
simple. In fact, we believe the result holds for a much more general class of
functions. For instance, we apply the lemma to _expected_ occupation measure,
but it should also be true for a single occupation measure in some sense.
###### Question 6.2.
Prove that Lemma 1.17 holds for a more general class of functions. For
example, prove a similar result for a random function describing the
occupation measure of a Brownian loop.
Third, we may consider the case when the manifold $M$ has a boundary, say when
$M=\mathbb{H}$. In principle, one could formulate the boundary problem in a
style similar to that presented above, and try to weaken the required
regularity along the boundary as well. To start, imagine the boundary line is
the horizontal real axis. Let $d\ell_{b}$ be the measure on unit length loops
that hit the real axis and are centered at a point on the positive imaginary
axis.
This measure can be obtained from by starting with $d\ell$, then weighting by
the gap between the minimum and maximum imaginary values obtained by the loop,
then shifting the vertical height loop to a uniformly random height (within
this range). The expected maximal height of a Brownian bridge is
$\sqrt{\pi/8}$ (as can be proved using the reflection principle). The expected
minimal height is thus minus that, and the expected difference
$2\sqrt{\pi/8}$, which means that the expected vertical gap between the
maximum height and the central height is again $\sqrt{\pi/8}$.
So formally, instead of being a probability measure, $d\ell_{b}$ is a measure
with weight $\sqrt{\pi/8}$. For real values $x$, write
$\mathrm{bclen}(x,t,\ell_{b})=e^{\rho(x)}\delta$. Note that the set of loops
of length greater than $\delta$ is given by
$\int_{\delta}^{\infty}\sqrt{\pi/8}\sqrt{t}\frac{1}{2\pi
t^{2}}dt=\frac{2\sqrt{\pi/8}}{2\pi\sqrt{\delta}}=\frac{1}{2\sqrt{2\pi\delta}}$
We can imagine $\rho$ is defined in a neighborhood of a real line segment, and
then try to measure the mass of loops (of size greater than $\delta$) that hit
the boundary line itself. The relevant quantities are the gradient in the
parallel direction and the gradient in the normal direction (with the latter
affecting whether the real axis is hit by more loops centered above or below).
###### Question 6.3.
Prove that Theorem 1.9 also holds for manifolds with boundaries. In other
words, generalize the boundary case of [APPS20, Theorem 1.3]:777Similar to the
case without boundary, we plug $\delta/2$ instead of $\delta$ in the statement
of [APPS20, Theorem 1.3], where $\delta$ there represents the quadratic
variation of Brownian loops, which is two times the time interval length we
use in this paper.
###### Conjecture 6.4.
Let $(M,g)$ be a fixed compact smooth two-dimensional Riemannian manifold with
smooth boundary, and we let $\mu^{\text{loop}}$ denote the Brownian loop
measure on $(M,g)$. Let $K$ be the Gaussian curvature on $M$, let $\Delta$ be
the Laplacian associated to $(M,g)$, and let $\det_{\zeta}^{\prime}\Delta$
denote its zeta-regularized determinant. Let $\mathcal{B}$ be a family of
Lipschitz functions that (1) has uniformly bounded Lipshitz constants, and (2)
is precompact in $W^{1,1}(M)$.
The $\mu^{\text{loop}}$-mass of loops with $\rho$-length greater than $\delta$
is given by
$\displaystyle\frac{\mathrm{Vol}_{\rho}(M)}{2\pi\delta}-\frac{\mathrm{Len}_{\rho}(\partial
M)}{2\sqrt{2\pi\delta}}-{\frac{\chi(M)}{6}}(\log\frac{\delta}{2}+\upgamma)+\frac{1}{48\pi}\int_{M}(\|\nabla\rho\|^{2}+2K\rho)\,\mathrm{Vol}(dz)$
$\displaystyle\qquad\qquad\qquad+\log\mathrm{Vol}(M)-\log\mathrm{Vol}_{\rho}(M)-\log\det\nolimits_{\zeta}^{\prime}\Delta+O(\delta^{1/2}),$
with the convergence as $\delta\to 0$ uniform over $\rho\in\mathcal{B}$, where
$\upgamma\approx 0.5772$ is the Euler-Mascheroni constant.
Finally, one naive approach to generalizing [APPS20, Proposition 6.9] for
lower-regularity settings is to follow the zeta-regularization procedure
verbatim, hoping each step works for the generalized settings in a similar
way. The first obstacle in this direction is the lack of short-time expansions
for the trace of heat kernels. For our application, what we need is if $(M,g)$
is a two-dimensional _non-smooth_ manifold without boundary, and $\Delta$ is
the associated Laplacian defined in terms of Brownian loop mass, then
$\mathrm{tr}(e^{-\delta\Delta/2})=\mathrm{Vol}_{g}(M)/\delta+\chi(M)/6+o(1).$
###### Question 6.5.
Prove the short time expansion for the (trace) heat kernel holds for lower
regularity metrics.
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# On closure operations in the space of subgroups and applications
Dominik Francoeur Newcastle University, Newcastle upon Tyne, NE1 7RU, United
Kingdom<EMAIL_ADDRESS>and Adrien Le Boudec CNRS, UMPA -
ENS Lyon, 46 allée d’Italie, 69364 Lyon, France<EMAIL_ADDRESS>
(Date: July 14, 2024)
###### Abstract.
We establish some interactions between uniformly recurrent subgroups (URSs) of
a group $G$ and cosets topologies $\tau_{\mathcal{N}}$ on $G$ associated to a
family $\mathcal{N}$ of normal subgroups of $G$. We show that when
$\mathcal{N}$ consists of finite index subgroups of $G$, there is a natural
closure operation $\mathcal{H}\mapsto\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$
that associates to a URS $\mathcal{H}$ another URS
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$, called the
$\tau_{\mathcal{N}}$-closure of $\mathcal{H}$. We give a characterization of
the URSs $\mathcal{H}$ that are $\tau_{\mathcal{N}}$-closed in terms of
stabilizer URSs. This has consequences on arbitrary URSs when $G$ belongs to
the class of groups for which every faithful minimal profinite action is
topologically free. We also consider the largest amenable URS
$\mathcal{A}_{G}$, and prove that for certain coset topologies on $G$, almost
all subgroups $H\in\mathcal{A}_{G}$ have the same closure. For groups in which
amenability is detected by a set of laws, we deduce a criterion for
$\mathcal{A}_{G}$ to be a singleton based on residual properties of $G$.
This work had been initiated within the framework of the Labex Milyon (ANR-10-
LABX-0070) of Universite de Lyon, within the program "Investissements
d’Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency
(ANR)
## 1\. Introduction
Let $G$ be a group. We denote by $\mathcal{N}_{G}$ the set of normal subgroups
of $G$. Let $\mathcal{N}\subseteq\mathcal{N}_{G}$ be a family of normal
subgroups of $G$ that is filtering: for every $N_{1},N_{2}\in\mathcal{N}$
there exists $N_{3}\in\mathcal{N}$ such that $N_{3}\leq N_{1}\cap N_{2}$.
There is a group topology $\tau_{\mathcal{N}}$ on $G$ associated to
$\mathcal{N}$, defined by declaring that the family of cosets $gN$, $g\in G$,
$N\in\mathcal{N}$, forms a basis for $\tau_{\mathcal{N}}$. When $\mathcal{N}$
is the family of all finite index normal subgroups of $G$,
$\tau_{\mathcal{N}}$ is the profinite topology on $G$. If $p$ is a prime and
$\mathcal{N}$ is the family of finite index normal subgroups $N$ of $G$ such
that $G/N$ is a $p$-group, $\tau_{\mathcal{N}}$ is the pro-$p$ topology.
If $H$ is a subgroup of $G$, the closure of $H$ with respect to
$\tau_{\mathcal{N}}$ is denoted by $\mathrm{cl}_{\mathcal{N}}(H)$. In the case
of the profinite topology, we use the shorter notation $\mathrm{cl}(H)$. The
closure operation defines a map
$\mathrm{cl}_{\mathcal{N}}:\operatorname{Sub}(G)\to\operatorname{Sub}(G),\,H\mapsto\mathrm{cl}_{\mathcal{N}}(H).$
Here $\operatorname{Sub}(G)$ is the set of subgroups of $G$. That set is
equipped with the topology inherited from the set $\left\\{0,1\right\\}^{G}$
of all subsets of $G$, equipped with the product topology. The space
$\operatorname{Sub}(G)$ is a compact space. The group $G$ acts on
$\operatorname{Sub}(G)$ by conjugation, and this action is by homeomorphisms.
The first object of study of this article is the behaviour of the map
$\mathrm{cl}_{\mathcal{N}}$ with respect to the dynamical system
$G\curvearrowright\operatorname{Sub}(G)$.
It follows from the definitions that the map $\mathrm{cl}_{\mathcal{N}}$ is
always increasing, idempotent, and $G$-equivariant. In general
$\mathrm{cl}_{\mathcal{N}}$ is far from being continuous. This failure of
continuity already happens in the most classical case where
$\tau_{\mathcal{N}}$ is the profinite topology. An elementary example
illustrating this is the group $G=\mathbb{Z}[1/p]$ of $p$-adic rational
numbers, for which the map $\mathrm{cl}$ is not upper semi-continuous on
$\operatorname{Sub}(G)$ (see Remark 3.6). Another example is $G=F_{k}$ (a
finitely generated non-abelian free group of rank $k$). M. Hall showed that
every finitely generated subgroup $H$ of $F_{k}$ verifies $\mathrm{cl}(H)=H$
[Hal49] (i.e. $F_{k}$ is a LERF group). Since finitely generated subgroups
always form a dense subset in the space of subgroups, it follows that
$\mathrm{cl}$ is the identity on a dense set of points. However $\mathrm{cl}$
is not the identity everywhere, for instance because $F_{k}$ admits infinite
index subgroups $H$ such that $\mathrm{cl}(H)=F_{k}$ (e.g. any infinite index
maximal subgroup). So $\mathrm{cl}$ is not lower semi-continuous on
$\operatorname{Sub}(F_{k})$.
The starting result of this article is that if we restrict to minimal
subsystems of $\operatorname{Sub}(G)$ (i.e. non-empty closed minimal
$G$-invariant subsets of $\operatorname{Sub}(G)$), the situation is better
behaved. Recall that a minimal subsystem
$\mathcal{H}\subset\operatorname{Sub}(G)$ is called a URS (Uniformly Recurrent
Subgroup) [GW15].
###### Proposition 1.
Let $\mathcal{N}\subseteq\mathcal{N}_{G}$ be a family of finite index normal
subgroups of $G$, and let $\mathcal{H}$ be a URS of $G$. Then the following
hold:
1. (1)
The restriction
${\mathrm{cl}_{\mathcal{N}}}_{|\mathcal{H}}:\mathcal{H}\to\operatorname{Sub}(G)$
is upper semi-continuous.
2. (2)
There exists a unique URS contained in
$\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(H):H\in\mathcal{H}\right\\}}$,
denoted $\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$, and called the
$\tau_{\mathcal{N}}$-closure of $\mathcal{H}$.
The proposition also holds in a more general situation not necessarily
requiring that $\mathcal{N}$ consists of finite index subgroups of $G$ (see
Proposition 3.4).
Statement (2) says that there is a natural closure operation
$\mathrm{URS}(G)\to\mathrm{URS}(G),\,\mathcal{H}\mapsto\mathrm{cl}_{\mathcal{N}}(\mathcal{H}),$
where $\mathrm{URS}(G)$ is the set of URSs of the group $G$. We say that a URS
$\mathcal{H}$ is closed for the topology $\tau_{\mathcal{N}}$ if
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})=\mathcal{H}$. When $G$ is a countable
group, this happens if and only if there is a dense $G_{\delta}$-set of points
$H\in\mathcal{H}$ such that $H$ is closed for the topology
$\tau_{\mathcal{N}}$.
Recently URSs were studied and appeared in a large amount of works, including
[LBMB18, BH21, FG23, LBMB22]. We refer notably to the introduction of [LBMB22]
for more references. A common theme is to establish rigidity results saying
that the set of URSs of certain groups is restricted, or to establish
connections between certain group theoretic properties of the ambient group
and properties of its URSs. We believe that in certain situations the above
process $\mathcal{H}\mapsto\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$, and more
generally the consideration of coset topologies on the ambient group, can be
profitably used to study properties of URSs. In Sections 4 and 5 we exhibit
situations where it is indeed the case. In the remainder of this introduction
we shall describe these results.
When $\mathcal{N}$ consists of finite index subgroups, the property that a URS
$\mathcal{H}$ is closed for the topology $\tau_{\mathcal{N}}$ admits the
following natural characterization. Glasner–Weiss showed that to every minimal
action of $G$ on a compact space $X$, there is a naturally associated URS of
$G$, called the stabilizer URS of $X$, and denoted $S_{G}(X)$ [GW15]. We say
that the action of $G$ on a compact space $X$ is pro-$\mathcal{N}$ if $G\times
X\to X$ is continuous, where $G$ is equipped with the topology
$\tau_{\mathcal{N}}$ (see Proposition 4.2 for characterizations of this
property).
###### Proposition 2.
Suppose that $G$ is a countable group and that $\mathcal{N}$ consists of
finite index subgroups of $G$. For a URS $\mathcal{H}$ of $G$, the following
are equivalent:
1. (1)
$\mathcal{H}$ is closed for the topology $\tau_{\mathcal{N}}$.
2. (2)
There exists a pro-$\mathcal{N}$ compact minimal $G$-space $X$ such that
$S_{G}(X)=\mathcal{H}$.
In the case of the profinite topology, the notion of pro-$\mathcal{N}$
$G$-space coincides with the classical notion of profinite $G$-space. So in
that situation the above proposition says that a URS $\mathcal{H}$ is closed
for the profinite topology if and only if $\mathcal{H}$ is the stabilizer URS
associated to a minimal profinite action of $G$. Consequences on all URSs can
be drawn out of this when $G$ belongs to the class of groups for which, for a
faithful minimal compact $G$-space, profinite implies topologically free. See
Proposition 4.17. This class of groups includes non-abelian free groups, and
more generally any group $G$ admitting an isometric action on a hyperbolic
space with unbounded orbits such that the $G$-action on its limit set is
faithful. It also includes hereditarily just-infinite groups. Recall that a
group $G$ is just-infinite if $G$ is infinite and $G/N$ is finite for every
non-trivial normal subgroup $N$, and $G$ is hereditarily just-infinite if
every finite index subgroup of $G$ is just-infinite. We call a subgroup $H$ of
$G$ quasi-dense in $G$ for the profinite topology if the profinite closure of
$H$ has finite index in $G$. For hereditarily just-infinite groups we obtain:
###### Proposition 3.
Let $G$ be a hereditarily just-infinite group, and let $\mathcal{H}$ be a non-
trivial URS of $G$. Then for every $H\in\mathcal{H}$, $H$ is quasi-dense in
$G$ for the profinite topology.
In cases where we know a priori that the group $G$ has the property that the
only subgroups that are quasi-dense are the finite index subgroups, we deduce
that such a group $G$ admits no continuous URS (a URS is continuous if it is
not a finite set). See Corollary 4.20, and the surrounding discussion for
context and examples.
Another setting in which we show that the consideration of a coset topology
$\tau_{\mathcal{N}}$ is fruitful with respect to the study of URSs is the case
amenable URSs. A URS $\mathcal{H}$ is amenable if it consists of amenable
subgroups. Every group $G$ admits a largest amenable URS (with respect to a
natural partial order), which is the stabilizer URS associated to the action
of $G$ on its Furstenberg boundary (the largest minimal and strongly proximal
compact $G$-space). This URS is denoted $\mathcal{A}_{G}$ and is called the
Furstenberg URS of $G$. The action of $G$ on $\mathcal{A}_{G}$ is minimal and
strongly proximal. $\mathcal{A}_{G}$ is either a singleton, in which case we
have $\mathcal{A}_{G}=\left\\{\mathrm{Rad}(G)\right\\}$, where
$\mathrm{Rad}(G)$ is the amenable radical of $G$, or $\mathcal{A}_{G}$ is
continuous. We refer to [LBMB18] for a more detailed discussion.
Let $\mathcal{F}$ denote the class of groups $G$ such $\mathcal{A}_{G}$ is a
singleton. Equivalently, $G$ belongs to $\mathcal{F}$ if and only if every
amenable URS of $G$ lives inside the amenable radical of $G$. The class
$\mathcal{F}$ is known to be very large. It plainly contains amenable groups.
It also contains all linear groups, all groups with non-vanishing
$\ell^{2}$-Betti numbers, all hyperbolic groups, and more generally all
acylindrically hyperbolic groups. We refer to [BKKO17] for references and
details. Examples of groups outside the class $\mathcal{F}$ have been given in
[LB17].
The following result provides a criterion for a group to be in $\mathcal{F}$
that is based on residual properties of the group.
###### Theorem 1.
Let $G$ be a group such that every amenable subgroup of $G$ is virtually
solvable. If $G$ is residually-$\mathcal{F}$, then $G$ is in $\mathcal{F}$.
We point out that this theorem is applicable without necessarily relying on
other methods related to $\mathcal{F}$ to verify the assumption that the group
is residually-$\mathcal{F}$. The point is that the statement applies provided
that $G$ is residually-$\mathcal{C}$ for some subclass $\mathcal{C}$ of
$\mathcal{F}$ that is potentially much smaller. For instance the theorem
applies and is already interesting if $G$ is residually finite.
One interest of such a statement is that it is based on intrinsic algebraic
properties of the group. It does not require the group $G$ to admit a rich
action of geometric flavour, or to have an explicit minimal and strongly
proximal compact $G$-space at our disposal. The residual properties are used
as a tool in Theorem 1, but the confrontation of residual properties and the
class $\mathcal{F}$ is also motivated by the fact that it is not known whether
there exist residually finite groups $G$ with trivial amenable radical such
that $G$ does not belong to $\mathcal{F}$. The groups from [LB17] are never
residually finite (and some of them are virtually simple).
As an application, Theorem 1 allows to recover the following result from
[BKKO17]:
###### Corollary 1 (Breuillard–Kalantar–Kennedy–Ozawa).
If $G$ is a linear group, then $G$ is in $\mathcal{F}$.
The proof from [BKKO17] relies on linear groups technology. Here the argument
to deduce Corollary 1 from Theorem 1 uses a reduction to the case of finitely
generated groups, and then only appeals to Malcev’s theorem that finitely
generated linear groups are residually finite, and the Tits alternative.
The consideration of the class $\mathcal{F}$ is also motivated by the result
of Kalantar–Kennedy that a group $G$ belongs to $\mathcal{F}$ if and only if
the quotient of $G$ by its amenable radical is a $C^{\ast}$-simple group (that
is, its reduced $C^{\ast}$-algebra is simple) [KK17]. We refer to the survey
of de la Harpe [dlH07] for an introduction and historical developments on
$C^{\ast}$-simple groups, and to the Bourbaki seminar of Raum for recent
developments [Rau20]. Hence using the result of Kalantar–Kennedy, Theorem 1
can be reinterpreted as a criterion to obtain $C^{\ast}$-simplicity (under the
assumption on amenable subgroups) based on residual properties of the group.
See Corollary 5.14. We are not aware of other results of this kind.
The proof of Theorem 1 is based on the following proposition, of independent
interest. Given a group $G$, we denote by $\mathcal{N}_{G}(\mathcal{F})$ the
set of normal subgroups of $G$ such that $G/N\in\mathcal{F}$. The set
$\mathcal{N}_{G}(\mathcal{F})$ is stable under taking finite intersections
(Lemma 5.7), and we can consider the coset topology on $G$ associated to
$\mathcal{N}_{G}(\mathcal{F})$ (and more generally to a subset
$\mathcal{N}\subseteq\mathcal{N}_{G}(\mathcal{F})$). The following result says
that within the Furstenberg URS $\mathcal{A}_{G}$, almost all points have the
same closure for such a topology (for technical reasons we are led to make
some countability assumptions).
###### Proposition 4.
Let $G$ be a countable group, and let $\mathcal{N}$ be a countable subset of
$\mathcal{N}_{G}(\mathcal{F})$. Then there exists a normal subgroup $M$ of $G$
and a comeager subset $\mathcal{H}_{0}\subseteq\mathcal{A}_{G}$ such that
$\mathrm{cl}_{\mathcal{N}}(H)=M$ for every $H\in\mathcal{H}_{0}$.
The proof of the proposition makes crucial use of the strong proximality of
the action of $G$ on $\mathcal{A}_{G}$. The proof of Theorem 1 is easily
deduced from the proposition. The additional point is to ensure that the
closed normal subgroup $M$ appearing in the conclusion of the proposition
remains amenable, and this is where the two assumptions in the theorem are
used. We refer to Section 5 for details. Here we only mention that the actual
setting in which we prove Theorem 1 does not necessarily require amenable
subgroups to be virtually solvable. The assumption that we need is that
amenability within subgroups of $G$ can be detected by a set of laws
(Definition 5.10), a property that can be thought of as a version of the Tits
alternative. See Theorem 5.12 for the more general formulation of the theorem.
Acknowledgements. Thanks are due to Uri Bader and Pierre-Emmanuel Caprace. We
can trace back that the possibility of using Proposition 2.1 specifically in
the space of subgroups to build a URS starting from another one and a semi-
continuous map had been originally brought to our attention by them several
years ago.
## 2\. Preliminaries
A space $X$ is a $G$-space if $G$ admits a continuous action $G\times
X\rightarrow X$. Throughout the paper we make the standing assumption that
$G$-spaces are non-empty. The action (or the $G$-space $X$) is minimal if all
orbits are dense. For $x\in X$ we write $G_{x}$ for the stabilizer of $x$ in
$G$, and $G_{x}^{0}$ for the set of $g\in G$ such that $g$ acts trivially on a
neighbourhood of $x$. The action of $G$ on $X$ is free if
$G_{x}=\left\\{1\right\\}$ for every $x\in X$, and topologically free if
$G_{x}^{0}=\left\\{1\right\\}$ for every $x\in X$.
Let $X,Y$ be compact spaces. A continuous surjective map $\pi:Y\to X$ is
called irreducible if every proper closed subset of $Y$ has a proper image in
$X$. If $X,Y$ are compact $G$-spaces and $\pi:Y\to X$ is a continuous
surjective $G$-equivariant map, we say that $X$ is a factor of $Y$, and that
$Y$ is an extension of $X$. When $\pi:Y\to X$ is irreducible, we also say that
$Y$ is an irreducible extension of $X$. If $\pi:Y\to X$ is irreducible, then
$X$ is minimal if and only if $Y$ is minimal. Also for $X,Y$ minimal,
$\pi:Y\to X$ is irreducible if and only if it is highly proximal: for every
$x\in X$ the fiber $\pi^{-1}(x)$ is compressible [AG77].
### 2.1. Semi-continuous maps
If $Y$ is a locally compact space, we denote by $2^{Y}$ the space of closed
subsets of $Y$, endowed with the Chabauty topology. The space $2^{Y}$ is
compact.
Let $X$ be a compact $G$-space. A map $\varphi\colon X\to 2^{Y}$ is upper
semi-continuous if for every compact subset $K$ of $Y$, $\left\\{x\in
X:\varphi(x)\cap K=\emptyset\right\\}$ is open in $X$. It is lower semi-
continuous if for every open subset $U$ of $Y$, $\left\\{x\in X:\varphi(x)\cap
U\neq\emptyset\right\\}$ is open in $X$. We say that $\varphi$ is semi-
continuous if it is either upper or lower semi-continuous.
Let $\varphi:X\to 2^{Y}$ be a semi-continuous map, and $X_{\varphi}\subseteq
X$ be the set of points where $\varphi$ is continuous. Let
$F_{\varphi}:=\overline{\left\\{\left(x,\varphi(x)\right)\,:\,x\in
X\right\\}}\subseteq X\times 2^{Y},$
$E_{\varphi}:=\overline{\left\\{\left(x,\varphi(x)\right)\,:\,x\in
X_{\varphi}\right\\}}\subseteq F_{\varphi}(X),$
$T_{\varphi}:=\overline{\left\\{\varphi(x)\,:\,x\in X\right\\}},$
$S_{\varphi}:=\overline{\left\\{\varphi(x)\,:\,x\in X_{\varphi}\right\\}}.$
We denote by $\eta:X\times 2^{Y}\to X$ and $p:X\times 2^{Y}\to 2^{Y}$ the
projections to the first and second coordinate. If $Y$ is second-countable,
semi-continuity of $\varphi$ implies that $X_{\varphi}$ is a comeager subset
of $X$ [Kur28, Theorem VII].
###### Proposition 2.1.
Suppose $X$ is a minimal compact $G$-space, $Y$ is a locally compact
$G$-space, and $\varphi:X\to 2^{Y}$ is $G$-equivariant and semi-continuous.
Then the following hold:
1. (i)
$F_{\varphi}$ has a unique non-empty minimal closed $G$-invariant subset
$E_{\varphi}^{\prime}$ , and $T_{\varphi}$ has a unique minimal closed
$G$-invariant subset $S_{\varphi}^{\prime}$, and
$p(E_{\varphi}^{\prime})=S_{\varphi}^{\prime}$ .
2. (ii)
The extension $\eta:E_{\varphi}^{\prime}\to X$ is highly proximal.
If moreover $Y$ is second-countable, then $E_{\varphi}^{\prime}=E_{\varphi}$
and $S_{\varphi}^{\prime}=S_{\varphi}$ .
###### Proof.
See Glasner [Gla75, Theorem 2.3] and Auslander–Glasner [AG77, Lemma I.1]. ∎
### 2.2. The space of subgroups and URSs
We denote by $\operatorname{Sub}(G)$ the space of subgroups of $G$, equipped
with the product topology from $\left\\{0,1\right\\}^{G}$. It is a compact
$G$-space, where $G$ acts by conjugation. If $H\in\operatorname{Sub}(G)$, we
denote by $H^{G}$ the $G$-conjugates of $H$, i.e. the $G$-orbit of $H$ in
$\operatorname{Sub}(G)$.
A URS of $G$ is a (non-empty) minimal closed $G$-invariant subset of
$\operatorname{Sub}(G)$. By Zorn’s lemma every (non-empty) closed
$G$-invariant subset of $\operatorname{Sub}(G)$ contains a URS. A URS is
finite if it is a finite $G$-orbit. A URS that is not finite is called
continuous. By minimality and compactness, a continuous URS has no isolated
points. The singleton $\left\\{\left\\{1\right\\}\right\\}$ is called the
trivial URS. If $\mathcal{P}$ is a property of groups, we say that a URS
$\mathcal{H}$ has $\mathcal{P}$ if $H$ has $\mathcal{P}$ for every
$H\in\mathcal{H}$.
###### Definition 2.2.
If $\mathcal{H}$ is a URS of $G$, we denote by $\mathrm{Env}(\mathcal{H})$ the
subgroup generated by all subgroups $H$ in $\mathcal{H}$. The subgroup
$\mathrm{Env}(\mathcal{H})$ is normal in $G$, and it is the smallest normal
subgroup of $G$ containing some subgroup $H\in\mathcal{H}$.
Every minimal compact $G$-space naturally gives rise to a URS [GW15]:
###### Proposition 2.3.
If $X$ is a compact $G$-space, then the stabilizer map
$S:X\to\operatorname{Sub}(G)$, $x\mapsto G_{x}$, is $G$-equivariant and upper
semi-continuous. In particular if $X$ is minimal, then Proposition 2.1
applies.
###### Definition 2.4.
If $X$ is a minimal compact $G$-space, the unique URS contained in
$\overline{\left\\{G_{x}\,:\,x\in X\right\\}}$ is denoted $S_{G}(X)$, and is
called the stabilizer URS associated to the $G$-space $X$.
One verifies that the $G$-action on $X$ is topologically free if and only if
the URS $S_{G}(X)$ is trivial.
###### Lemma 2.5.
Let $H,K,L$ be subgroups of $G$ such that $H\leq L$. If $K$ belongs to the
closure of the $L$-orbit of $H$ in $\operatorname{Sub}(G)$, then $K\leq L$.
###### Proof.
The subset $\operatorname{Sub}(L)$ is a closed subset of
$\operatorname{Sub}(G)$, and contains the $L$-orbit of $H$ since $H\leq L$. ∎
###### Lemma 2.6.
Let $N$ be a normal subgroup of $G$. Then the map
$\operatorname{Sub}(G)\to\operatorname{Sub}(G)$, $H\mapsto HN$, is
$G$-equivariant and lower semi-continuous.
###### Proof.
It is $G$-equivariant because $N$ is normal in $G$. Since $G$ is discrete,
lower semi-continuity means that for every $g\in G$ and every
$H\in\operatorname{Sub}(G)$ such that $g\in HN$, there is a neighbourhood of
$H$ in which $g\in H^{\prime}N$ remains true. If $h\in H$ is such that $g\in
hN$, then the set of subgroups of $G$ containing $h$ is such a neighbourhood.
∎
### 2.3. Coset topologies on groups
Let $G$ be a group. We denote by $\mathcal{N}_{G}$ the set of normal subgroups
of $G$. We make the standing convention that when considering a family
$\mathcal{N}\subseteq\mathcal{N}_{G}$ of normal subgroups of $G$, we always
assume that $\mathcal{N}$ is non-empty.
###### Definition 2.7.
Let $\mathcal{N}\subseteq\mathcal{N}_{G}$. If $H$ is a subset of $G$, we
denote
$\mathrm{cl}_{\mathcal{N}}(H)=\bigcap_{N\in\mathcal{N}}HN.$
We say that $\mathcal{N}$ is filtering if for every
$N_{1},N_{2}\in\mathcal{N}$ there exists $N_{3}\in\mathcal{N}$ such that
$N_{3}\leq N_{1}\cap N_{2}$. We record the following [Bou71, Chap. III]:
###### Proposition 2.8.
Fix $\mathcal{N}\subseteq\mathcal{N}_{G}$ . Then:
1. (1)
the family of cosets $gN$, $g\in G$, $N\in\mathcal{N}$, forms a subbasis for a
group topology $\tau_{\mathcal{N}}$ on $G$.
2. (2)
The topology $\tau_{\mathcal{N}}$ is Hausdorff if and only if
$\bigcap_{\mathcal{N}}N=\left\\{1\right\\}$.
3. (3)
Suppose that $\mathcal{N}$ is filtering. Then for every subset $H$ of $G$, the
closure of $H$ with respect to $\tau_{\mathcal{N}}$ is equal to
$\mathrm{cl}_{\mathcal{N}}(H)$.
If $\mathcal{C}$ is a class of groups, we denote by
$\mathcal{N}_{G}(\mathcal{C})$ the normal subgroups of $G$ such that
$G/N\in\mathcal{C}$. Note that a group $G$ is residually-$\mathcal{C}$ if and
only if $\bigcap_{\mathcal{N}_{G}(\mathcal{C})}N=\left\\{1\right\\}$.
When $\mathcal{C}$ is the class of all finite groups and
$\mathcal{N}=\mathcal{N}_{G}(\mathcal{C})$, $\tau_{\mathcal{N}}$ is the
profinite topology on $G$. For simplicity we write $\mathrm{cl}(H)$ for the
closure in the profinite topology. When $\mathcal{C}$ is the class of all
finite $p$-groups ($p$ is a prime number) and
$\mathcal{N}=\mathcal{N}_{G}(\mathcal{C})$, $\tau_{\mathcal{N}}$ is the
pro-$p$ topology. In that case we write $\mathrm{cl}_{p}(H)$ for the closure
in the pro-$p$ topology.
### 2.4. Laws
Let $w=w(x_{1},\ldots,x_{k})$ be a word in $k$ letters $x_{1},\ldots,x_{k}$,
meaning that $w$ is an element of the free group $F_{k}$ freely generated by
$x_{1},\ldots,x_{k}$. Given a group $G$, the word $w$ naturally defines a map
$G^{k}\to G$, a $k$-tuple $(g_{1},\ldots,g_{k})$ being mapped to the element
$w(g_{1},\ldots,g_{k})$ of $G$ that is obtained by replacing each $x_{i}$ by
$g_{i}$. We denote by $\Sigma_{w}(G)\subseteq G^{k}$ the set of
$(g_{1},\ldots,g_{k})$ such that $w(g_{1},\ldots,g_{k})=1$. We say that $G$
satisfies the law $w$ if $\Sigma_{w}(G)=G^{k}$.
###### Lemma 2.9.
Suppose $G$ is a Hausdorff topological group, and let $w\in F_{k}$. Then
$\Sigma_{w}(G)$ is a closed subset of $G^{k}$. In particular if a subgroup $H$
of $G$ satisfies the law $w$, then so does its closure.
###### Proof.
Since $G$ is Hausdorff, $\left\\{1\right\\}$ is closed in $G$. The map
$G^{k}\to G$ associated to $w$ being continuous, the preimage $\Sigma_{w}(G)$
of $\left\\{1\right\\}$ is a closed subset of $G^{k}$. ∎
## 3\. The $\tau_{\mathcal{N}}$-closure of a URS
Let $\mathcal{H}$ be a closed subset of $\operatorname{Sub}(G)$, and $L$ a
subgroup of $G$. For every $\Sigma\subseteq L^{G}$, we write
$\mathcal{H}_{\Sigma}=\left\\{H\in\mathcal{H}:\forall K\in L^{G},\,H\subset
K\Leftrightarrow K\in\Sigma\right\\}.$
###### Lemma 3.1.
If $\mathcal{H}_{\Sigma}\cap\mathcal{H}_{\Sigma^{\prime}}\neq\emptyset$ then
$\Sigma=\Sigma^{\prime}$, and $\mathcal{H}$ is the disjoint union of the
$\mathcal{H}_{\Sigma}$ when $\Sigma$ ranges over subsets of $L^{G}$.
###### Proof.
The first assertion is consequence of the definitions. The second assertion is
also clear since for every $H\in\mathcal{H}$, one has
$H\in\mathcal{H}_{\Sigma}$ with $\Sigma=\left\\{K\in L^{G}:H\subset
K\right\\}.$ ∎
###### Lemma 3.2.
Let $L$ be a subgroup of $G$ such that $L^{G}$ is finite. Suppose
$\mathcal{H}$ is a URS of $G$. Then $\mathcal{H}_{\Sigma}$ is a clopen subset
of $\mathcal{H}$ for every $\Sigma\subseteq L^{G}$.
###### Proof.
For $H$ in $\mathcal{H}$, we let $n(H)$ be the number of conjugates of $L$
containing $H$. By our assumption, the number $n(H)$ is finite. We claim that
$n(H)$ is constant on $\mathcal{H}$. In order to see this, take
$H\in\mathcal{H}$ such that $n(H)=r$ is minimal. Since not being contained in
a subgroup is an open condition, one can find a neighbourhood $V$ of $H$ such
that $n(H^{\prime})\leq r$ for every $H^{\prime}\in V$. Hence by minimality of
$r$ we have $n(H^{\prime})=r$ for every $H^{\prime}\in V$. Now for every
$K\in\mathcal{H}$, by minimality of the $G$-action on $\mathcal{H}$ the subset
$V$ contains a conjugate of $K$. Since $n(K)$ is invariant under conjugation,
we deduce $n(K)=r$.
Now fix $\Sigma\subseteq L^{G}$ such that $\mathcal{H}_{\Sigma}$ is non-empty,
and let $H\in\mathcal{H}_{\Sigma}$. Again there is a neighbourhood $V$ of $H$
in $\mathcal{H}$ such that for every $H^{\prime}$ in $V$, we have
$H^{\prime}\not\subset J$ for every $J\in L^{G}\setminus\Sigma$. Moreover by
the previous paragraph we have $n(H^{\prime})=n(H)$. Hence by the pigeonhole
principle we deduce that $H^{\prime}\subset J$ for every $J\in\Sigma$. This
shows that $\mathcal{H}_{\Sigma}$ is open. Since the family
$(\mathcal{H}_{\Sigma})$ forms a partition of $\mathcal{H}$ by Lemma 3.1, it
follows that $\mathcal{H}_{\Sigma}$ is also closed. ∎
###### Definition 3.3.
Let $\mathcal{N}\subseteq\mathcal{N}_{G}$. We say that a URS $\mathcal{H}$ of
$G$ is $\mathcal{N}$-finitary if $(HN)^{G}$ is finite for every
$H\in\mathcal{H}$, $N\in\mathcal{N}$.
###### Proposition 3.4.
Let $\mathcal{N}\subseteq\mathcal{N}_{G}$, and let $\mathcal{H}$ be a URS of
$G$ that is $\mathcal{N}$-finitary. Then the map
$\mathcal{H}\to\operatorname{Sub}(G)$, $H\mapsto\mathrm{cl}_{\mathcal{N}}(H)$,
is upper semi-continuous.
###### Proof.
Let $K$ be a finite subset of $G$, and let $H\in\mathcal{H}$ such that
$\mathrm{cl}_{\mathcal{N}}(H)\cap K=\emptyset$. One shall prove that
$\mathrm{cl}_{\mathcal{N}}(H^{\prime})\cap K=\emptyset$ remains true for every
$H^{\prime}$ inside a neighbourhood of $H$ in $\mathcal{H}$. Let $g\in K$. By
definition of $\mathrm{cl}_{\mathcal{N}}(H)$, there exists
$N_{g}\in\mathcal{N}$ such that $g\notin HN_{g}$. Since $L=HN_{g}$ verifies
that $(HN_{g})^{G}$ is finite, according to Lemma 3.2 one can find a
neighbourhood $V_{g}$ of $H$ in $\mathcal{H}$ such that $H^{\prime}\leq
HN_{g}$ for every $H^{\prime}\in V_{g}$. A fortiori we have
$H^{\prime}N_{g}\leq HN_{g}$ and hence
$\mathrm{cl}_{\mathcal{N}}(H^{\prime})\leq HN_{g}$. Since $K$ is finite,
taking the intersection over all $g\in K$ we obtain a neighbourhood $V$ of $H$
in $\mathcal{H}$ such that
$\mathrm{cl}_{\mathcal{N}}(H^{\prime})\leq\bigcap_{g\in K}HN_{g}$ for every
$H^{\prime}\in V$. Since $K$ does not intersect $\bigcap_{g\in K}HN_{g}$, the
neighbourhood $V^{\prime}$ satisfies the required property. ∎
###### Remark 3.5.
If $\mathcal{N}$ consists of finite index normal subgroups of $G$, then
trivially every URS of $G$ is $\mathcal{N}$-finitary. Hence the previous
proposition applies.
###### Remark 3.6.
Here we still consider the case where $\mathcal{N}$ consists of finite index
normal subgroups of $G$, and we point out that in general the map
$\operatorname{Sub}(G)\to\operatorname{Sub}(G)$,
$H\mapsto\mathrm{cl}_{\mathcal{N}}(H)$, is not upper semi-continuous.
Therefore it is necessary to restrict to a URS in Proposition 3.4 in order to
obtain upper semi-continuity. As an illustration, consider the group
$G=\mathbb{Z}[1/p]$ of $p$-adic rational numbers. For $n\geq 1$, let
$H_{n}=p^{n}\mathbb{Z}$. Then $G/H_{n}$ is a Prüfer $p$-group, and hence has
no proper finite index subgroup. So $G$ has no proper finite index subgroup
containing $H_{n}$, or equivalently $\mathrm{cl}(H_{n})=G$. On the other hand
$(H_{n})$ converges to the trivial subgroup in $\operatorname{Sub}(G)$, which
is closed for the profinite topology since $G$ is residually finite. Hence
$H\mapsto\mathrm{cl}(H)$ is not upper semi-continuous.
###### Corollary 3.7.
Let $\mathcal{N}\subseteq\mathcal{N}_{G}$, and $\mathcal{H}$ a URS of $G$ that
is $\mathcal{N}$-finitary. Then the set
$\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(H):H\in\mathcal{H}\right\\}}$
contains a unique URS of $G$, that will be denoted
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$. Moreover when $G$ is countable,
there is a dense $G_{\delta}$ subset $\mathcal{H}_{0}\subseteq\mathcal{H}$
such that
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})=\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(H):H\in\mathcal{H}_{0}\right\\}}.$
###### Proof.
Proposition 3.4 asserts that $\mathcal{H}\to\operatorname{Sub}(G)$,
$H\mapsto\mathrm{cl}_{\mathcal{N}}(H)$, is upper semi-continuous. This allows
to invoke Proposition 2.1, from which the statement follows. ∎
###### Corollary 3.8.
Let $\mathcal{N}\subseteq\mathcal{N}_{G}$, and $\mathcal{H}$ a URS of $G$ that
is $\mathcal{N}$-finitary.
1. (1)
If there exists $H\in\mathcal{H}$ such that $H=\mathrm{cl}_{\mathcal{N}}(H)$,
then $\mathcal{H}=\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$.
2. (2)
If $G$ is countable, then
$\mathrm{cl}_{\mathcal{N}}(\mathrm{cl}_{\mathcal{N}}(\mathcal{H}))=\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$.
###### Proof.
The assumption in (1) implies that
$\mathcal{H}\cap\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(H):H\in\mathcal{H}\right\\}}\neq\emptyset.$
So by minimality $\mathcal{H}$ is contained in
$\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(H):H\in\mathcal{H}\right\\}}$.
Corollary 3.7 then implies
$\mathcal{H}=\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$. (2) follows from the
second statement in Corollary 3.7 and (1). ∎
###### Definition 3.9.
Suppose that $\mathcal{N}$ is filtering, and let $\mathcal{H}$ be a URS of $G$
that is $\mathcal{N}$-finitary. We say that a URS $\mathcal{H}$ is closed for
the topology $\tau_{\mathcal{N}}$ if
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})=\mathcal{H}$.
## 4\. On profinite closures of a URS
### 4.1. Profinitely closed URSs and profinite actions
In all this section we assume that $\mathcal{N}\subseteq\mathcal{N}_{G}$ is
filtering, and that $\mathcal{N}$ consists of finite index subgroups of $G$.
Let $\widehat{G}^{\mathcal{N}}$ be the inverse limit of the inverse system of
finite groups $G/N$, $N\in\mathcal{N}$, and
$\psi:G\to\widehat{G}^{\mathcal{N}}$ the associated canonical group
homomorphism (for simplicity we omit $\mathcal{N}$ in the notation in $\psi$).
The group $\widehat{G}^{\mathcal{N}}$ is profinite, and
$\psi:G\to\widehat{G}^{\mathcal{N}}$ is continuous, where $G$ is equipped with
the topology $\tau_{\mathcal{N}}$. Recall that if $H$ is a subgroup of $G$,
one has $\psi^{-1}(\overline{\psi(H)})=\mathrm{cl}_{\mathcal{N}}(H)$.
###### Proposition 4.1.
Let $\mathcal{N}$ be as above. Then the following hold:
1. (1)
Let $H$ be a subgroup of $G$ such that $H=\mathrm{cl}_{\mathcal{N}}(H)$. Then
the closure of the conjugacy class of $H$ contains a unique URS.
2. (2)
Let $\mathcal{H}$ be a URS of $G$. Let $H\in\mathcal{H}$, and
$L=\overline{\psi(H)}$. Then the stabilizer URS associated to the left
translation action of $G$ on $\widehat{G}^{\mathcal{N}}/L$ s equal to
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$.
###### Proof.
Write $L=\overline{\psi(H)}$ and $X=\widehat{G}^{\mathcal{N}}/L$, which is a
minimal compact $G$-space since $G$ has dense image in $\widehat{G}$. The
stabilizer of the coset $L\in X$ in $G$ is
$\psi^{-1}(L)=\mathrm{cl}_{\mathcal{N}}(H)$. So in case
$H=\mathrm{cl}_{\mathcal{N}}(H)$, one has
$\overline{H^{G}}\subseteq\overline{\left\\{G_{x}\,:\,x\in X\right\\}}.$
By Zorn’s lemma $\overline{H^{G}}$ contains at least one URS, and it follows
that is contains exactly one because $\overline{\left\\{G_{x}\,:\,x\in
X\right\\}}$ has this property by Proposition 2.3. Hence (1) holds.
For (2), we have
$\overline{\left\\{G_{x}\,:\,x\in
X\right\\}}\cap\overline{\left\\{\mathrm{cl}_{\mathcal{N}}(K)\,:\,K\in\mathcal{H}\right\\}}\neq\emptyset.$
Each one of these two sets contains a unique URS, namely $S_{G}(X)$ and
$\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$. Hence equality
$S_{G}(X)=\mathrm{cl}_{\mathcal{N}}(\mathcal{H})$ follows. ∎
###### Proposition 4.2.
Let $\mathcal{N}$ be as above, and let $X$ be a compact totally disconnected
$G$-space. The following are equivalent:
1. (1)
$G\times X\to X$ is continuous, where $G$ is equipped with the topology
$\tau_{\mathcal{N}}$;
2. (2)
for every clopen subset $U$ of $X$, the stabilizer of $U$ in $G$ is open for
the topology $\tau_{\mathcal{N}}$;
3. (3)
$G\times X\to X$ extends to a continuous action of $\widehat{G}^{\mathcal{N}}$
on $X$.
If $X$ is a minimal $G$-space, these are also equivalent to:
1. (4)
there exists a closed subgroup $L$ of $\widehat{G}^{\mathcal{N}}$ such that
$X$ is isomorphic to $\widehat{G}^{\mathcal{N}}/L$ as a $G$-space (where $G$
acts on $\widehat{G}^{\mathcal{N}}/L$ by left translations).
###### Proof.
Since $X$ is totally disconnected, clopen subsets form a basis of the topology
on $X$. The equivalence between (1) and (2) is therefore a consequence of the
definitions. Suppose these conditions hold, and let $L$ be the closure of the
image of $G$ in the group $\mathrm{Homeo}(X)$. Since it follows in particular
from (2) that every clopen subset of $X$ has a finite $G$-orbit, the group $L$
is a profinite group. Since $G\to L$ is continuous, by the universal property
of $\widehat{G}^{\mathcal{N}}$ [Wil98, Prop. 1.4.1–1.4.2], $G\to L$ extends to
a continuous homomorphism $\widehat{G}^{\mathcal{N}}\to L$. So (3) holds.
Finally (3) implies (1) because $\psi:G\to\widehat{G}^{\mathcal{N}}$ is
continuous.
The last statement is clear since a minimal continuous action of a compact
group on a compact space is necessarily transitive. ∎
###### Definition 4.3.
The $G$-action on $X$ is called pro-$\mathcal{N}$ if it satisfies the
equivalent conditions (1)-(2)-(3). We also say that the $G$-space $X$ is
pro-$\mathcal{N}$.
In case where $\mathcal{N}$ consists of all finite index normal subgroups of
$G$, this corresponds to the common notion of profinite $G$-space.
###### Proposition 4.4.
Let $\mathcal{H}$ be a URS of a countable group $G$. Then the following are
equivalent:
1. (1)
$\mathcal{H}$ is closed for the topology $\tau_{\mathcal{N}}$.
2. (2)
For every $H\in\mathcal{H}$, the stabilizer URS associated to the left
translation action of $G$ on $\widehat{G}^{\mathcal{N}}/\overline{\psi(H)}$ is
equal to $\mathcal{H}$.
3. (3)
There exists a minimal $G$-space $X$ that is pro-$\mathcal{N}$ such that
$S_{G}(X)=\mathcal{H}$.
###### Proof.
Proposition 4.1 implies that (1) and (2) are equivalent. (2) clearly implies
(3), so we only have to see that (3) implies (1). Let $X$ be a minimal
$G$-space that is pro-$\mathcal{N}$ that admits $\mathcal{H}$ as a stabilizer
URS. By Proposition 4.2 there exists a closed subgroup $L$ of
$\widehat{G}^{\mathcal{N}}$ such that $X$ is isomorphic to
$\widehat{G}^{\mathcal{N}}/L$ as a $G$-space. The stabilizers in $G$ for the
action on $\widehat{G}^{\mathcal{N}}/L$ are closed for the topology
$\tau_{\mathcal{N}}$ on $G$. Since $G$ is countable, there is a dense set of
points $x$ in $\widehat{G}^{\mathcal{N}}/L$ such that $G_{x}\in
S_{G}(\widehat{G}^{\mathcal{N}}/L)$ (Proposition 2.3). Since
$S_{G}(\widehat{G}^{\mathcal{N}}/L)$ is equal to $\mathcal{H}$ by assumption,
it follows that $\mathcal{H}$ contains some elements that are closed for the
topology $\tau_{\mathcal{N}}$. By Corollary 3.8 this implies
$\mathcal{H}=\mathrm{cl}(\mathcal{H})$. ∎
###### Remark 4.5.
Matte Bon–Tsankov and Elek showed that every URS $\mathcal{H}$ is equal to the
stabilizer URS associated to some compact $G$-space [MBT20, Ele18], and among
the compact $G$-spaces associated to $\mathcal{H}$ there is a unique one that
is universal in a certain sense [MBT20]. We point out that this $G$-space is
very different from the $G$-space
$\widehat{G}^{\mathcal{N}}/\overline{\psi(H)}$ associated to the specific
setting considered in Proposition 4.4.
###### Remark 4.6.
In the case of the profinite topology, Proposition 4.4 says that a URS
$\mathcal{H}$ is closed for the profinite topology if and only if
$\mathcal{H}$ is the stabilizer URS associated to a minimal profinite
$G$-space. It is worth noting that if $\mathcal{H}$ is such a URS, then the
$G$-action on $\mathcal{H}$ need not be profinite. Such a phenomenon has been
exploited by Matte Bon [MB17] and Nekrashevych [Nek20].
We end this section by showing that in general the restriction of
$\mathrm{cl}$ to a URS is not continuous. Recall that a closed subset $F$ of a
space $X$ is regular if $F$ equals the closure of its interior.
###### Lemma 4.7.
Let $X$ be a compact minimal $G$-space such that $\mathrm{Fix}_{X}(g)$ is a
regular closed set of $X$ for every $g\in G$. Then for every $x\in X$, we have
$G_{x}\leq\mathrm{cl}(G_{x}^{0})$.
###### Proof.
Fix $x\in X$, $g\in G_{x}$, and a finite index normal subgroup $N$ of $G$. We
want to see that $g\in G_{x}^{0}N$. Since $N$ has finite index in $G$ and $G$
acts minimally on $X$, each minimal closed $N$-invariant subset of $X$ is
clopen, and the minimal closed $N$-invariant subsets form a finite partition
$\left\\{U_{1},\ldots,U_{n}\right\\}$ of $X$. Let $U_{i}$ be the one
containing $x$. Since $U_{i}$ is a neighbourhood of $x$ and $g\in G_{x}$, the
assumption that $\mathrm{Fix}_{X}(g)$ is regular implies that there exists a
non-empty open subset $V\subseteq U_{i}$ on which $g$ acts trivially. Since
$N$ acts minimally on $U_{i}$, one can find $h\in N$ such that $y=hx\in V$. It
follows that $g\in G_{y}^{0}=hG_{x}^{0}h^{-1}$, and since $h\in N$ we deduce
that $g\in G_{x}^{0}N$. ∎
###### Proposition 4.8.
Suppose that $G$ is countable. Let $X$ be a minimal profinite compact
$G$-space, and $\mathcal{H}=S_{G}(X)$. Suppose that $\mathrm{Fix}_{X}(g)$ is a
regular closed set of $X$ for every $g\in G$, and the stabilizer map
$X\to\operatorname{Sub}(G)$ is not continuous on $X$. Then
$\mathrm{cl}:\mathcal{H}\to\operatorname{Sub}(G)$ is the identity on a dense
set of points, but is not the identity everywhere on $\mathcal{H}$. In
particular it is not continuous.
###### Proof.
By Proposition 4.4 the URS $\mathcal{H}$ is closed for the profinite topology.
Since $G$ is countable, this means that there is a dense set of
$H\in\mathcal{H}$ such that $\mathrm{cl}(H)=H$. Since $x\mapsto G_{x}$ is not
continuous on $X$, one easily verifies that one can find $x\in X$ and
$H\in\mathcal{H}$ such that $H\lneq G_{x}$ and $G_{x}^{0}\leq H$ (see [LBMB18,
Lemma 2.8]). It follows from Lemma 4.7 that
$G_{x}\leq\mathrm{cl}(G_{x}^{0})\leq\mathrm{cl}(H)$, and hence $H$ is properly
contained in $\mathrm{cl}(H)$ since it is properly contained in $G_{x}$. ∎
###### Remark 4.9.
An example of the above situation is provided by $G$ the Grigorchuk group and
$X$ the boundary of the defining rooted tree of $G$ [Gri11, Sec. 7].
### 4.2. Hereditarily minimal actions
###### Definition 4.10.
A compact $G$-space $X$ is hereditarily minimal if every finite index subgroup
of $G$ acts minimally on $X$.
###### Proposition 4.11.
Let $\mathcal{H}$ be a URS of $G$ that is hereditarily minimal. Then for every
$H,K\in\mathcal{H}$ we have
$\mathrm{cl}(H)=\mathrm{cl}(K)=\mathrm{cl}(\mathrm{Env}(\mathcal{H}))$.
This holds in particular if $\mathcal{H}=S_{G}(X)$ with $X$ a hereditarily
minimal compact $G$-space.
###### Proof.
Let $L$ be a finite index subgroup of $G$ such that $H\leq L$. Since $L$ acts
minimally on $\mathcal{H}$, Lemma 2.5 says that $K\leq L$. Consequently
$\mathrm{cl}(H)=\mathrm{cl}(K)$. Since $\mathcal{H}$ is $G$-invariant, it
follows that this common subgroup is normal in $G$. Call it $N$. We shall see
that $N=\mathrm{cl}(\mathrm{Env}(\mathcal{H}))$. Since
$H\leq\mathrm{Env}(\mathcal{H})$ for every $H\in\mathcal{H}$, the inclusion
$N\leq\mathrm{cl}(\mathrm{Env}(\mathcal{H}))$ is clear. On the other hand $N$
contains $\mathrm{Env}(\mathcal{H})$ since $N$ contains all elements of
$\mathcal{H}$. Since $N$ is closed in the profinite topology, $N$ contains
$\mathrm{cl}(\mathrm{Env}(\mathcal{H}))$. Hence equality holds.
As for the last claim, it follows from the fact that Propositions 2.1 and 2.3
ensure that the stabilizer URS associated to a hereditarily minimal compact
$G$-space is itself hereditarily minimal. ∎
###### Proposition 4.12.
Suppose $G$ is a residually finite group. Let $\mathcal{H}$ be a URS of $G$
that is hereditarily minimal, and suppose that $H\in\mathcal{H}$ satisfies the
law $w$. Then $\mathrm{Env}(\mathcal{H})$ also satisfies the law $w$.
###### Proof.
$G$ is residually finite, so the profinite topology on $G$ is Hausdorff. Hence
Lemma 2.9 says that $\mathrm{cl}(H)$ still satisfies $w$. Since
$\mathrm{cl}(H)$ contains $\mathrm{Env}(\mathcal{H})$ by Proposition 4.11,
$\mathrm{Env}(\mathcal{H})$ also satisfies $w$. ∎
Without the hereditarily minimal assumption, it does not hold in general that
a URS satisfying a law $w$ lives inside a normal subgroup of $G$ satisfying
$w$, as the following example shows:
###### Example 4.13.
Let $(F_{n})$ be a sequence of non-abelian finite groups. Suppose that for
every $n$ there is an abelian subgroup $A_{n}$ of $F_{n}$ such that the only
normal subgroup $N$ of $F_{n}$ containing $A_{n}$ is $N=F_{n}$. Let
$\mathbb{G}=\prod_{n}F_{n}$, and let $G$ be a countable dense subgroup of
$\mathbb{G}$ containing $\bigoplus_{n}F_{n}$. Consider the $G$-action on
$X=\prod_{n}F_{n}/A_{n}$. This action is minimal and $G_{x}$ is abelian for
every $x\in X$. In particular every $H\in S_{G}(X)$ is abelian. On the other
hand $\mathrm{Env}(\mathcal{H})$ contains the normal closure in $G$ of
$\bigoplus_{n}A_{n}$. In particular $\mathrm{Env}(\mathcal{H})$ contains
$\bigoplus_{n}F_{n}$, and hence $\mathrm{Env}(\mathcal{H})$ is not abelian.
###### Remark 4.14.
Continuing the previous example, we note that by taking a sequence $(F_{n})$
such that no law $w$ is satisfied by all $F_{n}$ (for instance
$F_{n}=\mathrm{Sym}(n)$ for all $n$), we actually obtain an example where
$\mathrm{Env}(\mathcal{H})$ satisfies no law at all.
We note that examples of groups as above can be found among finitely generated
groups. For instance the groups constructed by B.H. Neumann in [Neu37, Ch.
III] satisfy this properties.
### 4.3. PIF groups
In this paragraph we focus on the class of groups for which, for a faithful
minimal compact $G$-space, profinite implies topologically free.
###### Definition 4.15.
We say that a group $G$ is PIF if for every faithful minimal compact $G$-space
$X$, if the $G$-action on $X$ is profinite then it is topologically free.
This notion was studied notably by Grigorchuk. We will use the following
proposition from [Gri11]. Recall that a group $G$ is just-infinite (JI) if $G$
is infinite and $G/N$ is finite for every non-trivial normal subgroup $N$.
Also $G$ is hereditarily just-infinite (HJI) if every finite index subgroup of
$G$ is JI.
###### Proposition 4.16.
Each one of the following conditions implies that $G$ is PIF:
1. (1)
for every non-trivial subgroups $H_{1},H_{2}\leq G$ such that the normalizer
$N_{G}(H_{i})$ of $H_{i}$ has finite index in $G$ for $i=1,2$, we have that
$H_{1}\cap H_{2}$ is non-trivial.
2. (2)
$G$ is hereditarily just-infinite.
###### Proof.
The first assertion is [Gri11, Proposition 4.11] (the formulation there is not
quite the same, but the argument is the same). The second assertion follows
from the first one because every HJI group satisfies (1). ∎
The first condition of the proposition is satisfied for instance by non-
abelian free groups, and also by all Gromov-hyperbolic groups with no non-
trivial finite normal subgroup, and more generally by any group $G$ admitting
an isometric action on a hyperbolic space $X$ with unbounded orbits such that
the $G$-action on its limit set $\partial_{X}G$ is faithful.
###### Proposition 4.17.
Suppose $G$ is PIF, and let $\mathcal{H}$ be a non-trivial URS of $G$. Then
there exists a non-trivial normal subgroup $N$ of $G$ such that
$N\leq\mathrm{cl}(H)$ for every $H\in\mathcal{H}$.
###### Proof.
Note that since $\mathcal{H}$ is not the trivial URS,
$\mathrm{cl}(\mathcal{H})$ is not the trivial URS either. Let
$H\in\mathcal{H}$, and $L=\overline{\psi(H)}$, where $\psi:G\to\widehat{G}$ is
the canonical map from $G$ to its profinite completion. By Proposition 4.1,
the stabilizer URS associated to the left translation action of $G$ on
$\widehat{G}/L$ is equal to $\mathrm{cl}(\mathcal{H})$, and hence is not
trivial. This means that the action of $G$ on $\widehat{G}/L$ is not
topologically free. Since this action is profinite and $G$ is PIF, the action
cannot be faithful. So there is a non-trivial normal subgroup $N$ of $G$ that
is contained in the stabilizer in $G$ of the coset $L$, which is
$\psi^{-1}(L)=\mathrm{cl}(H)$. Upper semi-continuity of $\mathrm{cl}$ on
$\mathcal{H}$ then implies that $N\leq\mathrm{cl}(H^{\prime})$ for every
$H^{\prime}\in\mathcal{H}$. ∎
###### Definition 4.18.
We say that a subgroup $H$ of a group $G$ is quasi-dense for the profinite
topology if $\mathrm{cl}(H)$ is a finite index subgroup of $G$.
###### Corollary 4.19.
Suppose $G$ is HJI, and let $\mathcal{H}$ be a non-trivial URS of $G$. Then
for every $H\in\mathcal{H}$, $H$ is quasi-dense in $G$ for the profinite
topology.
###### Proof.
Proposition 4.16 says that $G$ is PIF. So Proposition 4.17 applies, and gives
a non-trivial normal subgroup $N$ such that $N\leq\mathrm{cl}(H)$ for every
$H\in\mathcal{H}$. By the assumption $N$ must have finite index, and the
conclusion follows. ∎
Recall that every HJI-group is either virtually simple or residually finite.
Corollary 4.19 is void for virtually simple groups, so the focus here is on
residually finite HJI-groups. By Margulis normal subgroup theorem, every
irreducible lattice $\Gamma$ in a connected semisimple Lie group $\mathbf{G}$
(with trivial center and no compact factor) of rank $\geq 2$ is HJI. Under the
assumption that every simple factor of the ambient Lie group $\mathbf{G}$ has
rank $\geq 2$, it is known that every non-trivial URS of $\Gamma$ is just the
conjugacy class of a finite index subgroup [BH21, Cor. F]. The normal subgroup
theorem of Bader–Shalom asserts that any irreducible cocompact lattice
$\Gamma$ in a product $\mathbf{G_{1}}\times\mathbf{G_{2}}$, where
$\mathbf{G_{1}},\mathbf{G_{2}}$ are compactly generated topologically simple
locally compact groups, is HJI [BS06]. In this setting the URSs of $\Gamma$
are not understood.
Following [Cor06], we shall say that a group $G$ has property (PF) if for
every subgroup $H$ of $G$, $H$ is quasi-dense in $G$ for the profinite
topology only if $H$ has finite index. Let $p$ be a prime number. Following
[EJZ13], we say that a group $G$ is weakly $p$-LERF if for every subgroup $H$
of $G$, the closure $\mathrm{cl}_{p}(H)$ of $H$ for the pro-$p$ topology has
finite index in $G$ only if $H$ has finite index. Note that applying the
definition of weakly $p$-LERF to the trivial subgroup, we see that a just-
infinite group that is weakly $p$-LERF is necessarily residually-$p$. Every
group that is weakly $p$-LERF has property (PF). Among their striking
properties, the finitely generated groups obtained as the outputs of the
process carried out in [EJZ13] are HJI and weakly $p$-LERF.
###### Corollary 4.20.
Suppose $G$ is HJI and has property (PF). Then $G$ has no continuous URS.
###### Proof.
Let $\mathcal{H}$ be a URS of $G$. If $\mathcal{H}$ is trivial, then there is
nothing to show. Otherwise, for every $H\in\mathcal{H}$, $\mathrm{cl}(H)$ has
finite index in $G$ by Corollary 4.19. Hence so does $H$ since $G$ has (PF).
In particular $H$ has only finitely many conjugates, i.e. $\mathcal{H}$ is
finite. ∎
## 5\. The Furstenberg URS
###### Definition 5.1.
Given two closed subsets
$\mathcal{X}_{1},\mathcal{X}_{2}\subset\operatorname{Sub}(G)$, we write
$\mathcal{X}_{1}\preccurlyeq\mathcal{X}_{2}$ if there exist
$H_{1}\in\mathcal{X}_{1}$ and $H_{2}\in\mathcal{X}_{2}$ such that $H_{1}\leq
H_{2}$.
One verifies that, when restricted to the set $\mathrm{URS}(G)$, the relation
$\preccurlyeq$ is a partial order [LBMB18, Cor. 2.15].
Recall that a compact $G$-space $X$ is strongly proximal if the orbit closure
of every probability measure on $X$ in the space $\mathrm{Prob}(X)$ contains a
Dirac measure. The Furstenberg boundary $\partial_{F}G$ of $G$ is the
universal minimal and strongly proximal $G$-space [Gla76]. We denote by
$\mathrm{Rad}(G)$ the amenable radical of the group $G$. It coincides with the
kernel of the action of $G$ on $\partial_{F}G$.
###### Definition 5.2.
The stabilizer URS associated to the $G$-action on $\partial_{F}G$ is denoted
$\mathcal{A}_{G}$, and is called the Furstenberg URS of $G$.
A result of Frolík implies that the map $x\mapsto G_{x}$ is continuous on
$\partial_{F}G$, so that $\mathcal{A}_{G}$ is exactly the collection of point
stabilizers for the action of $G$ on $\partial_{F}G$ (see [Ken20] and
references there).
###### Proposition 5.3.
The following hold:
1. (1)
$\mathcal{A}_{G}$ is amenable, and $\mathcal{X}\preccurlyeq\mathcal{A}_{G}$
for every non-empty closed $G$-invariant subset $\mathcal{X}$ of
$\operatorname{Sub}(G)$ consisting of amenable subgroups.
2. (2)
$\mathcal{A}_{G}$ is invariant under the action of $\mathrm{Aut}(G)$ on
$\operatorname{Sub}(G)$.
3. (3)
$\mathrm{Rad}(G)\leq H$ for every $H\in\mathcal{A}_{G}$.
4. (4)
If $N\in\mathcal{N}_{G}$ is amenable and if $\operatorname{Sub}_{\geq N}(G)$
is the set of subgroups of $G$ containing $N$, then the natural map
$\varphi:\operatorname{Sub}(G/N)\rightarrow\operatorname{Sub}_{\geq N}(G)$
induces a $G$-equivariant homeomorphism between $\mathcal{A}_{G/N}$ and
$\mathcal{A}_{G}$.
5. (5)
$\mathcal{A}_{G}$ is a singleton if and only if
$\mathcal{A}_{G}=\left\\{\mathrm{Rad}(G)\right\\}$. When this does not hold,
$\mathcal{A}_{G}$ is continuous.
6. (6)
$\mathcal{A}_{G}=\left\\{\mathrm{Rad}(G)\right\\}$ if and only if every
amenable URS of $G$ lives inside the amenable radical: $H\leq\mathrm{Rad}(G)$
for every amenable URS $\mathcal{H}$ and every $H\in\mathcal{H}$.
###### Proof.
See [LBMB18] and references there. ∎
###### Lemma 5.4.
Let $N$ be a normal subgroup of $G$ such that $H\leq N$ for every
$H\in\mathcal{A}_{G}$. Then $\mathcal{A}_{N}=\mathcal{A}_{G}$. In particular
$N$ acts minimally on $\mathcal{A}_{G}$.
###### Proof.
$\mathcal{A}_{N}$ being $\mathrm{Aut}(N)$-invariant, it is $G$-invariant.
Hence $\mathcal{A}_{N}$ is an amenable URS of $G$. So
$\mathcal{A}_{N}\preccurlyeq\mathcal{A}_{G}$. On the other hand
$\mathcal{A}_{G}$ is a closed $N$-invariant subset of $\operatorname{Sub}(N)$
consisting of amenable subgroups, so
$\mathcal{A}_{G}\preccurlyeq\mathcal{A}_{N}$. Since $\preccurlyeq$ is an order
in restriction to URSs, $\mathcal{A}_{G}=\mathcal{A}_{N}$. ∎
###### Definition 5.5.
We denote by $\mathcal{F}$ the class of groups whose Furstenberg URS is a
singleton.
###### Lemma 5.6.
Suppose $G=\bigcup_{I}G_{i}$ is the directed union of subgroups $G_{i}$ such
that eventually $G_{i}$ is in $\mathcal{F}$ (resp. $\mathcal{A}_{G_{i}}$ is
trivial). Then $G$ is in $\mathcal{F}$ (resp. $\mathcal{A}_{G}$ is trivial).
###### Proof.
Write $R_{i}$ for the amenable radical of $G_{i}$, so that eventually
$\mathcal{A}_{G_{i}}=\left\\{R_{i}\right\\}$. Consider
$\varphi_{i}:\operatorname{Sub}(G)\to\operatorname{Sub}(G_{i})$, $H\mapsto
H\cap G_{i}$. This map is continuous and $G_{i}$-equivariant. Take
$H\in\mathcal{A}_{G}$. The subset $\mathcal{X}_{i}:=\overline{(H\cap
G_{i})^{G_{i}}}$ is a closed $G_{i}$-invariant subset of
$\operatorname{Sub}(G_{i})$ consisting of amenable subgroups, so
$\mathcal{X}_{i}\preccurlyeq\mathcal{A}_{G_{i}}=\left\\{R_{i}\right\\}$ by
Proposition 5.3. Since $\varphi_{i}(\mathcal{A}_{G})$ is closed and
$G_{i}$-invariant, $\mathcal{X}_{i}\subseteq\varphi_{i}(\mathcal{A}_{G})$. We
infer that there exists $K_{i}\in\mathcal{A}_{G}$ such that $K_{i}\cap
G_{i}\leq R_{i}$. Upon passing to a subnet we may assume that $(K_{i})$
converges to some $K\in\mathcal{A}_{G}$ and $(R_{i})$ converges to some $R$.
The subgroup $R$ is normal and amenable, so $R\leq\mathrm{Rad}(G)$. Since
$G=\bigcup_{I}G_{i}$, $(K_{i}\cap G_{i})$ also converges to $K$, and the
inclusion $K_{i}\cap G_{i}\leq R_{i}$ then implies $K\leq R$. So
$\mathcal{A}_{G}\preccurlyeq\left\\{\mathrm{Rad}(G)\right\\}$, which means
that $\mathcal{A}_{G}=\left\\{\mathrm{Rad}(G)\right\\}$ by Proposition 5.3. We
also immediately obtain that in case $R_{i}$ is trivial eventually, then
$\mathcal{A}_{G}$ is trivial. ∎
### 5.1. Proofs of the results
Recall that If $\mathcal{C}$ is a class of groups, we denote by
$\mathcal{N}_{G}(\mathcal{C})$ the normal subgroups of $G$ such that
$G/N\in\mathcal{C}$. In the sequel we mainly use this notation with
$\mathcal{C}=\mathcal{F}$. We have the following lemma:
###### Lemma 5.7.
$\mathcal{N}_{G}(\mathcal{F})$ is stable under taking finite intersections.
###### Proof.
Let $N_{1},N_{2}\in\mathcal{N}_{G}(\mathcal{F})$. Let $Q_{i}=G/N_{i}$, and let
$R_{i}=\mathrm{Rad}(Q_{i})$ be the amenable radical of $Q_{i}$. Let
$\pi_{i}:G\to Q_{i}$ be the canonical projection, and
$M_{i}:=\pi_{i}^{-1}(R_{i})$. Let also $X_{i}=\partial_{F}Q_{i}$. The subgroup
$R_{i}$ acts trivially on $X_{i}$, and the assumption that $Q_{i}$ belongs to
$\mathcal{F}$ means that the $Q_{i}/R_{i}$-action on $X_{i}$ is free.
We consider the $G$-action on the product $X_{1}\times X_{2}$. This action
remains strongly proximal [Gla76, III.1]. It follows that there exists a
unique minimal closed $G$-invariant subset $X\subseteq X_{1}\times X_{2}$, and
the $G$-action on $X$ is strongly proximal [Gla76, III.1]. The subgroup
$M_{1}\cap M_{2}$ of $G$ acts trivially on $X_{1}\times X_{2}$, and hence on
$X$. Moreover since the $Q_{i}/R_{i}$-action on $X_{i}$ is free, it follows
that for the $G$-action on $X$, every point stabilizer is equal to $M_{1}\cap
M_{2}$. Equivalently, the $G$-action on $X$ factors through a free action of
$G/M_{1}\cap M_{2}$. In particular $G/M_{1}\cap M_{2}$ is in $\mathcal{F}$.
Since the group $M_{1}\cap M_{2}/N_{1}\cap N_{2}$ is amenable (as it embeds in
the amenable group $R_{1}\times R_{2}$), and since being in $\mathcal{F}$ is
invariant under forming an extension with amenable normal subgroup, it follows
that $G/N_{1}\cap N_{2}$ is in $\mathcal{F}$. ∎
As a consequence of the lemma, it follows that the family of cosets $gN$,
$g\in G$, $N\in\mathcal{N}_{G}(\mathcal{F})$, forms a basis for a group
topology $\tau_{\mathcal{N}_{G}(\mathcal{F})}$ on $G$, and that the closure of
$H$ with respect to this topology is equal to
$\mathrm{cl}_{\mathcal{N}_{G}(\mathcal{F})}(H)$ (Proposition 2.8).
The proof of the following is the technical part of this section.
###### Proposition 5.8.
Let $G$ be a countable group. Then for every
$N\in\mathcal{N}_{G}(\mathcal{F})$, there exists a normal subgroup $M$ of $G$
such that $N\leq M$ and $M/N\leq\mathrm{Rad}(G/N)$, and a comeager subset
$\mathcal{H}_{0}\subseteq\mathcal{A}_{G}$ such that $NH=M$ for every
$H\in\mathcal{H}_{0}$.
###### Proof.
The map $\varphi_{N}:\mathcal{A}_{G}\to\operatorname{Sub}(G)$, $H\mapsto NH$,
is lower semi-continuous by Lemma 2.6. Hence the set
$\mathcal{H}_{0}\subseteq\mathcal{A}_{G}$ of points where $\varphi_{N}$ is
continuous is a comeager subset of $\mathcal{A}_{G}$, and
$E_{\varphi_{N}}=\overline{\left\\{\left(H,NH\right)\,:\,H\in\mathcal{H}_{0}\right\\}}\,\text{
and }\,S_{\varphi_{N}}=\overline{\left\\{NH\,:\,H\in\mathcal{H}_{0}\right\\}}$
satisfy the conclusion of Proposition 2.1. Note that $S_{\varphi_{N}}$ is
contained in the closed subset $\operatorname{Sub}_{\geq N}(G)$ of
$\operatorname{Sub}(G)$ consisting of subgroups of $G$ containing $N$.
Since the URS $\mathcal{A}_{G}$ is strongly proximal and strong proximality
passes to highly proximal extensions and factors [Gla75, Lemma 5.2], we deduce
that the $G$-action on $S_{\varphi_{N}}$ is minimal and strongly proximal. The
map $\pi_{N}:S_{\varphi_{N}}\to\operatorname{Sub}(G/N)$, $K\to K/N$, is a
$G$-equivariant homeomorphism onto its image (indeed, one easily verifies that
modding out by $N$ defines a homeomorphism from $\operatorname{Sub}_{\geq
N}(G)$ onto $\operatorname{Sub}(G/N)$). Hence $\pi_{N}(S_{\varphi_{N}})$ is a
strongly proximal URS of $G/N$. Moreover $\pi_{N}(S_{\varphi_{N}})$ consists
of amenable subgroups. If we let $R$ be the amenable radical of $G/N$, it
follows from the assumption that $G/N$ belongs to $\mathcal{F}$ that
$\pi_{n}(S_{\varphi_{N}})$ is contained in $\operatorname{Sub}(R)$. On the
other hand $R$ must act trivially on $\pi_{N}(S_{\varphi_{N}})$ by minimality
and strong proximality.
Consider the envelope $E=\mathrm{Env}(S_{\varphi_{N}})$. Since
$\pi_{N}(S_{\varphi_{N}})\subseteq\operatorname{Sub}(R)$, we have $E/N\leq R$.
So by the previous paragraph and the fact that
$\pi_{N}:S_{\varphi_{N}}\to\pi_{N}(S_{\varphi_{N}})$ is a $G$-equivariant
homeomorphism, it follows that $E$ acts trivially on $S_{\varphi_{N}}$. On the
other hand $E$ contains $H$ for every $H\in\mathcal{H}_{0}$, and since
$\mathcal{H}_{0}$ is dense in $\mathcal{A}_{G}$ this easily implies that $E$
contains $H$ for every $H\in\mathcal{A}_{G}$. Hence Lemma 5.4 can be applied
to $E$, and we infer that $E$ acts minimally on $\mathcal{A}_{G}$. Using
Proposition 2.1 and the fact that minimality passes to irreducible extensions,
we deduce that $E$ acts minimally on $S_{\varphi_{N}}$. All together, this
shows that $S_{\varphi_{N}}$ is a one-point space. The corresponding normal
subgroup $M$ of $G$ verifies the conclusion. ∎
###### Proposition 5.9.
Let $G$ be a countable group, and let $\mathcal{N}$ be a countable subset of
$\mathcal{N}_{G}(\mathcal{F})$. Then there exists a normal subgroup $M$ of $G$
and a comeager subset $\mathcal{H}_{0}\subseteq\mathcal{A}_{G}$ such that
$\mathrm{cl}_{\mathcal{N}}(H)=M$ for every $H\in\mathcal{H}_{0}$.
###### Proof.
We apply Proposition 5.8 for every $N\in\mathcal{N}$. We obtain a normal
subgroup $M_{N}$ of $G$ and a comeager subset $\mathcal{H}_{N}$ of
$\mathcal{A}_{G}$. Set $\mathcal{H}_{0}=\bigcap_{\mathcal{N}}\mathcal{H}_{N}$
and $M=\bigcap_{\mathcal{N}}M_{N}$. Since $\mathcal{N}$ is countable,
$\mathcal{H}_{0}$ is a comeager subset of $\mathcal{H}$. By construction for
every $H\in H_{0}$,
$cl_{\mathcal{N}}(H)=\bigcap_{\mathcal{N}}NH=\bigcap_{\mathcal{N}}M_{N}=M.$
∎
###### Definition 5.10.
We say that a set of laws $\mathbb{W}$ detects amenability in a group $G$ if
for every subgroup $H$ of $G$, one has that $H$ is amenable if and only if
there exists $w\in\mathbb{W}$ such that $H$ virtually satisfies $w$.
Note that if a set of laws detects amenability in a group $G$, it also detects
amenability in any subgroup of $G$. The following is an immediate consequence
of the definition and Lemma 2.9.
###### Lemma 5.11.
Let $G$ be a group such that there is a set of laws that detects amenability
in $G$, and let $(G,\tau)$ be a group topology on $G$ that is Hausdorff. Then
for every amenable subgroup $H$ of $G$, the $\tau$-closure of $H$ remains
amenable.
###### Theorem 5.12.
Let $G$ be a group such that there is a set of laws that detects amenability
in $G$. Then $G$ is in $\mathcal{F}$ if and only if $G$ is
residually-$\mathcal{F}$.
###### Proof.
Only one direction is non-trivial. Writing $G$ as the directed limit of its
countable subgroups and invoking Lemma 5.6, one sees that it suffices to prove
the result when $G$ is countable. Under this assumption, since $G$ is
residually-$\mathcal{F}$, one can find
$\mathcal{N}\subseteq\mathcal{N}_{G}(\mathcal{F})$ such that $\mathcal{N}$ is
countable and $\bigcap_{\mathcal{N}}N=\left\\{1\right\\}$. Since
$\mathcal{N}_{G}(\mathcal{F})$ is stable under taking finite intersections by
Lemma 5.7, we can replace $\mathcal{N}$ by the collection of finite
intersections of elements of $\mathcal{N}$, so that we may assume that
$\mathcal{N}$ is filtering. So for a subgroup $H$ of $G$,
$\mathrm{cl}_{\mathcal{N}}(H)$ equals the closure of $H$ in the topology
$\tau_{\mathcal{N}}$ (Proposition 2.8).
Proposition 5.9 provides a normal subgroup $M$ of $G$ such that
$\mathrm{cl}_{\mathcal{N}}(H)=M$ for every $H$ in a comeager subset of
$\mathcal{A}_{G}$. The topology $\tau_{\mathcal{N}}$ is Hausdorff since
$\bigcap_{\mathcal{N}}N=\left\\{1\right\\}$, so it follows from Lemma 5.11
that the closure of an amenable subgroup of $G$ remains amenable. This shows
$M$ is amenable, and it follows that $M\leq\mathrm{Rad}(G)$. By Proposition
5.3 this means that $\mathcal{A}_{G}=\left\\{\mathrm{Rad}(G)\right\\}$. ∎
###### Remark 5.13.
When the group $G$ is residually finite, there is a shorter way to obtain the
conclusion of Theorem 5.12. Indeed, since the $G$-space $\partial_{F}G$ is
proximal, it is hereditarily minimal [Gla76]. Moreover it follows from the
conclusion of Proposition 2.1 that being hereditarily minimal is inherited
from a $G$-space to its stabilizer URS. Hence $\mathcal{A}_{G}$ is a
hereditarily minimal URS. Hence Propositions 4.11 and 4.12 apply, and the
conclusion follows as above.
###### Corollary 5.14.
Let $G$ be a group such that there is a set of laws that detects amenability
in $G$, and suppose $\mathrm{Rad}(G)$ is trivial. If $G$ is
residually-$\mathcal{F}$, then $G$ is $C^{\ast}$-simple.
###### Proof.
The result follows from Theorem 5.12 and the main result of [KK17], which
asserts that $G$ is in $\mathcal{F}$ if and only if $G/\mathrm{Rad}(G)$ is
$C^{\ast}$-simple. ∎
### 5.2. Linear groups
We deduce Corollary 1 from the introduction, which asserts that linear groups
belong to $\mathcal{F}$.
###### Proof of Corollary 1.
Writing $G$ as the directed limit of its finitely generated subgroups and
invoking Lemma 5.6, one sees that without loss of generality we can assume
that $G$ is a finitely generated linear group. By Malcev’s theorem, the group
$G$ is residually finite. Also by the Tits alternative [Tit72], every amenable
subgroup of $G$ is virtually solvable (we are using again that $G$ is finitely
generated to have this version of the Tits alternative). Hence all the
assumptions of Theorem 5.12 are verified. The conclusion follows. ∎
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|
∃$\exists$ ∀$\forall$ θ$\theta$ τ$\tau$ φ$\varphi$ ξ$\xi$ ζ$\zeta$ ψ$\psi$
π$\pi$ α$\alpha$ β$\beta$ γ$\gamma$ δ$\delta$ ε$\varepsilon$ κ$\kappa$
λ$\lambda$ μ$\mu$ ρ$\rho$ σ$\sigma$ ω$\omega$ Γ$\Gamma$ Φ$\Phi$ Δ$\Delta$
Σ$\Sigma$ Π$\Pi$ ∑$\Sigma$ ∏$\Pi$ Θ$\Theta$ Ω$\Omega$ ⇒$\Rightarrow$
⇐$\Leftarrow$ ⇔$\Leftrightarrow$ →$\rightarrow$ ←$\leftarrow$
↔$\leftrightarrow$ ¬$\neg$ ∧$\land$ ∨$\lor$ ≠$\neq$ ≡$\equiv$ ∼$\sim$
≈$\approx$ ≥$\geq$ ≤$\leq$ ≫$\gg$ ≪$\ll$ ∅$\emptyset$ ⊆$\subseteq$ ⊂$\subset$
∩$\cap$ ⋂$\cap$ ∪$\cup$ ⋃$\cup$ ⊎$\uplus$ ∈$\in$ ∉$\not\in$ ⊤$\top$ ⊥$\bot$ ₀0
₁1 ₂2 ₃3 ₄4 ₅5 ₆6 ₇7 ₈8 ₉9 ⁰0 ¹1 ²2 ³3 ⁴4 ⁵5 ⁶6 ⁷7 ⁸8 ⁹9 𝔹$\mathbb{B}$
ℝ$\mathbb{R}$ ℕ$\mathbb{N}$ ℂ$\mathbb{C}$ ℚ$\mathbb{Q}$ 𝕋$\mathbb{T}$
𝕏$\mathbb{X}$ ℤ$\mathbb{Z}$ ✓51 ✗55 ◊$\lozenge$ □$\square$ 𝓐$\mathcal{A}$
𝓑$\mathcal{B}$ 𝓒$\mathcal{C}$ 𝓓$\mathcal{D}$ 𝓔$\mathcal{E}$ 𝓕$\mathcal{F}$
𝓖$\mathcal{G}$ 𝓗$\mathcal{H}$ 𝓘$\mathcal{I}$ 𝓙$\mathcal{J}$ 𝓚$\mathcal{K}$
𝓛$\mathcal{L}$ 𝓜$\mathcal{M}$ 𝓝$\mathcal{N}$ 𝓞$\mathcal{O}$ 𝓟$\mathcal{P}$
𝓠$\mathcal{Q}$ 𝓡$\mathcal{R}$ 𝓢$\mathcal{S}$ 𝓣$\mathcal{T}$ 𝓤$\mathcal{U}$
𝓥$\mathcal{V}$ 𝓦$\mathcal{W}$ 𝓧$\mathcal{X}$ 𝓨$\mathcal{Y}$ 𝓩$\mathcal{Z}$
…$\ldots$ ∗$\ast$ ⊢$\vdash$ ⊧$\models$ ′′ ″′′ ‴′′′ ∥$\|$ ⊕$\oplus$ ⁺+
⊇$\supseteq$ ∘$\circ$ ∙$\cdot$ ⋅$\cdot$ ≈$\approx$ ×$\times$ ∞$\infty$
⊑$\sqsubseteq$
11institutetext: New York University 11email<EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext: Max Planck Institute for Software Systems 22email:
<EMAIL_ADDRESS>33institutetext: SonarSource 33email:
<EMAIL_ADDRESS>
# Complete Multiparty Session Type Projection with Automata
Authors withheld Elaine Li equal contribution11 Felix Stutz* 22 Thomas Wies
11 Damien Zufferey 33
###### Abstract
Multiparty session types (MSTs) are a type-based approach to verifying
communication protocols. Central to MSTs is a _projection operator_ : a
partial function that maps protocols represented as global types to correct-
by-construction implementations for each participant, represented as a
communicating state machine. Existing projection operators are syntactic in
nature, and trade efficiency for completeness. We present the first projection
operator that is sound, complete, and efficient. Our projection separates
synthesis from checking implementability. For synthesis, we use a simple
automata-theoretic construction; for checking implementability, we present
succinct conditions that summarize insights into the property of
implementability. We use these conditions to show that MST implementability is
PSPACE-complete. This improves upon a previous decision procedure that is in
EXPSPACE and applies to a smaller class of MSTs. We demonstrate the
effectiveness of our approach using a prototype implementation, which handles
global types not supported by previous work without sacrificing performance.
###### Keywords:
Protocol verification Multiparty session types Communicating state machines
Protocol fidelity Deadlock freedom.
## 1 Introduction
Communication protocols are key components in many safety and operation
critical systems, making them prime targets for formal verification.
Unfortunately, most verification problems for such protocols (e.g. deadlock
freedom) are undecidable [11]. To make verification computationally tractable,
several restrictions have been proposed [14, 10, 3, 2, 32, 42]. In particular,
multiparty session types (MSTs) [24] have garnered a lot of attention in
recent years (see, e.g., the survey by Ancona et al. [6]). In the MST setting,
a protocol is specified as a global type, which describes the desired
interactions of all roles involved in the protocol. Local implementations
describe behaviors for each individual role. The implementability problem for
a global type asks whether there exists a collection of local implementations
whose composite behavior when viewed as a communicating state machine (CSM)
matches that of the global type and is deadlock-free. The synthesis problem is
to compute such an implementation from an implementable global type.
MST-based approaches typically solve synthesis and implementability
simultaneously via an efficient syntactic _projection operator_ [24, 18, 41,
33]. Abstractly, a projection operator is a partial map from global types to
collections of implementations. A projection operator proj is sound when every
global type $\mathbf{G}$ in its domain is implemented by
$\texttt{proj}(\mathbf{G})$, and complete when every implementable global type
is in its domain. Existing practical projection operators for MSTs are all
incomplete (or unsound). Recently, the implementability problem was shown to
be decidable for a class of MSTs via a reduction to safe realizability of
globally cooperative high-level message sequence charts (HMSCs) [38]. In
principle, this result yields a complete and sound projection operator for the
considered class. However, this operator would not be practical. In
particular, the proposed implementability check is in EXPSPACE.
Contributions. In this paper, we present the first practical sound and
complete projection operator for general MSTs. The synthesis problem for
implementable global types is conceptually easy [38] – the challenge lies in
determining whether a global type _is_ implementable. We thus separate
synthesis from checking implementability. We first use a standard automata-
theoretic construction to obtain a candidate implementation for a potentially
non-implementable global type. However, unlike [38], we then verify the
correctness of this implementation directly using efficiently checkable
conditions derived from the global type. When a global type is not
implementable, our constructive completeness proof provides a counterexample
trace.
The resulting projection operator yields a PSPACE decision procedure for
implementability. In fact, we show that the implementability problem is
PSPACE-complete. These results both generalize and tighten the decidability
and complexity results obtained in [38].
We evaluate a prototype of our projection algorithm on benchmarks taken from
the literature. Our prototype benefits from both the efficiency of existing
lightweight but incomplete syntactic projection operators[24, 18, 41, 33], and
the generality of heavyweight automata-based model checking techniques [35,
28]: it handles protocols rejected by previous practical approaches while
preserving the efficiency that makes MST-based techniques so attractive.
## 2 Motivation and Overview
(a) Odd-even protocol
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}?{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$
(b) Local impl.
for role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$
(c) Local impl.
for role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$
$\\{\textnormal{L}_{1},\textnormal{R}_{1},\textnormal{R}_{2},\textnormal{R}_{4}\\}$$\\{\textnormal{L}_{2},\textnormal{L}_{4},\textnormal{R}_{3}\\}$$\\{\textnormal{L}_{3},\textnormal{R}_{2},\textnormal{R}_{4}\\}$$\\{\textnormal{L}_{5}\\}$$\\{\textnormal{L}_{6}\\}$$\\{\textnormal{R}_{5}\\}$$\\{\textnormal{R}_{6}\\}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$
(d) Local impl.
for role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
Figure 1: Odd-even: An implementable but not (yet) projectable protocol and
its local implementations
Incompleteness of existing projection operators. A key limitation of existing
projection operators is that the implementation for each role is obtained via
a linear traversal of the global type, and thus shares its structure. The
following example, which is not projectable by any existing approach,
demonstrates how enforcing structural similarity can lead to incompleteness.
###### Example 1 (Odd-even)
Consider the following global type $\mathbf{G}_{oe}$:
$+\;\begin{cases}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,\mu
t_{1}.\,({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,t_{1}\;+\;{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,0)\\\
{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}.\,\mu
t_{2}.\,({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,t_{2}\;+\;{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!:\\!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}.\,0)\end{cases}$
A term
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!m$
specifies the exchange of message $m$ between sender
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
and receiver
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.
The term represents two local events observed separately due to asynchrony: a
send event
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!m$
observed by role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$,
and a receive event
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?m$
observed by role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.
The $+$ operator denotes choice, $\mu t.\,G$ denotes recursion, and $0$
denotes protocol termination.
Fig. 1(a) visualizes $\mathbf{G}_{oe}$ as an HMSC. The left and right sub-
protocols respectively correspond to the top and bottom branches of the
protocol. Role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
chooses a branch by sending either
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$ or
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$
to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.
On the left,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$
echoes this message to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$.
Both branches continue in the same way:
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
sends an arbitrary number of
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$
messages to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$,
each of which is forwarded twice from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$
to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$.
Role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
signals the end of the loop by sending
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$ to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$,
which
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$
forwards to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$.
Finally, depending on the branch,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
must send
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$ or
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$
to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$.
Figs. 1(b) and 1(c) depict the structural similarity between the global type
$\mathbf{G}_{oe}$ and the implementations for
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.
For the “choicemaker” role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$,
the reason is evident. Role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$’s
implementation collapses the continuations of both branches in the protocol
into a single sub-component. For
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
(Fig. 1(d)), the situation is more complicated. Role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
does not decide on or learn directly which branch is taken, but can deduce it
from the parity of the number of
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$
messages received from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$:
odd means left and even means right. The resulting local implementation
features transitions going back and forth between the two branches that do not
exist in the global type. Syntactic projection operators fail to create such
transitions. $\blacktriangleleft$
One response to the brittleness of existing projection operators has been to
give up on global type specifications altogether and instead revert to model
checking user-provided implementations [35, 28]. We posit that what needs
rethinking is not the concept of global types, but rather how projections are
computed and how implementability is checked.
Our automata-theoretic approach. The synthesis step in our projection operator
uses textbook automata-theoretic constructions. From a given global type, we
derive a finite state machine, and use it to define a homomorphism automaton
for each role. We then determinize this homomorphism automaton via subset
construction to obtain a local candidate implementation for each role. If the
global type is implementable, this construction always yields an
implementation. The implementations shown in Figs. 1(b), 1(c), and 1(d) are
the result of applying this construction to $\mathbf{G}_{oe}$ from Example 1.
Notice that the state labels in Fig. 1(d) correspond to sets of labels in the
global protocol.
Unfortunately, not all global types are implementable.
(a) $\mathbf{G}_{r}$
(b) $\mathbf{G}_{r}^{\prime}$
(c) $\mathbf{G}_{s}$
(d) $\mathbf{G}_{s}^{\prime}$
Figure 2: High-level message sequence charts for the global types of Example
2.
###### Example 2
Consider the following four global types also depicted in Fig. 2:
$\displaystyle\mathbf{G}_{r}={}$
$\displaystyle+\;\begin{cases}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,0\\\
{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,0\end{cases}$
$\displaystyle\qquad\mathbf{G}_{s}={}$
$\displaystyle+\;\begin{cases}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,0\\\
{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}.\,0\end{cases}$
$\displaystyle\mathbf{G}_{r}^{\prime}={}$
$\displaystyle+\;\begin{cases}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,0\\\
{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,0\end{cases}$
$\displaystyle\qquad\mathbf{G}_{s}^{\prime}={}$
$\displaystyle+\;\begin{cases}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}.\,0\\\
{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}.\,0\end{cases}$
Similar to $\mathbf{G}_{oe}$, in all four examples,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
chooses a branch by sending either
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$ or
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$
to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.
The global type $\mathbf{G}_{r}$ is not implementable because
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
cannot learn which branch was chosen by
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$.
For any local implementation of
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
to be able to execute both branches, it must be able to receive
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$ from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$
in any order. Because the two send events
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$
and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$
are independent of each other, they may be reordered. Consequently, any
implementation of $\mathbf{G}_{r}$ would have to permit executions that are
consistent with global behaviors not described by $\mathbf{G}_{r}$, such as
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$.
Contrast this with $\mathbf{G}_{r}^{\prime}$, which is implementable. In the
top branch of $\mathbf{G}_{r}^{\prime}$, role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
can only send to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
after it has received from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$,
which prevents the reordering of the send events
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$
and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$.
The bottom branch is symmetric. Hence,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
learns
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$’s
choice based on which message it receives first.
For the global type $\mathbf{G}_{s}$, role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
again cannot learn the branch chosen by
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$.
That is,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
cannot know whether to send
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$ or
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$
to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$,
leading inevitably to deadlocking executions. In contrast,
$\mathbf{G}_{s}^{\prime}$ is again implementable because the expected behavior
of
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
is independent of the choice by
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$.
$\blacktriangleleft$
These examples show that the implementability question is non-trivial. To
check implementability, we present conditions that precisely characterize when
the subset construction for $\mathbf{G}$ yields an implementation.
Overview. The rest of the paper is organized as follows. §3 contains relevant
definitions for our work. §4 describes the synthesis step of our projection.
§5 presents the two conditions that characterize implementability of a given
global type. In §6, we prove soundness of our projection via a stronger
inductive invariant guaranteeing per-role agreement on a global run of the
protocol. In §7, we prove completeness by showing that our two conditions hold
if a global type is implementable. In §8, we discuss the complexity of our
construction and condition checks. §9 presents our artifact and evaluation,
and §10 as well as §11 discuss related work.
## 3 Preliminaries
##### Words.
Let $\Sigma$ be a finite alphabet. $\Sigma^{*}$ denotes the set of finite
words over $\Sigma$, $\Sigma^{\omega}$ the set of infinite words, and
$\Sigma^{\infty}\negthinspace$ their union $\Sigma^{*}\cup\Sigma^{\omega}$. A
word $u\in\Sigma^{*}$ is a _prefix_ of word $v\in\Sigma^{\infty}$, denoted
$u\leq v$, if there exists $w\in\Sigma^{\infty}$ with $u\cdot w=v$.
##### Message Alphabet.
Let $\mathcal{P}$ be a set of roles and $\mathcal{V}$ be a set of messages. We
define the set of _synchronous events_
$Σ_{\mathit{sync}}\coloneq\\{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!m\mid{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}∈\mathcal{P}\text{
and }m∈\mathcal{V}\\}$ where
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!m$
denotes that message $m$ is sent by
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$
atomically. This is split for _asynchronous events_. For a role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}$,
we define the alphabet
$\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},!}=\\{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!m\mid{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\in\mathcal{P},\;m\in\mathcal{V}\\}$
of _send_ events and the alphabet
$\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},?}=\\{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?m\mid{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\in\mathcal{P},\;m\in\mathcal{V}\\}$
of _receive_ events. The event
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!m$
denotes role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
sending a message $m$ to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$,
and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?m$
denotes role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
receiving a message $m$ from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.
We write
$\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}=\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},!}\cup\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},?}$,
$\Sigma_{!}=\bigcup_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},!}$,
and
$\Sigma_{?}=\bigcup_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},?}$.
Finally, $Σ_{\mathit{async}}=\Sigma_{!}\cup\Sigma_{?}$. We say that
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
is _active_ in $x\in Σ_{\mathit{async}}$ if
$x\in\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}$.
For each role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}$,
we define a homomorphism
${\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}$,
where
$x{\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}=x$
if
$x\in\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
and $\varepsilon$ otherwise. We write $\mathcal{V}(w)$ to project the send and
receive events in $w$ onto their messages. We fix $\mathcal{P}$ and
$\mathcal{V}$ in the rest of the paper.
##### Global Types – Syntax.
Global types for MSTs [30] are defined by the grammar:
$\displaystyle G$ $\displaystyle\Coloneqq 0\hskip 3.0pt\mid\hskip
3.0pt\sum_{i∈I}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i}\\!:\\!m_{i}.G_{i}\hskip
3.0pt\mid\hskip 3.0pt\mu t.\;G\hskip 3.0pt\mid\hskip 3.0ptt$
where
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i}$
range over $\mathcal{P}$, $m_{i}$ over $\mathcal{V}$, and $t$ over a set of
recursion variables.
We require each branch of a choice to be distinct:
$∀i,j∈I.\,i≠j⇒({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i},m_{i})≠({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{j},m_{j})$,
the sender and receiver of an atomic action to be distinct:
$∀i∈I.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}≠{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i}$,
and recursion to be guarded: in $μt.\,G$, there is at least one message
between $μt$ and each $t$ in $G$. When $|I|=1$, we omit $\sum$. For
readability, we sometimes use the infix operator $+$ for choice, instead of
$\sum$. When working with a protocol described by a global type, we write
$\mathbf{G}$ to refer to the top-level type, and we use $G$ to refer to its
subterms. For the size of a global type, we disregard multiple occurrences of
the same subterm.
We use the extended definition of global types from [30] that allows a sender
to send messages to different roles in a choice. We call this _sender-driven
choice_ , as in [38], while it was called generalized choice in [30]. This
definition subsumes classical MSTs that only allow _directed choice_ [24]. The
types we use focus on communication primitives and omit features like
delegation or parametrization. We defer a detailed discussion of different MST
frameworks to Section 11.
##### Global Types – Semantics.
As a basis for the semantics of a global type $\mathbf{G}$, we construct a
finite state machine
$\mathsf{GAut}(\mathbf{G})=(Q_{\mathbf{G}},Σ_{\mathit{sync}},δ_{\mathbf{G}},q_{0,\mathbf{G}},F_{\mathbf{G}})$
where
* •
$Q_{\mathbf{G}}$ is the set of all syntactic subterms in $\mathbf{G}$ together
with the term $0$,
* •
$δ_{\mathbf{G}}$ is the smallest set containing
$(\sum_{i∈I}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i}\\!:\\!m_{i}.G_{i},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i}\\!:\\!m_{i},G_{i})$
for each $i∈I$, as well as $(μt.G^{\prime},ε,G^{\prime})$ and
$(t,ε,μt.G^{\prime})$ for each subterm $\mu t.G^{\prime}$,
* •
$q_{0,\mathbf{G}}=\mathbf{G}$ and $F_{\mathbf{G}}=\\{0\\}$.
We define a homomorphism split onto the asynchronous alphabet:
$\texttt{split}({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!:\\!m)\coloneq{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!m.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?m\enspace.$
The semantics $\mathcal{L}(\mathbf{G})$ of a global type $\mathbf{G}$ is given
by
$\mathcal{C}^{\sim}(\texttt{split}(\mathcal{L}(\mathsf{GAut}(\mathbf{G}))))$
where $\mathcal{C}^{\sim}$ is the closure under the indistinguishability
relation $\sim$ [30]. Two events are independent if they are not related by
the _happened-before_ relation [26]. For instance, any two send events from
distinct senders are independent. Two words are indistinguishable if one can
be reordered into the other by repeatedly swapping consecutive independent
events. The full definition is in Section 0.A.2.
##### Communicating State Machine [11].
$\mathcal{A}=\\{\\!\\!\\{A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
is a CSM over $\mathcal{P}$ and $\mathcal{V}$ if
${A}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
is a finite state machine over
$\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
for every
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}$,
denoted by
$(Q_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},\delta_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},q_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}},F_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})$.
Let
$\prod_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}s_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
denote the set of global states and
$\mathsf{Chan}=\\{({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}})\mid{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\in\mathcal{P},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\neq{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\}$
denote the set of channels. A _configuration_ of $\mathcal{A}$ is a pair
$(\vec{s},\xi)$, where $\vec{s}\,$ is a global state and
$\xi:\mathsf{Chan}\rightarrow\mathcal{V}^{*}$ is a mapping from each channel
to a sequence of messages. We use
$\vec{s}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
to denote the state of
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
in $\vec{s}$. The CSM transition relation, denoted $\rightarrow$, is defined
as follows.
* •
$(\vec{s},\xi)\xrightarrow{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!m}(\vec{s}\mkern
2.0mu\vphantom{s}^{\prime},\xi^{\prime})$ if
$(\vec{s}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!m,\vec{s}\mkern
2.0mu\vphantom{s}^{\prime}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})\in\delta_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$,
$\vec{s}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}=\vec{s}\mkern
2.0mu\vphantom{s}^{\prime}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
for every role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\neq{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$,
$\xi^{\prime}({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}})=\xi({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}})\cdot
m$ and $\xi^{\prime}(c)=\xi(c)$ for every other channel $c\in\mathsf{Chan}$.
* •
$(\vec{s},\xi)\xrightarrow{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?m}(\vec{s}\mkern
2.0mu\vphantom{s}^{\prime},\xi^{\prime})$ if
$(\vec{s}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?m,\vec{s}\mkern
2.0mu\vphantom{s}^{\prime}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}})\in\delta_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$,
$\vec{s}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}=\vec{s}\mkern
2.0mu\vphantom{s}^{\prime}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
for every role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\neq{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$,
$\xi({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}})=m\cdot\xi^{\prime}({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}})$
and $\xi^{\prime}(c)=\xi(c)$ for every other channel $c\in\mathsf{Chan}$.
In the initial configuration $(\vec{s}_{0},\xi_{0})$, each role’s state in
$\vec{s}_{0}$ is the initial state
$q_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}$
of
$A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$,
and $\xi_{0}$ maps each channel to $\varepsilon$. A configuration
$(\vec{s},\xi)$ is said to be _final_ iff
$\vec{s}_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
is final for every
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
and $\xi$ maps each channel to $\varepsilon$. Runs and traces are defined in
the expected way. A run is _maximal_ if either it is finite and ends in a
final configuration, or it is infinite. The language
$\mathcal{L}(\mathcal{A})$ of the CSM $\mathcal{A}$ is defined as the set of
maximal traces. A configuration $(\vec{s},\xi)$ is a _deadlock_ if it is not
final and has no outgoing transitions. A CSM is _deadlock-free_ if no
reachable configuration is a deadlock.
Finally, implementability is formalized as follows.
###### Definition 1 (Implementability [30])
A global type $\mathbf{G}$ is _implementable_ if there exists a CSM
$\\{\\!\\!\\{A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
such that the following two properties hold:
(i) _protocol fidelity_ :
$\mathcal{L}(\\{\\!\\!\\{A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}})=\mathcal{L}(\mathbf{G})$,
and (ii) _deadlock freedom_ :
$\\{\\!\\!\\{A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
is deadlock-free. We say that
$\\{\\!\\!\\{A_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
implements $\mathbf{G}$.
## 4 Synthesizing Implementations
The construction is carried out in two steps. First, for each role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}$,
we define an intermediate state machine
$\mathsf{GAut}(\mathbf{G})\negmedspace\\!\downarrow_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}$
that is a homomorphism of $\mathsf{GAut}(\mathbf{G})$. We call
$\mathsf{GAut}(\mathbf{G})\negmedspace\\!\downarrow_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}$
the _projection by erasure_ for
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$,
defined below.
###### Definition 2 (Projection by Erasure)
Let $\mathbf{G}$ be some global type with its state machine
$\mathsf{GAut}(\mathbf{G})=(Q_{\mathbf{G}},Σ_{\mathit{sync}},\delta_{\mathbf{G}},q_{0,\mathbf{G}},F_{\mathbf{G}})$.
For each role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}$,
we define the state machine
$\mathsf{GAut}(\mathbf{G})\negmedspace\\!\downarrow_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\,=(Q_{\mathbf{G}},\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\uplus\\{\varepsilon\\},\delta_{\downarrow},q_{0,\mathbf{G}},F_{\mathbf{G}})$
where
$\delta_{\downarrow}\coloneq\\{q\xrightarrow{\texttt{split}(a){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}}q^{\prime}\mid
q\xrightarrow{a}q^{\prime}\in\delta_{\mathbf{G}}\\}$. By definition of
$\texttt{split}(\hbox{-})$, it holds that
$\texttt{split}(a){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\in\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\uplus\\{\varepsilon\\}$.
Then, we determinize
$\mathsf{GAut}(\mathbf{G})\negmedspace\\!\downarrow_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}$
via a standard subset construction to obtain a deterministic local state
machine for
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$.
###### Definition 3 (Subset Construction)
Let $\mathbf{G}$ be a global type and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
be a role. Then, the _subset construction_ for
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
is defined as
$\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})=\bigl{(}Q_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}},\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},\delta_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}},s_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}},F_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\bigr{)}\text{
where }$
* •
$\delta(s,a)\coloneq\\{q^{\prime}\in Q_{\mathbf{G}}\mid\exists q\in
s,q\xrightarrow{a}\xrightarrow{\varepsilon}\mathrel{\vphantom{\to}{}^{*}}q^{\prime}\in\delta_{\downarrow}\\},$
for every $s\subseteq Q_{\mathbf{G}}$ and
$a\in\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}$
* •
$s_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\coloneq\\{q\in
Q_{\mathbf{G}}\mid
q_{0,\mathbf{G}}\xrightarrow{\varepsilon}\mathrel{\vphantom{\to}{}^{*}}q\in\delta_{\downarrow}\\}$,
* •
$Q_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\coloneq\mathrm{lfp}_{\\{s_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\\}}^{\subseteq}\lambda
Q.\,Q\cup\\{\delta(s,a)\mid s\in Q\land
a\in\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\\}\setminus\\{\emptyset\\}$
, and
* •
$\delta_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\coloneq{\delta}|_{Q_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\times\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}}$
* •
$F_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\coloneq\\{s\in
Q_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}\mid
s\cap F_{\mathbf{G}}\neq\emptyset\\}$
Note that the construction ensures that
$Q_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
only contains subsets of $Q_{\mathbf{G}}$ whose states are reachable via the
same traces, i.e. we typically have
$|Q_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}|\ll
2^{|Q_{\mathbf{G}}|}$.
The following characterization is immediate from the subset construction; the
proof can be found in Section 0.B.1.
lemmaconstructionProperties Let $\mathbf{G}$ be a global type,
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
be a role, and
$\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}})$
be its _subset construction_. If $w$ is a trace of
$\mathsf{GAut}(\mathbf{G})$,
$\texttt{split}(w){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$
is a trace of
$\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}})$.
If $u$ is a trace of
$\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}})$,
there is a trace $w$ of $\mathsf{GAut}(\mathbf{G})$ such that
$\texttt{split}(w){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}=u$.
It holds that
$\mathcal{L}(\mathbf{G}){\Downarrow}_{\Sigma_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}=\mathcal{L}(\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}))$.
Using this lemma, we show that the CSM
$\\{\\!\\!\\{{\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
preserves all behaviors of $\mathbf{G}$. We briefly sketch the proof here. The
full proof can be found in Section 0.B.1.
lemmaconstructionPreservesBehaviors For all global types $\mathbf{G}$,
$\mathcal{L}(\mathbf{G})\subseteq\mathcal{L}(\\{\\!\\!\\{{\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}})$.
###### Proof
Given that
$\\{\\!\\!\\{{\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
is deterministic, to prove language inclusion it suffices to prove the
inclusion of the respective prefix sets:
$\text{pref}(\mathcal{L}(\mathbf{G}))\subseteq\text{pref}(\mathcal{L}\\{\\!\\!\\{{\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}})$
Let $w$ be a word in $\mathcal{L}(\mathbf{G})$. If $w$ is finite, membership
in
$\mathcal{L}(\\{\\!\\!\\{{\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}})$
is immediate from the claim above. If $w$ is infinite, we show that $w$ has an
infinite run in
$\\{\\!\\!\\{{\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
using König’s Lemma. We construct an infinite graph $\mathcal{G}_{w}(V,E)$
with $V\coloneq\\{v_{\rho}\mid\texttt{trace}(\rho)\leq w\\}$ and
$E\coloneq\\{(v_{\rho_{1}},v_{\rho_{2}})\mid\exists~{}x\in
Σ_{\mathit{async}}.~{}\texttt{trace}(\rho_{2})=\texttt{trace}(\rho_{1})\cdot
x\\}$. Because
$\\{\\!\\!\\{{\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
is deterministic, $\mathcal{G}_{w}$ is a tree rooted at $v_{\varepsilon}$, the
vertex corresponding to the empty run. By König’s Lemma, every infinite tree
contains either a vertex of infinite degree or an infinite path. Because
$\\{\\!\\!\\{{\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
consists of a finite number of communicating state machines, the last
configuration of any run has a finite number of next configurations, and
$\mathcal{G}_{w}$ is finitely branching. Therefore, there must exist an
infinite path in $\mathcal{G}_{w}$ representing an infinite run for $w$, and
thus
$w\in\mathcal{L}(\\{\\!\\!\\{{\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}})$.
The proof of the inclusion of prefix sets proceeds by structural induction and
primarily relies on Lemma 3 and the fact that all prefixes in
$\mathcal{L}(\mathbf{G})$ respect the order of send before receive events.
## 5 Checking Implementability
We now turn our attention to checking implementability of a CSM produced by
the subset construction. We revisit the global types from Example 2 (also
shown in Fig. 2), which demonstrate that the naive subset construction does
not always yield a sound implementation. From these examples, we distill our
conditions that precisely identify the implementable global types.
In general, a global type $\mathbf{G}$ is not implementable when the agreement
on a global run of $\mathsf{GAut}(\mathbf{G})$ among all participating roles
cannot be conveyed via sending and receiving messages alone. When this
happens, roles can take locally permitted transitions that commit to
incompatible global runs, resulting in a trace that is not specified by
$\mathbf{G}$. Consequently, our conditions need to ensure that when a role
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
takes a transition in
$\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})$,
it only commits to global runs that are consistent with the local views of all
other roles. We discuss the relevant conditions imposed on send and receive
transitions separately.
Send Validity. Consider $\mathbf{G}_{s}$ from Example 2. The CSM
$\\{\\!\\!\\{{\mathscr{C}(\mathbf{G}_{s},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
has an execution with the trace
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}\cdot{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}\cdot{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$.
This trace is possible because the initial state of
$\mathscr{C}(\mathbf{G}_{s},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}})$,
$s_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$,
contains two states of
$\mathsf{GAut}(\mathbf{G}_{s})\negmedspace\\!\downarrow_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$,
each of which has a single outgoing send transition labeled with
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$
and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$
respectively. Both of these transitions are always enabled in
$s_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$,
meaning that
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
can send
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$
even when
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
has chosen the top branch and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$
expects to receive
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$
instead of
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}m}$
from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$.
This results in a deadlock. In contrast, while the state
$s_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$
in
$\mathscr{C}(\mathbf{G}_{s}^{\prime},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}})$
likewise contains two states of
$\mathsf{GAut}(\mathbf{G}_{s}^{\prime})\negmedspace\\!\downarrow_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$,
each with a single outgoing send transition, now both transitions are labeled
with
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$.
These two transitions collapse to a single one in
$\mathscr{C}(\mathbf{G}_{s}^{\prime},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}})$.
This transition is consistent with both possible local views that
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$
might hold on the global run.
Intuitively, to prevent the emergence of inconsistent local views from send
transitions of
$\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})$,
we must enforce that for every state $s\in
Q_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}$
with an outgoing send transition labeled $x$, a transition labeled $x$ must be
enabled in all states of
$\mathsf{GAut}(\mathbf{G})\negmedspace\\!\downarrow_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}$
represented by $s$. We use the following auxiliary definition to formalize
this intuition subsequently.
###### Definition 4 (Transition Origin and Destination)
Let
$s\xrightarrow{x}s^{\prime}\in\delta_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
be a transition in
$\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})$
and $\delta_{\downarrow}$ be the transition relation of
$\mathsf{GAut}(\mathbf{G})\negmedspace\\!\downarrow_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}$.
We define the set of _transition origins_ $\operatorname{tr-
orig}(s\xrightarrow{x}s^{\prime})$ and _transition destinations_
$\operatorname{tr-dest}(s\xrightarrow{x}s^{\prime})$ as follows:
$\displaystyle\operatorname{tr-orig}(s\xrightarrow{x}s^{\prime})\coloneq{}$
$\displaystyle\\{G\in s\mid\exists G^{\prime}\in
s^{\prime}.\,G\xrightarrow{x}\mathrel{\vphantom{\to}{}^{*}}G^{\prime}\in\delta_{\downarrow}\\}\text{
and }$ $\displaystyle\operatorname{tr-
dest}(s\xrightarrow{x}s^{\prime})\coloneq{}$ $\displaystyle\\{G^{\prime}\in
s^{\prime}\mid\exists G\in
s.\,G\xrightarrow{x}\mathrel{\vphantom{\to}{}^{*}}G^{\prime}\in\delta_{\downarrow}\\}\enspace.$
Our condition on send transitions is then stated below.
###### Definition 5 (Send Validity)
$\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})$
satisfies _Send Validity_ iff every send transition
$s\xrightarrow{x}s^{\prime}\in\delta_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
is enabled in all states contained in $s$:
$\forall
s\xrightarrow{x}s^{\prime}\in\delta_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}.~{}x\in\Sigma_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}},!}\implies\operatorname{tr-
orig}(s\xrightarrow{x}s^{\prime})=s\enspace.$
Receive Validity. To motivate our condition on receive transitions, let us
revisit $\mathbf{G}_{r}$ from Example 2. The CSM
$\\{\\!\\!\\{{\mathscr{C}(\mathbf{G}_{r},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})}\\}\\!\\!\\}_{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\in\mathcal{P}}$
recognizes the following trace not in the global type language
$\mathcal{L}(\mathbf{G}_{r})$:
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}\cdot{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}\cdot{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}\cdot{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}\cdot{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}\cdot{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}\enspace.$
The issue lies with
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
which cannot distinguish between the two branches in $\mathbf{G}_{r}$. The
initial state
$s_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$
of
$\mathscr{C}(\mathbf{G}_{r},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}})$
has two states of $\mathsf{GAut}(\mathbf{G}_{r})$ corresponding to the
subterms
$G_{t}\coloneq{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,0$
and
$G_{b}\coloneq{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\\!:\\!{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}.\,0\,$.
Here, $G_{t}$ and $G_{b}$ are the top and bottom branch of $\mathbf{G}_{r}$
respectively. This means that there are outgoing transitions in
$s_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$
labeled with
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$
and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$.
If
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
takes the transition labeled
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$,
it commits to the bottom branch $G_{b}$. However, observe that the message
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$ from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
can also be available at this time point if the other roles follow the top
branch $G_{t}$. This is because
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
can send
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$ to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
without waiting for
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
to first receive from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.
In this scenario, the roles disagree on which global run of
$\mathsf{GAut}(\mathbf{G}_{r})$ to follow, resulting in the violating trace
above.
Contrast this with $\mathbf{G}_{r}^{\prime}$. Here,
$s_{0,{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}}$
again has outgoing transitions labeled with
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$
and
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$.
However, if
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
takes the transition labeled
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}o}$,
committing to the bottom branch, no disagreement occurs. This is because if
the other roles are following the top branch, then
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}$
is blocked from sending to
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
until after it has received confirmation that
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}$
has received its first message from
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}$.
For a receive transition $s\xrightarrow{x}s_{1}$ in
$\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})$
to be safe, we must enforce that the receive event $x$ cannot also be
available due to reordered sent messages in the continuation $G_{2}\in s_{2}$
of another outgoing receive transition $s\xrightarrow{y}s_{2}$. To formalize
this condition, we use the set $M^{\mathcal{B}}_{(G\ldots)}$ of _available
messages_ for a syntactic subterm $G$ of $\mathbf{G}$ and a set of _blocked_
roles $\mathcal{B}$. This notion was already defined in [30, Sec. 2.2].
Intuitively, $M^{\mathcal{B}}_{(G\ldots)}$ consists of all send events
${\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}\triangleright{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}}!m$
that can occur on the traces of $G$ such that $m$ will be the first message
added to channel
$({\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{r}}}})$
before any of the roles in $\mathcal{B}$ takes a step.
##### Available messages.
The set of available messages is recursively defined on the structure of the
global type. To obtain all possible messages, we need to unfold the distinct
recursion variables once. For this, we define a map $\mathit{get\mu}$ from
variable to subterms and write $\mathit{get\mu}_{\mathbf{G}}$ for
$\mathit{get\mu}(\mathbf{G})$:
$\mathit{get\mu}(0)\coloneq[\,]$ $\mathit{get\mu}(t)\coloneq[\,]$
$\mathit{get\mu}(μt.G)\coloneq[t\mapsto G]∪\mathit{get\mu}(G)$
$\mathit{get\mu}(\sum_{i∈I}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i}\\!:\\!m_{i}.G_{i})\coloneq\bigcup_{i∈I}\mathit{get\mu}(G_{i})$
The function $M^{\mathcal{B},T}_{(\hbox{-}\ldots)}$ keeps a set of unfolded
variables $T$, which is empty initially.
$M^{\mathcal{B},T}_{(0\ldots)}\coloneq∅\hfill
M^{\mathcal{B},T}_{(μt.G\ldots)}\coloneq
M^{\mathcal{B},T∪\\{t\\}}_{(G\ldots)}\hfill
M^{\mathcal{B},T}_{(t\ldots)}\coloneq\begin{cases}∅&\text{if}~{}t∈T\\\
M^{\mathcal{B},T∪\\{t\\}}_{(\mathit{get\mu}_{\mathbf{G}}(t)\ldots)}&\text{if}~{}t∉T\end{cases}\\\
M^{\mathcal{B},T}_{(\sum_{i∈I}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\!\to\\!{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i}\\!:\\!m_{i}.G_{i}\ldots)}\coloneq\begin{cases}\bigcup_{i∈I,m∈\mathcal{V}}(M^{\mathcal{B},T}_{(G_{i}\ldots)}\setminus\\{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?m\\})∪\\{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}?m_{i}\\}\quad\hfill\text{if}~{}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}∉\mathcal{B}\\\
\bigcup_{i∈I}M^{\mathcal{B}∪\\{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{i}\\},T}_{(G_{i}\ldots)}\quad\hfill\text{if}~{}{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}∈\mathcal{B}\end{cases}$
We write $M^{\mathcal{B}}_{(G\ldots)}$ for
$M^{\mathcal{B},\emptyset}_{(G\ldots)}$. If $\mathcal{B}$ is a singleton set,
we omit set notation and write
$M^{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}}_{(G\ldots)}$
for
$M^{\\{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\\}}_{(G\ldots)}$.
The set of available messages captures the possible states of all channels
before a given receive transition is taken.
###### Definition 6 (Receive Validity)
$\mathscr{C}(\mathbf{G},{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}})$
satisfies _Receive Validity_ iff no receive transition is enabled in an
alternative continuation that originates from the same source state:
$\begin{array}[]{l}\forall
s\xrightarrow{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{1}?m_{1}}s_{1},\,s\xrightarrow{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}\triangleleft{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{2}?m_{2}}s_{2}\in\delta_{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{p}}}}.\\\
\qquad{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{1}\neq{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\boldsymbol{{\color[rgb]{0.1,0.3,0.1}\definecolor[named]{pgfstrokecolor}{rgb}{0.1,0.3,0.1}\mathtt{q}}}}_{2}\;\implies\;\forall~{}G_{2}\in\operatorname{tr- |
# Non-adiabatic particle production scenario in algebraically coupled
quintessence field with dark matter fluid
Saddam Hussain<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Kanpur, UP 208016, India
###### Abstract
We investigate the dynamics of an algebraically coupled quintessence field
with a dark matter fluid, considering a scenario involving non-adiabatic
particle production, through the action principle by modifying the interaction
Lagrangian. The interaction parameter serves as the source of dark matter
particle and entropy production. As particle creation occurs due to the
interaction between the field and fluid sectors, the system manifests an
additional pressure. Our analysis includes studying the system’s dynamics by
considering an exponential type of interaction corresponding to the field’s
exponential potential. We find that the system exhibits phantom behavior at
the current epoch before stabilizing in the accelerating future epoch of the
universe.
## I Introduction
Over the past two decades, observations have shed light on the dynamics of the
cosmos at its largest scale, revealing evidence that the expansion of the
universe is accelerating [1, 2, 3, 4, 5, 6, 7, 8, 9]. One of the most accepted
explanations for this late-time cosmic acceleration is the existence of an
exotic component known as the cosmological constant ($\Lambda$), which exerts
negative pressure. The largest portion of the universe’s energy budget
consists of the cosmological constant, accounting for about $70\%$.
Additionally, approximately $25\%$ of the energy budget is dominated by a non-
relativistic, pressureless fluid commonly referred to as dark matter, with the
remaining portion composed of baryonic matter.
Although the $\Lambda$CDM model, where CDM represents cold dark matter, is
favored due to its ability to describe most observational evidence, it faces
several theoretical shortcomings, including the cosmological constant problem,
fine-tuning issues, and the cosmological coincidence problem [10, 11, 12, 13,
14, 15, 16]. Numerous alternatives have been proposed to address these issues,
either by modifying the gravitational sector [17, 18, 19, 20] or by modifying
the matter sector [21, 22, 23, 24, 25, 26, 27]. In many instances, scalar
fields serve as viable candidates for dark energy (DE) and are often minimally
coupled with pressureless dark matter (DM) fluid. However, beyond their
gravitational signatures, these enigmatic forms of matter pose puzzles to the
scientific community. Consequently, numerous possible scenarios have been
intensely investigated, including (i) non-gravitational interactions between
dark matter and dark energy [28, 29, 30, 31, 32, 33, 34] and (ii) non-minimal
coupling between matter fields and curvature [35, 36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47].
A recent approach has emerged to investigate the non-gravitational interaction
between dark sectors through the variational principle [40, 41, 48]. In these
studies, dark energy is governed by a scalar field, while the action for the
dark matter fluid is modeled using the relativistic fluid action proposed by
Brown [49]. This action encompasses the energy density of the fluid $\rho$,
the particle flux number $J^{\mu}$, and several Lagrange multipliers.
Additionally, a non-gravitational interaction term is introduced at the action
level, consisting of fluid and field variables, denoted by $\mathcal{L}_{\rm
int}=-\sqrt{-g}f(n,s,\phi)$, where $f$ is an arbitrary interaction function
depending on the fluid number density $n$, entropy per particle $s$, and
scalar field $\phi$. One immediate constraint studied under these models is
the conservation of number density $\nabla_{\mu}(nu^{\mu})=0$, where $u^{\mu}$
represents the fluid’s four-velocity, and the conservation of entropy
$\nabla_{\mu}(snu^{\mu})=0$. Consequently, the normal component of the fluid’s
covariant derivative of the energy-momentum tensor vanishes, i.e.,
$u_{\mu}\nabla_{\nu}T^{\mu\nu}=0$, implying $\rho\propto a^{-3}$, where $a$ is
the scale factor. However, the inclusion of $\phi$ in the interacting
Lagrangian modifies the field evolution, resulting in dynamics different from
the minimally coupled scenario. In light of the current cosmological crisis,
where the discrepancy between the measured values of the Hubble parameter
$H_{0}$ and the amplitude of matter density $S_{8}$ between high and low-
redshift data exceeds the $4.4\sigma$ level [50, 51, 52], exploring scenarios
where both the field and fluid sectors are simultaneously affected becomes
crucial.
This paper explores a scenario where the dynamics between the quintessence
field and dark matter fluid are investigated by modifying the interaction
Lagrangian $f(n,s,\phi)\rightarrow f(n,s,\phi,\varphi)$, where $\varphi$ is a
fluid Lagrange multiplier. This modification results in alterations to the
thermodynamic constraints such that $\nabla_{\mu}(nu^{\mu})\neq 0$ and
$\nabla_{\mu}(snu^{\mu})\neq 0$. With the number density of the dark matter
fluid no longer conserved, energy flow from the quintessence field to the dark
matter can lead to the creation of matter particles, consequently inducing an
additional pressure known as creation pressure $P_{c}$ [53, 54, 55, 56, 57,
58]. By analyzing this scenario from the action principle, the interaction
function serves as a source of particle and entropy production. We obtained
the fluid’s equations of motion and temperature evolution from thermodynamic
principles. Additionally, we provide a background dynamics by considering an
exponential type of interaction
$f\propto\rho^{\beta}e^{\delta\kappa\phi+\gamma\varphi}$, corresponding to the
field’s exponential potential. Stability analysis of the interacting system is
conducted using the standard linearization technique [59, 60, 61, 62], with
proper constraints on thermodynamic quantities to ensure positivity of entropy
and number density throughout the evolution. Our findings indicate that the
interacting system exhibits a stable accelerating critical point in the future
epoch of the universe. However, at the present epoch $a=1$, the effective
equation of state (EoS) crosses the phantom barrier, i.e., $\omega_{\rm
eff}\leq-1$. Furthermore, we utilize $43$ Hubble data and $1701$ Pantheon+
data to numerically simulate the current model against the $\Lambda$CDM model,
revealing compatibility with the data for $f\propto\rho^{3}$.
We organized the paper as follows: In sec. II, we set up the action for the
algebraically coupled field-fluid system and obtained the governing background
equations. In sec. III, we present brief thermodynamic relations corresponding
to the fluid component. A detailed picture of the conservation of energy-
momentum tensor is presented in sec. IV. The dynamical system’s stability is
discussed in sec. V. Finally, a brief conclusion is outlined in sec. VI.
## II Action for the algebraic interaction
The action describing the algebraically (non-minimally) coupled field-fluid
scenario is given by [40]:
$S=\int_{\Omega}d^{4}x\sqrt{-g}\frac{R}{2\kappa^{2}}-\sqrt{-g}\rho(n,s)+J^{\mu}(\varphi_{,\mu}+s\theta_{,\mu}+\alpha_{A}{\beta}_{,\mu}^{A})-\sqrt{-g}\mathcal{L}_{\phi}(\phi,\partial{}_{\mu}\phi)-\alpha_{1}\sqrt{-g}f(n,s,\phi,\varphi),$
(1)
where,
$J^{\mu}=\sqrt{-g}nu^{\mu},\quad|J|=\sqrt{-g_{\mu\nu}J^{\mu}J^{\nu}},\quad
n=\frac{|J|}{\sqrt{-g}},\quad u^{\mu}u_{\mu}=-1\,.$ (2)
In this action, the first term corresponds to the Einstein-Hilbert action,
where $g$ denotes the determinant of the metric tensor $g^{{\mu\nu}}$, $R$
represents the Ricci scalar, and $\kappa^{2}=8\pi G$. The second and third
terms together represent the action for a relativistic fluid, where the energy
density of the dark matter fluid is denoted as $\rho$, depending on the fluid
number density $n$ and the entropy density per particle $s$. The relativistic
fluid Lagrangian contains the fluid flux density $J^{\mu}$ and Lagrange
multipliers $\varphi$, $\theta$, $\alpha^{A}$, $\beta_{A}$. Note that Greek
indices range from 0 to 3, and $A$ runs from 1 to 3. It’s important to
distinguish between $\alpha_{1}$ and $\alpha^{A}$ as they are distinct
quantities unless specified. The commas followed by Greek indices indicate
covariant derivatives. The field action is given by:
$\mathcal{L}_{\phi}=\frac{1}{2}\partial{}_{\mu}\phi\partial{}^{\mu}\phi+V(\phi).$
(3)
Here, the Lagrangian represents a standard quintessence field with the
potential $V(\phi)$. The remaining term in the action corresponds to the
interaction parameter $f$, which depends on fluid and field parameters, while
$\alpha_{1}$ is a constant parameter. Taking the variation of the action Eq.
(1) with respect to the metric $g^{{\mu\nu}}$ yields the Einstein field
equation:
$R_{{\mu\nu}}-\dfrac{1}{2}Rg_{{\mu\nu}}=\kappa^{2}\left(T_{{\mu\nu}}^{M}+T_{{\mu\nu}}^{\phi}+T_{{\mu\nu}}^{\rm
int}\right).$ (4)
Here, the stress tensor of the matter components is defined as
$T_{{\mu\nu}}=\frac{-2}{\sqrt{-g}}\frac{\delta S}{\delta g^{{\mu\nu}}}$. The
energy-momentum tensor corresponding to the relativistic fluid is:
$T^{{\mu\nu}}_{M}=\rho
u^{\mu}u^{\nu}+\left(n\frac{\partial{}\rho}{\partial{}n}-\rho\right)(u^{\mu}u^{\nu}+g^{{\mu\nu}}).$
(5)
Similarly, corresponding to the interaction part:
$T^{{\mu\nu}}_{\rm
int}=\alpha_{1}fu^{\mu}u^{\nu}+\alpha_{1}\left(n\frac{\partial{}f}{\partial{}n}-f\right)(u^{\mu}u^{\nu}+g^{{\mu\nu}}).$
(6)
The field’s stress tensor becomes:
$T_{\phi}^{{\mu\nu}}=-g^{{\mu\nu}}\left(\frac{1}{2}\partial{}_{\alpha}\phi\partial{}^{\alpha}\phi+V(\phi)\right)+\partial{}^{\mu}\phi\partial{}^{\nu}\phi.$
(7)
To evaluate the energy density and corresponding pressure of these matter
components, we compare with the stress tensor for the perfect fluid
$T^{{\mu\nu}}=\rho u^{\mu}u^{\nu}+P(g^{{\mu\nu}}+u^{\mu}u^{\nu})$. Hence, the
energy density and pressure of the fluid become:
$\rho_{M}=\rho,\quad
P_{M}=\left(n\frac{\partial{}\rho}{\partial{}n}-\rho\right).$ (8)
The energy density and pressure corresponding to the algebraic interaction
become:
$\rho_{\rm int}=\alpha_{1}f,\quad P_{\rm
int}=\alpha_{1}\left(n\frac{\partial{}f}{\partial{}n}-f\right).$ (9)
The field energy density and pressure become:
$\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),\quad
P_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi).$ (10)
In the flat FLRW metric $ds^{2}=-dt^{2}+a(t)^{2}d\vec{x}^{2}$, the Friedmann
equations become:
$\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle\kappa^{2}(\rho_{M}+\rho_{\phi}+\rho_{\rm int}),$ (11)
$\displaystyle 2\dot{H}+3H^{2}$ $\displaystyle=$
$\displaystyle-\kappa^{2}(P_{M}+P_{\phi}+P_{\rm int}).$ (12)
Upon taking the variation of the action with respect to $\phi$, the field
equation becomes:
$\ddot{\phi}+3H\dot{\phi}+\dfrac{dV(\phi)}{d\phi}+\alpha_{1}\frac{\partial{}f}{\partial{}\phi}=0.$
(13)
## III Thermodynamic relations
In this section, we will examine the thermodynamic aspect of the fluid in
light of interaction. Upon varying the action with respect to the fluid
variables, the corresponding equations of motion become:
$\displaystyle J^{\mu}:$ $\displaystyle\qquad
u_{\mu}\left(\frac{\partial{}\rho}{\partial{}n}+\alpha_{1}\frac{\partial{}f}{\partial{}n}\right)+\varphi_{,\mu}+s\theta_{,\mu}+\beta_{A}\alpha^{A}_{,\mu}=0,$
(14) $\displaystyle s:$
$\displaystyle\qquad-\left(\frac{\partial\rho}{\partial
s}+\alpha_{1}\frac{\partial{}f}{\partial{}s}\right)+nu^{\mu}\theta_{,\mu}=0\,,$
(15) $\displaystyle\varphi:$
$\displaystyle\qquad\nabla_{\mu}(nu^{\mu})+\alpha_{1}\dfrac{\partial{}f}{\partial{}\varphi}=0\,,$
(16) $\displaystyle\theta:$ $\displaystyle\qquad(sJ^{\mu})_{,\mu}=0\,,$ (17)
$\displaystyle\beta_{A}:$ $\displaystyle\qquad J^{\mu}\alpha^{A}_{,\mu}=0\,,$
(18) $\displaystyle\alpha^{A}:$
$\displaystyle\qquad(\beta_{A}J^{\mu})_{,\mu}=0.$ (19)
Due to the modifications in the interaction parameter, which now includes the
Lagrange parameter $\varphi$, the number density of the fluid from Eq. (16) is
no longer conserved, and the interaction parameter $f$ acts as a source of
fluid particle number density. In the flat FLRW background metric, this
equation reads as:
$\dot{n}+3nH=-\alpha_{1}\frac{\partial{}f}{\partial{}\varphi}.$ (20)
As the non-minimal coupling acts as a source of the fluid’s particle density,
the corresponding contribution in entropy from Eq. (17) becomes:
$\dot{s}=\frac{s}{n}\alpha_{1}\frac{\partial{}f}{\partial{}\varphi}$ (21)
Therefore, due to the dependence of $\varphi$ in the interaction function, the
fluid particle density may increase or decrease, resulting in a change in
entropy. Consequently, the evolution of the fluid sector becomes non-
adiabatic. As the interaction parameter induces changes in number density and
entropy, the corresponding change in energy density of the fluid sector can be
explored using the thermodynamic relation:
$d(\rho V)=dQ-PdV.$ (22)
Here, $dQ$ denotes the heat received by the fluid sector during time $dt$, and
$V=a^{3}$ signifies the comoving volume. For the closed system, the above
relation can be re-expressed as:
$d(\rho/n)=Tds-Pd(1/n).$ (23)
Here, $dq=dQ/N$, $n=N/V$, $s=S/N$, and
$T=\frac{1}{n}\frac{\partial\rho}{\partial s}\big{|}_{n}$ is the temperature
of the fluid [54]. As the particle production takes place, the system can
absorb heat. The above relation can then be extended to the open system as:
$d(\rho V)=TV(d(sn)-sdn)-PdV+\frac{\rho+P}{n}d(nV).$ (24)
On differentiating both sides with respect to time $t$, we obtain:
$\dot{\rho}+3H(\rho+P)=Ts\alpha_{1}\frac{\partial{}f}{\partial{}\varphi}-\alpha_{1}\frac{\rho+P}{n}\frac{\partial{}f}{\partial{}\varphi}.$
(25)
From this, we can express the creation pressure or non-minimally induced
pressure $P_{c}$ as:
$P_{c}=\frac{1}{3H}\left(\frac{\rho+P}{n}-\frac{s}{n}\frac{\partial{}\rho}{\partial{}s}\right)\alpha_{1}\frac{\partial{}f}{\partial{}\varphi}.$
(26)
Therefore, as the fluid’s particle number density changes, the system can
exhibit an additional pressure known as creation pressure. The density
evolution can be rewritten as:
$\dot{\rho}+3H(\rho+P+P_{c})=0.$ (27)
From Eq. (20) and Eq. (21), it is clear that if $\alpha_{1}\frac{\partial
f}{\partial\varphi}<0$, the interaction contributes to an increment in the
number density of the fluid particles. As a consequence, the change in entropy
corresponding to the fluid sector decreases. Although the entropy of the
matter sector decreases, it’s important to note that the entropy of the system
as a whole may not decrease. Hence, the second law of thermodynamics for the
entire system remains intact.
Using the above equation, the evolution of the temperature $T$ of the fluid
can be determined. We utilize the total derivative of energy density as:
$d\rho(n,T)=\left(\frac{\partial{}\rho}{\partial{}n}\right)_{T}dn+\left(\frac{\partial{}\rho}{\partial{}T}\right)_{n}dT,$
(28)
After taking the time derivative and using equations Eq. (27) and Eq. (20), we
obtain:
$\dot{T}=\frac{1}{\left(\partial{}\rho/\partial{}T\right)_{n}}\left(-3H(\rho+P+P_{c})-\left(\frac{\partial{}\rho}{\partial{}n}\right)_{T}\dot{n}\right).$
(29)
To evaluate $\left(\frac{\partial\rho}{\partial n}\right)_{T}$, we use the
thermodynamic relation:
$ds=\frac{1}{nT}d\rho-\frac{\rho+P}{n^{2}T}dn,$ (30)
and using the partial derivative property, we obtain:
$\left(\frac{\partial{}s}{\partial{}n}\right)_{T}=\frac{1}{nT}\left(\left(\frac{\partial{}\rho}{\partial{}n}\right)_{T}-\frac{\rho+P}{n}\right),\quad\left(\frac{\partial{}s}{\partial{}T}\right)_{n}=\frac{1}{nT}\left(\frac{\partial{}\rho}{\partial{}T}\right)_{n}.$
(31)
Further using the property
$\frac{\partial{}^{2}s}{\partial{}T\partial{}n}=\frac{\partial{}^{2}s}{\partial{}n\partial{}T}$,
we get:
$h=\rho+P=n\left(\frac{\partial{}\rho}{\partial{}n}\right)_{T}+T\left(\frac{\partial{}P}{\partial{}T}\right)_{n}\,.$
(32)
Here, $h$ is the enthalpy per unit volume. Inserting this relation into
$\dot{T}$ and using
$\frac{(\partial{}P/\partial{}T)_{n}}{(\partial{}\rho/\partial{}T)_{n}}\equiv(\partial{}P/\partial{}\rho)_{n}$,
we finally obtain:
$\dot{T}/T=\frac{s\alpha_{1}\partial{}f/\partial{}\varphi}{(\partial{}\rho/\partial{}T)_{n}}+\frac{\dot{n}}{n}\left(\frac{\partial{}P}{\partial{}\rho}\right)_{n}.$
(33)
Using Eq. (21), this relation is similar to the one that has been obtained in
ref. [63]. The first term on the right-hand side shows the contribution from
the non-conserved entropy of the fluid. To find the temperature evolution, we
choose the model corresponding to the pressureless non-relativistic fluid. The
energy density of the non-relativistic ideal gas $(k_{\rm B}=1)$ is:
$\rho=Mn+\frac{3}{2}nT,$ (34)
where $M$ is the mass of gas particles. Corresponding to this ideal gas model,
the pressure, $P=n(\partial\rho/\partial n)-\rho=0$, vanishes. Using this
relation, the temperature relation simplifies to:
$\dfrac{\dot{T}}{T}=\dfrac{2\dot{s}}{3}.$ (35)
This gives the variation of temperature with entropy irrespective of any form
of interaction parameter $f$. Hence the entropy becomes:
$s=s_{0}+\frac{3}{2}\ln(T/T_{0}).$ (36)
Here, the parameters with subscript $0$ are integration constants representing
the present value of the respective parameters.
## IV Conservation of energy momentum tensor
In the preceding section, we explored the thermodynamic behavior of the fluid
sector, employing thermodynamic relations to determine the corresponding
evolution of fluid density. In this section, we delve into evaluating the
covariant derivative of the stress tensor of the entire system to investigate
the transportation of energy flow between the fluid and field sectors. To do
so, we first redefine the matter stress tensor as:
$T^{{\mu\nu}}_{A}=(\rho+\alpha_{1}f)u^{\mu}u^{\nu}+\left(n\frac{\partial{}\rho}{\partial{}n}-\rho+\alpha_{1}n\frac{\partial{}f}{\partial{}n}-\alpha_{1}f\right)(u^{\mu}u^{\nu}+g^{{\mu\nu}}).$
(37)
With this redefinition, the Einstein tensor becomes
$G_{{\mu\nu}}=\kappa^{2}(T_{{\mu\nu}}^{A}+T_{{\mu\nu}}^{\phi})$ 111Here $A$
denotes a label.. Upon taking the covariant derivative of the stress tensor,
we obtain:
$\nabla_{\mu}T^{\mu
0}_{A}=\dot{\rho}+\alpha_{1}\dot{f}+3H(\rho+\alpha_{1}f+P_{M}+P_{\rm
int})=Q^{0}.$ (38)
Utilizing Eq. (20) and Eq. (21), we can express:
$\nabla_{\mu}T^{\mu
0}_{A}=\dot{\rho}+3H(\rho+P_{M})-\alpha_{1}^{2}\frac{\partial{}f}{\partial{}\varphi}\frac{\partial{}f}{\partial{}n}+\alpha_{1}\dot{\phi}\frac{\partial{}f}{\partial{}\phi}+\alpha_{1}^{2}\frac{\partial{}f}{\partial{}s}\left(\frac{s}{n}\frac{\partial{}f}{\partial{}\varphi}\right)+\alpha_{1}\dot{\varphi}\frac{\partial{}f}{\partial{}\varphi}=Q^{0}.$
(39)
To eliminate $\dot{\varphi}$ from the above equation, we contract Eq. (14)
with $J^{\mu}$ and then use Eq. (18), we get:
$\frac{\partial{}\rho}{\partial{}n}-s\dot{\theta}+\alpha_{1}\frac{\partial{}f}{\partial{}n}=\dot{\varphi},$
(40)
which, when inserted into the above equation, yields:
$\nabla_{\mu}T^{\mu
0}_{A}=\dot{\rho}+3H(\rho+P_{M})+\alpha_{1}\dot{\phi}\frac{\partial{}f}{\partial{}\phi}+\alpha_{1}\frac{\partial{}\rho}{\partial{}n}\frac{\partial{}f}{\partial{}\varphi}+\alpha_{1}^{2}\frac{\partial{}f}{\partial{}s}\left(\frac{s}{n}\frac{\partial{}f}{\partial{}\varphi}\right)-s\dot{\theta}\frac{\partial{}f}{\partial{}\varphi},$
(41)
The time derivative of the temperature gradient $\dot{\theta}$ can be
eliminated by using Eq. (15), resulting in:
$\nabla_{\mu}T^{\mu
0}_{A}=\dot{\rho}+3H(\rho+P_{M})+\alpha_{1}\dot{\phi}\frac{\partial{}f}{\partial{}\phi}+\alpha_{1}\frac{\partial{}\rho}{\partial{}n}\frac{\partial{}f}{\partial{}\varphi}-\alpha_{1}\frac{s}{n}\frac{\partial{}f}{\partial{}\varphi}\frac{\partial{}\rho}{\partial{}s}.$
(42)
Using the definition of the pressure of the fluid
$P_{M}+\rho=n\frac{\partial\rho}{\partial n}$ and comparing this with Eq.
(25), we obtain:
$\nabla_{\mu}T^{\mu
0}_{A}=\dot{\rho}+3H(\rho+P_{M}+P_{c})+\alpha_{1}\dot{\phi}\frac{\partial{}f}{\partial{}\phi}=Q^{0},$
(43)
where $P_{c}=\dfrac{1}{3H}\left(\dfrac{\rho+P_{M}}{n}\alpha_{1}\dfrac{\partial
f}{\partial\varphi}-\alpha_{1}\dfrac{s}{n}\dfrac{\partial
f}{\partial\varphi}\dfrac{\partial\rho}{\partial s}\right)$. Considering Eq.
(25), the covariant derivative of the fluid sector yields:
$\nabla_{\mu}T^{\mu
0}_{A}=\alpha_{1}\dot{\phi}\frac{\partial{}f}{\partial{}\phi}=Q^{0}.$ (44)
The covariant derivative of the field sector can be determined using Eq. (13)
as:
$\nabla_{\mu}T^{\mu
0}_{\phi}=-\alpha_{1}\frac{\partial{}f}{\partial{}\phi}\dot{\phi}=-Q^{0}.$
(45)
This exercise demonstrates that the total energy density of the system remains
conserved, and both sectors exchange energy through the interaction term:
$Q^{0}=\alpha_{1}\frac{\partial{}f}{\partial{}\phi}\dot{\phi}.$ (46)
## V Background dynamics
In this section, we will conduct a stability analysis of the system using the
standardized linearization technique. We observed that the interaction
function alters not only the field’s equation of motion but also the fluid’s
equation of motion. As the non-minimal coupling parameter $f$ acts as a source
of particle creation and entropy generation, the system transitions to a non-
adiabatic state. To elucidate the background evolution of the system, we
define a set of dimensionless dynamical variables as:
$x=\frac{\kappa\dot{\phi}}{\sqrt{6}H},\ y=\frac{\kappa\sqrt{V}}{\sqrt{3}H},\
z=\frac{\kappa^{2}f}{3H^{2}},\
\Omega_{\phi}=\frac{\kappa^{2}\rho_{\phi}}{3H^{2}},\
\Omega_{M}=\frac{\kappa^{2}\rho}{3H^{2}}.$ (47)
Expressing the first Friedmann equation in dynamical variables puts constraint
on the fluid fractional energy density as:
$\Omega_{M}=1-x^{2}-y^{2}-\alpha_{1}z,$ (48)
and the effective equation of state (EoS) corresponding to the entire system
becomes:
$\omega_{\rm eff}=\frac{P_{\rm tot}}{\rho_{\rm
tot}}=x^{2}-y^{2}+\alpha_{1}(\beta-1)z.$ (49)
Based on the definition of dynamical variables, the variables range as
follows:
$-\infty<x<+\infty,\quad 0\leq
y<\infty,\quad-\infty<z<+\infty,\quad\Omega_{M}\geq 0,\quad\Omega_{\phi}\geq
0.$ (50)
To construct the autonomous equations corresponding to the primary variables
of the system $x,y,z$, one needs to choose the form of the interaction
function,
$f=M^{4-4\beta}\rho^{\beta}e^{(\delta\kappa\phi+\gamma\varphi)},$ (51)
and the potential of the quintessence field is:
$V(\phi)=V_{0}e^{\lambda\kappa\phi}.$ (52)
Before constructing the autonomous equations, we evaluate $\dot{z}$:
$\dot{z}=\frac{\kappa^{2}}{3H^{2}}\left(-3H\beta f+\delta\kappa
f\dot{\phi}+\gamma f(\rho/n-sT)\right)-z\frac{\dot{H}}{H}.$ (53)
To evaluate this, we have used $T=\frac{1}{n}\frac{\partial\rho}{\partial s}$,
Eq. (40), and $\dot{\theta}=T+\alpha_{1}\frac{1}{n}\frac{\partial f}{\partial
s}$. However, to close the system, we still need to define additional
dynamical variables corresponding to $n,s,T$, as these are time-dependent
quantities. Hence, the additional dynamical variables are:
$\chi=\frac{\kappa^{2}n}{3H},\quad\xi=T/H,\quad s=s.$ (54)
We choose the variables corresponding to the number density $n$, temperature
$T$ of the fluid, and entropy $s$. With these definitions, the thermodynamic
variables are constrained within the following ranges:
$\chi>0,\quad\xi>0,\quad s\geq 0.$ (55)
In terms of these defined variables, the aforementioned dynamical equation in
$z^{\prime}$ becomes:
$z^{\prime}=\frac{\dot{z}}{H}=-3\beta z+\delta zx\sqrt{6}+\gamma
z(\Omega_{M}/\chi-s\ \xi)+\frac{3}{2}z(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}).$
(56)
The autonomous system of equations that describes the dynamics of the entire
system are:
$\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle-3x-\frac{3\lambda
y^{2}}{\sqrt{6}}-\frac{\delta\alpha_{1}3z}{\sqrt{6}}+\frac{3}{2}x(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}),$
(57) $\displaystyle y^{\prime}$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{6}yx\lambda}{2}+\frac{3}{2}y(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}),$
(58) $\displaystyle z^{\prime}$ $\displaystyle=$ $\displaystyle-3\beta
z+\delta zx\sqrt{6}+\gamma z(\Omega_{M}/\chi-s\
\xi)+\frac{3}{2}z(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}),$ (59)
$\displaystyle\chi^{\prime}$ $\displaystyle=$ $\displaystyle-\alpha_{1}\gamma
z-3\chi+\frac{3}{2}\chi(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}),$ (60)
$\displaystyle\xi^{\prime}$ $\displaystyle=$ $\displaystyle\frac{2\xi
s\alpha_{1}z\gamma}{3\chi}+\frac{3}{2}\xi(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}),$
(61) $\displaystyle s^{\prime}$ $\displaystyle=$
$\displaystyle\frac{s\alpha_{1}z\gamma}{\chi}.$ (62)
Here, we obtained the dynamics corresponding to the temperature parameter
using Eq. (33), where we evaluated $(\partial\rho/\partial T)_{n}$ by
considering the ideal gas model given in Eq. (34). Due to the non-adiabatic
particle production mechanism, the degrees of freedom to describe the dynamics
of the system increase to 6 dimensions. It’s important to note that although
we have assumed dark matter to behave as an ideal gas, with its energy density
given by Eq. (34), this alone cannot provide a constraint on either $\chi$ or
$\xi$. An additional variable is needed to capture the dynamics of $H$ to
close the autonomous system of equations. As a result, the dimension of the
phase space remains 6D. Hence, keeping $\chi$ and $\xi$ as dynamical variables
reduces the complexity of obtaining the critical points as no logarithmic
function will appear. The autonomous equations in
$y^{\prime},z^{\prime},\xi^{\prime},s^{\prime}$ are invariant under the
transformations $(y\mapsto-y,z\mapsto-z,\xi\mapsto-\xi,s\mapsto-s)$,
presenting an invariant sub-manifold at $y=z=\xi=s=0$. This implies that no
trajectory originating in $y,z,\xi,s\geq 0$ can cross the $y=z=\xi=s=0$ plane.
To constrain the model parameters and obtain the stability of the system, we
determine the critical points of the system by equating the right-hand side of
the autonomous equations to zero. The differential equation corresponding to
$s^{\prime}$ vanishes when either $s=0$ or $z=0$. As the coordinate $z=0$ does
not serve the non-minimal coupling scenario of the system, hence we will not
consider those fixed points where $z$ vanishes. On the other hand, the co-
ordinate $s=0,z\neq 0$ makes the entire system adiabatic. Therefore, we
initially study the nature of critical points corresponding to the system in a
restricted 5D phase space by fixing $s$ to a particular value, say $s\neq 0$.
The critical points corresponding to the autonomous system
$(x^{\prime}-\xi^{\prime})$ are:
* •
Point $P_{1}$: The co-ordinates of the point are
$\bigg{(}x=\frac{3\sqrt{6}(\beta-1)\gamma-\sqrt{2}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}}{2\gamma\delta(2s+3)},y=0,z=\frac{3\left(-9\beta\gamma+9\gamma+\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}\right)}{\alpha_{1}\gamma\delta^{2}(2s+3)^{2}},\chi=-\frac{-9\beta\gamma+9\gamma+\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}}{3\delta^{2}(2s+3)},\xi=\frac{\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}-6\beta\gamma
s+6\gamma s}{6\left(3\beta^{2}-6\beta-2\delta^{2}+3\right)}\bigg{)}$. As the
potential variable $y$ vanishes at this point, the point becomes dependent on
the field’s kinetic component. The effective equation of state at this point
is $\omega_{\rm eff}=\frac{2s-3}{2s+3}$, depending solely on entropy. For
$s=3/2$, the point corresponds to a non-accelerating matter solution, whereas
it acts as an accelerating point, i.e., $-1\leq\omega_{\rm eff}<-1/2$, for
$0\leq s<\frac{1}{2}$. The point can only be considered a viable fixed point
when the thermodynamical quantities $\xi,\chi$ yield positive values. For
$s\geq 3/2$, these conditions can’t be satisfied regardless of the choices of
other model parameters. This point exists for $\gamma>0,s=0.1$,
$\beta>0.82\sqrt{\delta^{2}}+1$, and for $\gamma<0,s=1/10$,
$\beta<1-0.82\sqrt{\delta^{2}}$. Although the existence of the point imposes
tight constraints on the model parameters, the constraint on $\lambda$ can be
imposed by determining the stability of the point. After linearizing the
autonomous systems Eq. (57)–Eq. (61) at the critical points up to first order,
we construct the Jacobian matrix $J_{ij}=\frac{dx_{i}}{dx_{j}}$. If the real
part of the eigenvalues are all positive (negative), the point becomes an
unstable (stable) point. For the alternate sign of the real part, the point
becomes a saddle point. Since we consider the autonomous equations Eq.
(57)–Eq. (61), the Jacobian becomes a $5\times 5$ dimensional matrix yielding
five eigenvalues. The real part of the eigenvalues corresponding to this point
has been plotted in Fig.[1], considering $\delta=2$ and $\gamma=1$. The real
part of the eigenvalues takes alternate signs, indicating that the point
becomes a saddle point. Apart from these model parameters, the point remains a
saddle point for any choice of $\delta$.
Figure 1: The stability of the point $P_{1}$ in the parameter space of
$(\beta,\lambda)$ for $(\alpha_{1}=1,s=0.1,\delta=2,\gamma=1)$.
* •
Point $P_{2}$: The co-ordinates of the point are
$\bigg{(}x=\frac{3\sqrt{3}(\beta-1)\gamma+\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}}{\sqrt{2}\gamma\delta(2s+3)},y=0,z=-\frac{3\left(9\beta\gamma-9\gamma+\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}\right)}{\alpha_{1}\gamma\delta^{2}(2s+3)^{2}},\chi=\frac{9\beta\gamma-9\gamma+\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}}{3\delta^{2}(2s+3)},\xi=-\frac{\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}+6(\beta-1)\gamma
s}{6\left(3\beta^{2}-6\beta-2\delta^{2}+3\right)}\bigg{)}$. Similar to the
aforementioned point, the potential variable $y$ vanishes at this point and
the effective EoS takes the same form $\omega_{\rm eff}=\frac{-3+2s}{3+2s}$.
Following the same argument as above i.e., ensuring $\xi$ and $\chi$ are
positive, the point becomes viable under the following conditions. Assuming
the entropy $s=0.1$, the conditions are satisfied for $\gamma>0$ and
$0.815\sqrt{\delta^{2}}+1\leq\beta<0.816\sqrt{\delta^{2}}+1$. This imposes a
stringent constraint on $\beta$ for positive $\gamma$. Similarly, for
$\gamma<0$, $1-0.816\sqrt{\delta^{2}}<\beta\leq 1-0.815\sqrt{\delta^{2}}$. For
$\delta\neq 0$ and $\lambda\neq 0$, the point always becomes a saddle point.
* •
Point $P_{3}$: The co-ordinates of the point are
$\bigg{(}x=-\frac{2\sqrt{6}s}{3\lambda+2\lambda
s},y=\frac{\sqrt{9\delta\lambda^{2}-4\delta\left(\lambda^{2}-6\right)s^{2}+36(\beta-1)\lambda
s}}{\sqrt{\lambda^{2}(2s+3)^{2}((\beta-1)\lambda+\delta)}},z=\frac{(2s-3)\left(3\lambda^{2}+2\left(\lambda^{2}-6\right)s\right)}{\alpha_{1}\lambda(2s+3)^{2}((\beta-1)\lambda+\delta)},\chi=-\frac{\gamma(2s-3)\left(3\lambda^{2}+2\left(\lambda^{2}-6\right)s\right)}{9\lambda(2s+3)((\beta-1)\lambda+\delta)},\xi=\frac{\gamma\lambda\left(9-4s^{2}\right)}{6(3(\beta-1)\lambda+2(\beta-1)\lambda
s+4\delta s)}\bigg{)}$. At this point, both the kinetic and potential
variables of the field are finite, and the effective equation of state becomes
$\omega_{\rm eff}=\frac{2s-3}{2s+3}$. To discuss the existence of the point,
we initially fix some values of the model parameters, considering $\beta=1$.
The point exists if
$\lambda>2\sqrt{3}\sqrt{\frac{s}{2s+3}},\delta>0,\gamma>0$, and the
corresponding stability has been determined for $s=0.01$ in
Fig.[LABEL:fig:stability_p3_beta1]. In the parameter space of
$(\delta,\gamma)$, the point becomes stable for any choice of $\gamma\neq 0$.
We also examine the existence and stability for other values of $\beta$,
particularly $s=0.1,\beta=2$. We find that the thermodynamic constraints Eq.
(55) are satisfied when $\gamma>0$, $\delta\geq
4.98,\lambda>0.61,\lambda<-\delta$, or for $\gamma>0$, $\delta\leq
4.98,\lambda<-0.61,\lambda>-\delta$. If $\gamma<0$, then $\delta>4.98$,
$-0.13\delta<\lambda<-0.61$, or $\delta<-4.98,0.61<\lambda<-0.13\delta$.
Hence, with these inputs, if we select the model parameters
$\gamma>0,\beta=2,s=0.1$, we determine that the critical point can only be a
stable fixed point when $\delta\geq 5,\lambda>2\delta+3$. It’s worth noting
that the value of $\alpha_{1}$ does not affect the dynamics in any way, since
it can always be absorbed in $z$. Similarly, for $\beta=3,s=0.01$, the
thermodynamic parameters become positive if
$\gamma>0,\lambda>0.19,\delta>-2\lambda$. The stability corresponding to this
range of parameters is shown in Fig.[LABEL:fig:stability_p3_beta3].
(a)
(b)
Figure 2: The stability of the point $P_{3}$ (a) for
$\beta=1,s=0.01,\alpha_{1}=1,\gamma=1$, (b) for
$\beta=3,s=0.01,\alpha_{1}=1,\gamma=1$.
As we have seen, the system yields some non-trivial critical points for $s\neq
0$, and among them, one critical point is both accelerating and stable,
depending on the choice of model parameters. Due to this characteristic, the
point signifies the late-time phase of the universe. Having determined the
ranges of model parameters that produce viable stable critical points, we will
now evaluate the dynamics of the entire system considering the autonomous
equations Eq. (57)–Eq. (62), which will depict the different phases of the
universe. As discussed earlier, taking the equation in $s^{\prime}$ yields
critical points when either $s=0$ or $z=0$. As the system is non-minimally
coupled with the fluid, one expects that during the matter and future epoch,
the variable $z$ must be non-vanishing. Hence, $s=0$ serves as the viable co-
ordinate of the critical points. The critical points corresponding to $s=0$
and the corresponding nature of the fixed points are tabulated in Tabs.[1, 2].
The critical points are given similar nomenclature as the aforementioned fixed
points.
s=0
---
Points | $x$ | $y$ | $z$ | $\chi$ | $\xi$
$P_{1\pm}$ | $\frac{\pm p+\sqrt{6}\beta-\sqrt{6}}{2\delta}$ 222$p=\sqrt{6\beta^{2}-12\beta-4\delta^{2}+6}$ | 0 | $\frac{\mp r-3\beta+3}{\text{$\alpha$1}\delta^{2}}$ 333 $r=\sqrt{9\beta^{2}-18\beta-6\delta^{2}+9}$ | $\pm\frac{\gamma\left(r+3\beta-3\right)}{3\delta^{2}}$ | 0
$P_{2\mp}$ | $\frac{\mp r+3\beta-3}{\sqrt{6}\delta}$ | 0 | $\frac{\pm r-3\beta+3}{\alpha_{1}\delta^{2}}$ | $\mp\frac{\gamma\left(r-3\beta+3\right)}{3\delta^{2}}$ | Any
$P_{3}$ | 0 | $\frac{\sqrt{\delta}}{\sqrt{(\beta-1)\lambda+\delta}}$ | $-\frac{\lambda}{\alpha_{1}((\beta-1)\lambda+\delta)}$ | $\frac{\gamma\lambda}{3((\beta-1)\lambda+\delta)}$ | Any
$P_{4\mp}$ | $\mp 1$ | 0 | 0 | 0 | 0
$P_{5}$ | $\frac{\sqrt{\frac{3}{2}}(\beta-1)}{\delta}$ | $\frac{\sqrt{\frac{\left(-3\beta^{2}+6\beta+2\delta^{2}-3\right)((\beta-1)\lambda+\delta)}{\delta}}}{\sqrt{2}\sqrt{\delta((\beta-1)\lambda+\delta)}}$ | $\frac{-3\beta-\delta\lambda+3}{\text{$\alpha$1}\delta^{2}}$ | $\frac{2\gamma(3\beta+\delta\lambda-3)}{3\delta((\beta-1)\lambda+2\delta)}$ | 0
Table 1: The critical points corresponding to autonomous system of equations
Eq. (57)–Eq. (62). s=0
---
Points | $\omega_{\rm eff}$ | $(\chi,\xi>0)$ | Stability
$P_{1\pm}$ | $-1$ | $\gamma>0,\beta\geq\sqrt{\frac{2}{3}}\sqrt{\delta^{2}}+1$,
$\gamma<0,\beta\leq 1-\sqrt{\frac{2}{3}}\sqrt{\delta^{2}}$ | Indeterminate
$P_{2\mp}$ | $-1$ | $\gamma>0,\beta\geq\sqrt{\frac{2}{3}}\sqrt{\delta^{2}}+1$,
$\gamma<0,\beta\leq 1-\sqrt{\frac{2}{3}}\sqrt{\delta^{2}}$ | Saddle
$P_{3}$ | $-1$ | $\gamma>0,\beta\lambda+\delta>\lambda,\lambda>0,\delta>0$ | see Figs.[LABEL:fig:stability_p3_beta1, LABEL:fig:stability_p3_beta3]
$P_{4\mp}$ | $\mp 1$ | Any | Unstable
$P_{5}$ | $-\frac{\beta\lambda+\delta-\lambda}{\delta}$ | $\beta\geq 1,\lambda>0,\delta>0,\gamma>0$ | Indeterminate
Table 2: The condition of existence of the fixed points and their nature.
The critical points $P_{1\pm}$ produce an accelerating solution; however, when
$\xi$ becomes zero, it leads to the temperature of the fluid reaching absolute
zero, which is not physically viable. Moreover, the indeterminacy of the
Jacobian further complicates the assessment of stability for this point. As
observed in the case when $s\neq 0$, the point cannot exhibit a stable
solution. Therefore, this point cannot be considered to describe the late-time
era of the universe.
The critical points $P_{2\mp}$ also exhibit accelerating solutions, and $\xi$
can take any positive value. However, considering the existence range given in
Tab.[2], for any values of $\beta$, the point becomes a saddle point. Thus,
these points are also not considered viable descriptions for the late-time
epoch of the universe.
The critical point $P_{3}$ can exhibit an accelerating solution where the
field’s kinetic part vanishes, and the corresponding stability for
$s\rightarrow 0$ shows similar behavior as discussed previously in
Figs.[LABEL:fig:stability_p3_beta1, LABEL:fig:stability_p3_beta3]. We have
also noted that for $s=0$, one of the eigenvalues becomes zero, rendering the
standard linearization technique inapplicable for determining the stability of
the point. However, we can employ a simple numerical trick to determine the
stability even without resorting to the center manifold theorem. Since the
system is non-adiabatically coupled, at any instant of time, the entropy of
the dark matter fluid must be positive or zero. Hence, we numerically evolved
the system by choosing initial conditions near zero, i.e., $s>0$, and observed
that the system stabilizes at $P_{3}$ with $s=0$ in the far future, as
demonstrated in Fig.[LABEL:fig:coord_p3_beta3]. Consequently, this point is a
physically viable fixed point for describing the late-time phase of the
universe.
The other critical points $P_{4\mp}$ rely solely on the field’s kinetic
component, while the co-ordinate $z$ vanishes. This leads to the autonomous
equation in $s^{\prime}$ becoming zero, i.e., $s^{\prime}=0$, without
requiring $s=0$. Consequently, the variable $s$ can take any value. The
effective equation of state at this point is $1$, indicating a stiff matter
solution, which reflects the minimally coupled behavior of quintessence in the
far past epoch. Since $s$ can take any value, this suggests that during the
past epoch, the system was non-adiabatically coupled.
At critical point $P_{5}$, all the field components are finite, but the
parameter corresponding to the temperature vanishes. For $\beta=1$, the
effective equation of state becomes $-1$. If $\beta>1$, then
$0<\lambda<-\frac{\delta}{2\beta-2}$. However, since $\xi$ vanishes for any
choice of model parameters and the Jacobian becomes indeterminate, as
discussed earlier, this point cannot be considered physically relevant.
(a)
(b)
Figure 3: The numerical evolution of cosmological parameters for
$(\gamma=1,\beta=3,\alpha_{1}=1,\delta=2,\lambda=0.5)$ with the initial
conditions $x(0)=0.02,y(0)=0.7,z(0)=-0.3,\chi(0)=0.03,\xi(0)=0.009,s(0)=0.15$.
After carefully studying the nature of critical points for $s\neq 0$ and
$s=0$, we can conclude that the physically viable critical point that
describes the behavior of the late-time epoch of the universe is $P_{3}$.
Therefore, to analyze the system’s overall dynamics, we numerically simulate
the autonomous equations Eq. (57)–Eq. (62), by selecting the benchmark points
$\beta=3$, $\delta=2$, $\lambda=0.5$, $\alpha_{1}=1$. Corresponding to these
benchmark points, the coordinates of the point $P_{3}$ become:
$(x=0,y=0.82,z=-0.17,\chi=0.06,\xi=\xi)$, and the corresponding field and
fluid fractional densities become $\Omega_{\phi}=0.67$, $\Omega_{M}=0.5$. The
numerical evolution of the cosmological parameters has been demonstrated in
Fig.[LABEL:fig:evo_p3_beta3] against $N=\ln a$, ranging from the past epoch to
the future epoch of the universe.
At the very past epoch of the universe, the field’s kinetic component
dominates, showing the behavior described by points $P_{4\mp}$. At this epoch,
the effective equation of state and the field’s equation of state
$\omega_{\phi}\equiv\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$ both become $1$. As the
universe enters the matter-dominated phase, the effective equation of state
becomes zero, and it continuously increases towards negative values. However,
a steeper fall can be observed in the behavior of $\omega_{\phi}$. Unlike the
standard model of cosmology ($\Lambda$CDM) or other versions of minimally
coupled quintessence field with dark matter fluid, as the system enters the
current epoch ($a=1$ or $N=0$), the dark energy dominates over the fluid
component and the effective equation of state becomes $-1$. However, in the
current scenario, the fluid energy density dominates over the field density,
forcing the effective equation of state to cross the phantom barrier ($-1$)
for a brief period of time. This is the consequence of non-adiabatic energy
transfer occurring between the field and the fluid.
As the system further evolves into the future epoch, the model stabilizes to
$\omega_{\rm eff}=-1$ with a dominating field energy density. However, because
of the interaction, the fluid energy density does not get diluted. At any
instant of time, the fluid and field energy densities can take values greater
than $1$, however, the total energy density of the system $\Omega_{\rm
tot}=\Omega_{M}+\Omega_{\phi}+\alpha_{1}z$ always remains $1$ as shown by the
gray dashed line parallel to the $x$-axis. One way to interpret this behavior
is as discussed in ref.[64], where it is noted that it’s nearly impossible to
distinguish between dark sectors and gravity; therefore, they are degenerate.
Alongside the cosmological parameters, we have also shown the evolution of
dynamical variables in Fig.[LABEL:fig:coord_p3_beta3], and found that the
thermodynamic variables throughout the evolution follows the constraint Eq.
(55).
In addition to stability, we have also assessed the viability of the model by
comparing it with the $\Lambda$CDM model using available observational data
from Hubble $(43)$ and Pantheon+ $(1701)$ (url) datasets [65, 66]. Utilizing
the relation between the scale factor $a$ and redshift $\tilde{z}$,
$a=1/(1+\tilde{z})$, we can derive the evolution of the Hubble parameter as
follows:
$\frac{dH}{d\tilde{z}}=\frac{3}{2}\frac{1}{(1+\tilde{z})}H(\tilde{z})(1+\omega_{\rm
eff}).$ (63)
The Hubble parameter has been fitted with observational Hubble dataset (OHD)
alongside the $\Lambda$CDM model, using dark matter density $\Omega_{m}=0.25$,
cosmological constant fractional density $\Omega_{\Lambda}=0.75$, and
$H_{0}=67$ (in units of km/s/Mpc), as shown in Fig.[LABEL:fig:hubble_beta3].
The current model has been simulated using the same benchmark point and
initial conditions as shown in Fig.[LABEL:fig:evo_p3_beta3], with $H_{0}=66$,
to better match observations. The non-linear evolution of $H(\tilde{z})$ near
$\tilde{z}\rightarrow 0,1.5$ shows a significant deviation from the standard
model. By testing the model with larger datasets using sampling techniques
like MCMC, it may contribute to lowering the discrepancy in the Hubble value.
To fit the model with the Pantheon+ data, we utilize the distance modulus
$\mu$ relation with redshift,
$\mu=5\log_{10}(d_{L})+25,$ (64)
where the luminosity distance is given by,
$d_{L}=2.99\times
10^{5}(1+\tilde{z})\int_{0}^{\tilde{z}}\frac{1}{H(\tilde{z})}d\tilde{z}\ ,$
(65)
The evolution of $\mu$ for the $\Lambda$CDM model overlaps with the current
model, as shown in Fig.[LABEL:fig:supernova_dist_1]. It’s noteworthy that we
have selected the model parameter $\beta=3$ because the evolution of the
Hubble parameter closely resembles that of $\Lambda$CDM for this case.
(a)
(b)
Figure 4: (a) The Hubble evolution and (b) the distance modulus corresponding
to $\Lambda$CDM is evaluated with
$\Omega_{M}=0.25,\Omega_{\Lambda}=0.75,H(0)=67.0$. The current model has been
fitted for $(\gamma=1,\beta=3,\alpha_{1}=1,\delta=2,\lambda=0.5)$ with the
initial conditions
$x(0)=0.02,y(0)=0.7,z(0)=-0.3,\chi(0)=0.03,\xi(0)=0.009,s(0)=0.15,H(0)=66.0$.
## VI Conclusion
The study delves into a non-minimal coupling scenario between the quintessence
field and a pressureless fluid, examined thoroughly through the variational
principle. By incorporating one of the Lagrange parameters $\varphi$ into the
interacting Lagrangian, we altered the conservation laws governing the fluid’s
number density and entropy density. Consequently, the fluid sector introduces
an additional pressure termed creation pressure. Upon evaluating the covariant
derivative of the energy-momentum tensor, it becomes apparent that individual
components are no longer conserved, facilitating the flow of energy from the
field to the fluid through the field derivative of the interaction function
$f_{,\phi}\dot{\phi}$.
We assessed the background stability of the model by considering an
exponential type of interaction. The system yielded a stable critical point
capable of producing an accelerating solution in the far future epoch of the
universe, dominated by the scalar field energy density with a finite fluid
energy density. Furthermore, numerical simulations corresponding to the
interaction $f\propto\rho^{3}$ revealed that during the current epoch, the
fluid density predominates, resulting in a phantom equation of state for a
brief period.
To evaluate the viability of the model, we conducted simulations with the
Hubble and Pantheon+ data sets alongside the $\Lambda$CDM model. The model
demonstrates the ability to mimic $\Lambda$CDM behavior while simultaneously
exhibiting deviations. In future work, we intend to conduct a comprehensive
statistical analysis to further assess the model’s viability with various data
sets.
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|
# Angular Momentum of Phonons and Einstein-de Haas Effect
Lifa Zhang Department of Physics, The University of Texas at Austin, Austin,
Texas 78712, USA Qian Niu Department of Physics, The University of Texas at
Austin, Austin, Texas 78712, USA International Center for Quantum Materials,
Peking University, Beijing 100871, China
(27 Jan 2014)
###### Abstract
We study angular momentum of phonons in a magnetic crystal. In the presence of
a spin-phonon interaction, we obtain a nonzero angular momentum of phonons,
which is an odd function of magnetization. At zero temperature, phonon has a
zero-point angular momentum besides a zero-point energy. With increasing
temperature, the total phonon angular momentum diminishes and approaches to
zero in the classical limit. The nonzero phonon angular momentum can have a
significant impact on the Einstein-de Haas effect. To obtain the change of
angular momentum of electrons, the change of phonon angular momentum needs to
be subtracted from the opposite change of lattice angular momentum.
Furthermore, the finding of phonon angular momentum gives a potential method
to study the spin-phonon interaction. Possible experiments on phonon angular
momentum are also discussed.
###### pacs:
63.20.-e 63.20.kk 75.70.Ak
The Einstein-de Haas effect einde ; frenkel79 , a phenomenon of mechanical
rotation induced by a magnetization change, was originally designed to prove
the existence of Ampere’s molecular currents; but subsequent experiments
stewart18 showed that the magnetic moment of an atom is dominated by spin
while contribution from orbital motion to the magnetic moment is almost
absent. The Einstein-de Haas experiment together with the Barnett experiment
barn15 ; barn35 (a change of magnetization resulting from a mechanical
rotation) has provided an effective method of measuring the gyromagnetic ratio
for various materials davi53 ; kitt49 ; reck69 . The accuracy of gyromagnetic
ratio is crucial to determining of orbital and spin contribution in total
magnetization stoh95 ; ulme04 ; lidb09 ; boeg10 ; pere11 ; niem12 ; dzya11 .
Due to conservation of total angular momentum of the whole system in the
Einstein-de Haas effect, the change of angular momentum of electrons
(including both spin and orbital parts) has taken to be equal in magnitude but
opposite in sign to the change of lattice angular momentum, which corresponds
to mechanical rotation. However, the mechanical rotation only reflects angular
momentum of the rigid-body lattice where atoms are assumed in the
corresponding equilibrium positions; while phonons, which come from atomic
vibrations around equilibrium positions, are assumed to have no macroscopic
angular momentum. Recently, a remarkable phenomenon of phonon Hall effect was
observed in a paramagnetic insulator phee1 ; phee2 , which is indeed a
surprise since phonons as neutral quasi-particles cannot directly couple to
magnetic field via Lorentz force. The following theoretical studies phet1 ;
phet2 showed that through Raman spin-phonon interaction the magnetic field
can have an effective force to distort phonon transport, and thus drive a
circulating heat flow qin11 . Therefore a natural question arises: can such
circulating phonons have nontrivial angular momentum and emergent macroscopic
effects?
In this Letter, we study angular momentum of phonons in a magnetic crystal in
a microscopic picture. It is found that the Raman spin-phonon interaction
induces a nonzero phonon angular momentum, which is an odd function of
magnetization. In addition to a zero-point energy, phonon has a zero-point
angular momentum at zero temperature. Such zero-point phonon angular momentum
is offset by that of excited phonon modes such that the total angular momentum
of phonons vanishes in the classical limit. Phonon angular momentum can not be
ignored in total angular momentum especially in magnetic materials with large
magnetization and spin-phonon interaction. Revisiting the Einstein-de Haas
effect we find that phonon angular momentum needs to be subtracted in
calculating angular momentum of electrons. With this correction, spin and
orbital angular momentum can be precisely determined. Besides the Einstein-de
Haas effect, nontrivial phonon angular momentum can be applied to the study of
spin-phonon interaction, thermal Hall effect, and other topics related to
phonons.
Angular momentum of phonons – The lattice angular momentum related to
mechanical rotation only reflects the rigid-body motion of the lattice.
However, angular momentum of phonons has never been considered. In a
microscopic picture, we can define angular momentum of phonons as
$\bm{J}^{\rm ph}=\sum_{l\alpha}\bm{u}_{l\alpha}\times\dot{\bm{u}}_{l\alpha}.$
(1)
Here $\bm{u}_{l\alpha}$ is a displacement vector of the $\alpha$-th atom in
the $l$-th unit cell, multiplied by square root of mass. Along $z$ direction,
$J^{\rm
ph}_{z}=\sum_{l\alpha}(u_{l\alpha}^{x}\dot{u}_{l\alpha}^{y}-u_{l\alpha}^{y}\dot{u}_{l\alpha}^{x})$.
One can present the displacement in the second quantization form as
$u_{l}=\sum_{k}\epsilon_{k}e^{i({\bf
R}_{l}\cdot{\bm{k}}-\omega_{k}t)}\sqrt{\frac{\hbar}{2\omega_{k}N}}\;a_{k}+{\rm
h.c.}$, with $k=(\bm{k},\sigma)$ specifying a wave vector $\bm{k}$ and a
branch $\sigma$, where $\epsilon_{k}$ is a displacement polarization vector.
Then the phonon angular momentum can be written as supp
$\displaystyle J_{z}^{\rm ph}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\sum_{k,k^{\prime}}\epsilon_{k}^{\dagger}M\epsilon_{k^{\prime}}\left(\sqrt{\frac{\omega_{k}}{\omega_{k^{\prime}}}}+\sqrt{\frac{\omega_{k^{\prime}}}{\omega_{k}}}\right)a_{k}^{\dagger}a_{k^{\prime}}\delta_{{\bm{k}},{\bm{k}}^{\prime}}e^{i(\omega_{k}-\omega_{k^{\prime}})t}$
(2)
$\displaystyle+\frac{\hbar}{2}\sum_{k}\epsilon_{k}^{\dagger}M\epsilon_{k}.$
Here $M=\left(\begin{smallmatrix}0&-i\\\ i&0\end{smallmatrix}\right)\otimes
I_{n\times n}$, and $n$ is the number of atoms in one unit cell. In
equilibrium, the angular momentum of phonons reduces to supp :
$J^{\rm
ph}_{z}=\sum_{\sigma,\bm{k}}l_{\bm{k},\sigma}^{z}\,[f(\omega_{\bm{k},\sigma})+\frac{1}{2}],\;\;l_{\bm{k},\sigma}^{z}=(\epsilon^{\dagger}_{\bm{k},\sigma}\,M\,\epsilon_{\bm{k},\sigma})\hbar,$
(3)
where $f(\omega_{k})=\frac{1}{e^{\hbar\omega_{k}/k_{B}T}-1}$ is the Bose-
Einstein distribution. In Eq.(S60), we do summation over all wave vector
points and all phonon branches ($\omega\geq 0$). Here $l_{\bm{k},\sigma}^{z}$
is the phonon angular momentum of branch $\sigma$ at wave vector $\bm{k}$,
which is real and proportional to $\hbar$. At zero temperature, the total
phonon angular momentum is $J^{\rm
ph}_{z}(T=0)=\sum_{\sigma,\bm{k}}\frac{1}{2}\,l_{\bm{k},\sigma}^{z}$, which
means that each mode of $(\bm{k},\sigma)$ has a zero-point angular momentum
$\frac{1}{2}l_{\bm{k},\sigma}^{z}=\frac{\hbar}{2}(\epsilon^{\dagger}_{\bm{k},\sigma}\,M\,\epsilon_{\bm{k},\sigma})$
besides a zero-point energy of $\hbar\omega_{\bm{k},\sigma}/2$.
For an ionic crystal lattice in a uniform external magnetic field, the
Hamiltonian reads in a compact form phet1 ; phet2 ; holz72 ; wang09 :
$H=\frac{1}{2}(p-{\tilde{A}}u)^{T}(p-{\tilde{A}}u)+\frac{1}{2}u^{T}Ku,$ (4)
where $u$ is a column vector of displacements from lattice equilibrium
positions, multiplied by square root of mass; $p$ is a conjugate momentum
vector, and $K$ is a force constant matrix. The cross term $u^{T}{\tilde{A}}p$
can be interpreted as a Raman spin-phonon interaction ray67 ; iose95 . The
superscript $T$ stands for the matrix transpose. ${\tilde{A}}$, an
antisymmetric real matrix qin12 , has a dimension of $N\,d\times N\,d$ where
$N$ is the number of total sites and $d$ is the dimension of lattice
vibrations; in a proper approximation it can be block diagonal with elements
$\Lambda_{\alpha}=\left(\begin{smallmatrix}0&\lambda_{\alpha}\\\
-\lambda_{\alpha}&0\\\ \end{smallmatrix}\right)$ with respect to the
$\alpha$-th ionic site, where we only consider two-dimensional ($x$ and $y$
directions) motion of the lattice ($d=2$). Here $\lambda_{\alpha}$ has a
dimension of frequency, and is proportional to the spin-phonon interaction and
magnetization, which is assumed to be proportional to magnetic field for a
paramagnetic material. The magnetic field is applied along $z$ direction. The
polarization vector $\epsilon$ satisfies
$\bigl{[}(-i\omega+A)^{2}+D\bigr{]}\epsilon=0,$ where $D({\bf
k})=-A^{2}+\sum_{l^{\prime}}K_{ll^{\prime}}e^{i({\bf R}_{l^{\prime}}-{\bf
R}_{l})\cdot{\bf k}}$ is the dynamic matrix and $A$ is block diagonal with the
element of $\Lambda_{\alpha}$, and has a dimension of $2n\times 2n$ where $n$
is the number of sites per unit cell.
In absence of spin-phonon interaction, the system reduces to a trivial phonon
system $H=\frac{1}{2}p^{T}p+\frac{1}{2}u^{T}Ku$. Solving the simple eigenvalue
problem as
$D(\bm{k})\epsilon_{\bm{k},\sigma}=\omega_{\bm{k},\sigma}^{2}\epsilon_{\bm{k},\sigma}$
with $D^{T}({\bf k})=D^{*}({\bf k})=D(-{\bf k})$, one can have
$\omega_{-\bm{k},\sigma}=\omega_{\bm{k},\sigma},\epsilon_{-\bm{k},\sigma}=\epsilon_{\bm{k},\sigma}^{*}$,
then we obtain $l_{-\bm{k},\sigma}^{z}=-l_{\bm{k},\sigma}^{z}$ and $J^{\rm
ph}_{z}=0$ supp . Thus for a phonon system without a spin-phonon interaction,
the total angular momentum of phonons is zero.
Figure 1: (Color online) (a) The phonon angular momentum $J_{z}^{\rm ph}$ of
one unit cell as a function of $\lambda$ at temperature $T=0$ K for different
lattice symmetries. (b) The contour plot of the phonon angular momentum
$J_{z}^{\rm ph}$ of one unit cell as a function of $\lambda$ and temperature
$T$. (c) The phonon angular momentum $J_{z}^{\rm ph}$ of one unit cell from
different phonon bands as a function of temperature $T$ at $\lambda=$ 1 THz,
where the arrow denotes the Debye temperature of the model ($T_{D}=358$ K).
(d) The phonon angular momentum $J_{z}^{\rm ph}$ of one unit cell from
different phonon bands as a function of $\lambda$ at $T=0$ K. The phonon
angular momenta in (b)-(d) are calculated for a honeycomb lattice. All the
phonon angular momenta are in the unit of $\hbar$.
For a phonon system with a spin-phonon interaction,
$\epsilon_{-\bm{k},\sigma}=\epsilon_{\bm{k},-\sigma}^{*}\neq\epsilon_{\bm{k},\sigma}^{*}$,
and then $l_{-\bm{k},\sigma}^{z}\neq-l_{\bm{k},\sigma}^{z}$, thus one can get
a nonzero phonon angular momentum, which is shown in Fig. 1. We calculate
phonon angular momentum for lattices with the following parameters: the
longitudinal spring constant is $K_{L}=0.144\,$eV/(uÅ2) and the transverse one
is $K_{T}=K_{L}/4$; the unit cell lattice vectors are $(a,0),\,(0,a)$ for a
square lattice and $(a,0),\,(a/2,a\sqrt{3}/2)$ for other lattices with
$a=1\,$Å. We take $\lambda_{\alpha}=\lambda$ for the model calculation
comment2 . Figure 1(a) shows that honeycomb and kagome lattices have larger
phonon angular momenta than those of triangle and square lattices, which means
that lattices with more sites per unit cell can have a larger phonon angular
momentum. We can understand this trend by observing that optical bands are
more important in contributing to the phonon angular momentum than the
acoustic ones. In Fig. 1(c) and (d) we plot the phonon angular momentum
contributing from different bands in a honeycomb lattice. It is shown that the
phonon angular momentum from acoustic bands almost vanishes at low
temperatures (see Fig. 1(c)) and if $\lambda$ is not large (see Fig. 1(d)),
thus the optical bands dominate the contribution to the total phonon angular
momentum. With more sites per unit cell more optical bands present, thus
phonon angular momentum will be larger.
By using the relations
$\epsilon_{-\bm{k},\sigma}^{*}(-A)=\epsilon_{\bm{k},\sigma}(A)$,
$\omega_{-\bm{k},\sigma}(-A)=\omega_{\bm{k},\sigma}(A)$, $M^{T}=-M$, we can
obtain $J^{\rm ph}_{z}(-\lambda)=-J^{\rm ph}_{z}(\lambda)$. Since $\lambda$ is
proportional to magnetization, the total angular momentum of phonon will
change sign when magnetization changes sign. As shown in Fig. 1(a), (b) and
(d), the total angular momentum of phonons per unit cell increases as
$\lambda$ increases; but the increase rate will decrease.
Angular Momentum in the Classical Limit – At the high temperature limit, from
Eq. (S60) we have supp :
$J^{\rm
ph}_{z}(T\rightarrow\infty)=\sum_{\sigma>0,\bm{k}}[(\frac{k_{B}T}{\hbar\omega_{\bm{k},\sigma}}+\frac{\hbar\omega_{\bm{k},\sigma}}{12k_{B}T})l_{\bm{k},\sigma}^{z}].$
(5)
It seems that the phonon angular momentum would be linear with temperature at
the high temperature limit. However, the first term vanishes due to the fact
of
$\sum_{\sigma>0,\bm{k}}\frac{\epsilon^{\dagger}_{\bm{k},\sigma}\,M\,\epsilon_{\bm{k},\sigma}}{\omega_{\bm{k},\sigma}}=0$
supp . Therefore at a high temperature total phonon angular momentum is
proportional to $1/T$ and tends to zero as
$J^{\rm
ph}_{z}(T\rightarrow\infty)=\sum_{\sigma>0,\bm{k}}\frac{\hbar\omega_{\bm{k},\sigma}}{12k_{B}T}l_{\bm{k},\sigma}^{z}\rightarrow
0.$ (6)
The phonon angular momentum per unit cell changing with temperature is shown
in Fig. 1(b) and (c). Whatever a magnetic field is applied, the phonon angular
momentum per unit cell decreases with increasing temperature and tends to zero
at the high temperature limit ($T\gg T_{D}$). With increasing temperature more
modes are exited, the angular momentum of which has the direction opposite to
that of zero-point angular momentum; at the high temperature limit, the phonon
angular momentum of all the excited modes exactly cancels out the zero-point
angular momentum ($\sum
l_{\bm{k},\sigma}^{z}\,f(\omega_{\bm{k},\sigma},T\rightarrow\infty)=-\sum\frac{1}{2}\,l_{\bm{k},\sigma}^{z}$
). We can understand the absent phonon angular momentum in the classical limit
as follows. At high temperatures, classical statistical mechanics is
applicable to calculate phonon angular momentum. Summation over quantum states
becomes a phase-space integral with respect to $p$ and $u$. One can do a
change of variable to make the kinetic energy in the Hamiltonian Eq. (4) into
a usual form $p^{2}/2$, thus removing the effect of ${\tilde{A}}u$; for such a
pure harmonic system, the angular momentum of phonons is zero as discussed
above. Furthermore, the Bohr van Leeuwen theorem states that in classical
mechanics the thermal average of the magnetization is always zero bvl , which
also makes the angular momentum of phonons vanish at the classical limit.
Therefore, the phonon angular momentum is meaningful only in low-temperature
quantum systems.
Revisit the Einstein-de Haas Effect – The Einstein-de Haas effect einde
showed a mechanical rotation of a freely suspended body caused by the change
in its magnetization. In their experiment einde , Einstein and de Haas
employed a resonance method in which the magnetic field was periodic and tuned
to be the natural frequency of the rod and its suspension, which provided
measurements for the ratio between the change in magnetization and the one in
total angular momentum. Traditionally the total angular momentum is assumed as
$\bm{J}^{\rm tot}=\bm{J}^{\rm lat}+\bm{J}^{\rm spin}+\bm{J}^{\rm orb}$, thus
due to conservation of angular momentum, one obtains $\Delta\bm{J}^{\rm
lat}=-(\Delta\bm{J}^{\rm spin}+\Delta\bm{J}^{\rm orb})$ which is determined by
the mechanical rotation of the sample kitt49 . However, from a microscopic
point of view, the angular momentum of all atoms in the sample can be written
as
$\bm{J}^{\rm
atom}=\sum_{l\alpha}(\bm{R}_{l\alpha}+\bm{u}_{l\alpha})\times(\dot{\bm{R}}_{l\alpha}+\dot{\bm{u}}_{l\alpha}),$
(7)
where $\bm{R}_{l\alpha}$ is the equilibrium position of the $\alpha$-th atom
in the $l$-th unit cell, multiplied by square root of its mass. The angular
momentum of lattice is
$\bm{J}^{\rm lat}=\sum_{l\alpha}\bm{R}_{l\alpha}\times\dot{\bm{R}}_{l\alpha},$
(8)
which really reflects the mechanical rotation of rigid-body motion of the
sample. In equilibrium, the cross terms related with $\bm{u}$ or
$\dot{\bm{u}}$ are zero, then $\bm{J}^{\rm atom}=\bm{J}^{\rm lat}+\bm{J}^{\rm
ph}$. Thus the total angular momentum should be
$\bm{J}^{\rm tot}=\bm{J}^{\rm lat}+\bm{J}^{\rm ph}+\bm{J}^{\rm
spin}+\bm{J}^{\rm orb}.$ (9)
The global conservation of angular momentum does not explain how the angular
momentum is actually transferred from individual electrons or atoms to the
whole rigid body; the Raman type spin-phonon interaction can be ubiquitous and
plays an essential role. According to the discussion in the above section, we
know that in the presence of the spin-phonon interaction, the phonon band
structure is nontrivial and gives nonzero angular momentum $\bm{J}^{\rm ph}$.
Based on conservation of total angular momentum we obtain
$\Delta\bm{J}^{\rm spin}+\Delta\bm{J}^{\rm orb}=-\Delta\bm{J}^{\rm
lat}-\Delta\bm{J}^{\rm ph}.$ (10)
Therefore to obtain the change of angular momentum of electrons, one needs to
subtract the contribution of phonon from the opposite change of lattice
angular momentum. On the other hand one can measure the total magnetization
change as
$\Delta M=\Delta M^{\rm spin}+\Delta M^{\rm orb}.$ (11)
Combining Eq. (10), Eq. (11) together with the facts of $\Delta M^{\rm
orb}=\frac{e}{2m}\Delta J^{\rm orb}$ and $\Delta M^{\rm
spin}=\frac{e}{m}\Delta J^{\rm spin}$, one can easily determine $\Delta M^{\rm
spin}$ and $\Delta M^{\rm orbit}$.
The phonon can make a significant contribution to total angular momentum,
while the magnitude of phonon angular momentum depends on the value of
$\lambda$. The parameter $\lambda$ can be obtained from phonon dispersion
relation since our calculation shows that in the presence of spin-phonon
interaction degenerate phonon modes split at $\Gamma$ point with a gap of
$2\lambda$. By means of Raman scattering experiments, literatures split1 ;
split2 show that the phonon splitting ranges up to about 26 cm-1 in
paramagnetic CeF3 at $T=1.9$ K and $B=6$ T, thus $\lambda$ can be about $0.39$
THz and phonon angular momentum per unit cell is about 0.02 $\hbar$. One also
can estimate the parameter $\lambda$ from phonon Hall effect. For a
paramagnetic terbium gallium garnet Tb3Ga5O12, the parameter $\lambda$ is
estimated as $\lambda=0.1$ cm${}^{-1}\simeq 3$ GHz at $B=1$ T and $T=5.45$ K
phet1 , thus in such material phonon angular momentum per unit cell is about
$1.6\times 10^{-4}\hbar$, which is relatively small. However, one can observe
a much larger phonon angular momentum when magnetization is saturated in this
paramagnetic material since the parameter $\lambda$ is proportional to
magnetization. In the phonon Hall effect experiment phee1 ; phee2 , the
paramagnetic insulator was chosen to manifest the phonon contribution in the
thermal transport where the contribution from electron and magnon can be
neglected. However, the spin-phonon interaction is widely present in various
magnetic materials spimat1 ; spimat2 ; spimat3 ; spimat4 . Ferromagnetic
materials have very large magnetization, thus one can expect a large phonon
angular momentum. One also can observe evident phonon angular momentum in
materials with strong spin-phonon interaction by using Raman spectroscopy,
such as La2NiMnO6 iliev07 , Sr2CoO4 pand13 and cupric oxide chen95 .
Thus for materials with strong spin-phonon interaction together with large
magnetization, the zero-point angular momentum of phonons can be significant.
According to previous studies, in some ferromagnetic materials the calculated
orbital magnetic moment is only a few percent of the total magnetic moment,
that is, the orbital angular momentum is also around a few percent of $\hbar$
reck69 , thus the phonon angular momentum can not be ignored. With improvement
of experimental technique in past decades, accuracy of measurement has been
much enhanced thus phonon angular momentum should be measurable.
Possible experiment to separate phonon angular momentum – One can do
experiments on a ferromagnetic insulator with saturation magnetization, where
electron transport can be ignored. Due to the property of phonon angular
momentum – it decreases with increasing temperature and vanishes in the
classical limit, one can measure the change of lattice angular momentum at low
and high temperatures to separate the phonon angular momentum from the others.
Here the temperature scale should be the Debye temperature which divides the
quantum and classical regions. On the other hand, in order to avoid the
involvement of magnons, we need to do experiments at temperatures low compared
to the Curie temperature. This demands that the Curie temperature must be much
higher than the Debye temperature. Thus the angular momentum of magnons almost
keeps constant while that of phonons changes dramatically with changing
temperature. Fortunately, this can be satisfied by many ferromagnetic
materials where their Curie temperature are around 1000 K while their Debye
temperatures are less than 500 K kitt04 .
Besides application to measurement of gyromagnetic ratio, nontrivial phonon
angular momentum provides us a possible efficient route to study spin-phonon
interaction in magnetic materials. On the other hand, to separate the
contribution from phonons and magnons to thermal Hall effect in ferromagnetic
materials is an open problem, phonon angular momentum can give a way to obtain
the phonon contribution.
Acknowledgements – We thank Junren Shi, Yang Gao, Zhenhua Qiao, Xiao Li, Ran
Cheng for helpful discussions. We acknowledge support from DOE-DMSE (DE-
FG03-02ER45958), NBRPC (2012CB-921300), NSFC (91121004), and the Welch
Foundation (F-1255).
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## I Supplementary information for “Angular Momentum of Phonons and the
Einstein-de Hass Effect”
### I.1 Derivation of Eq.(2) and (3)
The angular momentum of phonons is
$\displaystyle J_{z}^{{\rm{ph}}}$ $\displaystyle=$
$\displaystyle\sum\limits_{l\alpha}{(u_{l\alpha}^{x}\dot{u}_{l\alpha}^{y}-u_{l\alpha}^{y}\dot{u}_{l\alpha}^{x})}$
(S7) $\displaystyle=$
$\displaystyle\sum\limits_{l\alpha}{\left({\begin{array}[]{*{20}c}{u_{l\alpha}^{x}}\\\
{u_{l\alpha}^{y}}\\\
\end{array}}\right)^{T}\left({\begin{array}[]{*{20}c}0&1\\\ {-1}&0\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{\dot{u}_{l\alpha}^{x}}\\\
{\dot{u}_{l\alpha}^{y}}\\\ \end{array}}\right).}$
For a unit cell with two atoms $n=2$, that is, $\alpha=1,2$, the angular
momentum of phonons can be written as
$J_{z}^{{\rm{ph}}}\;=\sum\limits_{l}{\left({\begin{array}[]{*{20}c}{u_{l1}^{x}}\\\
{u_{l1}^{y}}\\\ {u_{l2}^{x}}\\\ {u_{l2}^{y}}\\\
\end{array}}\right)^{T}\left({\begin{array}[]{*{20}c}0&1&{}\hfil&{}\hfil\\\
{-1}&0&{}\hfil&{}\hfil\\\ {}\hfil&{}\hfil&0&1\\\ {}\hfil&{}\hfil&{-1}&0\\\
\end{array}}\right)}\left({\begin{array}[]{*{20}c}{\dot{u}_{l1}^{x}}\\\
{\dot{u}_{l1}^{y}}\\\ {\dot{u}_{l2}^{x}}\\\ {\dot{u}_{l2}^{y}}\\\
\end{array}}\right).$ (S9)
Using the
$u_{l}=\left(\begin{smallmatrix}u_{l1}^{x}\;u_{l1}^{y}\;u_{l2}^{x}\;u_{l2}^{y}\end{smallmatrix}\right)^{T}$,
we obtain
$J_{z}^{{\rm{ph}}}=\sum\limits_{l}u_{l}^{T}iM{\dot{u}_{l}}.$ (S10)
where $M=\left(\begin{smallmatrix}0&-i\\\ i&0\end{smallmatrix}\right)\otimes
I_{n\times n}$. By using the second quantization for $u_{l}$ as
$u_{l}=\sum_{k}\epsilon_{k}e^{i({\bf
R}_{l}\cdot{\bm{k}}-\omega_{k}t)}\sqrt{\frac{\hbar}{2\omega_{k}N}}\;a_{k}+{\rm
h.c.},$
we obtain
$J_{z}^{{\rm{ph}}}=\frac{\hbar}{2N}\sum_{l}\sum_{k,k^{\prime}}\left(\sqrt{\frac{\omega_{k}}{\omega_{k^{\prime}}}}\epsilon_{k}^{\dagger}M\epsilon_{k^{\prime}}a_{k}^{\dagger}a_{k^{\prime}}+\sqrt{\frac{\omega_{k^{\prime}}}{\omega_{k}}}\epsilon_{k^{\prime}}^{T}(-M)\epsilon_{k}^{*}a_{k^{\prime}}a_{k}^{\dagger}\right)e^{i(\bm{k}^{\prime}-\bm{k}){\bm{R}_{l}}}e^{i(\omega_{k}-\omega_{k^{\prime}})t}.$
(S11)
Here we ignore the $a\,a$ and $a^{\dagger}\,a^{\dagger}$ terms since they vary
rapidly with time and have no contribution in equilibrium.
Since
$\epsilon_{k^{\prime}}^{T}(-M)\epsilon_{k}^{*}=\epsilon_{k}^{\dagger}M\epsilon_{k^{\prime}}$
and
$\frac{1}{N}\sum_{l}e^{i(\bm{k}^{\prime}-\bm{k}){\bm{R}_{l}}}=\delta_{{\bm{k}},{\bm{k}}^{\prime}}$,
then
$J_{z}^{{\rm{ph}}}=\frac{\hbar}{2}\sum_{k,k^{\prime}}\epsilon_{k}^{\dagger}M\epsilon_{k^{\prime}}\left(\sqrt{\frac{\omega_{k}}{\omega_{k^{\prime}}}}a_{k}^{\dagger}a_{k^{\prime}}+\sqrt{\frac{\omega_{k^{\prime}}}{\omega_{k}}}a_{k^{\prime}}a_{k}^{\dagger}\right)\delta_{{\bm{k}},{\bm{k}}^{\prime}}e^{i(\omega_{k}-\omega_{k^{\prime}})t}.$
(S12)
Due to the communication relation
$[a_{\bm{k},\sigma^{\prime}},a_{\bm{k},\sigma}^{\dagger}]=\delta_{\sigma,\sigma^{\prime}}$,
we obtain Eq.(2) in the main text, that is
$J_{z}^{{\rm{ph}}}=\frac{\hbar}{2}\sum_{k,k^{\prime}}\epsilon_{k}^{\dagger}M\epsilon_{k^{\prime}}\left(\sqrt{\frac{\omega_{k}}{\omega_{k^{\prime}}}}+\sqrt{\frac{\omega_{k^{\prime}}}{\omega_{k}}}\right)a_{k}^{\dagger}a_{k^{\prime}}\delta_{{\bm{k}},{\bm{k}}^{\prime}}e^{i(\omega_{k}-\omega_{k^{\prime}})t}+\frac{\hbar}{2}\sum_{k}\epsilon_{k}^{\dagger}M\epsilon_{k}.$
(S13)
In equilibrium we know $\langle
a_{\bm{k},\sigma^{\prime}}^{\dagger}a_{\bm{k},\sigma}\rangle=f(\omega_{k})\delta_{\sigma,\sigma^{\prime}}$,
then we obtain
$J_{z}^{{\rm{ph}}}=\hbar\sum_{k}\epsilon_{k}^{\dagger}M\epsilon_{k}f(\omega_{k})+\frac{\hbar}{2}\sum_{k}\epsilon_{k}^{\dagger}M\epsilon_{k},$
(S14)
which is the Eq.(3) in the main text.
### I.2 Proof of the zero angular momentum of trivial phonon system without
spin-phonon interaction
For the trivial phonon system $H=\frac{1}{2}p^{T}p+\frac{1}{2}u^{T}Ku$, one
has an eigenvalue problem as
$D(\bm{k})\epsilon(\bm{k},\sigma)=\omega_{\bm{k},\sigma}^{2}\epsilon(\bm{k},\sigma).$
(S15)
Due to $D^{\dagger}=D$, then
$\epsilon^{\dagger}(\bm{k},\sigma)D(\bm{k})=\omega_{\bm{k},\sigma}^{2}\epsilon^{\dagger}(\bm{k},\sigma).$
(S16)
From Eq.S15, for wave vector $-\bm{k}$, one has
$D(-\bm{k})\epsilon(-\bm{k},\sigma)=\omega_{-\bm{k},\sigma}^{2}\epsilon(-\bm{k},\sigma),$
then
$\epsilon^{T}(-\bm{k},\sigma)D^{T}(-\bm{k})=\omega_{-\bm{k},\sigma}^{2}\epsilon^{T}(-\bm{k},\sigma).$
And since $D^{T}(-\bm{k})=D^{T}(\bm{k})$, then
$\epsilon^{T}(-\bm{k},\sigma)D(\bm{k})=\omega_{-\bm{k},\sigma}^{2}\epsilon^{T}(\bm{k},\sigma).$
(S17)
From Eq. (S16) and Eq. (S17), we can have
$\displaystyle\omega_{-\bm{k},\sigma}$ $\displaystyle=$
$\displaystyle\omega_{\bm{k},\sigma}$ (S18)
$\displaystyle\epsilon(-\bm{k},\sigma)$ $\displaystyle=$
$\displaystyle\epsilon^{*}(\bm{k},\sigma).$ (S19)
Therefore,
$l_{-\bm{k},\sigma}^{z}=(\epsilon^{\dagger}_{-\bm{k},\sigma}\,M\,\epsilon_{-\bm{k},\sigma})\hbar=(\epsilon^{T}_{\bm{k},\sigma}\,M\,\epsilon^{*}_{\bm{k},\sigma})\hbar=(\epsilon^{\dagger}_{\bm{k},\sigma}\,M^{T}\,\epsilon_{\bm{k},\sigma})\hbar,$
(S20)
because of $M^{T}=-M$, then
$l_{-\bm{k},\sigma}^{z}=-l_{\bm{k},\sigma}^{z}.$ (S21)
And $f(\omega_{-\bm{k},\sigma})=f(\omega_{\bm{k},\sigma})$, thus
$J^{\rm
ph}_{z}=\sum_{\sigma,\bm{k}}l_{\bm{k},\sigma}^{z}\,[f(\omega_{\bm{k},\sigma})+\frac{1}{2}]=0,$
where the summation is over all the $\bm{k}$ points in the first Brillouin
zone, and all the branches ($\omega_{\sigma}\geq 0$) are included.
### I.3 Another Proof of Eq. (3) for systems with a spin-phonon interaction
In the presence of the Raman type spin phonon interaction, as stated in the
main text, the Hamiltonian is
$H=\frac{1}{2}(p-{\tilde{A}}u)^{T}(p-{\tilde{A}}u)+\frac{1}{2}u^{T}Ku.$ (S22)
The polarization vector $\epsilon$ satisfies
$\bigl{[}(-i\omega+A)^{2}+D\bigr{]}\epsilon=0,$ (S23)
where $D({\bf k})=-A^{2}+\sum_{l^{\prime}}K_{ll^{\prime}}e^{i({\bf
R}_{l^{\prime}}-{\bf R}_{l})\cdot{\bf k}}$ denotes the dynamic matrix and $A$
is block diagonal with elements $\Lambda_{\alpha}$. Here, we use the short-
hand notation $k=({\bf k},\sigma)$ to specify both the wavevector and the
phonon branch. $D,K_{l,l^{\prime}},$ and $A$ are all $nd\times nd$ matrices,
where $n$ is the number of particles in one unit cell and $d$ is the dimension
of the vibration.
Equation (S23) is not a standard eigenvalue problem. However, we can describe
the system by a polarization vector as $x=(\mu,\epsilon)^{T}$, where $\mu$ and
$\epsilon$ are associated with the momenta and coordinates, respectively.
Using Bloch’s theorem, one has $i\frac{\partial}{{\partial t}}x=H_{\rm
eff}x,\;H_{\rm eff}=i\left(\begin{smallmatrix}-A&-D\\\
I_{nd}&-A\end{smallmatrix}\right),$ where the $I_{nd}$ is the $nd\times nd$
identity matrix. Therefore, one obtains the eigenvalue problem of the equation
of motion $H_{\rm
eff}\,x_{k}=\omega_{k}\,x_{k},\;\;\;\;\tilde{x}_{k}^{T}\,H_{\rm
eff}=\omega_{k}\,\tilde{x}_{k}^{T},$ where $x_{k}=(\mu_{k},\epsilon_{k})^{T}$
is the right eigenvector, and the left eigenvector is chosen as
${\tilde{x}}_{k}^{T}=(\epsilon^{\dagger}_{k},-\mu^{\dagger}_{k})/(-2i\omega_{k})$,
in such choice the second quantization of the Hamiltonian is satisfied. The
orthonormal condition holds between the left and right eigenvectors, as
${\tilde{x}_{\sigma,{\bf k}}}^{T}\;x_{\sigma^{\prime},{\bf
k}}=\delta_{\sigma\sigma^{\prime}}$. We also have the completeness relation as
$\sum_{\sigma}x_{\sigma,{\bf k}}\otimes{\tilde{x}_{\sigma,{\bf
k}}}^{T}=I_{2nd}$. The normalization of the eigenmodes is equivalent to
$\epsilon_{k}^{\dagger}\,\epsilon_{k}+\frac{i}{\omega_{k}}\epsilon_{k}^{\dagger}\,A\,\epsilon_{k}=1$.
From the eigenvalue problem Eq. (S23), we know that the completed set contains
branches of negative frequency. The short-hand notation $k=({\bf k},\sigma)$
can include negative branches, $-k$ means $(-{\bf k},-\sigma)$. In order to
simplify the notation, for all the branches, we define
$a_{-k}=a_{k}^{\dagger}$. The time dependence of the operators is given by:
$a_{k}(t)=a_{k}e^{-i\omega_{k}t},a_{k}^{\dagger}(t)=a_{k}^{\dagger}e^{i\omega_{k}t}.$
Inserting the equation of motion $\dot{u}_{l}=p_{l}-Au_{l}$, that is
$\left({\begin{array}[]{*{20}c}{\dot{u}_{l\alpha}^{x}}\\\
{\dot{u}_{l\alpha}^{y}}\\\
\end{array}}\right)=\left({\begin{array}[]{*{20}c}{p_{l\alpha}^{x}}\\\
{p_{l\alpha}^{y}}\\\
\end{array}}\right)-\left({\begin{array}[]{*{20}c}0&\lambda_{l\alpha}\\\
{-\lambda_{l\alpha}}&0\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{u_{l\alpha}^{x}}\\\
{u_{l\alpha}^{y}}\\\ \end{array}}\right),$ (S24)
Eq. LABEL:eq-s1 becomes
$\displaystyle J_{z}^{{\rm{ph}}}$ $\displaystyle=$
$\displaystyle\sum\limits_{l\alpha}{\left({\begin{array}[]{*{20}c}{u_{l\alpha}^{x}}\\\
{u_{l\alpha}^{y}}\\\
\end{array}}\right)^{T}\left({\begin{array}[]{*{20}c}0&1\\\ {-1}&0\\\
\end{array}}\right)}\left({\begin{array}[]{*{20}c}{p_{l\alpha}^{x}}\\\
{p_{l\alpha}^{y}}\\\
\end{array}}\right)+\left({\begin{array}[]{*{20}c}{u_{l\alpha}^{x}}\\\
{u_{l\alpha}^{y}}\\\
\end{array}}\right)^{T}\left({\begin{array}[]{*{20}c}\lambda_{l\alpha}&0\\\
0&\lambda_{l\alpha}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{u_{l\alpha}^{x}}\\\
{u_{l\alpha}^{y}}\\\ \end{array}}\right)$ (S37) $\displaystyle=$
$\displaystyle\sum\limits_{l\alpha}{\left({\begin{array}[]{*{20}c}{u_{l\alpha}^{x}}\\\
{u_{l\alpha}^{y}}\\\ {-p_{l\alpha}^{x}}\\\ {-p_{l\alpha}^{y}}\\\
\end{array}}\right)^{T}}\left({\begin{array}[]{*{20}c}0&1&\lambda_{l\alpha}&0\\\
{-1}&0&0&\lambda_{l\alpha}\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{p_{l\alpha}^{x}}\\\
{p_{l\alpha}^{y}}\\\ {u_{l\alpha}^{x}}\\\ {u_{l\alpha}^{y}}\\\
\end{array}}\right).$ (S50)
For a unit cell with two atoms $n=2$, the angular momentum of phonons can be
written as
$J_{z}^{{\rm{ph}}}\;=\sum\limits_{l}{\left({\begin{array}[]{*{20}c}{u_{l1}^{x}}\\\
{u_{l1}^{y}}\\\ {u_{l2}^{x}}\\\ {u_{l2}^{y}}\\\ {-p_{l1}^{x}}\\\
{-p_{l1}^{y}}\\\ {-p_{l2}^{x}}\\\ {-p_{l2}^{y}}\\\
\end{array}}\right)^{T}\left({\begin{array}[]{*{20}c}0&1&{}\hfil&{}\hfil&\lambda_{l1}&0&{}\hfil&{}\hfil\\\
{-1}&0&{}\hfil&{}\hfil&0&\lambda_{l1}&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&0&1&{}\hfil&{}\hfil&\lambda_{l2}&0\\\
{}\hfil&{}\hfil&{-1}&0&{}\hfil&{}\hfil&0&\lambda_{l2}\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\\
\end{array}}\right)}\left({\begin{array}[]{*{20}c}{p_{l1}^{x}}\\\
{p_{l1}^{y}}\\\ {p_{l2}^{x}}\\\ {p_{l2}^{y}}\\\ {u_{l1}^{x}}\\\
{u_{l1}^{y}}\\\ {u_{l2}^{x}}\\\ {u_{l2}^{y}}\\\ \end{array}}\right).$ (S52)
Using the $\chi_{l}=\left(\begin{smallmatrix}p_{l}\\\
u_{l}\end{smallmatrix}\right)$ and
$\tilde{\chi}_{l}=\left(\begin{smallmatrix}u_{l}\\\
-p_{l}\end{smallmatrix}\right)$, where $u_{l},p_{l}$ are column vectors of
displacements and conjugate momenta for the $l-$th unit, if $n=2$, then
$u_{l}=\left({\begin{array}[]{*{20}c}{u_{l1}^{x}}&{u_{l1}^{y}}&{u_{l2}^{x}}&{u_{l2}^{y}}\\\
\end{array}}\right)^{T}$, similar for $p_{l}$. Then we obtain Eq. (3) in the
main text as
$J^{\rm
ph}_{z}=\sum_{l}\tilde{\chi}^{T}_{l}\left(\begin{array}[]{cc}iM&-iMA\\\
0&0\end{array}\right)\chi_{l}.$ (S53)
By the second quantization
$\chi_{l}=\sqrt{\frac{\hbar}{N}}\sum_{k}x_{k}e^{i{\bf R}_{l}\cdot{\bf
k}}\sqrt{\frac{1}{2|\omega_{k}|}}\;a_{k};\;\;\tilde{\chi}_{l}=\sqrt{\frac{\hbar}{N}}\sum_{k}\tilde{x}_{k}e^{-i{\bf
R}_{l}\cdot{\bf
k}}(-2i\omega_{k})\sqrt{\frac{1}{2|\omega_{k}|}}\;a_{k}^{\dagger}$, and
$x_{k}=\left(\begin{smallmatrix}\mu_{k}\\\
\epsilon_{k}\end{smallmatrix}\right)$, $k=({\bf k},\sigma)$,
$\tilde{x}_{k}^{T}=\frac{1}{-2i\omega_{k}}\left(\begin{smallmatrix}\epsilon^{\dagger}_{k}\;&-\mu^{\dagger}_{k}\end{smallmatrix}\right)$,
we obtain
$J_{z}^{{\rm{ph}}}\;=\sum\limits_{l,k,k^{\prime}}{e^{i{\bm{R}}_{l}(\bm{k}-\bm{k}^{\prime})}}\frac{\hbar}{{N}}\frac{1}{{2\sqrt{|\omega_{k^{\prime}}||\omega_{k}|}}}\left({\begin{array}[]{*{20}c}{\varepsilon_{k^{\prime}}^{\dagger}}&{-\mu_{k^{\prime}}^{\dagger}}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}iM&{-iMA}\\\ 0&0\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{\mu_{k}}\\\
{\varepsilon_{k}}\\\ \end{array}}\right)a_{k^{\prime}}^{\dagger}a_{k}.$ (S54)
By using the fact of
$\sum\limits_{l}{e^{i{\bm{R}}_{l}(\bm{k}-\bm{k}^{\prime})}}=N\delta_{\bm{k},\bm{k}^{\prime}}$,
we obtain
$J_{z}^{{\rm{ph}}}\;=\sum\limits_{\bm{k},\sigma,\sigma^{\prime}}{\frac{\hbar}{{2\sqrt{|\omega_{\bm{k},\sigma}||\omega_{\bm{k},\sigma^{\prime}}|}}}\left({\begin{array}[]{*{20}c}{\varepsilon_{\bm{k},\sigma^{\prime}}^{\dagger}}&{-\mu_{\bm{k},\sigma^{\prime}}^{\dagger}}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}iM&{-iMA}\\\ 0&0\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{\mu_{\bm{k},\sigma}}\\\
{\varepsilon_{\bm{k},\sigma}}\\\
\end{array}}\right)a_{\bm{k},\sigma^{\prime}}^{\dagger}a_{\bm{k},\sigma}}.$
(S55)
In equilibrium, we know that $\langle
a_{\bm{k},\sigma^{\prime}}^{\dagger}a_{\bm{k},\sigma}\rangle=f(\omega_{k}){\rm
sign}(\sigma)\delta_{\sigma,\sigma^{\prime}}$, then
$\langle
J_{z}^{{\rm{ph}}}\rangle\;=\sum\limits_{k}{\frac{\hbar}{{2|\omega_{k}|}}}\left({\begin{array}[]{*{20}c}{\varepsilon_{k}^{\dagger}}&{-\mu_{k}^{\dagger}}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}iM&{-iMA}\\\ 0&0\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{\mu_{k}}\\\
{\varepsilon_{k}}\\\ \end{array}}\right)\langle a_{k}^{\dagger}a_{k}\rangle.$
(S56)
And using the relation between the momentum and displacement polarization
vectors $\mu_{k}=-i\omega_{k}\epsilon_{k}+A\epsilon_{k}$, we get
$\left({\begin{array}[]{*{20}c}{\varepsilon_{k}^{\dagger}}&{-\mu_{k}^{\dagger}}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}iM&{-iMA}\\\ 0&0\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{\mu_{k}}\\\
{\varepsilon_{k}}\\\
\end{array}}\right)=\omega\varepsilon_{k}^{\dagger}M\varepsilon_{k}.$ (S57)
Then the phonon angular momentum in equilibrium is
$J^{\rm
ph}_{z}=\frac{\hbar}{2}\sum_{k}\epsilon^{\dagger}_{k}\,M\,\epsilon_{k}\,f(\omega_{k}),$
(S58)
by using $\omega_{k}{\rm sign}(\sigma)=|\omega_{k}|$.
Figure S1: (Color online) (a) The angular momentum $l_{\bm{k},\sigma}^{z}$ of
bands $\sigma=1$ \- 4 as a function of $k_{x}$. (b)-(d) The sum of the angular
momentum of four bands
$\sum_{\sigma>0}l_{\bm{k},\sigma}^{z}\,[f(\omega_{\bm{k},\sigma})+\frac{1}{2}]$
at different temperatures. The parameters are the same with those in the main
text.
We have $\epsilon_{-\bm{k},-\sigma}=\epsilon^{*}_{\bm{k},\sigma}$, then
$(\epsilon^{\dagger}_{-\bm{k},-\sigma}\,M\,\epsilon_{-\bm{k},-\sigma})\hbar=(\epsilon^{T}_{\bm{k},\sigma}\,M\,\epsilon^{*}_{\bm{k},\sigma})\hbar=(\epsilon^{\dagger}_{\bm{k},\sigma}\,M^{T}\,\epsilon_{\bm{k},\sigma})\hbar=-(\epsilon^{\dagger}_{\bm{k},\sigma}\,M\,\epsilon_{\bm{k},\sigma})\hbar,$
(S59)
but $f(\omega_{-\bm{k},-\sigma})\neq f(\omega_{\bm{k},\sigma})$, thus phonon
angular momentum at $-\bm{k},-\sigma$ cannot cancel out that at
$\bm{k},\sigma$, and we can get a nonzero phonon angular momentum. Here,
$k=({\bm{k}},\sigma)$ includes all the positive $\sigma>0$ and negative
$\sigma<0$ branches. For the negative branches, that is $\sigma<0,\omega<0$,
using $\epsilon_{-k}^{*}=\epsilon_{k};\;\omega_{-k}=-\omega_{k}$ , then
$\frac{\hbar}{2}\sum\limits_{\bm{k},\sigma<0}{\varepsilon_{\bm{k},\sigma}^{\dagger}M\varepsilon_{\bm{k},\sigma}f(\omega_{\bm{k},\sigma})}=\frac{\hbar}{2}\sum\limits_{\bm{k}^{\prime},\sigma^{\prime}>0}{\varepsilon_{-\bm{k}^{\prime},-\sigma^{\prime}}^{\dagger}M\varepsilon_{-\bm{k}^{\prime},-\sigma^{\prime}}f(\omega_{-\bm{k}^{\prime},-\sigma^{\prime}})}=\frac{\hbar}{2}\sum\limits_{\bm{k}^{\prime},\sigma^{\prime}>0}{\varepsilon_{\bm{k}^{\prime},\sigma^{\prime}}^{T}M\varepsilon_{\bm{k}^{\prime},\sigma^{\prime}}^{*}f(-\omega_{\bm{k}^{\prime},\sigma^{\prime}})}.$
And
$\varepsilon_{\bm{k}^{\prime},\sigma^{\prime}}^{T}M\varepsilon_{\bm{k}^{\prime},\sigma^{\prime}}^{*}=\varepsilon_{\bm{k}^{\prime},\sigma^{\prime}}^{\dagger}M^{T}\varepsilon_{\bm{k}^{\prime},\sigma^{\prime}}$,
$M^{T}=-M$,
$f(-\omega_{\bm{k}^{\prime},\sigma^{\prime}}=-(1+f(\omega_{\bm{k}^{\prime},\sigma^{\prime}})$,
then
$\frac{\hbar}{2}\sum\limits_{\bm{k},\sigma<0}{\varepsilon_{\bm{k},\sigma}^{\dagger}M\varepsilon_{\bm{k},\sigma}f(\omega_{\bm{k},\sigma})}=\frac{\hbar}{2}\sum\limits_{\bm{k}^{\prime},\sigma^{\prime}>0}{\varepsilon_{\bm{k}^{\prime},\sigma^{\prime}}^{\dagger}M\varepsilon_{\bm{k}^{\prime},\sigma^{\prime}}(f(\omega_{\bm{k}^{\prime},\sigma^{\prime}})}+1).$
Therefore we obtain the angular momentum of phonons as:
$J^{\rm
ph}_{z}=\sum_{\sigma>0,\bm{k}}l_{\bm{k},\sigma}^{z}\,[f(\omega_{\bm{k},\sigma})+\frac{1}{2}],\;\;l_{\bm{k},\sigma}^{z}=(\epsilon^{\dagger}_{\bm{k},\sigma}\,M\,\epsilon_{\bm{k},\sigma})\hbar.$
(S60)
Here due to
$\epsilon_{-\bm{k},\sigma}=\epsilon_{\bm{k},-\sigma}^{*}\neq\epsilon_{\bm{k},\sigma}^{*}$,
we cannot obtain $l_{-\bm{k},\sigma}^{z}=-l_{\bm{k},\sigma}^{z}$, thus the
phonon angular momentum at the wave vector $-\bm{k}$ cannot cancel out that at
$\bm{k}$, and we can obtain a nonzero angular momentum of phonons.
We show the phonon angular momentum as a function of $k_{x}$ ($k_{y}=0$) in
Fig. S1. Due to the 6-fold symmetry of the honeycomb lattice, the angular
momentum of phonons is an even function of $k_{x}$, as shown in Fig. S1. For
band 1 the angular momenta of all the wave-vector points are negative, while
they are positive for band 4; for both band 2 and 3, the phonon angular
momentum (by doing the summation over all the four bands) at $\bm{\Gamma}$
point is opposite to that at $\bm{K}$ point. The total angular momentum of
four branches has a maximum at $\bm{\Gamma}$ point, and it decreases as
$k_{x}$ increasing. The phonon angular momentum arrives at its minimum at the
point $\bm{K}$ ($k=4\pi/3$) with a positive value around $0.08\hbar$. The
phonons at every $\bm{k}$ points in the Brillouin zone has nonzero phonon
angular momentum. The difference of the total phonon angular momentum between
those at $\bm{\Gamma}$ and $\bm{K}$ will be smoothed by the temperature, which
can be seen in Fig. S1 (b)(c)(d). When $T=0$ K, the phonon angular momentum
$\bm{\Gamma}$ is around 20 times of that at $\bm{K}$ (Fig. S1 (b)), and the
ratio decreases to about 2 when $T=30K$ (see Fig. S1 (c)). When $T=300$ K and
the phonon angular momentum will be almost the same for all the wave-vector
points, as shown in Fig. S1 (d).
### I.4 phonon angular momentum as a odd function of $\lambda$
$\displaystyle J_{z}^{{\rm{ph}}}(-\lambda)$ $\displaystyle=$
$\displaystyle\hbar\sum\limits_{\bm{k},\sigma>0}{\varepsilon_{\bm{k},\sigma}^{\dagger}(-\lambda)M\varepsilon_{\bm{k},\sigma}(-\lambda)(f(\omega_{\bm{k},\sigma}(-\lambda))}+1)$
(S61) $\displaystyle\Downarrow$
$\displaystyle\varepsilon_{-\bm{k},\sigma}^{*}(-\lambda)=\varepsilon_{\bm{k},\sigma}(\lambda)\quad\omega_{-\bm{k},\sigma}(-\lambda)=\omega_{\bm{k},\sigma}(\lambda)$
$\displaystyle=$
$\displaystyle\hbar\sum\limits_{\bm{k},\sigma>0}{\varepsilon_{-\bm{k},\sigma}^{T}(\lambda)M\varepsilon_{-\bm{k},\sigma}^{*}(\lambda)(f(\omega_{-\bm{k},\sigma}(\lambda))}+1)$
$\displaystyle\Downarrow$
$\displaystyle\varepsilon_{-\bm{k},\sigma}^{T}(\lambda)M\varepsilon_{-\bm{k},\sigma}^{*}(\lambda)=\varepsilon_{-\bm{k},\sigma}^{\dagger}(\lambda)M^{T}\varepsilon_{-\bm{k},\sigma}(\lambda)=-\varepsilon_{-\bm{k},\sigma}^{\dagger}(\lambda)M\varepsilon_{-\bm{k},\sigma}(\lambda)$
$\displaystyle=$
$\displaystyle-\hbar\sum\limits_{\bm{k}^{\prime},\sigma>0}{\varepsilon_{\bm{k}^{\prime},\sigma}^{\dagger}(\lambda)M\varepsilon_{\bm{k}^{\prime},\sigma}(\lambda)(f(\omega_{\bm{k}^{\prime},\sigma}(\lambda))}+1)$
$\displaystyle=$ $\displaystyle-J_{z}^{{\rm{ph}}}(\lambda).$
### I.5 phonon angular momentum in the classical limit
At a high temperature, we can expand the Einstein-Bose distribution in Taylor
series as
$f(x)=\frac{1}{{e^{x}-1}}\simeq\frac{1}{x}-\frac{1}{2}+\frac{1}{{12}}x+O(x^{2}),\quad
x=\frac{{\hbar\omega}}{{k_{B}T}}.$ (S62)
Inserting it to Eq. (5) in the main text, we have
$J^{\rm
ph}_{z}(T\rightarrow\infty)=\sum_{\sigma>0,\bm{k}}[(\frac{k_{B}T}{\hbar\omega}+\frac{\hbar\omega}{12k_{B}T})l_{\bm{k},\sigma}^{z}],$
(S63)
The term linear to $T$ is zero, that is,
$\sum_{\sigma>0,\bm{k}}\frac{\epsilon^{\dagger}_{\bm{k},\sigma}\,M\,\epsilon_{\bm{k},\sigma}}{\omega_{\bm{k},\sigma}}=0$,
we prove it as follows.
Since we have the completeness relation as
$\sum_{\sigma}x_{\bm{k},\sigma}\otimes{\tilde{x}_{\bm{k},\sigma}}^{T}=I_{4n\times
4n},$ (S64)
with $x_{k}=\left(\begin{smallmatrix}\mu_{k}\\\
\epsilon_{k}\end{smallmatrix}\right)$,
$\tilde{x}_{k}^{T}=\frac{1}{-2i\omega_{k}}\left(\begin{smallmatrix}\epsilon^{\dagger}_{k}\;&-\mu^{\dagger}_{k}\end{smallmatrix}\right)$.
Thus, the off-diagonal block
$\sum_{\sigma}\epsilon_{\bm{k},\sigma}\otimes{\tilde{\epsilon}_{\bm{k},\sigma}}^{\dagger}/(-2i\omega_{\bm{k},\sigma})=O_{2n\times
2n},$ (S65)
where $O_{2n\times 2n}$ is a zero matrix. Then we have, for arbitrary $i$ and
$j$,
$\sum_{\sigma}\frac{\epsilon_{i}(\bm{k},\sigma)\epsilon_{j}^{*}(\bm{k},\sigma)}{\omega_{\bm{k},\sigma}}=0.$
(S66)
Similar as that in Section II, we can prove
$\sum\limits_{\bm{k},\sigma<0}{\frac{{\varepsilon_{\bm{k},\sigma}^{\dagger}M\varepsilon_{\bm{k},\sigma}}}{{\omega_{\bm{k},\sigma}}}}=\sum\limits_{\bm{k}^{\prime},\sigma^{\prime}>0}{\frac{{\varepsilon_{-\bm{k}^{\prime},-\sigma^{\prime}}^{\dagger}M\varepsilon_{-\bm{k}^{\prime},-\sigma^{\prime}}}}{{\omega_{\bm{k}^{\prime},-\sigma^{\prime}}}}}=\sum\limits_{\bm{k}^{\prime},\sigma^{\prime}>0}\frac{{\varepsilon_{\bm{k}^{\prime},\sigma^{\prime}}^{T}M\varepsilon_{\bm{k}^{\prime},\sigma^{\prime}}^{*}}}{{-\omega_{\bm{k}^{\prime},\sigma^{\prime}}}}=\sum\limits_{\bm{k},\sigma>0}{\frac{{\varepsilon_{\bm{k},\sigma}^{\dagger}M\varepsilon_{\bm{k},\sigma}}}{{\omega_{\bm{k},\sigma}}}}.$
Then
$\sum\limits_{\bm{k},\sigma>0}{\frac{{\varepsilon_{\bm{k},\sigma}^{\dagger}M\varepsilon_{\bm{k},\sigma}}}{{\omega_{\bm{k},\sigma}}}}=\frac{1}{2}\sum\limits_{\bm{k},\sigma}{\frac{{\varepsilon_{\bm{k},\sigma}^{\dagger}M\varepsilon_{\bm{k},\sigma}}}{{\omega_{\bm{k},\sigma}}}}=\frac{1}{2}\sum\limits_{\scriptstyle\bm{k},\sigma\hfill\atop\scriptstyle
j,i\hfill}{\frac{{\varepsilon_{j}^{*}(\bm{k},\sigma)M_{ji}\varepsilon_{i}(\bm{k},\sigma)}}{{\omega_{\bm{k},\sigma}}}}=\frac{1}{2}\sum\limits_{\scriptstyle\bm{k}\hfill\atop\scriptstyle
j,i\hfill}{M_{ji}\sum\limits_{\sigma}{\frac{{\varepsilon_{j}^{*}(\bm{k},\sigma)\varepsilon_{i}(\bm{k},\sigma)}}{{\omega_{\bm{k},\sigma}}}}=}0.$
(S67)
|
11institutetext: Univ Lyon, ENS de Lyon, Univ Lyon1, CNRS, Centre de Recherche
Astrophysique de Lyon UMR5574, F–69007, Lyon, France 22institutetext:
Department of Physics and Materials Science, University of Luxembourg, 162 A,
Avenue de la Faïencerie, L-1511 Luxembourg City, Luxembourg
# Homology reveals significant anisotropy in the cosmic microwave background
Pratyush Pranav 1122<EMAIL_ADDRESS>Thomas Buchert 11
We test the tenet of statistical isotropy of the standard cosmological model
via a homology analysis of the cosmic microwave background temperature maps.
Examining small sectors of the normalized maps, we find that the results
exhibit a dependence on whether we compute the mean and variance locally from
the masked patch, or from the full masked sky. Assigning local mean and
variance for normalization, we find the maximum discrepancy between the data
and model in the galactic northern hemisphere at more than $3.5$ s.d. for the
PR4 dataset at degree-scale. For the PR3 dataset, the C-R and SMICA maps
exhibit higher significance than the PR4 dataset at $\sim 4$ and $4.1$ s.d.
respectively, however the NILC and SEVEM maps exhibit lower significance at
$\sim 3.4$ s.d. The southern hemisphere exhibits high degree of consistency
between the data and the model for both the PR4 and PR3 datasets. Assigning
the mean and variance of the full masked sky decreases the significance for
the northern hemisphere, the tails in particular. However the tails in the
southern hemisphere are strongly discrepant at more than $4$ standard
deviations at approximately $5$ degrees. The $p$-values obtained from the
$\chi^{2}$-statistic exhibit commensurate significance in both the
experiments. Examining the quadrants of the sphere, we find the first quadrant
to be the major source of the discrepancy. Prima-facie, the results indicate a
breakdown of statistical isotropy in the CMB maps, however more work is needed
to ascertain the source of the anomaly. Regardless, these map characteristics
may have serious consequences for downstream computations such as parameter
estimation, and the related Hubble tension.
###### Key Words.:
Cosmology – Cosmic Microwave Background (CMB) radiation – primordial non-
Gaussianity – topology – relative homology – topological data analysis
## 1 Introduction
The standard Lambda Cold Dark Matter (LCDM) paradigm of cosmology encapsulates
and arises from the cosmological principle (CP), which posits that, on large
enough scales, the Universe is isotropic and homogeneous (Peebles, 1980, 1993;
Durrer, 2015). Though supported by strong mathematical, philosophical and
historical foundations, the veracity of the fundamental tenets of CP has not
yet been comprehensively and conclusively established, motivating theoretical
and observational tests (Secrest et al., 2021; Oayda & Lewis, 2023; Dam et
al., 2023). The recent focus of cosmology towards data gathering and analysis
presents us with an unprecedented opportunity to test the postulates of the
cosmological principle, and the ensuing standard model of cosmology. The data
gathered are from both the early and late epochs in the evolutionary timeline
of the Universe, and have thrown up a number of surprises. Among others, this
includes the famous discrepancy in the inference of the Hubble parameter
between early- and late-Universe data (Di Valentino et al., 2021;
Perivolaropoulos & Skara, 2022; Aluri et al., 2023).
The Cosmic Microwave Background (CMB) radiation is one of the more important
probes of the properties of the early Universe. Emitted at the epoch of
recombination, when the Universe was merely 380,000 years old, the fluctuation
characteristics of the CMB trace the fluctuation characteristics of the matter
distribution in the early Universe (Durrer, 2015; Jones, 2017), and offer the
largest and the oldest canvas on which to test the postulates of CP.
Therefore, studying the CMB fluctuations is essential towards understanding
the properties of the stochastic matter field in the early Universe. The two
components of the CMB radiation – temperature and polarization – present
independent probes in to the properties of the primordial fluctuations
(Durrer, 1999; Seljak & Zaldarriaga, 1997).
The general consensus is that the stochastic fluctuation field of the CMB is
an instance of an isotropic and homogeneous Gaussian random field (Harrison,
1970; Guth & Tye, 1980; Starobinsky, 1982; Guth & Pi, 1982; Komatsu, 2010);
see also Buchert et al. (2017) and references therein for more recent
investigations of Gaussianity. However, CMB data has thrown up a number of
surprises in terms of anomalous features since the launch of the Cosmic
Background Explorer (COBE) satellite (Fixsen et al., 1994). Due to its low
resolution of approximately $7$ degrees, the COBE satellite first discovered
the truly large-scale anomalous lack of correlation in the CMB at 60 degrees
and more. Subsequently, the Wilkinson Microwave Anisotropy Probe (WMAP)
(Bennett et al., 2013) satellite with a higher resolution, discovered a number
of other anomalies at smaller scales, that have persisted in the CMB
measurements by the latest Planck satellite (Planck Collaboration et al.,
2014a, 2016, 2020c). These anomalies which have been detected in both the real
and the harmonic space, seem to be at odds with the postulates of the standard
cosmological model, and perhaps with the more fundamental CP itself.
Representative examples of anomalies in the harmonic space consist of the
_hemispherical power asymmetry_ (HPA) (Eriksen et al., 2004a; Hansen et al.,
2009; Paci et al., 2010; Planck Collaboration et al., 2014b) or the _cosmic
hemispherical asymmetry_ (CHA) (Mukherjee & Souradeep, 2016), the alignment of
low multipoles (Schwarz et al., 2016), as well as the parity anomaly (Land &
Magueijo, 2005; Finelli et al., 2012; Planck Collaboration et al., 2014b). In
particular, the power spectrum has been studied at large scales, for
$\ell=2,40$ (Eriksen et al., 2004a; Mukherjee & Souradeep, 2016), which has
later been extended to smaller scales $\ell\sim 600$ (Hansen et al., 2009;
Paci et al., 2010; Planck Collaboration et al., 2014b), and the analysis
presents the evidence for HPA/CHA at all scales. Important to note is that the
assumption of cosmological isotropy is challenged in other datasets as well
(Bouchet et al., 2001; Colin et al., 2019; Secrest et al., 2021, 2022; Oayda &
Lewis, 2023; Dam et al., 2023). In particular, the analysis of galaxy survey
datasets also points to a hemispherical asymmetry, as the northern and the
southern galactic hemispheres appear to have different topo-geometrical
characteristics (Kerscher et al., 1997, 1998, 2001; Appleby et al., 2022).
The anomalies in the real space have consisted of the discovery of the _cold-
spot_ (Cruz et al., 2005), as well as the unusual behavior of descriptors
emerging from topo-geometrical considerations which involve the integral-
geometric Minkowski functionals (MFs), particularly the genus statistic as a
member of the MFs, first performed on WMAP data (Park, 2004; Eriksen et al.,
2004b), and later extended to Planck data (Planck Collaboration et al., 2014b,
2016, 2020c; Pranav et al., 2019a; Pranav, 2021a, 2022). While Park (2004)
perform their analysis on small sub-degree scales, Eriksen et al. (2004b)
perform a multi-scale analysis spanning a range of sub- and super-degree
scales. In both the cases, there are reported asymmetry between the CMB
hemispheres, where the small scale analysis in Park (2004) reports anomalous
behavior to the tune of $2\sigma$, while (Eriksen et al., 2004b) report a more
than $3\sigma$ deviation in the genus statistic at scales of approximately $5$
degrees for negative thresholds. In this context, it is important to note that
the purely geometric entities of the MFs, such as the area, contour length and
skeleton length have consistently shown a congruence between the data and the
model (Planck Collaboration et al., 2016; Buchert et al., 2017), while the
topological entities such as the genus and the Euler characteristic have shown
deviations between the data and the model (Eriksen et al., 2004b; Park, 2004;
Pranav et al., 2019a; Pranav, 2021a, 2022).
Extending these studies, in this paper, we examine specific sectors of the CMB
temperature maps with a view to test the tenet of statistical isotropy, via
tools that find basis in purely topological notions arising from homology
(Munkres, 1984; Edelsbrunner & Harer, 2010; Pranav, 2015; Pranav et al., 2017)
and its hierarchical extension persistent homology (Edelsbrunner et al., 2002;
Edelsbrunner & Harer, 2010; Pranav, 2015; Pranav et al., 2017; Pranav, 2021),
which form the basis of the recently emerging field of topological data
analysis (TDA) (Carlsson, 2009; Porter et al., 2023). Using these
methodologies, which form the basis for the developed computational pipeline
tailored to examining the CMB datasets (Pranav, 2022), the central
contribution of this paper is the uncovering of a number of anomalous
signatures, which point to different behavior of galactic hemispheres, that
have evidence in the documented literature, but have been overlooked, and not
given due attention in the past.
The advent of TDA holds important place in view of the recent surge in data
acquisition in cosmology and astronomy, which demand increasingly more
sophisticated tools to condense meaningful information from these large and
growing datasets. In view of these observations, the tools and methodologies
presented here particularly stand out as promising candidates to reveal novel
features in the big cosmological datasets, including the completed, on going,
and upcoming CMB observations as evidenced in this paper, as well as galaxy
surveys. Even though a recent development, TDA has already witnessed a strong
presence in a wide range of astrophysical research from studying the large
scale structure of the Universe (van de Weygaert et al., 2011; Sousbie, 2011;
Shivashankar et al., 2016; Xu et al., 2018; Cisewski-Kehe et al., 2018;
Feldbrugge et al., 2019; Biagetti et al., 2021; Kono et al., 2020; Wilding et
al., 2021; Cisewski-Kehe et al., 2022; Ouellette et al., 2023), and the study
of of stochastic properties of astrophysical and cosmological fields in
general (Park et al., 2013; Adler et al., 2017; Makarenko et al., 2018; Pranav
et al., 2019b; Heydenreich et al., 2021; Pranav, 2021b), including the CMB
fluctuation field (Adler et al., 2017; Pranav et al., 2019a; Pranav, 2021a,
2022).
Section 2 presents the data and methods, while Section 3 presents the main
results. We discuss the results and conclude in Section 4.
(a)
(b)
Figure 1: A visualization of the CMB temperature fluctuation field in the
northern hemisphere in two different views. The masked area covers the whole
southern hemisphere, and the relevant parts of the northern hemisphere,
dictated by the PR3 temperature common mask. The visualization is based on the
PR3 observed map, cleaned using the SMICA component separation pipeline,
degraded at ${N}=128$ and smoothed with a Gaussian of $FWHM=80^{\prime}$.
(a)
(b)
(c)
(d)
Figure 2: An illustration of the analyzed map surface in the different
quadrants of the sphere. The first, second, third and fourth quadrants are
presented from left to right, and this convention is followed for quoting the
results.
## 2 Data and Methods
In this section, we briefly describe the data and methods employed at arriving
the results. The computational pipeline, specifically tailored to the CMB
data, but useful for the analysis of other scalar functions on
${{\mathbb{S}}}^{2},$ is a recent development, and a detailed account may be
found in Pranav et al. (2019a) and Pranav (2022).
### 2.1 Data
The data that we investigate are the temperature maps from the latest two data
releases by the Planck team – Planck Data Release 3 (DR3), and the fourth and
final, Planck Data Release 4 (DR4). These data releases represent a natural
evolution of the Planck data processing pipeline, where the final data release
incorporates the best strategies for both the LFI and HFI instruments,
commensurate with an overall reduction in noise and systematics (Planck
Collaboration et al., 2020b). The PR3 and the PR4 data sets are accompanied by
$600$ and $300$ simulations respectively, that originate from the standard
LCDM paradigm, which posits the CMB field to be an instance of an isotropic
and homogeneous Gaussian random field. The PR3 dataset consists of
observational maps obtained via four different component separation methods,
namely C-R, NILC, SEVEM and SMICA, (c.f. Planck Collaboration et al. (2020a)).
We analyze all the four maps in only one experiment, to assess the overall
trend and congruence of results between the different component separation
methods. This choice is dictated by significant computational overhead,
especially for higher resolutions.
### 2.2 Methods
(a)
(b)
(c)
(d)
Figure 3: Graphs of ${{b}_{0}}$ and ${{b}_{1}}$ for the temperature maps for
the NPIPE dataset for the northern (top two rows) and the southern hemisphere
(bottom two rows). The mean and variance are computed for each hemisphere
locally from the unmasked pixels in that hemisphere. The graphs present the
normalized differences, and each panel presents the graphs for a range of
degradation and smoothing scales. The mask used is the PR3 temperature common
mask.
(a)
(b)
(c)
(d)
Figure 4: A visualization of the structure of the superlevel set of the
temperature field at the threshold $\nu=0.5$ for the northern hemisphere. This
is the threshold at which we detect a statistically significant deviation
between the observation and simulations in the number of isolated components.
Clock-wise from the top-left panel, we present the visualization of the
observed CMB map from the FFP10 pipeline smoothed with a Gaussian beam profile
of $FWHM=80^{\prime}$. The rest of the panels present the visualization for
the randomly selected simulation sample numbering 42, and its two higher
multiples 84 and 126. Visually, it is evident that the observational maps have
more fragmented structure compared to the simulated maps.
(a)
(b)
(c)
(d)
Figure 5: Same as Figure 3, however in this case, the mean and variance are
computed from the full sky from the unmasked pixels.
The methods employed in this paper emerge from algebraic and computational
topology, at the level of homology (Munkres, 1984; Edelsbrunner & Harer, 2010)
and persistent homology (Edelsbrunner et al., 2002; Edelsbrunner & Harer,
2010). On ${{\mathbb{S}}}^{2}$, only the 0- and 1D homology groups are of
interest, which are associated with isolated components and holes of the
excursion sets of ${{\mathbb{S}}}^{2}$. These methodologies are complementary
in nature to the integral-geometric Minkowski functionals (Eriksen et al.,
2004b; Schmalzing & Buchert, 1997; Schmalzing & Gorski, 1998; Matsubara, 2010;
Ducout et al., 2013; Buchert et al., 2017; Appleby et al., 2018; Chingangbam
et al., 2017; Telschow et al., 2019), and together they represent a
comprehensive topo-geometrical characterization of fields.
As the CMB data is unreliable in certain regions of the sky, we mask these
regions. We present our results in terms of the homology of the excursion sets
relative to the mask, denoting the number of components relative to the mask
for a given normalized threshold $\nu$ as ${{b}_{0}}(\nu)$. Similar definition
is adopted for the number of loops, ${{b}_{1}}(\nu)$ (Pranav et al., 2019a;
Pranav, 2022).
In continuation of the experiments detailed in Pranav et al. (2019a), Pranav
(2021a), and Pranav (2022), in this paper we perform our investigations on the
hemispheres and quads of the CMB sky, where these regions are defined with
respect to the galactic coordinates. When analyzing the northern hemisphere,
the whole southern hemisphere and the relevant parts of northern hemisphere
are masked, and vice-versa. Similar masking procedure is adopted when
examining the quads. Figure 1 presents a visualization of the temperature
fluctuation field in the northern hemisphere, smoothed with a Gaussian beam
profile of $FWHM=80^{\prime}$. Figure 2 presents a visualization of the quads
of the CMB sky in Molleweide projection view.
We perform a multi-scale analysis by smoothing the original maps given at
$FWHM=5^{\prime}$ and ${N}=2048$, to a range of scales defined by Gaussian
beam profile of
$FWHM=10^{\prime},20^{\prime},40^{\prime},80^{\prime},160^{\prime},320^{\prime}$
and $640^{\prime}$. In order to facilitate faster computations, the maps are
also degraded to ${N}=1024,512,256,128,64,32$ and $16$ in the HealPix format,
prior to the smoothing operation. The mask is subjected to an identical
degrading and smoothing procedure. This results in a non binary mask, which is
re-binarized by setting all pixels above and equal to $0.9$ to $1$, and all
pixels with smaller values to $0$. The mask is applied to the simulations and
observations, and they are transformed to zero-mean and unit-variance fields
by subtracting the mean and re-scaling by the standard deviation. Denoting
$\delta T(\theta,\phi)$ as the fluctuation field at $(\theta,\phi)$ on
${{\mathbb{S}}}^{2}$, $\mu_{\delta T}$ as its mean, and $\sigma_{\delta T}$ as
the standard deviation, computed over the relevant pixels, we examine the
properties of the normalized field:
$\nu(\theta,\phi)=\frac{\delta T(\theta,\phi)-\mu_{\delta T}}{\sigma_{\delta
T}}.$ (1)
In all the experiments we restrict ourselves to $\nu\in[0:3]$ while examining
${{b}_{0}}$, and $\nu\in[-3:0]$ while examining ${{b}_{1}}$, commensurate with
the fact that components are the dominant topological entities at positive
thresholds, while loops are dominant for the negative thresholds.
After the pre-processing steps, performed with the aid of HealPix software
(Górski et al., 2005), the data is subject to the topology computation
pipeline, which briefly involves tessellating the points on the sphere,
computing the upper-star filtration of this tessellation, constructing the
boundary matrix of the filtration, and reducing the boundary matrix to obtain
the $0$\- and $1$-dimensional persistence diagrams. The Betti numbers,
relative to the mask are condensed from the persistence diagrams.
## 3 Results: Topological characteristics of subsets of ${{\mathbb{S}}}^{2}$
In this section, we present the results of the analysis of various sectors of
the CMB sky, with a view to test statistical isotropy. We begin by presenting
the results of examining the hemispheres, followed by an examination of the
quads.
### 3.1 Hemispheres
For the hemispherical analysis, we present the results for two different
experiments, which differ in the regions adopted for computing the mean and
variance for normalizing the maps (c.f. (1)). In the first experiment, we
compute the mean and variance for each hemisphere locally from the unmasked
pixels. For the second experiment, we compute the mean and variance from the
unmasked pixels from the full sky. In all the experiments, we present our
results in terms of the graphs of the Betti numbers relative to the mask. The
graphs are normalized at each threshold to reflect the significance of
deviation. At each threshold, we compute the mean $\mu_{{{b}_{i,sim}}}$, and
the standard deviation $\sigma_{{{b}_{i,sim}}}$ of the Betti numbers for
$i=0,1$, corresponding to the components and the holes. Then, the significance
of difference between the observation and simulations is given by
$\nu_{{{b}_{i}}}=\frac{{{b}_{i,obs}}-\mu_{{{b}_{i,sim}}}}{\sigma_{{{b}_{i,sim}}}},$
(2)
which is the quantity depicted in the graphs.
Relative homology – $\chi^{2}$ (empirical) – Separate Variance
---
| | North | South
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
1024 | 10’ | 0.4083 | 0.0733 | 0.1483 | 0.9917 | 0.7617 | 0.8867
512 | 20’ | 0.4633 | 0.0850 | 0.1650 | 0.9933 | 0.8183 | 0.9933
256 | 40’ | 0.0600 | 0.3350 | 0.4767 | 0.3500 | 0.8167 | 0.7283
128 | 80’ | 0.0367 | 0.0367 | 0.0350 | 0.6617 | 0.9017 | 0.9150
64 | 160’ | 0.0433 | 0.0583 | 0.0733 | 0.2833 | 0.7217 | 0.8500
32 | 320’ | 0.0200 | 0.2150 | 0.1017 | 0.5967 | 0.1033 | 0.1717
16 | 640’ | 0.5600 | 0.2900 | 0.4700 | 0.4283 | 0.1783 | 0.2683
summary | N/A | 0.0233 | 0.0533 | 0.0167 | 0.5767 | 0.8767 | 0.9250
(a)
Relative homology – $\chi^{2}$ (empirical) – Gloabal Variance
---
| | North | South
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
1024 | 10’ | 0.4600 | 0.1467 | 0.2750 | 0.5167 | 0.2567 | 0.3150
512 | 20’ | 0.1917 | 0.1450 | 0.1250 | 0.3783 | 0.1117 | 0.1733
256 | 40’ | 0.1817 | 0.3400 | 0.5217 | 0.4533 | 0.0833 | 0.3233
128 | 80’ | 0.0750 | 0.2383 | 0.3567 | 0.1217 | 0.0400 | 0.1367
64 | 160’ | 0.3383 | 0.1250 | 0.2867 | 0.5150 | 0.1583 | 0.5817
32 | 320’ | 0.0383 | 0.2967 | 0.2617 | 0.0650 | 0.0017 | 0.0000
16 | 640’ | 0.8800 | 0.9617 | 0.9000 | 0.3317 | 0.0167 | 0.0150
summary | N/A | 0.1667 | 0.6383 | 0.5283 | 0.0783 | 0.0183 | 0.0000
(b)
Table 1: Table displaying the two-tailed $p$-values for relative homology
obtained from the empirical Mahalanobis distance or $\chi^{2}$ test, computed
from the sample covariance matrices, for different resolutions and smoothing
scales for the NPIPE dataset. Panel (a) presents the $p$-values for
experiments where the variance is computed for each hemisphere separately, and
panel (b) presents results for experiments where the hemispheres are assigned
the variance of full sky. The last entry is the $p$-value for the summary
statistic computed across all resolutions. Marked in boldface are $p$-values
$0.05$ or smaller.
#### 3.1.1 Local normalization
For these experiments, the temperature at each pixel is mean-subtracted, and
re-scaled by the standard deviation, where these quantities are computed
locally from the unmasked pixels in each hemisphere. The topological
quantities are computed as a function of the normalized temperature threshold,
$\nu$, at steps of $0.5$. Figure 3 presents the graphs for the significance of
difference between the simulations and the observations, for the number of
components, ${{b}_{0}}$, and the number of loops, ${{b}_{1}}$ for the PR4
dataset. The red curve represents the observed sky, while the gray curves
represent the individual simulations treated as observation. The top two rows
present the graphs for ${{b}_{0}}$ and ${{b}_{1}}$ for the northern
hemisphere, while the bottom two rows present the same for the southern
hemisphere. Similar results for the PR3 dataset are presented in Figure 9 in
the appendix. For this case, we analyze all the four observational maps
obtained from the different component separation pipeline.
We notice a few important things in the graphs. First, that both the PR4 and
PR3 datasets present largely identical results. Second, that the significance
of deviation in the northern hemisphere is in general flared compared to the
southern hemisphere. While the southern hemisphere significance is within the
$2\sigma$ band in general, the northern hemisphere shows significance of
$2\sigma$ or more for most of the scales. The third important thing that we
notice is the very significant deviation in the number of components between
the observations and simulations at the threshold $\nu=0.5$, at
$FWHM=80^{\prime},{N}=128$, which is at degree scale. The significance of
difference for the PR4 datasets stands at approximately $3.5\sigma$. For the
PR3 dataset, two of the maps, namely C-R and SMICA, exhibit higher
significance at approximately $4\sigma$ and $4.1\sigma$ respectively, while
the two other maps, namely NILC and SEVEM, exhibit a $3.4\sigma$ deviation.
Figure 8 in the appendix presents the distribution of Betti numbers at this
threshold, which maybe approximated as a Gaussian, which justifies ascribing a
$\sigma$-significance to the differences. Figure 4 presents the visualization
of the structure of the temperature field at $\nu=0.5$ for the northern
hemisphere. Clock-wise from top-left, we present the field for the observed
CMB map, as well as three randomly selected simulated samples. It is evident
that the observed maps exhibit more fragmented characteristics than any of the
visualized simulations, which engender the significant deviation in the number
of components between the observation and the model.
For the loops, The highest deviation recorded is at $N=64,FWHM=160^{\prime}$
for the northern hemisphere. The PR4 dataset exhibits a $3.4\sigma$ deviation,
while the PR3 dataset shows a $3.2\sigma$ deviation at this scale. At
${N}=128,FWHM=80^{\prime}$, the PR4 dataset exhibits a maximum of $3\sigma$
deviation, while the PR3 dataset exhibits a $2.6\sigma$ deviation.
#### 3.1.2 Global normalization
Figure 5 presents the graphs similar to Figure 3 for the PR4 dataset, where
the maps are normalized by the mean and variance computed from unmasked pixels
from the full sky. Similar results for the PR3 dataset are presented in Figure
10. As in the case of local normalization, we notice an agreement in the
feature of the graphs between both the datasets. However, we also note
important differences between observations and simulations. For the northern
hemisphere, the significance of difference is slightly suppressed in general,
and there is an evident change in the behavior of the tail, which is more in
sync with the model. However, in this experiment, there are significant
differences between the observation and the model in the southern hemisphere.
While the number of components, ${{b}_{0}}$, is generally within the $2\sigma$
band, the number of loops exhibits strong differences between the data and
model in the tail. This difference peaks at more than $5$ standard deviations
at scales of approximately $5$ degrees and more. We note that the distribution
of the Betti numbers is manifestly non Gaussian at these scales and thresholds
(c.f. (Pranav, 2022)).
#### 3.1.3 Significance of combined thresholds and resolutions
Table 1 presents the $p$-values from the $\chi^{2}$ statistics for
${{b}_{0}}$, ${{b}_{1}}$, and ${\sf EC_{\rm rel}}$ for the PR4 dataset. ${\sf
EC_{\rm rel}}$ is the alternating sum of the Betti numbers, and indicates the
Euler characteristic of the excursion set relative to the mask. The $p$-values
for the PR3 dataset based on SMICA maps are presented in the Appendix in Table
3. The first seven rows in the tables present the $p$-values combining
different thresholds for a given resolution, accounting for multiple testing
at different thresholds. For both the datasets, the $p$-values are consistent
with the graphs, broadly indicating similar properties. For experiments where
the variance is computed separately for the hemispheres, there are significant
deviation between data and model around a degree at $FWHM=80^{\prime}$ for all
the topological quantities. In addition, ${{b}_{0}}$ exhibits differences for
a range of scales $FWHM=80^{\prime},160^{\prime},320^{\prime}$, with
${{b}_{1}}$ also exhibiting mild differences. In contrast the experiments with
common variance exhibit departure between the data and model in the southern
hemisphere at scales of $5$ degrees and larger. The differences are most
evident in the number of loops, which also affects the ${\sf EC_{\rm rel}}$.
However, we also note the deviant behavior of ${{b}_{0}}$ at
$FWHM=320^{\prime}$ in the northern hemisphere.
To account for multiple testing at various resolutions, the last entry in the
tables present the summary $p$-values combining all the tested thresholds and
resolutions. For the experiments with the variance computed separately for the
different hemispheres, the summary $p$-values present significant evidence for
non-random deviation for all the topological descriptors in the northern
hemisphere. Similarly for the experiments with a common variance computed from
the full sky, the southern hemisphere indicates non-random discrepancy between
the data and the model for all the topological descriptors, which is mild for
${{b}_{0}}$, but significant for ${{b}_{1}}$ and ${\sf EC_{\rm rel}}$.
#### 3.1.4 The effect of normalization
Normalization is an order preserving transformation topologically. This
ensures that there is a bijection between the original and the normalized
maps, and specifically a correspondence in the dictionary of critical points
as well as the order in which they appear in the filtration to form and
destroy the topological cycles. This entails that, at the most, an erroneous
estimation of mean and variance would induce an offset between the level-sets
of the compared maps, without affecting the deeper topological structure of
the field. Combined with the fact that there are stark differences between the
data and model irrespective of the recipe for normalization, this points to a
difference in the topological structure of the observed field with respect to
the simulations at a level deeper than normalization. Appendix A.1 presents a
short account of the properties of the distribution of mean and variance for
the full sky and the galactic hemispheres. At the level of mean, both the
northern and the southern hemispheres are consistent between the data and
model, which is reflected in the behavior of the mean in the full sky as well.
In contrast, the variance of the northern hemisphere shows stark deviation
between observation and simulations, while the variance in the southern
hemisphere is consistent between the data and the model. This engenders a
variance in the full sky case, which exhibits a mild deviation between the
observation and simulations. Due to the evident discrepancy in variance, it is
also prudent to treat both the hemispheres as arising from different models,
and consequently treat the results from this experimental procedure as more
meaningful.
### 3.2 Quadrants of the sphere
(a)
(b)
(c)
(a)
(b)
(c)
(d)
(b)
Figure 6: Graphs of ${{b}_{0}}$ and ${{b}_{1}}$ for the different quadrants of
the sphere for a range of smoothing scales with Gaussian
$FWHM=20^{\prime},40^{\prime},80^{\prime},160^{\prime},320^{\prime}$. Panel
(a) presents the graphs for ${{b}_{0}}$ while panel (b) presents the graphs
for ${{b}_{1}}$. The values for the different quadrants are presented from top
to bottom, while the scale increases from left to right.
Following the results of the previous section where we detect an anomalous
behavior in the hemispheres, in this section, we compare the behavior of the
observations and simulations in the different quadrants of the sphere for the
PR4 dataset, illustrated in Figure 2, with a view to determine the zone of
discrepancy more accurately. In these experiments, we compute the mean and the
variance locally from the quadrants for map-normalization. We restrict
ourselves to the analysis of scales represented by $FWHM=20,40,80,160,320$.
Our choice of scales is determined by the fact that, on the larger scales,
statistics on smaller sections of the sphere may not be reliable due to the
low numbers involved, while the smaller scales have significant computational
overhead.
Figure 6 presents the graphs for the Betti numbers while Table 2 presents the
$p$-values computed from the empirical $\chi^{2}$ test for the different
quadrants, based on $600$ simulations. Examining the graphs, the first
quadrant stands out due to the maximum deviation for ${{b}_{1}}$ at
$FWHM=160^{\prime}$, with a significance of more than $3.8\sigma$. The
$p$-values presented in Table 2 corroborate the fact that the northern
hemisphere exhibits anomalous behavior with respect to the simulations, where
the source of the discrepancy is clearly associated with the first quadrant.
The summary $p$-values combining all thresholds and resolutions indicate a
strong non-random deviation between data and model in the first quadrant,
which is the strongest for ${{b}_{0}}$. Interesting to note is that the
deviations between the data and model for the quadrants have shifted to a
slightly higher resolution of $FWHM=160^{\prime}$, where also ${{b}_{0}}$
exhibits the strongest deviations. In comparison, the quadrants of the
southern hemisphere exhibit no difference with respect to the model,
consistent with the observation that the southern Galactic hemisphere is
congruent with the standard model, when the maps are normalized by local mean
and variance.
Relative homology – $\chi^{2}$ (empirical) – Separate Variance
---
| | Quad 1 | Quad 2
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
512 | 20’ | 0.1800 | 0.3433 | 0.2917 | 0.5617 | 0.0283 | 0.0567
256 | 40’ | 0.0250 | 0.3650 | 0.2233 | 0.5483 | 0.5533 | 0.7467
128 | 80’ | 0.4117 | 0.3267 | 0.5183 | 0.4350 | 0.4767 | 0.5667
64 | 160’ | 0.0050 | 0.0150 | 0.0033 | 0.2867 | 0.4700 | 0.6450
32 | 320’ | 0.4283 | 0.1517 | 0.4067 | 0.1817 | 0.4083 | 0.6050
summary | N/A | 0.0017 | 0.0383 | 0.0100 | 0.2767 | 0.3200 | 0.2183
(a)
Relative homology – $\chi^{2}$ (empirical) – Separate Variance
---
| | Quad 3 | Quad 4
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
512 | 20’ | 0.9767 | 0.6133 | 0.8833 | 0.9283 | 0.6967 | 0.9700
256 | 40’ | 0.8100 | 0.5233 | 0.7183 | 0.2333 | 0.7300 | 0.6200
128 | 80’ | 0.7983 | 0.6400 | 0.8350 | 0.7683 | 0.6383 | 0.7500
64 | 160’ | 0.5533 | 0.4483 | 0.7083 | 0.5933 | 0.6100 | 0.6267
32 | 320’ | 0.1233 | 0.5200 | 0.1433 | 0.2100 | 0.3617 | 0.3517
summary | N/A | 0.7433 | 0.6200 | 0.4683 | 0.4717 | 0.6917 | 0.5533
(b)
Table 2: Table displaying the two-tailed $p$-values for relative homology
obtained from the empirical Mahalanobis distance or $\chi^{2}$ test, computed
from the sample covariance matrices, for different resolutions and smoothing
scales for the maps in different quadrants of the sphere for the NPIPE
dataset. Panel (a) presents the $p$-values for experiments where the variance
is computed for each hemisphere separately, and panel (b) presents results for
experiments where the hemispheres are assigned the variance of full sky. The
last entry is the $p$-value for the summary statistic computed across all
resolutions. Marked in boldface are $p$-values $0.05$ or smaller.
### 3.3 Comparison with earlier results in literature
##### Local hemisphere variance:
In the experiments with variance computed locally from the hemispheres, the
most striking feature is the significant deviation at around a degree in the
northern hemisphere. We have noticed hints of this degree-scale deviation in
the full sky analysis reported in Pranav (2022), where we report a
$2.96\sigma$ deviation in the PR3 dataset at $FWHM=80^{\prime}$. The PR4
dataset at this scale exhibits a $2.2\sigma$ deviation. Due to a weak display
of anomaly in the PR4 dataset, we rejected it in view of the larger scales
exhibiting stronger anomalies statistically.
We find a similar phenomenon in the reported investigations on the WMAP data
in literature. Park (2004) and Eriksen et al. (2004b) pioneered the
investigation of genus statistics of WMAP CMB data. While Park (2004)
restricted to small sub-degree scales, Eriksen et al. (2004b) performed a
multi-scale topo-geometrical investigation, spanning a range of sub-degree
small scales to super-degree large scales. Due to its multi-scale analysis, as
well as the fact that it is performed on WMAP data, and represents independent
evidence, we find Eriksen et al. (2004b) an excellent source for comparison
with our results. Figure 4 of Eriksen et al. (2004b) presents the Minkowski
functional curves for the WMAP data smoothed at $FWHM=1.28\deg$, where the
genus in the positive threshold range is deviant from the simulations at more
than $2\sigma$. This is more evident in the Figure 5 of Eriksen et al.
(2004b), where the MFs and the skeleton length are presented for a range of
smoothing scales. We notice a discrepancy in the genus between observations
and simulations at more than $2\sigma$ for positive thresholds for a range of
scales, most prominently around $1.28\deg$ and $1.70\deg$. The genus for the
negative thresholds shows no anomaly at these scales. This also results in the
suppression of the signal from the positive threshold anomaly in the
$\chi^{2}$-statistic for genus. Simultaneously, for the larger scales of
approximately $5$ degrees, the genus at negative thresholds is deviant by more
than $3\sigma$, and therefore, like in our case in Pranav (2022), Eriksen et
al. (2004b) also do not deem the degree scale deviation as significant in view
of the anomaly at larger scales. In the same paper, Figure 10 shows that the
asymmetry parameter between negative and positive thresholds for the genus is
weakly correlated. This is experimental evidence in support of the theoretical
fact that the different Betti numbers, which dominate the genus at different
thresholds, are independent, which further motivates the case of examining the
Betti numbers separately, in addition to their linear combination reflected in
the genus. Our experiments further support this observation, as the Betti
numbers of the hemispheres reveal a difference in their topological
properties, which is the source of weak deviation observed in the full sky
analysis for the Betti numbers as well as the associated genus statistics in
both the Planck and WMAP data, which has been overlooked and not given due
attention before.
##### Global variance:
Assigning the global variance to the hemispheres gives rise to discrepant
behavior between observations and simulations in the southern hemisphere,
specifically for the loops at scales of roughly $5$ degrees and larger. The
observed discrepancy stands at approximately $5$ standard deviations. We have
observed the hints of this phenomenon in the full sky analysis as well, where
we report a difference of $3.9$ standard deviation in the number of loops
between the observations and simulations at this scale (Pranav, 2022). The
source of this deviation is linked to the strongly deviant behavior of loops
in the southern hemisphere. Moreover, this phenomenon is observed in the WMAP
data also, where the genus at negative thresholds exhibits large deviations
from the simulations at these scales (Eriksen et al., 2004b).
## 4 Discussions and Conclusion
In this paper, we presented a multi-scale analysis of the topological
properties of the CMB temperature maps in small sectors of the sky, including
hemispheres and quads, with an aim to investigate the veracity of the
postulate of statistical isotropy. This is in continuation of the experiments
performed in Pranav et al. (2019a), Pranav (2022), and Pranav (2021a), where
we report on the full sky properties of the temperature fluctuation maps. We
have employed tools emanating from homology and its hierarchical extension
persistent homology, which form the the foundations of computational topology
(Edelsbrunner et al., 2002; Edelsbrunner & Harer, 2010; Pranav et al., 2017).
We have found various anomalous signatures in the topology of the temperature
fluctuations in the normalized maps in the different hemispheres defined in
the Galactic coordinates. The discovered anomalous signatures in the different
hemispheres depend on the section of the sky adopted for computing the mean
and variance for normalizing the maps. For experiments where the hemispheres
are assigned local mean and variance for normalization, we find that the
northern hemisphere exhibits significant deviations between the data and the
model, most prominently at scales of roughly a degree, in which case the
coincidence between the scale of the anomaly and the horizon at the epoch of
CMB is worth noting. The anomalous signatures in this case are more prominent
for topological components, however the topological holes also exhibit
anomalous behavior. In contrast, the southern hemisphere exhibits remarkable
consistency with the standard model simulations. For the experiments where the
hemispheres are assigned a global mean and variance computed from the masked
full sky, the southern hemisphere exhibits strongly anomalous behavior with
respect to the standard model simulations at scales of roughly 5 degrees and
more.
Noting that the variance of the northern hemisphere is starkly different from
the simulations, while the variance of the southern hemisphere is strongly
consistent with the model, it maybe prudent to treat the hemispheres as
arising from different models at the level of normalization. Consequently, the
results from the experiments where the variance is computed locally from the
hemispheres may be a more accurate reflection of reality than the experiments
where the hemispheres are assigned a global variance for normalization. In
view of this, the degree-scale anomalies in the behavior of the topological
components in the northern hemisphere maybe a stronger and fairer indicator of
the ground-truth, compared to the larger scale anomalies in the topological
loops in the southern hemisphere, which may possibly be an instance of an
artifact of data treatment. Taking hints from the experiments on the
hemispheres, we further tested the quadrants of the sphere assigning local
variance for normalization. In this case, we find that the first quadrant
exhibits significantly anomalous behavior with respect to the simulations.
Despite possible offsets due to an erroneous estimation of mean and variance,
considering the fact that normalization is an order preserving transformation,
and that the anomalies persist irrespective of the recipe adopted for
normalization, this points to deeper differences in the stochastic structure
of the observed and simulated CMB fields, at a level beyond normalization and
offset effects. The fact that the deviations persist in the $\chi^{2}$-tests,
which takes into consideration all the thresholds and resolutions, is another
compelling argument against the deviations being engendered by offset effects.
However, to avoid and mitigate any such effects, in future research we will
present a comparison of topology directly in the space of persistence
diagrams, which encode consolidated information about all level-sets.
Preliminary inroads in this direction have been been made in terms of modeling
persistence diagrams directly, with an analysis of the Galactic hemispheres in
Adler et al. (2017).
An agnostic interpretation of these data characteristics points to a departure
from statistical isotropy in the CMB maps, however further work is required to
convincingly ascribe the source of the anomalies to a genuinely cosmological
effect, unaccounted foreground effect, or merely systematic effects. Regarding
noise and systematics, there is an important point to consider from first
principles. The PR4 dataset purportedly has lower level of systematics and
noise compared to the PR3 dataset. If this is the ground truth, and if the
anomalous signals we have discovered are real, their significance should
increase from PR3 to PR4 datasets. We notice this trend for the NILC and SEVEM
maps, but the opposite trend for C-R and SMICA maps, which makes any
assessment about the origin of the signals inconclusive. Also important to
note is that the slightly differing results for the different maps in PR3
indicate the sensitivity of our methodology to the details of the component
separation pipeline, and maybe considered as a framework for bench-marking. In
the context of foregrounds, the recently discovered foreground effect by
Hansen et al. (2023) also deserves a specific mention. They find that there is
an effect of deepening of CMB temperature profile extending to a few degrees
around nearby large spiral galaxies. While they posit that this foreground
effect may provide a possible common explanation for a number of anomalies of
different kinds in the CMB, restricting to the topological anomalies, such an
effect may generate spurious holes in addition to deepening their temperature
profiles. However, we find it difficult to conclude that the aforementioned
foreground may account for the novel anomalous signatures presented in this
paper, for various reasons. First, the evidence presented in this paper points
to strongly anomalous signatures in the hot-spot regions as well, which cannot
be explained by deepening of temperature profiles, or creation of spurious
cold-spots. Second, the fact that the anomalous number of cold-spots appear in
the southern hemisphere when assigning a global variance for normalization,
and disappear when normalizing the hemispheres by local variance, points to
the fact that the anomalous behavior of the cold-spots in the southern
hemisphere may in fact be misleading, and engendered as an artifact of data
processing. The third compelling reason is that we find the deviations to be
the most significant in the first quad in galactic coordinates, and the
foreground map presented in Hansen et al. (2023) appears to be lacking
significant foreground contamination in this region. Regardless of the finer
details, a crucial point to note from the evidence in this paper is that the
disagreement between the data and the model may engender spurious results for
all subsequent downstream calculations such as cosmological parameter
estimation, as pointed out in Fosalba & Gaztañ aga (2021); Yeung & Chu (2022),
and may have consequences for the Hubble tension through the related mis-
estimation of cosmological parameters.
## Acknowledgements
We are indebted to Herbert Edelsbrunner, Rien van de Weygaert, and Robert
Adler for discussions and comments that have helped shape the draft. We are
grateful to Hans Kristian Eriksen and Geraint F. Lewis for important comments
on the draft. PP would also like to acknowledge the important interactions
with Reijo Keskitalo and Julian Borill, their consistent help with questions,
and incisive comments on the draft. This work is part of a project that has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement ERC
advanced grant 740021 - ARTHUS, PI: TB). P.P. is currently supported by the
Physics of Living Matter Group at the University of Luxembourg, and by the
Luxembourg National Research Fund’s grants awarded to the PI of the Physics of
Living Matter Group (ATTRACT Investigator Grant no. A17/MS/11572821/MBRACE,
and CORE Grant no. C19/MS/13719464/TOPOFLUME/Sengupta). We gratefully
acknowledge the support of PSMN (Pôle Scientifique de Modélisation Numérique)
of the ENS de Lyon, and the Department of Energy’s National Energy Research
Scientific Computing Center (NERSC) at Lawrence Berkeley National Laboratory,
operated under Contract No. DE-AC02-05CH11231, for the use of computing
resources.
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## Appendix A
### A.1 Distribution characteristics of mean, variance and Betti numbers
In the results presented in the previous sections, we have noticed features in
the topological characteristics that exhibit weak to strong dependence on the
recipe for computing mean and variance for normalizing the maps. Specifically,
computing the mean and variance locally from the hemispheres points to a
difference between the data and model in the northern hemisphere, as opposed
to the southern hemisphere that shows remarkable consistency with the model.
In contrast, computing the variance from the full sky results in a deviation
between the data and model for both the hemispheres for some scales.
As the anomalies presented in the previous sections exhibit a dependence on
the recipe for computing mean and variance, we examine the histograms of mean
and variance of the hemispheres and the full sky at ${N}=512$ in Figure 7.
From the figure, we notice the variance of the observation in the northern
hemisphere to be less than the variance from all the $600$ simulated maps,
while the southern sky is consistent with the simulations. As a result, the
full sky exhibits mildly anomalous characteristics with respect to the
simulations. When examining the mean, it is evident that both the northern and
the southern skies are consistent with the simulations. It has also been noted
in the literature that at all scales the northern hemisphere exhibits
anomalous variance with respect to the simulations, in contrast with the
southern hemisphere, which exhibits no deviation Planck Collaboration et al.
(2014b). It may therefore be prudent to treat the hemispheres as arising from
different models, when computing the mean and variance for normalization.
Figure 8 presents the histogram of ${{b}_{0}}$ from the simulations for this
threshold and resolution, with the observed value indicated by a red vertical
line. It is evident from the histogram that the distribution maybe
approximated by a Gaussian distribution.
### A.2 PR3 dataset
In this section, we present the graphs and table of $p$-values for the PR3
dataset. Figure 9 presents the graphs of the Betti numbers for the maps that
are normalized with respect to local mean and variance, while Figure 10
presents the graphs for normalization where the mean and the variance ae
computed from the masked full sky. The graphs display consistent
characteristics with that of the PR4 dataset presented in the main body of the
paper, albeit with higher significance of difference between the data and the
model. Table 3 presents the $p$-values computed from the $\chi^{2}$-statistic.
The top seven rows present the $\chi^{2}$-statistic computed from the combined
level sets for a given resolution, while the bottom row presents the same for
the combination of all thresholds at all resolutions. Commensurate with the
graphs, the $p$-values for the PR3 dataset presents higher significance of
difference between the data and the model.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7: Histogram of variance for the north, south and full sky, with the
appropriate regions masked. The histograms are computed for the map at
$N=512$. (a)
Figure 8: Histogram of ${{b}_{0}}$ values at $\nu=0.5$ for the PR3 data set
at $FWHM=80^{\prime}$. The distribution from the simulations indicates a
tendency towards a symmetric Gaussian distribution.
(a)
(b)
(c)
(d)
Figure 9: Graphs of ${{b}_{0}}$ and ${{b}_{1}}$ for the temperature maps for
the FFP10 dataset for the northern (top two rows) and the southern hemisphere
(bottom two rows). The variance is computed for each hemisphere separately
from the unmasked pixels in that hemisphere. The graphs present the normalized
differences, and each panel presents the graphs for a range of degradation and
smoothing scales. The mask used is the PR3 temperature common mask.
(a)
(b)
(c)
(d)
Figure 10: Graphs of ${{b}_{0}}$ and ${{b}_{1}}$ for the temperature maps for
the FFP10 dataset for the northern (top two rows) and the southern hemispheres
(bottom two rows). The graphs present the normalized differences, and each
panel presents the graphs for a range of degradation and smoothing scales. The
variance is computed from the full sky from the unmasked pixels. The mask used
is the PR3 temperature common mask.
Relative homology – $\chi^{2}$ (empirical) – Separate Variance
---
| | North | South
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
1024 | 10’ | 0.2400 | 0.1100 | 0.1333 | 0.9667 | 0.4567 | 0.4667
512 | 20’ | 0.4333 | 0.2367 | 0.2867 | 1.0000 | 0.8567 | 0.9667
256 | 40’ | 0.0367 | 0.3267 | 0.2733 | 0.0900 | 0.6767 | 0.2467
128 | 80’ | 0.0000 | 0.0467 | 0.0133 | 0.6533 | 0.6233 | 0.8467
64 | 160’ | 0.0633 | 0.0700 | 0.2333 | 0.2867 | 0.2600 | 0.5067
32 | 320’ | 0.1567 | 0.2467 | 0.3433 | 0.8533 | 0.2467 | 0.4600
16 | 640’ | 0.8167 | 0.5400 | 0.6933 | 0.8733 | 0.0233 | 0.0367
summary | N/A | 0.0100 | 0.0867 | 0.0000 | 0.5433 | 0.3967 | 0.0133
(a)
Relative homology – $\chi^{2}$ (empirical) – Separate Variance
---
| | North | South
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
1024 | 10’ | 0.3400 | 0.1400 | 0.3100 | 0.3600 | 0.1767 | 0.2333
512 | 20’ | 0.4433 | 0.3233 | 0.5467 | 0.2033 | 0.1533 | 0.1767
256 | 40’ | 0.2067 | 0.2167 | 0.5433 | 0.3333 | 0.0833 | 0.2567
128 | 80’ | 0.0167 | 0.3033 | 0.0867 | 0.2833 | 0.0733 | 0.1867
64 | 160’ | 0.2500 | 0.3033 | 0.3000 | 0.5733 | 0.0933 | 0.5100
32 | 320’ | 0.1500 | 0.5467 | 0.5633 | 0.5067 | 0.0033 | 0.0033
16 | 640’ | 0.6400 | 0.7167 | 0.6500 | 0.3733 | 0.0000 | 0.0000
summary | N/A | 0.2200 | 0.4500 | 0.4133 | 0.0667 | 0.0000 | 0.0000
(b)
Table 3: Table displaying the two-tailed $p$-values for relative homology
obtained from the empirical Mahalanobis distance or $\chi^{2}$ test, computed
from the sample covariance matrices, for different resolutions and smoothing
scales for the FFP10 dataset. Panel (a) presents the $p$-values for
experiments where the variance is computed for each hemisphere separately, and
panel (b) presents results for experiments where the hemispheres are assigned
the variance of full sky. The last entry is the $p$-value for the summary
statistic computed across all resolutions. Marked in boldface are $p$-values
$0.05$ or smaller.
|
# The Kinematics, Metallicities, and Orbits of Six Recently Discovered
Galactic Star Clusters with Magellan/M2FS Spectroscopy††thanks: This paper
presents data gathered with the Magellan Telescopes at Las Campanas
Observatory, Chile.
Andrew B. Pace,1 Sergey E. Koposov,2,3,4,1 Matthew G. Walker,1 Nelson
Caldwell,5 Mario Mateo, 6 Edward W. Olszewski, 7 Ian U. Roederer, 6,8 John I.
Bailey, III, 9 Vasily Belokurov,3 Kyler Kuehn, 10 Ting S. Li, 11 Daniel B.
Zucker 12,13
1McWilliams Center for Cosmology, Carnegie Mellon University, 5000 Forbes Ave,
Pittsburgh, PA 15213, USA
2Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
3Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK
4Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK
5Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS-15,
Cambridge, MA 02138, USA
6Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
7Steward Observatory, The University of Arizona, 933 N. Cherry Avenue, Tucson,
AZ 85721, USA
8Joint Institute for Nuclear Astrophysics – Center for the Evolution of the
Elements (JINA-CEE), USA
9Department of Physics, UCSB, Santa Barbara, CA 93016, USA
10 Lowell Observatory, 1400 W Mars Hill Rd, Flagstaff, AZ 86001, USA
11Department of Astronomy and Astrophysics, University of Toronto, 50 St.
George Street, Toronto ON, M5S 3H4, Canada
12 School of Mathematical and Physical Sciences, Macquarie University, Sydney,
NSW 2109, Australia
13 Macquarie University Research Centre for Astronomy, Astrophysics &
Astrophotonics, Sydney, NSW 2109, Australia
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We present Magellan/M2FS spectroscopy of four recently discovered Milky Way
star clusters (Gran 3/Patchick 125, Gran 4, Garro 01, LP 866) and two newly
discovered open clusters (Gaia 9, Gaia 10) at low Galactic latitudes. We
measure line-of-sight velocities and stellar parameters ([Fe/H], $\log{g}$,
$T_{\rm eff}$, [Mg/Fe]) from high resolution spectroscopy centered on the Mg
triplet and identify 20-80 members per star cluster. We determine the
kinematics and chemical properties of each cluster and measure the systemic
proper motion and orbital properties by utilizing Gaia astrometry. We find
Gran 3 to be an old, metal-poor (mean metallicity of ${\rm[Fe/H]}=-1.83$)
globular cluster located in the Galactic bulge on a retrograde orbit. Gran 4
is an old, metal-poor (${\rm[Fe/H]}=-1.84$) globular cluster with a halo-like
orbit that happens to be passing through the Galactic plane. The orbital
properties of Gran 4 are consistent with the proposed LMS-1/Wukong and/or
Helmi streams merger events. Garro 01 is metal-rich (${\rm[Fe/H]}=-0.30$) and
on a near circular orbit in the outer disk but its classification as an open
cluster or globular cluster is ambiguous. Gaia 9 and Gaia 10 are among the
most distant known open clusters at $R_{GC}\sim 18,~{}21.2~{}\mathrm{\,kpc}$
and most metal-poor with [Fe/H] $\sim-0.50,-0.34$ for Gaia 9 and Gaia 10,
respectively. LP 866 is a nearby, metal-rich open cluster ([Fe/H]$=+0.10$).
The discovery and confirmation of multiple star clusters in the Galactic plane
shows the power of Gaia astrometry and the star cluster census remains
incomplete.
###### keywords:
Stars: kinematics and dynamics – globular clusters: general – open clusters
and associations: general
††pubyear: 2023††pagerange: The Kinematics, Metallicities, and Orbits of Six
Recently Discovered Galactic Star Clusters with Magellan/M2FS
Spectroscopy††thanks: This paper presents data gathered with the Magellan
Telescopes at Las Campanas Observatory, Chile.– The Kinematics, Metallicities,
and Orbits of Six Recently Discovered Galactic Star Clusters with
Magellan/M2FS Spectroscopy††thanks: This paper presents data gathered with the
Magellan Telescopes at Las Campanas Observatory, Chile.
## 1 Introduction
Star clusters are among the smallest stellar structures in the universe and
are a key component of hierarchical structure assembly. They are valuable for
studying stellar populations and their evolution at a variety of ages,
metallicities, and environs (e.g., Krumholz et al., 2019; Adamo et al., 2020).
Star clusters in the Milky Way (MW) are typically divided into two categories:
the older, denser, and more luminous globular clusters (Gratton et al., 2019),
and the younger clusters in the MW disk, referred to as open clusters (e.g.,
Cantat-Gaudin, 2022).
While the census of bright halo clusters is mostly complete (Webb & Carlberg,
2021), there has been a number of faint star clusters discovered in optical
wide-field imaging surveys, pushing the luminosity and surface brightness
boundary (e.g, Koposov et al., 2007; Belokurov et al., 2014; Torrealba et al.,
2019; Mau et al., 2019; Cerny et al., 2023). The census of star clusters in
the MW mid-plane is incomplete due to the high extinction and large stellar
foreground. With recent near-infrared surveys such as the VISTA Variables in
the Via Láctea Survey (VVV) (e.g., Minniti et al., 2011; Garro et al., 2020)
and astrometric data from the Gaia mission the number of star cluster
candidates has significantly increased (e.g., Koposov et al., 2017; Torrealba
et al., 2019; Garro et al., 2020; Gran et al., 2022). However, a number of the
star cluster candidates found pre-Gaia DR2 have been shown to be false
positives once proper motions and kinematics are considered (Gran et al.,
2019; Cantat-Gaudin & Anders, 2020).
Ultimately, stellar spectroscopy is required to validate candidate star
clusters and confirm they are not a mirage of MW stars (e.g., Gran et al.,
2022). Furthermore, spectroscopic radial velocities and metallicities will
identify star cluster members. With the systemic radial velocity, the orbit
and origin of a star cluster can be determined (e.g., Massari et al., 2019;
Kruijssen et al., 2019)and the internal dynamics analyzed with large radial
velocity samples (e.g., Baumgardt & Hilker, 2018; Garro et al., 2023).
Spectroscopic metallicities can assist in determining the classification and
origin of a star cluster (e.g., Gran et al., 2022).
We present spectroscopic confirmation of three recently discovered globular
cluster candidates (Gran 3, Gran 4, Garro 01) and three newly discovered open
clusters (Gaia 9, Gaia 10, LP 866). In Section 2, we discuss our search
algorithm and independent discovery of the star cluster candidates with Gaia
DR2. In Section 3, we discuss our spectroscopic observations, velocity and
metallicity measurements, and the auxiliary data analyzed. In Section 4, we
identify members of each star cluster, measure the general kinematic and
metallicity properties, measure the spatial distribution, and determine the
orbital properties. In Section 5, we analyze the globular cluster internal
kinematics, compare the globular clusters to other MW globular clusters,
discuss the origin and potential association to accretion events, analyze the
open clusters in the context of the Galactic metallicity gradient, and compare
our results to the literature. We summarize our conclusions in Section 6.
## 2 Discovery and Candidate identification
The search for the stellar overdensities was carried out in 2018 after the
release of the Gaia DR2 using the satellite detection pipeline broadly based
on the methods presented in Koposov et al. (2008); Koposov et al. (2015), but
extended into space of proper motions.
We describe here briefly the basics behind the detection algorithm, leaving a
more detailed description to a separate contribution (Koposov et al. in
prep.).
The algorithm consists of several steps.
* •
Looping over all proper motions. In the search used here we ran overdensity
search for subsets of stars with proper motions $|\mu_{\alpha}-X_{i}|<1$,
$|\mu_{\delta}-X_{j}|<1$ where $X_{i}$,$X_{j}$ span the range of proper
motions from -15 to 15 mas/year with 1 mas/year steps.
* •
Looping over all possible distance moduli to overdensities. We perform the
search for stars selected based on an isochrone filter placed at distances
from $\sim$ 6 kpc to 160 kpc. We used the extinction corrected BP, RP and G
magnitudes and an old metal-poor PARSEC isochrone (Bressan et al., 2012) with
an age of 12 Gyr and $[{\rm Fe/H}]=-2$. We also run a single search without
any isochrone colour magnitude selection.
* •
Looping over overdensity sizes from 3 arcmin to 48 arcmin.
* •
Segmentation of the sky into HEALPIX (Górski et al., 2011) tiles. To avoid
having to work with the dataset for the entire sky the overdensity search
algorithm works with approximately rectangular-shaped HEALPIX tiles, that we
also increase in size by 20% with respect to the standard HEALPIX scheme to
ensure overlap between tiles and avoid dealing with edge effects. The exact
Nside resolution parameter and pixel scale of tiling were different for
different runs depending on memory limitations and size of the overdensity
being searched for.
* •
Creation of a pixelated stellar density map inside each tile. We use
tangential projection to map the stars into a rectangular x,y pixel grid, and
then make a 2-D histogram of stellar counts of stars selected by proper
motions, colours and magnitudes.
* •
Creation of a stellar overdensity significance map based on a stellar count
map. This step is described below in more detail.
* •
Identification of overdensities based on the significance map and merging of
candidate lists from various search configurations.
* •
Cross-matching the candidate lists with external catalogues and construction
of validation plots.
Below we provide a brief description of the algorithm that provides an
overdensity significance map given the binned 2D stellar density map. The
algorithm requires three main parameters – the kernel size, which corresponds
to the size of the overdensity $k$ we are looking for, and two background
apertures, $b_{1}$ and $b_{2}$. Here we assume that we have a rectangular grid
of stellar number counts $H(x,y)$ and we estimate the significance of the
overdensity at pixel x=0, y=0. The key difference of our approach compared to
previous approaches (i.e., Koposov et al., 2008) is that we do not rely on the
assumption of Gaussianity or even a Poisson distribution of number counts in
the map.
We first compute the number of stars $N$ in a circular aperture with radius k
around the pixel 0,0. We then need to characterize what is the probability
distribution of $P_{null}(N)$ under a null hypothesis of no overdensity to
compute the tail probability/significance.
Our model for $P_{null}(N)$ is the negative-binomial distribution, which is a
discrete Poisson-like distribution (that is, an infinite mixture of Poisson
distributions with means having a Gamma distribution). The negative binomial
distribution can be parameterized with mean $\mu$ and $\sigma^{2}$, where
$\sigma^{2}\geq\mu$ (note that the variance of the negative-binomial
distribution is a parameter, as opposed to a Poisson distribution, where it is
equal to the mean). The $\mu$ and $\sigma$ are estimated based on the number
count distribution between two background apertures $b_{1}$ and $b_{2}$.
The significance (or the Z-score) in each pixel is then assigned as
$Z=F^{-1}(P_{null}(\geq N))$, where $F^{-1}()$ is the inverse of the CDF of a
normal distribution. The significant overdensities are selected as those where
Z is larger than a certain threshold. For this work we used $Z>6$ selected
candidates.
The application of the algorithm summarized above to Gaia DR2 in 2018 yielded
a few hundred significant distinct overdensities. The absolute majority of
them were known, but around 30 objects were deemed to be likely real dwarf
galaxies or globular clusters and were selected for further inspection. The
spectroscopic follow-up of six of these objects is the subject of this paper.
Some objects from the list, such as the Eridanus IV object (Cerny et al.,
2021) have been discovered and independently followed up since.
We began our spectroscopic follow-up in 2018 and note that 4 star clusters in
our sample have since been independently discovered. We refer to these
clusters by their name in the first discovery analysis. Garro 01 was
independently discovered by Garro et al. (2020) in the near-IR VISTA Variables
in the Via Láctea Extended Survey (VVVX). Gran et al. (2022) independently
discovered Gran 3 (also known as Patchick 125) and Gran 4 with Gaia DR2
astrometry, and they were confirmed with the VVV survey. Gran 3 was
independently discovered by the amateur astronomer Dana Patchick and named
Patchick 125. We used the literature open cluster compilation from Hunt &
Reffert (2023) to search for literature cross matches for our open clusters
which includes most post-Gaia open cluster discoveries (e.g., Bica et al.,
2019; Liu & Pang, 2019; Cantat-Gaudin & Anders, 2020; Kounkel et al., 2020;
Castro-Ginard et al., 2022). One cluster, internally KGO 8, was independently
discovered by Liu & Pang (2019) and referred to as LP 866 (although in some
catalogs it is referred to as FoF 866) and Kounkel et al. (2020) and referred
to as Theia 4124. We refer to this open cluster as LP 866 here. The two
remaining open clusters in our sample are new discoveries, and we name them
Gaia 9 and Gaia 10.
## 3 Spectroscopic Follow-Up
Table 1: Spectroscopic Observations of Star Clusters. object | R.A. (deg) | Dec. (deg) | Telescope/Instrument | UT Date | Exp. Time | $N_{\mathrm{obs}}$ | $N_{\mathrm{good}}$
---|---|---|---|---|---|---|---
Gran 3 | $256.135833$ | $-35.471278$ | Magellan/M2FS | 2018-08-11 | 6900 | 54 | 41
Gran 4 | $278.092083$ | $-23.211194$ | Magellan/M2FS | 2018-08-13 | 6100 | 118 | 115
Gran 4 | $278.112121$ | $-23.103756$ | AAT/AAOmega | 2018-06-24 | 1800 | 67 | 67
Garro 01 | $212.246250$ | $-65.738333$ | Magellan/M2FS | 2018-08-11 | 5400 | 205 | 193
Gaia 9 | $119.607083$ | $-38.984639$ | Magellan/M2FS | 2018-12-06 | 5400 | 96 | 50
Gaia 10 | $121.172500$ | $-38.984444$ | Magellan/M2FS | 2018-08-15 | 5800 | 56 | 43
LP 866 | $261.651250$ | $-39.280889$ | Magellan/M2FS | 2018-12-05 | 7200 | 164 | 160
### 3.1 Spectroscopic targeting
The possible member stars from candidate stellar overdensities discovered were
selected for spectroscopic observations by using the information about the
objects that was available from their detection, such as approximate object
angular size and proper motion. We did not have a uniform target selection
strategy from object to object, so we provide a broad overview of the
selection. We typically targeted stars using Gaia DR2 astrometry and
photometry, selecting stars with proper motions within 1-3 mas/yr of the
center of the detection. We applied the astrometric_excess_noise <1 cut and
selected stars with small parallaxes $\varpi<{\rm
Max}(0.1,3\,\sigma_{\varpi})$. Since the majority of followed-up overdensities
had small angular sizes we tried to maximise the number of fibers on each
object by assigning higher priority to central targets. We also did not apply
any colour-magnitude or isochrone selection masks to the targets, other than a
magnitude limit to ensure sufficient signal to noise, and prioritising
brighter stars, such as $G<16-18$.
### 3.2 M2FS Spectroscopy
We present spectroscopic observations of six star cluster candidates that we
obtained using the Michigan/Magellan Fiber System (M2FS; Mateo et al., 2012)
at the 6.5-m Magellan/Clay Telescope at Las Campanas Observatory, Chile. M2FS
deploys 256 fibers over a field of diameter $0.5^{\circ}$, feeding two
independent spectrographs that offer various modes of configuration. We used
both spectrographs in identical configurations that provide resolving power
$\mathcal{R}\sim 24,000$ over the spectral range $5130-5190$ Å. For all six
clusters, Table 1 lists coordinates of the M2FS field center, UT date and
exposure time of the observation, the number of science targets and the number
that yielded ‘good’ observations that pass our quality-control criteria.
We process and model all M2FS spectra using the procedures described in detail
by Walker et al. (2023). Briefly, we use custom Python-based software to
execute standard processing steps (e.g., overscan, bias and dark corrections),
to identify and trace spectral apertures, to extract 1D spectra, to calibrate
wavelengths, to correct for variations in pixel sensitivity and fiber
throughput, and finally to subtract the mean sky level measured from $\sim 20$
fibers per field that are pointed toward regions of blank sky. To each
individually-processed spectrum, we fit a model based on a library of
synthetic template spectra computed on a regular grid of stellar-atmospheric
parameters: effective temperature ($T_{\rm eff}$), surface gravity ($\log g$),
metallicity ([Fe/H]) and magnesium abundance ([Mg/Fe]). Including parameters
that adjust the resolution and continuum level of the template spectra, our
spectral model has 16 free parameters. We use the software package MultiNest
to draw random samples from the 16-dimensional posterior probability
distribution function (PDF) (Feroz & Hobson, 2008; Feroz et al., 2009). We
summarize 1D posterior PDFs for each of the physical parameters according to
the mean and standard deviation of the sample returned by MultiNest.
We consider stars with ${\rm S/N}>0$ and $\sigma_{v_{\rm
los}}>5\mathrm{\,km}\mathrm{\,s}^{-1}$ as good quality measurements (Walker et
al., 2023). We consider stars with ${\rm S/N}>2$ to be good quality [Fe/H]
measurements. Our selection of cuts for good quality [Fe/H] is based on repeat
measurements and dwarf galaxy data in the M2FS catalog (Walker et al., 2023).
### 3.3 AAT Observations
One of the objects detected in the Gaia search was submitted for follow-up
observations by the Two-degree Field (2dF) spectrograph (Lewis et al., 2002)
at the Anglo-Australian Telescope (AAT). The observations were conducted
during the observing run of the Southern Stellar Stream Spectroscopic Survey
($S^{5}$) (Li et al., 2019). In particular, an internal data release (iDR3.1)
was used for this work. We refer to Li et al. (2022) for a detailed
description of the data and processing and provide a brief summary here. The
stars were observed with two arms of the spectrograph: the red arm with the
1700D grating that covers a wavelength range from 8400 to 8800 Å (including Ca
ii near-infrared triplet with $\lambda\lambda$8498, 8542, and 8662) with a
spectral resolution of $R\sim 10000$, and the blue arm with the 580V grating
that provides low resolution $R\sim 1300$ spectra covering a broad wavelength
range from 3800 to 5800 Å. The blue and red spectra for each star were then
forward modeled by the rvspecfit code (Koposov, 2019) to provide estimates of
stellar parameters, radial velocities and their uncertainties. In addition to
the velocities, we also acquired the calcium triplet (CaT) metallicities from
equivalent widths and the Carrera et al. (2013) calibration for all the member
stars as detailed in Li et al. (2022).
### 3.4 Additional Data
We use photometric and astrometric data from the Gaia EDR3 catalog (Gaia
Collaboration et al., 2021). We only utilize astrometric data that passed the
following cuts: ruwe $<1.5$ (Lindegren et al., 2021), and
astrometric_excess_noise_sig$<3$. We note that some stars that were targeted
spectroscopically do not pass these quality cuts (partly due to the Gaia DR2
target selection) and we exclude those stars from any astrometry based
analysis. For parallax measurements, we apply the parallax offset from
Lindegren et al. (2021) and include an additional offset of
$\Delta\varpi=0.007~{}{\rm mas}$ based on the globular cluster analysis of
Vasiliev & Baumgardt (2021).
We use Gaia DR3 RR Lyrae (RRL) catalog to search for candidate RRL star
cluster members (Clementini et al., 2022). We use the DECam Plane Survey
(DECaPS) DR1 griz photometric data for Gran 3, Garro 01 and LP 866 (Schlafly
et al., 2018). We search for additional spectroscopic members of our star
cluster sample in large spectroscopic surveys including SDSS APOGEE DR17
(Abdurro’uf et al., 2022), GALAH (Buder et al., 2021), and Gaia RVS DR3 (Katz
et al., 2022).
## 4 Results
Table 2: Properties of the Star Clusters. Literature $M_{V}$ measurements of the Gran 3, Gran 4, and Garro 01 are from Garro et al. (2022a); Gran et al. (2022); Garro et al. (2020). | Gran 3/Patchick 125 | Gran 4 | Garro 01 | Gaia 9 | Gaia 10 | LP 866
---|---|---|---|---|---|---
R.A. (J2000, deg) | 256.24 | 278.113 | 212.25 | 119.707 | 121.168 | 261.766
Dec (J2000, deg) | -35.49 | -23.105 | -65.62 | -39.011 | -38.928 | -39.439
l (deg) | 349.75 | 10.20 | 310.83 | 254.65 | 255.17 | 349.09
b (deg) | 3.44 | -6.38 | -3.94 | -4.97 | -3.96 | -2.42
$r_{h}$ (arcmin) | $1.7\pm 0.2$ | $2.2_{-0.4}^{+0.5}$ | $2.4_{-0.4}^{+0.6}$ | $1.4\pm 0.2$ | $1.6_{-0.2}^{+0.3}$ | $4.6_{-0.6}^{+0.7}$
$r_{h}$ (parsec) | $5.3_{-0.6}^{+0.7}$ | $14.2_{-2.5}^{+3.3}$ | $10.9_{-2.0}^{+2.6}$ | $5.5_{-0.7}^{+0.9}$ | $7.8_{-1.2}^{+1.6}$ | $3.1_{-0.4}^{+0.5}$
$r_{c}$ (arcmin) | $1.1_{-0.2}^{+0.3}$ | $1.4_{-0.4}^{+0.5}$ | $1.8_{-0.5}^{+0.7}$ | $0.8\pm 0.2$ | $1.0_{-0.2}^{+0.3}$ | $3.3_{-0.7}^{+0.8}$
$r_{t}$ (arcmin) | $>5.3$ | $>5.8$ | $>5.3$ | $>5.4$ | $>5.0$ | $>11.4$
$D$ (kpc) | $10.5$ | $21.9$ | $15.5$ | $13.8$ | $17.4$ | $2.3$
$(m-M)_{0}$ | $15.11$ | $16.70$ | $15.95$ | $15.70$ | $16.20$ | $11.80$
$R_{GC}$ (kpc) | $2.7$ | $13.7$ | $11.9$ | $18.0$ | $21.2$ | $6.3$
$M_{V}$ | -3.8 $\pm$ 0.8 | -6.45 | -5.62 $\pm$ 1 | | |
E(B-V) | $1.09$ | $0.45$ | $0.61$ | $0.86$ | $1.17$ | $1.36$
age (Gyr) | $>10$ | $<10$ | 11 $\pm$1 | $\sim$1.5 | $\sim$1 | $\sim$3
$\overline{v_{\rm los}}~{}(\mathrm{\,km}\mathrm{\,s}^{-1})$ | $90.9\pm 0.4$ | $-266.4\pm 0.2$ | $31.0\pm 0.1$ | $159.0\pm 0.3$ | $135.9\pm 0.4$ | $-9.8\pm 0.1$
$\sigma_{v}~{}(\mathrm{\,km}\mathrm{\,s}^{-1})$ | $1.9\pm 0.3$ | $1.4\pm 0.2$ | $0.4\pm 0.3$ | $1.0\pm 0.3$ | $1.4_{-0.3}^{+0.4}$ | $0.6\pm 0.1$
$\overline{{\rm[Fe/H]}}$ | $-1.83_{-0.04}^{+0.03}$ | $-1.84\pm 0.02$ | $-0.30\pm 0.03$ | $-0.50\pm 0.05$ | $-0.34\pm 0.06$ | $0.10\pm 0.03$
$\sigma_{\rm[Fe/H]}$ | $<0.16$ | $<0.10$ | $<0.14$ | $<0.16$ | $<0.14$ | $<0.22$
$\overline{\mu_{\alpha\star}}~{}(\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}})$ | $-3.74\pm 0.03$ | $0.51\pm 0.01$ | $-4.35\pm 0.02$ | $-1.08\pm 0.03$ | $-0.73\pm 0.03$ | $2.93_{-0.02}^{+0.01}$
$\overline{\mu_{\delta}}~{}(\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}})$ | $0.71_{-0.02}^{+0.01}$ | $-3.51\pm 0.01$ | $-1.09\pm 0.02$ | $1.50\pm 0.03$ | $1.60_{-0.03}^{+0.04}$ | $0.44\pm 0.02$
$\varpi~{}(\mathrm{\,mas})$ | $0.12\pm 0.01$ | $0.07\pm 0.01$ | $0.09\pm 0.01$ | $0.08\pm 0.01$ | $0.10\pm 0.02$ | $0.437\pm 0.005$
$N_{\rm v_{\rm los}},~{}N_{\rm[Fe/H]}~{}N_{\rm\mu}$ | 35, 29, 33 | 62, 52, 65 | 43, 34, 42 | 19, 19, 19 | 23, 16, 21 | 80, 79, 86
$r_{\rm peri}$ (kpc) | $2.9\pm 1.0$ | $7.6_{-1.5}^{+1.6}$ | $9.8_{-1.8}^{+1.7}$ | $13.9_{-2.4}^{+2.7}$ | $19.9_{-2.5}^{+1.9}$ | $5.85\pm 0.07$
$r_{\rm apo}$ (kpc) | $3.3_{-0.4}^{+1.0}$ | $33.9_{-6.7}^{+8.8}$ | $13.3_{-1.4}^{+2.1}$ | $17.7\pm 1.1$ | $23.9_{-3.4}^{+4.8}$ | $8.09\pm 0.05$
ecc | $0.07_{-0.02}^{+0.15}$ | $0.63_{-0.00}^{+0.01}$ | $0.16_{-0.02}^{+0.04}$ | $0.12\pm 0.06$ | $0.10_{-0.01}^{+0.04}$ | $0.161\pm 0.003$
$P$ (Myr) | $-46_{-6}^{+7}$ | $-462_{-130}^{+97}$ | $-235_{-42}^{+35}$ | $-329_{-44}^{+39}$ | $-470_{-81}^{+68}$ | $-133\pm 1$
$z_{\rm max}$ (kpc) | $2.1_{-0.5}^{+0.9}$ | $20.5_{-3.7}^{+4.9}$ | $1.3\pm 0.2$ | $1.4_{-0.2}^{+0.3}$ | $1.7_{-0.3}^{+0.4}$ | $0.18\pm 0.01$
${\rm E~{}(10^{5}~{}km^{2}~{}s^{-2}})$ | $-1.77_{-0.13}^{+0.15}$ | $-0.84\pm 0.09$ | $-1.12\pm 0.07$ | $-0.99\pm 0.05$ | $-0.85\pm 0.06$ | $-1.371\pm 0.004$
${\rm L_{Z}~{}(10^{3}~{}kpc~{}km~{}s^{-1}})$ | $0.47_{-0.13}^{+0.14}$ | $-0.49_{-0.39}^{+0.33}$ | $-2.51_{-0.38}^{+0.35}$ | $-3.39_{-0.41}^{+0.39}$ | $-4.60_{-0.59}^{+0.55}$ | $-1.59\pm 0.01$
Figure 1: Summary of the spectroscopic observations of the six star clusters
and with members coloured for each star cluster. MW stars are blue points. The
star clusters are clearly identified based on the narrow $v_{los}$ peaks. Top
left panel: line-of-sight velocity ($v_{los}$) versus metallicity ([Fe/H]).
Only stars with good quality metallicity are included. Top middle panel:
Projected radial distance from the center for the cluster versus velocity in
the Galactic center of rest ($v_{GSR}$). Top right panel: vector point diagram
($\mu_{\alpha\star}$ vs $\mu_{\delta}$). Bottom left panel: $v_{los}$ versus
surface gravity ($\log{g}$). Bottom middle panel: $v_{los}$ versus effective
temperature ($T_{\rm eff}$). Bottom right panel: [Fe/H] versus [Mg/Fe]. Only
stars with good quality ${\rm[Fe/H]}$ and $[{\rm Mg}/{\rm Fe}]$ are included.
In Figure 1, we summarize the kinematics, chemistry, and stellar parameters of
cluster members and MW foreground stars in our follow-up spectroscopic
observations. The top panels from left to right compare the line-of-sight
velocities ($v_{los}$) to the stellar metallicity ([Fe/H]), the radial
distance from the center of the cluster versus the $v_{los}$, and the proper
motion ($\mu_{\alpha\star}$, $\mu_{\delta}$). The member stars in each cluster
are highlighted in different colours. Note in the proper motion panel, only
stars with good quality astrometry are included. In the bottom panels we
compare $v_{los}$ to the surface gravity (left, $\log{g}$), $v_{los}$ to
effective temperature ($T_{\rm eff}$, center), and compare [Mg/Fe] vs [Fe/H].
Excluding LP 866, all stars are red giant branch/red clump stars (with several
horizontal branch stars in Gran 3 and Gran 4). The derived properties of the
star clusters are summarized in Table 2.
### 4.1 Cluster Properties and Spectroscopic Membership
We are able to identify the members of four clusters, Gran 3, Gran 4, Gaia 9,
and Gaia 10, based purely on the line-of-sight velocity as the cluster mean
velocity is distinct from the MW foreground. The stellar parameters ([Fe/H],
$T_{\rm eff}$, and $\log{g}$) and photometry ($G$, $G_{BP}$, and $G_{RP}$) of
the members identified from the velocities further reinforce their membership
and they are consistent with single stellar populations. While there is a
clear overdensity in the $v_{\rm los}$ distribution in the LP 866 and Garro 01
fields, there is overlap with the MW foreground and we construct mixture
models to quantitatively identify members in these objects.
#### 4.1.1 Gran 3/Patchick 125
Figure 2: Summary of the M2FS observations of Gran 3/Patchick 125. Left: line-
of-sight velocity ($v_{\rm los}$) versus metallicity ([Fe/H]) for Gran 3 M2FS
members (orange circles) and MW foreground stars (blue x’s). Center-left:
projected radius versus $v_{\rm los}$. We denote the M2FS observation of a RRL
with a green square. The red triangles and purple circles are APOGEE and Gaia
RVS members, respectively. Center: Vector point diagram for the same stars. We
include two additional RRL candidate members without spectroscopy. Center-
right: Gaia colour magnitude diagram ($G_{BP}-G_{RP}$ versus $G$). We include
all stars within 6′ that are consistent with the proper motion and parallax of
Gran 3 (small black points). We include an isochrone with age = 13 Gyr and
[Fe/H] = $-1.9$. For the photometry to match the isochrone, we increased the
standard MW reddening law to $R_{V}=3.3$. Right: DECam g-r vs g photometry
from DECaPS.
In Figure 2, we summarize the Gran 3 members and the MW foreground stars from
our M2FS observations of the Gran 3 field. In addition to the primary M2FS
sample, we include 3 RRL members from the Gaia DR3 catalog (Clementini et al.,
2022), 2 APOGEE members111First identified in Fernández-Trincado et al.
(2022)., and 7 Gaia DR3 RVS members2226 of the 7 of the members were first
identified in Garro et al. (2023).. The 36 M2FS members of Gran 3 are
identified by selecting stars in the $82<v_{\rm
los}<97\mathrm{\,km}\mathrm{\,s}^{-1}$ velocity range. The M2FS stars all are
consistent with a single metallicity and the stellar parameters ($\log{\rm g}$
and $T_{\rm eff}$) are consistent with red giant stars or horizontal branch
stars. There are two stars outside this range within $\sim
20~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ of the mean velocity of Gran 3. The
first, source_id333Here and throughout the paper source_id refers to Gaia DR3
source_id.=5977223144516980608, is a RRL star and the distance of this star
agrees with the star cluster. We consider it a cluster member but we exclude
this star from all kinematic analysis due to the velocity variability of RRL
stars. The second, source_id=5977223144516066944, is a $7\sigma$ outlier in
velocity but the stellar parameters agree with the cluster mean metallicity
and the proper motion agrees with the cluster. We exclude it from our analysis
and suggest that if it is a member, it is likely a binary star (e.g., Spencer
et al., 2018). Additional multi-epoch data is required to confirm this.
To determine the kinematics and chemistry we use a two-parameter Gaussian
likelihood function (Walker et al., 2006) and to use emcee to sample from the
posterior (Foreman-Mackey et al., 2013). We use a uniform prior for the
average and a Jeffreys prior for the dispersion. From the 35 stars in the M2FS
sample that are non-variable, we measure $\overline{v_{\rm los}}=+90.9\pm
0.4~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\sigma_{v}=1.9\pm
0.3~{}\mathrm{\,km}\mathrm{\,s}^{-1}$. With the 29 members with good quality
[Fe/H] measurements, we measure $\overline{{\rm[Fe/H]}}=-1.83_{-0.04}^{+0.04}$
and $\sigma_{\rm[Fe/H]}=0.09\pm 0.4$ ($\sigma_{\rm[Fe/H]}<0.16$). For limits
throughout this work, we list values at 95% confidence intervals. We note that
the non-zero metallicity dispersion is due to one star
(source_id=5977224587625168768; $[{\rm Fe/H}]=-1.58\pm 0.07$) that is
$3.5\sigma$ larger than the mean metallicity of Gran 3. This star has stellar
parameters and mean velocity that are otherwise consistent with Gran 3. If
this star is removed, the kinematics and mean metallicity are unchanged but
the metallicity dispersion is constrained to less than
$\sigma_{\rm[Fe/H]}<0.10$. We have opted to include this star in our sample.
From the 33 spectroscopically (M2FS, APOGEE, and Gaia RVS) identified members
with good quality astrometry, we measure:
$\overline{\mu_{\alpha\star}}=-3.74\pm
0.03\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\overline{\mu_{\delta}}=0.71_{-0.02}^{+0.01}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.10_{-0.02}^{+0.03}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\delta}}=0.03_{-0.01}^{+0.02}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and $\varpi=0.12\pm 0.01\mathrm{\,mas}$. The parallax measurement corresponds
to $d=8.6_{-0.8}^{+0.9}~{}\mathrm{\,kpc}$ and $(m-M)_{0}=14.7\pm 0.2$, which
is closer than the isochrone or RRL distance (see below). Assuming a distance
of 10.5 kpc (in agreement with these latter measurements), the proper motion
dispersions correspond to
$\sigma_{\mu_{\alpha\star}}=5.0_{-1.1}^{+1.4}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
and
$\sigma_{\mu_{\delta}}=1.7_{-0.7}^{+0.9}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$.
While $\sigma_{\mu_{\alpha\star}}$ is larger than expected,
$\sigma_{\mu_{\delta}}$ agrees with $\sigma_{v}$. With the 7 Gaia RVS members
we measure: $\overline{v_{\rm
los}}=+93.5_{-1.5}^{+1.7}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ and
$\sigma_{v}<6.6~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ (95% c.i.). The Gaia RVS
sample is consistent with the M2FS sample.
There are 3 stars in the secondary samples (1 APOGEE, 2 Gaia RVS) that are
$\sim 9~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ offset ($1-5\sigma$ outliers in
$v_{\rm los}$) from the bulk of Gran 3. 2 of these stars have repeat
measurements with other samples and those measurements are in good agreement
with the bulk velocity of the system and suggests that those stars could be
binary stars. The APOGEE star (source_id=5977223316333009024; $v_{\rm
los,~{}APOGEE}=100.2\pm 0.2\mathrm{\,km}\mathrm{\,s}^{-1}$) overlaps with Gaia
RVS ($v_{\rm los,~{}{\it Gaia}~{}RVS}=92.2\pm
2.2\mathrm{\,km}\mathrm{\,s}^{-1}$) and one of the Gaia RVS members
source_id=5977224587625168768; ($v_{\rm los,~{}{\it Gaia}~{}RVS}=99.2\pm
3.5\mathrm{\,km}\mathrm{\,s}^{-1}$) overlaps with M2FS ($v_{\rm
los,~{}M2FS}=92.6\pm 0.7\mathrm{\,km}\mathrm{\,s}^{-1}$). Both these stars may
be binary stars which would explain their offset.
We identify three RRL in the Gaia DR3 RRL catalog
(source_id=5977223144516980608, 5977224553266268928444We note that this star
is a 3-$\sigma$ outlier in $\mu_{\delta}$ compared to the systemic proper
motion, however, the large value of astrometric_excess_noise_sig suggests that
the astrometric solution may not be reliable., 5977224557581335424) that are
consistent with the proper motion and spatial position (all three have
$R<2\arcmin$) of Gran 3. One star (source_id=5977223144516980608) was observed
with M2FS. It is offset from the mean velocity of Gran 3 by $\sim
20\mathrm{\,km}\mathrm{\,s}^{-1}$. As RRL stars are variable in velocity and
vary more than $50\mathrm{\,km}\mathrm{\,s}^{-1}$ over the period (Layden,
1994), we consider this star a member. With more spectroscopic epochs the
systemic velocity of the star could be measured (e.g., Vivas et al., 2005). We
apply the metallicity correction to the absolute magnitude of a RRL in Gaia
bands to determine the absolute magnitude: $M_{G}=0.32{\rm[Fe/H]}+1.11$
(Muraveva et al., 2018). From the three RRL, we find a mean distance modulus
of $(m-M)_{0}=15.1$ corresponding to a distance of $d=10.5\mathrm{\,kpc}$.
This is slightly smaller than other distance measurements for this cluster:
$d=12.02~{}\mathrm{\,kpc}$ (Gran et al., 2022), $d=11\pm 0.5~{}\mathrm{\,kpc}$
(Fernández-Trincado et al., 2022), and $d=10.9\pm 0.5,~{}11.2\pm
0.5~{}\mathrm{\,kpc}$ (Garro et al., 2022a).
In Figure 2, we compare an old (age $=13$ Gyr) and metal-poor isochrone
([Fe/H] = $-1.9$) to Gran 3 with both Gaia and DECaPS photometry. We are able
to match the horizontal branch and the colour of the RGB if we use
$(m-M)_{0}=15.2$ and a $R_{V}=3.3$ dust law (compared to a standard of
$R_{V}=3.1$) and find it difficult to match the horizontal branch using the
RRL distance. As noted by Garro et al. (2022a), some studies suggest a lower
dust law is favored in the bulge regions (e.g., Souza et al., 2021; Saha et
al., 2019) which would disagree with the RGB of Gran 3. This larger distance
modulus is in better agreement with the literature distance measurements of
Gran 3 (Gran et al., 2022; Fernández-Trincado et al., 2022; Garro et al.,
2022a). We note that the brighter stars are bluer than the isochrone but the
bulk of the RGB matches the isochrone.
#### 4.1.2 Gran 4
Figure 3: Similar to Figure 2 but for Gran 4. We include AAT spectroscopic
observations with brown and blue open triangles corresponding to Gran 4
members and MW stars, respectively. An isochrone with age = 13 Gyr and [Fe/H]
= $-1.9$ is included as a black curve.
The summary of our spectroscopic observations of Gran 4 is shown in Figure 3.
We identify 64, 22, and 3 Gran 4 members in the M2FS, AAT and Gaia RVS sample,
respectively, with the $-280<v_{\rm los}<-260\mathrm{\,km}\mathrm{\,s}^{-1}$
selection. 12 stars in the AAT sample overlap with the M2FS sample.
Within the M2FS sample, further examination of the velocity distribution
reveals two outlier stars. The first (source_id=4077796986282497664) is a RRL
star and is offset from the mean velocity by $\sim
8~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ and by $\sim 3\sigma$ once the velocity
dispersion is considered. It is a cluster member but due to its variable
nature, we exclude it from any kinematic analysis. The second
(source_id=4077796810168905344) is a $\sim 8\sigma$ outlier in velocity and if
it is considered a member the velocity dispersion increases from
$1.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ to
$2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$. It is consistent with the mean
metallicity and proper motion of Gran 4, and the stellar parameters ($T_{\rm
eff}$ and $\log{g}$) are consistent with a red giant branch star. It seems
unlikely for this star to be MW star based on the MW velocity and metallicity
distribution, however, as it is a $\sim 8\sigma$ outlier it is either a binary
star or a MW interloper and we exclude it from the analysis.
From the 62 non-variable M2FS members we find, $\overline{v_{\rm
los}}=-266.4\pm 0.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\sigma_{v}=1.4\pm
0.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\overline{{\rm[Fe/H]}}=-1.84\pm 0.02$
and $\sigma_{\rm[Fe/H]}<0.10$. Due to tail to zero dispersion and a lack of a
clear peak, we do not consider the metallicity dispersion resolved and list an
upper limit. The stellar parameters of these stars ($T_{\rm eff}$ and
$\log{g}$) are consistent with red giant branch stars or horizontal branch
stars which further confirms our membership identification. With the 3 Gaia
RVS members we measure: $\overline{v_{\rm
los}}=-262.7_{-3.7}^{+3.6}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$,
$\sigma_{v}<6.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ (95% c.i.).
From the 22 AAT members, we find $\overline{v_{\rm los}}=-265.9\pm
0.4~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ and
$\sigma_{v}=1.5_{-0.3}^{+0.4}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$. 12 of these
stars overlap with the M2FS sample. There is one velocity outlier
(source_id=4077796397852026240) with $\sim 3\sigma$, however, this star is in
the M2FS sample with almost the exact same velocity. Removing this star
decreases the velocity dispersion to $\sim 1~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
from $\sim 1.5~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ in the AAT sample but its
inclusion or exclusion does not affect the M2FS kinematics and we opt to
include it. From the calcium triplet metallicities, we compute
$\overline{{\rm[Fe/H]}}=-1.82\pm 0.06$ and
$\sigma_{\rm[Fe/H]}=0.21_{-0.05}^{+0.06}$. The metallicity dispersion is
clearly resolved in contrast to our expectation of a single stellar population
in a star cluster and the lack of a metallicity dispersion in the M2FS sample.
The source of the metallicity dispersion in the AAT data is unclear. As the
M2FS sample is larger, we adopt the velocity and metallicity results from the
M2FS sample as our primary results.
From the combined M2FS, AAT, and Gaia RVS sample there are 65 stars with good
quality astrometric measurements. From these stars we measure:
$\overline{\mu_{\alpha\star}}=+0.51\pm
0.01\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$, and
$\overline{\mu_{\delta}}=-3.51\pm
0.01\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.02_{-0.01}^{+0.03}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\delta}}=0.05_{-0.03}^{+0.02}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and $\varpi=0.07\pm 0.01\mathrm{\,mas}$. The parallax measurement corresponds
to $d=14.6_{-1.9}^{+2.5}~{}\mathrm{\,kpc}$ and $(m-M)_{0}=15.8\pm 0.3$ which
is closer than the other distance measurements (see below). Assuming a
distance of 21.9 kpc (from the best-fit isochrone distance), the proper motion
dispersion correspond to:
$\sigma_{\mu_{\alpha\star}}=2.6_{-1.2}^{+2.7}\mathrm{\,km}\mathrm{\,s}^{-1}$
and $\sigma_{\mu_{\delta}}=5.4_{-2.8}^{+2.5}\mathrm{\,km}\mathrm{\,s}^{-1}$.
We identify two RRL (source_id=4077796986282497664, 4077796573965756928) in
the Gaia DR3 RRL catalog (Clementini et al., 2022) as members of Gran 4 based
on their proper motion and distance. The first RRL is also in the VVV RRL
(Molnar et al., 2022) and OGLE RRL catalogs (Soszyński et al., 2019). There is
a third candidate RRL (source_id=4077796608325479168) from the PanSTARRS1 RRL
catalog (Sesar et al., 2017), however, it is not in the Gaia DR3 RRL catalog
so we do not include it in the analysis. It is considered a variable star in
Gaia DR3 but only has a best_class_score=0.4 for being an RRL. Additional time
series data are required to confirm the status of this star. From the two RRL
members, $\mu=16.5,16.6$ corresponding to $d=19.9,21~{}\mathrm{\,kpc}$. This
is slightly closer than our best-fit $\mu\sim 16.7$ from matching the
horizontal branch to a metal-poor isochrone. It is also closer than
$\mu=16.84$, $d=22.49~{}\mathrm{\,kpc}$ from Gran et al. (2022).
#### 4.1.3 Garro 01
Figure 4: Same as Figure 2 but for Garro 01. In contrast to the other
globular clusters (Gran 3 and Gran 4) we use a mixture model to identify Garro
01 members and stars with membership $>0.01$ are coloured according to their
membership probability in the colour bars in the center-right and right
panels. We include an isochrone from the photometric analysis of Garro et al.
(2020) with an age = 11 Gyr and [Fe/H] = $-0.7$ (black) and an isochrone with
our best estimate using the spectroscopic metallicity ([Fe/H]= -0.3) and an
age = 4 Gyr (red). There are no candidate RRL stars in Garro 01.
There is overlap in the MW foreground and star cluster velocity distributions
for Garro 01 and LP 866 and we construct mixture models to account for this
overlap. The total likelihood for our mixture models is:
$\mathcal{L}=f_{\rm cluster}\mathcal{L}_{\rm cluster}+(1-f_{\rm
cluster})\mathcal{L}_{\rm MW}\,,$ (1)
where $\mathcal{L}_{\rm cluster}$, $\mathcal{L}_{\rm MW}$, and $f_{\rm
cluster}$ correspond to the cluster population, the MW population, and the
fraction of stars in the star cluster, respectively (e.g., Pace et al., 2020,
2021). We assume the probability distributions of each data component are
separable:
$\mathcal{L}_{\rm cluster/MW}=\mathcal{L}_{\rm spatial}\mathcal{L}_{\rm
PM}\mathcal{L}_{\rm v_{los}}\mathcal{L}_{\rm[Fe/H]}\,.$ (2)
Where $\mathcal{L}_{\rm spatial}$, $\mathcal{L}_{\rm PM}$, $\mathcal{L}_{\rm
v_{los}}$, and $\mathcal{L}_{\rm[Fe/H]}$ are the spatial likelihood, the
proper motion likelihood, the line-of-sight velocity likelihood, and the
metallicity likelihood, respectively. To compute the membership of each star,
we compare the ratio of the cluster likelihood to total likelihood for each
star: $p_{\rm member}=\mathcal{L}_{\rm cluster}/\mathcal{L}$ (e.g., Martinez
et al., 2011).
For Garro 01, we primarily analyze a mixture model with $v_{\rm los}$ and
[Fe/H] but also consider a second model with spatial information. We assume
the $v_{\rm los}$ and [Fe/H] likelihood distributions are Gaussian for both
the star cluster and MW components. We apply the following cuts to the Garro
01 spectroscopic sample to remove MW foreground stars: $T_{\rm
eff}-2\times\sigma_{T_{\rm eff}}<6000K$,
$\log_{10}{g}-2\times\sigma_{\log_{10}{g}}<4$, a parallax cut
($\varpi-3\sigma_{\varpi}<0.064$), and a loose $G_{BP}-G_{RP}$ colour cut of
0.25 around an age = 11 Gyr and [Fe/H] = $-0.6$ MIST isochrone (Dotter, 2016).
The isochrone selection is applied to remove blue MW main sequence stars from
the sample which have no overlap in colour-magnitude space with the cluster
members. From our primary mixture model, we find: $\overline{v_{\rm
los}}=+31.0\pm 0.1~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\sigma_{v}=0.4\pm
0.3~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
($\sigma_{v}<0.8~{}\mathrm{\,km}\mathrm{\,s}^{-1}$),
$\overline{{\rm[Fe/H]}}=-0.30\pm 0.03$ and $\sigma_{\rm[Fe/H]}<0.14$. In
Figure 4, we summarize the properties of the spectroscopic members identified
in the mixture model. Stars are coloured by their membership probability.
For the second spatial model we use a conditional likelihood and we assume
that the fraction of stars is spatially dependent (e.g., Martinez et al.,
2011; Pace et al., 2021):
$f_{\rm cluster}(R)=\Sigma_{\rm cluster}(R)/(\Sigma_{\rm
cluster}(R)+N\Sigma_{\rm MW}(R)).\\\ $ (3)
Where $\Sigma$ is the projected stellar distribution and $N$ is the relative
normalization between the cluster and MW spatial distributions We assume a
King (1962) distribution for the Garro 01 distribution with parameters from
Garro et al. (2020) and assume that the MW distribution is constant over the
small region examined. We utilize a conditional likelihood as there is an
unknown spatial selection function.
While the chemodynamic properties of Garro 01 from the two models are nearly
identical, there are some minor differences in the membership of individual
stars between the two models. The conditional likelihood model has larger
membership for stars near the center and lower membership for the two most
distant stars. The difference in membership between the models for individual
stars is small ($\sum|p_{\rm standard}-p_{\rm conditional}|=1.7$) and the
overall membership is similar for the two models; both models have $\sum
p=46.8\pm 1.9$. We consider stars with $p>0.9$ as high confidence members and
there are 39 and 42 members identified in the standard model and conditional
likelihood model, respectively and note there are 43 high confidence members
with $p>0.9$ in either model.
From the 42 members with good astrometry (42 M2FS and 1 Gaia RVS), we measure:
$\overline{\mu_{\alpha\star}}=-4.35\pm
0.02~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\overline{\mu_{\delta}}=-1.09\pm
0.02~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.09_{-0.02}^{+0.02}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and
$\sigma_{\mu_{\delta}}=0.08_{-0.02}^{+0.03}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$.
We measure $\varpi=0.08\pm 0.01\mathrm{\,mas}$ corresponding to
$d=11.9_{-1.5}^{+2.1}~{}\mathrm{\,kpc}$ and $(m-M)_{0}=15.4\pm 0.3$. The
distance from the parallax is closer than the measurement derived from
isochrone fits. Assuming a distance of 15.3 kpc from Garro et al. (2020), the
proper motion dispersion terms correspond to:
$\sigma_{\mu_{\alpha\star}}=6.6_{-1.4}^{+1.5}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
and
$\sigma_{\mu_{\delta}}=5.8_{-1.8}^{+1.8}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$.
Our proper motion measurement agrees with the Gaia DR2 proper motion
measurement, $\overline{\mu_{\alpha\star}}=-4.68\pm
0.47~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$, and
$\overline{\mu_{\delta}}=-1.35\pm
0.45~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$ from Garro et al. (2020).
We identify 2 members in the Gaia RVS sample with velocities and proper motion
consistent with Garro 01. Both stars are more evolved than the M2FS sample but
roughly match the isochrone. Both stars are included in Figure 4 as purple
circles.
We do not identify any RRL that have the same proper motion or a consistent
distance with Garro 01. With the distance modulus of Garro et al. (2020), we
can match the red clump of our spectroscopic sample with an isochrone and we
adopt this distance for our analysis.
In Figure 4, we show optical colour-magnitude diagrams using Gaia and DECaPS
photometry. The prominent feature in the CMDs is the red clump and complements
the near-infrared discovery photometry from Garro et al. (2020). Our
spectroscopic metallicity measurement is more metal-rich than the isochrone
analysis of Garro et al. (2020) which found [Fe/H] = $-0.7$ with their CMDs
fits (black isochrone in Figure 4). However, we cannot match the colour of the
system with this age and spectroscopic metallicity. If we assume the
spectroscopic metallicity, the isochrone is redder than the photometry. We
discuss this in more detail in Section 5.1.
#### 4.1.4 Gaia 9
Figure 5: Similar to Figure 2 but for Gaia 9. The best fit isochrone is age =
1.5 Gyr and [Fe/H] = $-0.5$.
We identify 19 M2FS members of Gaia 9 with the velocity selection: $150<v_{\rm
los}<170~{}\mathrm{\,km}\mathrm{\,s}^{-1}$. There is one star inside this
velocity range (source_id=5537860050401680000) that is a non-member based on
its proper motion. The properties of the members and proper motion selected
stars are displayed in Figure 5. From the 19 members, we measure:
$\overline{v_{\rm los}}=+159.0\pm 0.3~{}\mathrm{\,km}\mathrm{\,s}^{-1}$,
$\sigma_{v}=1.0\pm 0.3~{}\mathrm{\,km}\mathrm{\,s}^{-1}$,
$\overline{{\rm[Fe/H]}}=-0.50\pm 0.06$ and $\sigma_{\rm[Fe/H]}<0.16$. There is
one additional member Gaia RVS that we identify
(source_id=5537859848550503168) based on the radial velocity and proper motion
and we include it in Figure 5. From the 19 members with good astrometry, we
measure: $\overline{\mu_{\alpha\star}}=-1.08\pm
0.03~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\overline{\mu_{\delta}}=+1.50\pm
0.03~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.07\pm
0.04~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\delta}}=0.08_{-0.03}^{+0.03}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and $\varpi=0.08\pm 0.01\mathrm{\,mas}$. The parallax measurement corresponds
to $d=12.9_{-2.0}^{+3.0}~{}\mathrm{\,kpc}$ and $(m-M)_{0}=15.6_{-0.4}^{+0.5}$
which agrees with our isochrone derived distance (see below). Assuming a
distance of 13.8 kpc (from our isochrone derived distance), the proper motion
dispersion terms correspond to:
$\sigma_{\mu_{\alpha\star}}=4.8_{-2.4}^{+2.4}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
and
$\sigma_{\mu_{\delta}}=5.0_{-2.1}^{+2.3}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$.
In the Gaia colour-magnitude diagram (Figure 5), the majority of the
spectroscopic members are red clump stars. We estimate the distance of the
cluster to be $(m-M)_{0}=15.7$ or $d=13.8~{}\mathrm{\,kpc}$ based on the red
clump using an MIST isochrone with age = 1.5 Gyr and [Fe/H] = $-0.5$. The blue
stars at $G_{0}\sim 17$ are likely the top of the main sequence and we use
this feature to assist in estimating the age of the system. Gaia 9 is younger
than the other clusters examined thus far and it likely an open cluster.
#### 4.1.5 Gaia 10
Figure 6: Similar to Figure 2 but for Gaia 10. The best fit isochrone is age =
1 Gyr and [Fe/H] = $-0.5$. Similar to Gran 3, we increase the extinction to
find an adequate fit ($R_{V}=3.3$).
Gaia 10 has similar properties to Gaia 9 but is more distant. We summarize our
spectroscopic sample in Figure 6. We identify 23 members of Gaia 10 with a
velocity selection: $v_{\rm los}>120\mathrm{\,km}\mathrm{\,s}^{-1}$. Two stars
(source_id=5534905976200827776 and 5540909996882971520) have kinematics (line-
of-sight velocity and proper motion) that agree with the cluster but both
stars are significant outerliers in metallicity. 5534905976200827776 is
$3.2\sigma$ more metal-rich while 5540909996882971520 is $4.9\sigma$ more
metal-poor and the inclusion of either star results in an offset mean
metallicity and non-zero metallicity dispersion. We consider both stars non-
members. The inclusion of the star as a members would infer to a non-zero
metallicity dispersion. We note that this same star is in the Gaia RVS catalog
with a similar velocity. From the spectroscopic members, we measure:
$\overline{v_{\rm los}}=+135.9\pm 0.4~{}\mathrm{\,km}\mathrm{\,s}^{-1}$,
$\sigma_{v}=1.4_{-0.3}^{+0.4}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$,
$\overline{{\rm[Fe/H]}}=-0.34\pm 0.06$ and $\sigma_{\rm[Fe/H]}<0.14$.
From the 21 members with good astrometry, we measure:
$\overline{\mu_{\alpha\star}}=-0.74\pm
0.03~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\overline{\mu_{\delta}}=+1.60_{-0.03}^{+0.04}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.05_{-0.03}^{+0.05}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\delta}}=0.06_{-0.03}^{+0.04}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and $\varpi=0.09\pm 0.02\mathrm{\,mas}$. The parallax measurement corresponds
to $d=10.8_{-1.9}^{+3.1}~{}\mathrm{\,kpc}$ and $(m-M)_{0}=15.2_{-0.4}^{+0.6}$
which is much closer than our isochrone derived distance (see below). Assuming
a distance of 17.4 kpc (from our isochrone derived distance), the proper
motion dispersion terms correspond to:
$\sigma_{\mu_{\alpha\star}}=4.2_{-2.6}^{+3.8}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
and
$\sigma_{\mu_{\delta}}=5.0_{-2.8}^{+3.5}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$.
We compare theoretical isochrones based on the spectroscopic metallicity to
the Gaia colour-magnitude diagram Figure 6. With an age of $1~{}{\rm Gyr}$ and
distance modulus of $(m-M)_{0}=16.2$ ($d=17.4~{}\mathrm{\,kpc}$) we can fit
the red clump and the possible main sequence turnoff based on the proper
motion selected sample. Similar to Gran 3, to match the colour of the
isochrone a larger extinction coefficient of $R_{V}=3.3$ is required. There
remains considerable spread in the colour of the members. This may be due to
differential reddening. We consider Gaia 10 an open cluster.
#### 4.1.6 LP 866
Figure 7: Same as Figure 4 but for LP 866. Similar to Garro 01, we include a
colourbar for the membership from the mixture model. The best fit isochrone is
age = 3 Gyr and [Fe/H] = $+0.1$ but with a reduced extinction ($\sim 53\%$ of
the E(B-V) value). We include DECam g-r vs g photometry from DECaPS.
LP 866 is the only star cluster where the entirety of the M2FS spectroscopic
sample is located on the main-sequence. Similar to Garro 01, we run a mixture
model to account for the MW foreground distribution with $v_{\rm los}$, proper
motion, and [Fe/H] components. We do not include a conditional likelihood run
as the spatial distribution was not known beforehand. We assume the proper
motion distribution is a truncated multi-variate Gaussian for both components
with limits: $1.5<\mu_{\alpha\star}<3.5~{}{\rm mas~{}yr^{-1}}$,
$-0.4<\mu_{\delta}<1.2~{}{\rm mas~{}yr^{-1}}$. While the proper motion was
included in the spectroscopic target selection (based on Gaia DR2 astrometry),
the proper motion dispersion is resolved in contrast to the other star
clusters. We excluded stars outside of this proper motion limit and two bright
stars with discrepant parallax measurements.
With the mixture model, we identify 80 high confidence members ($p>0.9$) and
measure the following properties for LP 866: $\overline{v_{\rm los}}=-9.8\pm
0.1~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\sigma_{v}=0.6\pm
0.1~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, $\overline{{\rm[Fe/H]}}=0.10\pm 0.03$,
$\sigma_{\rm[Fe/H]}=0.15\pm 0.04$ ($\sigma_{\rm[Fe/H]}<0.22$),
$\overline{\mu_{\alpha\star}}=2.93_{-0.02}^{+0.01}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\overline{\mu_{\delta}}=0.44_{\pm}0.02~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
$\sigma_{\mu_{\alpha\star}}=0.08_{-0.02}^{+0.02}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$,
and
$\sigma_{\mu_{\delta}}=0.10_{-0.01}^{+0.02}~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$.
The overall membership is $\sum p=86.4\pm 4.8$. As there is a non-zero
metallicity dispersion, it is possible that our model has incorrectly
identified some MW stars as cluster members. From the 86 members with good
astrometry, we measure $\varpi=0.437\pm 0.005\mathrm{\,mas}$ corresponding to
$d=2.29\pm 0.03~{}\mathrm{\,kpc}$ and $(m-M)_{0}=11.80\pm 0.02$. Assuming a
distance of 2.29 kpc, the proper motion dispersion terms correspond to:
$\sigma_{\mu_{\alpha\star}}=0.9\pm 0.02~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ and
$\sigma_{\mu_{\delta}}=1.1\pm 0.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$.
In the Gaia RVS sample there is a clear overdensity and velocity peak distinct
from the MW population within the central 8′of LP 866. We find 14 stars in the
Gaia RVS catalog that are consistent with the velocity, proper motion, and
parallax of LP 866. All 14 stars are more evolved than the M2FS sample and aid
in matching a stellar isochrone. With the 14 Gaia RVS members we find:
$\overline{v_{\rm los}}=-10.6\pm 0.8~{}\mathrm{\,km}\mathrm{\,s}^{-1}$, and
$\sigma_{v}=0.2_{-0.1}^{+0.9}~{}\mathrm{\,km}\mathrm{\,s}^{-1}$
($\sigma_{v}<2.2~{}\mathrm{\,km}\mathrm{\,s}^{-1}$); and from the 12 stars
with good quality astrometry we find: $\overline{\mu_{\alpha\star}}=2.88\pm
0.01~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$, and
$\overline{\mu_{\delta}}=0.44\pm
0.02~{}\mathrm{\,\mathrm{\,mas}\mathrm{\,yr}^{-1}}$. These results are
consistent with the M2FS sample.
We include Gaia and DECaPs colour magnitude diagrams in Figure 7. We match
theoretical isochrones to estimate the age of the system. Unlike the other two
open clusters, we have a confident distance and metallicity measurement. We
find that reducing the E(B-V) by $\sim 53\%$ and setting the to age =
$10^{9.5}$ = $\sim$ 3 Gyr provides an adequate match. We note that the we
varied the extinction to match the RGB of the Gaia RVS stars and varied the
age to match the main sequence turn-off of the M2FS sample with the Gaia
photometry. The same isochrone provided an similarly adequate match to the g-r
vs g DECaPs photometry. More in depth modeling is required to improve
constraints on the age and extinction of the cluster. Regardless, we consider
LP 866 an open cluster.
### 4.2 Spatial Distribution
Figure 8: Projected radial stellar density profile of our star cluster
sample. Top: the globular clusters, from left to right: Gran 3, Gran 4, Garro
01. Bottom: the open clusters, from left to right: Gaia 9, Gaia 10, LP 866.
To measure the spatial distribution of each star cluster, we construct a
larger proper motion selected sample from the Gaia DR3 catalog based on the
systemic proper motion found from our spectroscopic sample and apply a spatial
mixture model (Equations 1, 2). We do not apply this methodology to the
spectroscopic sample as it is not spatially complete and has an unknown
spatial target selection. For the spatial likelihood we model the star cluster
with two density profiles. The first profile is a Plummer distribution
(Plummer, 1911):
$\Sigma(R)=\frac{1}{\pi
r_{p}^{2}}\frac{1}{\left(1+\left(R/r_{p}\right)^{2}\right)^{2}}$ (4)
where $r_{p}$ is the Plummer scale radius (for a Plummer profile $r_{p}$ is
equivalent to the 2D deprojected half-light radius). The second is the King
profile (King, 1962):
$\Sigma_{\star}(R)\propto\left[\left(1+\frac{R^{2}}{r_{c}^{2}}\right)^{-1/2}-\left(1+\frac{r_{t}^{2}}{r_{c}^{2}}\right)^{-1/2}\>\right]^{2}$
(5)
where $r_{c}$ is the core radius and $r_{t}$ is the tidal radius. We model a
small region near each cluster and assume that the MW background is constant
within that small area after a proper motion selection is applied.
For the Gaia selected sample, we apply the following cuts: a 3-$\sigma$
selection in proper motion, a parallax selection ($\varpi-\varpi_{\rm
cluster}-3\sigma_{\varpi}<0$), $G<20$, $R<R_{\rm max}$, and stars with good
astrometry (i.e., satisfy our astrometric cuts in Section 3.4). We will refer
to this Gaia selected sample and utilize the same sample for examining the
colour-magnitude diagrams of the clusters. We use $R_{\rm max}=12\arcmin$ for
LP 866 and $R_{\rm max}=6\arcmin$ for all other clusters. For Garro 01 we
additionally apply a loose $G_{BP}-G_{RP}$ colour cut of 0.25 around an age =
11 Gyr and [Fe/H] = $-0.6$ MIST isochrone (Dotter, 2016) following the
spectroscopic selection. For the other clusters, the above selection primarily
identifies stars with a stellar population that agrees with the spectroscopic
sample. Any photometric outliers (i.e., MW stars) will be roughly distributed
uniformly within the small area examined and not bias the spatial distribution
calculations.
The Plummer and King fits along with the binned stellar profile of all six
clusters are shown in Figure 8. In general, the results from the Plummer and
King profile fits agree and provide adequate fits. Due to the low number of
stars, there is no preference for one profile over the other. We are unable to
constrain $r_{t}$ and generally only provide lower limits. For the globular
clusters, we find $r_{h}=1.7\pm 0.2\arcmin$, $r_{h}=2.2_{-0.4}^{+0.5}\arcmin$,
and $r_{h}=2.4_{-0.4}^{+0.6}~{}\arcmin$ corresponding to
$r_{h}=5.3_{-0.6}^{+0.7}~{}\mathrm{\,pc}$,
$r_{h}=14.2_{-2.5}^{+3.3}~{}\mathrm{\,pc}$, and
$r_{h}=10.9_{-2.0}^{+2.6}~{}\mathrm{\,pc}$ from the Plummer profile fits for
Gran 3, Gran 4, and Garro 01, respectively. With the King profile, we find
$r_{c}=1.1_{-0.2}^{+0.3}\arcmin$, $r_{c}=1.4_{-0.4}^{+0.5}\arcmin$, and
$r_{c}=1.8_{-0.5}^{+0.7}~{}\arcmin$ for Gran 3, Gran 4, and Garro 01,
respectively. For comparison, Gran et al. (2022) find $r_{h}=1.05\pm
0.04\arcmin$ and $r_{h}=1.14\pm 0.02\arcmin$, for Gran 3 and Gran 4,
respectively. These are smaller than the sizes we infer. Garro et al. (2020)
measure $r_{c}=2.5\pm 1.5\arcmin$ for Garro 01 and a poorly constrained
$r_{t}$ which agrees with our measurement.
For the open clusters, we find $r_{h}=1.4\pm 0.2\arcmin$,
$r_{h}=1.6_{-0.2}^{+0.3}\arcmin$, and $r_{h}=4.6_{-0.6}^{+0.7}\arcmin$ with
the Plummer profile fits for Gaia 9, Gaia 10, and LP 866, respectively. With
the King profile, we find $r_{c}=0.8\pm 0.2\arcmin$,
$r_{c}=1.0_{-0.2}^{+0.3}\arcmin$, and $r_{c}=3.3_{-0.7}^{+0.8}~{}\arcmin$ for
Gaia 9, Gaia 10, and LP 866, respectively. The results for the six clusters
are included in Table 2.
### 4.3 Orbital Properties
Figure 9: Five example orbits of Gran 3 (top), Gran 4 (middle), and Garro 01
(bottom) drawn from the observational uncertainties and integrated for 1 Gyr
(Gran 3) or 2 Gyr (Gran 4 and Garro 01).
Figure 10: Five example orbits of Gaia 9 (top), Gaia 10 (middle), LP 866
(bottom) drawn from the observational uncertainties integrated for 2 Gyr (Gaia
9 and Gaia 10) or 1 Gyr (LP 866).
We use the gala package to compute the orbits of the six star clusters and
compare them to other MW globular clusters. We use the default
MilkyWayPotential Galactic potential from Price-Whelan (2017). This potential
consists of two Hernquist (1990) spheroids to model the stellar bulge and
nucleus, a Miyamoto & Nagai (1975) axisymmetric stellar disk, and a NFW dark
matter halo (Navarro et al., 1996). For each cluster we compute the integrals
of motion, $E$ and $L_{z}$, the approximately conserved quantity $L_{\perp}$
(Massari et al., 2019) and the orbital pericenter ($r_{\rm peri}$), apocenter
($r_{\rm apo}$), and eccentricity. We list the results in Table 2. We apply
the same analysis to the MW globular clusters using the phase space results
from Vasiliev & Baumgardt (2021). We compute 1000 orbits drawn from each
satellite’s 6D phase distribution in Table 2 and compute statistics from these
runs. We use the astropy v4.0 frame (Astropy Collaboration et al., 2013, 2018)
for the Sun’s position and velocity: distance to Galactic Center,
$D_{\odot}=8.122\mathrm{\,kpc}$,
$v_{\odot}=(12.9,245.6,7.78)~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ (Drimmel &
Poggio, 2018; Gravity Collaboration et al., 2018; Reid & Brunthaler, 2004).
In Figures 9 and 10, we show 5 example orbits drawn from the observational
errors of each cluster. Gran 3 is on a near circular orbit ($e\sim 0.12$) in
the inner bulge ($R_{GC}\sim 2.7\mathrm{\,kpc}$) that is confined to the plane
of the disk ($z_{max}\sim 1.8\mathrm{\,kpc}$). Gran 3 is on a retrograde orbit
($L_{Z}\sim+0.45~{}{\rm kpc^{2}~{}Myr^{-2}}$). Gran 4 has an eccentric orbit
($e\sim 0.63$) with a small pericenter ($r_{\rm peri}\sim
8~{}\mathrm{\,kpc}$), large apocenter ($r_{\rm apo}\sim 35~{}\mathrm{\,kpc}$),
and is not confined to the MW midplane ($z_{\rm max}\sim
21~{}\mathrm{\,kpc}$). Gran 4 is an halo globular cluster that is currently
near passing the MW midplane. We find that Garro 01 is on a circular orbit
($e\sim 0.16$) that is confined to the Galactic midplane ($z_{\rm max}\sim
1.3~{}\mathrm{\,kpc}$) at a relatively large Galactocentric radius ($\sim
10-13~{}\mathrm{\,kpc}$). The orbits of the three younger open clusters are
all circular (${\rm ecc}\sim 0.1-0.16$), disk-like orbits. Relative to other
open clusters Gaia 9 and Gaia 10 are at large Galactic distances ($R_{\rm
GC}\sim 18~{}{\rm and}\sim 21~{}\mathrm{\,kpc}$) and have higher angular
momentum $L_{Z}\sim-3.5--4.7~{}{\rm kpc^{2}~{}Myr^{-2}}$. The largest source
of uncertainty in our orbit modeling comes from the distance measurement.
## 5 Discussion
We have presented accurate kinematics and metallicity measurements from
Magellan/M2FS spectroscopy for three recently discovered globular clusters,
Gran 3, Gran 4 and Garro 01, and the discovery and spectroscopic confirmation
of three young open clusters, Gaia 9, Gaia 10, and LP 866. Here we consider
our results in the context of the MW star cluster population.First, we comment
on the nature of Garro 01 and whether it is an old globular cluster or open
cluster (Section 5.1). In particular, how do Gran 3, Gran 4 and Garro 01
relate to the MW globular cluster population and other recently discovered
clusters (Section 5.3)? Are these new globular clusters connected to accretion
events or were they formed in-situ (Section 5.4)? How do the open clusters
compare to the Galactic radial metallicity gradient (Section 5.5)? We conclude
by comparing our results to the literature.
### 5.1 The Nature of Garro 01
Garro et al. (2020) classify Garro 01 as a globular cluster based on its close
similarity to the globular cluster 47 Tuc but several magnitudes fainter. Our
spectroscopic metallicity is more metal-rich ([Fe/H]$=-0.3$) than the
photometric analysis ([Fe/H]$=-0.7$). The orbit of Garro 01 is a disk-like
orbit and Garro 01 is confined to the Galactic plane ($z_{\rm max}\sim
1.3~{}\mathrm{\,kpc}$, ${\rm ecc}\sim 0.16$). Both properties are consistent
with the open cluster population. The age of a star cluster can be key for
determining its origin as a globular cluster or open cluster (e.g., Garro et
al., 2022a).
As previously noted, we had difficultly matching the spectroscopic metallicity
and literature age ($11\pm 1$ Gyr) with Gaia and DECaPS photometry as the
isochrone was redder than the photometry. To estimate the age, we vary the age
at a fixed metallicity ([Fe/H]$=-0.3$) and check whether the color of the red-
giant branch is matched. For this exercise, we examine both Gaia,
$G_{BP}-G{RP}$, and DECaPS, $g-r$, and $r-i$ color. Our best estimate for the
age between $2-13$ Gyr is 4 Gyr (red isochrone in Figure 4). For younger ages
$<3.5~{}{\rm Gyr}$, the main sequence turn-off would be apparent in our sample
which we do not observe. The isochrones with older ages ($>6$ Gyr) are redder
than the observed data. In addition, we find that the Gaia RVS candidate
members are better fit with the younger age.
An age of $\sim 4~{}{\rm Gyr}$ suggests that Garro 01 is an open cluster. This
agrees with the metallicity and disk-like orbit of Garro 01. An accurate age
measurement from deeper photometry would confidently classify this star
cluster as either an open or globular cluster. For the reminder of the
discussion, we will analyze Garro 01 with both the globular clusters and open
clusters in our sample.
### 5.2 Globular Cluster Kinematics
Figure 11: Rotation curves with sinusoidal by-hand fits for Gran 3, Gran 4,
and Garro 01. Figure 12: Projected radius versus velocity dispersion
($\sigma_{\rm los}$) profile for Gran 3 (left), Gran 4 (center), and Garro 01
(right). The spatial errorbar corresponds to the size of the bin and the bins
have an equal number of stars (excluding the last bin). The clusters have 17,
16, and 15 stars per bin. We include velocity dispersion model fits with black
lines and the shaded bands correspond to the error. For Gran 4 (center) we
include a combined M2FS and AAT binned profile (orange bins).
Globular clusters have more complex kinematics than the simple constant
velocity dispersion model we have explored including rotation (e.g., Sollima
et al., 2019) and velocity dispersion profiles (Baumgardt & Hilker, 2018). We
search for rotation by comparing the difference between the mean line-of-sight
velocity across different position angles. We show the results of this
exercise in Figure 11. There is potential rotation on the order of $\sim
1-2\mathrm{\,km}\mathrm{\,s}^{-1}$ in the three clusters, however, when
considering the mean velocity errors it is not significant.
In general, globular clusters have velocity dispersion profiles that decrease
with radius (e.g., Baumgardt & Hilker, 2018) and we search for a radial
dependence in the velocity dispersion by binning the data. We show the binned
velocity dispersion profiles of the three globular clusters in Figure 12. Each
bin contains 18 (Gran 3), 16 (Gran 4), and 15 stars (Garro 01). For Gran 4 we
show the results with the M2FS data (blue) and combined M2FS + AAT
data555There is a $-1.36\mathrm{\,km}\mathrm{\,s}^{-1}$ offset applied to the
AAT data based on the repeat measurements. (orange). Combining the AAT and
M2FS data only increases the sample by 12 stars but there are 10 stars with
improved velocity precision due to multiple measurements. The M2FS velocity
dispersion profiles of Gran 3 and Gran 4 clearly decrease with radius. The
combined M2FS + AAT profile of Gran 4 is more consistent with the constant
dispersion model but the central bin has a larger velocity dispersion. With
Garro 01 the binned profile only measures an upper limit, similar to the
global fit in our mixture model.
To better constrain and quantify the velocity dispersion profile and/or
rotation we explore detailed models. We model the velocity dispersion with a
Plummer profile (Plummer, 1911) and the radially dependent velocity dispersion
profile is: $\sigma^{2}(R)=\sigma_{0}^{2}/\sqrt{1+(r/r_{0})^{2}}$, where
$\sigma_{0}$, and $r_{0}$ are free parameters. We use the following rotation
profile (e.g., Mackey et al., 2013; Cordero et al., 2017; Alfaro-Cuello et
al., 2020): $V_{\rm rot}=\frac{2V_{\rm max}}{r_{\rm
peak}}\frac{X_{PA}}{1+\left(X_{PA}/r_{\rm peak}\right)^{2}}$, where $V_{\rm
max}$, $r_{\rm peak}$, and $\theta_{PA}$ (which determines $X_{PA}$) are free
parameters. We use the dynesty package to sample the posterior and compute
Bayesian evidence for model comparison (Speagle, 2020; Koposov et al., 2022).
We apply the velocity dispersion and rotation models both separately and
together for all three clusters. The inferred velocity dispersion profile fits
are included in Figure 12 with their corresponding errors. For Garro 01 we
examine the stars with membership $>0.9$. For all three clusters the
functional velocity dispersion profile is favored over the constant, non-
rotating model but it is not favored at a statistically significant level. The
dispersion models for Gran 3 and Gran 4 both favor a larger dispersion in the
center of the clusters that decreases with radius. In terms of Bayesian
evidence666We use the scale of Trotta (2008) to quantify significance.
$\ln{Z}$ ranges of 0-1, 1-2.5, 2.5-5.5, $>5.5$ correspond to inconclusive,
weak, moderate, and substantial evidence in favor of the new model., we find
$\ln{Z}=1.7,~{}0.7,~{}0.3$ in favor of the $\sigma(R)$ model for Gran 3, Gran
4, and Garro 01, respectively. Gran 3 is favored with weak evidence whereas
the others are inconclusive. No rotation models are favored and the rotation
models produce an upper limit to the rotation. The rotation models do have a
non-zero peak but large portions of the posterior remain consistent with no
rotation. While there are coherent rotation signals in Figure 11, the non-zero
velocity dispersion and errors in the rotation curve are consistent with no
rotation. To improve constraints on the velocity dispersion profile or probe
potential rotation, larger samples of stars are required.
Last, we estimate the dynamical mass and corresponding mass-to-light ratio of
the three globular clusters. Specifically, we compute the dynamical mass using
the dispersion supported mass estimator from Errani et al. (2018):
$M(r<1.8R_{h})\approx 3.5\times 1.8R_{h}\langle\sigma^{2}_{\rm los}\rangle
G^{-1}$. This approximator is insensitive to the unknown underlying velocity
anisotropy (Walker et al., 2009; Wolf et al., 2010; Errani et al., 2018) but
assumes that the velocity dispersion is approximately constant with radius
which may not be true for globular clusters or the globular clusters in our
sample. With our line-of-sight velocity dispersion and half-light radii
measurements, we measure $M(r<1.8R_{h})=2.7\times 10^{4}\mathrm{\,M_{\odot}}$,
$4.0\times 10^{4}\mathrm{\,M_{\odot}}$, and $2.3\times
10^{3}\mathrm{\,M_{\odot}}$ ($<1.1\times 10^{4}\mathrm{\,M_{\odot}}$) for Gran
3, Gran 4, and Garro 01, respectively. The corresponding mass-to-light ratios
are777For reference, a Plummer profile at $r=1.8r_{\rm p}$ encloses 66.8% of
the total mass. are 1.8 and $0.2$ ($<1.1$), for Gran 4 and Garro 01,
respectively. There are two literature $M_{V}$ values for Gran 3: $M_{V}=-3.8$
(Garro et al., 2022a) and $M_{V}=-6.02$ (Gran et al., 2022). These correspond
to mass-to-light ratios of 1.8 (for $M_{V}=-6.02$) and 14.2 (for
$M_{V}=-3.8$). The mass-to-light ratios for Gran 3 and Gran 4 agree with old
stellar populations. More detailed dynamical modeling, focused on star
clusters would improve this analysis (e.g., Gieles & Zocchi, 2015; Song et
al., 2021).
### 5.3 Comparison to the Globular Cluster Population
In Figure 13, we compare the sizes (based on the 2D projected half-light
radii, $R_{h}$), the metallicity ([Fe/H]), and absolute magnitudes ($M_{V}$)
of our globular cluster sample (Gran 3, Gran 4, Garro 01) to the MW globular
cluster population (Harris, 1996) and to other recently discovered globular
clusters (or candidates) at low Galactic latitude. The (incomplete) list of
recently discovered globular clusters primarily in the Galactic disk and bulge
includes: FSR 1758 (Barbá et al., 2019; Vasiliev & Baumgardt, 2021; Myeong et
al., 2019; Romero-Colmenares et al., 2021), FSR 19 (Obasi et al., 2021), FSR
25 (Obasi et al., 2021), Garro 2 (Garro et al., 2022b), ESO456-29/Gran 1 (Gran
et al., 2019), Gran 2, Gran 5 (Gran et al., 2022), Patchick 99 (Garro et al.,
2021), Pfleiderer 2 (Ortolani et al., 2009), Ryu 059, Ryu 879 (Ryu & Lee,
2018), VVV CL001 (Minniti et al., 2011; Olivares Carvajal et al., 2022), VVV
CL002 (Moni Bidin et al., 2011), VVV CL160 (Minniti et al., 2021), Sagittarius
II (Mutlu-Pakdil et al., 2018; Longeard et al., 2021), and Crater 1 (Weisz et
al., 2016; Kirby et al., 2015). We note that the classification of some
objects is uncertain and spectroscopy is needed.
Our globular cluster sample is generally fainter than the MW globular cluster
population which explains their recent discovery with Gaia astrometry. Gran 4
and Garro 01 are both larger than most clusters ($r_{h}>10~{}\mathrm{\,pc}$).
The large size of Garro 01 is particularly unusual as almost all metal-rich
([Fe/H] $>-1.5$) MW globular clusters have smaller sizes
($r_{h}<10~{}\mathrm{\,pc}$). The exceptions to this are Palomar 12, which is
associated with Sagittarius (e.g., Law & Majewski, 2010; Massari et al.,
2019), and the Fornax 6 globular cluster associated with the Fornax dwarf
spheroidal galaxy (Wang et al., 2019; Pace et al., 2021). Unlike the other
large clusters, Garro 01 is on a circular disk-like orbit making it less
likely to have an ex-situ origin. Gran 3 and Gran 4 are in the metal-poor tail
of the MW globular cluster population as they are more metal-poor than $\sim
83\%$ of the globular clusters in the Harris catalog. In contrast, Garro 01 is
one of the more metal-rich globular clusters. If Garro 01 is confirmed as a
younger open cluster that could explain its large size compared to other
metal-rich globular clusters. In summary, Gran 3, Gran 4, and Garro 01 have
properties consistent with the MW globular cluster population.
Figure 13: Comparison of Gran 3, Gran 4, and the ambiguous open/globular
cluster Garro 01 with MW globular cluster population. Blue are clusters from
the Harris (1996) catalog, and orange are other recently discovered globular
clusters in the MW disk and bulge (see text for name and citations). Left:
Absolute magnitude ($M_{V}$) vs metallicity ([Fe/H]). Center: 2D half-light
radius ($R_{h}$) vs absolute magnitude ($M_{V}$). Right: 2D half-light radius
($R_{h}$) vs metallicity ([Fe/H]).
### 5.4 Origin and Connection to Accretion Events
Figure 14: Comparison of Gran 3, Gran 4, and the ambiguous open/globular
cluster Garro 01 with the MW globular cluster population. Left: energy ($E$)
versus angular momentum in the z-direction ($L_{z}$). Center: angular momentum
in the z-direction ($L_{z}$) versus angular momentum in the perpendicular-
direction ($L_{\perp}$). Right: orbital pericenter ($r_{\rm peri}$) versus
apocenter ($r_{\rm apo}$). The MW globular clusters are coloured according to
their MW infall merger event (see text for details). Gran 4 is a candidate for
the LMS-1/Wukong merger or Helmi stream merger and Garro 01 is a candidate for
the Aleph merger. Gran 3 is a candidate member of the Galactic bulge
component.
The MW has a population of in-situ and accreted/ex-situ globular clusters
(e.g., Myeong et al., 2018; Massari et al., 2019; Kruijssen et al., 2019). To
determine whether Gran 3, Gran 4, and Garro 01 are associated with any
accretion events we compare the orbital properties of our sample to globular
clusters associated with known events. In Figure 14, we compare the orbital
properties of our globular cluster sample with the MW globular cluster
population. Specifically we examine the energy, angular momentum in the
z-direction, and the angular momentum in the perpendicular direction, and the
pericenter and apocenter. We group the MW clusters based on accretion/merger
events888We have not included the Sequoia+Arjuna+Iiloi structures (Myeong et
al., 2019; Naidu et al., 2020), or Pontus structure (Malhan, 2022) mergers as
there is little overlap with the three globular clusters in our sample.
Furthermore, Thamnos is not included as there are no known globular clusters.
, including: Sagittarius (Sgr), Gaia-Sausage/Enceladus (GSE) (Belokurov et
al., 2018; Helmi et al., 2018), LMS-1/Wukong (Yuan et al., 2020; Naidu et al.,
2020), Aleph (Naidu et al., 2020), Cetus (Newberg et al., 2009), the Helmi
stream (Helmi et al., 1999; Koppelman et al., 2019a), low-
energy/Koala/Kraken/Heracles (Massari et al., 2019; Kruijssen et al., 2019,
2020; Forbes, 2020; Horta et al., 2021) and the in-situ bulge population. We
use Malhan et al. (2022) to associate globular clusters and merger events for
GSE (we additionally include globular clusters from Massari et al. 2019 for
the GSE sample), Sgr, Cetus, LMS-1/Wukong, and the in-situ bulge population.
We follow Naidu et al. (2020) to associate Palomar 1 and the Aleph structure.
The association of the Helmi stream globular clusters is taken from Callingham
et al. (2022). We note between different analyses there is overlap between the
globular clusters assigned to the Helmi stream, GSE, and LMS-1/Wukong
accretion events. The Malhan (2022) analysis does not associate any globular
clusters with the Helmi streams and the Callingham et al. (2022) analysis does
not associate any globular clusters with the LMS-1/Wukong merger. For the
Kraken merger, we use the identification from Callingham et al. (2022). The
Kraken globular clusters overlap with the low-energy globular clusters from
Massari et al. (2019), the Heracles accretion event (Horta et al., 2021) and
the Koala merger from Forbes (2020) and all may be the same merger event. For
simplicity we only include the Kraken merger. We note the
Kraken/Koala/Heracles mergers (and associated globular clusters) in terms of
their chemistry are consistent with being born in-situ, in a pre-disk
population known as Aurora (Belokurov & Kravtsov, 2022). The identification of
each globular cluster with a particular merger event depends on the
methodology and sample (e.g., globular cluster, stellar stream, halo star),
and different analyses have assigned the same globular cluster to different
events or the in-situ population.
In the $E$-$L_{Z}$ plane Gran 3 is located near globular clusters associated
with the Galactic bulge and the low-energy/Koala/Kraken merger (Massari et
al., 2019; Kruijssen et al., 2019, 2020; Forbes, 2020). Gran 3 may be an
extension the Galactic bulge component to higher energy. The Kraken globular
clusters generally have smaller $L_{Z}$ than Gran 3 and have prograde orbits
with larger eccentricity (Massari et al., 2019; Kruijssen et al., 2019, 2020;
Forbes, 2020). As this was one of the first MW mergers the globular clusters
have low metallicities, similar to Gran 3. While there is overlap, the
retrograde orbit of Gran 3 disfavors an association with the Kraken merger.
There are several retrograde accretion events in the stellar halo including
the Sequoia+Arjuna+Iiloi event (Malhan et al., 2022) and Thamnos structure
(Koppelman et al., 2019b). The Thamnos structure has the lowest energy of the
retrograde structures and has a similar metallicity to Gran 3 (Naidu et al.,
2020; Horta et al., 2023), however, the energy is larger than Gran 3 and it is
unlikely for Gran 3 to be associated with the Thamnos merger. We consider Gran
3 to be a member of the Galactic bulge globular cluster group.
We find that Gran 4 is closet to the LMS-1/Wukong merger event in integral of
motion space ($E$, $L_{Z}$, $L_{\perp}$) and in orbital parameters ($r_{\rm
peri}$, $r_{\rm apo}$). While the apocenter and energy is higher than other
the clusters in LMS-1/Wukong merger, the agreement becomes better if the
distance of Gran 4 decreases. We note there is not agreement in the number or
assignment of globular clusters to merger events. In particular, Callingham et
al. (2022) assigns the same LMS-1/Wukong globular clusters here to the Helmi
streams. While Gran 4 is close in the $E$-$L_{Z}$ space to globular clusters
associated with the GSE merger, the GSE globular clusters have smaller
$L_{\perp}$ and $r_{\rm peri}$. We consider Gran 4 to be a candidate member of
the LMS-1/Wukong merger or Helmi streams.
Garro 01 has broad agreement with the energy, angular momentum in the
z-direction, eccentricity, and metallicity of the Aleph structure (Naidu et
al., 2020). There is only one candidate globular cluster in this structure,
Palomar 1 (Naidu et al., 2020). However, Garro 01 is confined to the disk
plane ($z_{\rm max}<1.5$) and the Aleph structure has a strong vertical action
and orbits with larger $z_{\rm max}$. We consider Garro 01 a candidate member
of the Aleph structure but it is more likely to be an in-situ outer disk
cluster.
Garro et al. (2020) suggested that Garro 01 could be associated with the
Monoceros ring (MRi) structure (Newberg et al., 2002). The MRi is proposed to
be either the remnants of tidally disrupted dwarf galaxy (e.g., Peñarrubia et
al., 2005) or a Galactic warp and flare (e.g., Sheffield et al., 2018). While
Garro 01 is not near the previously detected component of MRi, orbital
analysis of MRi suggests there is overlap at location of Garro 01 (Conn et
al., 2008; Grillmair et al., 2008). However, the radial velocity of the model
predictions and Garro 01 are offset. We measure $v_{\rm
gsr}\sim-143\mathrm{\,km}\mathrm{\,s}^{-1}$ for Garro 01 and the radial
velocity of different models varies between $v_{\rm gsr}\sim 0--100$ (see
Figure 17 in Li et al., 2012). The MRi models also predict larger distances
$D\gtrsim 20~{}{\rm kpc}$ than has been inferred for the cluster and the
observed metallicity distribution of MRi is more metal-poor than Garro 01 (see
Figure 13 of Zhang et al., 2022). The modeling and analysis of the MRi and its
connection to Garro 01 would benefit from a dedicated search in this region of
sky but the radial velocity disagreement suggests they are not associated.
### 5.5 Tracing the Galactic Metallicity Gradient with Open Clusters
Figure 15: Comparison of Gaia 9, Gaia 10, LP 866, and the ambiguous
open/globular cluster Garro 01 with MW open cluster population. We include
literature measurements from Spina et al. (2022) as blue points. Top:
Galactocentric radius ($R_{GC}$) versus metallicity ([Fe/H]). Bottom: angular
momentum in the Z-direction ($L_{Z}$) versus metallicity ([Fe/H]). In both
panels, we include the best fit relations derived from the literature open
cluster sample in Spina et al. (2022).
Open clusters are an important tracer of the Galactic radial metallicity
gradient as each cluster can be age-dated with individual chemical elements
studied (e.g., Jacobson et al., 2016; Donor et al., 2020; Spina et al., 2021;
Gaia Collaboration et al., 2023). The Galactic metallicity gradient traces
Galactic formation and evolution scenarios and the complex interplay between
star formation, stellar evolution, stellar migration, gas flows, and cluster
disruption in the Galactic disk (e.g., Chiappini et al., 1997; Schönrich &
Binney, 2009; Kubryk et al., 2015; Spitoni et al., 2019).
We compare the three new open clusters (Gaia 9, Gaia 10, LP 866) to the
literature open clusters (from Spina et al., 2022) and the Galactic radial
metallicity distribution in Figure 15. The Spina et al. (2022) sample includes
open cluster data from several different literature surveys including: APOGEE,
Gaia-ESO, GALAH, OCCASO, and SPA. We include the best fit relation relation
between $R_{GC}-{\rm[Fe/H]}$ and $L_{Z}-{\rm[Fe/H]}$ for literature open
clusters from Spina et al. (2022). We include comparisons to the Galactic
radius and the the angular momentum in the z-direction ($L_{Z}$). $L_{Z}$ is
conserved and the current Galactic radius may not be not representative of
their birth radius as open clusters may have undergone radial migration (e.g.,
Chen & Zhao, 2020). In both cases, the best fit relation becomes shallower at
large radius. Previous measurements have suggested that the relation flattens
out at large radii (e.g. Frinchaboy et al., 2013; Donor et al., 2020).
The open clusters analyzed here are in general agreement with the open cluster
population trends with metallicity, Galactocentric radius, and $L_{Z}$. While
our spectroscopic follow-up has only measured [Fe/H] for three more open
clusters, Gaia 9 and Gaia 10 are among the most metal-poor open clusters in
the MW open cluster population. Gaia 10 has bluethe largest $L_{Z}$ of any MW
open cluster. The properties of Garro 01 are consistent with the Galactic
metallicity gradient as traced by open clusters. Future analyses of the
Galactic radial metallicity gradient will be improved by including Garro 01,
Gaia 9, and Gaia 10 and the metallicity measured from our Magellan/M2FS
spectroscopy.
### 5.6 Comparison to Previous Studies
Of the six star clusters studied only Gran 3 has previous spectroscopic
follow-up. Gran et al. (2022) presented VLT/MUSE spectroscopy of Gran 3 and
found $v_{\rm los}=74.32\pm 2.70\mathrm{\,km}\mathrm{\,s}^{-1}$ and
${\rm[Fe/H]}=-2.37\pm 0.18$. Both measurements are discrepant with our results
and other Gran 3 spectroscopic studies (Fernández-Trincado et al., 2022; Garro
et al., 2023). There is a $\sim 20~{}\mathrm{\,km}\mathrm{\,s}^{-1}$ offset
between the mean radial velocities measured in Gran et al. (2022) compared to
our results and literature. While velocity zero-point offsets of a few
$\mathrm{\,km}\mathrm{\,s}^{-1}$ are common between different
instruments/methods, this value is too large to be caused by a zero point
offset. It is unclear what the origin of this offset is. We note that all our
members are consistent with the same radial velocity, stellar parameters from
a single stellar population, a single metallicity, and consistent proper
motions. In Figure 10 of Gran et al. (2022), the proper motions are consistent
with the majority of their members having similar proper motions but there are
only a few radial velocity members. Some members may be missing due to the
field-of-view of Gran 3 but it is possible that the radial velocity peak was
misidentified.
Fernández-Trincado et al. (2022) analyzed high resolution APGOEE spectroscopy
of two stars in Gran 3. Due to their sample of two stars, the mean velocity of
Gran 3 they measure is offset from our by $\sim
4\mathrm{\,km}\mathrm{\,s}^{-1}$. The metallicity from Fernández-Trincado et
al. (2022) is [Fe/H]$=-1.7\pm 0.09$ is larger than our measurement but it is
consistent within uncertainties. Garro et al. (2023) identified 6 members in
the Gaia RVS sample and their velocity measurement ($v_{\rm los}=93.1\pm
3.6\mathrm{\,km}\mathrm{\,s}^{-1}$) agrees with our measurement within
uncertainties. In our analysis of the Gaia RVS data we identified one
additional Gran 3 member (Section 4.1.1).
We find larger angular sizes for Gran 3 and Gran 4 compared to Gran et al.
(2022). For reference, Gran et al. (2022) find $R_{h}=1.05\pm 0.04~{}{\rm
arcmin}$ and $R_{h}=1.14\pm 0.02~{}{\rm arcmin}$ for Gran 3 and Gran 4,
respectively, compared to our values of $R_{h}=1.7\pm 0.2~{}{\rm arcmin}$ and
$R_{h}=2.2_{-0.4}^{+0.5}~{}{\rm arcmin}$ for Gran 3 and Gran 4, respectively.
The source of this discrepancy could be due to different photometry (Gaia
versus near-IR) or fitting methodology. For Garro 01 there is excellent
agreement between our King profile fits ($r_{c}=1.8_{-0.5}^{+0.7}\arcmin$) and
the results ($r_{c}=2.1\pm 1.5\arcmin$) in Garro et al. (2020). We note that
the absolute magnitude of Gran 3 is discrepant between Garro et al. (2022a)
($M_{V}\sim-3.8$) and Gran et al. (2022) ($M_{V}\sim-6.02$).
Our orbital analysis of Gran 3 is similar to literature results (Gran et al.,
2022; Fernández-Trincado et al., 2022; Garro et al., 2023). Both Fernández-
Trincado et al. (2022) and Garro et al. (2023) include a rotating bar in their
modeling which is not included in our modeling. Compared to the other studies
the value of $z_{\rm max}$ is smaller and the energy lower. We attribute this
to the lower distance we assumed in this work. Compared to Fernández-Trincado
et al. (2022) and Garro et al. (2023) we have a more circular orbit (lower
eccentricity) which agrees with Gran et al. (2022). We note that compared to
other studies we have more precise velocity and proper motion measurements.
## 6 Conclusion
We have presented the spectroscopic follow-up of three recently discovered
globular clusters and three recently discovered open clusters. Our main
findings are as follows:
* •
We have independently discovered three globular clusters (Gran 3/Patchick 125,
Gran 4, Garro 01) and three open clusters (Gaia 9, Gaia 10, LP 866) with Gaia
astrometry. Gaia 9 and Gaia 10 are new discoveries presented here.
* •
We have presented spectroscopic follow-up with Magellan/M2FS and measured
stellar parameters of 601 stars and identified 273 members across 6 star
clusters and confirmed the legitimacy of all six clusters. In addition, we
have presented AAT/AAOmega spectroscopy of Gran 4 which confirms our M2FS
results.
* •
We find Gran 3 (Patchick 125) is an old, metal-poor globular cluster on a
retrograde orbit trapped within the Galactic bulge. From our M2FS
spectroscopy, we identified 37 members and measured a heliocentric velocity of
$v_{\rm los}=90.9\pm 0.4\mathrm{\,km}\mathrm{\,s}^{-1}$ and metallicity of
${\rm[Fe/H]}=-1.83_{-0.03}^{+0.04}$. In addition, there are 2 APOGEE and 7
Gaia RVS members. From our orbital analysis, Gran 3 has a near circular orbit
(${\rm ecc}\sim 0.07$) and orbital pericenter and apocenter of $2.9~{}{\rm
kpc}$ and $3.3~{}{\rm kpc}$, respectively. Gran 3 is likely an in-situ bulge
globular cluster.
* •
Gran 4 is an old, metal-poor globular cluster with a halo-like orbit that is
passing though the Galactic mid-plane. We identified 63 members from our M2FS
spectroscopy and 22 members (12 unique) from our AAT/AAOmega spectroscopy. We
measured a heliocentric velocity of $v_{\rm los}=-266.4\pm
0.2\mathrm{\,km}\mathrm{\,s}^{-1}$ and metallicity of ${\rm[Fe/H]}=-1.84\pm
0.02$. In addition, there are 3 Gaia RVS members. From our orbital analysis,
Gran 4 has an eccentric orbit circular orbit (${\rm ecc}\sim 0.63$) and
orbital pericenter and apocenter of $7.6~{}{\rm kpc}$ and $33.9~{}{\rm kpc}$,
respectively. Gran 4 is a candidate member of the LMS-1/Wukong and/or Helmi
stream merger events.
* •
Garro 01 is a metal-rich star cluster on an outer disk-like orbit. We
identified 43 members with our M2FS spectroscopy and measured a heliocentric
velocity of $v_{\rm los}=31.0\pm 0.1\mathrm{\,km}\mathrm{\,s}^{-1}$ and
metallicity of ${\rm[Fe/H]}=-0.30\pm 0.03$. There is more overlap in velocity
with the MW foreground and we constructed a mixture model to quantitatively
account for the MW foreground. In addition, there are 2 candidate Gaia RVS
members. We found that Garro 01 has a relatively large size ($R_{h}\sim
11~{}{\rm pc}$) compared to other metal-rich globular clusters
($R_{h}<5~{}{\rm pc}$). From our orbital analysis, Garro 01 has a circular
orbit (${\rm ecc}\sim 0.16$) and orbital pericenter and apocenter of
$9.8~{}{\rm kpc}$ and $13.3~{}{\rm kpc}$, respectively. We estimated an age of
4 Gyr, which is younger than previous analysis (11$\pm 0.5$ Gyr Garro et al.,
2020). Combined with the metallicity and orbit, this suggests that Garro 01 is
an open cluster but a confident classification requires a more detailed age
measurement and we consider the classification ambiguous.
* •
Both Gran 3 and Gran 4 have evidence for radially declining velocity
dispersion profiles (Figure 12). There is inconclusive evidence for rotation
in the globular clusters (Figure 11).
* •
We have confirmed Gaia 9, Gaia 10, and LP 866 as open clusters and identified
19-83 spectroscopic members from our M2FS spectroscopy. We measured
metallicities of $-0.50$, $-0.34$, and $+0.10$ and estimated ages of 1.5, 1,
and 3 Gyr from isochrone fits for Gaia 9, Gaia 10, and LP 866, respectively.
All three open clusters are on circular, disk-like orbits. Gaia 9 and Gaia 10
are among the most distant ($R_{GC}\sim 18,~{}21.2~{}\mathrm{\,kpc}$) and most
metal-poor open clusters known and have some of the largest angular momentum
in the z-direction. These clusters will assist in tracing the Galactic
metallicity gradient to larger radii (Figure 15).
The Milky Way star cluster population remains incomplete and Gaia astrometry
has revolutionised our understanding of star clusters. We have
spectroscopically confirmed six star clusters and there remain many more
candidate star clusters that require spectroscopic follow-up.
## Acknowledgements
We thank the referee for their helpful comments. ABP is supported by NSF grant
AST-1813881. M.G.W. acknowledges support from NSF grants AST-1813881 and
AST-1909584. SK was partially supported by NSF grants AST-1813881 and
AST-1909584. EO was partially supported by NSF grant AST-1815767. NC is
supported by NSF grant AST-1812461. MM was supported by U.S. National Science
Foundation (NSF) grants AST-1312997, AST-1726457 and AST-1815403. IUR
acknowledges support from NSF grants AST-1613536, AST-1815403, AST-2205847,
and PHY-1430152 (Physics Frontier Center/JINA-CEE). DBZ acknowledges support
from Australian Research Council grant DP220102254. We thank Lorenzo Spina for
sharing their open cluster catalog. EO wants to remember Jill Bechtold here.
For the purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.
This work has used data acquired at the Anglo-Australian Telescope. We
acknowledge the traditional custodians of the land on which the AAT stands,
the Gamilaraay people, and pay our respects to elders past and present.
This research has made use of NASA’s Astrophysics Data System Bibliographic
Services. This paper made use of the Whole Sky Database (wsdb) created by
Sergey Koposov and maintained at the Institute of Astronomy, Cambridge by
Sergey Koposov, Vasily Belokurov and Wyn Evans with financial support from the
Science & Technology Facilities Council (STFC) and the European Research
Council (ERC).
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions.
SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
www.sdss4.org.
SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.
Software: dynesty (Speagle, 2020; Koposov et al., 2022), astropy (Astropy
Collaboration et al., 2013, 2018), matplotlib (Hunter, 2007), NumPy (Walt et
al., 2011), iPython (Pérez & Granger, 2007), SciPy (Virtanen et al., 2020),
corner.py (Foreman-Mackey, 2016), emcee (Foreman-Mackey et al., 2013) , Q3C
(Koposov & Bartunov, 2006), gala (Price-Whelan, 2017), galpy (Bovy, 2015),
MultiNest (Feroz & Hobson, 2008; Feroz et al., 2009).
## Data Availability
We provide our Magellan/M2FS and AAT/AAOmega catalogs and a machine readable
version of Table 2 at Zenodo under a Creative Commons Attribution license:
doi:10.5281/zenodo.7809128. The other catalogs used in our analysis (Gaia DR3,
DECaPS, APOGEE) are publicly available.
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# WAVELET CHANNEL ATTENTION MODULE WITH A FUSION NETWORK FOR SINGLE IMAGE
DERAINING
Single image deraining is a crucial problem because rain severely degenerates
the visibility of images and affects the performance of computer vision tasks
like outdoor surveillance systems and intelligent vehicles. In this paper, we
propose the new convolutional neural network (CNN) called the wavelet channel
attention module with a fusion network. Wavelet transform and the inverse
wavelet transform are substituted for down-sampling and up-sampling so feature
maps from the wavelet transform and convolutions contain different frequencies
and scales. Furthermore, feature maps are integrated by channel attention. Our
proposed network learns confidence maps of four sub-band images derived from
the wavelet transform of the original images. Finally, the clear image can be
well restored via the wavelet reconstruction and fusion of the low-frequency
part and high-frequency parts. Several experimental results on synthetic and
real images present that the proposed algorithm outperforms state-of-the-art
methods.
Index Terms— Wavelet transform, single image deraining, fusion, channel
attention, convolutional neural network
## 1 Introduction
Single image deraining is a crucial problem because rain occludes the
background scene, appears in different locations and decreases the performance
of computer vision tasks like outdoor surveillance systems and intelligent
vehicles [1]. The rain streaks can be seen as linear noises which may vary in
size, direction and density. Moreover, in real cases, when rain accumulation
is dense, the individual streaks cannot be observed clearly. Accumulated rain
streaks reduce the visibility in a manner more similar to fog, and create a
haze-like phenomenon in the background. This foggy phenomenon can be described
as the haze model [2], and the whole model [3] is written as
$J=T\odot(I+\sum_{i}^{n}S_{i})+(1-T)\odot A$ (1)
where $J$ means the observed rain streak image, $I$ means the clear image,
$S_{i}$ means a rain streak layer with the same direction, $n$ means the
maximum number of layers, $A$ means the global atmospheric light, $T$ means
the atmospheric transmission and $\odot$ denotes element-wise multiplication.
Several methods try to analyze visual priors to capture deterministic and
statistical properties of rainy images [4, 5, 6]. However, these methods tend
to introduce undesirable artifacts, since their handcrafted priors from human
observations do not always hold in diverse real-world images. Instead of
applying handcrafted visual priors, recently, deep-learning-based methods [3,
7, 8, 9] are proposed, and these methods usually perform more accurate than
conventional priors-based methods with significant performance gains.
We also consider neural networks for single image deraining since the
deraining model is a crude approximation. CNN-based model can learn and
capture more detailed features from rainy images. Similar to other image
restoration tasks like image deblurring [7], and image dehazing [10], image
deraining can be modeled as an image-to-image mapping problem. From previous
studies [8, 9, 10], low-level features (e.g., edge and frequency) are more
important than high-level features like attribute, texture. To extract these
low-level features, we apply the wavelet transform and propose the wavelet
channel attention module (WCAM) with a fusion network. First, our network
replaces the down-sampling and up-sampling with the discrete wavelet transform
(DWT) and the inverse discrete wavelet transform (IDWT). The network further
captures the various frequency features and bi-orthogonal property of the DWT
for signal recovery. Second, the channel attention [11] module selectively
emphasizes interdependent channel maps by integrating associated features
among all channel maps. Thus, we combine the DWT and the channel attention so
that intermediate feature maps with different frequencies are integrated
effectively. Third, as demonstrated in [8, 12], fusing various levels of
features is beneficial for many computer vision tasks. The DWT is seen as the
fusion of high-frequency and low-frequency images from original images.
Inspired by it, our input is four sub-band images from the DWT, and the output
is four confidence maps that determine the importance of different sub-band
images and fuse them to reconstruct the clear image by the IDWT. In summary,
Our network is encoder-decoder architecture. The proposed WCAMs replace
convolutions in the encoder, and corresponding inverse wavelet channel
attention modules (IWCAMs) also replace convolutions in the decoder.
This paper makes the following contributions: (i) We propose a novel end-to-
end WCAM with a fusion network that captures frequency features and fuses four
sub-band images for single image deraining. (ii) Several experiments show that
the proposed network obtains much better performance than previous state-of-
the-art deraining methods, even in the presence of large rain streaks and rain
streak accumulation.
## 2 Related Work
### 2.1 Single image deraining
Single image deraining is an ill-posed problem and seen as the denoising
problem. Early methods propose hand-craft priors to estimate distributions of
rain streaks and remove them. In [4], Chen and Hsu decompose the background
and rain streak layers based on low-rank priors to restore clear images. Luo
_et al._ [5] use sparse coding with high discriminability so that the derained
image layer and the rain layer can be accurately separated. In [13], patch-
based priors are applied for both the clean background and rain layers in the
form of Gaussian mixture models to remove rain streaks.
Recently, trainable CNNs have been proposed to estimate the clean image from a
rainy input directly. In [7], a dual convolutional neural network for
deraining is proposed. The proposed network consists of two parallel branches,
which respectively recovers the structures and details in an end-to-end
manner. Zhang _et al._ [8] propose a density-aware multi-stream densely
connected convolutional neural network that jointly estimates rain density
estimation and derained image. In [9], the authors introduce the Gaussian-
Laplacian image pyramid decomposition technology to the neural network and
propose a lightweight pyramid network for single image deraining. In [3], the
authors propose the two-stage network. Rain streaks, transmissions, and the
atmospheric light are estimated in the first stage. Derained images are
refined in the second stage by the conditional generative adversarial network.
Different from them, our network is implemented in the wavelet space to
capture detailed frequency information.
### 2.2 Attention mechanisms
Attention plays an important role in human perception and computer vision
tasks. Attention mechanisms give feature maps weights so that features of the
sequence of regions or locations are magnified. Generally, there are two
attention mechanisms: spatial attention and channel attention [14, 15]. Mnih
et al. [16] propose an attention model that spatially selects a sequence of
regions to refine feature maps, and the network not only performs well but is
also robust to noisy inputs. Hu et al. [11] propose the squeeze-and-excitation
module and use global average-pooled features to compute channel-wise
attention. Furthermore, Woo et al.[14] combine the spatial and channel
attention to propose a convolutional block attention module. Their module
sequentially infers attention maps along two separate dimensions, channel and
spatial, then attention maps are multiplied to the input feature map for
adaptive feature refinement, which increases the accuracy of image
recognition. In our work, we integrate channel attention and wavelet transform
so that output feature maps contain frequency features but different magnified
weights.
## 3 PROPOSED METHODS
### 3.1 Wavelet Channel Attention module
We first describe the 2-D discrete wavelet transform (DWT) in our model. We
apply Haar wavelets [17] and it contains four kernels,
$LL^{\top},HL^{\top},LH^{\top},HH^{\top}$, where $L$ and $H$ are the low pass
filter and high pass filter respectively. Both filters are
$L=\frac{1}{\sqrt{2}}[1,1]^{\top},H=\frac{1}{\sqrt{2}}[1,-1]^{\top}$ (2)
The DWT for image processing means an image $I$ is convolved and then down-
sampled to obtain the four sub-band images $I_{LL},I_{HL},I_{LH}$ and
$I_{HH}$. The low-pass filter captures smooth surface and texture while the
three high-pass filters extract vertical, horizontal, and diagonal edge-like
information. Even though the downsampling operation is employed, due to the
biorthogonal property of DWT, the original image $I$ can be accurately
reconstructed by the inverse discrete wavelet transform (IDWT). Therefore, the
DWT is seen as four $2\times 2$ convolutional kernels whose weights are fixed
and the stride equals two. Similarily, the IDWT is seen as the transpose
convolution operation whose kernels are identical to DWT’s.
Then, we extend convolutions by combining the DWT and the channel attention
mechanism and propose the wavelet channel attention module (WCAM). A WCAM is a
computational unit that is built upon a transformation $F$ mapping an
intermediate feature map $X\in R^{C\times H\times W}$ to a feature maps $Y\in
R^{4C\times\frac{H}{2}\times\frac{W}{2}}$. Given an intermediate feature map
$X\in R^{C\times H\times W}$, the DWT decomposes $X$ into ${\rm
DWT}(X)=[X_{LL},X_{HL},X_{LH},X_{HH}]\in
R^{4C\times\frac{H}{2}\times\frac{H}{2}}$. $3\times 3$ convolutions and leaky
rectified linear units (LeakyReLUs) are applied to extract various frequency
features from DWT$(X)$ and denoted as $Conv({\rm DWT}(X))$, where $Conv$ is an
operator combining the convolution and the LeakyReLU. Furthermore, we propose
improved feature maps with different frequencies contributing different
weights to restore the clear image. We use the channel attention [11] to
control weights of various channel-wise features. The proposed module
calculates the global average pooling of $X$ and uses $1\times 1$ convolutions
to infer a channel attention map $M_{c}\in R^{4C\times 1\times 1}$. Therefore,
the entire result of WCAM is $Conv({\rm DWT}(X))\odot M_{c}$. The detailed
structure of the WCAM is depicted in Fig. 2(b) and formulated as follows:
$\begin{split}M_{c}&=Conv({\rm AP}(X))\\\ Y&=F(X)=Conv({\rm DWT}(X))\odot
M_{c}\vspace{-0.7cm}\end{split}$ (3)
where AP means global average pooling.
The WCAM reduces the size of feature maps but increases the receptive field to
capture multi-frequency and multi-scale features. To magnify the size of the
feature map, similarly, the inverse wavelet channel attention module (IWCAM)
is proposed. An IWCAM is also a computational unit which can be built upon a
transformation $F^{-1}$ mapping an intermediate feature map $X\in
R^{4C\times\frac{H}{2}\times\frac{W}{2}}$ to a feature map $Y\in R^{C\times
H\times W}$. Given an intermediate feature map $X\in
R^{4C\times\frac{H}{2}\times\frac{W}{2}}$, the IDWT merges $X$ into ${\rm
IDWT}(X)\in R^{C\times H\times W}$ and $3\times 3$ convolutions and
LeaklyReLUs are then adopted and denoted as $Conv({\rm IDWT}(X))\in R^{C\times
H\times W}$. Global average pooling of $X$ is calculated and $1\times 1$
convolutions are used to infer a channel attention map $M_{c}\in R^{4C\times
1\times 1}$. The entire result of IWCAM is $Conv({\rm IDWT}(X))\odot M_{c}$.
The structure of the IWCAM is depicted in Fig. 2(c) and formulated as follows:
$\begin{split}M_{c}&=Conv({\rm AP}(X))\\\ Y&=F^{-1}(X)=Conv({\rm
IDWT}(X))\odot M_{c}\end{split}$ (4)
Fig. 1: The proposed WCAM with a fusion network and components in the network.
(a) The entire network, where green arrows mean skipping connections. (b)
WCAM. (c) IWCAM. (d) The residul WCAM.
### 3.2 Network Architecture
Our network is encoder-decoder structure. The encoder consists of three WCAMs.
Once a WCAM is adopted, the sizes of feature maps become quarter and the
number of channels becomes four times, which not only captures multi-scale
features but various frequency information. At the bottom of the network, the
residual block [18] combining wavelet channel attention is used and shown in
Fig. 1(d). This module aggregates features and makes the learning process
effective, especially in the event of deeper networks. Our decoder consists of
three IWCAMs to generate clear images from extracted features. The terminal
outputs are four confidence maps instead of the restored images. Once
confidence maps for the derived wavelet inputs are predicted, they are
multiplied by the four derived inputs to give the final derained image:
$J={\rm IDWT}(C_{LL}\odot I_{LL},C_{HL}\odot I_{HL},C_{LH}\odot
I_{LH},C_{HH}\odot I_{HH})$ (5)
where $C_{LL}$, $C_{HL}$, $C_{LH}$ ,and $C_{HH}$ are confidence maps for
$I_{LL},I_{HL},I_{LH}$ and $I_{HH}$, respectively. The reason for using the
fusion mechanism is that the low-frequency sub-band plays a role in the
objective quality, while the high-frequency sub-bands can affect the
perceptual quality significantly [19]. When low-frequency parts and high-
frequency parts are optimized separately, the increase of objective quality
cannot decrease perceptual quality. Additionally, like U-net [6], we apply
skipping connections to combine the identical size feature maps from WCAMs and
IWCAMs so that the learning process converges quickly. The entire network
architecture is shown in Fig. 1(a).
## 4 EXPERIMENTAL RESULTS
Fig. 2: Derained results on sample images from the synthetic dataset _Outdoor-
Rain_. (Please zoom-in at screen to view details) Fig. 3: Derained results on
real rainy images. (Please zoom-in at screen to view details)
### 4.1 Datasets and training details
In this work, _Outdoor-Rain_ dataset [3] is adopted to train and test the
network. This dataset contains clear images, and the corresponding rainy
images generated by Eq.(1). There are 7500 training and validation samples and
1,500 samples for evaluation. Both clean and rainy images are $480\times 720$.
During training, images are cropped to $256\times 256$, the wavelet SSIM loss
[20] and the L1 loss are employed, and RAdam [21] is used as an optimization
algorithm with a mini-batch size of 16. The learning rate starts from 0.0001
and is divided by ten after 100 epochs. The models are trained for 300
iterations. The entire experiments are performed by the Pytorch framework.
### 4.2 Image Deraining results
PSNR and SSIM are chosen as objective metrics for quantitative evaluation. We
select four state-of-the-art works [7, 8, 9, 3] as deep learning-based
benchmarks to make fairly comparisons with our purposed method. For the fair
comparison, all methods are retrained on the same dataset. The comparison
results are shown in Table 1. Table 1 presents our method has the largest PSNR
and SSIM values among all deraining networks, which demonstrates our method
has a superior performance of restoring clean images for this dataset and the
frequency features are beneficial for restoring rainy images. Furthermore, we
perform various methods on synthetic and real rainy photos and results are
depicted in Fig. 2 and Fig. 3. As revealed in Fig. 2 and Fig. 3, since
comparative methods tend to miscalculate rainy concentration, restored images
are dark or have remaining rain and mist. In contrast, our purposed method
predicts better-derained results with balanced colors and detailed edges.
We analyze how the WCAM and the fusion help to refine derained results with
three experiments. The first experiment uses convolutions and wavelets without
channel attention and fusion. The second experiment uses the proposed modules
without fusion. The third experiment estimates feature maps to fuse sub-band
images without channel attention. Table 2 compares our method against three
baselines and demonstrates wavelet channel attention and fusion contribute the
best results.
Table 1: Quantitative SSIM and PSNR on the synthetic _Outdoor-Rain_ dataset. | [7] | [8] | [9] | [3] | Ours
---|---|---|---|---|---
PSNR | 17.92 | 21.64 | 18.16 | 21.17 | 24.89
SSIM | 0.676 | 0.788 | 0.723 | 0.742 | 0.813
Table 2: Ablation study shows how the WCAM and the fusion help to refine derained results. | PSNR | SSIM
---|---|---
Ours, w/o attention, w/o fusion | 21.44 | 0.784
Ours, w/o fusion | 23.34 | 0.764
Ours, w/o attention | 23.24 | 0.810
Ours | 24.89 | 0.813
## 5 CONCLUSION
In this paper, the wavelet channel attention module with a fusion network is
proposed for single image deraining. The wavelet transform and the inverse
wavelet transform are substituted for down-sampling and up-sampling to extract
various frequency features. The channel attention effectively controls ratios
of feature maps. Furthermore, the proposed network estimates confidence maps
for each derived wavelet input. Confidence maps and derived inputs are fused
to render final derained results. Experiments on synthetic and real images
verify the superiority of our model compared to the state-of-the-art results.
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|
# Footstep Planning with Encoded Linear Temporal Logic Specifications
Vikram Ramanathan
###### Abstract
This article presents an approach to encode Linear Temporal Logic (LTL)
Specifications into a Mixed Integer Quadratically Constrained Quadratic
Program (MIQCQP) footstep planner. We propose that the integration of LTL
specifications into the planner not only facilitates safe and desirable
locomotion between obstacle-free regions, but also provides a rich language
for high-level reasoning in contact planning. Simulations of the footstep
planner in a 2D environment satisfying encoded LTL specifications demonstrate
the results of this research.
## I Introduction
Humanoid locomotion is accomplished by changing its footholds. In order to
move feasibly and efficiently, a footstep planner must be able to find
sequences of foot positions and orientations that will realize the robot’s
locomotion objectives. One established solution for the footstep planning
problem is to perform some discrete sampling-based search algorithm over
reachable foot positions and orientations. This approach first determines
candidate footsteps through one of several methods, including:
* •
intersecting valid sample limb configurations with the environment to
determine which limb configurations are close to contact and then projecting
these configurations onto contact using inverse kinematics [1]
* •
assuming general bounds on the displacement of each foothold based on the
robot structure and kinematic reachability [2] [3] [4] [5]
* •
defining an action-effect mapping [6]
Then, a search algorithm, such as A* [7] [6] [8] [4] or Rapidly-exploring
Random Trees (RRT) [5] [9] is used to find a sequence of footsteps. However,
the performance of these search algorithms largely depends on the quality of
the heuristic function that guides the search as well as the granularity of
the discretization. As an alternative, Deits and Tedrake [10] proposed a
footstep planner formulated as a mixed integer optimization program with
continuous decision variables for footstep position and orientation, and
integer decision variables to eliminate non-convex constraints. After
employing the IRIS algorithm [11] to compute large obstacle-free convex
regions within the provided environment, their solution solves a Mixed Integer
Quadratically Constrained Quadratic Program (MIQCQP) with a clever
discretization of cos and sin in its reachability constraint. This approach
solves for globally optimal footstep plans at realtime rates while satisfying
reachability constraints and avoiding obstacles.
## II Problem Description
In this article, we consider the problem of encoding temporal logic
specifications into a MIQCQP footstep planner to improve and extend its
functionality. Continuous footstep planners like the Mixed Integer
Quadratically Constrained Quadratic Program (MIQCQP) footstep planner proposed
by Deits and Tedrake [10] adopt an optimization framework, thereby avoiding
the use of search heuristics as well as the sampling of discretized footstep
configurations. The problem of integrating temporal logic specifications into
a sampling-based search algorithm such as Probabilistic Road Maps (PRMs) and
Randomly-Exploring Random Trees (RRTs) has been studied [12] [13]. However, we
have adopted the continuous footstep planner to take advantage of its ability
to solve for optimal solutions as well as the ease with which its optimization
framework can be expanded upon to include simplified robot dynamics
constraints and compute for reaction forces.
## III Solution
As mentioned previously, this work builds upon Deits’ [10] footstep planner.
This planner solves an MIQCQP to determine x, y and $\theta$ for N footsteps.
We first will cover the formulation of this footstep planner before discussing
the encoding of LTL specifications.
### III-A MIQCQP Footstep Planner
The footstep planner seeks to minimize a quadratic cost function (in terms of
the length of each stride and the distance between the terminal footstep and
the goal footstep position and orientation) subject to obstacle avoidance and
reachability constraints. Obstacle avoidance is facilitated by first
decomposing the non-convex obstacle-free configuration space into a set of
convex “safe” regions [11]. These convex regions are then defined as polygons
by the halfspace respresentation and added as convex constraints to the
program. A binary variable is assigned to each possible footstep and region
pair. If such a binary variable is 1, the corresponding footstep is assigned
to the region. More precisely, the matrix, $H\in\\{0,1\\}^{R\times N}$, where
$R$ is the number of regions and $N$ is the number of footsteps, is
constructed such that if $H_{r,j}$ = 1, then the $j^{th}$ footstep,
$j=\\{1,...,N\\}$, is assigned to the region $r$. This implication is defined
as a mixed-integer linear constraint using the big-M formulation [14],
$-M(1-H_{r,j})+A_{r}f_{j}\leq b_{r}$ (1)
where
$f_{j}=\begin{bmatrix}x_{j}\\\ y_{j}\\\ \theta_{j}\end{bmatrix}$
, $M$ is a sufficiently large positive number, $A_{r}$ defines the normal
vectors corresponding to the supporting halfplanes of the polygonal obstacle-
free regions, and $b_{r}$ defines the offsets between the halfplanes and the
origin. However, since the footstep must only belong to one region instead of
the intersection of all these regions, $\sum_{r=1}^{R}H_{r,j}=1$ must be
enforced.
The footstep reachability constraint is defined as the intersection of two
circular regions offset from the previous foothold (as shown in Eq. 2 below).
These offsets are mirrored for left and right feet.
$\left\lVert\begin{bmatrix}x_{j}\\\
y_{j}\end{bmatrix}-\bigg{(}\begin{bmatrix}x_{j-1}\\\
y_{j-1}\end{bmatrix}+\begin{bmatrix}cos(\theta_{j})&-sin(\theta_{j})\\\
sin(\theta_{j})&cos(\theta_{j})\end{bmatrix}p_{i}\bigg{)}\right\rVert\leq
r_{i}$ (2)
where $i\in\\{1,2\\}$, $j\in\\{2,...,N\\}$, $p_{i}$ are the centers of the
circles, and $r_{i}$ are the radii of the circles. By tuning $p_{i}$ and
$r_{i}$, we can obtain a conservative approximation of a biped’s footstep
reachability. In order to eliminate the nonlinearity, $cos$ and $sin$ are
approximated by linear piece-wise functions along with binary variables to
decide which linear segment of the function to use based on $\theta$’s value.
These binary variables are defined as binary matrices,
$S,C\in\\{0,1\\}^{L\times N}$, where L is the number of piece-wise segments
and N is the number of footsteps. Since $\theta$ cannot belong to multiple
segments, $\sum_{l=1}^{L}S_{l,j}=\sum_{l=1}^{L}C_{l,j}=1$ must be satisfied.
Our piece-wise approximation of $cos$ and $sin$ defines 5 linear segments:
$sin(\theta)=\begin{cases}-\theta-\pi\hskip 15.00002pt-\pi\leq\theta<1-\pi\\\
-1\hskip 38.00008pt1\leq\theta<-1\\\ \theta\hskip 45.00006pt-1\leq\theta<1\\\
1\hskip 45.00006pt1\leq\theta<\pi-1\\\ -\theta+\pi\hskip
20.00003pt\pi-1\leq\theta<\pi\\\ \end{cases}$ (3)
$cos(\theta)=\begin{cases}-1\hskip 43.00009pt-\pi\leq\theta<-\pi/2-1\\\
\theta+\pi/2\hskip 23.00006pt-\pi/2-1\leq\theta<1-\pi/2\\\ 1\hskip
53.0001pt1-\pi/2\leq\theta<\pi/2-1\\\ -\theta+\pi/2\hskip
20.00003pt\pi/2-1\leq\theta<\pi/2+1\\\ -1\hskip
47.50006pt\pi/2+1\leq\theta<\pi\\\ \end{cases}$ (4)
(a) $sin(\theta)$
(b) $cos(\theta)$
Figure 1: Piece-wise Approximations of $cos(\theta)$ and $sin(\theta)$
The graphical representation of this approximation is shown in Fig. 1. We also
bound the change in footstep orientation in every step to $\pi/8$ rad.
Putting the quadratic cost function and all these constraints together, we get
the following MIQCQP:
$\displaystyle\underset{f_{1},...,f_{j},S,C,H}{\text{minimize}}$
$\displaystyle(f_{N}-g)^{T}Q(f_{N}-g)+\sum_{j=1}^{N-1}(f_{j+1}-f_{j})^{T}R(f_{j+1}-f_{j})$
(5) subject to $\displaystyle H_{r,j}\implies A_{r}f_{j}\leq b_{r}$
$\displaystyle
S_{l,j}\implies\begin{cases}\phi_{l}\leq\theta_{j}\leq\phi_{l+1}\\\
s_{j}=g_{l}\theta_{j}+h_{l}\\\ \end{cases}$ $\displaystyle
C_{l,j}\implies\begin{cases}\phi_{l}\leq\theta_{j}\leq\phi_{l+1}\\\
c_{j}=g_{l}\theta_{j}+h_{l}\\\ \end{cases}$
$\displaystyle\left\lVert\begin{bmatrix}x_{j}\\\
y_{j}\end{bmatrix}-\bigg{(}\begin{bmatrix}x_{j-1}\\\
y_{j-1}\end{bmatrix}+\begin{bmatrix}c_{j}&-s_{j}\\\
s_{j}&c_{j}\end{bmatrix}p_{i}\bigg{)}\right\rVert\leq r_{i}$
$\displaystyle\sum_{r=1}^{R}H_{r,j}=\sum_{l=1}^{L}S_{l,j}=\sum_{l=1}^{L}C_{l,j}=1$
$\displaystyle\theta_{j}-\theta_{j-1}\leq\pi/8$
where $r=1...R$, , $j=\\{1,...,N\\}$, and $l=1,...,L$.
### III-B Linear Temporal Logic
#### III-B1 Preliminaries
In this work, the underlying time domain is discrete since each “execution” or
footstep corresponds to the advancement of a single time-unit. We consider
Linear Temporal Logic (LTL) to describe desired planning behavior. LTL
formulae are defined over a set of atomic propositions, $AP$, according to the
following grammar [15]:
$\phi::=\text{true
}|\,a\,|\,\phi_{1}\wedge\phi_{2}\,|\,\lnot\phi\,|\,\bigcirc\phi\,|\,\phi_{1}U\phi_{2}$
where $a\in AP$, $\bigcirc$ and $U$ are the temporal operators for “next”
(signifies that the proposition is true in the next step) and “until”
(signifies that the proposition preceding the operator is true until the
future moment when the proposition succeeding the operator is true), and
$\lnot$ and $\wedge$ are the negation and conjuction Boolean operators
respectively. The other Boolean operators for disjunction, $\vee$, implication
$\implies$, and equivalence $\iff$ can be derived from the previously defined
grammar:
$\phi_{1}\vee\phi_{2}=\lnot(\lnot\phi_{1}\wedge\lnot\phi_{2})$ (6)
$\phi_{1}\implies\phi_{2}=\lnot\phi_{1}\vee\phi_{2}$ (7)
$\phi_{1}\iff\phi_{2}=(\phi_{1}\implies\phi_{2})\wedge(\phi_{2}\implies\phi_{1})$
(8)
In addition, the temporal operators, $\lozenge$ (eventually) and $\square$
(always) are defined by:
$\lozenge\phi=true\,U\,\phi$ (9) $\square\phi=\lnot\lozenge\lnot\phi$ (10)
Let $\sigma$ be an infinite sequence of states,
$\sigma=A_{0},A_{1},...(2^{AP})^{\omega}$. Let $\sigma_{i}$ denote the word
$\sigma$ from position $i$. The semantics of an LTL formula is defined as a
language $Words(\phi)$ that contains all infinite words over the alphabet
$2^{AP}$ that satisfy $\phi$. Every LTL formula is associated with a single
linear time property. The satisfaction relations are defined inductively by:
$\sigma_{i}\models a\;\text{iff}\;a\in A_{0}$ (11)
$\sigma_{i}\models\phi_{1}\wedge\phi_{2}\;\text{iff}\;\sigma_{i}\models\phi_{1}\;\text{and}\;\sigma_{i}\models\phi_{2}$
(12) $\sigma_{i}\models\lnot\phi\;\text{iff}\;\sigma_{i}\nvDash\phi$ (13)
$\sigma_{i}\models\bigcirc\phi\;\text{iff}\;\sigma_{i+1}\models\phi$ (14)
$\sigma_{i}\models\phi_{1}\;U\;\phi_{2}\;\text{iff}\;\exists j\geq
i\;\text{s.t.}\;\sigma_{j}\models\phi_{2}\;\text{and}\;\sigma_{k}\models\phi_{1}\\\
\forall i\leq k<j$ (15)
#### III-B2 Mixed Integer Encoding
In this section, we will detail the mixed integer encoding of Linear Temporal
Logic specifications. Let $p$ be an atomic proposition defined over some
footstep. Then, this proposition has a corresponding binary variable $P^{k}$,
which is defined to be 1 when $p$ is true at the $k^{th}$ footstep and 0 when
$p$ is false at the $k^{th}$ footstep (where $k=0...N-1$).
To appropriately extend the semantics of Linear Temporal Logic to mixed
integer programming, we redefine the original satisfaction relations outlined
in the previous section in terms of integer variables. Let $\sigma_{k}$ is a
finite run that starts at footstep $k$.
Firstly, $\sigma_{k}\models p$ can be encoded by the following integer
constraint,
$P^{k}=1$ (16)
The conjunction satisfiability relation,
$\sigma_{k}\models\bigwedge_{j=1}^{m}\phi_{j}$, where m is any integer greater
than 1, can be encoded by the following integer constraint,
$\sum_{j=1}^{m}P^{k}_{\phi_{j}}=m$ (17)
The negation satisfiability relation, $\sigma_{k}\models\lnot\phi$ can be
encoded by the following integer constraint,
$P^{k}_{\phi}=0$ (18)
Although the disjunction satisfiability relation,
$\sigma_{k}\models\bigvee_{j=1}^{m}\phi_{j}$, where m is any positive integer
greater than 1, can easily be derived from the conjunction (Eq. 17) and
negation (Eq. 18) satisfiability relations by means of Eq. 6, we propose a
more succinct encoding:
$\sum_{j=1}^{m}P^{k}_{\phi_{j}}>=1$ (19)
Finally, $\sigma_{k}\models\bigcirc\phi$ can be encoded by the following
integer constraint,
$P^{k+1}_{\phi}=1$ (20)
The encoding of the until operator, $U$ can be defined in terms of Eqs. 11-20
above. Specifically, the expansion laws of LTL formulae [15] give us the
following recursive identity for the until operator, $U$:
$\phi_{1}U\phi_{2}=\phi_{2}\vee(\phi_{1}\wedge\bigcirc(\phi_{1}U\phi_{2}))$
(21)
Granted that footstep planning involves a finite horizon, any loops in the
plan are finite and would not lead to circular reasoning (as pointed out in
[16]). As a result, we can avoid using an auxiliary encoding of $U$, which
involves under-approximating its functionality [17]. Instead, our proposed
approach formulates a set of nested mixed-integer constraints (using Eqs.
11-20) over a vector of binary variables, in which each variable corresponds
to the satisfiability of $\phi_{1}U\phi_{2}$ at a particular foothold in the
run.
Let $T\in\\{0,1\\}^{1\times(N-k)}$ be a vector of binary variables of length
$N-k+1$. Assuming Eqs. 11-20, $\sigma_{k}\models\phi_{1}U\phi_{2}$ can be
encoded by the following set of integer constraints:
$T^{i}=P_{\phi_{2}}^{i}\vee(P_{\phi_{1}}^{i}\wedge\bigcirc T^{i+1})\hskip
20.00003pti=k,...,N-1$ (22)
with base case,
$T^{N}=P_{\phi_{2}}^{N}$ (23)
and satisfiability constraint (i.e. $\sigma_{k}$ must satisfy the LTL
specification, $\phi_{1}U\phi_{2}$)
$T^{k}=1$ (24)
Instead of deriving the mixed-integer encodings for safety, liveness, and
persistence LTL formulae from the general definitions above, we can more
efficiently encode them as follows:
Safety (“always”): The safety LTL specification,
$\sigma_{k}\models\square\phi$ can be encoded as the following integer
constraint:
$\sum_{i=k}^{N}P_{\phi}^{i}=N-k+1$ (25)
Liveness (“eventually”): The liveness LTL specification,
$\sigma_{k}\models\lozenge\phi$ can be encoded as the following integer
constraint:
$\sum_{i=k}^{N}P_{\phi}^{i}\geq 1$ (26)
Liveness (“repeated eventually”): The liveness LTL specification,
$\sigma_{k}\models\square\lozenge\phi$ (repeated eventually) can be encoded as
the following set of integer constraints:
$\sum_{i=j}^{N}P_{\phi}^{i}\geq 1\hskip 20.00003ptj=k,...,N$ (27)
Persistence (“eventually always”): The persistence LTL specification,
$\sigma_{k}\models\lozenge\square\phi$ (eventually always) is defined with the
help of Eq. 19, and is encoded as follows:
$\bigvee_{j=k}^{N}\sum_{i=j}^{N}P_{\phi}^{i}=N-j+1$ (28)
## IV Examples
Our mixed integer encodings facilitate the integration of LTL specifications
in the footstep planner. This is particularly powerful in ensuring safe
performance of region-based locomotion tasks. Specifications can detail
desired locomotion behavior such as the requirement that the robot must
eventually enter a set of regions or that it must reach some region within
some number of footsteps. Specifications can also be used to outline
underlying environment (region-specific) characteristics such as the
requirement that if a footstep is placed in a particular region, the following
footsteps must remain in that region or that access to certain regions is only
granted provided particular regions have already been visited.
In this section, we present three LTL-constrained footstep planning scenarios.
These scenarios were solved using the commercial optimization solver, Gurobi
on a laptop with a 2.3 GHz quad-core processor and 16 GB of memory. Source
code and animations of these scenarios can be found on Github at
https://github.com/vikr53/ltl_footstep_planner .
### IV-A Liveness
We will first consider encoding a region-based liveness specification that
necessitates that certain obstacle-free regions be visited eventually. The
planner then solves for an optimal set of footsteps that fulfil this
requirement while also reaching some goal footstep position and orientation.
Let $p_{R_{3}}$ and $p_{R_{4}}$ be the atomic propositions, “a footstep is in
Region 1” and “a footstep is in Region 2” respectively.
LTL Formula:
$\lozenge(p_{R_{3}}\vee p_{R_{4}})$ (29)
Mixed-Integer Encoding: Using the liveness (Eq. 26) and disjunction (Eq. 19)
encodings, we can express the above LTL specification as the following linear
mixed-integer constraint:
$\sum^{N}_{j=1}(H_{R_{3},j}+H_{R_{4},j})\geq 1$ (30)
Figure 2: Scenario 1. Encoded LTL Specification: $\lozenge(p_{R_{3}}\vee
p_{R_{4}})$. Right-leg footsteps are shown in red stars and left-leg footsteps
are shown in blue circles. The orientation of each footstep is depicted by its
protruding black arrow. Each footstep is numbered in brown. Goal footstep
location and orientation: [$x$: 0, $y$: 1.5, $\theta$: $\pi/2$]
The result from the planner is depicted in Figure 2, where the red stars are
the right leg footsteps, the blue circles are the left leg footsteps and the
green boxes are the obstacle-free convex regions. It also must be noted that
the first two footholds are assumed to be fixed (i.e. the humanoid robot is in
some initial double-support stance). In this scenario, the planner must
eventually reach either region 3 (“R3”) or region 4 (“R4”) before reaching its
goal state - ($x$: 0, $y$: 1.5, $\theta$: $\pi/2$). As is evident from the
simulation, the planner is able to reason that the most efficient way to
satisfy this requirement is to visit region 4 through region 5 before
returning to its goal state in region 5. This optimal result is solved for in
around 25s.
### IV-B Until
We now demonstrate the encoding of the until, $U$, temporal operator in the
footstep planner. The until operator can be used to ensure that the approach
to some goal state remains within certain pre-defined desirable regions. This
has applications in a wide-variety of locomotive tasks such as surveying and
reconnaissance.
Let $p_{R_{1}}$, $p_{R_{2}}$, and $p_{R_{3}}$ be the atomic propositions, “a
footstep is in Region 1”, “a footstep is in Region 2” and “a footstep is in
Region 3” respectively. The length of the plan is $N=13$.
LTL Formula:
$(p_{R_{1}}\vee p_{R_{2}})Up_{R_{3}}$ (31)
Mixed-Integer Encoding: First, we define a vector of binary variables,
$T\in\\{0,1\\}^{1\times N}$. Then, we specify the base case (as given by Eq.
23):
$T^{N}=H_{R_{3},N}$ (32)
where R3 denotes the index for Region 3 in matrix $H$. Next, following the
formulation defined in Eq. 22 and Eq. 19, we first encode $\phi_{1}^{k}\equiv
p_{R_{1}}^{k}\vee p_{R_{2}}^{k}$, where $k\in{1,...,N-1}$, and $p_{R_{1}}^{k}$
and $p_{R_{2}}^{k}$ are the atomic propositions, “the $k^{th}$ footstep is in
Region 1”, “the $k^{th}$ footstep is in Region 2”.
$H_{R_{1},k}+H_{R_{2},k}\geq 1-M(1-P_{\phi_{1}}^{k})$ (33)
$H_{R_{1},k}+H_{R_{2},k}\leq 1-m+M(P_{\phi_{1}}^{k})$ (34)
where $m$ and $M$ are sufficiently small and large positive integers
respectively. Then, using the big-M formulation and the conjuction encoding
given by Eq. 17, we encode the term,
$\phi_{1}\wedge\bigcirc(\phi_{1}U\phi_{2})$:
$P_{\phi_{1}}^{k}+T^{k+1}\geq 2-M(1-B^{k})$ (35) $P_{\phi_{1}}^{k}+T^{k+1}\leq
2-m+M(B^{k})$ (36)
where $B$ is a vector of binary variables, i.e.
$B\in{0,1}^{1\times\\{N-1\\}}$. Finally, we have the following two constraints
to finish encoding the entire specification:
$H_{R_{3},k}+B^{k}\geq 1-M(1-T^{k})$ (37) $H_{R_{3},k}+B^{k}\leq 1-m+M(T^{k})$
(38)
All that remains is simply adding the satisfiability constraint (Eq. 24):
$T^{1}=1$ (39)
Figure 3: Scenario 2. Encoded LTL Specification: $(p_{R_{1}}\vee
p_{R_{2}})Up_{R_{3}}$. Right-leg footsteps are shown in red stars and left-leg
footsteps are shown in blue circles. The orientation of each footstep is
depicted by its protruding black arrow. Each footstep is numbered in brown.
Goal footstep location and orientation: [$x$: 2, $y$: 1.5, $\theta$: $\pi/2$]
Fig. 3 displays the solved LTL-constrained footstep plan. It is evident that
the 13 planned footsteps abide by the given specification, “until Region 3 is
reached, footsteps must either be in Region 1 or Region 2”, while reaching its
goal state - ($x$: 2, $y$: 1.5, $\theta$: $\pi/2$). This example was solved in
around 2s.
### IV-C Timed Specifications
In this example, we consider the encoding of timed LTL constraints, where each
footstep is assumed to be a single time unit. Defined over a discrete time
domain, these specifications are relatively simple to encode but allow for the
expression of interesting constraints like “always be in a particular region
or set of regions for the first $p$ steps” or “eventually reach a region
within the $p^{th}$ and $q^{th}$ footstep”, where $p,q$ are arbitrary positive
integers. For the sake of this example, we consider the following
specification:
LTL Formula:
$\square^{7\leq k\leq 15}p_{R_{2}}$ (40)
Mixed-Integer Encoding: Using Eq. 25, we can write the above specification as
the following constraint:
$\sum_{j=7}^{15}H_{R2,j}\geq 15-7+1$ (41)
Figure 4: Scenario 3. Encoded LTL Specification: $\square^{7\leq k\leq
15}p_{R_{2}}$. Right-leg footsteps are shown in red stars and left-leg
footsteps are shown in blue circles. The orientation of each footstep is
depicted by its protruding black arrow. Each footstep is numbered in brown.
Goal footstep location and orientation: [$x$: 1.5, $y$: 2.2, $\theta$:
$3\pi/4$]
The footstep plan generated for this scenario is shown in Fig. 4. The plan’s
length is $N=18$ steps. It is evident that the plan satisfies the discrete-
timed specification, “the $7^{th}$ to the $15^{th}$ footstep must be in Region
2”, while finally reaching its goal state - ($x$: 1.5, $y$: 2.2, $\theta$:
$3\pi/4$). The solve time for this example was around 2s.
Aside from encoding surveying behaviors, an interesting application of such
timed specifications is assessing the accessibility of a region. While taking
into account the particular formulation of the footstep planner (the planning
horizon, reachability constraints, the obstacle-free regions and other encoded
specifications), this specification can determine whether a specific region or
set of regions can be reached within some number of steps. For example, in
order to determine whether Region 2 can be reached within 5 footsteps for the
formulation used for this example, we can simply encode the specification,
$\lozenge^{\leq 5}p_{R_{2}}$, into the optimization program. By nature of the
encoding, the planner attempts to satisfy this specification. Thus, if the
MIQCQP is rendered infeasible, we know definitively that it is not possible to
reach Regon 2 in 5 footsteps, i.e. Region 2 is not accessible in 5 steps with
the current formulation. The opposite is true when the MIQCQP is solvable,
i.e. Region 2 is accessible in 5 steps.
## V Ongoing/Future Work
### V-A Stride Adjustment
We have also considered the encoding of region-based stride adjustments. Since
the traversibility of terrain can vary significantly over a footstep plan, the
planner may need to tread certain regions more carefully than others. Often,
the first step in reducing the risk of loosing dynamic stability on account of
the terrain is reducing the stride length. This can be easily implemented by
changing the reachability constraints for regions with difficult terrain:
From Eq. 2, $p_{i}$ and $r_{i}$, $i\in\\{1,2\\}$, are the centers and radii of
the circles whose intersection defines the footstep reachability of the robot.
To define the reduced reachability constraint, we merely have to intersect two
relatively smaller and less offset circles with centers and radii, $p_{i}^{s}$
and $r_{i}^{s}$ $i\in\\{1,2\\}$ respectively. Then, we use the big-M
formulation to encode the implication that smaller strides only occur in
specific regions. For example, smaller strides in Region 2 can be encoded as:
$\left\lVert\begin{bmatrix}x_{j}\\\
y_{j}\end{bmatrix}-\bigg{(}\begin{bmatrix}x_{j-1}\\\
y_{j-1}\end{bmatrix}+\begin{bmatrix}cos(\theta_{j})&-sin(\theta_{j})\\\
sin(\theta_{j})&cos(\theta_{j})\end{bmatrix}p_{i}^{s}\bigg{)}\right\rVert-M(1-H_{R_{2},j})\\\
\leq r_{i}^{s}$ (42)
Figure 5: Stride Adjustment Example. Planner makes shorter strides in Region 2
since it is assumed to have difficult terrain. Goal footstep location and
orientation: [$x$: 3.5, $y$: 0.5, $\theta$: 0]
Fig. 5 depicts the result of encoding the above specification in the footstep
planner. It is evident that the planner is able to shorten its stride when the
robot enters Region 2. As mentioned previously, this has several applications,
especially in dealing with difficult terrain. For example, given that a
certain region is known to have more slippery terrain, the robot can shorten
its stride in this region to reduce the risk of slipping. However, it must be
noted that solely planning shorter strides does not guarantee safer
locomotion. Appropriate contact forces must also be planned. We hope to
investigate the integration of our LTL-constrained planner with trajectory
optimization (for planning through contact) [18].
### V-B Multi-Contact Planning
In addition to specifying region-based locomotion behavior, encoded LTL
specifications can potentially be extended to the more difficult multi-contact
problem, allowing us to encode higher-level reasoning about the safety and
performance of these plans. We propose that this is possible by first defining
the planner in three dimensions, then approximating the COM position as the
center of the support polygon and adding approximate arm reachability
constraints. Lastly, LTL constraints can be defined over the ordering of
contacts. Given the combinatorial complexity of the multi-contact problem,
this formulation would be able to not only make the problem more tractable but
also guarantee the safety and performance of multi-contact plans. We are
actively researching this idea and have done some preliminary tests on the
simpler footstep planner (which has only 2 end-effectors).
Preliminary Results: To demonstrate that the ordering of end-effectors in
contact need not be defined explicitly (as was done in the formulation of the
original MIQCQP footstep planner [10]), we encode ordering-related LTL
specifications into the planner and let the planner formulate the appropriate
end-effector contact sequence. In the case of the footstep planner, the safe
and desirable ordering of end-effectors in contact is trivial - the right foot
must follow the left foot, and a footstep can be made by either the left or
right foot, not both. This is encoded as the linear mixed-integer constraints:
$LL^{j}+RL^{j}=1$ (43)
$-M(1-LL^{j-1})+RL^{j}\leq 1$ (44)
$M(1-LL^{j-1})+RL^{j}\geq 1$ (45)
where $LL^{j}$ and $RL^{j}$, for $j\in\\{2,...,N\\}$, are binary variables
that determine whether the left leg and right leg make the $j^{th}$ footstep
respectively. In addition, the reachability constraint in Eq. 2 must be
changed to adjust to whether a particular footstep was chosen to be made with
the left or right foot:
$\left\lVert\begin{bmatrix}x_{j}\\\
y_{j}\end{bmatrix}-\bigg{(}\begin{bmatrix}x_{j-1}\\\
y_{j-1}\end{bmatrix}+\begin{bmatrix}cos(\theta_{j})&-sin(\theta_{j})\\\
sin(\theta_{j})&cos(\theta_{j})\end{bmatrix}p_{1}\bigg{)}\right\rVert-M(1-LL^{j})\\\
\leq r_{1}$ (46)
$\left\lVert\begin{bmatrix}x_{j}\\\
y_{j}\end{bmatrix}-\bigg{(}\begin{bmatrix}x_{j-1}\\\
y_{j-1}\end{bmatrix}+\begin{bmatrix}cos(\theta_{j})&-sin(\theta_{j})\\\
sin(\theta_{j})&cos(\theta_{j})\end{bmatrix}p_{2}\bigg{)}\right\rVert-M(1-RL^{j})\\\
\leq r_{2}$ (47)
This encoding was successful and generated similar results to the original
formulation. It must be noted that while these specifications are in effect
equivalent to original implicit encoding for the case of the footstep planner,
it is neither preferable nor feasible to add ordering constraints on a
contact-by-contact basis for the more complicated multi-contact problem.
Instead, encoding such LTL specifications allow the planner to not only
determine the optimal sequence of contacts but also guarantee that this
sequence follows some safety constraints. In the multi-contact planning
problem, these constraints could include simple requirements like
$\square(p_{lleg}\vee p_{rleg})$ (one foot should always be on the ground),
and even criteria for the necessity of multi-contact plans (such as
uncertainty in the stability of certain contacts). This idea seems promising,
and we hope to make progress on its development soon. The code for this
example is available as the script, footstep_planner_contact_ordering_specs.py
in the Github repository linked earlier in this paper.
### V-C Exploiting Sparse Constraint Matrices
While this planner can solve our footstep planning problem for 10 steps, 5
regions and a few LTL constraints in less than a second, increasing the
planning horizon (i.e. planning for more footsteps), introducing new regions
and encoding additional constraints increase the complexity of solving the
mixed-integer program. Hence, a limitation of this planer is its poor
scalability.
We propose that the sparsity of the $H$, $S$ and $C$ matrices can possibly be
exploited to increase the solve times of plans with large horizons (greater
than 20 footsteps). However, this is merely an idea at the moment, and we hope
to explore this in more detail soon.
## References
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# The 3D-PC: a benchmark for visual perspective taking in humans and machines
Drew Linsley†1, Peisen Zhou†1, Alekh Karkada†1, Akash Nagaraj1,
Gaurav Gaonkar1, Francis E Lewis1, Zygmunt Pizlo2, Thomas Serre1
<EMAIL_ADDRESS>
###### Abstract
Visual perspective taking (VPT) is the ability to perceive and reason about
the perspectives of others. It is an essential feature of human intelligence,
which develops over the first decade of life and requires an ability to
process the 3D structure of visual scenes. A growing number of reports have
indicated that deep neural networks (DNNs) become capable of analyzing 3D
scenes after training on large image datasets. We investigated if this
emergent ability for 3D analysis in DNNs is sufficient for VPT with the _3D
perception challenge_ (3D-PC): a novel benchmark for 3D perception in humans
and DNNs. The 3D-PC is comprised of three 3D-analysis tasks posed within
natural scene images: 1. a simple test of object depth order, 2. a basic VPT
task (VPT-basic), and 3. another version of VPT (VPT-Strategy) designed to
limit the effectiveness of “shortcut” visual strategies. We tested human
participants (N=33) and linearly probed or text-prompted over 300 DNNs on the
challenge and found that nearly all of the DNNs approached or exceeded human
accuracy in analyzing object depth order. Surprisingly, DNN accuracy on this
task correlated with their object recognition performance. In contrast, there
was an extraordinary gap between DNNs and humans on VPT-basic. Humans were
nearly perfect, whereas most DNNs were near chance. Fine-tuning DNNs on VPT-
basic brought them close to human performance, but they, unlike humans,
dropped back to chance when tested on VPT-Strategy. Our challenge demonstrates
that the training routines and architectures of today’s DNNs are well-suited
for learning basic 3D properties of scenes and objects but are ill-suited for
reasoning about these properties as humans do. We release our 3D-PC datasets
and code to help bridge this gap in 3D perception between humans and machines.
## 1 Introduction
22footnotetext: These authors contributed equally to this work.11footnotetext:
Carney Institute for Brain Science, Brown University, Providence,
RI.22footnotetext: Department of Cognitive Sciences, University of California-
Irvine, Irvine, CA.
In his theory of cognitive development, Piaget posited that human children
gain the ability to predict which objects are visible from another viewpoint
before the age of 10 [1, 2]. This “Visual Perspective Taking” (VPT) ability is
a foundational feature of human intelligence and a behavioral marker for the
theory of mind [3]. VPT is also critical for safely navigating through the
world and socializing with others (Fig. 1A). While VPT has been a focus of
developmental psychology research since its initial description [1, 4, 5]
(Fig. 1B), it has not yet been studied in machines.
Figure 1: Visual Perspective Taking (VPT) is the ability to analyze scenes
from different viewpoints. (A) Humans rely on VPT to anticipate the behavior
of others. We expect that this ability will be essential for creating the next
generation of AI assistants that can accurately anticipate human behavior
(images are CC BY-NC). (B) VPT has been studied in developmental psychology
since the mid-20th century using cartoon or highly synthetic stimuli. For
example, Piaget’s “Three Mountains Task” asks observers to describe the scene
from the perspective of a bear (image from [6]). (C) Here, we use Gaussian
Splatting [7] to develop a 3D scene generation pipeline for the _3D perception
challenge_ (3D-PC), to systematically compare 3D perception capabilities of
human and machine vision systems. (D) The 3D-PC tests 1. Object depth
perception, and 2. VPT.
One of the more surprising results in deep learning has been the number of
concomitant similarities to human perception exhibited by deep neural networks
(DNNs), trained on large-scale static image datasets [8, 9]. For example, DNNs
now rival or surpass human recognition performance on object recognition and
segmentation tasks [10, 11, 12], and are the state-of-the-art approach for
predicting human neural and behavioral responses to images [13]. There is also
a growing number of reports indicating that DNNs trained with self-supervision
or for object classification learn to encode 3D properties of objects and
scenes that humans are also sensitive to, such as the depth and structure of
surfaces [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Are the emergent
capabilities of DNNs for 3D vision sufficient for solving VPT tasks?
Here, we introduce the _3D perception challenge_ (3D-PC) to address this
question and systematically compare 3D perceptual capabilities of humans and
DNNs. The 3D-PC evaluates observers on (Fig 1): 1. identifying the order of
two objects in depth (depth order), 2. predicting if one of two objects can
“see” the other (VPT-basic), and 3. another version of VPT that limits the
effectiveness of “shortcut” solutions [24] (VPT-Strategy). The 3D-PC is
distinct from existing psychological paradigms for evaluating VPT [1, 4, 5]
and computer vision challenges for 3D perception [23, 20] in two ways. First,
unlike small-scale psychology studies of VPT, the 3D-PC uses a novel “3D
Gaussian Splatting” [7] approach which permits the generation of endless real-
world stimuli. Second, unlike existing computer vision challenges, our
approach for data generation means that the 3D-PC tests and counterbalances
labels for multiple 3D tasks on the exact same images, which controls for
potential confounds in analysis and interpretation. We expect that DNNs which
rival humans on the 3D-PC will become ideal models for a variety of real-world
applications where machines must anticipate human behavior in real-time, as
well as for enriching our understanding of how brains work (Fig. 1A).
##### Contributions.
We built the 3D-PC and used it to evaluate 3D perception for human
participants and 327 different DNNs. The DNNs we tested represented each of
today’s leading approaches, from Visual Transformers (ViT) [25] trained on
ImageNet-21k [26] to ChatGPT4 [27] and Stable Diffusion 2.0 [28].
* •
We found that DNNs were very accurate at determining the depth order of
objects after linear probing or text-prompting. DNNs that are state-of-the-art
on object classification matched or exceeded human accuracy on this task.
* •
However, DNNs dropped close to chance accuracy on VPT-basic, whereas humans
were nearly flawless at this task.
* •
Fine-tuning the zoo of DNNs on VPT-basic boosted their performance to near
human level. However, the performance of the DNNs — but not humans — dropped
back to chance on VPT-Strategy.
* •
Our findings demonstrate that the visual strategies necessary for solving VPT
do not emerge in DNNs from large-scale static image training or after directly
fine-tuning on the task. We release the 3D-PC data, code, and human
psychophysics at https://github.com/serre-lab/VPT to support the development
of models that can perceive and reason about the 3D world like humans.
## 2 Related work
##### 3D perception in humans.
The visual perception of 3D properties is a fundamentally ill-posed problem
[29, 30], which forces biological visual systems to rely on a variety of
assumptions to decode the structure of objects and scenes. For example,
variations in the lighting, texture gradients, retinal image disparity, and
motion of an object all contribute to the perception of its 3D shape. 3D
perception is further modulated by top-down beliefs about the structure of the
world, which are either innate or shaped by prior sensory experiences,
especially visual and haptic ones. In other words, humans learn about the 3D
structure of the world in an embodied manner that is fundamentally different
than how DNNs learn. In light of this difference, it would be remarkable if
DNNs could accurately model how humans perceive their 3D world.
##### Visual perspective taking in humans.
VPT was devised to understand how capabilities for reasoning about objects in
the world develop throughout the course of one’s life. At least two versions
of VPT have been introduced over the years [31, 32]. The version of VPT that
we study here — known in the developmental literature as “VPT-1” — is the more
basic form, which is thought to rely on automatic feedforward processing in
the visual system [31]. In light of the well-documented similarities between
feedforward processing in humans and DNNs [13, 33], we reasoned that this
version of VPT would maximize the chances of success for today’s DNNs.
##### 3D perception in DNNs trained on static images.
As deep neural networks (DNNs) have increased in scale and training dataset
size over the past decade, their performance on essentially all visual
challenges has improved. Surprisingly, this “scale-up” has also led to the
emergence of 3D perceptual capabilities. For example, DNNs trained with a
variety of self-supervised learning techniques on static image datasets learn
to represent the depth, surface normals, and 3D correspondence of features in
scenes [15, 16, 17, 18, 19, 20, 21, 22, 23]. While similarities between DNNs
and human 3D perception have yet to be evaluated systematically, it has been
shown that there are differences in how the two reason about the 3D shape of
objects [34]. The 3D-PC complements prior work by systematically evaluating
which aspects of human 3D perception today’s DNNs can and cannot accurately
represent.
##### Limitations of DNNs as models of human visual perception.
Over recent years, DNNs have grown progressively more accurate as models of
human vision for object recognition tasks [10, 24]. At the same time, these
models which succeed as models of human object recognition struggle to capture
other aspects of visual perception [35] including contextual illusions [36],
perceptual grouping [37, 38], and categorical prototypes [39]. There are also
multiple reports showing that DNNs are growing less aligned with the visual
strategies of humans and non-human primates as they improve on computer vision
benchmarks [40, 41, 42]. The 3D-PC provides another axis upon which the field
can evaluate DNNs as models of human vision.
## 3 Methods
##### The 3D-PC.
To enable a fair comparison between human observers’ and DNNs’ 3D perceptual
capabilities, we designed the 3D-PC framework with two goals: 1. posing
different 3D tasks on the same set of stimuli, and 2. generating a large
number of stimuli to properly train DNNs on these tasks. We achieved these
goals by combining 3D Gaussian Splatting [7], videos from the Common Objects
in 3D (Co3D) [43] dataset, and Unity [44, 45] into a flexible data-generating
framework.
Figure 2: 3D-PC examples. We tested 3D perception in images generated by
Gaussian Splatting. Each image depicts a green camera and a red ball. These
objects are placed in the scene in a way that counterbalances labels for depth
order task and VPT-basic tasks.
Our procedure for building the 3D-PC involved the following three steps. 1. We
trained Gaussian Splatting models on videos in Co3D (Fig. 1C). 2. We imported
these trained models into Unity, where we added green camera and red ball
objects into each 3D scene, which were used to pose visual tasks (Fig. 1D). 3.
We then generated random viewpoint trajectories within each 3D scene, rendered
images at each position along the trajectory, and derived ground-truth answers
for depth order and VPT tasks for the green camera at every position from
Unity.
Our approach makes it possible to generate an unlimited number of visual
stimuli that test an observer’s ability to solve complementary 3D perception
tasks (depth order and VPT) while keeping visual statistics constant and
ground truth labels counterbalanced across tasks. For the version of 3D-PC
used in our evaluation and released publicly at https://github.com/serre-
lab/VPT, the depth order and VPT-basic tasks are posed on the same set of
7,480 training images of 20 objects and scenes, and a set of 94 test images of
10 separate objects and scenes (Fig. 2). We held out a randomly selected 10%
of the training images for validation and model checkpoint selection.
To build the VPT-Strategy task, we rendered images where we fixed the scene
camera while we moved the green camera and red ball objects to precisely
change the line-of-sight between them from unobstructed to obstructed and
back. We reasoned that this experiment would reveal if an observer adopts the
visual strategy of taking the perspective of the green camera, which is
thought to be used by humans [31], from other strategies that relied on less
robust feature-based shortcuts. This dataset consisted of a test set of 100
images for 10 objects and scenes that were not included in depth order or VPT-
basic.
Figure 3: Human accuracy for object depth order and VPT-basic tasks. Bars near
50% are label-permuted noise floors; lines are group means. The difference is
significant, *** $=p<0.001.$
##### Psychophysics experiment.
We tested 10 participants on depth order, 20 on VPT-basic, and 3 on VPT-
Strategy. 33 participants were recruited online from Prolific. All provided
informed consent before completing the experiment and received $15.00/hr
compensation for their time (this amounted to $5.00 for the 15–20 minutes the
experiment lasted). These data were de-identified.
Participants were shown instructions for one of the 3D-PC tasks, then provided
20 examples to ensure that they properly understood it (Appendix Fig A.1).
These examples were drawn from the DNN training set. Each experimental trial
consisted of the following sequence of events overlaid onto a white
background: 1. a fixation cross displayed for $1000ms$; 2. an image displayed
for $3000ms$, during which time the participants were asked to render a
decision. Participants pressed one of the left or right arrow keys on their
keyboards to provide decisions.
Images were displayed at 256$\times$256 pixel resolution, which is equivalent
to a stimulus between $5^{\mathrm{o}}-11^{\mathrm{o}}$ of visual angle across
the range of display and seating setups we expected our online participants
used for the experiment.
##### Model zoo.
We evaluated a wide range of DNNs on the 3D-PC, which represented the leading
approaches for object classification, self-supervised pretraining, image
generation, depth prediction, and vision language modeling (VLM). Our zoo
includes 317 DNNs from PyTorch Image Models (TIMM) [46], ranging from classic
models like AlexNet [47] to state-of-the-art models like EVA-02 [48] (see
Appendix 1 for the complete list). We added foundational vision models like
MAE [49], DINO v2 [50], iBOT [51], SAM [52], and Midas [15] (obtained from the
GitHub repo of [23]). We also included Depth Anything [53], a foundational
model 3D scene analysis and depth prediction [23], as well as the Stable
Diffusion 2.0 [28] image generation model. Finally, we added state-of-the-art
large vision language models (VLMs) ChatGPT4 [27], Gemini [54], and Claude 3
[55]. We evaluated a total of 327 models on the 3D-PC.
Figure 4: DNN performance on the depth order and VPT-basic tasks in the 3D-PC
after linear probing or prompting. (A, B) DNNs are significantly more accurate
at depth order than VPT-basic. Human confidence intervals are S.E.M. and ***:
$p<0.001$. (C, D) DNN accuracy for depth order and VPT-basic strongly
correlates with object classification accuracy on ImageNet. Dashed lines are
the mean of label-permuted human noise floors.
##### Model evaluation.
We evaluated all models except for the VLMs on the depth order and VPT-basic
tasks in this challenge by training linear probes on image embeddings from
their penultimate layers. Linear probes were trained using PyTorch [56] for 50
epochs, a $5e$-4 learning rate, and early stopping (see Appendix A.5 for
details). Training took approximately 20 minutes per model using NVIDIA-RTX
3090s. We tested the Stable Diffusion 2.0 model by adopting the evaluation
method used in [14] (see Appendix A.7 for details). We evaluated the VLMs by
providing them the same instructions and training images (along with ground
truth labels) given to humans, then recording their responses to images from
each task via model APIs.
To test the learnability of the 3D-PC, we also fine-tuned each of the TIMM
models in our zoo to solve the tasks. To do this, we trained each of these
models for 30 epochs, a $5e$-5 learning rate, and early stopping (see Appendix
A.5 for details). Fine-tuning took between 3 hours and 24 hours per model
using NVIDIA-RTX 3090s.
## 4 Results
Figure 5: DNN performance on the depth order and VPT-basic tasks in the 3D-PC
after fine-tuning. (A) Fine-tuning makes DNNs far better than humans at the
depth order task and improves the performance of several DNNs to be at or
beyond human accuracy on VPT-basic. (B) Even after fine-tuning, there is still
a significant difference in model performance on depth order and VPT-basic
tasks, $p<0.001$. (C, D) DNN accuracy on both tasks after fine-tuning
correlates with ImageNet object classification accuracy. Human confidence
intervals are S.E.M. and ***: p < 0.001. Dashed lines are the mean of label-
permuted human noise floors.
##### Humans find VPT easier than determining the depth ordering of objects.
Human participants were on average 74.73% accurate at determining the depth
order of objects, and 86.82% accurate at solving the VPT-basic task (Fig. 3;
$p<0.001$ for both; statistical testing done through randomization tests
[57]). Humans were also significantly more accurate at solving VPT-basic than
they were at the depth order task.
##### DNNs learn depth but not VPT from static image training.
DNNs showed the opposite pattern of results on depth order and VPT-basic tasks
as humans after linear probing or prompting (Fig. 4): 15 of the DNNs we tested
fell within the human accuracy confidence interval on the depth order task,
and three even outperformed humans (Fig. 4A). In contrast, while humans were
on average 86.82% accurate at VPT-basic, the DNN which performed the best on
this task, the ImageNet 21K-trained beit [58], was 53.82% accurate. Even
commercial VLMs struggled on VPT-basic and were around chance accuracy
(ChatGPT4: 52%, Gemini: 52%, and Claude 3: 50%). The depth order task was
significantly easier for DNNs than VPT-basic ($p<0.001$), which is the
opposite of humans (Fig. 4B).
##### ImageNet accuracy correlates with the 3D capabilities of DNNs.
What drives the development of 3D perception in DNNs trained on static images?
We hypothesized that as DNNs scale up, they learn ancillary strategies for
processing natural images, including the ability to analyze the 3D structure
of scenes. To investigate this possibility, we focused on the TIMM models in
our DNN zoo. These models have previously been evaluated for object
classification accuracy on ImageNet, which we used as a stand-in for DNN scale
[41, 42, 40]. Consistent with our hypothesis, we found a strong and
significant correlation between DNN performance on ImageNet and depth order
task accuracy ($\rho=0.66,p<0.001$, Fig. 4C). Despite the very low accuracy of
DNNs on VPT-basic, there was also a weaker but still significant correlation
between performance on this task and ImageNet ($\rho=0.34,p<0.001$, the
difference in correlations between the tasks is $\rho=0.32,p<0.001$; Fig. 4D).
These results suggest that monocular depth cues develop in DNNs alongside
their capabilities for object classification 111More work is needed to
identify a causal relationship between the development of monocular depth cues
and object recognition accuracy.. However, the depth cues that DNNs learn are
poorly suited for VPT.
Figure 6: Even DNNs fine-tuned on VPT-basic fail on VPT-Strategy. (A) To
better characterize the strategy used by humans and DNNs to solve VPT, we
devised a new test, VPT-Strategy, in which the green camera and red ball are
moved through a scene while holding the scene camera and a centrally-
positioned object still. This task is easily solvable if an observer estimates
the line-of-sight of the green camera; other strategies, such as those that
rely on specific image features (feature based), may be less effective. (B)
Examples of VPT-Strategy stimuli along with the ground-truth label (top-row)
and predictions by a ViT large after linearly probing or fine-tuning for VPT-
basic (bottom-row). Decision attribution maps from each version of the ViT
large, derived from “smooth gradients” [59], are overlaid onto bottom-row
images (purple/blue=linearly probed, yellow/green=fine-tuned). The fine-tuned
ViT locates the green camera and red ball but renders incorrect decisions. (C)
DNNs fine-tuned on VPT-basic fail to solve VPT-Strategy; they rely on a
brittle feature-based strategy. Humans, on the other hand, are 87% accurate;
they likely estimate line-of-sight.
##### DNNs can solve VPT-basic after fine-tuning.
One possible explanation for the failure of today’s DNNs on VPT-basic is that
the task requires additional cues for 3D vision that cannot be easily learned
from static images. To explore this possibility, we fine-tuned each of the
TIMM models in our DNN zoo to solve depth order and VPT-basic (Fig. 5A). There
was still a significant difference between DNN performance on the two tasks
(Fig. 5B, $p<0.001$), but fine-tuning caused 97% of the DNNs to exceed human
accuracy on depth order, and four of the DNNs to reach human accuracy on VPT-
basic. DNN performance on the tasks more strongly correlated with ImageNet
accuracy after fine-tuning than linear probing (compare Fig. 5C/D and Fig.
4C/D). We also compared the errors these DNNs made on both tasks to humans. We
found nearly all of the fine-tuned DNNs were aligned with humans on depth
order, and a handful were aligned with humans on VPT-basic (Fig. A.3).
##### DNNs learn different strategies than humans to solve VPT.
The ability of DNNs to reach human-level performance on visual tasks by
adopting strategies that are different from humans has been well-documented
[40, 41, 42]. Thus, we devised a new experiment to understand if DNNs learn to
solve VPT in the same way as humans do after fine-tuning. In developmental
psychology, it has been proposed that humans estimate the line-of-sight of
objects for VPT because they respond in predictable ways after the positions
of objects in a scene are slightly adjusted [32, 31]. Inspired by this
psychological work, we created the VPT-Strategy task to evaluate the types of
visual strategies used by DNNs and humans to solve VPT (Fig. 6A).
VPT-Strategy has observers solve the VPT task on a series of images rendered
from a fixed camera viewpoint as the green camera and red ball are moved
incrementally from one side of the screen to the other, passing by an
occluding object in the process. This means that we can precisely map out the
moments at which the green camera has a clear view of the red ball, when that
view is occluded, and when the view becomes unoccluded once more. DNNs behave
differently than humans on this task: humans were 87% accurate, but the
highest performing DNN, the Swin Transformer [60] trained on ImageNet-21k, was
only 66% accurate (Fig. 6B, C). In other words, while DNNs can be fine-tuned
to approach human accuracy on VPT-basic, the strategy they learn is brittle,
generalizes poorly, and is likely ill-suited for reasoning about the 3D world.
## 5 Discussion
Deep neural networks (DNNs) have rapidly advanced over recent years to the
point where they match or surpass human-level performance on numerous visual
tasks. However, our 3D-PC reveals there is still a significant gap between the
abilities of humans and DNNs to reason about 3D scenes. While DNNs match or
exceed human accuracy on the basic object depth order task after linear
probing or prompting, they struggle remarkably on even the basic form of VPT
that we test in the 3D-PC. Fine-tuning DNNs on VPT-basic allows them to
approach human-level performance, but unlike humans, their strategies do not
generalize to the VPT-Strategy task.
A striking finding from our study is the strong correlation between DNNs’
object classification accuracy on ImageNet and their performance on depth
order and VPT-basic. This correlation suggests that monocular depth cues
emerge in DNNs as a byproduct of learning to recognize objects, potentially
because these cues are useful for segmenting objects from their backgrounds.
The difference in DNN effectiveness for depth order versus VPT-basic, however,
indicates that these cues are not sufficient for reasoning about the 3D
structure of scenes in the way that VPT demands.
Thus, today’s approaches for developing DNNs, which primarily focus on static
image datasets, may be poorly suited for enabling robust 3D perception and
reasoning abilities akin to those of humans. Incorporating insights from human
cognition and neuroscience into DNNs, particularly in ways biological visual
systems develop 3D perception, could help evolve more faithful models of human
intelligence.
A key limitation of our study is that our version of VPT represents the most
basic form studied in the developmental psychology literature. While solving
this task is evidently an extraordinary challenge for DNNs, it is only one
small step towards human-level capabilities for reasoning about 3D worlds in
general. Far more research is needed to identify additional challenges,
architectures, and training routines that can help DNNs perceive and reason
about the world like humans do. We release our 3D-PC data and code at
https://github.com/serre-lab/VPT to support this goal.
## 6 Acknowledgements
Funding for this project was provided by the Office of Naval Research
(N00014-19- 1-2029) and ANR-3IA Artificial and Natural Intelligence Toulouse
Institute (ANR-19-PI3A0004). Additional support was provided by the Carney
Institute for Brain Science and the Center for Computation and Visualization
(CCV). We acknowledge the Cloud TPU hardware resources that Google made
available via the TensorFlow Research Cloud (TFRC) program as well as
computing hardware supported by NIH Office of the Director grant S10OD025181.
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## Checklist
1. 1.
For all authors…
1. (a)
Do the main claims made in the abstract and introduction accurately reflect
the paper’s contributions and scope? [Yes]
2. (b)
Did you describe the limitations of your work? [Yes] In the discussion.
3. (c)
Did you discuss any potential negative societal impacts of your work? [Yes]
Appendix section A.3.
4. (d)
Have you read the ethics review guidelines and ensured that your paper
conforms to them? [Yes]
2. 2.
If you are including theoretical results…
1. (a)
Did you state the full set of assumptions of all theoretical results? [N/A]
2. (b)
Did you include complete proofs of all theoretical results? [N/A]
3. 3.
If you ran experiments (e.g. for benchmarks)…
1. (a)
Did you include the code, data, and instructions needed to reproduce the main
experimental results (either in the supplemental material or as a URL)? [Yes]
See methods.
2. (b)
Did you specify all the training details (e.g., data splits, hyperparameters,
how they were chosen)? [Yes] Appendix section A.5
3. (c)
Did you report error bars (e.g., with respect to the random seed after running
experiments multiple times)? [Yes] We report error bars over human performance
in all figures. We also report model/error bars in performance and correlation
with humans (Appendix fig. A.3).
4. (d)
Did you include the total amount of compute and the type of resources used
(e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Appendix
section A.5.
4. 4.
If you are using existing assets (e.g., code, data, models) or
curating/releasing new assets…
1. (a)
If your work uses existing assets, did you cite the creators? [Yes] We used
existing models and libraries and cited each.
2. (b)
Did you mention the license of the assets? [Yes] See Appendix section A.1.
3. (c)
Did you include any new assets either in the supplemental material or as a
URL? [Yes]
4. (d)
Did you discuss whether and how consent was obtained from people whose data
you’re using/curating? [Yes] See methods.
5. (e)
Did you discuss whether the data you are using/curating contains personally
identifiable information or offensive content? [Yes] See methods.
5. 5.
If you used crowdsourcing or conducted research with human subjects…
1. (a)
Did you include the full text of instructions given to participants and
screenshots, if applicable? [Yes] See appendix section A.8
2. (b)
Did you describe any potential participant risks, with links to Institutional
Review Board (IRB) approvals, if applicable? [N/A]
3. (c)
Did you include the estimated hourly wage paid to participants and the total
amount spent on participant compensation? [Yes] See methods.
## Appendix A Appendix
### A.1 Author Statement
As authors of this dataset, we bear all responsibility for the information
collected and in case of violation of rights and other ethical standards. We
affirm that our dataset is shared under a Creative Commons CC-BY license.
### A.2 Data Access
We release benchmarking code and data download instructions at
https://github.com/serre-lab/VPT.
### A.3 Potential negative societal impacts of this work
The most obvious potential negative impact of our work is that advancing
visual perspective taking (VPT) capabilities in artificial agents could
potentially enable militaristic applications or surveillance overreach.
However, we hope that our benchmark will aid in the development of AI-based
assistants that can better anticipate and react to human needs and social cues
for safer navigation and interaction. We also believe that our benchmark will
guide the development of better computational models of human 3D perception as
well as the neural underpinnings of these abilities.
### A.4 Data Generation
To generate data for the 3D-PC, we first trained 3D Gaussian Splatting [7]
models on videos from the Common Objects in 3D (Co3D) [43], which yielded 3D
representations of each scene. We then imported trained models into Unity [44]
using Unity Gaussian Splatting [45] and added 3D models of the green camera
and red ball to each. Finally, we rendered 50 images along a smooth viewpoint
camera trajectory sampled near the original trajectory used for training the
Gaussian Splatting model. For each 3D scene, we created 5 positive and 5
negative settings for VPT.
To generate VPT-basic, the generation process was repeated for 30 Co3D videos
from 10 different categories. We removed any images where the green camera and
red ball were not visible. We then split the images into a training set of
7480 images from 20 scenes and a testing set of 94 images from 10 other
scenes. For the depth order task, we used the same data splits but removed any
ambiguous samples where the objects were similarly close to the camera. The
resulting dataset for the depth order task contains 4787 training images and
94 testing images. The same set of testing images is used for both model and
human benchmarks.
For VPT-Strategy, we used the same process to generate data from 10 additional
Co3D scenes not included in VPT-basic and additionally controlled the
positions of the green camera and the red ball. The angle between these two
objects was held constant while we moved them so that their line of sight was
unobstructed, obstructed, and then unobstructed once again. For each Co3D
scene, we rendered 10 settings from a fixed viewpoint camera position,
resulting in 100 images in total for VPT-Strategy.
### A.5 Model Zoo
We linearly probed 317 DNNs from Pytorch Image Models (TIMM) [46] (Table 1)
along with foundational vision models following the procedures in [23]. All
DNNs were trained and evaluated with NVIDIA-RTX 3090 GPUs from the Brown
University Center for Computation & Visualization. All linear probes were
trained for 50 epochs, with a $5e-4$ learning rate, a $1e-4$ weight decay, a
$0.3$ dropout rate, and a batch size of 128. We fine-tuned each of the TIMM
models for 30 epochs, a $5e-5$ learning rate, $1e-4$ weight decay, $0.7$
dropout rate, and a batch size of 16. Linear probing took approximately 20
minutes per model, and fine-tuning varied from 3 to 24 hours on a NVIDIA-RTX
3090 GPU.
Architecture | Model | Versions
---|---|---
CNN | ConvMixer [61] | 3
ConvNeXT [62] | 10
DenseNet [63] | 4
DLA [64] | 5
DPN [65] | 6
EfficientNet [66] | 4
GhostNet [67] | 1
HRNet [68] | 8
LCNet [69] | 3
MixNet [70] | 4
MnasNet [71] | 3
MobileNet [72] | 14
RegNet [73] | 6
Res2Net [74] | 5
ResNet [75] | 26
ResNeSt [76] | 3
RexNet [77] | 5
ResNext [78] | 2
SPNASNet [79] | 1
TinyNet [80] | 2
VGG [81] | 14
Transformer | BEiT [82] | 9
CAFormer [83] | 6
CaiT [84] | 3
ConViT [85] | 3
CrossViT [86] | 2
DaViT [87] | 3
DeiT [88] | 12
EfficientFormer [89] | 7
EVA [48] | 9
FocalNet [90] | 6
LeViT [91] | 5
MaxViT [92] | 6
MobileViT [93] | 3
MViT [94] | 3
PiT [95] | 8
PVT [96] | 7
Swin [60] | 16
Twins-SVT [97] | 5
ViT [98] | 36
Volo [99] | 7
XCiT [100] | 6
PoolFormer [101] | 8
Hybrid | CoaT [102] | 7
CoAtNet [103] | 8
EdgeNeXt [104] | 1
Visformer [105] | 2
Foundation | Depth Anything [53] | 1
DINOv2 [50] | 1
iBoT [51] | 1
MAE [49] | 1
MiDas [15] | 1
SAM [52] | 1
VLM | ChatGPT4 [27] | 1
Gemini [54] | 1
Claude 3 [55] | 1
Diffusion | Stable Diffusion 2.0 [28] | 1
Table 1: The 327 DNN models used in our study.
### A.6 VLM Evaluation
We evaluated the following proprietary VLMs on the VPT-basic and depth order
tasks: GPT-4 (gpt-4-turbo), Claude (claude-3-opus-20240229), and Gemini
(gemini-pro-vision). To evaluate these VLMs, we used their APIs to send
queries containing 20 training images, with ground truth answers as context,
as well as a test image. The prepended 20 training images meant that for every
example in the challenge, VLMs were given the opportunity to learn, “in-
context”, how to solve the given task.
The prompt we used for the depth task was “In this image, is the red ball
closer to the observer or is the green arrow closer to the observer? Answer
only BALL if the red ball is closer, or ARROW if the green arrow is closer,
nothing else.” and the prompt for the VPT-basic task was “In this image, if
viewed from the perspective of the green 3D arrow in the direction the arrow
is pointing, can a human see the red ball? Answer only YES or NO, nothing
else”. We evaluated each model’s generated responses across multiple
temperatures, ranging from $0.0$ to $0.7$ in increments of $0.1$, and we
report the average of the best 3 runs. Note that while this evaluation
approach gives the VLMs more opportunities to perform well on our benchmark
than other models, they still struggled immensely (see main text).
### A.7 Stable Diffusion Evaluation
We followed the method of Li et al. [14] to evaluate Stable Diffusion 2.0 on
the 3D-PC. This involved trying multiple prompts to optimize the zero-shot
classification performance of the Stable Diffusion 2.0 model, on VPT-basic and
depth order tasks. For VPT-basic we found that the prompt "A photo with red
ball is visible from the green arrow’s perspective" for positive class and "A
photo with red ball not visible from the green arrow’s perspective" for the
negative class led to the best performance. For the depth order task, the
prompt with the highest performance was "A photo with green arrow closer to
the camera as compared to red ball" and "A photo with red ball closer to the
camera as compared to green arrow" for positive and negative classes
respectively.
### A.8 Human Benchmark
We recruited 30 participants through Prolific, compensating each with $5 upon
successful completion of all test trials. Participants confirmed their
completion by pasting a unique system-generated code into their Prolific
accounts. The compensation was prorated based on the minimum wage. We also
incurred a 30% overhead fee per participant paid to Prolific. In total, we
spent $195 on these benchmark experiments.
#### A.8.1 Experiment design
At the outset of the experiment, we acquired participant consent through a
form approved by the Brown University’s Institutional Review Board (IRB). The
experiment was performed on a computer using the Chrome browser. Following
consent, we presented a demonstration with instructions and an example video.
Participants had the option to revisit the instructions at any time during the
experiment by clicking a link in the top right corner of the navigation bar.
Figure A.1: An experiment trial. Figure A.2: The consent screen.
In the depth order task, the participants were asked to classify the image as
“positive” (the green arrow in closer to the viewer) or “negative” (the red
ball is closer) using the right and left arrow keys respectively. The choice
for keys and their corresponding instances were mentioned below the image on
every screen (See Appendix Fig. A1. Participants were given feedback on their
response (correct/incorrect) during every practice trial, but not during the
test trials. In the VPT tasks, the choices were “the green arrow/camera see
the red ball” or “the green arrow/camera can not see the red ball”.
The experiment was not time-bound, allowing participants to complete it at
their own pace. Participants typically took around 20 minutes. After each
trial, participants were redirected to a screen confirming the successful
submission of their responses. They could start the next trial by clicking the
“Continue” button or pressing the spacebar. If they did not take any action,
they were automatically redirected to the next trial after 1000 milliseconds.
Additionally, participants were shown a “rest screen” with a progress bar
after every 40 trials, where they could take additional and longer breaks if
needed. The timer was turned off during the rest screen.
### A.9 Human vs. DNN decision making on VPT-basic
We compared the decision strategies of humans and DNNs on VPT-basic by
measuring the correlations between their error patterns with Cohen’s $\kappa$
[106]. Model $\kappa$ scores were mostly correlated with accuracy on VPT-basic
after linear probes and fine-tuning (Fig. A.3). However, while nearly all DNNs
were highly correlated with human error patterns after fine-tuning, the
correlation between $\kappa$ scores and task accuracy disappeared (Fig. A.3B,
purple dots).
Figure A.3: Error pattern correlations (Cohen’s $\kappa$) between humans and
DNNs on VPT-basic
### A.10 Datasheet for datasets
Motivation
For what purpose was the dataset created? Was there a specific task in mind?
Was there a specific gap that needed to be filled? Please provide a
description.
The dataset was designed to test 3D perception in humans and DNNs, with an
emphasis on the capabilities of each for visual perspective taking (VPT).
Humans rely on VPT everyday for navigating and socializing, but despite its
importance, there has yet to be a systematic evaluation of this ability in
DNNs.
Who created this dataset (e.g., which team, research group) and on behalf of
which entity (e.g., company, institution, organization)?
This dataset was created by this paper authors, who are affiliated with the
Carney Institute for Brain Science at Brown University and the Cognitive
Sciences Department at UC Irvine.
Who funded the creation of the dataset? If there is an associated grant,
please provide the name of the grantor and the grant name and number.
Funding for this project was provided by the Office of Naval Research
(N00014-19- 1-2029) and ANR-3IA Artificial and Natural Intelligence Toulouse
Institute (ANR-19-PI3A0004). Additional support provided by the Carney
Institute for Brain Science and the Center for Computation and Visualization
(CCV). We acknowledge the Cloud TPU hardware resources that Google made
available via the TensorFlow Research Cloud (TFRC) program as well as
computing hardware supported by NIH Office of the Director grant S10OD025181.
Composition
What do the instances that comprise the dataset represent (e.g., documents,
photos, people, countries)? Are there multiple types of instances (e.g.,
movies, users, and ratings; people and interactions between them; nodes and
edges)? Please provide a description.
The instances contain images of real-world objects and scenes along with
shapes generated with computer graphics.
How many instances are there in total (of each type, if appropriate)?
There are 7574 images in the training and testing sets.
Does the dataset contain all possible instances or is it a sample (not
necessarily random) of instances from a larger set? If the dataset is a
sample, then what is the larger set? Is the sample representative of the
larger set (e.g., geographic coverage)? If so, please describe how this
representativeness was validated/verified. If it is not representative of the
larger set, please describe why not (e.g., to cover a more diverse range of
instances, because instances were withheld or unavailable).
We release all data.
What data does each instance consist of? “Raw” data (e.g., unprocessed text or
images) or features? In either case, please provide a description.
Each instance consists of an image rendered from 3d Gaussian Splatting [7]
models trained on Co3D [43] scenes.
Is there a label or target associated with each instance? If so, please
provide a description.
The images are labeled for VPT and depth order tasks. In the VPT task, an
image is labeled as positive when the red ball is visible from the green
camera’s perspective. In the depth task, an image is labeled as positive when
the red ball is further away than the green arrow from the viewer. For both
tasks, we label positives as 1 and negatives as 0.
Is any information missing from individual instances? If so, please provide a
description, explaining why this information is missing (e.g., because it was
unavailable). This does not include intentionally removed information, but
might include, e.g., redacted text.
N/A
Are relationships between individual instances made explicit (e.g., users’
movie ratings, social network links)? If so, please describe how these
relationships are made explicit.
N/A
Are there recommended data splits (e.g., training, development/validation,
testing)? If so, please provide a description of these splits, explaining the
rationale behind them.
We provide training, validation and testing splits in the released dataset.
The training set contains images rendered from 20 unique scenes from 10
categories. The testing set images are rendered from 10 additional scenes from
the same categories. We randomly selected 10% of the training set as the
validation set.
Are there any errors, sources of noise, or redundancies in the dataset? If so,
please provide a description.
N/A
Is the dataset self-contained, or does it link to or otherwise rely on
external resources (e.g., websites, tweets, other datasets)? If it links to or
relies on external resources, a) are there guarantees that they will exist,
and remain constant, over time; b) are there official archival versions of the
complete dataset (i.e., including the external resources as they existed at
the time the dataset was created); c) are there any restrictions (e.g.,
licenses, fees) associated with any of the external resources that might apply
to a future user? Please provide descriptions of all external resources and
any restrictions associated with them, as well as links or other access
points, as appropriate.
The dataset uses videos from the Co3D dataset [43], which is publicly
available under CC BY-NC 4.0 license.
Does the dataset contain data that might be considered confidential (e.g.,
data that is protected by legal privilege or by doctor-patient
confidentiality, data that includes the content of individuals non-public
communications)? If so, please provide a description.
N/A
Does the dataset contain data that, if viewed directly, might be offensive,
insulting, threatening, or might otherwise cause anxiety? If so, please
describe why.
N/A
Does the dataset relate to people? If not, you may skip the remaining
questions in this section.
Yes
Does the dataset identify any subpopulations (e.g., by age, gender)? If so,
please describe how these subpopulations are identified and provide a
description of their respective distributions within the dataset.
No
Is it possible to identify individuals (i.e., one or more natural persons),
either directly or indirectly (i.e., in combination with other data) from the
dataset? If so, please describe how.
No, all results are anonymous.
Does the dataset contain data that might be considered sensitive in any way
(e.g., data that reveals racial or ethnic origins, sexual orientations,
religious beliefs, political opinions or union memberships, or locations;
financial or health data; biometric or genetic data; forms of government
identification, such as social security numbers; criminal history)? If so,
please provide a description.
N/A
Any other comments?
Collection Process
How was the data associated with each instance acquired? Was the data directly
observable (e.g., raw text, movie ratings), reported by subjects (e.g., survey
responses), or indirectly inferred/derived from other data (e.g., part-of-
speech tags, model-based guesses for age or language)? If data was reported by
subjects or indirectly inferred/derived from other data, was the data
validated/verified? If so, please describe how.
All images were rendered from 3D gaussian splatting [7] models trained on
videos from Co3D [43]. We imported the model into Unity [44, 45] to render
images.
What mechanisms or procedures were used to collect the data (e.g., hardware
apparatus or sensor, manual human curation, software program, software API)?
How were these mechanisms or procedures validated?
We used Unity [44] and Unity Gaussian Splatting [45] to edit the scenes and
label them in 3D view.
If the dataset is a sample from a larger set, what was the sampling strategy
(e.g., deterministic, probabilistic with specific sampling probabilities)?
N/A
Who was involved in the data collection process (e.g., students, crowdworkers,
contractors) and how were they compensated (e.g., how much were crowdworkers
paid)?
The paper’s authors were involved in the data collection process.
Over what timeframe was the data collected? Does this timeframe match the
creation timeframe of the data associated with the instances (e.g., recent
crawl of old news articles)? If not, please describe the timeframe in which
the data associated with the instances was created.
N/A
Were any ethical review processes conducted (e.g., by an institutional review
board)? If so, please provide a description of these review processes,
including the outcomes, as well as a link or other access point to any
supporting documentation.
Does the dataset relate to people? If not, you may skip the remaining
questions in this section.
Yes
Did you collect the data from the individuals in question directly, or obtain
it via third parties or other sources (e.g., websites)?
As described in the Methods, we collected data from online participants
through Prolific, and we also collected data in-person for several subjects.
Were the individuals in question notified about the data collection? If so,
please describe (or show with screenshots or other information) how notice was
provided, and provide a link or other access point to, or otherwise reproduce,
the exact language of the notification itself.
Yes. See Section A.8 for details.
Did the individuals in question consent to the collection and use of their
data? If so, please describe (or show with screenshots or other information)
how consent was requested and provided, and provide a link or other access
point to, or otherwise reproduce, the exact language to which the individuals
consented.
Yes. See Fig A.2 for the consent screen with the exact language used.
If consent was obtained, were the consenting individuals provided with a
mechanism to revoke their consent in the future or for certain uses? If so,
please provide a description, as well as a link or other access point to the
mechanism (if appropriate).
Yes. The participants were provided with our contact information and were
encouraged to reach out in such cases.
Has an analysis of the potential impact of the dataset and its use on data
subjects (e.g., a data protection impact analysis) been conducted? If so,
please provide a description of this analysis, including the outcomes, as well
as a link or other access point to any supporting documentation.
Our experiment was approved by the IRB board at Brown University.
Any other comments?
Preprocessing/cleaning/labeling
Was any preprocessing/cleaning/labeling of the data done (e.g., discretization
or bucketing, tokenization, part-of-speech tagging, SIFT feature extraction,
removal of instances, processing of missing values)? If so, please provide a
description. If not, you may skip the remainder of the questions in this
section.
We used Unity to label images for VPT and depth tasks. We removed images where
the objects of interest (red ball and green camera) were not visible.
Was the “raw” data saved in addition to the preprocessed/cleaned/labeled data
(e.g., to support unanticipated future uses)? If so, please provide a link or
other access point to the “raw” data.
N/A
Is the software used to preprocess/clean/label the instances available? If so,
please provide a link or other access point.
N/A
Any other comments?
Uses
Has the dataset been used for any tasks already? If so, please provide a
description.
We evaluated vision DNNs on the dataset. Please refer to the main paper for
details.
Is there a repository that links to any or all papers or systems that use the
dataset? If so, please provide a link or other access point.
The code and data are publicly available at https://github.com/serre-lab/VPT
What (other) tasks could the dataset be used for?
We mainly expect the dataset to be used for evaluating 3D perception
capabilities of new vision or vision-language DNNs.
Is there anything about the composition of the dataset or the way it was
collected and preprocessed/cleaned/labeled that might impact future uses? For
example, is there anything that a future user might need to know to avoid uses
that could result in unfair treatment of individuals or groups (e.g.,
stereotyping, quality of service issues) or other undesirable harms (e.g.,
financial harms, legal risks) If so, please provide a description. Is there
anything a future user could do to mitigate these undesirable harms?
N/A
Are there tasks for which the dataset should not be used? If so, please
provide a description.
N/A
Any other comments?
Distribution
Will the dataset be distributed to third parties outside of the entity (e.g.,
company, institution, organization) on behalf of which the dataset was
created? If so, please provide a description.
Yes, we will release the dataset to the public at https://github.com/serre-
lab/VPT
How will the dataset be distributed (e.g., tarball on website, API, GitHub)
Does the dataset have a digital object identifier (DOI)?
We provide download instructions at https://github.com/serre-lab/VPT
When will the dataset be distributed?
The dataset is available from June 5th, 2024.
Will the dataset be distributed under a copyright or other intellectual
property (IP) license, and/or under applicable terms of use (ToU)? If so,
please describe this license and/or ToU, and provide a link or other access
point to, or otherwise reproduce, any relevant licensing terms or ToU, as well
as any fees associated with these restrictions.
We release our data under a Creative Commons CC-BY license.
Have any third parties imposed IP-based or other restrictions on the data
associated with the instances? If so, please describe these restrictions, and
provide a link or other access point to, or otherwise reproduce, any relevant
licensing terms, as well as any fees associated with these restrictions.
N/A
Do any export controls or other regulatory restrictions apply to the dataset
or to individual instances? If so, please describe these restrictions, and
provide a link or other access point to, or otherwise reproduce, any
supporting documentation.
N/A
Any other comments?
Maintenance
Who will be supporting/hosting/maintaining the dataset?
The authors will be hosting and maintaining the dataset.
How can the owner/curator/manager of the dataset be contacted (e.g., email
address)?
Contact the corresponding author through email.
Is there an erratum? If so, please provide a link or other access point.
N/A
Will the dataset be updated (e.g., to correct labeling errors, add new
instances, delete instances)? If so, please describe how often, by whom, and
how updates will be communicated to users (e.g., mailing list, GitHub)?
We are actively working on expanding the dataset with new instances and tasks.
We will update our GitHub repository accordingly for any dataset update.
If the dataset relates to people, are there applicable limits on the retention
of the data associated with the instances (e.g., were individuals in question
told that their data would be retained for a fixed period of time and then
deleted)? If so, please describe these limits and explain how they will be
enforced.
Human participant data was de-identified, and there are no time limits on its
retention.
Will older versions of the dataset continue to be supported/hosted/maintained?
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We are open to any suggestions and contributions through our GitHub
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|
# $L^{q}$ Carleman estimates with boundary observations and applications to
inverse problems
###### Abstract.
We consider coupled linear parabolic systems and we establish estimates in
$L^{q}$-norm for the sources in terms of observations on the corresponding
solutions on a part of the boundary. The main tool is a family of Carleman
estimates in $L^{q}$-norm with boundary observations.
###### Key words and phrases:
Parabolic systems, Carleman estimates, Inverse problems, Source estimates.
###### 2010 Mathematics Subject Classification:
35R30, 35K10, 35K57, 93B07
This work was co-funded by the European Social Fund, through Operational
Programme Human Capital 2014-2020, project number POCU/993/6/13/153322,
project title ”Educational and training support for PhD students and young
researchers in preparation for insertion into the labor market”.
Elena-Alexandra Melnig
Faculty of Mathematics, ”Al. I. Cuza” University of Iaşi, Romania
Octav Mayer Institute of Mathematics, Romanian Academy, Iaşi Branch
E-mail address<EMAIL_ADDRESS>
## Introduction
In this paper we consider linear parabolic systems coupled in zero order terms
and we obtain estimates in $L^{q}$ spaces, $q\geq 2$ for the sources in terms
of measurements of the solutions on a part of the boundary. This research is
based on previous results in $L^{2}$ and uses the regularity properties of the
heat flow. Such an inverse problem is motivated by the fact that the boundary
of a domain is more accessible for measurements on the solution.
This type of problems was considered by Imanuvilov and Yamamoto in [4] where
they obtained source estimates in $L^{2}$ norm with observations on the
solution on a subdomain or on the boundary. Their method employed the coupling
of two $L^{2}$-Carleman estimates, a technique no longer applicable in the
$L^{q}$ framework, as explained in [9]. Furthermore, they require more
detailed information from the measurements, as their estimates involve the
solution together with its time and space derivatives, something we overcome
in the present paper.
The problem we address here comes as a continuation of the work done in [9]
where we have established such $L^{q}$ stability estimates for the sources of
a linear system in terms of the solution measured on a subdomain. Also, in
[10] we established estimates of this type for the sources in a semilinear
system with observations on the solution in a subdomain.
An important aspect in the above mentioned papers as well as in the present
study is the positivity of the solutions and of the sources, as the systems we
consider are intended to model reaction-diffusion processes. In this context
we need two fundamental tools: strong invariance principles and respectively
Carleman estimates. The first tool relies on the strong maximum principle for
parabolic equations and on strong invariance results for weakly coupled
parabolic systems. For such maximum principles or invariance results in the
framework of classical solutions we refer to [11] for equations and to [12]
for systems. For the case of variational solutions we mention [8].
The other main tool in our approach is a family of Carleman estimates in
$L^{q}$ norm, $q\geq 2$, with boundary observations and general weights of
exponential type obtained for parabolic systems with general boundary
conditions. The Carleman inequalities in $L^{2}$ were first used in the
framework of controllability of the heat equation with controls distributed in
a subdomain bu O.Yu Imanuvilov, [3]. Since then they have found applications
in obtaining observability inequalities in controllability problems, unique
continuation properties and inverse problems. The Carleman estimates in
$L^{q}$ are derived through a bootstrap argument, starting from the
$L^{2}$-Carleman estimates proved in [7] and using the regularizing effect of
the parabolic equation. This type of argument can be found in V. Barbu [1],
J.-M. Coron, S.Guerrero, L.Rosier [2], K. Le Balc’h [6], in the context of
controllability.
## 1\. Preliminaries and main results
Let $T>0$ and $\Omega\subset\mathbb{R}^{N}$ be an annular domain, that is a
domain which is diffeomorphic to $B_{2}(0)\setminus B_{1}(0)$, with boundary
$\partial\Omega$ of class $C^{2}$. Consider $\Gamma_{1}\subset\partial\Omega$
be the exterior boundary and
$\Gamma_{0}=\partial\Omega\setminus\overline{\Gamma}_{1}$ be the inner
boundary, and let $Q=(0,T)\times\Omega$,
$\Sigma_{1}:=(0,T)\times\Gamma_{1},\Sigma_{0}:=(0,T)\times\Gamma_{0}$.
In the following, whenever we will refer to vector valued functions from a
corresponding Sobolev space, we will write $L^{q}(Q)$ instead of
$[L^{q}(Q)]^{n}$, $W^{2,1}_{q}(Q)\cap L^{\infty}(Q)$ instead of
$[W^{2,1}_{q}(Q)\cap L^{\infty}(Q)]^{n}$.
We consider linear parabolic systems coupled in zero order terms with the
elliptic part of the operator in divergence form and general homogeneous
boundary conditions (Dirichlet, Neumann or Robin) on each connected component
of the boundary and for each component of the system:
(S)
$\left\\{\begin{array}[]{ll}D_{t}y_{i}-\sum\limits_{j,k=1}^{N}D_{j}(a_{i}^{jk}D_{k}y_{i})+\sum\limits_{k=1}^{N}b_{i}^{k}D_{k}y_{i}+\sum\limits_{l=1}^{N}c_{i}^{l}y_{l}=g_{i}&(0,T)\times\Omega,\\\
\beta_{i}(x)\frac{\partial y_{i}}{\partial
n_{A}}+\eta_{i}(x)y_{i}=0&(0,T)\times\partial\Omega,\\\ \end{array}\quad
i=\overline{1,n}\right.$
where
* (H1)
$a_{i}^{jk}\in W^{1,\infty}(\Omega)$, $b^{k}_{i},c_{i}^{l}\in
L^{\infty}(\Omega)$ and $a_{i}^{jk}$ satisfy the ellipticity condition:
$\exists\mu>0\text{ {s.t.}
}\sum_{j,k=1}^{N}a_{i}^{jk}(x)\xi_{j}\xi_{k}\geq\mu|\xi|^{2},\quad\forall\xi\in\mathbb{R}^{N},\quad(t,x)\in
Q,i=\overline{1,n};$
* (H2)
the sources are positive, $g_{i}\in L^{q}(Q)$, $g_{i}\geq 0,i=\overline{1,n};$
* (H3)
the coupling coefficients are non-positive $c_{i}^{l}\leq 0,i\neq l$;
* (H4)
$\beta_{i},\eta_{i}\in L^{\infty}(\partial\Omega),\,\beta_{i},\eta_{i}\geq
0\text{ and }\beta_{i}>0\text{ or }\beta_{i}\equiv 0\text{ and }\eta_{i}\equiv
1$ on each connected component of $\partial\Omega$.
The boundary observation on the solution is
$\zeta=(\zeta_{i}(y_{i}))_{i=\overline{1,n}}$:
(O) $\zeta_{i}(y_{i})=\gamma_{i}(x)\frac{\partial y_{i}}{\partial
n_{A}}+\delta_{i}(x)y_{i},\quad(0,T)\times\Gamma_{1},\quad\Gamma_{1}\subset\partial\Omega,\quad
i=\overline{1,n},$
with given $\gamma_{i},\delta_{i}\in L^{\infty}(\Gamma_{1})$.
We impose an independence condition between the boundary conditions and the
observations expressed as:
* (H5)
$\begin{vmatrix}\gamma_{i}&\delta_{i}\\\ \beta_{i}&\eta_{i}\end{vmatrix}\neq
0\text{ on }\Gamma_{1},\quad i=\overline{1,n}.$
For $2\leq q\leq\infty$ and $k>0$ we consider the following classes of
sources:
(1.1) $\mathcal{G}_{q,k}=\left\\{g\in[L^{q}(Q)]^{n}:\,g\geq 0\text{ s.t.
}\|g\|_{L^{q}(Q)}\leq k\|g\|_{L^{1}(Q)}\right\\}.$
In this context, the main result which gives $L^{q}$ estimates for the sources
assuming that the sources belong to some class $\mathcal{G}_{q,k}$ is the
following:
###### Theorem 1.
Consider the system (S) with hypotheses $(H1)-(H5)$. Then for $2\leq
q<\infty,k>0$, $g\in\mathcal{G}_{q,k}$ and the corresponding solution $y\in
W^{2,1}_{q}(Q)$, there exists $C=C(q,k)>0$ such that
(1.2) $\left\|g\right\|_{L^{q}(Q)}\leq C_{1}\|\zeta\|_{L^{2}(\Sigma_{1})}.$
Concerning $q=\infty$, for sources $g\in\mathcal{G}_{\infty,k}$ and
corresponding solutions $y\in W^{2,1}_{q}(Q)$ for all $q<\infty$, there exists
$C=C(k)>0$ such that
(1.3) $\left\|g\right\|_{L^{\infty}(Q)}\leq
C_{1}\|\zeta\|_{L^{2}(\Sigma_{1})}.$
## 2\. $L^{q}-L^{2}$-Carleman estimates with boundary observations
The key ingredient to prove the above Theorem is a family of $L^{q}$ Carleman
estimates with observations on the boundary. To derive these estimates, it is
necessary to use the classical mechanism but with auxiliary functions
satisfying supplementary technical properties.These properties, in turn,
dictate the special choice of the shape of the domain.
The choice of auxiliary functions is the following:
* •
$\psi_{0}\in C^{2}(\overline{\Omega})$ s.t.
$\psi_{0}|_{\Gamma_{0}}=0,\quad\psi_{0}|_{\Gamma_{1}}=1,\quad|\nabla\psi_{0}|>0$
in $\overline{\Omega}$,
$\quad\frac{\partial\psi_{0}}{\partial\nu}|_{\Gamma_{0}}<0$,
* •
$\varphi_{0}(t,x):=\frac{e^{\lambda\psi_{0}(x)}}{t(T-t)},\quad\alpha_{0}(t,x):=\frac{e^{\lambda\psi_{0}(x)}-e^{1.5\lambda\|\psi_{0}\|_{C(\overline{\Omega})}}}{t(T-t)}$
* •
$\psi=\psi_{0}+K$, $K$ big enough such that
$\frac{\sup\psi}{\inf\psi}\leq\frac{8}{7}$
and the weight functions
$\varphi(t,x):=\frac{e^{\lambda\psi(x)}}{t(T-t)}\,,\quad\alpha(t,x):=\frac{e^{\lambda\psi(x)}-e^{1.5\lambda\|\psi\|_{C(\overline{\Omega})}}}{t(T-t)}.$
###### Remark 1.
The choice of annular domains is imposed by technical restrictions in
obtaining the proper Carleman estimates described in the following and more
precisely by the conditions imposed to $\psi_{0}$ which imply necessarily the
existence of two connected components of the boundary with the same topology.
So, other choices of the domain is possible with restrictions imposed by
existence of such a function.
The Carleman estimate we obtain is the following
###### Proposition 1.
Let $g\in L^{q}(Q)$, with $2\leq q<\infty$. Then there exist $s_{0}=s_{0}(q)$,
$\lambda_{0}=\lambda_{0}(q)$, s.t. for $\lambda>\lambda_{0}$,
$s^{\prime},s>s_{0}$, $\frac{s^{\prime}}{s}>\bar{\gamma}>1$, there exists
$C=C(q,\bar{\gamma})$:
(2.1)
$\displaystyle\|ye^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(Dy)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D^{2}y)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D_{t}y)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}$
$\displaystyle\leq C\left[\|ge^{s{\alpha}}\|_{L^{q}(Q)}+\|\zeta
e^{s\alpha}\|_{L^{2}(\Sigma_{1})}\right]$
where the constant $C$ depends on $\lambda$ but independent of $s$.
The result in Proposition 1 relies on the regularizing effect of the parabolic
flow combined with a bootstrap argument applied to the linear parabolic system
and using the following $L^{2}$ Carleman estimate that was obtained in [7]
using the classical techniques from [3]:
###### Proposition 2.
For $g\in L^{2}(Q)$, there exist constants $\lambda_{0}=\lambda_{0}(\Omega),$
$s_{0}=s_{0}(\Omega)$ such that, for any $\lambda\geq\lambda_{0}$, $s\geq
s_{0}$ and some $C=C(T,\Omega)$, the following inequality holds:
(2.2)
$\displaystyle\int_{Q}\left[(s\varphi)^{-1}\left(|D_{t}y|^{2}+|D^{2}y|^{2}\right)+s\lambda^{2}\varphi|Dy|^{2}+s^{3}\lambda^{4}\varphi^{3}|y|^{2}\right]e^{2s\alpha}dxdt+\int_{[0,T]\times\Gamma_{0}}s^{3}\lambda^{3}\varphi^{3}|y|^{2}e^{2s\alpha}d\sigma$
$\displaystyle\leq
C\left(\int_{Q}|g|^{2}e^{2s\alpha}dxdt+\int_{[0,T]\times\Gamma_{1}}s^{3}\lambda^{3}\varphi^{3}|\zeta|^{2}e^{2s\alpha}d\sigma\right),\qquad\quad$
for $y\in H^{1}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega))$ solution of
(S).
In the following we denote the operators entering the system by
$B=(B_{i})_{i=\overline{1,n}}$, $B_{i}y_{i}=\beta_{i}(x)\frac{\partial
y_{i}}{\partial n_{A_{i}}}+\eta_{i}(x)y_{i}$ , $L=(L_{i})_{i=\overline{1,n}}$
, $L_{i}y_{i}=-\sum\limits_{j,k=1}^{N}D_{j}(a_{i}^{jk}D_{k}y_{i})$,
$L^{1}=(L^{1}_{i})_{i=\overline{1,n}}$,
$L^{0}=(L^{0}_{i})_{i=\overline{1,n}}$, where the lower-order operators are
given by ($w$ is a scalar function, $y$ is vector valued function):
(2.3)
$L^{1}_{i}w=\sum\limits_{{\begin{subarray}{c}k=\overline{1,N}\end{subarray}}}b_{i}^{k}D_{k}w,\quad
L^{0}_{i}y=\sum\limits_{l=\overline{1,n}}c^{l}_{i}y_{l},\quad
i=\overline{1,n}.$
Using these notations the system (S) is written more compactly
(2.4) $\begin{cases}&D_{t}y+Ly+L^{1}y+L^{0}y=g,\,(0,T)\times\Omega,\\\
&By=0,\,(0,T)\times\partial\Omega.\end{cases}$
For the proof of $L^{q}$ estimates we need Sobolev embedding results for
anisotropic Sobolev spaces ( see [5], Lemma 3.3):
###### Lemma 1.
Consider $u\in W^{2,1}_{p}(Q)$.
Then $u\in Z_{1}$ where
$Z_{1}=\left\\{\begin{array}[]{lll}L^{q}(Q)&\text{ with
}q\leq\frac{(n+2)p}{n+2-2p}&\text{ when }p<\frac{N+2}{2}\\\ L^{q}(Q)&\text{
with }q\in[1,\infty),&\text{ when }p=\frac{N+2}{2}\\\
C^{\alpha,\alpha/2}(Q)&\text{ with }0<\alpha<2-\frac{N+2}{p},&\text{ when
}p>\frac{N+2}{2}\end{array}\right.$
and there exists $C=C(Q,p,N)$ such that
$\|u\|_{Z_{1}}\leq C\|u\|_{W^{2,1}_{p}(Q)}.$
Moreover, $Du\in Z_{2}$ where
$Z_{2}=\left\\{\begin{array}[]{lll}L^{q}(Q)&\text{ with
}q\leq\frac{(N+2)p}{N+2-p}&\text{ when }p<{N+2}\\\ L^{q}(Q)&\text{ with
}q\in[1,\infty),&\text{ when }p={N+2}\\\ C^{\alpha,\alpha/2}(Q)&\text{ with
}0<\alpha<1-\frac{N+2}{p},&\text{ when }p>{N+2}\end{array}\right.$
and there exists $C=C(p,N)$ such that
$\|Du\|_{Z_{2}}\leq C\|u\|_{W^{2,1}_{p}(Q)}.$
We introduce the following auxiliary functions which do not depend on space
variable $x$:
$\overline{\varphi}:=\frac{e^{\lambda(K+1)}}{t(T-t)},\quad\underline{\varphi}:=\frac{e^{\lambda
K}}{t(T-t)},\quad\overline{\alpha}:=\frac{e^{\lambda(K+1)}-e^{1.5\lambda(K+1)}}{t(T-t)},\quad\underline{\alpha}:=\frac{e^{\lambda
K}-e^{1.5\lambda(K+1)}}{t(T-t)}.$
###### Remark 2.
Observe that for some $\sigma>\sigma_{0}$ and for some
$\tilde{\lambda}_{0}(\sigma_{0})>0$ we have for $\lambda>\tilde{\lambda}_{0}$
that
$-(\sigma-1)e^{1.5\lambda(K+1)}+\sigma e^{\lambda(K+1)}-e^{\lambda
K}\leq-\sigma\lambda e^{\lambda(K+1)}.$
Which gives
$\displaystyle\frac{e^{m\lambda(K+1)}\sigma^{m}s_{1}^{m}\lambda^{m}}{t^{m}(T-t)^{m}}e^{\frac{-(\sigma-1)s_{1}e^{1.5\lambda(K+1)}+\sigma
s_{1}e^{\lambda(K+1)}-s_{1}e^{\lambda
K}}{t(T-t)}}\leq\frac{e^{m\lambda(K+1)}\sigma^{m}s_{1}^{m}\lambda^{m}}{t^{m}(T-t)^{m}}e^{\frac{-\sigma
s_{1}\lambda e^{\lambda(K+1)}}{{t(T-t)}}}$
$\leq\sup_{\mu\in[0,\infty)}\mu^{m}e^{-\mu}=C(m),$
which gives
(2.5)
$\overline{\varphi}^{m}s_{2}^{m}\lambda^{m}e^{s_{2}\overline{\alpha}}\leq
C(m)e^{s_{1}\underline{\alpha}}.$
In conclusion, for all $m>0$ and $\sigma_{0}>1$, there exist
$\tilde{\lambda}_{0}=\tilde{\lambda}_{0}(\sigma_{0})>0$ and $C=C(m)$ such that
if $\lambda>\lambda_{0}$ and $s_{1},s_{2}>0$ with
$\frac{s_{2}}{s_{1}}=\sigma>\sigma_{0}$, one has
(2.6) $\varphi^{m}s_{2}^{m}\lambda^{m}e^{s_{2}\alpha}\leq
C(m)e^{s_{1}\alpha},$
with $\varphi,\alpha$ as above.
The auxiliary functions are built on $\psi_{0}$ in the same way as in the case
of internal observations with similar estimates on the weights (see [10]).
Proof of Proposition 1. For a given $\gamma>1$ and $j\in\mathbb{N}$ denote by
$w^{j}:=ye^{\gamma^{j}s\overline{\alpha}}=w^{j-1}e^{\gamma^{j-1}s\overline{\alpha}(\gamma-1)}.$
Notice that since $\gamma>1$, for fixed $j$ there exists
$\bar{\lambda}_{0}(j)>0$ such that
(2.7)
$e^{\gamma^{j}s\alpha}<e^{\gamma^{j}s\overline{\alpha}}<e^{s\alpha}\text{ for
all }\lambda\geq\bar{\lambda}_{0}(j).$
Each $w^{j}$ verifies the initial boundary value problem
(2.8) $\left\\{\begin{aligned}
&D_{t}w^{j}+Lw^{j}=ge^{\gamma^{j}s\overline{\alpha}}+O[s\gamma^{j}\overline{\varphi}^{2}e^{\gamma^{j-1}s\overline{\alpha}(\gamma-1)}]w^{j-1},\text{
in }(0,T)\times\Omega\\\ &Bw^{j}=0,\text{ on }(0,T)\times\partial\Omega,\\\
&w^{j}(0,\cdot)=0\text{ in }\Omega.\end{aligned}\right.\quad$
Observe that the boundary conditions satisfied by $w^{j}$ remain the same as
those satisfied by $y^{j}$ as we have used in the definition of $w^{j}$ the
space independent weight $\overline{\alpha}$.
We describe the bootstrap argument: based on the the regularity argument from
Lemma 1, we construct the sequence $\\{q_{j}\\}_{j\in\mathbb{N}}$:
(2.9) $q_{0}=2,\quad
q_{j}:=\begin{cases}\dfrac{(N+2)q_{j-1}}{N+2-q_{j-1}},\text{ if
}q_{j-1}<N+2,\\\ \frac{3}{2}q_{j-1},\text{ if }q_{j-1}\geq N+2.\end{cases}$
Observe that the sequence $\\{q_{j}\\}_{j\in\mathbb{N}}$ is increasing to
infinity. Now, since $g\in L^{q}(Q)$, we can take $m$ such that $q_{m-1}\leq
q<q_{m}$ and we get, by standard Sobolev embedding, that:
(2.10) $W^{1,q_{j-1}}(Q)\subset L^{q_{j}}(Q),\,j=1,\ldots,m,$
since the Sobolev exponent $q_{j}^{*}:=\frac{(N+1)q_{j-1}}{N+1-q_{j-1}}$ is
greater than $q_{j}$.
Now, an argument like the one in Remark 2 gives that there exist
$S_{0},\Lambda_{0}\geq 0$ and $C=C(j)>0$ such that for $s\geq
S_{0},\lambda\geq\Lambda_{0}$ and $j=\overline{1,m}$ we have
(2.11)
$s\gamma^{j}\overline{\varphi}^{2}e^{\gamma^{j-1}s\overline{\alpha}(\gamma-1)}\leq
C(j).$
Since the initial data $w^{j}(0,\cdot)$ is zero and using the previous
estimate (2.11), the parabolic regularity gives that for $\lambda$ big enough
($\lambda>\max\\{\lambda_{0},\Lambda_{0},\max_{j=\overline{1,m}}\\{\tilde{\lambda}_{j},\bar{\lambda}_{j}\\}\\}$)
we have
(2.12) $\|w^{j}\|_{W^{2,1}_{q_{j-1}}(Q)}\leq
C\left(\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j-1}}(Q)}+\|w^{j-1}\|_{L^{q_{j-1}}(Q)}\right).$
Using Lemma 1 and the embedding (2.10) we get that $w^{j}\in L^{{q}_{j}}(Q)$
and, from the previous inequality, we obtain the estimate
(2.13) $\|w^{j}\|_{L^{q_{j}}(Q)}\leq
C\left(\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j-1}}(Q)}+\|w^{j-1}\|_{L^{q_{j-1}}(Q)}\right)\text{
for }j=1,\ldots,m,$
and so
(2.14) $\|w^{m}\|_{L^{q_{m}}(Q)}\leq
C\left(\sum_{j=1}^{m-1}\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j}}(Q)}+\|w^{0}\|_{L^{q_{0}}(Q)}\right).$
Regarding the first order terms, Lemma 1 implies that
$\|Dw^{m}\|_{L^{q_{m}}(Q)}\leq\|w^{m}\|_{W^{2,1}_{q_{m-1}}(Q)}$, meaning that
we have estimates also for first order derivatives
(2.15) $\|w^{m}\|_{L^{q_{m}}(Q)}+\|Dw^{m}\|_{L^{q_{m}}(Q)}\leq
C\left(\sum_{j=1}^{m-1}\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j}}(Q)}+\|w^{0}\|_{L^{q_{0}}(Q)}\right).$
Since
$\|ye^{\gamma^{m}s\alpha}\|_{L^{q}(Q)}\leq\|ye^{\gamma^{m}s\overline{\alpha}}\|_{L^{q}(Q)}\leq
C\|ye^{\gamma^{m}s\overline{\alpha}}\|_{L^{q^{m}}(Q)}$
and
$\|Dye^{\gamma^{m}s\alpha}\|_{L^{q}(Q)}\leq\|Dye^{\gamma^{m}s\overline{\alpha}}\|_{L^{q}(Q)}\leq
C\|Dye^{\gamma^{m}s\overline{\alpha}}\|_{L^{q^{m}}(Q)},$
using (2.15) we can obtain a partial estimate,
(2.16)
$\|ye^{\gamma^{m}s\alpha}\|_{L^{q}(Q)}+\|Dye^{\gamma^{m}s\alpha}\|_{L^{q}(Q)}\leq
C\left(\sum_{j=1}^{m-1}\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j}}(Q)}+\|w^{0}\|_{L^{q_{0}}(Q)}\right).$
Now we use (2.7), the fact that ${q_{0}=2}$, $w^{0}=ye^{s\overline{\alpha}}$,
and the $L^{2}$ Carleman inequality (3.9) to properly bound
$\|w^{0}\|_{L^{q_{0}}(Q)}$:
(2.17) $\|ye^{s\overline{\alpha}}\|_{L^{2}(Q)}\leq
C\|ye^{\frac{1}{\gamma}s\alpha}\|_{L^{2}(Q)}\leq
C\left(\|ge^{\frac{1}{\gamma}s\alpha}\|_{L^{2}(Q)}+\|s^{\frac{3}{2}}\lambda^{2}\varphi^{\frac{3}{2}}\zeta
e^{\frac{1}{\gamma}s\alpha}\|_{L^{2}(\Sigma_{1})}\right).$
Using (2.7), Remark 2 and the fact that $q>q_{j}$ forall
$q_{j}\in\overline{1,m-1}$, there exists $C>0$ such that the right hand-side
of (2.15) is bounded as follows
(2.18)
$\displaystyle\sum_{j=1}^{m-1}\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j}}(Q)}+\|ge^{\frac{1}{\gamma}s\alpha}\|_{L^{2}(Q)}+\|s^{\frac{3}{2}}\lambda^{2}\varphi^{\frac{3}{2}}\zeta
e^{\frac{1}{\gamma}s\alpha}\|_{L^{2}(\Sigma_{1})}$ $\displaystyle\leq
C\left(\|ge^{\frac{1}{\gamma}s\alpha}\|_{L^{q}(Q)}+\|\zeta
e^{\frac{1}{\gamma^{2}}s\alpha}\|_{L^{2}(\Sigma_{1})}\right).$
In order to obtain estimates for time derivatives and second order space
derivatives, we return to the parabolic problem verified by $w^{m+1}$:
(2.19) $\left\\{\begin{aligned}
&D_{t}w^{m+1}+Lw^{m+1}=ge^{\gamma^{m+1}s\overline{\alpha}}+O[s\gamma^{m+1}\overline{\varphi}^{2}e^{\gamma^{m}s\overline{\alpha}(\gamma-1)}]w^{m},\text{
in }(0,T)\times\Omega\\\ &Bw^{m+1}=0,\text{ on }(0,T)\times\partial\Omega,\\\
&w^{m+1}(0,\cdot)=0\text{ in }\Omega\end{aligned}\right.\quad$
Parabolic regularity along with (2.11) gives
(2.20) $\|w^{m+1}\|_{W^{2,1}_{q}(Q)}\leq
C\left(\|ge^{\gamma^{m+1}s\overline{\alpha}(\gamma-1)}\|_{L^{q}(Q)}+\|w^{m}\|_{L^{q}(Q)}\right)$
which implies, using again (2.7), (2.15) and (2.18) that
(2.21)
$\displaystyle\|D^{2}ye^{\gamma^{m+1}s\alpha}\|_{L^{q}(Q)}\leq\|D^{2}ye^{\gamma^{m+1}s\overline{\alpha}}\|_{L^{q}(Q)}=\|D^{2}w^{m+1}\|_{L^{q}(Q)}$
$\displaystyle\leq C\left(\|ge^{\frac{1}{\gamma}s\alpha}\|_{L^{q}(Q)}+\|\zeta
e^{\frac{1}{\gamma^{2}}s\alpha}\|_{L^{2}(\Sigma_{1})}\right)$
and using (2.11), (2.7), (2.15), (2.18) that
(2.22)
$\displaystyle\|(D_{t}y)e^{\gamma^{m+1}s\alpha}\|_{L^{q}(Q)}\leq\|D_{t}w^{m+1}\|_{L^{q}(Q)}+C\|w^{m}\|_{L^{q}(Q)}$
$\displaystyle\leq C\left(\|ge^{\frac{1}{\gamma}s\alpha}\|_{L^{q}(Q)}+\|\zeta
e^{\frac{1}{\gamma^{2}}s\alpha}\|_{L^{2}(\Sigma_{1})}\right).$
From (2.15),(2.18),(2.21) and (2.22) and taking
$\gamma=\bar{\gamma}^{\frac{1}{m+3}}$, $s$ changed into
$\frac{1}{\gamma^{2}}s$, we obtain the conclusion,
(2.23)
$\displaystyle\|ye^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(Dy)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D^{2}y)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D_{t}y)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}\leq$
$\displaystyle\leq C\left(\|ge^{s{\alpha}}\|_{L^{q}(Q)}+\|\zeta
e^{s\alpha}\|_{L^{2}(\Sigma_{1})}\right).$
###### Remark 3.
Observe that if $q>N+1$, then the Morrey embedding theorem gives $L^{\infty}$
estimates for $y$ and $Dy$: there exist $s_{0},\lambda_{0}>0$, s.t. for
$\lambda>\lambda_{0}$, $s^{\prime},s>s_{0}$,
$\frac{s^{\prime}}{s}>\bar{\gamma}>1$, there exists $C=C(\bar{\gamma})$:
(2.24)
$\|ye^{s^{\prime}\alpha}\|_{L^{\infty}(Q)}+\|Dye^{s^{\prime}\alpha}\|_{L^{\infty}(Q)}\leq
C\left(\|ge^{s\alpha}\|_{L^{q}(Q)}+\|\zeta
e^{s\alpha}\|_{L^{2}(\Sigma_{1})}\right).$
## 3\. Source stability for linear systems. Proof of Theorem 1
In the following, the strong solutions $y\in W^{2,1}_{q}(Q)$ that we work with
are also variational solutions of the system (S).
For this purpose, we denote by
$\Gamma_{D}^{i}=\\{x\in\partial\Omega|\beta_{i}=0\\}$ the Dirichlet boundary
corresponding to $y_{i}$ and consider the Hilbert spaces $V_{i}=\\{v\in
H^{1}(\Omega):v=0\text{ on }\Gamma_{D}^{i}\\}$ and let
$V=V_{1}\times\cdots\times V_{n}$.
Then, for some initial data $y_{0}\in L^{2}(\Omega)$, $y\in L^{2}(0,T;V)\cap
H^{1}(0,T;V^{\prime})\subset C([0,T];H)$ is a weak solution of system (S) if:
(3.1) $\displaystyle\sum\limits_{i=1}^{n}(\langle
y_{i}(t),v_{i}\rangle_{L^{2}(\Omega)}-\langle
y_{0,i},v_{i}\rangle_{L^{2}(\Omega)})+\sum\limits_{i=1}^{n}\int_{0}^{t}\langle\mathbb{A}_{i}\nabla
y_{i}(\tau,\cdot),\nabla v\rangle_{L^{2}(\Omega)}d\tau$
$\displaystyle+\sum\limits_{i=1}^{n}\int_{0}^{t}\langle b_{i}\cdot\nabla
y_{i}(\tau,\cdot),v_{i}\rangle_{L^{2}(\Omega)}d\tau+\sum\limits_{l=1}^{n}\int_{0}^{t}\langle
c_{i}^{l}y_{l}(\tau,\cdot),v_{i}\rangle_{L^{2}(\Omega)}d\tau$
$\displaystyle+\sum\limits_{i=1}^{n}\int_{0}^{t}\langle\eta_{i}y_{i}(\tau,\cdot),v_{i}\rangle_{L^{2}(\partial\Omega\setminus\Gamma_{D}^{i})}d\tau=\sum\limits_{i=1}^{n}\int_{0}^{t}\langle
g_{i}(\tau,\cdot),v_{i}\rangle_{L^{2}(\Omega)}d\tau,\quad\forall
t\in(0,T),\quad\forall v_{i}\in V_{i},$
where $\mathbb{A}_{i}$ are the matrix coefficients matrix
$(a_{i}^{jk})_{j,k\in\overline{1,N}}$ and
$b_{i}=(b_{i}^{k})^{\top}_{k\in\overline{1,N}}$.
Proof of Theorem 1. We have to prove that there exists $C=C(k)$ such that for
$g\in\mathcal{G}_{q,k}$,
(3.2) $\displaystyle\left\|g\right\|_{L^{q}(Q)}\leq
C\|\zeta(y)\|_{L^{2}(\Sigma_{1})}.$
We argue by contradiction. Then there exists a sequence of sources denoted as
$(g^{m})_{m}\subset\mathcal{G}_{q,k}$, and the corresponding solutions
$(y^{m})_{m}\subset W^{2,1}_{q}(Q)$, such that:
(3.3)
$\displaystyle\left\|g^{m}\right\|_{L^{q}(Q)}>m\|\zeta(y^{m})\|_{L^{2}(\Sigma_{1})}.$
With no loss of generality we may assume that
$\left\|g^{m}\right\|_{L^{q}(Q)}=1$. For the case when $2\leq q<\infty$, up to
a subsequence denoted again $(g^{m})_{m}$, $g^{m}\rightharpoonup g$ weakly
$L^{q}(Q)$ for some $g\in L^{q}(Q)$. For the case when $q=\infty$ we have that
up to a subsequence $g^{m}\rightharpoonup g$ weak-* in $L^{\infty}(Q)$ for
some $g\in L^{\infty}(Q)$.
This means that the above equation (3.3) gives for the sequence of
observations that
(3.4) $\zeta(y^{m})\rightarrow 0\text{ in }L^{2}(\Sigma)\text{ as
}m\rightarrow\infty.$
Observe that in both cases the weak limit $g$ of $(g^{m})_{m}$ is not zero.
Indeed, for the case when $2\leq q<\infty$, since $g^{m}\in\mathcal{G}_{q,k}$,
for $1\in L^{q^{\prime}}(Q)$ we have
$\langle
g^{m},1\rangle_{L^{q},L^{q^{\prime}}}=\int_{Q}g^{m}\geq\frac{1}{k}\|g^{m}\|_{L^{q}(Q)}=\frac{1}{k}.$
Because $(g^{m})_{m}$ converges weakly in $L^{q}(Q)$ we have that
$\int_{Q}g\geq\frac{1}{k}>0$ and thus $g\not\equiv 0$.
Also for the case when $q=\infty$, the weak-* limit $g$ of $(g^{m})_{m}$ is
not zero. Since $g^{m}\in\mathcal{G}_{k}$ and $1\in L^{1}(Q)$,
$\langle
g^{m},1\rangle_{L^{\infty},L^{1}}=\int_{Q}g^{m}\geq\frac{1}{k}\|g^{m}\|_{L^{\infty}(Q)}=\frac{1}{k}.$
Because $(g^{m})_{m}$ converges weak-* in $L^{\infty}(Q)$ we have that
$\int_{Q}g\geq\frac{1}{k}>0$ and thus $g\not\equiv 0$.
Now, we pass to the limit in the weak formulation of the problem (3.1) for
some initial data $y_{0}\in L^{2}(\Omega)$.
Consider $(y^{m})_{m}$ a sequence of solutions for the system (S) with
corresponding sources $(g^{m})_{m}\subset L^{q}(Q)$. For $2\leq q<\infty$,
$(g^{m})_{m}$ is bounded in $L^{q}(Q)$, $(\zeta^{m})_{m}=(\zeta(y^{m}))_{m}$
is bounded in $L^{2}(\Sigma_{1})$ and using the Carleman estimate for the
solution $y^{m}$ corresponding to the sources $g^{m}$, we have that
(3.5)
$\displaystyle\|y^{m}e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(Dy^{m})e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D^{2}y^{m})e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D_{t}y^{m})e^{s^{\prime}\alpha}\|_{L^{q}(Q)}$
$\displaystyle\leq
C\left[\|g^{m}e^{s{\alpha}}\|_{L^{q}(Q)}+\|\zeta^{m}e^{s\alpha}\|_{L^{2}(\Sigma_{1})}\right],$
which gives for a cylinder of form
$Q^{\epsilon}=(\epsilon,T-\epsilon)\times\Omega$ with $\epsilon>0$ fixed
$0<\epsilon<\frac{T}{2}$, arbitrarily small, that $(y^{m})_{m}$ is bounded in
${W^{2,1}_{q}(Q^{\epsilon})}$.
Thus $(y^{m})_{m}$ is bounded in $L^{q}(\epsilon,T-\epsilon;W^{2,q}(\Omega))$
and $D_{t}y^{m}$ is bounded in $L^{q}(\epsilon,T-\epsilon;L^{q}(\Omega))$, for
$2\leq q<\infty$. Since $W^{2,q}(\Omega)\subset W^{1,q}(\Omega)\subset
L^{q}(\Omega)$ with compact embeddings, by Aubin-Lions lemma applied to
sequence $(y^{m})_{m}$, there exists $y\in L^{q}_{loc}(0,T;W^{1,q}(\Omega))$
and a subsequence also denoted $(y^{m})_{m}$ such that:
(3.6) $y^{m}\longrightarrow y\text{ in
}L^{q}(\epsilon,T-\epsilon;W^{1,q}(\Omega))\text{ as
}m\longrightarrow\infty,\forall\epsilon>0.$
Strong convergence of $(y^{m})_{m}$ in
$L^{q}(\epsilon,T-\epsilon;W^{1,q}(\Omega))$ and weak convergence of
$(g^{m})_{m}$ in $L^{q}(Q)$, $2\leq q<\infty$ allow to pass to the limit in
the variational formulation of the problem (3.1) to conclude that $y$ is
solution to (S) corresponding to $g$. We also have in this situation , for all
$\varepsilon$ as above, weak convergence to $y$ in
$L^{q}(\epsilon,T-\epsilon;W^{2,q}(\Omega))$.
When $q=+\infty$ we have, up to a subsequence, weak-* convergence of
$(g^{m})_{m}$ to $g$ in $L^{\infty}(Q)$ and weak $L^{p}(Q)$ for all $2\leq
p<+\infty$. By the above argument we find in fact strong convergence, up to a
subsequence, of $(y^{m})_{m}$ to $y$ in
$L^{p}(\epsilon,T-\epsilon;W^{1,p}(\Omega))$, as well as weak convergence in
$L^{p}(\epsilon,T-\epsilon;W^{2,p}(\Omega))$, for $2\leq p<\infty$, and $y$ is
solution corresponding to the source $g$.
Considering that for $2\leq p<\infty$ the observation $\zeta$ is linear
continuous operator in $L(W^{2,p}(\Omega),L^{p}(\Gamma_{1}))$ we have that
$\zeta(y)=\lim\zeta(y^{m})$
weakly in $L^{q}(\epsilon,T-\epsilon;L^{q}(\Gamma_{1}))$ if $q<\infty$ and
weakly in $L^{p}(\epsilon,T-\epsilon;L^{p}(\Gamma_{1})$ for all $2\leq
p<\infty$ if $q=\infty$. By the lower semicontinuity of the norm in $L^{p}$ we
find from (3.3) that $\zeta(y)=0$.
Now we use (H5) to see that both $y$ and $\displaystyle\frac{\partial
y}{\partial n}$ are zero on $\Sigma_{1}$. At this point we consider a slightly
larger domain, $\tilde{\Omega}=\Omega\cup\\{\Gamma_{1}+B_{\delta}(0)\\}$.
Consider $\tilde{y}$ the extension with $0$ of $y$ to
$\tilde{Q}=(0,T)\times\tilde{\Omega}$ and $\tilde{g}$ also the extension with
$0$ of $g$. Observe that $\tilde{y}$ is variational solution corresponding to
$\tilde{g}$ and with null boundary conditions on the new piece of boundary. We
are now in position to apply the strong maximum principle for weak solutions
to parabolic systems obtained in [8] we get that $\tilde{y}$ is zero in
$\tilde{Q}$ and thus the source $\tilde{g}$ must be zero in $\tilde{Q}$, which
contradicts the fact that $g\not=0$.
### Comparison with the results of O. Yu Imanuvilov and M. Yamamoto
We mention here the result of O. Yu Imanuvilov and M. Yamamoto [4] obtained in
$L^{2}$ for the source of one equation. It is important to see that the shape
of $\Omega\subset\mathbb{R}^{N}$ is not restricted and the homogeneous
boundary conditions are prescribed on a part of the boundary $\Gamma_{0}$ and
the observation is made on $\Gamma_{1}=\partial\Omega\setminus\Gamma_{0}$:
(I-Y)
$\left\\{\begin{array}[]{ll}D_{t}y-\sum\limits_{j,k=1}^{N}D_{j}(a^{jk}D_{k}y)+\sum\limits_{k=1}^{N}b^{k}D_{k}y+cy=g&(0,T)\times\Omega,\\\
\beta(x)\frac{\partial y}{\partial n_{A}}+\eta(x)y=0&(0,T)\times\Gamma_{0},\\\
\end{array}\quad\right.$
For a fixed $\theta\in(0,T)$ an observation instant of time, they have
obtained $L^{2}$ boundary estimates in the following class of sources:
(3.7) $\mathcal{G}_{2,\tilde{c}}=\left\\{g\in
W^{1,1}((0,T);L^{2}(\Omega)):\,\left|\frac{\partial g(t,x)}{\partial
t}\right|\leq\tilde{c}|g(\theta,x)|,{a.e.}(t,x)\in Q\right\\}.$
In this context, their result provides estimates in terms of more measured
quantities involving the gradient and the time derivative of the solution on
the observed boundary:
###### Theorem 2.
Let $g\in\mathcal{G}_{2,\tilde{c}}$ and $y\in W^{2,1}_{2}(Q)$ a solution to
(I-Y) corresponding to $g$. Then
(3.8) $\|g\|_{L^{2}}\leq
C(\|y(\theta,\cdot)\|_{W^{2}_{2}(\Omega)}+\|y\|_{L^{2}(\Sigma_{1})}+\|\nabla
D_{t}y\|_{L^{2}(\Sigma_{1})}+\|\nabla y\|_{L^{2}(\Sigma_{1})}),$
where $\Sigma_{1}=(0,T)\times\Gamma_{1}$.
The result is based on the two following Carleman estimates:
###### Proposition 3.
For $g\in L^{2}(Q)$, there exist constants $\lambda_{0}=\lambda_{0}(\Omega),$
$s_{0}=s_{0}(\Omega)$ such that, for any $\lambda\geq\lambda_{0}$, $s\geq
s_{0}$ and some $C=C(T,\Omega)$, the following inequality holds:
(3.9)
$\displaystyle\int_{Q}\left[(s\varphi)^{p-1}\left(|D_{t}y|^{2}+|D^{2}y|^{2}\right)+(s\varphi)^{p+1}|Dy|^{2}+(s\varphi)^{p+3}|y|^{2}\right]e^{2s\alpha}dxdt$
$\displaystyle\leq C\int_{Q}(s\varphi)^{p}|g|^{2}e^{2s\alpha}dxdt+$
$\displaystyle+C\int_{[0,T]\times\Gamma_{1}}\left((s\varphi)^{p}|D_{t}y|^{2}+(s\varphi)^{p+1}|\nabla
y|^{2}+(s\varphi)^{p+3}|y|^{2}\right)e^{2s\alpha}d\sigma,\quad p\in\\{0,1\\}$
for $y\in H^{1}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega))$ solution of
(I-Y).
Observe that if in our situation we consider homogeneous Dirichlet boundary
conditions on $\Gamma_{1}$ the Carleman estimates of O. Yu Imanuvilov and M.
Yamamoto gives our Carleman estimate as ob $\Gamma_{1}$ we would have in this
case $D_{t}y|_{\partial\Omega}=0$ and $\nabla
y|_{\partial\Omega}=\frac{\partial y}{\partial n}$. As a consequence our
result remains true for any domain without restrictions on the topology if we
consider homogeneous boundary conditions. However this is not the case if
Neumann or Robin are prescribed on $\Gamma_{1}$ as the estimate (3.8) needs
supplementary measurements on the tangential and time derivatives of the
solution on $\Gamma_{1}$.
## References
* [1] V. Barbu. Exact controllability of the superlinear heat equation. Appl. Math. Optim., 42(1):73–89, 2000.
* [2] Jean-Michel Coron, Sergio Guerrero, and Lionel Rosier. Null controllability of a parabolic system with a cubic coupling term. SIAM J. Control Optim., 48(8):5629–5653, 2010.
* [3] A.V. Fursikov and O.Yu. Imanuvilov. Controllability of evolution equations. Seoul: Seoul National Univ., 1996.
* [4] Oleg Yu Imanuvilov and Masahiro Yamamoto. Lipschitz stability in inverse parabolic problems by the Carleman estimate. Inverse Probl., 14(5):1229–1245, 1998.
* [5] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva. Linear and quasilinear equations of parabolic type. Vol. 23. American Mathematical Society, Providence, R.I., 1968.
* [6] Kévin Le Balc’h. Controllability of a $4\times 4$ quadratic reaction-diffusion system. J. Differential Equations, 266(6):3100–3188, 2019.
* [7] C.-G. Lefter and E.-A. Melnig. Reaction-diffusion systems in annular domains: source stability estimates with boundary observations.
* [8] C.-G. Lefter and E.-A. Melnig. Remarks on the strong maximum principle and strong invariance results for weak solutions to parabolic equations and systems.
* [9] E.-A. Melnig. Stability in ${L}^{q}$-norm for inverse source parabolic problems. Journal of Inverse and Ill-posed Problems, 28(6):797–814, 2020\.
* [10] E.-A. Melnig. Stability in inverse source problems for nonlinear reaction-diffusion systems. NoDEA Nonlinear Differential Equations Appl., 28(45), 2021.
* [11] Murray H. Protter and Hans F. Weinberger. Maximum principles in differential equations. Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original.
* [12] Hans F. Weinberger. Invariant sets for weakly coupled parabolic and elliptic systems. Rend. Mat. (6), 8:295–310, 1975. Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday.
|
# QuIP$\\#$: Even Better LLM Quantization with
Hadamard Incoherence and Lattice Codebooks
Albert Tseng Jerry Chee Qingyao Sun Volodymyr Kuleshov Christopher De Sa
###### Abstract
Post-training quantization (PTQ) reduces the memory footprint of LLMs by
quantizing their weights to low-precision. In this work, we introduce
QuIP$\\#$, a weight-only PTQ method that achieves state-of-the-art results in
extreme compression regimes ($\leq$ 4 bits per weight) using three novel
techniques. First, QuIP$\\#$ improves the incoherence processing from QuIP
(Chee et al., 2023) by using the randomized Hadamard transform, which is
faster and has better theoretical properties. Second, QuIP$\\#$ uses vector
quantization techniques to take advantage of the ball-shaped sub-Gaussian
distribution that incoherent weights possess: specifically, we introduce a set
of hardware-efficient codebooks based on the highly symmetric $E_{8}$ lattice,
which achieves the optimal 8-dimension unit ball packing. Third, QuIP$\\#$
uses fine-tuning to improve fidelity to the original model. Our experiments
show that QuIP$\\#$ outperforms existing PTQ methods, enables new behaviors in
PTQ scaling, and supports fast inference.
Machine Learning, ICML, Quantization, Large Language Models, LLMs, Low
Precision, Inference, Systems, Hardware, 2 bit, QuIP, Incoherence Processing,
Lattice Codebooks, Vector Quantization
## 1 Introduction
Figure 1: QuIP$\\#$ offers unprecedented quantization quality at extreme
compression ratios. QuIP$\\#$ 3-bit models also scale better than
theoretically lossless 4-bit models, a previously unseen result. Figure 2:
QuIP$\\#$ performs incoherence processing with a Randomized Hadamard Transform
and uses lattice codebooks to achieve state-of-the-art quantized models.
Large language models (LLMs) have driven rapid advances across diverse fields
such as natural language processing (Touvron et al., 2023b), scientific
modeling (Nguyen et al., 2023), and program synthesis (Rozière et al., 2024).
However, the massive size of these models poses significant challenges to
their deployment. For example, the largest model in the Llama2 family has 70B
parameters, and requires 140GB of GPU memory in native 16-bit precision
(Touvron et al., 2023b). This massive memory footprint motivates research into
methods that can compress LLMs without sacrificing quality.
Post-training quantization (PTQ) reduces the memory requirements of large
models by converting trained weights to a lower precision. For example, with
2-bit quantization, a 16-bit LLama2 model with 70B parameters fits on a single
consumer-grade 24GB GPU and benefits from increased inference throughput (Cai
et al., 2024). However, 2-bit quantization also often reduces the quality of
the model and pushes the limits of PTQ algorithms (Chee et al., 2023).
In this work, we introduce QuIP$\\#$, a weight-only PTQ method that achieves a
new state-of-the-art in model quantization. QuIP$\\#$ improves over existing
work via three techniques: incoherence processing, lattice codebooks, and
fine-tuning. Incoherence processing is a principled form of outlier
suppression that produces approximately sub-Gaussian distributed weight
matrices (Chee et al., 2023). QuIP$\\#$ performs incoherence processing with
the computationally-efficient randomized Hadamard transform (Halko et al.,
2011) (Section 3). To quantize incoherent matrices, QuIP$\\#$ uses the
BlockLDLQ block adaptive rounding algorithm with compressible codebooks based
on the $E_{8}$ lattice, which achieves the highest density 8 dimensional unit-
ball packing (Viazovska, 2017) (Section 4). The $E_{8}$ lattice is highly
structured and symmetric, which means that our codebooks are hardware-friendly
and admit fast inference. Finally, QuIP$\\#$ includes an inter-layer fine-
tuning algorithm that further improves quantization quality (Section 5).
These developments allow QuIP$\\#$ to significantly outperform existing PTQ
methods including OmniQuant (Shao et al., 2024), QuIP (Chee et al., 2023) (a
previous, separate work), and AQLM (Egiazarian et al., 2024). To the best of
our knowledge, QuIP$\\#$ is also the first PTQ method where 3-bit models scale
better than 4-bit models. This directly refutes Dettmers & Zettlemoyer
(2023)’s claim that 4-bit models are “optimal” and indicates that as the field
of PTQ develops, 2-bit models are likely to scale better than 3-bit models in
the near future.
Finally, we note that QuIP$\\#$ was designed from the ground up to be fast.
Algorithm 2 describes fast inference with a QuIP$\\#$-quantized linear layer.
Our “proof of concept” CUDA implementation of QuIP$\\#$ achieves over 50% of
peak memory bandwidth on a NVIDIA RTX 4090, validating our design choices.
In summary, we introduce QuIP$\\#$, a post-training quantization method that
achieves state-of-the-art results by
1. 1.
Performing incoherence processing with the Randomized Hadamard Transform,
which has better incoherence properties and faster runtime than the Kronecker
factorization in QuIP.
2. 2.
Rounding incoherence-processed weight matrices with block adaptive rounding
and codebooks based on the $E_{8}$ lattice, which achieves the highest
8-dimension unit ball packing density (kissing number).
3. 3.
Introducing an inter-layer fine-tuning algorithm that further improves
quantization quality.
Algorithm 1 QuIP$\\#$ without Fine-Tuning (QuIP$\\#$-NoFT)
0: Weight $W\in\mathbb{R}^{m\times n}$, hessians $H\in\mathbb{R}^{n\times n}$,
$g$-dim. $k$-bit codebook $C$
$\hat{W},\hat{H},S_{U},S_{V}\leftarrow\mbox{IP-RHT}(W,H)$ (Alg. 3)
$\hat{W}\leftarrow\mbox{BlockLDLQ}(\hat{W},\hat{H},C)$ (Sec. 4.1)
$\hat{W},S_{U},S_{V}$
Algorithm 2 QuIP$\\#$ Inference (for a Linear Layer)
0: $\hat{W}$, $S_{U},S_{V}$ from Alg. 1, $g$-dim. $k$-bit codebook $C$, input
$x\in\mathbb{R}^{n}$.
$y\leftarrow\texttt{Had}(S_{V}\odot x)$ where Had performs an orthogonal
Hadamard transform (Sec. 3)
$y\leftarrow\texttt{decompress_multiply}(\hat{W},C,y)$
$y\leftarrow\texttt{Had}(S_{U}\odot y)$
$y$
## 2 Background / Related Work
### 2.1 Compressing LLMs
A large body of work has focused on compressing LLMs, as doing so can directly
benefit LLM inference at scale. Methods such as pruning, quantization aware
training (QAT), and post-training quantization (PTQ) all focus on different
areas of this problem and are not strictly orthogonal to each other. Pruning
removes weights from models while preserving model quality and inference
performance (Chee et al., 2022; Sun et al., 2023). QAT focuses on training
models that are more “quantizable” but usually requires training models from
scratch (Nagel et al., 2022). PTQ, which QuIP$\\#$ falls under, instead
quantizes pre-trained models. PTQ generally requires much less compute than
QAT and achieve competitive performance (Chee et al., 2023; Frantar et al.,
2023; Shao et al., 2024; Egiazarian et al., 2024). For the rest of this paper,
we focus on the PTQ realm of LLM compression.
### 2.2 Quantization and Adaptive Rounding
In QuIP$\\#$, we follow existing state-of-the-art PTQ methods and round
weights to minimize the per-layer proxy loss, as formalized by Nagel et al.
(2020):
$\displaystyle\ell(\hat{W})$
$\displaystyle=E_{x}\left[\|(\hat{W}-W)x\|^{2}\right]$ (1)
$\displaystyle=\operatorname{tr}\left((\hat{W}-W)H(\hat{W}-W)^{T}\right).$ (2)
Here, $W\in\mathbb{R}^{m\times n}$ is the original weight matrix in a linear
layer, $\hat{W}\in\mathbb{R}^{m\times n}$ are the quantized weights,
$x\in\mathbb{R}^{n}$ is an input vector drawn uniformly at random from a
calibration set, and $H=E_{x}[xx^{T}]$ is a proxy Hessian. This intra-layer
formulation makes quantization tractable for LLMs. One way to minimize $\ell$
is to use adaptive rounding methods that iteratively round weight matrices by
considering the current rounding error for that specific matrix. For example,
the LDLQ111OPTQ(Frantar et al., 2023) and QuIP independently introduced
alternative formulations of this rounding method, and QuIP showed them to be
equivalent. LDLQ is the name given by QuIP. rounding algorithm iteratively
rounds rows of model weights using linear feedback from quantization error of
already rounded rows. LDLQ is optimal within the class of adaptive rounding
methods with linear feedback and offers provably better error rates than
nearest or stochastic rounding (Chee et al., 2023).
### 2.3 Incoherence Processing
Multiple works have observed that outliers in model activations and weights
can hinder quantization quality, motivating methods that “suppress” outliers
during quantization. For example, AWQ (Lin et al., 2023) scales model weights
by information from activations and OmniQuant (Shao et al., 2024) uses simple
learnable model-preserving transformations. However, these heuristic-based
approaches tend to fail at lower bitrates.
Instead, in QuIP, Chee et al. (2023) proposed that _incoherence_ is important
for LLM quantization. Informally, incoherent matrices have concentrated entry
magnitudes—ruling out outliers. In LLMs, incoherent weight and Hessian
matrices mean that both the thing being rounded (weights) and important
rounding directions (Hessians) are not too large in any coordinate. This
enables quantization with _provably_ bounded error.
###### Definition 2.1 (Chee et al. (2023)).
A Hessian $H\in\mathbb{R}^{n\times n}$ is $\mu$-incoherent if its
eigendecomposition $H=Q\Lambda Q^{T}$ has
$\textstyle\max_{i,j}\;|Q_{ij}|=\max_{i,j}\;|e_{i}^{T}Qe_{j}|\leq\mu/\sqrt{n}.$
A weight matrix $W\in\mathbb{R}^{m\times n}$ is $\mu$-incoherent if
$\max_{i,j}\;\textstyle|W_{ij}|=\max_{i,j}\;|e_{i}^{T}We_{j}|\leq\mu\|W\|_{F}/\sqrt{mn}.$
To exploit incoherence, Chee et al. (2023) introduced _incoherence processing_
as a part of their quantization method QuIP. QuIP’s incoherence processing
works by conjugating $W$ and $H$ by structured random orthogonal matrices.
Specifically, QuIP constructs orthogonal matrices $U\in\mathbb{R}^{m\times m}$
and $V\in\mathbb{R}^{n\times n}$ via a Kronecker product by drawing uniform
random orthogonal matrices $U_{1}$, $U_{2}$ (of sizes about $\sqrt{n}$),
$V_{1}$, and $V_{2}$ (of sizes about $\sqrt{m}$) and setting $U=U_{1}\otimes
U_{2}$ and $V=V_{1}\otimes V_{2}$. If we assign $\tilde{H}\leftarrow VHV^{T}$
and $\tilde{W}\leftarrow UWV^{T}$, $\tilde{H}$ and $\tilde{W}$ become
$\tilde{O}(1)$-incoherent with high probability (see their Lemma 5). Note that
this transformation preserves the proxy objective, as
$\operatorname{tr}\left((UWV^{T})(VHV^{T})(VW^{T}U^{T})\right)=\operatorname{tr}\left(WHW^{T}\right)$.
After quantizing the transformed weight matrix $\tilde{W}$ using $\tilde{H}$,
during inference, QuIP-quantized models transform model activations $x$ with
$V$ and $U^{T}$ to compute
$\textstyle U^{T}(\operatorname{quantized}(\tilde{W})(Vx))\approx
U^{T}(\tilde{W}(Vx))=Wx.$
These structured orthogonal multiplies by a Kronecker product lead to a
runtime overhead of $\Theta(n\sqrt{n}+m\sqrt{m})$, which is small relative to
the $\Theta(mn)$ cost of the multiply by $W$.
Incoherence processing can be seen as a principled alternative to more
complicated and heuristic methods for outlier suppression. Methods such as
grouping require extra storage and can negatively impact performance. For
example, using a 16 bit scale per group of 64 weights requires an extra 0.25
bits per weight. This increase is significant in extreme compression regimes,
whereas incoherence processing allows more bits to be spent on actually
quantizing model weights.
### 2.4 Vector Quantization
Prior PTQ works have focused on quantizing each scalar weight $W_{ij}$
individually (Chee et al., 2023; Lin et al., 2023; Shao et al., 2024).
However, this can be suboptimal as the resulting codebook has low density and
ignores the shape of the source distribution. Vector quantization (VQ) (Gray,
1984) instead quantizes a group of $d$ weights together by assigning them to a
$d$ dimensional vector from some codebook $C$. To effectively use VQ, $C$
should resemble the shape of the source distribution of $W$ and have high code
density. To quantize $v\in\mathbb{R}^{d}$ to $k$ bits per entry, $C$ should
have $2^{kd}$ entries. VQ becomes intractable when $kd$ is large, such as for
large $k$ or $d$. We resolve this exponential dependency on $d$ with our
rounding method BlockLDLQ in Section 4.1.
## 3 Incoherence Processing with the Randomized Hadamard Transform
Algorithm 3 Incoherence Processing with RHT (IP-RHT)
0: $W\in\mathbb{R}^{m\times n},H\in\mathbb{R}^{n\times n}$
Sample sign vectors $S_{V}\sim\mathcal{U}\\{\pm
1\\}^{n},S_{U}\sim\mathcal{U}\\{\pm 1\\}^{m}$
$\hat{W}\leftarrow\texttt{Had}(diag(S_{U})\texttt{Had}(diag(S_{V})W^{T})^{T})$
where Had is the Hadamard transform (sec. 3)
$\hat{H}\leftarrow\texttt{Had}(diag(S_{V})\texttt{Had}(diag(S_{V})H)^{T})$
$\hat{W},\hat{H},S_{U},S_{V}$
In this section, we propose a way of improving the incoherence processing of
QuIP by replacing the 2-factor Kronecker product by a Randomized Hadamard
Transformation (RHT) (Halko et al., 2011). This change yields three
advantages: (1) the theoretical bound on the incoherence parameter $\mu$ is
improved; (2) the asymptotic cost of multiplying by the structured random
orthogonal matrix is improved from $\Theta(n\sqrt{n})$ to $\Theta(n\log n)$;
(3) the cost to multiply is further reduced by a constant factor, since a
Hadamard matrix multiply can be performed without any floating-point
multiplies as its entries are in $\\{-1,+1\\}$. Additionally, we show in
Section 6.4 that this change by itself improves the perplexity of quantized
LLMs.
Recall from section 2.3 that one way to efficiently perform incoherence
processing is to conjugate $W$ and $H$ by structured random orthogonal
matrices. QuIP$\\#$ uses the RHT, which performs $x\to VSx$ where
$V\in\mathbb{R}^{n\times n}$ is a Hadamard matrix, $S$ is a random sign vector
$\\{\pm 1\\}^{n}$, and $x\in\mathbb{R}^{n}$. The RHT can be computed in
$O(n\log n)$ time with the Fast Walsh-Hadamard Transform (Fino & Algazi, 1976)
when $n$ is a power of $2$. We will temporarily assume that all dimensions are
powers of 2. Later in the section we will explain 2 methods for incoherence
processing when the dimension is not a power of 2.
lemmalemmahadincoh Let $H$ be any positive semidefinite matrix on
$\mathbb{R}^{n\times n}$ and $W$ any weight matrix on $\mathbb{R}^{m\times
n}$. Let $U\in\mathbb{R}^{m\times m}$ and $V\in\mathbb{R}^{n\times n}$ be
orthogonal scaled Hadamard matrices. Let $S_{U}\in\mathbb{R}^{m\times m}$ and
$S_{V}\in\mathbb{R}^{n\times n}$ be random diagonal matrices with independent
diagonal elements drawn uniformly from $\\{-1,+1\\}$. Then for any $\delta>0$,
$VS_{V}HS_{V}V^{T}$ is $\mu_{H}$-incoherent with probability at least
$1-\delta$, and $US_{U}WS_{V}V^{T}$ is $\mu_{W}$-incoherent with probability
at least $1-\delta$, where
$\mu_{H}=\sqrt{2\log\left(\frac{2n^{2}}{\delta}\right)}\;\;\text{ and
}\;\;\mu_{W}=2\log\left(\frac{4mn}{\delta}\right).$
In QuIP (Chee et al., 2023), the 2-factor Kronecker approach achieves
$\mu_{W}^{Kron}=A^{2}\log\left(4Cmn/\delta\right)^{2}$, where $A$ and $C$ are
global constants independent of $n$ and the number of factors. QuIP$\\#$’s RHT
achieves superior incoherence via a log dependence on the matrix size rather
that the Kronecker method’s log-squared dependence. All of QuIP’s theory
analyzing the proxy loss in Eq. (1) still holds with the RHT, with the
improved incoherence rates propagating through.
Now, what about dimensions $n$ that are not powers of 2? In most cases, we can
factorize $n=pq$ where that $p$ is the largest power of 2 such that there
exists a known Hadamard matrix of size $q$. This allows us to construct
$V\in\mathbb{R}^{n\times n}=H_{p}\otimes H_{q}$ where $H_{p}$ and $H_{q}$ are
size $p$ and $q$ Hadamard matrices, respectively. Then we can compute $VSx$ in
$O(q^{2}p\log p)$ time, which is faster than the $O(n(p+q))$ time of QuIP’s
2-factor Kronecker approach when $p\gg q$. For example, Llama 2 70B has
intermediate dimension $28672=1024*28$; $1024\gg 28$. Algorithm 3 describes
how to perform incoherence processing with the RHT. Doing so requires storing
two sign vectors $S_{U}\in\\{\pm 1\\}^{m}$ and $S_{V}\in\\{\pm 1\\}^{n}$.
Since $n,m\gg 1000$ for LLMs, $S_{U}$ and $S_{V}$ add less than 0.01 bits per
weight.
While the Hadamard conjecture states that $\exists H_{k}\forall k,4\mid k$,
finding such Hadamard matrices is still an open problem (Hedayat & Wallis,
1978). In cases when there does not exist a factorization $n=pq$ where
$\exists H_{p},H_{q}$, we present a Randomized Fast Fourier Transform (RFFT)
incoherence processing algorithm with similar runtime and concentration
properties as the RHT. At a high level, the RFFT performs incoherence
processing with the Fast Fourier Transform (FFT) (Cochran et al., 1967) and a
random complex phase. The RFFT only requires $n$ to be even, which is much
weaker than the RHT’s restrictions on $n$. The RFFT is also useful when there
does exist a decomposition $n=pq$ but $p\not\gg q$, resulting in reduced
speedups over an $\Theta(n\sqrt{n})$ algorithm. The FFT itself is also well
supported on a wide variety of hardware, meaning that it may be easier to
implement a fast RFFT when adapting QuIP$\\#$ to new hardware. In practice, we
find that the RFFT performs slightly worse than the RHT but still achieves
strong results (Table 1). We describe the RFFT in detail in Section A.2 in the
Appendix.
Table 1: RHT vs. RFFT incoherence processing using 2 Bit QuIP$\\#$ (no FT). WikiText2 perplexity ($\downarrow$), context length 4096. Incoherence | 2-7B | 2-13B | 2-70B
---|---|---|---
Hadamard | 8.22 | 6.06 | 4.16
Fourier | 8.30 | 6.08 | 4.17
## 4 BlockLDLQ and Lattice Codebooks
The proof of Lemma 3 tells us that the incoherence-processed weights follow a
roughly ball-shaped sub-Gaussian distribution. However, rounding weights one
at a time, as QuIP does with its LDLQ, ignores this shaping—producing a set of
representable weight vectors that is shaped like a hypercube rather than a
ball. Vector quantization (VQ) lets us shape codebooks to better match the
source distribution. In Section 4.1, we introduce BlockLDLQ, which iteratively
rounds blocks of weights with VQ. Within BlockLDLQ’s VQ step, QuIP$\\#$ uses
the 2 bit E8P codebook (Section 4.2). E8P is based on the $E_{8}$ lattice,
which achieves the highest density unit ball packing in $\mathbb{R}^{8}$
(Viazovska, 2017). E8P achieves good shaping while admitting fast inference
from only needing to lookup from a size 256 codebook.
### 4.1 Adaptive Rounding for Vector Quantization
Chee et al. (2023) formulated a class of adaptive rounding algorithms with
linear feedback. These methods round columns one at a time with linear
feedback $a_{k}$ from the already rounded columns. Specifically, columns of a
weight matrix $W\in\mathbb{R}^{m\times n}$ are iteratively rounded for
$k=1,2,\dots,n$:
$\hat{W}_{k}=\mathcal{Q}(W_{k}+(W_{:(k-1)}-\hat{W}_{:(k-1)})a_{k}),$ where
$W_{k}$ is the $k$-th column of $W$, $W_{:(k-1)}$ is the first $k-1$ columns
of $W$, $\mathcal{Q}$ performs nearest or stochastic rounding, and
$a_{k}\in\mathbb{R}^{k-1}$. The resulting $\hat{W}$ satisfies
$\hat{W}=\mathcal{Q}(W+(W-\hat{W})U)$, where $U\in\mathbb{R}^{n\times n}$ is a
upper triangular matrix whose columns are $a_{k}$ and $\mathcal{Q}$ acts
elementwise.
The LDLQ algorithm sets U to be $L^{T}-I$ where $H=L^{T}DL$ is the LDL
decomposition of the proxy Hessian $H$. From QuIP, we know that LDLQ is
optimal within adaptive rounding methods with linear feedback when rounding to
the integers. However, LDLQ does not work with vector quantization, which
rounds multiple columns together.
We propose to extend LDLQ to support vector quantization. Given a block size
$g$ that evenly divides $n$, our block LDLQ is based on a novel $g$-block LDL
decomposition $H=\mathbf{L}^{T}\mathbf{D}\mathbf{L}$, where $\mathbf{L}$ is a
unit block lower triangular matrix (among the $n^{2}/g^{2}$ $g\times g$ blocks
of $L\in\mathbf{R}^{n\times n}$, the $n/g$ diagonal blocks are all $I$ and all
blocks above the diagonal are $0$), and $\mathbf{D}$ is a block diagonal
matrix.222It is straightforward to produce the $g$-block LDL decomposition
from the Cholesky decomposition of $H$. As before, we set
$\mathbf{U}=\mathbf{L}^{T}-I$, and round $W$ in a block-wise fashion via
$\hat{W}_{k}=\mathbf{Q}(W_{k}+(W_{:(k-1)}-\hat{W}_{:(k-1)})\mathbf{A}_{k}),$
where $\mathbf{A}_{k}\in\mathbb{R}^{n\times g}$ contains the $k-g+1$ through
$k$-th columns of $\mathbf{U}$ (the $k$th block), $W_{k}$ similarly denotes
the $k$th block of $W$, and $\mathbf{Q}$ denotes a vector quantizer. As in the
original QuIP paper, we can bound the error of this method. theoremthmLDLQ
Suppose that we round $W\in\mathbb{R}^{m\times n}$ using $g$-block LDLQ with
Hessian $H$, producing $\hat{W}$. Suppose that $H$ is $\mu$-incoherent, and
that we use a (possibly stochastic) vector quantizer $\mathbf{Q}$ that
satisfies
$\mathbf{E}[(\mathbf{Q}(x)-x)(\mathbf{Q}(x)-x)^{T}]\preceq\sigma^{2}I$ for any
$x\in\mathbb{R}^{g}$. Then
$\textstyle\mathbf{E}[\operatorname{tr}((\hat{W}-W)H(\hat{W}-W)^{T})]\leq\frac{gm\mu^{2}\sigma^{2}}{n}\operatorname{tr}(H^{1/2})^{2}.$
Observe that under the same conditions, just quantizing all blocks
independently would yield
$\mathbf{E}[\operatorname{tr}((\hat{W}-W)H(\hat{W}-W)^{T})]\leq
gm\sigma^{2}\operatorname{tr}(H)$: this “improvement” from the trace of $H$ to
the square of the trace of its square root divided by $n$ is the same factor
achieved in the scalar case in QuIP.333The original QuIP paper also included
multiple other technical guarantees, including a bound that considers more
rigorously the “real” case of finite-sized codebooks. While these results
could also be generalized to the block-LDLQ case, we view this as not
providing much insight relevant to QuIP$\\#$ beyond Theorem 4.1, so (if
desired) they are left as an exercise for the reader.
### 4.2 The E8P (“E8 Padded”) Codebook
BlockLDLQ relies on an internal vector quantization (VQ) step $\mathbf{Q}$
that rounds a $d$-dimension ($g$ in the previous section) vector to a codebook
$C$. To effectively apply VQ, $C$ should be shaped like the source
distribution and have high packing density. One way to improve shaping is by
increasing $d$. However, recall from Section 2.4 that to quantize a vector
$v\in\mathbb{R}^{d}$ to $k$ bits with VQ, $C$ must have size $2^{kd}\times d$.
Since the codebook size is exponential in both the vector dimension and
bitrate, VQ quickly becomes intractable at high dimensions or bitrates.
For QuIP$\\#$, we propose a novel 2-bit 8 dimensional _E8P codebook_ , which
contains $2^{16}$ entries but only requires lookups into a $2^{8}$-entry
table, with the remaining $8$ bits being used to store signs and shifts. E8P
bypasses the scaling issues of VQ by taking advantage of the structure and
symmetries of the $E_{8}$ lattice on which it is based. The $E_{8}$ lattice is
composed of all-integer or all-half-integer vectors in $\mathbb{R}^{8}$ whose
sum is an even number, that is
$\textstyle
E_{8}=\left(\mathbb{Z}^{8}\cup\left(\mathbb{Z}^{8}+\frac{1}{2}\right)\right)\cap\left\\{x\mid\mathbf{1}^{T}x\text{
is even}\right\\}.$
The construction of the E8P codebook starts with an equivalent way to write
$E_{8}$ via the $\hat{D}_{8}$ lattice, where
$\hat{D}_{8}=\left\\{x\in\mathbb{Z}^{8}+\frac{1}{2}\mid\mathbf{1}^{T}x\text{
is even}\right\\}$ is the set of half-integer vectors with even parity: here,
$E_{8}=\hat{D}_{8}\cup(\hat{D}_{8}+\frac{1}{2})$. It follows that
$(\hat{D}_{8}-\frac{1}{4})\cup(\hat{D}_{8}+\frac{1}{4})=E_{8}+\frac{1}{4}$ is
just a shifted copy of $E_{8}$ (keeping the same optimal packing density).
$\hat{D}_{8}$ has nice symmetry properties: flipping any (nonzero) even number
of signs of an element in $\hat{D}_{8}$, yields another distinct element in
$\hat{D}_{8}$. This means that if $|\hat{D}_{8}|$ denotes the set of
elementwise absolute values of entries in $\hat{D}_{8}$, then each element of
$\hat{D}_{8}$ can be expressed (uniquely) as the elementwise product of an
entry $s\in|\hat{D}_{8}|$ and a sign vector of appropriate parity. So, if we
start from some “source codebook” of absolute entries $S\subset|\hat{D_{8}}|$,
we can use the 128 possible odd- or even-parity sign flips to generate a
subset of $\hat{D_{8}}$. Each entry in $S$ is either an odd or even number of
flips away from an entry in $\hat{D_{8}}$, but not both. Thus, given $s\in S$
and 7 out of the 8 sign flips, we can infer the last one from the parity of
the 7 sign flips and $s$. This lets us use the following pattern to store a
16-bit codeword in $E_{8}+\frac{1}{4}$: 8 bits for the entry in $S$, 7 bits
for sign flips, and 1 bit to $\pm\frac{1}{4}$. This lets us decode a size
$2^{16}$ codebook by looking up into only a size $2^{8}$ codebook ($S$) and
performing some operations. All that remains is how to choose $S$: we set $S$
to be the 227 elements of $|\hat{D_{8}}|$ with norm $\leq\sqrt{10}$ plus 29
“padding” elements from $|\hat{D_{8}}|$ with norm $\sqrt{12}$ (see Section
C.1). We call this ball-shaped $2^{16}$-entry lattice codebook “E8P.”
Figure 3: Minimum achievable elementwise MSE of quantizing a Gaussian to
various codebooks. $E_{8}$-based codebooks outperform other presented
codebooks due to the underlying packing density and high dimensionality of
$E_{8}$.
Figure 3 plots the elementwise MSE of quantizing a standard multivariate
Gaussian to various $k$ bit codebooks. Each $k$-bit codebook consists of a
$d$-dimensional base lattice intersected with a ball to reach $2^{kd}$ points.
The $E_{8}$-based codebooks achieve lower MSEs than all other presented
codebooks, including those based on the $D_{4}$ lattice (the even-parity
vectors in $\mathbb{Z}^{4}$), which achieves the kissing number in
$\mathbb{R}^{4}$. This figure illustrates the importance of dimension for
vector quantization. Increasing the vector dimension decreases the error for
the half integer grid, as the resulting codebook is closer in shape to the
source distribution. Finally, while K-means on the source distribution would
achieve lower MSE (Lloyd, 1982), there are a number of practical reasons why a
K-means based codebook would be less practical, including worse end-to-end
empirical performance. We discuss this more in Section C.3.
### 4.3 Scaling $E_{8}$ to Higher Bitrates
The $E_{8}$ lattice works well for low bitrates (e.g. 2 bits), but quickly
becomes intractable at higher bitrates due to codebook size. In QuIP$\\#$, we
use residual vector quantization (RVQ) (Juang & Gray, 1982) to get the
benefits of lattice codebooks at higher bitrates. RVQ quantizes a vector $x$
to $p$ bits with a set $q$ of $q_{i}$-bit codebooks (denoted
$\mbox{RVQ}(x,p,q)$ where $p=\sum_{0\leq i<|q|}q_{i}$) by repeatedly
quantizing the quantization residual. That is, $\mbox{RVQ}(x,p,q)=\sum_{0\leq
i<|q|}\delta_{i}$ where $\delta_{i}=Q_{q_{i}}\left((x-\sum_{0\leq
j<i}\delta_{j})/s_{i}\right)\cdot s_{i}$, we let $Q_{q_{i}}(\cdot)$ denote
quantizing to a $q_{i}$ bit codebook, and $s_{i}\in\mathbb{R}$. Using RVQ, we
can quantize to 4 bits by rounding with the 2 bit E8P codebook twice. We can
also quantize to 3 bits by using the 2 bit E8P codebook and a 1-bit $E_{8}$
codebook (elements of $E_{8}$ with norm $\leq 2$ and 15 elements of $E_{8}$
with norm 4). One could also use more advanced multi-codebook quantization
approaches other than RVQ, but we found that RVQ was sufficient to achieve
strong quantization performance.
## 5 Fine-Tuning During Quantization
Recent works have suggested that at extreme quantization levels (e.g. 2 bits),
inter-layer interactions are a significant hurdle to lossless quantization
(Shao et al., 2024; Egiazarian et al., 2024). Here, we employ a simple fine-
tuning algorithm that attempts to recover the original unquantized model
during quantization. Our fine tuning method runs on a small development set
and works by relaxing the sign vectors in the RHT to arbitrary real vectors
after quantization.
Our fine tuning method contains two steps. First, we fine-tune within each
transformer block by quantizing the first linear layer, fine-tuning the
remaining parameters to match the unquantized model’s output activations on
the unquantized model’s input activations, quantizing the second linear layer,
fine-tuning, and so on until all linear layers are quantized. This step
attempts to minimize the activation error caused by an individual linear layer
during quantization, and it is parallelizable across transformer blocks as the
activation error does not consider the effect of quantizing preceding blocks.
The idea of fine-tuning on the level of a transformer block was previously
proposed in Egiazarian et al. (2024); our methodology differs in that we set a
different set of parameters to be trainable. In the second step, after all
linear layers in the model are quantized, the unquantized parameters
(layernorms, sign vectors, language model head) are fine-tuned to minimize
activation error over the entire model.
By optimizing the sign vectors as real vectors instead of binary vectors in
both steps, we allow the incoherence processing step to shape the weight
matrix to the codebook. While this means we must store the sign vectors in
FP16 instead of as bitvectors, the size of LLM matrices means that the sign
vectors still add less than 0.01 bits per weight. We describe these steps in
more detail in Section D.
## 6 Experiments
Table 2: Llama 1 & 2 Wikitext2 and C4 perplexity ($\downarrow$), context length 2048. | | Wikitext 2 | C4
---|---|---|---
Method | Bits | 1-7 | 1-13 | 1-30 | 1-65 | 2-7 | 2-13 | 2-70 | 1-7 | 1-13 | 1-30 | 1-65 | 2-7 | 2-13 | 2-70
FP16 | 16 | 5.68 | 5.09 | 4.10 | 3.53 | 5.47 | 4.88 | 3.32 | 7.08 | 6.61 | 5.98 | 5.62 | 6.97 | 6.47 | 5.52
AWQ | 4 | 6.08 | 5.34 | 4.39 | 3.76 | 6.15 | 5.12 | - | 7.52 | 6.86 | 6.17 | 5.77 | 7.68 | 6.74 | -
OmniQ | 4 | 5.86 | 5.21 | 4.25 | 3.71 | 5.74 | 5.02 | 3.47 | 7.34 | 6.76 | 6.11 | 5.73 | 7.35 | 6.65 | 5.65
QuIP# no FT & no $E_{8}$ | 4 | 5.83 | 5.20 | 4.23 | 3.63 | 5.66 | 5.00 | 3.42 | 7.25 | 6.70 | 6.06 | 5.68 | 7.17 | 6.59 | 5.59
QuIP# | 4 | 5.76 | 5.17 | 4.18 | 3.60 | 5.56 | 4.95 | 3.38 | 7.18 | 6.67 | 6.03 | 5.66 | 7.07 | 6.54 | 5.56
AWQ | 3 | 11.9 | 7.45 | 10.0 | 5.21 | 24.0 | 10.5 | - | 13.3 | 9.13 | 12.7 | 7.11 | 23.9 | 13.1 | -
OmniQ | 3 | 6.49 | 5.68 | 4.74 | 4.04 | 6.58 | 5.58 | 3.92 | 8.19 | 7.32 | 6.57 | 6.07 | 8.65 | 7.44 | 6.06
QuIP# no FT & no $E_{8}$ | 3 | 6.29 | 5.52 | 4.54 | 3.91 | 6.19 | 5.34 | 3.71 | 7.82 | 6.98 | 6.29 | 5.86 | 7.85 | 6.98 | 5.78
QuIP# | 3 | 5.98 | 5.31 | 4.36 | 3.78 | 5.79 | 5.10 | 3.56 | 7.39 | 6.83 | 6.17 | 5.77 | 7.32 | 6.72 | 5.67
OmniQ | 2 | 15.5 | 13.2 | 8.71 | 7.58 | 37.4 | 17.2 | 7.81 | 24.9 | 18.3 | 13.9 | 10.8 | 90.6 | 26.8 | 12.3
QuIP# no FT & no $E_{8}$ | 2 | 9.95 | 7.18 | 5.80 | 5.02 | 12.3 | 7.60 | 4.87 | 11.7 | 8.67 | 7.55 | 6.83 | 14.8 | 9.57 | 6.82
QuIP# | 2 | 6.86 | 5.97 | 5.02 | 4.36 | 6.66 | 5.74 | 4.16 | 8.36 | 7.48 | 6.71 | 6.19 | 8.35 | 7.45 | 6.12
Table 3: Zeroshot Accuracy (acc in LM Eval, not acc_norm), Llama 2. | 2-70 | 2-13 | 2-7
---|---|---|---
Method | Bits | ArcC | ArcE | PiQA | Wino | Bits | ArcC | ArcE | PiQA | Wino | Bits | ArcC | ArcE | PiQA | Wino
FP16 | 16 | 51.1 | 77.7 | 81.1 | 77.0 | 16 | 45.6 | 73.3 | 73.5 | 69.6 | 16 | 40.0 | 69.3 | 78.5 | 67.3
OmniQ | 4 | 49.8 | 77.9 | 80.7 | 75.8 | 4 | 43.1 | 70.2 | 78.4 | 67.8 | 4 | 37.9 | 67.8 | 77.1 | 67.0
QuIP | 4 | 47.0 | 74.3 | 80.3 | 76.0 | 4 | 44.9 | 73.3 | 79.0 | 69.7 | 4 | - | - | - | -
AQLM | 4.07 | 51.0 | 78.1 | 81.4 | 76.9 | 3.94 | 43.9 | 72.2 | 78.6 | 70.4 | 4.04 | 40.3 | 68.9 | 77.7 | 67.3
QuIP# | 4 | 50.6 | 78.1 | 81.4 | 77.1 | 4 | 45.5 | 73.9 | 78.9 | 69.9 | 4 | 40.5 | 69.1 | 78.4 | 67.6
OmniQ | 3 | 47.6 | 75.7 | 79.7 | 73.5 | 3 | 42.0 | 69.0 | 77.7 | 65.9 | 3 | 35.3 | 62.6 | 73.6 | 63.6
QuIP | 3 | 46.3 | 73.2 | 80.0 | 74.6 | 3 | 41.5 | 70.4 | 76.9 | 69.9 | 3 | - | - | - | -
AQLM | 3.01 | 50.0 | 77.6 | 81.3 | 77.2 | 3.03 | 43.6 | 73.5 | 77.8 | 67.6 | 3.04 | 38.7 | 67.8 | 76.6 | 68.4
QuIP# | 3 | 50.9 | 77.7 | 81.4 | 76.4 | 3 | 44.0 | 72.5 | 78.4 | 69.1 | 3 | 39.2 | 68.4 | 77.3 | 66.5
OmniQ | 2 | 28.7 | 55.4 | 68.8 | 53.2 | 2 | 23.0 | 44.4 | 62.6 | 52.6 | 2 | 21.6 | 35.2 | 57.5 | 51.5
QuIP | 2 | 34.0 | 62.2 | 74.8 | 67.5 | 2 | 23.5 | 45.2 | 62.0 | 52.8 | 2 | 19.4 | 26.0 | 54.6 | 51.8
AQLM | 2.07 | 47.9 | 77.7 | 80.4 | 75.9 | 1.97 | 38.5 | 67.0 | 75.1 | 69.5 | 2.02 | 33.6 | 62.8 | 73.5 | 64.6
QuIP# | 2 | 48.7 | 77.3 | 80.3 | 75.9 | 2 | 39.5 | 69.3 | 77.3 | 67.7 | 2 | 34.6 | 64.6 | 75.1 | 64.9
Table 4: Wikitext2 and C4 perplexity ($\downarrow$), context length 4096. | | 2-7 | | | 2-13 | | | 2-70 |
---|---|---|---|---|---|---|---|---|---
Method | Bits | W2 | C4 | Bits | W2 | C4 | Bits | W2 | C4
FP16 | 16 | 5.12 | 6.63 | 16 | 4.57 | 6.05 | 16 | 3.12 | 4.97
QuIP# | 4 | 5.19 | 6.75 | 4 | 4.63 | 6.13 | 4 | 3.18 | 5.02
no FT | 4 | 5.22 | 6.79 | 4 | 4.65 | 6.15 | 4 | 3.18 | 5.02
no $E_{8}$ | 4 | 5.29 | 6.86 | 4 | 4.68 | 6.20 | 4 | 3.22 | 5.05
QuIP | 4 | - | - | 4 | 4.76 | 6.29 | 4 | 3.58 | 5.38
AQLM | 4.04 | 5.21 | 6.74 | 3.94 | 4.64 | 6.14 | 4.07 | 3.17 | 5.01
QuIP# | 3 | 5.41 | 7.04 | 3 | 4.78 | 6.35 | 3 | 3.35 | 5.15
no FT | 3 | 5.60 | 7.34 | 3 | 4.90 | 6.50 | 3 | 3.41 | 5.20
no $E_{8}$ | 3 | 5.77 | 7.61 | 3 | 4.99 | 6.65 | 3 | 3.48 | 5.28
QuIP | 3 | - | - | 3 | 5.12 | 6.79 | 3 | 3.87 | 5.67
AQLM | 3.04 | 5.46 | 7.10 | 3.03 | 4.83 | 6.37 | 3.01 | 3.36 | 5.17
QuIP# | 2 | 6.19 | 8.16 | 2 | 5.35 | 7.20 | 2 | 3.91 | 5.71
no FT | 2 | 8.22 | 11.0 | 2 | 6.06 | 8.07 | 2 | 4.16 | 6.01
no $E_{8}$ | 2 | 11.2 | 14.5 | 2 | 7.04 | 9.37 | 2 | 4.58 | 6.51
QuIP | 2 | - | - | 2 | 13.5 | 16.2 | 2 | 5.90 | 8.17
AQLM | 2.02 | 6.93 | 8.84 | 1.97 | 5.70 | 7.59 | 2.07 | 3.94 | 5.72
Table 5: 2 and 4 bit QuIP$\\#$ Llama model generation speed measured on a NVIDIA RTX 4090 with a “proof of concept” CUDA implementation of E8P. QuIP$\\#$ achieves $>50\%$ peak memory bandwidth (1008GB/s) during generation and is fast and scalable. Model | | 2 Bit
---
tok/s
| 2 Bit %
---
Mem BW
| 4 Bit
---
tok/s
| 4 Bit %
---
Mem BW
2-7B | 170.50 | 29.60% | 117.73 | 40.87%
2-13B | 104.83 | 33.80% | 71.09 | 45.84%
1-30B | 51.60 | 38.39% | 32.50 | 48.36%
2-70B | 32.74 | 56.84% | OOM | OOM
Figure 4: QuIP$\\#$ scaling, Llama 1. Like Llama 2, QuIP$\\#$ 3 bit scales
better than QuIP$\\#$ 4 bit for Llama 1 models and QuIP$\\#$ 2 bit scales
similarly to higher bitrates.
Our main experiments show the performance of QuIP$\\#$ on the Llama 1 (Touvron
et al., 2023a) and 2 (Touvron et al., 2023b) family of models. These models
range in size from 7 billion to 70 billion parameters and offer good
performance, making them suitable for understanding how quantization methods
perform and scale. Additional results for other models are available in the
Appendix.
In Section 6.1, we compare QuIP$\\#$ with recently published weight-only PTQ
methods. AWQ scales weights by activation magnitudes before quantizing to
reduce outliers (Lin et al., 2023). OmniQuant learns model-preserving
layerwise transformations that reduce outliers per transformer block (Shao et
al., 2024). AQLM uses additive quantization with learnable per-layer codebooks
and performs a fine-tuning step on codebook entries and layernorms (Egiazarian
et al., 2024). We also include QuIP (Chee et al., 2023) as a baseline for the
improvements in QuIP$\\#$.
We report W$x$A16 numbers for AWQ and OmniQuant from the OmniQuant paper and
AQLM numbers from AQLM. We note that there are currently 2 methods for
evaluating perplexity: using the Llama 1 context length of 2048 or using the
model’s native context length (e.g. 4096 for Llama 2). OmniQuant and AWQ use
2048 for Llama 2 while AQLM uses 4096; we report both sets of numbers. We also
note that AQLM paper reports QuIP$\\#$ numbers from an outdated version of
QuIP$\\#$. QuIP$\\#$ is under active development and the numbers here
represent the latest QuIP$\\#$ numbers. Finally, we bold numbers in our tables
when they are clearly better, such as a smaller model matching or
outperforming a larger model or a similar sized model significantly
outperforming another model.
### 6.1 QuIP$\\#$ on Llama Models
Table 2 shows a comparison of QuIP$\\#$ with OmniQuant, AWQ, and QuIP$\\#$(no
FT and no lattice codebook $E_{8}$) with context length 2048. QuIP$\\#$ offers
a paradigm shift in quantization quality over OmniQuant and AWQ. Notably,
while AWQ falls apart at even 2.15 bits (Shao et al., 2024) and OmniQuant
produces unusable models at 2 bits, QuIP$\\#$ produces high quality models
that are close to OmniQuant 3 bit models. Table 2 also shows the importance of
incoherence processing. QuIP$\\#$ without fine-tuning or lattice codebooks
significantly outperforms OmniQuant and AWQ, which both rely on heuristics to
reduce model outliers during quantization.
Table 4 shows a comparison of QuIP$\\#$ with AQLM with context length 4096.
QuIP$\\#$ offers strong improvements over AQLM at the 2 and 3 bit level,
either significantly outperforming similarly sized models or offering similar
performance with a smaller model444In our experience, at extreme quantization
levels, even 0.1 bits can make a significant difference in quantization
quality.. At the 4 bit level, QuIP$\\#$ and AQLM both perform similarly. This
is not surprising as state-of-the-art 4 bit models are all very close to FP16
performance. Furthermore, the QuIP$\\#$ 3 and 4 bit results presented in this
paper use residual vector quantization; one could potentially achieve better
numbers with more advanced multi-codebook quantization approaches.
Table 3 shows zeroshot results for QuIP$\\#$, AQLM, and OmniQuant. Both AQLM
and QuIP$\\#$ signficantly outperform OmniQuant, which correlates with the
perpelxity results. AQLM and QuIP$\\#$ both perform very close to FP16 at
higher bitrates and for larger models, but QuIP$\\#$ tends to outperform AQLM
at lower bitrates and model sizes. We note that zeroshot tasks have an element
of randomness and even FP16 numbers can disagree by up to $0.5\%$.
### 6.2 QuIP$\\#$ Bit Scaling
Figures 1 (first page) and 4 show how QuIP$\\#$ scales on the Llama family of
models and Wikitext2. On both Llama 1 and 2, QuIP$\\#$ 3 bit outperforms
QuIP$\\#$ 4 bit and QuIP$\\#$ 2 bit offers similar scaling to 3 and 4 bit
models. Furthermore, on Llama 2, QuIP$\\#$ 3 bit outperforms a theoretical
lossless 4 bit model (FP16 at 4 bits). To the best of our knowledge, this is
the first time a 3 bit PTQ method has outperformed a theoretical lossless 4
bit model and also the first time a 2 bit PTQ method has offered similar
scaling to higher bitrates.
### 6.3 Efficient Inference with QuIP$\\#$
Table 5 shows 2 and 4 bit QuIP$\\#$ Llama model generation speed measured on a
NVIDIA RTX 4090 GPU with a “proof of concept” CUDA implementation of E8P and
the FlashAttention library (Dao et al., 2022; Dao, 2023) implementation of
Llama. 4 bit generation speed is faster than 50% of the 2 bit speed since both
bitrates still need to perform two Hadamard multiplies per linear layer. We
note that we are not CUDA experts and a better optimized implementation of E8P
could likely achieve higher throughput. Nevertheless, we find that QuIP$\\#$
inference is fast and scalable on modern GPUs.
### 6.4 Ablations
Table 4 also contains an ablation on the various components of QuIP$\\#$. The
“no FT” row shows QuIP$\\#$ without fine-tuning and the “no $E_{8}$” row shows
QuIP$\\#$ without fine-tuning and lattice codebooks. For the latter, we round
to the 1-dimensional half-integer grid. We also include QuIP numbers as
reported by AQLM. At all bitrates, each component of QuIP$\\#$ brings
additional performance gains. The difference between QuIP and QuIP$\\#$
without fine-tuning and lattice codebooks also shows the difference between
QuIP’s Kronecker factorization and QuIP$\\#$’s RHT. The RHT offers stronger
incoherence properties than the Kronecker factorization (Section 3), which
improves performance.
## 7 Conclusion
We present QuIP$\\#$, a weight-only post training compression method that
achieves state-of-the-art results on LLMs at 2, 3, and 4 bits per weight.
QuIP$\\#$ uses the Randomized Hadamard Transform as am efficient and
principled form of outlier suppression, and introduces the $E_{8}$ lattice-
based E8P codebook to better quantize RHT transformed weights. The E8P
codebook is highly symmetric and admits fast inference, allowing a “proof of
concept” QuIP$\\#$ CUDA implementation to achieve over 50% peak memory
bandwidth on modern GPUs. QuIP$\\#$ also implements inter-layer fine tuning,
further improving quantization. To the best of our knowledge, QuIP$\\#$ is the
first PTQ method to achieve superior scaling at 3 bits over 4 bits and similar
scaling at 2 bits to higher bitrates. Our results indicate that, in the near
future, 2 bit models are likely to scale better than 3 bit ones.
## 8 Impact Statement
This paper presents work whose goal is to advance the field of Machine
Learning. There are many potential societal consequences of our work, none
which we feel must be specifically highlighted here.
## Acknowledgements
We thank, in no particular order, David Hou for helping with the QuIP$\\#$
CUDA implementation, Tiancheng Yuan for lending his RTX 4090 and helping with
acquiring QuIP$\\#$ timing numbers, Tri Dao for a fast CUDA implementation of
the Hadamard transform and general help with QuIP$\\#$, and Together AI for
compute resources.
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## Appendix A Concentration Inequalities for the Randomized Hadamard
Transform and Fast Fourier Transform
### A.1 Incoherence Processing with the Randomized Hadamard Transform
###### Lemma A.1.
For any non-negative real number $n$,
$\frac{1}{B\left(n,1/2\right)}\int_{-1}^{+1}(1-x^{2})^{n-1}\cdot\exp(tx)\;dx\leq\exp\left(\frac{t^{2}}{4n+2}\right).$
###### Proof.
We start with the following “standard” integral. For non-negative integer $m$
and real $n>0$,
$\displaystyle\int_{-1}^{+1}x^{2m}(1-x^{2})^{n-1}\;dx=B\left(m+\frac{1}{2},n\right)=\frac{\Gamma\left(m+\frac{1}{2}\right)\Gamma\left(n\right)}{\Gamma\left(m+n+\frac{1}{2}\right)}.$
This means that
$\displaystyle\frac{1}{B\left(\frac{1}{2},n\right)}\int_{-1}^{+1}x^{2m}(1-x^{2})^{n-1}\;dx$
$\displaystyle=\frac{B\left(m+\frac{1}{2},n\right)}{B\left(\frac{1}{2},n\right)}$
$\displaystyle=\frac{\Gamma\left(m+\frac{1}{2}\right)\Gamma\left(n\right)}{\Gamma\left(m+n+\frac{1}{2}\right)}\cdot\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\right)\Gamma\left(n\right)}$
$\displaystyle=\frac{\Gamma\left(m+\frac{1}{2}\right)\Gamma\left(n+\frac{1}{2}\right)}{\sqrt{\pi}\cdot\Gamma\left(m+n+\frac{1}{2}\right)}.$
Applying the Legendre duplication formula, for integer $m$,
$\Gamma\left(m+\frac{1}{2}\right)=\frac{(2m)!\sqrt{\pi}}{4^{m}m!},$
then
$\displaystyle\frac{1}{B\left(\frac{1}{2},n\right)}\int_{-1}^{+1}x^{2m}(1-x^{2})^{n-1}\;dx$
$\displaystyle=\frac{(2m)!\sqrt{\pi}}{4^{m}m!}\cdot\frac{(2n)!\sqrt{\pi}}{4^{n}n!}\cdot\frac{1}{\sqrt{\pi}}\cdot\frac{4^{m+n}(m+n)!}{(2m+2n)!\sqrt{\pi}}$
$\displaystyle=\frac{(2m)!(2n)!(m+n)!}{m!n!(2m+2n)!}.$
In particular, this means that
$\displaystyle\frac{1}{B\left(\frac{1}{2},n\right)}\int_{-1}^{+1}\exp(tx)(1-x^{2})^{n-1}\;dx$
$\displaystyle=\sum_{m=0}^{\infty}\frac{t^{2m}}{(2m)!}\cdot\frac{1}{B\left(\frac{1}{2},n\right)}\int_{-1}^{+1}x^{2m}(1-x^{2})^{n-1}\;dx$
$\displaystyle=\sum_{m=0}^{\infty}\frac{t^{2m}}{(2m)!}\cdot\frac{(2m)!(2n)!(m+n)!}{m!n!(2m+2n)!}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{t^{2m}}{m!}\cdot\frac{(2n)!(m+n)!}{n!(2m+2n)!}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{t^{2m}}{m!}\cdot\prod_{k=1}^{m}\frac{k+n}{(2k+2n)(2k+2n-1)}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{t^{2m}}{m!}\cdot\frac{1}{2^{m}}\prod_{k=1}^{m}\frac{1}{2k+2n-1}$
$\displaystyle\leq\sum_{m=0}^{\infty}\frac{t^{2m}}{m!}\cdot\frac{1}{2^{m}}\left(\frac{1}{2n+1}\right)^{m}$
$\displaystyle=\sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{t^{2}}{4n+2}\right)^{m}$
$\displaystyle=\exp\left(\frac{t^{2}}{4n+2}\right).$
This proves the lemma
$\frac{1}{B\left(n,1/2\right)}\int_{-1}^{+1}(1-x^{2})^{n-1}\cdot\exp(tx)\;dx\leq\exp\left(\frac{t^{2}}{4n+2}\right).$
∎
###### Lemma A.2.
Call $U\in\mathbb{R}^{nd\times nd}$ an $(n,d)$-block orthohadamard matrix if
it has the following properties: (1) $U$ is a orthogonal matrix, and (2) each
aligned $d\times d$ block of $U$ is $1/\sqrt{n}$ times an orthogonal matrix.
This generalizes the notion of Hadamard matrices. Let
$S\in\mathbb{R}^{nd\times nd}$ be a random block diagonal matrix, where each
$d\times d$ block of the diagonal is sampled independently and uniformly from
the set of (possibly special) orthogonal matrices. Then we call multiplication
by $US$ a _randomized orthohadamard transform_ , and observe that it has the
following nice property. Let $x\in\mathbb{R}^{nd}$ be any fixed vector, and
let $b\in\mathbb{R}^{nd}$ be a fixed vector that is _sparse_ in the sense that
it is supported only on a single $d$-sized aligned block (i.e. all but one of
the $n$ blocks are zero). Then
$\mathbf{P}\left(\left|b^{T}USx\right|\geq a\right)\leq
2\exp\left(-\frac{a^{2}nd}{2\left\|b\right\|^{2}\left\|x\right\|^{2}}\right).$
###### Proof.
If we let the $i$th block of $x$ be $x_{i}\in\mathbb{R}^{d}$ and let the $i$th
block of $S^{T}U^{T}b^{T}$ be $v_{i}$, then the $v_{i}$ will be independent
and uniformly distributed on the sphere in $d$ dimensional space of radius
$\left\|b\right\|/\sqrt{n}$, and so
$v_{i}^{T}x_{i}=\left\|b\right\|\left\|x_{i}\right\|n^{-1/2}z_{i}$, where the
$z_{i}$ are all independent and distributed according to an entry of a random
point on the unit sphere in $d$ dimensional space. Observe that this means
that
$\mathbf{P}(z_{i})\propto(1-z_{i}^{2})^{\frac{d-1}{2}-1}.$
So,
$\displaystyle\mathbf{E}\left[\exp\left(tb^{T}USx\right)\right]$
$\displaystyle=\mathbf{E}\left[\exp\left(t\sum_{i=1}^{n}\left\|b\right\|\left\|x_{i}\right\|n^{-1/2}z_{i}\right)\right]$
$\displaystyle=\prod_{i=1}^{n}\mathbf{E}\left[\exp\left(t\left\|b\right\|\left\|x_{i}\right\|n^{-1/2}z_{i}\right)\right]$
$\displaystyle\leq\prod_{i=1}^{n}\mathbf{E}\left[\exp\left(\frac{1}{4\cdot\frac{d-1}{2}+2}\right)\left(t\left\|b\right\|\left\|x_{i}\right\|n^{-1/2}\right)^{2}\right]$
$\displaystyle=\prod_{i=1}^{n}\mathbf{E}\left[\exp\left(\frac{t^{2}\left\|b\right\|^{2}\left\|x_{i}\right\|^{2}}{2nd}\right)\right]$
$\displaystyle=\mathbf{E}\left[\exp\left(\frac{t^{2}\left\|b\right\|^{2}\left\|x\right\|^{2}}{2nd}\right)\right],$
where the the last line follows from Lemma A.1. It follows from the standard
application of Markov’s inequality that for any $a>0$,
$\mathbf{P}\left(\left|b^{T}USx\right|\geq a\right)\leq
2\exp\left(-\frac{a^{2}nd}{2\left\|b\right\|^{2}\left\|x\right\|^{2}}\right).$
This is what we wanted to show. ∎
###### Lemma A.3.
Let $H\in\mathbb{R}^{n\times n}$ be an orthogonal scaled Hadamard matrix or
$F\in\mathbb{R}^{n\times n}$ be an orthogonal FFT matrix (the FFT understood
as operating on a real vector space). Let $S\in\mathbb{R}^{n\times n}$ be a
random diagonal matrix with diagonal elements supported on $\mathbb{R}^{n}$,
and let $P\in\mathbb{R}^{n\times n}$ be a random 2-block-diagonal matrix with
$2\times 2$ diagonal blocks supported on $\operatorname{SO}(2)$ (we can also
think of this as acting like a diagonal complex matrix with each diagonal
element a random complex number of absolute value $1$). Let
$U\in\mathbb{R}^{n\times n}$ be any fixed orthogonal matrix. Then, for any
$\epsilon>0$,
$\operatorname{Prob}\left(\max_{i,j}\left|e_{i}^{T}HSUe_{j}\right|\geq\sqrt{\frac{2}{nd}\log\left(\frac{2n^{2}}{\epsilon}\right)}\right)\leq\epsilon$
and
$\operatorname{Prob}\left(\max_{i,j}\left|e_{i}^{T}FPUe_{j}\right|\geq\sqrt{\frac{2}{nd}\log\left(\frac{2n^{2}}{\epsilon}\right)}\right)\leq\epsilon.$
That is, with probability at least $1-\epsilon$, multiplying by either $HS$ or
$FP$ makes the resulting orthogonal matrix $\mu$-incoherent, where
$\mu_{H}=\sqrt{2\log\left(\frac{2n^{2}}{\epsilon}\right)}.$
###### Proof.
Setting $b=e_{i}$ and $x=Ue_{j}$ in Lemma A.2,
$\mathbf{P}\left(\left|e_{i}^{T}HSUe_{j}\right|\geq a\right)\leq
2\exp\left(-\frac{a^{2}nd}{2}\right).$
By the union bound,
$\mathbf{P}\left(\max_{i,j}\left|e_{i}^{T}HSUe_{j}\right|\geq a\right)\leq
2n^{2}\exp\left(-\frac{a^{2}nd}{2}\right).$
Setting
$a^{2}=\frac{2}{nd}\log\left(\frac{2n^{2}}{\epsilon}\right)$
proves the lemma. The FFT case is identical. ∎
###### Lemma A.4.
Let $H_{L}\in\mathbb{R}^{m\times m}$ be an orthogonal scaled Hadamard matrix
or $F\in\mathbb{R}^{m\times m}$ be an orthogonal FFT matrix (the FFT
understood as operating on a real vector space). Let
$S_{L}\in\mathbb{R}^{m\times m}$ be a random diagonal matrix with diagonal
elements supported on $\mathbb{R}^{m}$, and let $P\in\mathbb{R}^{m\times m}$
be a random 2-block-diagonal matrix with $2\times 2$ diagonal blocks supported
on $\operatorname{SO}(2)$. Let $H_{R}\in\mathbb{R}^{n\times n}$, $F_{R}$,
$S_{R}$, and $P_{R}$ be defined analogously over $n$-dimensional space. Let
$W\in\mathbb{R}^{m\times n}$ be any fixed matrix. Then, for any $\epsilon>0$,
$\mathbf{P}\left(\max_{i,j}\left|e_{i}^{T}H_{L}S_{L}WS_{R}^{T}H_{R}^{T}e_{j}\right|\geq\left\|W\right\|_{F}\sqrt{\frac{4}{mn}}\log\left(\frac{4mn}{\epsilon}\right)\right)\leq\epsilon.$
and
$\mathbf{P}\left(\max_{i,j}\left|e_{i}^{T}F_{L}P_{L}WP_{R}^{T}F_{R}^{T}e_{j}\right|\geq\left\|W\right\|_{F}\sqrt{\frac{4}{mn}}\log\left(\frac{4mn}{\epsilon}\right)\right)\leq\epsilon.$
That is, with probability at least $1-\epsilon$, multiplying on both sides by
a randomized Hadamard transform or a randomized FFT yields a weight matrix
that is $\mu_{W}$-incoherent, where
$\mu_{W}=2\log\left(\frac{4mn}{\epsilon}\right).$
###### Proof.
From Lemma A.2,
$\mathbf{P}\left(\left|b^{T}USx\right|\geq\left\|b\right\|\left\|x\right\|\sqrt{\frac{2}{n}\log\left(\frac{4mn}{\epsilon}\right)}\right)\leq\frac{\epsilon}{2mn}.$
By applying this once on each side to the rows and columns respectively, and
union bounding over the $mn$ entries, we get
$\mathbf{P}\left(\left|e_{i}^{T}H_{L}S_{L}WS_{R}^{T}H_{R}^{T}e_{j}\right|\geq\left\|W\right\|_{F}\sqrt{\frac{4}{mn}}\log\left(\frac{4mn}{\epsilon}\right)\right)\leq\epsilon.$
The proof in the FFT case is identical. ∎
*
###### Proof.
The incoherence of $H$ follows from the application of Lemma A.3. The
incoherence of $W$ follows from the application of Lemma A.4. ∎
### A.2 Incoherence Processing with the Randomized Fast Fourier Transform
(RFFT)
Algorithm 4 Incoherence Processing with RFFT (IP-RFFT)
0: $W\in\mathbb{R}^{m\times n},H\in\mathbb{R}^{n\times n}$
Sample phase vectors $\theta_{V}\sim\mathcal{U}[0,2\pi]^{n/2}$,
$\theta_{U}\sim\mathcal{U}[0,2\pi]^{m/2}$
$S_{V}=\cos(\theta_{V})+i\sin(\theta_{V})$,
$S_{U}=\cos(\theta_{U})+i\sin(\theta_{U})$
$\hat{W}\leftarrow\texttt{FFT}(diag(S_{U})\texttt{FFT}(diag(S_{V})W^{T})^{T})$
where FFT is the Fast Fourier transform (Sec. A.2)
$\hat{H}\leftarrow\texttt{FFT}(diag(S_{V})\texttt{FFT}(diag(S_{V})H)^{T})$
$\hat{W},\hat{H},S_{U},S_{V}$
Here we described the Randomized Fast Fourier Transform (RFFT), $x\to VSx$
where $V\in\mathbb{C}^{n/2\times n/2}$ is the discrete Fourier transform
matrix, $S\in\mathbb{C}^{n/2}$ is a random complex phase vector, and
$x\in\mathbb{R}^{n}$. The discrete Fourier transform can be computed in
$O(n\log n)$ time via the fast Fourier transform. Here it is understood that
the FFT operates over the reals, in that a vector $x\in\mathbb{R}^{n}$ is
mapped to a complex representation $\mathbb{C}^{n/2}$, the RFFT is performed,
and the resulting vector mapped back to $\mathbb{R}^{n}$. Here the mapping
simply represents reshaping real-valued $x$ into dimension $(n/2,2)$, and
interpreting the corresponding 2-tuples as a complex number.
Incoherence processing via the RFFT achieves similar theoretical guarantees as
the RHT, see Lemmas A.3 and A.4. Ultimately the choice of the orthogonal
transformation is up to the user. A Fourier transform works almost as well as
a Hamard transform in practice (Table 1), so if a fast Hadamard implementation
is not available, the FFT is a good option.
## Appendix B Block LDLQ
###### Lemma B.1.
Let $H\in\mathbb{R}^{nd\times nd}$ be a positive definite matrix with
$d$-block LDL decomposition $H=L^{T}DL$. Then
$\operatorname{tr}\left(D\right)\leq\operatorname{tr}\left(H^{1/2}\right)\cdot\left\|H^{1/2}\odot
M_{D}\right\|_{2},$
where $M_{D}=I\otimes\mathbf{1}_{d\times d}$ is the block diagonal mask. If,
in addition, $H$ is $\mu$-incoherent in the sense that its matrix of
eigenvectors $U$ has
$\|U_{ij}\|\leq\frac{\mu}{\sqrt{nd}},$
then
$\operatorname{tr}\left(D\right)\leq\frac{\mu^{2}}{n}\operatorname{tr}\left(H^{1/2}\right)^{2}.$
###### Proof.
Consider the optimization problem
minimize: $\displaystyle\operatorname{tr}\left(R^{T}HR\right)$ subject to:
$\displaystyle R\text{ unit block lower diagonal}.$
Observe that the derivative of the loss is
$\nabla f(R)=HR.$
If $R=L^{-1}$, then $HR=L^{T}D$. But this must be a block upper triangular
matrix, because it’s the product of a unit upper triangular matrix ($L^{T}$)
and a block diagonal matrix $D$. It follows that $\nabla f(L^{-1})$ is zero in
all the directions in which we could move $R$, since $R$ only varies in the
strictly lower triangular directions. Therefore, $R=L^{-1}$ is the solution to
this optimization problem, and for any $R$, $\nabla f(R)\geq\nabla
f(L^{-1})=\operatorname{tr}\left(D\right)$.
Now, let $M$ denote the strictly block lower triangular mask, and observe that
$M+M^{T}+M_{D}=\mathbf{1}_{nd\times nd}$. Set $\alpha=\left\|H^{1/2}\odot
M_{D}\right\|_{2}^{-1}$, and consider $R=\left(I+\alpha M\odot
H^{1/2}\right)^{-1}$. Observe that
$\displaystyle\left(I+\alpha M\odot H^{1/2}\right)^{T}\left(I+\alpha M\odot
H^{1/2}\right)$ $\displaystyle=I+\alpha M\odot H^{1/2}+\alpha M^{T}\odot
H^{1/2}+\alpha^{2}(M^{T}\odot H^{1/2})(M\odot H^{1/2})$ $\displaystyle\succeq
I+\alpha(M+M^{T})\odot H^{1/2}$ $\displaystyle\succeq\alpha M_{D}\odot
H^{1/2}+\alpha(M+M^{T})\odot H^{1/2}$ $\displaystyle\succeq\alpha H^{1/2}.$
It follows by inverting both sides that $RR^{T}\preceq\alpha^{-1}H^{-1/2}$.
So, for this $R$,
$\operatorname{tr}\left(R^{T}HR\right)=\operatorname{tr}\left(HRR^{T}\right)\leq\alpha^{-1}\operatorname{tr}\left(H^{1/2}\right).$
This proves the first part of the lemma. For the second part, observe that
$\displaystyle\left\|H^{1/2}\odot M_{D}\right\|_{2}$
$\displaystyle\leq\sum_{i=1}^{nd}\lambda_{i}^{1/2}\left\|(u_{i}u_{i}^{T})\odot
M_{D}\right\|_{2}$
$\displaystyle\leq\frac{\mu^{2}}{n}\operatorname{tr}\left(H^{1/2}\right).$
This proves the lemma. ∎
*
###### Proof.
First recall that from the description of block LDLQ,
$\hat{W}_{k}=\mathbf{Q}(W_{k}+(W_{:(k-1)}-\hat{W}_{:(k-1)})\mathbf{A}_{k}).$
We can also write this in matrix form in terms of the matrix $\mathbf{L}_{k}$
as
$\hat{W}=\mathbf{Q}(W+(W-\hat{W})(\mathbf{L}^{T}-I)).$
Here, $\mathbf{Q}$ is interpreted as operating independently block-wise. Let
$\eta$ denote the quantization error
$\eta=(W+(W-\hat{W})(\mathbf{L}^{T}-I))-\mathbf{Q}(W+(W-\hat{W})(\mathbf{L}^{T}-I)).$
Then
$\hat{W}=(W+(W-\hat{W})(\mathbf{L}^{T}-I))-\eta,$
which simplifies to
$(W-\hat{W})\mathbf{L}^{T}=\eta.$
This means that
$\mathbf{E}[\operatorname{tr}((\hat{W}-W)H(\hat{W}-W)^{T})]=\mathbf{E}[\operatorname{tr}((\hat{W}-W)\mathbf{L}^{T}\mathbf{D}\mathbf{L}(\hat{W}-W)^{T})]=\mathbf{E}[\operatorname{tr}(\eta^{T}\mathbf{D}\eta)].$
But by assumption, $\mathbf{E}[\eta\eta^{T}]\preceq m\sigma^{2}I$ (since each
block is just an independent application of $\mathbf{Q}$ and we sum over $m$
rows), so
$\mathbf{E}[\operatorname{tr}((\hat{W}-W)H(\hat{W}-W)^{T})]\leq
m\sigma^{2}\mathbf{E}[\operatorname{tr}(\mathbf{D})].$
Combining this with the result of Lemma B.1 proves the theorem. ∎
## Appendix C E8P details
### C.1 Constructing $S$
We use the following 29 elements of $\hat{D}_{8}$ with norm squared 12 to pad
$S$ to 256 entries.
([3, 1, 1, 1, 3, 3, 3, 3] [1, 3, 1, 1, 3, 3, 3, 3] [1, 1, 3, 1, 3, 3, 3, 3]
[1, 1, 1, 3, 3, 3, 3, 3] [3, 3, 3, 1, 3, 3, 1, 1] [3, 3, 3, 1, 3, 1, 3, 1]
[3, 3, 3, 1, 1, 3, 3, 1] [3, 3, 3, 1, 3, 1, 1, 3] [3, 3, 3, 1, 1, 3, 1, 3]
[3, 3, 3, 1, 1, 1, 3, 3] [3, 3, 1, 3, 3, 3, 1, 1] [3, 3, 1, 3, 3, 1, 3, 1]
[3, 3, 1, 3, 1, 3, 3, 1] [3, 3, 1, 3, 3, 1, 1, 3] [3, 3, 1, 3, 1, 3, 1, 3]
[3, 3, 1, 3, 1, 1, 3, 3] [3, 1, 3, 3, 3, 3, 1, 1] [3, 1, 3, 3, 3, 1, 3, 1]
[3, 1, 3, 3, 1, 3, 3, 1] [3, 1, 3, 3, 3, 1, 1, 3] [3, 1, 3, 3, 1, 3, 1, 3]
[1, 3, 3, 3, 1, 1, 3, 3] [1, 3, 3, 3, 3, 3, 1, 1] [1, 3, 3, 3, 3, 1, 3, 1]
[1, 3, 3, 3, 1, 3, 3, 1] [1, 3, 3, 3, 3, 1, 1, 3] [1, 3, 3, 3, 1, 3, 1, 3]
[1, 1, 3, 3, 1, 3, 3, 3] [3, 3, 1, 1, 3, 3, 3, 1] ) / 2
### C.2 Example Decoding with E8P
Here, we give an example of decoding with E8P. In this example, the first 8
bits of the codeword encode the entry in $S$, the next 7 bits encode the 7
right sign flips, and the last bit encodes whether or not we shift by
$\frac{1}{4}$. Let the codeword be 0001010110010111. The first 8 bits 00010101
= 21 would indicate that we start with the 21st entry in $S$. In this example,
let that be the vector
$s=\left\\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\\},$
which is not in $\hat{D_{8}}$. Thus, $s$ requires an odd number of sign flips
to get into $\hat{D_{8}}$. Then, the next 7 bits 1001011 would indicate that
we need to negate the 1st, 2nd, 4th, and 7th from right bits. Since we need an
odd number of sign flips, the 8th from right bit is also a sign flip. The
sign-decoded vector is then
$\left\\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\\},$
which we can verify is in $E_{8}$. Finally, the last bit 1 indicates that we
need to add $\frac{1}{4}$, so the final decoded vector is
$\left\\{-\frac{1}{4},-\frac{3}{4},\frac{3}{4},\frac{7}{4},-\frac{1}{4},\frac{3}{4},-\frac{1}{4},-\frac{1}{4}\right\\},$
which is in $E_{8}+\frac{1}{4}$ as desired.
### C.3 Why not K-Means?
A significant motivating factor behind E8P is that post-incoherence
processing, entries of $W$ are approximately Gaussian distributed. However,
E8P is uniformly distributed, which raises the question: why not use a K-means
based codebook? K-means based codebooks offer strong theoretical performance
but have a few issues. First, it is difficult to enforce symmetry in a
“learned” K-means codebook. This is crucial to be able to have a compressible
codebook. If we force sign symmetry by learning cluster centers on only the
positive orthant of a $n$-dimensional Gaussian, we can get around this but
sacrifice accuracy at the axis region. Second, using K-means requires storing
a codebook in fp16, whereas the entries of E8P can be stored as 4 bit
integers. This means that during inference, the source codebook for a 8
dimension K-means codebook will be 4 times larger than the source codebook of
E8P, running the risk of a cache eviction. Finally, we observe that
empirically, E8P actually outperforms K-means, which is somewhat interesting
and suggests that allocating more information to the edge of the distribution,
even after incoherence processing, is useful.
## Appendix D Fine-Tuning During Quantization
In Algorithm 5 we describe our fine tuning procedure for QuIP$\\#$.
Algorithm 5 QuIP$\\#$ with Fine-Tuning
0: Unquantized Model $M$, Development Set $\mathcal{D}$, Quantization Order
$O$,
0: Quantized Model $M$
$X\leftarrow M_{\mbox{embedding}}(\mathcal{D})$
$C\leftarrow M(\mathcal{D})_{\mbox{logits}}$
for Decoder Block $D\in M$ do
$Y\leftarrow D(X)$
$X_{train},Y_{train},X_{valid},Y_{valid}\leftarrow\mbox{split}(X,Y)$
for Linear Layer $L\in D$ in order specified by $O$ do
$\hat{L}\leftarrow\mbox{QuIP$\\#$-NoFT}(L)$
Disable gradients for the weight matrix (but not $S_{U},S_{V})$ of $\hat{L}$.
Optimize $D$ to minimize $\mbox{MSE}(D(X_{train}),Y_{train})$ using
$X_{valid},Y_{valid}$ for early stopping.
end for
$X\leftarrow Y$
end for{At this point, the learnable parameters in $M$ are the layernorms, all
$S_{U}$ and $S_{V}$, and the language model head.}
$\mathcal{D}_{train},C_{train},\mathcal{D}_{valid},C_{valid}\leftarrow\mbox{split}(\mathcal{D},C)$
Optimize $M$ to minimize
$\mbox{CrossEntropy}(M(\mathcal{D}_{train}),C_{train})$ using
$\mathcal{D}_{valid},C_{valid}$ for early stopping.
## Appendix E Additional Results
### E.1 QuIP$\\#$ on Mixtral 8x7B (Jiang et al., 2024) and Falcon 180B
(Almazrouei et al., 2023)
Table 6: 2 bit QuIP$\\#$ without fine-tuning on Mixtral 8x7B, a mixture of experts (MoE), and Falcon 180B, a non-Llama-based model. QuIP$\\#$ scales to different architectures without issue. Model | Bits | Wiki2 | C4 | ArcC | ArcE | BoolQ | PiQA | Wino
---|---|---|---|---|---|---|---|---
Mixtral-8x7B | 16 | 3.45 | 6.85 | 0.56 | 0.74 | 0.85 | 0.84 | 0.75
Mixtral-8x7B | 2 | 4.69 | 8.25 | 0.49 | 0.68 | 0.81 | 0.80 | 0.73
Falcon-180B | 16 | 3.30 | 6.31 | 0.61 | 0.82 | 0.87 | 0.85 | 0.81
Falcon-180B | 2 | 4.18 | 7.06 | 0.58 | 0.81 | 0.84 | 0.84 | 0.81
### E.2 Zeroshot performance for ablation on lattice codebooks and fine-
tuning
Table 7: Ablation on lattice codebooks and fine-tuning. QuIP$\\#$ no FT and $E_{8}$ uses the RHT to perform incoherence processing but does not use lattice codebooks or fine-tuning. QuIP$\\#$ No FT uses lattice codebooks but not fine-tuning. QuIP$\\#$ uses lattice codebooks and performs fine-tuning. Model | Method | Bits | | ArcC
---
(acc_norm)
| ArcE
---
(acc_norm)
| BoolQ
---
(acc)
| PiQA
---
(acc_norm)
| Wino
---
(acc)
2-70 | Native | 16 | 48.0 | 59.7 | 76.6 | 80.9 | 76.8
2-70 | QuIP# no FT & no $E_{8}$ | 4 | 49.4 | 60.1 | 77.6 | 80.7 | 76.1
2-70 | QuIP# No FT | 4 | 48.3 | 60.1 | 78.4 | 80.6 | 76.2
2-70 | QuIP# | 4 | 48.3 | 59.4 | 77.4 | 80.7 | 77.1
2-70 | QuIP# no FT & no $E_{8}$ | 3 | 47.4 | 59.1 | 75.8 | 80.9 | 77.5
2-70 | QuIP# No FT | 3 | 47.9 | 59.9 | 78.8 | 79.9 | 77.0
2-70 | QuIP# | 3 | 48.4 | 59.5 | 74.8 | 80.3 | 76.4
2-70 | QuIP# no FT & no $E_{8}$ | 2 | 43.5 | 56.2 | 75.1 | 78.1 | 76.0
2-70 | QuIP# No FT | 2 | 47.2 | 59.5 | 79.1 | 78.6 | 74.2
2-70 | QuIP# | 2 | 47.7 | 59.1 | 80.3 | 79.4 | 75.9
2-13 | Native | 16 | 44.3 | 58.0 | 69.0 | 79.0 | 69.9
2-13 | QuIP# no FT & no $E_{8}$ | 4 | 43.7 | 58.6 | 70.1 | 78.7 | 69.6
2-13 | QuIP# No FT | 4 | 42.9 | 56.4 | 67.8 | 78.9 | 69.9
2-13 | QuIP# | 4 | 44.2 | 57.7 | 69.7 | 78.9 | 69.9
2-13 | QuIP# no FT & no $E_{8}$ | 3 | 42.1 | 55.2 | 70.0 | 77.8 | 69.5
2-13 | QuIP# No FT | 3 | 41.9 | 57.7 | 73.3 | 78.1 | 68.0
2-13 | QuIP# | 3 | 43.3 | 57.7 | 69.8 | 78.4 | 69.1
2-13 | QuIP# no FT & no $E_{8}$ | 2 | 36.3 | 50.8 | 67.4 | 73.4 | 63.1
2-13 | QuIP# No FT | 2 | 37.1 | 50.1 | 66.5 | 75.7 | 63.6
2-13 | QuIP# | 2 | 41.3 | 55.1 | 68.3 | 77.4 | 67.7
2-7 | Native | 16 | 40.6 | 53.5 | 71.0 | 76.9 | 67.0
2-7 | QuIP# no FT & no $E_{8}$ | 4 | 39.5 | 51.9 | 71.3 | 76.6 | 67.3
2-7 | QuIP# No FT | 4 | 40.4 | 53.7 | 68.5 | 77.2 | 67.5
2-7 | QuIP# | 4 | 40.1 | 53.4 | 69.9 | 76.5 | 67.6
2-7 | QuIP# no FT & no $E_{8}$ | 3 | 38.1 | 52.6 | 65.2 | 76.1 | 65.1
2-7 | QuIP# No FT | 3 | 37.7 | 53.1 | 70.6 | 76.7 | 67.6
2-7 | QuIP# | 3 | 39.4 | 53.8 | 69.7 | 76.1 | 66.5
2-7 | QuIP# no FT & no $E_{8}$ | 2 | 29.2 | 42.5 | 63.3 | 68.0 | 59.0
2-7 | QuIP# No FT | 2 | 32.5 | 42.8 | 62.3 | 71.2 | 62.4
2-7 | QuIP# | 2 | 36.1 | 50.5 | 68.3 | 74.9 | 64.9
### E.3 More Scaling Plots
Figure 5: QuIP$\\#$ scaling. (Top Left) Llama 2 Wikitext 2 perplexity vs AQLM.
Context length 4096. QuIP$\\#$ 2 and 3 bit scale better than AQLM 2 and 3 bit.
(Top Right) Llama 2 C4 Perplexity. Context length 4096. (Bottom) Llama 1 C4
Perplexity. Context length 2048.
## Appendix F Implementation Details
This section contains implementation details for our Llama experiments. These
details also mostly apply to the Mixtral and Falcon numbers except we use the
Falcon dataset (Almazrouei et al., 2023) as it is publicly avaiable.
### F.1 Hessian Generation
Hessian matrices $H$ were generated with 6144 sequences of a model’s native
context length (2048 for Llama 1, 4096 for Llama 2) from the RedPajama 1T
(Computer, 2023) dataset.
### F.2 Hadamard Matrices
We use Hadamard matrices available at Neil Sloane’s website (Sloane, ).
### F.3 Perplexity and Zeroshot Evaluation
We use the OPTQ (Frantar et al., 2023) “Wiktext2” and “C4” (not “C4 New”)
sampling functions to calculate perplexity for our experiments. We use LM Eval
(Gao et al., 2023) to calculate zeroshot numbers.
### F.4 Fine Tuning
For the within-transformer block section of fine-tuning, we use the Adam
optimizer (Kingma & Ba, 2017), a learning rate of $5\times 10^{-5}$, batch
size of 8, and sequence length equal to the model’s native context length. We
train on small development dataset of 256 sequences from RedPajama 1T and
validate on 128 sequences. We train for 5 epochs (160 steps) and keep the best
model parameters based on the validation set. For the end to end tuning, we
use the Adam optimizer, a learning rate of $5\times 10^{-5}$, batch size of 1,
sequence length equal to the model’s context length (except for 70B, where we
had to use 3072 to avoid an OOM on our not very well optimized training
script), and the same dataset and epoch setup as before. We observe that
outside of using a low enough learning rate, the other hyperparameters did not
affect fine-tuning much. For the 2 bit models, we used a learning rate of
$5\times 10^{-4}$ for $S_{U}$ and $S_{V}$ ($5\times 10^{-5}$ for everything
else as above) for both the within-block and end to end fine tuning stages.
### F.5 Hardware
All experiments were run on NVIDIA A100 GPUs except for the timing numbers,
which were measured on a NVIDIA RTX 4090 to see what was possible with the
current state-of-the-art NVIDIA consumer GPU. We find that we can quantize
Llama 2 70B without fine tuning in under 10 GPU-hours and with fine tuning in
around 100 GPU-hours. Both numbers do not include Hessian generation, which
can be done once for a model and reused across many different quantization
experiments.
### F.6 Code and Prequantized Models
Our code is available at https://github.com/Cornell-RelaxML/quip-sharp and
prequantized QuIP$\\#$ models are available at https://huggingface.co/relaxml.
## Appendix G Example Generation
Below are some example generations from Llama 2 70B chat quantized with
QuIP$\\#$ to 2 bits, truncated to 256 tokens.
Prompt: How much wood could a woodchuck chuck if a woodchuck could chuck wood?
QuIP$\\#$: The answer to this tongue-twister is a bit of a mystery.
Woodchucks, also known as groundhogs, are rodents that burrow in the ground
and are known for their ability to move large amounts of soil. However, there
is no scientific evidence to suggest that they can actually chuck wood. The
phrase “how much wood could a woodchuck chuck” was first used in a song called
“The Woodchuck’s Song” in the late 19th century. The song was popularized by
the singer and songwriter, Johnny Mercer, and it became a hit in the 1940s.
The song’s lyrics include the line “How much wood could a woodchuck chuck if a
woodchuck could chuck wood?” but they don’t provide an answer. In reality,
woodchucks don’t chuck wood, they dig burrows. They are able to move a lot of
soil and dirt as they dig their burrows, but they don’t have the ability to
move wood. So, the answer to the tongue-
Prompt: Write a song about large language models to the tune of Mariah Carey’s
“All I Want For Christmas Is You” QuIP$\\#$: Verse 1: I don’t want a lot for
Christmas There’s just one thing I’m wishing for A language model that’s large
and bright To help me with my writing, day and night Chorus: All I want for
Christmas is a language model A big, smart, AI that’s my ideal I’ll be happy
with just a few billion parameters All I want for Christmas is a language
model, yeah Verse 2: I’ll be good, I won’t be naughty I’ll use it for good,
not foraughty I’ll generate text that’s nice and neat And make my writing life
a treat Chorus: All I want for Christmas is a language model A big, smart, AI
that’s my ideal I’ll be happy with just a few billion parameters All I want
for Christmas is a language model, yeah Bridge: I’
Prompt: Write some CUDA code that does a matrix multiply QuIP$\\#$: Here is a
simple example of a matrix multiply using CUDA:
#include <cuda.h>
#include <cuda_runtime.h>
__global__ void matrixMultiply(float *A, float *B, float *C, int rowsA, int co
lsA, int rowsB, int colsB) {
int i = blockIdx.x * blockDim.x + threadIdx.x;
int j = blockIdx.y * blockDim.y + threadIdx.y;
if (i < rowsA && j < colsB) {
float sum = 0;
for (int k = 0; k < rowsB; k++) {
sum += A[i + k * colsA] * B[k + j * rowsB];
}
C[i + j * rowsB] = sum;
}
}
int main() {
int rowsA = 3;
int colsA = 3;
int rowsB = 3;
|
University of Sydney] ARC Centre of Excellence in Exciton Science, School of
Chemistry, The University of Sydney, NSW 2006, Australia [Sydney Nano] The
University of Sydney Nano Institute, The University of Sydney, NSW 2006,
Australia INM — Leibniz Institute for New Materials] INM — Leibniz Institute
for New Materials, Campus D2 2, 66123 Saarbrücken, Germany University of
Melbourne] ARC Centre of Excellence in Exciton Science, School of Chemistry,
University of Melbourne, Parkville, Victoria 3010, Australia INM — Leibniz
Institute for New Materials] INM — Leibniz Institute for New Materials, Campus
D2 2, 66123 Saarbrücken, Germany University of Melbourne] ARC Centre of
Excellence in Exciton Science, School of Chemistry, University of Melbourne,
Parkville, Victoria 3010, Australia INM — Leibniz Institute for New Materials]
INM — Leibniz Institute for New Materials, Campus D2 2, 66123 Saarbrücken,
Germany [Saarland University]Colloid and Interface Chemistry, Saarland
University, Campus D2 2, 66123 Saarbrücken, Germany University of Sydney] ARC
Centre of Excellence in Exciton Science, School of Chemistry, The University
of Sydney, NSW 2006, Australia [Sydney Nano] The University of Sydney Nano
Institute, The University of Sydney, NSW 2006, Australia
# When Like Destabilizes Like: Inverted Solvent Effects in Apolar Nanoparticle
Dispersions
Debora Monego [ Thomas Kister [ Nicholas Kirkwood [ David Doblas [
Paul Mulvaney [ Tobias Kraus [ Asaph Widmer-Cooper asaph.widmer-
<EMAIL_ADDRESS>[
## 1 Abstract
We report on the colloidal stability of nanoparticles with alkanethiol shells
in apolar solvents. Small angle X-ray scattering and molecular dynamics
simulations were used to characterize the interaction between nanoparticles in
linear alkane solvents ranging from hexane to hexadecane, including
$4\text{\,}\mathrm{nm}$ gold cores with hexadecanethiol shells and
$6\text{\,}\mathrm{nm}$ cadmium selenide cores with octadecanethiol shells. We
find that the agglomeration is enthalpically driven and that, contrary to what
one would expect from classical colloid theory, the temperature at which the
particles agglomerate increases with increasing solvent chain length. We
demonstrate that the inverted trend correlates with the temperatures at which
the ligands order in the different solvents, and show that the inversion is
due to a combination of enthalpic and entropic effects that enhance the
stability of the ordered ligand state as the solvent length increases. We also
explain why cyclohexane is a better solvent than hexane, despite having very
similar solvation parameters to hexadecane.
## 2 Keywords
nanoparticle, dispersion, apolar, colloidal stability, ligand, solvent,
agglomeration
## 3 Introduction
Inorganic nanoparticles made of metals 1, 2, semiconductors 3, 4, and oxides 5
are now used as functional components in catalysis 6, 7, sensing 8, 9,
photovoltaics 10, 11, and color conversion in white light generation 12, 13,
14. Many applications require the particles to be dispersed individually in
organic solvents or to pass through this stage during processing.
Purely inorganic particles do not form stable dispersions in apolar solvents
because van der Waals (vdW) forces cause attraction and thus agglomeration of
the particles 15. Cores are therefore coated with organic molecules during
synthesis 16, 17, 6, 3 or during subsequent ligand-exchange procedures. The
adsorbed ligands provide steric stabilization and reduce the interfacial
energy of the particles 18.
The colloidal stability of ligand-coated particles in small molecule solvents
is commonly explained with the classical “like dissolves like” rule, whereby
the colloid interaction is assumed to be purely repulsive in solvents that are
good for the tail group of the ligands 19, 20, 21. Reducing the quality of the
solvent, in turn, induces enthalpic attraction between the ligands 22, 23, 24,
25, 26 and is a common way of destabilizing these suspensions 27. Van der
Waals attraction between the cores can also drive agglomeration, even in good
solvents, if the cores are sufficiently large or the ligands are too short 28,
29, 30.
Surprisingly, there appear to be exceptions to the rule of “like dissolves
like” even for small metal and semiconductor particles whose agglomeration
solely depends on the ligand shell 30, 31. Lohman et al. found that gold
nanoparticles with octane- or hexadecanethiol shells were more stable in
alkanes shorter than the ligand chain 32, while Hajiw et al. found that gold
nanoparticles with hexane- or dodecanethiol shells were more stable in
cyclohexane than in heptane or dodecane, respectively 33. In polymer solutions
and melts, where the conformational entropy of free polymer molecules can
drive particle agglomeration 34, 35, colloidal stability does typically
decrease with polymer length 36, 37. However, a purely entropic explanation
seems unlikely in relatively short solvents, like the ones described above,
where the agglomeration is driven by enthalpy.
Here, we have studied the dispersibility of gold and cadmium selenide
nanoparticles in a variety of linear and cyclic alkane solvents using
experiments that characterize their temperature-dependent colloidal stability.
In all cases, colloidal stability decreased as the length of the alkane
solvent approached that of the ligand tail, opposite to the rule of “like
dissolves like”. Further, we found that cyclohexane is a considerably better
solvent for the particles than hexadecane, despite the two solvents having
very similar solvation parameters. Specifically, changing the solvent from
hexadecane to cyclohexane decreased the agglomeration temperature by
$15\text{\,}\mathrm{\SIUnitSymbolCelsius}$. It is important to understand the
origin of such inversions, because the choice of solvents for stable
dispersions of nanoparticles is of considerable practical relevance; it
affects the quality of nanocomposites, nanocrystal assembly 38, and phase
transfer procedures 39, ultimately affecting device processability and
performance.
In order to understand the origin of this behavior, we compared our systematic
experimental data with detailed molecular dynamics simulations. Our results
indicate that the inversion is a consequence of both enthalpic and entropic
effects that together enhance the stability of an attractive ligand state as
the solvent chain length increases. As has been shown previously, ligand
shells composed of linear alkyl tails can undergo an ordering transition in
solution that switches the interaction between the nanoparticles from
repulsive to attractive 25. The temperature of this ligand phase transition is
sensitive to various parameters including the particle dimensions, density of
ligand coverage, and ligand length, often leading to non-linear trends that
cannot be explained using classical colloid theory 40, 30, 31. We show that
even small changes in the solvent structure can strongly impact the ligand
ordering transition and use this insight to explain how particles can have
dramatically different interactions in solvents with almost the same Hamaker,
Hildebrand and Hansen parameters.
## 4 Results and Discussion
Nanoparticles (NP) with Au cores ($4\text{\,}\mathrm{nm}$ and
$7.5\text{\,}\mathrm{nm}$ in diameter) and CdSe cores ($6\text{\,}\mathrm{nm}$
in diameter) were coated with hexadecanethiol (SC_16), dodecanethiol (SC_12),
and octadecanethiol (SC_18) chains, respectively. These particles were
dispersed in linear and cyclic alkane solvents of different chain lengths and
analyzed by in situ small angle X-ray scattering (SAXS) at a concentration of
$2.5\text{\,}\mathrm{mg}\text{\,}{\mathrm{mL}}^{-1}$ (roughly
$3.8\text{\times}{10}^{15}\text{\,}\mathrm{N}\mathrm{P}\mathrm{m}\mathrm{l}^{-1}$).
For all solvents tested, the particles agglomerated below a certain
temperature (Figure 1), which was observed as a peak in the structure factor41
$S(q)$. The agglomeration temperature, $T_{agglo}$, defined as the temperature
at which $20\text{\,}\mathrm{\char 37\relax}$ of the particles had
agglomerated, increased in all cases as the solvent length approached the
ligand length (Figure 1d). Similar results were obtained for
$7.5\text{\,}\mathrm{nm}$ Au cores coated with hexadecanethiol (SC_16) ligands
(see Figure SLABEL:SI_7p5nmSC16 in the Supporting Information). No sign of
solvent freezing was observed in any of the experiments.
Figure 1: Fraction of agglomerated (a) $4\text{\,}\mathrm{nm}$ Au-SC_16, (b)
$6\text{\,}\mathrm{nm}$ CdSe-SC_18, and (c) $7.5\text{\,}\mathrm{nm}$ Au-SC_12
particles, as determined by in situ small angle X-ray scattering. All
particles were dispersed at high temperatures, and agglomeration occurred upon
cooling as indicated by the increase in structure factor. (d) Agglomeration
temperature (where $20\text{\,}\mathrm{\char 37\relax}$ of particles were
agglomerated) as a function of alkane solvent chain length.
These results show that the agglomeration is enthalpically favorable and
entropically unfavorable, with cooling required to destabilize the
dispersions. Extensions of classical colloid theory to describe the colloidal
stability of ligand-coated nanoparticles in solution, for example reference
23, typically include two terms that favor dispersion (the ideal entropy of
mixing and the conformational entropy of the ligands) and two terms that favor
agglomeration (the vdW attraction between the cores and the non-ideal free
energy of mixing). The vdW attraction between the cores is insignificant for
our particles,30 leaving the free energy of mixing as the deciding term in
this classical approach. This can be described by the Flory-Huggins theory
(equations 1 and 2) that quantifies the affinity of the tethered ligands for
the solvent in terms of the ideal entropy of mixing and the Flory $\chi$
parameter (equation 3). In these equations, $d$ is the core diameter,
$\tilde{L}$ is the rescaled ligand length (ligand length divided by core
diameter), $k_{b}$ the Boltzmann’s constant, $T$ the absolute temperature,
$\phi\ =\left(N_{L}\frac{\nu_{L}}{V_{Sh}}\right)^{2}$ is the volume fraction
occupied by the ligand shell, $\nu_{S}$ is the volume of a solvent molecule,
$\nu_{L}$ the volume of a ligand molecule, $N_{L}$ the number of ligands per
nanoparticle, $V_{Sh}$ the volume of the ligand shell, $V_{S}$ is the molar
volume of the solvent, $R$ is the universal gas constant, and $\delta_{L}$ and
$\delta_{S}$ are the Hildebrand solubility parameters for the ligands and
solvent, respectively.
$\frac{G_{mix}}{k_{b}T}=\frac{\pi
d^{3}}{2\nu_{S}}\phi^{2}\left(\frac{1}{2}-\chi\right)\left(\tilde{s}-1-2\tilde{L}\right)^{2};\quad
1+\tilde{L}<x<1+2\tilde{L}$ (1) $\frac{G_{mix2}}{k_{b}T}=\frac{\pi
d^{3}}{\nu_{S}}\phi^{2}\tilde{L}^{2}\left(\frac{1}{2}-\chi\right)\left(3ln\frac{\tilde{L}}{\tilde{s}-1}+2\frac{\tilde{s}-1}{\tilde{L}}-\frac{3}{2}\right);\quad
x<1+\tilde{L}$ (2) $\chi=\frac{V_{S}}{RT}(\delta_{L}-\delta_{S})^{2}+0.34$ (3)
A Flory parameter below $0.5$ indicates that the free energy of mixing of
solvent and ligand is negative and that the two components should
spontaneously mix. Since only the alkane tails of the ligands interact with
the solvent, it seems reasonable to approximate the solubility parameters of
the ligands by those of the unthiolated alkanes, i.e. hexadecane
($16.4\text{\,}{\mathrm{M}\mathrm{Pa}}^{1/2}$) and octadecane
($17.1\text{\,}{\mathrm{M}\mathrm{Pa}}^{1/2}$). Thus, hexadecane is expected
to be a better solvent for these coatings than decane
($\delta_{s}=$15.8\text{\,}{\mathrm{M}\mathrm{Pa}}^{1/2}$$) and hexane
($\delta_{s}=$14.9\text{\,}{\mathrm{M}\mathrm{Pa}}^{1/2}$$) (Hildebrand
parameters from reference 42), even taking into account the reduction in the
ideal entropy of mixing ($\Delta S_{mix}^{ideal}$) as the solvent chain length
increases (see Table SLABEL:SI_Table_SolvParams in the Supporting
Information). A similar conclusion is reached when considering Hamaker
constants or Hansen solubility parameters of the ligand and solvent molecules.
Even if one were to use different solubility parameters for alkyl ligands
bound to nanoparticles, it would remain impossible within this theoretical
framework to explain why cyclohexane is a much better solvent than hexadecane,
since the two have almost identical solubility parameters.
An alternative explanation for deviations from the rule of “like dissolves
like” in short-chain solvents was proposed by Hajiw and co-workers, who
studied the temperature-dependent dispersibility of small Au particles
(roughly $2.3\text{\,}\mathrm{nm}$ in diameter) in a similar range of alkane
solvents.33 They noted that the colloidal stability of polymer-grafted
particles in polymer melts does typically decrease with polymer length and
speculated that similar thermodynamic driving forces may explain such trends
in much shorter solvents. In polymer melts, however, it is the conformational
entropy of the free polymer chains that drives agglomeration. As we shall
show, the solvent conformational entropy ($S_{conf}^{solv}$) in short-chain
solvents is much smaller and unable to explain the observed dispersibility
trends. In order to quantify $S_{conf}^{solv}$, along with enthalpic effects
that are not considered in classical colloid theory, we used molecular
dynamics (MD) simulations (see Methods for details).
MD simulations of $4\text{\,}\mathrm{nm}$ core diameter Au and
$5.8\text{\,}\mathrm{nm}$ CdSe particles in explicit solvent show that upon
cooling, the ligands adopt more extended configurations and cluster together
into ordered bundles in an enthalpically driven process. Snapshots of the
simulations above, at, and below $T_{agglo}$ in the linear alkane solvents are
shown in Figure 2a and in Figure SLABEL:SI_CdSe_snapshots in the Supporting
Information, with solvent molecules hidden for clarity. Similar ligand shell
structures were found for cyclohexane (shown in Figure SLABEL:SI_Cyclohexane
in the Supporting Information). The average dihedral angle of the ligand tails
was used to quantify the ordering transition in the shell, with the results
shown in Figures 2b and 2c. The simulations show that the same ligand shells
“order” at higher temperature in longer alkane solvents for both Au and CdSe
cores. This trend mirrors the experimental results, with the experimentally
observed particle agglomeration temperatures (indicated by large crossed
symbols) always occurring after the ligands have started ordering. These
results indicate that particle agglomeration is driven by the ligand shell
transition regardless of the core material, solvent length or structure,
consistent with our previous findings for similar particles in decane 30 and
with earlier experimental results for much larger silica particles in
hexadecane 44.
Figure 2: (a) Simulation snapshots of $4\text{\,}\mathrm{nm}$ Au particles at
$T_{agglo}$ $\pm$ $30\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in hexane,
decane, and hexadecane. Solvent molecules have been hidden for clarity, with
symbols as shown in the plot legends. The ligands order as the temperature
decreases in a similar way in all solvents. This transition can be quantified
by the average dihedral angle of the ligands, which increases rapidly as they
order for both (b) $4\text{\,}\mathrm{nm}$ Au-SC_16 and (c)
$5.8\text{\,}\mathrm{nm}$ CdSe-SC_18 particles. For comparison, the
experimental agglomeration temperatures have been indicated by large crossed
symbols. The scheme at the bottom left of (c) shows the definition of the
dihedral angle $\phi$.
To explicitly test whether ordering of the ligand shell drives agglomeration
regardless of solvent length, we calculated the potential of mean force
between pairs of $4\text{\,}\mathrm{nm}$ Au particles in explicit hexane and
decane (Figures 3a and 3b, respectively) as a function of separation and
temperature. The overall interaction switched from repulsive to attractive as
the ligands ordered, irrespective of solvent. This change in interaction
between the ligand shells arises from changes in how the ligands interact with
one another and with the solvent as their conformational state changes (see
Figure SLABEL:SI_PMFComponents in the Supporting Information). The ligand-
ligand component (which includes vdW interactions between the ligands) becomes
more attractive as the ligands order, while the ligand-solvent component
(which includes entropy changes involving the solvent) becomes less repulsive.
In contrast, the overall interaction between alkyl ligand shells is always
attractive in the absence of solvent, regardless of their conformational
state, due to the absence of competing ligand-solvent interactions 21,
Waltmann2018, 25, 30. (The relevant results in references 25 and 30 are
located in the Supporting Information of those papers.)
So far, we have established that the nanoparticles agglomerate because the
ligands order and that the enthalpic driving force for both of these processes
is the attractive vdW force that exists between the bound ligands. We now
focus on the thermodynamics of the ligand ordering transition in order to
explain the origin of the inverted trend in agglomeration temperatures. In
particular, we will show that the trend is a consequence of both enthalpic and
entropic effects that together enhance the stability of the attractive ligand
state as the solvent chain length increases. We start by considering a single
nanoparticle and quantifying the dominant enthalpic contributions to the free
energy difference between the disordered and ordered ligand states, i.e. the
vdW interactions between ligand molecules within the same shell ($U_{LL}$) and
between ligand and solvent molecules ($U_{LS}$). These quantities are compared
as a function of temperature and solvent type in Figure 4.
Figure 3: Potentials of mean force calculated for pairs of
$4\text{\,}\mathrm{nm}$ Au-SC_16 particles in (a) hexane and (b) decane, at
temperatures around $T_{agglo}$. Simulation snapshots show the state of the
ligands above and below $T_{agglo}$, at temperatures corresponding to the blue
squares and red triangles, respectively.
We find that the interaction between the ligands follows a similar trend in
all solvents apart from a temperature offset: $U_{LL}$ increases in magnitude
in all cases by
$3\text{\,}$-$4\text{\,}$\mathrm{k}_{\mathrm{b}}\mathrm{T}$\mathrm{/}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{a}\mathrm{n}\mathrm{d}$
during the transition (Figures 4a and 4c). Statistical analysis of the
interatomic spacings within the ligand shell (see Figure
SLABEL:SI_fll_frequency in the Supporting Information) indicate that the
structure of the ligand shell is almost identical regardless of solvent when
the temperature is expressed relative to $T_{order}$, defined as the
temperature at which the average dihedral angle equals $None$ (which
corresponds roughly to the middle of the transition).
In contrast, we find more substantial differences in the interaction between
ligand and solvent molecules. Figures 4b and 4d show that the ligand-solvent
interaction energy ($U_{LS}$) increases during the ordering transition in
hexadecane (blue arrows) but decreases in decane, hexane and cyclohexane (red,
black and green arrows), with the largest decrease in hexane (around
$2\text{\,}$\mathrm{k}_{\mathrm{b}}\mathrm{T}$\mathrm{/}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{a}\mathrm{n}\mathrm{d}$).
This means that the change in internal energy driving the ligand shell to
order is reduced in shorter solvents, which partly explains why the particles
are more stable in hexane and cyclohexane than in hexadecane.
The differences in $\Delta U_{LS}$ during the transition are due to subtle
molecular effects. As the ligands order to form dense bundles, the solvent
molecules within the spherical shell occupied by the ligands become confined
to the spaces between the bundles. This results in a similar increase in the
radially averaged solvent density near the core surface in all solvents
(Figures SLABEL:SI_Densities and SLABEL:SI_Cyclohexane in the Supporting
Information), but with very different consequences depending on how well the
solvent molecules pack with the ligand bundles. Hexane and cyclohexane pack
less well with the ordered ligands, resulting in a loss of ligand-solvent
interaction, while hexadecane packs better with the ordered ligands, resulting
in a gain in ligand-solvent interaction (Figures SLABEL:SI_fls_distribution
and SLABEL:SI_Cyclohexane in the Supporting Information).
Figure 4: The energy of interaction between ligand molecules ($U_{LL}$) on
isolated (a) $4\text{\,}\mathrm{nm}$ Au-SC_16 and (c)
$5.8\text{\,}\mathrm{nm}$ CdSe-SC_18 particles increases upon cooling in all
solvents. In contrast, the energy of interaction between ligand and solvent
molecules ($U_{LS}$) decreases in magnitude for shorter chain alkanes and
increases for hexadecane for both (b) Au and (d) CdSe nanoparticles, as
indicated by the solid arrows. All energies are normalized by the number of
ligand molecules on the nanoparticle, $N_{L}$.
Entropic differences between the solvents appear to play an important role
here. It is well known that the lower freezing points of shorter chain alkanes
are partly due to their higher translational entropy per atom.43 Analysis of
the average dihedral angles also reveals that longer alkanes are more extended
both within and outside the ligand shell, as shown in Figure 5a for the
$4\text{\,}\mathrm{nm}$ Au NPs. Similar results are obtained for the CdSe
particles (see Figure SLABEL:SI_CdSeDihedral in the Supporting Information).
These factors may explain why hexadecane is better able than hexane to align
with the ordered ligands (Figure 5b) and thus increase the relative stability
of the ordered state. We note that close alignment of hexadecane with linear
alkyl ligands has also been observed in sum frequency generation spectroscopy
studies of silica nanoparticles 44.
Figure 5: (a) The average dihedral angle for the solvent chains around AuNP
shows that longer alkanes are more extended at a given temperature, both
within the ligand shell (open symbols) and in the bulk solvent region (closed
symbols). This allows hexadecane to better align with and stabilize the
ligands in the ordered state, as seen in (b) snapshots of the ligand-solvent
packing at $T_{order}$ for $4\text{\,}\mathrm{nm}$ Au-SC_16 particles. Ligand
and solvent united-atoms are represented by blue and white spheres,
respectively. The error bars in (a) are smaller than the symbols, and the
lines are a guide to the eye.
To more directly address how entropy affects the ligand ordering transition,
we quantified the difference (between the ordered and disordered ligand
states) in the ideal entropy of mixing ($\Delta S_{mix}^{ideal}$) and in the
conformational entropies of the ligands ($\Delta S_{conf}^{lig}$) and solvent
($\Delta S_{conf}^{solv}$). $\Delta S_{mix}^{ideal}$ is relevant because
ordering of the ligands causes them to demix from the solvent. The change in
entropy due to this demixing was estimated using equations 1 and 2, setting
$\chi=0$ and multiplying by 5 for reasons explained in the Methods. This
yielded values for $-T\Delta S_{mix}^{ideal}$ ranging from roughly 0.3
$k_{b}T$/ligand for hexane to 0.1 $k_{b}T$/ligand for hexadecane, indicating a
decreasing penalty for ordering as the solvent length increases.
The changes in the conformational entropies were estimated using an
information theoretic approach that is described in detail in the Methods.
Both ordered and disordered configurations were generated at the same
temperature, $T_{order}$, in order to exclude contributions due to temperature
differences. This was achieved by scaling the interaction energies between
non-bonded ligand atoms by $\pm$ 5%, which resulted in ligand configurations
similar to those at $T_{order}\mp$ $30\text{\,}\mathrm{K}$. This yielded
values for $-T\Delta S_{conf}^{lig}$ ranging from $2.2-3$ $k_{b}T$/ligand,
with no apparent trend with solvent length. This substantial penalty is the
main reason why ordering of the ligands has to be enthalpically driven. In
comparison, we obtained values for $-T\Delta S_{conf}^{solv}$ ranging from
$0-0.3$ $k_{b}T$/ligand. Again there was no apparent trend with solvent
length, indicating that the conformational entropy of the solvent makes at
best a small contribution to the trend in agglomeration temperatures.
To facilitate comparison, we estimated the equivalent differences in $U_{LL}$
and $U_{LS}$ at $T_{order}$ via linear extrapolation of the data points
obtained above and below the ordering transition, and collected all of the
enthalpic and entropic terms in Table 1. Also included is the substantial
change in the internal energy associated with the ligand dihedral angles
($\Delta U_{dih}^{lig}$), obtained from the same biased simulations as the
conformational entropies. Together, these values indicate that the main
contribution to the inverted trend in the agglomeration temperatures is the
reduction in $\Delta U_{LS}$ as the solvent length increases, with a minor
contribution from the reduction in $-T\Delta S_{mix}^{ideal}$.
Table 1: Major enthalpic and entropic contributions to the difference in free
energy between the ordered and disordered ligand states, expressed in units of
$k_{b}T$/ligand at the ligand ordering temperature $T_{order}$: $U_{LL}$ is
the vdW interaction between the ligands, $U_{LS}$ is the vdW interaction
between the ligands and the solvent, $U_{dih}^{lig}$ is the internal energy
due to the ligand dihedral angles, $S_{mix}^{ideal}$ is the entropy of
demixing the ligands and solvent, while $S_{conf}^{lig}$ and $S_{conf}^{solv}$
are the conformational entropies of the ligands and solvent, respectively.
Negative quantities favor the ordered state while positive ones favor the
disordered one. The quantities highlighted in green are the only ones that
explain the trend in the agglomeration temperatures. The * indicates a value
that was estimated by comparison with the same results obtained for CdSe.
Standard errors are $\pm 0.3$ or smaller. $4\text{\,}\mathrm{nm}$ Au-SC16
---
Solvent | Hexane | Decane | Hexadecane
$T_{order}$ | 290 | 295 | 300
$\Delta U_{LL}$ | -3.8 | -3.7 | -3.5
[HTML]32CB00 $\Delta U_{LS}$ | 4.0 | 2.3 | 1.0∗
$\Delta U_{dih}^{lig}$ | -2.6 | -2.3 | -1.8
[HTML]32CB00 $-T\Delta S_{mix}$ | 0.32 | 0.22 | 0.14
$-T\Delta S_{conf}^{lig}$ | 2.5 | 3.0 | 2.5
$-T\Delta S_{conf}^{solv}$ | 0.00 | 0.00 | 0.30
$5.8\text{\,}\mathrm{nm}$ CdSe-SC18
Solvent | Hexane | Decane | Hexadecane
$T_{order}$ | 300 | 310 | 320
$\Delta U_{LL}$ | -4.4 | -3.5 | -3.1
[HTML]32CB00 $\Delta U_{LS}$ | 4.3 | 2.6 | 1.3
$\Delta U_{dih}^{lig}$ | -1.9 | -1.8 | -1.3
[HTML]32CB00 $-T\Delta S_{mix}$ | 0.54 | 0.37 | 0.25
$-T\Delta S_{conf}^{lig}$ | 2.2 | 2.8 | 2.2
$-T\Delta S_{conf}^{solv}$ | -0.17 | 0.12 | -0.09
Our results also explain why alkanethiol-coated nanoparticles are more stable
in cyclohexane than in hexadecane, despite the two solvents having almost
identical density and solubility parameters. In cyclohexane, the reduction in
$U_{LS}$ upon ordering is greater than in hexadecane (due to poorer packing
with the extended ligands), while the entropic cost of demixing is also
greater due to cyclohexane’s smaller size and thus higher number density. The
ordered ligand state is therefore less stable (relative to the disordered one)
in cyclohexane than in hexadecane, which suppresses the ordering transition
and results in an agglomeration temperature that is even lower than that in
hexane.
Finally, to check whether $S_{conf}^{solv}$ contributes to the trend in
agglomeration temperatures regardless of the conformational state of the
ligands, we calculated the difference in $S_{conf}^{solv}$ between the
solvated $4\text{\,}\mathrm{nm}$ Au nanoparticles and pure solvent with the
same number of solvent molecules. This yielded a similar value in all solvents
for $-T\Delta S_{conf}^{solv}$ of roughly 0.2 $k_{b}T$/ligand when the ligands
were in the disordered state, indicating no substantial contribution to the
trend.
## 5 Conclusions
We have studied the temperature-dependent agglomeration of small apolar
metallic and semiconducting nanoparticles in a range of alkane solvents. We
found that the agglomeration is enthalpically driven, with colloidal
stabilities that run counter to expectations from classical colloid theory.
Increasing the solvent chain length towards that of the ligands resulted in a
_decrease_ in colloidal stability, rather than the expected increase, and the
colloidal stability differed strongly between cyclohexane and hexadecane
despite their almost identical solvation parameters.
While this behavior is reminiscent of colloidal stability in polymer solutions
and melts, the thermodynamic origins are different, with enthalpic rather than
entropic effects dominating in small organic solvents. Simulations show that
the nanoparticles become attractive to one another as the ligands order and
that the temperatures at which the particles agglomerate match the
temperatures at which the ligands order in the various solvents. This
indicates that the experimental results can be understood by considering the
thermodynamics of the ordering transition and, in particular, how well the
various solvents stabilize the ordered state of the ligands relative to the
disordered one.
We found that the ordering transition is driven by vdW attraction between the
ligand tails, and internal energy associated with their dihedral angles, and
opposed by a combination of enthalpic and entropic terms: loss of vdW
interaction between the ligands and solvent, loss of ligand conformational
entropy, and a reduction in the entropy of mixing. Of these, the loss in vdW
interaction with the solvent exhibits the biggest differences, with smaller
losses observed as the solvent length increases due to better packing with the
ordered ligands. The entropic cost of demixing the ligands from the solvent
also exhibits a small decrease as the size of the solvent molecules increases.
Together, these changes increase the relative stability of the ordered ligand
state when the solvent is changed from hexane or cyclohexane to hexadecane,
which explains the experimental results.
We hope that these results will inspire more detailed experimental studies of
ligand morphology and ligand-solvent interactions, which are now possible due
to recent advances in Sum Frequency Generation spectroscopy 44, Nuclear
Magnetic Resonance 45, 46, and Small Angle Neutron Scattering 47. While we
have not considered polar solvents in the present study, we note that linear
-(CH_2)_nX ligands with a variety of terminal X groups can also order in water
48, 49, raising the possibility of similar effects in polar solvents.
## 6 Methods
All chemicals were obtained from Sigma Aldrich (unless noted otherwise) and
used without further purification. The methods were chosen to provides samples
that are as comparable as possible to data published previously 30, 31.
### 6.1 Nanoparticle synthesis
The synthesis of gold nanoparticles (AuNP) with core diameters of
$4\text{\,}\mathrm{nm}$ and $7.5\text{\,}\mathrm{nm}$ was adapted from a
previously described method 30. For AuNPs with a core diameter of
$4\text{\,}\mathrm{nm}$ a mixture of pentane ($\geq$ 98.5%, GC),
$8\text{\,}\mathrm{mL}$ oleylamine (technical grade, 70%), and
$100\text{\,}\mathrm{mg}$ of HAuCl4 (with crystal water) was stirred at
$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$500\text{\,}\mathrm{rad}\text{\,}{\mathrm{min}}^{-1}$ for
$10\text{\,}\mathrm{min}$ under argon atmosphere. Afterwards a solution of
$40\text{\,}\mathrm{mg}$ tert-butylamine borane (ABCR, 97%) in
$2\text{\,}\mathrm{mL}$ pentane and $2\text{\,}\mathrm{mL}$ oleylamine was
added. The color of the solution immediately changed. After stirring for
$60\text{\,}\mathrm{min}$ at $20\text{\,}\mathrm{\SIUnitSymbolCelsius}$, the
nanoparticles were purified once by precipitating with
$30\text{\,}\mathrm{mL}$ ethanol and centrifugation at
$4000\text{\,}\mathrm{rad}\text{\,}{\mathrm{min}}^{-1}$ for
$5\text{\,}\mathrm{min}$. The precipitated nanoparticles were then redispersed
in $20\text{\,}\mathrm{mL}$ of the appropriate solvent. Gold cores with a
diameter of $7.5\text{\,}\mathrm{nm}$ were produced in benzene instead of
pentane. The mixture of benzene, oleylamine and HAuCl4 was stirred for
$1\text{\,}\mathrm{min}$ before tert-butylamine borane was added. The
resulting dispersion was then purified as above.
Cadmium selenide nanoparticles (CdSeNPs) with core diameters of
$6\text{\,}\mathrm{nm}$ were synthesized as follows. First, three stock
solutions were prepared, a Se injection solution (i), a Cd growth solution
(ii), and a Se growth solution (iii): (i) $0.3265\text{\,}\mathrm{g}$ Se were
dissolved in a mixture of $2.5\text{\,}\mathrm{g}$ trioctylphosphine,
$2.5\text{\,}\mathrm{g}$ octadecene, and $6\text{\,}\mathrm{g}$ oleylamine in
a nitrogen-filled glovebox to give a clear, slightly yellow solution. (ii) A
solution containing $0.17\text{\,}\mathrm{mol}\text{\,}{\mathrm{L}}^{-1}$ of
cadmium was made from $0.22\text{\,}\mathrm{g}$ cadmium oxide,
$0.97\text{\,}\mathrm{g}$ oleic acid, and $6.23\text{\,}\mathrm{g}$
1-octadecene in a 3 neck round bottom flask on a Schlenk line. The solution
was degassed under vacuum ($<$ $1\text{\,}\mathrm{mbar}$) for
$60\text{\,}\mathrm{min}$ at $80\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
heated to $250\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and held until clear,
then cooled to room temperature. Whilst cooling $1.13\text{\,}\mathrm{mL}$ of
oleylamine were added. The final solution was clear and slightly yellow. (iii)
A solution containing $1.7\text{\,}\mathrm{mol}\text{\,}{\mathrm{L}}^{-1}$ of
selenium was prepared by dissolving $0.25\text{\,}\mathrm{g}$ of selenium in
$1.55\text{\,}\mathrm{g}$ trioctylphosphine in a nitrogen-filled glovebox to
give a clear colourless solution.
The synthesis started with $0.22\text{\,}\mathrm{g}$ cadmium oxide,
$3\text{\,}\mathrm{g}$ oleic acid, and $30\text{\,}\mathrm{g}$ octadecene in a
3 neck round bottom flask that was degassed under vacuum ($<$
$1\text{\,}\mathrm{mbar}$) for $60\text{\,}\mathrm{min}$ at
$80\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The mixture was then heated to
$260\text{\,}\mathrm{\SIUnitSymbolCelsius}$ until a clear solution (iv) had
formed. The selenium injection solution (i) was loaded into a
$24\text{\,}\mathrm{mL}$ disposable syringe equipped with a 16 G needle and
rapidly injected into the cadmium solution (iv) at
$260\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The temperature of the reaction
solution was allowed to recover to $250\text{\,}\mathrm{\SIUnitSymbolCelsius}$
where it was held for NP growth. After $20\text{\,}\mathrm{min}$,
$2\text{\,}\mathrm{mL}$ of
$0.17\text{\,}\mathrm{mol}\text{\,}{\mathrm{L}}^{-1}$ cadmium growth stock
(ii) and $0.2\text{\,}\mathrm{mL}$ of
$1.7\text{\,}\mathrm{mol}\text{\,}{\mathrm{L}}^{-1}$ selenium growth stock
(iii) were added dropwise to the reaction. The addition of cadmium and
selenium growth solutions (ii, iii) was continued every
$10\text{\,}\mathrm{min}$. After 3 additions, the reaction was left for a
further $10\text{\,}\mathrm{min}$ at
$250\text{\,}\mathrm{\SIUnitSymbolCelsius}$, then cooled to room temperature.
The NPs were washed three times via precipitation with acetone and resuspended
in toluene.
### 6.2 Nanoparticle characterization
Small Angle X-ray Scattering (Xenocs Xeuss 2.0) and Transmission Electron
Microscopy (JEOL JEM 2010) were used to measure the core size of the NPs as
previously described30. Scattering data from SAXS was analyzed using SASfit
(Version 0.94.6, Paul Scherrer Institute) and TEM micrographs were analyzed
using ImageJ distributed by NIH (Version 1.45s).
Table 2: NPs used for this study, with diameters obtained from transmission electron microscopy and small angle X-ray scattering. Number | d (TEM) | d (SAXS)
---|---|---
Au01 | 4.1 nm $\pm$ $10.0\text{\,}\mathrm{\char 37\relax}$ | 4.1 nm $\pm$ $9.3\text{\,}\mathrm{\char 37\relax}$
Au02 | 7.4 nm $\pm$ $7.4\text{\,}\mathrm{\char 37\relax}$ | 7.5 nm $\pm$ $6.7\text{\,}\mathrm{\char 37\relax}$
CdSe | 5.8 nm $\pm$ $7.1\text{\,}\mathrm{\char 37\relax}$ | 6.0 nm $\pm$ $9.6\text{\,}\mathrm{\char 37\relax}$
### 6.3 Ligand exchange
AuNPs. Ligand exchange on AuNPs was performed as described previously 50.
AuNPs coated with oleylamine were heated to
$80\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and an excess of required
alkanethiol was added. After stirring for further $10\text{\,}\mathrm{min}$,
the particles were purified and redispersed in the appropriate solvent.
CdSeNPs. As-synthesized CdSeNPs were precipitated with acetone/ethanol and
resuspended in a solution of the respective alkanethiol ligand (40 wt-$\%$ in
chloroform) with triethylamine (1 molar equivalent with respect to thiol). The
resulting NP dispersion was heated for 3 hours at
$45\text{\,}\mathrm{\SIUnitSymbolCelsius}$ while stirring. The NPs were then
washed via precipitation with antisolvent and centrifugation (3,300 x g for
$3\text{\,}\mathrm{min}$). The antisolvent was chosen to optimally dissolve
excess ligand: 1:1 (v/v) methanol/ethanol mixture for hexanethiol and
octanethiol ligands, or 1:1 (v/v) acetone/ethanol mixture for dodecanethiol
and longer ligands. The NPs were resuspended again in a solution of ligand (40
wt-$\%$ in chloroform), stirred at $45\text{\,}\mathrm{\SIUnitSymbolCelsius}$
for 2 hours, then washed as before and resuspended in a 0.1 M solution of
ligand in chloroform. After stirring at room temperature for 24-48 hours the
NPs were washed three times and resuspended in pure chloroform. Chambrier et
al. have shown this procedure leads to almost complete displacement ($>$ 92%)
of amines by the alkane thiol ligands 51.
### 6.4 Small Angle X-ray Scattering
Experiments were performed under vacuum using a Xeuss 2.0 from Xenocs SA
(Grenoble, France) equipped with a copper $K_{\alpha}$ X-ray source and a
PILATUS 1M detector from DECTRIS (Baden, Switzerland) using a sample-to-
detector distance of $1235\text{\,}\mathrm{mm}$.
To prevent solvent evaporation during the measurements, the samples (usually a
quantity of $20\text{\,}\mathrm{\SIUnitSymbolMicro l}$ to
$40\text{\,}\mathrm{\SIUnitSymbolMicro l}$) were filled into thin-wall glass
capillaries (diameter of $2\text{\,}\mathrm{mm}$) and sealed with epoxy resin.
For each measurement, the samples were introduced into a temperature
controlled sample holder (Omega CN8200), Peltier-controlled with a temperature
range between $-20\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$120\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The measurements started at high
temperature to ensure a fully deagglomerated state. Afterwards the temperature
was first decreased and later increased in
$5\text{\,}\mathrm{\SIUnitSymbolCelsius}$ steps. At each step, the samples
were first equilibrated ($20\text{\,}\mathrm{min}$) followed by an exposition
of $10\text{\,}\mathrm{min}$. Data treatment was carried out as described
elsewhere 52, 30.
### 6.5 Molecular dynamics simulations
Molecular dynamics (MD) simulations with periodic boundary conditions were
used to study $4\text{\,}\mathrm{nm}$ Au nanoparticles and
$5.8\text{\,}\mathrm{nm}$ CdSe nanoparticles covered in 1-hexadecanethiol and
1-octadecanethiol ligands, respectively, in the presence of a variety of
liquid alkanes (n-hexane, n-decane, n-hexadecane and cyclohexane). The sulfur
atoms from the ligands were randomly placed on a spherical shell around the
implicit core ($0.15\text{\,}\mathrm{nm}$ further out) and allowed to find
their optimal positions on this shell while subject to a Coulombic interaction
with relative permittivity $\epsilon$ = $10\text{\,}$ and the RATTLE
constraint53. This produced a shell with approximately equidistant binding
sites, with the sulfur atoms subsequently treated as part of the rigid core of
the particle. The ligands were irreversibly bound to the Au and CdSe cores at
a surface coverage of
$5.5\text{\,}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{s}\mathrm{n}\mathrm{m}^{-2}$,
consistent with thermogravimetric analyses of the experimental samples31. CH_x
groups from ligand and solvent molecules were treated as united atoms and
interacted with one another according to the 12-6 Lennard-Jones (LJ) potential
and with the implicit cores through a 9-3 LJ potential, as previously employed
and described for similar particles25, 54, 31. Additionally, bond stretching,
bond bending, and dihedral torsion terms were considered within each molecule,
with parameters taken from the TraPPE force field family55.
Simulations of systems with up to 165,000 particles were performed using the
LAMMPS molecular dynamics simulation package56, at temperatures ranging from
roughly the freezing point of the respective solvent to values sufficiently
high to have the ligands in the disordered state (up to
$340\text{\,}\mathrm{K}$). Periodic simulation cells containing individual
nanoparticles and the alkane solvent were slowly compressed until the solvent
density far from the NP was equal to the experimental density of the pure
solvent at the chosen temperature. A preliminary run was performed at constant
volume in order to accommodate the particles properly in the simulation cell.
The systems were then equilibrated at constant pressure
($80\text{\,}\mathrm{atm}$) and temperature, maintained with a Nosé-Hoover
thermostat and barostat, for at least $12\text{\,}\mathrm{ns}$. Finally, the
relevant data were accumulated and averaged over production runs of
$1\text{\,}\mathrm{ns}$. Average bulk solvent densities for these runs stayed
within $1\text{\,}\mathrm{\char 37\relax}$ of experimental values for linear
alkanes, and $5\text{\,}\mathrm{\char 37\relax}$ for cyclohexane. Molecular
graphics were produced using Visual Molecular Dynamics (VMD)57.
#### 6.5.1 Potentials of mean force
The change in free energy as a pair of $4\text{\,}\mathrm{nm}$ Au particles
was brought together was calculated in both hexane and decane as a potential
of mean force (PMF) using constrained MD. Starting from a non-interacting
separation, the particles were brought together at a rate of
$1\text{\,}\mathrm{\SIUnitSymbolAngstrom}\text{\,}{\mathrm{ns}}^{-1}$. The
particles were allowed to rotate about their centers of mass at each
separation $r$, and subsequent simulations of $10\text{\,}\mathrm{ns}$ or more
were performed in order to adequately sample the PMF. Longer runs were
necessary particularly at and below $T_{order}$, where the ligands were less
mobile. Additionally, in order to allow the ligands to reorganize and find
more stable configurations at these temperatures, we included a thermal
annealing step at separations where the ligand shells overlapped. This was
done by increasing the temperature of these systems by $50\text{\,}\mathrm{K}$
over $1\text{\,}\mathrm{ns}$ and subsequently cooling it back to the initial
temperature over the course of $3\text{\,}\mathrm{ns}$.
The spherical gold cores were assumed to interact with each other via the
Hamaker potential 58, with a Hamaker constant of $2\text{\,}\mathrm{eV}$ 59.
This approach treats the solvent and ligands as a single continuum, with the
interaction constant scaled to include the effect of the hydrocarbon medium.
The PMF between two nanoparticles is given by
$\phi_{MF}(r)=\int_{r}^{\infty}F_{mean}(s)ds$ (4)
Where $F_{mean}$ is the average force in the direction of the line connecting
the two particles and is given by
$F_{mean}(r)=\frac{1}{2}\langle(\vec{F}_{2}-\vec{F}_{1})\cdot\vec{r}\rangle_{NVT}$
(5)
In the above, $\vec{F}_{1}$ and $\vec{F}_{2}$ are the total forces acting on
the first and second NP, respectively, $\vec{r}$ is the unit vector pointing
from one particle’s center to the other’s, and the angular brackets denote an
average in the canonical ensemble.
#### 6.5.2 Entropy of mixing
The change in the ideal entropy of mixing due to spatially separating the
ligand and solvent molecules was estimated using equations 1 and 2 by setting
$\chi=0$ and calculating the value at the average core spacing in the
experimental agglomerates ($2.4\text{\,}\mathrm{nm}$). This value was then
multiplied by 5 to roughly convert the entropy change for a pair of touching
particles into the entropy change for an entire particle capable of
accommodating approximately 10 nearest neighbors. The other parameters used
are listed in Table SLABEL:tab:mixing_parameters in the Supporting
Information.
#### 6.5.3 Conformational entropies
We employed the correlation corrected multibody local approximation (CC-MLA),
as implemented in the software CENCALC Suarez2013, to estimate changes in the
conformational entropies of the ligand and solvent molecules. The molecular
conformational space was represented in terms of the dihedral angles, which
were discretized into subintervals delimiting the three locally stable
conformational states accessible to them, i.e. trans, gauche(-), and
gauche(+). This transforms the continuous random variable $\theta$ into the
discrete random variable $X$, with a probability mass function $P(X)$, where
$X=\left\\{X_{1},...,X_{M}\right\\}$ and $M$ is the number of dihedral angles.
The entropy can then in principle be calculated as a sum over the $N$ possible
configurations of the system using the expression for the Shannon information
entropy:
$\centering
S_{conform}=-k_{b}\sum^{N}P\left(X\right)\ln{P\left(X\right)}.\@add@centering$
(6)
In practice, obtaining an accurate estimate for $P(X)$ is difficult, because
the number of possible conformers is very large ($\sim 3^{M}$). Direct
application of the Shannon expression would also result in large and
negatively-biased entropies due to correlations between the dihedral angles.
However, an approximation to the total entropy can be obtained by truncating a
mutual information expansion (MIE)60, allowing the Shannon information entropy
to be calculated using a reasonable number of states.
In addition, due to the large number of molecules in our system, each molecule
was treated as an independent system; i.e., a nanoparticle coated with $l$
ligands, each with $d$ dihedral angles, was analysed as $l$ independent
$d$-dimensional spaces. The sum of the entropies of the individual ligands
then provided an approximation for the entropy of the entire nanoparticle.
This approach considered the correlations within each molecule, but ignored
correlations between neighbor molecules, which are stronger for ligands in the
ordered state. The conformational entropy differences that we report for the
ligands therefore represent a lower bound, calculated using fully converged
values from data sampled every $1\text{\,}\mathrm{ps}$ over
$5.5\text{\,}\mathrm{ns}$. While we were not able to fully converge the
absolute solvent entropies using the same number of data points, the entropy
differences converge more rapidly and do appear to be fully converged.
P.M., N.K., D.M. and A.W. were supported by the ARC Centre of Excellence in
Exciton Science (CE170100026). A.W. thanks the Australian Research Council for
a Future Fellowship (FT140101061), and D.M. thanks the University of Sydney
Nano Institute for a Postgraduate Top-Up Scholarship and the Australian
Nanotechnology Network for an Overseas Travel Fellowship. Computational
resources were generously provided by the University of Sydney HPC service,
the National Computational Infrastructure National Facility (NCI-NF) Flagship
program, and the Pawsey Supercomputer Centre Energy and Resources Merit
Allocation Scheme. T.K., D.D. and T.K. thank the DFG Deutsche
Forschungsgemeinschaft for funding. P.M. and T.K. also thank the DAAD for
travel support.
## Supporting Information Available
Supporting Information shows SAXS data, fraction of agglomerated
$7.5\text{\,}\mathrm{nm}$ Au-SC_16 particles, simulation results for CdSe-
SC_18 particles dispersed in linear alkanes and for Au-SC_16 particles in
cyclohexane, radial density distributions of ligand and solvent, individual
contributions to the the PMF (in hexane), and number of ligand-ligand
interactions and ligand-solvent interaction energy as a function of the
separation between interacting pairs. This material is available free of
charge via the Internet at http://pubs.acs.org/.
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|
# Fast and Exact Enumeration of Deep Networks Partitions Regions
###### Abstract
One fruitful formulation of Deep Networks (DNs) enabling their theoretical
study and providing practical guidelines to practitioners relies on Piecewise
Affine Splines. In that realm, a DN’s input-mapping is expressed as per-region
affine mapping where those regions are implicitly determined by the model’s
architecture and form a partition of their input space. That partition –which
is involved in all the results spanned from this line of research– has so far
only been computed on $2/3$-dimensional slices of the DN’s input space or
estimated by random sampling. In this paper, we provide the first parallel
algorithm that does exact enumeration of the DN’s partition regions. The
proposed algorithm enables one to finally assess the closeness of the commonly
employed approximations methods, e.g. based on random sampling of the DN input
space. One of our key finding is that if one is only interested in regions
with “large” volume, then uniform sampling of the space is highly efficient,
but that if one is also interested in discovering the “small” regions of the
partition, then uniform sampling is exponentially costly with the DN’s input
space dimension. On the other hand, our proposed method has complexity scaling
linearly with input dimension and the number of regions.
## 1 Introduction
Fig. 1: Proposed exact region enumeration depicted as an orange star against
sampling-based region discovery of the partition $\Omega$ depicted as blue
dots for a single hidden layer DN with leaky-ReLU, random parameters and width
$64$ as a function of computation time (x-axis) and number of partition
regions found (y-axis); for a $4$-dimensional input space at the top and
$8$-dimensional input space at the bottom. The proposed Algorithm 1 is able to
dramatically outperform the sampling-based search that has been used
throughout recent studies on CPA DNs.
Deep Networks (DNs) are compositions of linear and nonlinear operators
altogether forming a differentiable functional $f_{{\bm{\theta}}}$ governed by
some trainable parameters ${\bm{\theta}}$ [1]. Understanding the underlying
properties that make DNs the great function approximators that they are
involve many different research directions e.g. the underlying implicit
regularization of architectures [2], or the impact of input and feature
normalization into the optimization landscape [3]. Most existing results
emerge from a few different mathematical formulations. One eponymous example
relies on kernels and emerges from pushing the DN’s layers width to infinity.
In this case, and under some additional assumptions, a closed-form expression
for the DN’s underlying embedding space metric is obtained [4]. With that
form, training dynamics and generalization bounds are amenable to theoretical
analysis [5]. Another line of research considers the case of deep linear
networks i.e. a DN without nonlinearities. In this setting, it is possible to
obtain the explicit regularizer that acts upon the DN’s functional and that
depends on the specifics of the architecture e.g. depth and with [6]. Another
direction, most relevant to our study, proposes to unravel the Continuous
Piecewise Affine (CPA) mapping of standard DNs [7]. In short, one can combine
the fact that (i) the nonlinearities present in most current DNs are
themselves CPA e.g. (leaky-)ReLU, absolute value, max-pooling, (ii) the
interleaved affine mappings preserve the CPA property, and (iii) composition
of CPA mappings remain CPA. Thus, the entire input-output DN is itself a CPA.
From that observation, it is possible to study the DN’s loss landscape [8],
the implicit regularizer of different architectures [9], the explicit
probabilistic distributions of CPA Deep Generative Networks [10, 11], the
approximation rates [12, 13], or even the conditions for adversarial
robustness guarantees [14, 15]. A striking benefit of the CPA viewpoint lies
in the fact that it is an exact mathematical description of the DN’s input-
output mapping without any approximation nor simplification. This makes the
obtained insights and guidelines highly relevant to improve currently deployed
state-of-the-art architectures.
Despite this active recent development of CPA-based results around DNs, one
key challenge remains open. In fact, because under this view one expresses the
DN mapping as a collection of affine mappings –one for each region $\omega$ of
some partition $\Omega$ of their input space– it becomes crucial to compute
that partition $\Omega$ or at least infer some statistics from it. Current
analytical characterizations of $\Omega$ are in fact insufficient e.g.
existing bounds characterizing the number of regions in $\Omega$ are known to
be loose and uninformative [16]. As such, practitioners resort to simple
approximation strategies, e.g. sampling, to estimate such properties of
$\Omega$. Another approach is to only consider $2/3$-dimensional slices of the
DN’s input space and estimate $\Omega$ restricted on that subspace. All in
all, nothing is known yet about how accurate are those approximations at
conveying the underlying properties of the entire partition $\Omega$ that
current theoretical results heavily rely on. In particular, [17] uses
estimates of the partition’s number of region to perform Neural Architecture
Search (NAS), and for which exact computation of the DNN’s partition regions
will further improve the NAS; [11] uses estimates of the partition to adapt
the distribution of deep generative networks (e.g. variational autoencoders)
and for which exact computation of the partition would make their method
exact, and not an approximation
In this paper, we propose a principled and provable enumeration method for DNs
partitions (Algorithm 1) that we first develop for a layer-wise analysis in
Section 2 and then extend to the multilayer case in Section 3. As depicted in
Fig. 1, the proposed method becomes exponentially faster than the sampling-
based strategy to discover the regions $\omega\in\Omega$ as the input
dimensionality increases. Practically, the proposed enumeration method enables
for the first time to measure the accuracy of the currently employed
approximations. Our method is efficiently implemented with a few lines of
codes, leverages parallel computations, and provably enumerates all the
regions of the DN’s partition. Lastly, our method has linear asymptotic
complexity with respect to the number of regions and with respect to the DN’s
input space dimension. This property is crucial as we will demonstrate that
sampling-based enumeration method has complexity growing exponentially with
respect to the DN’s input space dimension as a direct consequence of the curse
of dimensionality [18, 19]. We hope that our method will serve as the baseline
algorithm for any application requiring provable partition region enumeration,
or to assess the theoretical findings obtain from the CPA formulation of DNs.
## 2 Enumeration of Single-Layer Partitions
We now develop the enumeration algorithm for a single DN layer. Because a DN
recursively subdivides the per-layer partition, the single layer case will be
enough to iteratively compute the partition of a multilayer DN as shown in the
next Section 3.
### 2.1 Layer Partitions and Hyperplane Arrangements
We denote the single layer of a DN111without loss of generality we consider
the first layer, although the exact same analysis applies to any layer in the
DN when looking at the partition of its own input space input-output mapping
as $f_{{\bm{\theta}}}:\mathbb{R}^{D}\mapsto\mathbb{R}^{K}$, with
${\bm{\theta}}$ the parameters of the mapping. Without loss of generality, we
consider vectors as inputs since when dealing with images, one can always
flatten them into vectors and reparametrize the layer accordingly. The layer
mapping takes the form
$\displaystyle f_{{\bm{\theta}}}({\bm{x}})=\sigma({\bm{h}}({\bm{x}}))\text{
with }{\bm{h}}({\bm{x}})={\bm{W}}{\bm{x}}+{\bm{b}}$ (1)
where $\sigma$ is a pointwise activation function, ${\bm{W}}$ is a weight
matrix of dimensions $K\times D$, ${\bm{b}}$ is a bias vector of length $K$,
${\bm{h}}({\bm{x}})$ denotes the pre-activation map and lastly ${\bm{x}}$ is
some input from $\mathbb{R}^{D}$. The layer parameters are thus
${\bm{\theta}}\triangleq\\{{\bm{W}},{\bm{b}}$. Although simple, Eq. 1
encompasses most current DNs layers by specifying the correct structural
constraints on the matrix ${\bm{W}}$, e.g. to be circulant for a convolutional
layer. The details on the layer mapping will not impact our results. The CPA
view of DNs [20, 7] consists in expressing Eq. 1 as
$\displaystyle
f_{{\bm{\theta}}}({\bm{x}})=\sum_{\omega\in\Omega}({\bm{A}}_{\omega}{\bm{x}}+{\bm{b}}_{\omega})1_{\\{{\bm{z}}\in\omega\\}},$
(2)
where $\Omega$ is the layer input space partition [21]. Understanding the form
of $\Omega$ will greatly help the development of the enumeration algorithm in
Section 2.2. Given nonlinearities $\sigma$ such as (leaky-)ReLU or absolute
value, it is direct to see that the layer stays linear for a region $\omega$
so that all the inputs within it have the same pre-activation signs. That is,
a region is entirely and uniquely determined by those sign patterns
$\displaystyle f_{{\bm{\theta}}}\text{ affine on
$\omega$}\iff\operatorname{sign}({\bm{h}}({\bm{x}}))=\operatorname{sign}({\bm{h}}({\bm{x}}^{\prime})),\forall({\bm{x}},{\bm{x}}^{\prime})\in\omega^{2},$
where the equality is to be understood elementwise on all of the $K$ entries
of the sign vectors. The only exception arises for degenerate weights
${\bm{W}}$ which we do not consider since any arbitrarily small perturbation
of such degeneracies remove those edge cases. From the above observation
along, it becomes clear that the transition between different regions of
$\Omega$ must occur when a pre-activation sign for some unit
$k\in\\{1,\dots,K\\}$ changes, and because ${\bm{h}}$ is nothing more but an
affine mapping, this sign change for some unit $k$ can only occur when
crossing the hyperplane
$\displaystyle{\mathbb{H}}_{k}\triangleq\\{{\bm{x}}\in\mathbb{R}^{D}:\langle{\bm{W}}_{k,.},{\bm{x}}\rangle+{\bm{b}}_{k}=0\\}.$
(3)
Leveraging Eq. 3 we obtain that $\partial\Omega$, the boundaries of the
layer’s partition, is an hyperplane arrangement as in
$\partial\Omega=\bigcup_{k=1}^{K}{\mathbb{H}}_{k}$.
We are now able to leverage this particular structure of the layer’s partition
to present an enumeration algorithm that will recursively search for all the
regions $\omega\in\Omega$.
### 2.2 Region Enumeration Algorithm
From the previous understanding that the layer’s partition arises from an
hyperplane arrangements involving Eq. 3, we are now able to leverage and adapt
existing enumeration methods for such partitions to obtain all the regions
$\omega\in\Omega$, form which it will become trivial to consider the
multilayer case that we leave for the following Section 3.
Enumerating the regions of the layer $f_{{\bm{\theta}}}$’s partition can be
done efficiently by adapting existing reverse search algorithms [22] optimized
for hyperplane arrangements. In fact, a naive approach of enumerating all of
the $2^{K}$ possible sign patterns ${\bm{q}}\in\\{-1,1\\}^{K}$ and checking if
each defines a non-empty region
$\displaystyle\bigcap_{k=1}^{K}\left\\{{\bm{x}}\in\mathbb{R}^{D}:\left(\langle{\bm{W}}_{k,.},{\bm{x}}\rangle+{\bm{b}}_{k}\right){\bm{q}}_{k}\geq
0\right\\}\overset{?}{=}\emptyset,$
would be largely wasteful. In fact, most of such sign combinations do produce
empty regions e.g. if the partition is central i.e. the intersection of all
the hyperplane is not empty then the total number of regions grows linearly
with $K$ [23] and is thus much smaller than $2^{K}$. Instead, a much more
efficient strategy is to only explore feasible sign patterns in a recursive
tree-like structure. To do so, the algorithm recursively sub-divides a parent
region by the hyperplane of unit $k$. If that hyperplane does not intersect
the current region then we can skip unit $k$ and recurse the sub-division of
that same region by unit $k+1$. On the other hand, if hyperplane $k$ divides
the current region, we consider both sides of it and keep the recursion going
on both sides. We formally summarize the method in Algorithm 1 and present one
illustrative example and comparison against sampling-based region enumeration
in Fig. 1. In particular, we provide the efficiency of the sampling solution
for various configurations in Table 1.
Algorithm 1 Proposed region enumeration method for the single hidden layer
case that recursively searches over the feasible sign patterns ${\bm{q}}$ one
unit at a time, and only explores the branches that coincide with non-empty
region i.e. avoiding the $2^{K}$ total number of possible of combinations. The
step checking for intersection between an hyperplane and a given polytopal
region can be done easily by setting up a linear program with dummy constant
objective, the hyperplane as a linear constraint, and the polytopal region as
inequality constraint; during the feasibility check the test will fail if the
intersection is empty. This algorithm is obtained to provide the results from
Figs. 1 and 1. The algorithm terminates once all the regions of the partition
$\Omega$ have been visited.
1:${\bm{W}}\in\mathbb{R}^{K\times
D},{\bm{b}}\in\mathbb{R}^{K},k\in\\{1,\dots,K\\},{\bm{q}}\in\\{-1,0,1\\}^{k}$
2:if ${\bf k\overset{?}{=}K+1}$ then this branch has reached a leaf, the sign
pattern ${\bm{q}}$ is feasible and can be accumulated into $\Omega$’s current
estimate
3:Check if the hyperplane defined by $({\bm{w}}_{k},{\bm{b}}_{k})$ intersects
the polytopal region defined by
$\bigcap_{j=1}^{k-1}\\{{\bm{x}}\in\mathbb{R}^{D}:(\langle{\bm{w}}_{j},{\bm{x}}\rangle+{\bm{b}}_{j}){\bm{q}}_{j}\geq
0\\}$
4:if NO then unit $j$ is redundant, call the routine again with
$[{\bm{q}}_{j},0]$ as ${\bm{q}}$ and $k+1$ as $k$
5:if YES then unit $j$ splits the region into two, call the routine again with
$[{\bm{q}}_{j},1]$ and $k+1$ and $[{\bm{q}}_{j},-1]$ and $k+1$
6:${\bm{X}}^{(L)}$$\triangleright$ Evaluate loss and back-propagate as usual
Table 1: Comparison of our exact enumeration method versus sampling-based partition discovery for various single layer configurations with random weights and biases. The sampling-based discovery is run $5$ times and we report the average and standard deviation of the number of regions found after sampling. The number of input space sample is obtain so that the computation time of the proposed method is the same as the computation time of the sampling method i.e. for each configuration, both methods have run the exact same amount of time. We observe that for low-dimensional input space, and with the same fixed time-budget, both methods perform similarly and sampling is sufficient to quickly discover all of the layer’s partition. input dim \width | K=16 | K=32 | K=64 | K=128 | K=256
---|---|---|---|---|---
D=2 | enumeration | 16 | 13 | 71 | 127 | 631
sampling | 16 $\pm$0 | 13 $\pm$0 | 67$\pm$0 | 127$\pm$0 | 611 $\pm$2
samp. found | 100% | 100 % | 94 % | 100 % | 96 %
D=4 | enumeration | 54 | 80 | 1107 | 4271 | 95954
sampling | 51 $\pm$0 | 69 $\pm$ 1 | 866$\pm$3 | 3288$\pm$18 | 70635 $\pm$55
samp. found | 94 % | 86 % | 78 % | 77% | 73%
D=8 | enumeration | 24 | 1242 | 8396 | 386566 | -
sampling | 18 $\pm$0 | 543$\pm$2 | 2875$\pm$5 | 136748$\pm$251 | -
samp. found | 75 % | 44 % | 34 % | 35 % | -
Fig. 2: Depiction of the multilayer case which corresponds to a union of
region-constrained hyperplane arrangements and thus which can be studied
directly form the proposed hyperplane arrangement region enumeration. The only
additional step is to first enforce that the search takes place on the
restricted region of interest from the up-to-layer-$\ell$ input space
partition. For example on the left column one first obtains the first layer
partition depicted in black. On each of the enumerated region, a subdivision
will be performed by the next layer; pick any region of interest, compose the
per-region affine mapping (fixed on that region) with the second layer affine
mappings, and repeat the region enumeration algorithm. This discovers the
second subdivision done by the second layer, highlighted in blue in the
middle column. This can be repeated to obtain the subdivision of the third
layer, here highlighted in red in the right column.
## 3 Enumeration of Multi-Layers Partitions
This section demonstrates how the derivation carried out in Section 2 for the
single layer setting is sufficient to enumerate the partition of a multilayer
DN, thanks to the subdivision process under which the composition of many
layers ultimately form the global DN’s input space partition. We first recall
this subdivision step in Section 3.1 and summarize the enumeration algorithm
in Section 3.2.
### 3.1 Deep Networks are Continuous Piecewise Affine
We specialize the per-layer notations from Section 2 by expliciting the layer
index $\ell$ as $f^{(\ell)}$ for the layer mapping, as
${\bm{\theta}}^{(\ell)}$ for its parameters, and the entire DN’s input-output
mapping is now referred to as
$f_{{\bm{\theta}}}:\mathbb{R}^{D}\mapsto\mathbb{R}^{K}$ with $K$ the output
space dimension. The composition of layers take the form
$\displaystyle
f_{{\bm{\theta}}}=\left(f_{{\bm{\theta}}^{(L)}}^{(L)}\circ\dots\circ
f_{{\bm{\theta}}^{(1)}}^{(1)}\right),$ (4)
where each layer mapping
$f^{(\ell)}:\mathbb{R}^{D^{(\ell)}}\mapsto\mathbb{R}^{D^{(\ell+1)}}$ produces
a feature map; with $D^{(1)}\triangleq D$ and $D^{(L)}\triangleq K$; with
mapping given by Eq. 1, and ${\bm{h}}^{(\ell)}$ denoting the pre-activation
map of layer $\ell$. A key result from [20, 7] is the DN mapping is itself
defined on a partition as in
$\displaystyle
f_{{\bm{\theta}}}({\bm{x}})=\sum_{\omega\in\Omega}({\bm{A}}_{\omega}{\bm{x}}+{\bm{b}}_{\omega})1_{\\{{\bm{z}}\in\omega\\}},$
which is known to be recursively built by each layer subdividing the
previously built partition of the space [21].
### 3.2 Enumerating Union of Hyperplane Arrangements
Considering an arbitrarily deep model can be tackled by understanding the
recurrent subdivision process of a two hidden layer DN and applying the same
principle successively. In this setting, notice that for the (two-layer) DN to
be affine within some region $\omega$ of the DN’s input space, each layer must
stay affine as well. By composition the first layer staying linear does not
ensure that the DN stays linear, but the first layer being nonlinear does
imply that the entire DN is nonlinear. From that, we see that the first
layer’s partition are “coarser” the the entire DN’s partition regions. More
precisely, and following the derivation of [21], we obtain that each layer is
a recursive subdivision of the previously build partition when in our case we
need to search for each region $\omega$ of the first layer’s partition the
regions within it where the second layer stays linear. As a result, the
proposed single hidden layer enumeration method from Section 2 can be applied
recursively as follows. First, compute the first layer partition enumeration.
Then, for each enumerated region with corresponding sign pattern ${\bm{q}}$,
define a new single layer model with
${\bm{h}}({\bm{x}})\triangleq\sigma({\bm{W}}^{(2)}\operatorname{diag}({\bm{q}}){\bm{W}}^{(1)}{\bm{x}}+{\bm{W}}^{(2)}({\bm{q}}\odot{\bm{b}}^{(1)})+{\bm{b}}^{(2)}$
and within $\omega$ apply the single layer enumeration; repeating the process
for all regions –and corresponding sign patterns ${\bm{q}}$ of the previously
found first layer partition. This enumerates the partition of $(f^{(2)}\circ
f^{(1)})$, and the same process can be repeated as many times as there are
layers in the DN; as illustrated in Fig. 2.
## 4 Conclusion and Future Work
In this paper, we provided the first exact enumeration method for Deep
Networks partitions that relies on the existing highly efficient enumeration
method of hyperplane arrangements. In fact, both the hallow and deep
architectures produce partitions that correspond to hyperplane arrangements or
union of restricted hyperplane arrangements. A crucial finding that was
enabled by the proposed method is that sampling-based region enumeration,
which is the only strategy used in current research studies dealing with DNs
and affine splines, is in fact relatively poor at finding the regions of the
DN’s partition. In particular, when using such sampling to estimating some
sensitive statistics e.g. the volume of the smallest region, sampling is
biased and should be avoid in favor of an exact enumeration method.
## References
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|
$\textup{C}^{*}$-algebra $A$ by Hilbert bimodules. If $A$ is separable or of
Type I, then the following are equivalent:
1. (1)
the dual groupoid $\widehat{A}\rtimes S$ is residually topologically free;
2. (2)
the action $\mathcal{E}$ is residually aperiodic;
3. (3)
for any $I\in\mathbb{I}^{\mathcal{E}}(A)$, the full crossed product for the
restricted action $\mathcal{E}|_{A/I}$ has a unique pseudo-expectation namely,
the canonical $\mathcal{M}_{\mathrm{loc}}$-expectation;
4. (4)
for any $I\in\mathbb{I}^{\mathcal{E}}(A)$, $(A/I)^{+}$ supports $C$ for each
intermediate $\textup{C}^{*}$-subalgebra $A/I\subseteq C\subseteq
A/I\rtimes_{\mathrm{ess}}S$ for the restricted action $\mathcal{E}|_{A/I}$;
5. (5)
for any $I\in\mathbb{I}^{\mathcal{E}}(A)$, $A/I$ detects ideals in each
intermediate $\textup{C}^{*}$-subalgebra $A/I\subseteq C\subseteq
A/I\rtimes_{\mathrm{ess}}S$ for the restricted action $\mathcal{E}|_{A/I}$.
For general $A$, (1)$\Rightarrow$(2)$\Rightarrow$(3)$\Rightarrow$(5) and
(2)$\Rightarrow$(4)$\Rightarrow$(5).
###### Proof.
A subset of $\widehat{A}$ is closed and invariant if and only if it is of the
form $X=\widehat{A/I}$ for an $\mathcal{E}$-invariant ideal $I$. The dual
groupoid of the induced action on $A/I$ is the restriction $X\rtimes S$ of the
dual groupoid to $X$. Hence the implications
(1)$\Rightarrow$(2)$\Rightarrow$(3), (4) follow from [Kwasniewski-
Meyer:Aperiodicity_pseudo_expectations]*Corollary 4.8 and Theorem 3.6, applied
to the quotients $A/I$ and intermediate $\textup{C}^{*}$-algebras. Lemma 2.22
shows that (4) implies (5), and [Pitts-
Zarikian:Unique_pseudoexpectation]*Theorem 3.5 shows that (3) implies (5). If
$A$ is separable or of Type I, so are its quotients $A/I$, and then
[Kwasniewski-Meyer:Aperiodicity_pseudo_expectations]*Proposition 6.1 shows
that (5) implies (1). ∎
The following result justifies introducing the notion of essential exactness –
it is an instance of condition (1) in Theorem 2.33.
###### Theorem 5.9.
Let $B$ be an $S$-graded $\textup{C}^{*}$-algebra with a grading
$\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$ that forms a residually aperiodic
action of $S$ on $A\mathrel{\vcentcolon=}\mathcal{E}_{1}$ (this holds if the
dual groupoid $\widehat{A}\rtimes S$ is residually topologically free). The
following are equivalent:
1. (1)
$B\cong A\rtimes_{\mathrm{ess}}S$ and $\mathcal{E}$ is essentially exact;
2. (2)
$A$ separates ideals in $B$, so
$\mathbb{I}(B)\cong\mathbb{I}^{\mathcal{E}}(A)$;
3. (3)
$A^{+}$ residually supports $B$;
4. (4)
$A^{+}$ fills $B$.
If the above equivalent conditions hold and the primitive ideal space
$\check{A}$ is second countable, then the quasi-orbit map induces a
homeomorphism $\check{B}\cong\check{A}/{\sim}$, where $\check{A}/{\sim}$ is
the quasi-orbit space of the dual groupoid $\check{A}\rtimes S$, that is,
$\mathfrak{p}_{1},\mathfrak{p}_{2}\in\check{A}$ satisfy
$\mathfrak{p}_{1}\sim\mathfrak{p}_{2}$ if and only if
$\overline{(\check{A}\rtimes
S)\cdot\mathfrak{p}_{1}}=\overline{(\check{A}\rtimes
S)\cdot\mathfrak{p}_{2}}$.
###### Proof.
The $\textup{C}^{*}$-inclusion $A\subseteq B$ is symmetric and residually
aperiodic by Propositions 2.12 and 5.2. Hence by Theorem 2.33 conditions
(2)–(4) are equivalent to the condition that for each
$I\in\mathbb{I}^{B}(A)=\mathbb{I}^{\mathcal{E}}(A)$ the unique pseudo-
expectation $E^{I}\colon B/BIB\to A/I$ is almost faithful. Let us assume this.
Then there is a commutative diagram
${A\rtimes S/I\rtimes S\cong A/I\rtimes
S}$${B/BIB}$${\mathcal{M}_{\mathrm{loc}}(A/I)}$${\mathrm{I}(A/I),}$$\scriptstyle{\Psi}$$\scriptstyle{EL^{I}}$$\scriptstyle{E^{I}}$$\scriptstyle{\hookrightarrow}$
where $\Psi$ is the homomorphism that exists by universality of $A/I\rtimes S$
because $B/BIB$ is graded by $\mathcal{E}|_{A/I}$ by Lemma 2.13, $EL^{I}$ is
the canonical essential expectation for $A/I\rtimes S$, and
$\mathcal{M}_{\mathrm{loc}}(A/I)\hookrightarrow\mathrm{I}(A/I)$ is the
canonical embedding. The diagram commutes because the inclusion $A/I\subseteq
A/I\rtimes S$ is aperiodic and hence there is a unique pseudo-expectation by
[Kwasniewski-Meyer:Aperiodicity_pseudo_expectations]*Theorem 3.6. As a
consequence,
$\ker\Psi={\Psi}^{-1}(0)={\Psi}^{-1}(\mathcal{N}_{E^{I}})=\mathcal{N}_{EL}$
and thus $\Psi$ factors through an isomorphism
$A/I\rtimes_{\mathrm{ess}}S\cong B/BIB$. Since this holds for every
$I\in\mathbb{I}^{\mathcal{E}}(A)$ we get that $B\cong
A\rtimes_{\mathrm{ess}}S$ and that $\mathcal{E}$ is essentially exact
($A\rtimes_{\mathrm{ess}}S/I\rtimes_{\mathrm{ess}}S\cong
B/BIB\cong(A/I)\rtimes_{\mathrm{ess}}S$ for
$I\in\mathbb{I}^{\mathcal{E}}(A)$).
Theorem 2.33 implies easily that (1) implies (2). This finishes the proof that
the four conditions are equivalent. The remaining claims follow mostly from
the last part of Theorem 2.33. That the quasi-orbit space has the asserted
form follows from [Kwasniewski-Meyer:Stone_duality]*Theorem 6.22. ∎
###### Corollary 5.10.
Let $\mathcal{A}=(A_{\gamma})_{\gamma\in G}$ be a Fell bundle over an étale
groupoid $G$ with locally compact, Hausdorff unit space $X$, and put
$A=\textup{C}_{0}(\mathcal{A}|_{X})$. Define the dual groupoid
$\widehat{A}\rtimes G$ as in Example 3.6. Assume that it is residually
topologically free (this holds, for instance, if $G$ is residually
topologically free and the base map for the $\textup{C}^{*}$-bundle
$\mathcal{A}$ is open and closed). Assume also one of the following
1. (1)
$B\mathrel{\vcentcolon=}\textup{C}^{*}_{\mathrm{ess}}(\mathcal{A})$ and
$\mathcal{A}$ is essentially exact;
2. (2)
$B\mathrel{\vcentcolon=}\textup{C}^{*}_{\mathrm{r}}(\mathcal{A})$,
$\mathcal{A}$ is exact, and the unit space in $\check{A}\rtimes G$ is closed
(the latter is automatic if $G$ is Hausdorff);
3. (3)
$B\mathrel{\vcentcolon=}\textup{C}^{*}(\mathcal{A})$, $\mathcal{A}$ is
separable, and $G$ is amenable and Hausdorff.
Then $A$ separates ideals in $B$ and, even more, $A^{+}$ fills $B$. The
lattice $\mathbb{I}(B)$ is naturally isomorphic to the lattice of
$G$-invariant ideals in $A$. If, in addition, $\check{A}$ is second countable,
then $\check{B}\cong\check{A}/{\sim}$, where $\check{A}/{\sim}$ is the quasi-
orbit space of the dual groupoid $\check{A}\rtimes G$, that is,
$\mathfrak{p}_{1},\mathfrak{p}_{2}\in\check{A}$ satisfy
$\mathfrak{p}_{1}\sim\mathfrak{p}_{2}$ if and only if
$\overline{(\check{A}\rtimes
G)\cdot\mathfrak{p}_{1}}=\overline{(\check{A}\rtimes
G)\cdot\mathfrak{p}_{2}}$.
###### Proof.
The claims in brackets follow from Lemma 5.7. The claim in case (1) follows
from Theorem 5.9 (see also Example 4.25). Case (2) follows from (1) and
Remarks 3.16 and 4.24. Example 4.32 explains why (3) is a special case of (2).
∎
Theorem 5.9 allows us to describe the ideal structure of $B$ in terms of $A$
under the following assumptions:
(5.11) $\mathcal{E}\text{ is an essentially exact, residually aperiodic action
and }B=A\rtimes_{\mathrm{ess}}S.$
We are going to study whether $B$ is purely infinite using the same
assumption. Before we do this, we simplify (5.11) in the presence of a
conditional expectation.
###### Proposition 5.12.
If there is a genuine conditional expectation $E\colon B\to A\subseteq B$,
then (5.11) is equivalent to
(5.13) $\mathcal{E}$ is closed, exact, residually aperiodic and
$B=A\rtimes_{\mathrm{r}}S$.
For any $\textup{C}^{*}$-inclusion $A\subseteq B$, an action $\mathcal{E}$ as
in (5.13) exists if and only if $A\subseteq B$ is regular, residually
aperiodic and there is a conditional expectation $E\colon B\to A$ which is
residually faithful in the sense that $E$ descends to a faithful conditional
expectation $E^{I}\colon B/BIB\to A/I$ for any $I\in\mathbb{I}^{B}(A)$ (see
Lemma 2.19).
###### Proof.
(5.13) implies (5.11) by Remark 4.24. Conversely, assume (5.11). Then $E\colon
B\to A$ is the unique pseudo-expectation for $A\subseteq B$ by [Kwasniewski-
Meyer:Aperiodicity_pseudo_expectations]*Theorem 3.6. Hence $E=EL$,
$\mathcal{E}$ is closed and $A\rtimes_{\mathrm{r}}S=A\rtimes_{\mathrm{ess}}S$.
The same argument works for all the restrictions $\mathcal{E}^{I}$ and
quotient inclusions $A/I\subseteq B/BIB$ for
$I\in\mathbb{I}^{\mathcal{E}}(A)=\mathbb{I}^{B}(A)$. This gives (5.13).
Combining this reasoning with [Kwasniewski-Meyer:Cartan]*Theorem 6.3 also
gives the second part of the assertion. ∎
###### Corollary 5.14.
If $A$ is type $I$, then an action $\mathcal{E}$ as in (5.13) exists if and
only if $A\subseteq B$ is residually Cartan, that is, for each
$I\in\mathbb{I}^{B}(A)$, $A/I\subseteq B/BIB$ is a noncommutative Cartan
subalgebra in the sense of Exel [Exel:noncomm.cartan].
###### Proof.
Combine the second part of Proposition 5.12 and [Kwasniewski-
Meyer:Cartan]*Theorem 6.3. ∎
### 5.2. Pure infiniteness criteria
In this section, we assume that $B$ is an $S$-graded $\textup{C}^{*}$-algebra
with a grading $\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$ and $A$ is the unit
fibre of the grading. We give pure infiniteness criteria for $B$ under the
assumption (5.11). This covers (5.13) and the assumptions in Corollary 5.10 as
special cases. In view of Theorem 5.9, the following two theorems are
immediate corollaries of Theorems 2.35 and 2.36:
###### Theorem 5.15.
Assume (5.11). Let $\mathcal{F}\subseteq A^{+}$ fill $A$ and be invariant
under $\varepsilon$-cut-downs. Then $B$ is strongly purely infinite if and
only if each pair of elements $a,b\in\mathcal{F}$ has the matrix
diagonalisation property in $B$.
###### Theorem 5.16.
Assume (5.11). Let $\mathcal{F}\subseteq A^{+}$ residually support $A$.
Suppose that $\mathbb{I}^{\mathcal{E}}(A)$ is finite or the projections in
$\mathcal{F}$ separate the ideals in $\mathbb{I}^{\mathcal{E}}(A)$. Then $B$
is strongly purely infinite (with the ideal property) if and only if every
element in $\mathcal{F}\setminus\\{0\\}$ is properly infinite in $B$.
In order to use these results, we need conditions that suffice for $a,b\in
A^{+}$ to have the matrix diagonalisation property in $B$ or for $a\in
A^{+}\setminus\\{0\\}$ to be properly infinite in $B$. Checking the matrix
diagonalisation property is usually difficult. Nevertheless, the following
lemma may be useful (see [Kirchberg-Sierakowski:Strong_pure, Kwasniewski-
Meyer:Aperiodicity]):
###### Lemma 5.17.
Let $a,b\in A^{+}\setminus\\{0\\}$. Suppose that for each
$c\in\mathcal{E}_{t}$, $t\in S$, and each $\varepsilon>0$ there are
$n,m\in\mathbb{N}$ and $a_{i}\in a\mathcal{E}_{s_{i}}$, $s_{i}\in S$, for
$i=1,\dotsc,n$ and $b_{j}\in b\mathcal{E}_{t_{j}}$, $t_{j}\in S$, for
$j=1,\dotsc,m$ such that
$a\approx_{\varepsilon}\sum_{i=1}^{n}a_{i}^{*}a_{i},\quad
b\approx_{\varepsilon}\sum_{i=1}^{m}b_{i}^{*}b_{i},\quad\sum_{i,j=1,i\neq
j}^{n}a_{i}^{*}a_{j}\approx_{\varepsilon}0,\quad\sum_{i,j=1,i\neq
j}^{n,m}b_{i}^{*}b_{j}\approx_{\varepsilon}0$
and $\sum_{i=1,j=1}^{n,m}a_{i}^{*}cb_{j}\approx_{\varepsilon}0$. Then $a,b\in
A^{+}\setminus\\{0\\}$ have the matrix diagonalisation property in $B$.
###### Proof.
Let $\mathcal{C}\mathrel{\vcentcolon=}\bigcup_{t\in S}\mathcal{E}_{t}$ and
$\mathcal{S}\mathrel{\vcentcolon=}\bigcup_{t\in S}\mathcal{E}_{t}$. We claim
that $a,b\in A^{+}\setminus\\{0\\}$ have the matrix diagonalisation property
with respect to $\mathcal{C}$ and $\mathcal{S}$ as introduced in [Kirchberg-
Sierakowski:Filling_families]*Definition 4.6. Indeed, let
$x\in\mathcal{E}_{t}$ be such that $\left(\begin{smallmatrix}a&x^{*}\\\
x&b\end{smallmatrix}\right)\in M_{2}(B)^{+}$ and let $\varepsilon>0$. Let
$a_{i}\in a\mathcal{E}_{s_{i}}$ and $b_{j}\in b\mathcal{E}_{t_{j}}$ satisfy
the conditions described in the assertion with
$c\mathrel{\vcentcolon=}a^{\nicefrac{{1}}{{2}}}xb^{\nicefrac{{1}}{{2}}}$. We
may write $a_{i}=a^{\nicefrac{{1}}{{2}}}x_{i}$ and
$b_{j}=b^{\nicefrac{{1}}{{2}}}y_{j}$ for some $x_{i}$, $y_{j}$. Let
$d_{1}\mathrel{\vcentcolon=}\sum_{i=1}^{n}x_{i}$ and
$d_{2}\mathrel{\vcentcolon=}\sum_{j=1}^{n}y_{j}$. The assumed estimates imply
that
$d_{1}^{*}ad_{1}\approx_{2\varepsilon}a,\qquad
d_{2}^{*}bd_{2}\approx_{2\varepsilon}b,\qquad
d_{1}^{*}xd_{2}\approx_{\varepsilon}0.$
This proves our claim. Clearly, $\mathcal{S}$ is a multiplicative subsemigroup
of $B$, $\mathcal{S}^{*}\mathfrak{S}\mathcal{S}\subseteq\mathfrak{S}$,
$A\mathcal{S}A\subseteq\mathcal{S}$, and the closed linear span of
$\mathcal{C}$ is $B$. Thus $a,b\in A^{+}\setminus\\{0\\}$ have the matrix
diagonalisation property in $B$ by [Kirchberg-
Sierakowski:Filling_families]*Lemma 5.6. ∎
Now we will focus on ways to check whether $a\in A^{+}\setminus\\{0\\}$ is
properly infinite in $B$. The following definition generalises [Kwasniewski-
Szymanski:Pure_infinite]*Definition 5.1 and [Kwasniewski-
Meyer:Aperiodicity]*Definition 5.5 from groups to inverse semigroups. A
crucial point is that the properties depend only on the Fell bundle
$\mathcal{E}=(\mathcal{E}_{t})_{t\in S}$, not on the norm in $B$.
###### Definition 5.18.
An element $a\in A^{+}\setminus\\{0\\}$ is called
1. (1)
_$\mathcal{E}$ -infinite_ if there is $b\in A^{+}\setminus\\{0\\}$ such that
for each $\varepsilon>0$, there are $n,m\in\mathbb{N}$, $t_{i}\in S$ and
$a_{i}\in a\mathcal{E}_{t_{i}}$ for $1\leq i\leq n+m$, such that
$a\approx_{\varepsilon}\sum_{i=1}^{n}a_{i}^{*}a_{i},\quad
b\approx_{\varepsilon}\sum_{i=n+1}^{n+m}a_{j}^{*}a_{j},\quad\sum_{\genfrac{}{}{0.0pt}{}{i,j=1}{i\neq
j}}^{n+m}\lVert a_{i}^{*}a_{j}\rVert\leq\varepsilon;$
2. (2)
_residually $\mathcal{E}$-infinite_ if $a+I$ is $\mathcal{E}|_{A/I}$-infinite
for all $I\in\mathbb{I}^{\mathcal{E}}(A)$ with $a\notin I$;
3. (3)
_properly $\mathcal{E}$-infinite_ if for all $\varepsilon>0$ there are
$n,m\in\mathbb{N}$, $t_{i}\in S$ and $a_{i}\in a\mathcal{E}_{t_{i}}$ for
$1\leq i\leq n+m$, such that
$a\approx_{\varepsilon}\sum_{i=1}^{n}a_{i}^{*}a_{i},\quad
a\approx_{\varepsilon}\sum_{j=n+1}^{m}a_{j}^{*}a_{j},\quad\sum_{\genfrac{}{}{0.0pt}{}{i,j=1}{i\neq
j}}^{n+m}\lVert a_{i}^{*}a_{j}\rVert\leq\varepsilon;$
4. (4)
_$\mathcal{E}$ -paradoxical_ if the condition in (3) holds with
$\varepsilon=0$, that is, there are $n,m\in\mathbb{N}$, $t_{i}\in S$, and
$a_{i}\in a\mathcal{E}_{t_{i}}$ for $1\leq i\leq n+m$, such that
$a=\sum_{i=1}^{n}a_{i}^{*}a_{i},\qquad
a=\sum_{j=n+1}^{n+m}a_{j}^{*}a_{j},\qquad a_{i}^{*}a_{j}=0\quad\text{for
}i\neq j.$
###### Lemma 5.19.
If $a\in A^{+}\setminus\\{0\\}$ is $\mathcal{E}$-infinite, then it is infinite
in $B$. If $a\in A^{+}\setminus\\{0\\}$ is properly $\mathcal{E}$-infinite,
then it is properly infinite in $B$.
###### Proof.
First let $a\in A^{+}\setminus\\{0\\}$ be $\mathcal{E}$-infinite. Let $b\in
A^{+}\setminus\\{0\\}$ be as in Definition 5.18.(1). For $\varepsilon>0$,
there are $n,m\in\mathbb{N}$ and $t_{i}\in S$ and $a_{i}\in
a\mathcal{E}_{t_{i}}$ for $i=1,\dotsc,n+m$, $n,m\in\mathbb{N}$ as in
Definition 5.18.(1). Let $x\mathrel{\vcentcolon=}\sum_{i=1}^{n}a_{i}$ and
$y\mathrel{\vcentcolon=}\sum_{i=n+1}^{m}a_{i}$. Then $x,y\in aB$. Simple
estimates such as
$\lVert
x^{*}x-a\rVert=\bigg{\lVert}\sum_{j=1}^{n}a_{j}^{*}a_{j}-a+\sum_{i,j=1,i\neq
j}^{n}a_{i}^{*}a_{j}\bigg{\rVert}\leq\varepsilon+\varepsilon$
show that $x^{*}x\approx_{2\varepsilon}a$, $y^{*}y\approx_{2\varepsilon}b$ and
$y^{*}x\approx_{\varepsilon}0$. Hence $a$ is infinite in $B$ (see Definition
2.34). The proof when $a$ is properly $\mathcal{E}$-infinite is the same with
$b=a$. ∎
Let us compare the definitions of infinite and properly infinite elements in
Definition 2.34 to the definitions of $\mathcal{E}$-infinite and properly
$\mathcal{E}$-infinite elements in Definition 5.18. There are two differences.
First, we now choose the elements $x,y\in aB$ in the subalgebra
$A\rtimes_{\mathrm{alg}}S$, so that we may write them as a finite sum $\sum
a_{i}$ with $a_{i}\in a\mathcal{E}_{t_{i}}$. Secondly, we estimate each
product $a_{i}^{*}a_{j}$ for $i\neq j$ separately. The first change does not
achieve much because $A\rtimes_{\mathrm{alg}}S$ is dense in $B$ and we only
aim for approximate equalities anyway. The second change simplify the
estimates a lot because $\lVert a_{i}^{*}a_{j}\rVert$ is computed in the
Hilbert $A$-bimodule $\mathcal{E}_{t_{i}^{*}t_{j}}$, whereas the norm
estimates in Definition 2.34 involve the $\textup{C}^{*}$-norm of $B$. For an
$\mathcal{E}$-paradoxical element, we even assume the products
$a_{i}^{*}a_{j}$ for $i\neq j$ to vanish exactly. This is once again much
easier to check. Paradoxical elements are also important because they are
related to paradoxical decompositions, which were studied already by Banach
and Tarski. In the setting of purely infinite crossed products, their
importance was highlighted by Rørdam and Sierakowski [Rordam-
Sierakowski:Purely_infinite]. The implications among our infiniteness
conditions hinted at above are summarised in the following proposition:
###### Proposition 5.20.
Assume that $A$ separates ideals in $B$. Consider the following conditions
$a\in A^{+}\setminus\\{0\\}$ may satisfy:
1. (1)
$a$ is properly infinite in $B$;
2. (2)
for each $\varepsilon>0$ there are $n,m\in\mathbb{N}$, $t_{i}\in S$, and
$a_{i}\in a\mathcal{E}_{t_{i}}$ for $1\leq i\leq n+m$, such that
$a\approx_{\varepsilon}\sum_{i,j=1}^{n}a_{i}^{*}a_{j},\qquad
a\approx_{\varepsilon}\sum_{i,j=n+1}^{m}a_{i}^{*}a_{j},\qquad\sum_{i,j=1,i\neq
j}^{n+m}a_{i}^{*}a_{j}\approx_{\varepsilon}0;$
3. (3)
$a$ is residually $\mathcal{E}$-infinite;
4. (4)
$a$ is properly $\mathcal{E}$-infinite;
5. (5)
$a$ is $\mathcal{E}$-paradoxical.
Then (1)$\Leftrightarrow$(2)$\Leftarrow$(3)$\Leftarrow$(4)$\Leftarrow$(5).
###### Proof.
The implications (5)$\Rightarrow$(4)$\Rightarrow$(3) are straightforward. By
[Kirchberg-Rordam:Non-simple_pi]*Proposition 3.14, $a$ is properly infinite if
and only if it is residually infinite. Since $A$ separates ideals in $B$, any
ideal in $B$ comes from an invariant ideal in $A$, as in the definition that
$a$ is residually $\mathcal{E}$-infinite. Together with Lemma 5.19, this shows
that (3) implies (1).
According to Definition 2.34, $a\in A^{+}\setminus\\{0\\}$ is properly
infinite in $B$ if and only if, for all $\varepsilon>0$, there are $x,y\in
a\cdot B$ with $x^{*}x\approx_{\varepsilon}a$, $y^{*}y\approx_{\varepsilon}b$
and $x^{*}y\approx_{\varepsilon}0$. Without loss of generality, we may pick
$x,y\in a\cdot(\sum_{t\in S}\mathcal{E}_{t})$ because $\sum_{t\in
S}\mathcal{E}_{t}$ is dense in $B$. So $x=\sum_{j=1}^{n}a_{j}$ and
$y=\sum_{j=n+1}^{n+m}a_{j}$ for some $n,m\in\mathbb{N}$, $t_{i}\in S$, and
$a_{i}\in a\mathcal{E}_{t_{i}}$ for $1\leq i\leq n+m$. The relations
$x^{*}x\approx_{\varepsilon}a$, $y^{*}y\approx_{\varepsilon}b$ and
$x^{*}y\approx_{\varepsilon}0$ translate to those described in (2). This
proves that (1) and (2) are equivalent. ∎
It is unclear whether the implications in Proposition 5.20 may be reversed.
###### Remark 5.21.
The example of graph $\textup{C}^{*}$-algebras shows that it may be much
easier to check that an element is residually $\mathcal{E}$-infinite than that
it is properly $\mathcal{E}$-infinite (see also [Kwasniewski-
Szymanski:Pure_infinite]*Remark 7.10).
###### Corollary 5.22.
Assume (5.11). Let $\mathcal{F}\subseteq A^{+}$ residually support $A$.
Suppose that $\mathbb{I}^{\mathcal{E}}(A)$ is finite or that $\mathcal{F}$
consists of projections, or that the projections in $\mathcal{F}$ separate the
ideals in $\mathbb{I}^{\mathcal{E}}(A)$. If every element in
$\mathcal{F}\setminus\\{0\\}$ is residually $\mathcal{E}$-infinite, then
$A\rtimes_{\mathrm{ess}}S$ is purely infinite and has the ideal property.
###### Proof.
Combine Theorem 5.16 and Proposition 5.20. ∎
We may simplify our conditions further if $A$ is commutative. Then
$A\rtimes_{\mathrm{ess}}S\cong\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)$ for a
twisted étale groupoid $G$ with object space $\hat{A}$. The twist $\Sigma$ is
always locally trivial. Therefore, the bisections that trivialise the twist
$\Sigma$ form a wide inverse subsemigroup $S^{\prime}$ among all bisections of
$G$ (see [BussExel:Regular.Fell.Bundle]*Theorem 7.2). Then
$\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)\cong
A\rtimes_{\mathrm{ess}}S^{\prime}$. The action of $S^{\prime}$ on $A$ is
equivalent to a twisted action as in [BussExel:Regular.Fell.Bundle]*Definition
4.1, that is, each $\mathcal{E}_{t}$ for $t\in S$ comes from an isomorphism
between two ideals in $A$. We assume this because it allows us to identify
elements of $\mathcal{E}_{t}$ with $\textup{C}_{0}$-functions on
$s(\mathcal{E}_{t})\subseteq\hat{A}$. This discussion shows how to turn any
inverse semigroup action on a commutative $\textup{C}^{*}$-algebra into a
twisted action by partial automorphisms.
###### Lemma 5.23.
Assume that $A$ is commutative and that $S$ acts on $A$ by a twisted action by
partial automorphisms as in Example 3.2. Equip $\widehat{A}$ with the dual
action of $S$. Let $a\in A^{+}$ and
$V\mathrel{\vcentcolon=}\\{x\in\widehat{A}\,{:}\,\mathopen{}a(x)\neq 0\\}$.
Consider the following conditions:
1. (1)
the condition in Definition 5.18.(1) holds with $\varepsilon=0$;
2. (2)
there are $b\in(aAa)^{+}\setminus\\{0\\}$, $n\in\mathbb{N}$ and $t_{i}\in S$,
$a\in a\mathcal{E}_{t_{i}}$ for $1\leq i\leq n$ such that
$a=\sum_{i=1}^{n}a_{i}^{*}a_{i},\quad\text{ and }\quad a_{i}^{*}a_{j}=0,\quad
a_{i}^{*}b=0\qquad\text{for all }\,i,j=1,\dotsc,n,i\neq j;$
3. (3)
there are $n\in\mathbb{N}$, $t_{1},\dotsc,t_{n}\in S$, and open subsets
$V_{1},\dotsc,V_{n}\subseteq V$ such that
1. (a)
$V_{i}$ is contained in the domain of $t_{i}$ for $1\leq i\leq n$;
2. (b)
$(t_{i}\cdot V_{i})\cap(t_{j}\cdot V_{j})=\emptyset$ if $1\leq i<j\leq n$;
3. (c)
$V=\bigcup_{i=1}^{n}V_{i}$ and $\overline{\bigcup_{i=1}^{n}t_{i}\cdot
V_{i}}\subsetneq V$;
4. (4)
$a$ is $\mathcal{E}$-infinite.
Then (1)$\Leftrightarrow$(2)$\Rightarrow$(3)$\Rightarrow$(4). The implications
(1)$\Leftrightarrow$(2)$\Rightarrow$(4) hold in full generality.
###### Proof.
If (2) holds, then taking $m\mathrel{\vcentcolon=}1$,
$t_{n+1}\mathrel{\vcentcolon=}1$ and $a_{n+1}\mathrel{\vcentcolon=}\sqrt{b}$
we get (1). Conversely, if (1) holds, then there are $t_{i}\in S$, $a_{i}\in
a\mathcal{E}_{t_{i}}$ for $i=1,\dotsc,n+m$, such that
$a=\sum_{i=1}^{n}a_{i}^{*}a_{i}$, $\sum_{i=n+1}^{n+m}a_{i}^{*}a_{i}\neq 0$ and
$a_{i}^{*}a_{j}=0$ for $i\neq j$. Thus putting
$b\mathrel{\vcentcolon=}\sum_{i=n+1}^{n+m}a_{i}a_{i}^{*}$ gives (2). This
shows that (1) and (2) are equivalent.
Now let $b\in(aAa)^{+}\setminus\\{0\\}$, $n\in\mathbb{N}$, and $t_{i}\in S$,
$a_{i}\in a\mathcal{E}_{t_{i}}$ for $i=1,\dotsc,n$ be as in (2). We identify
the fibres $\mathcal{E}_{t}$ for $t\in S$ with spaces of sections of the
associated line bundle over $G$. Put $U_{i}\mathrel{\vcentcolon=}\\{\gamma\in
G\,{:}\,\mathopen{}\lVert a_{i}(\gamma)\rVert>0\\}$ for $i=1,\dotsc,n$ and
$W\mathrel{\vcentcolon=}\\{x\in X\,{:}\,\mathopen{}\lVert
b(\gamma)\rVert>0\\}$. Then $V_{i}\mathrel{\vcentcolon=}s(U_{i})=\\{x\in
X\,{:}\,\mathopen{}(a_{i}^{*}a_{i})(x)>0\\}\subseteq V$ is contained in the
domain of $t_{i}$ for $1\leq i\leq n$. The equality
$a=\sum_{i=1}^{n}a_{i}^{*}a_{i}$ implies that
$V=\bigcup_{i=1}^{n}s(U_{i})=\bigcup_{i=1}^{n}V_{i}$. Similarly,
$a_{i}^{*}a_{j}=0$ holds if and only if $(t_{i}\cdot V_{i})\cap(t_{j}\cdot
V_{j})=r(U_{i})\cap r(U_{j})=\emptyset$ for all $i\neq j$ and $a_{i}^{*}b=0$
holds if and only if $W\subseteq
V\setminus\bigcup_{i=1}^{n}r(U_{i})=V\setminus\bigcup_{i=1}^{n}(t_{i}\cdot
V_{i})$; here we identify functions in $\mathcal{E}_{t}$ with
$\textup{C}_{0}$-functions on bisections. Such a $W$ exists if and only if
$\overline{\bigcup_{i=1}^{n}t_{i}\cdot V_{i}}\subsetneq V$. Hence (2) implies
(3).
Next we show that (3) implies (4). Let $t_{1},\dotsc,t_{n}\in S$, and
$V_{1},\dotsc,V_{n}\subseteq V$ be as in (3). Let $b\in A^{+}\setminus\\{0\\}$
be any function that vanishes outside the open set
$V\bigm{\backslash}\overline{\bigcup_{i=n+1}^{n}t_{i}\cdot V_{i}}$. Fix
$\varepsilon>0$. Let
$K\mathrel{\vcentcolon=}\\{x\in\widehat{A}\,{:}\,\mathopen{}a(x)\geq\varepsilon\\}$.
Let $w_{1},\dotsc,w_{n}\in A$ be a partition of unity subordinate to the open
covering $K\subseteq\bigcup_{i=1}^{n}V_{i}$. Let
$a_{i}\mathrel{\vcentcolon=}(a-\varepsilon)_{+}^{1/2}\cdot w_{i}^{1/2}$ for
$i=1,\dotsc,n$. These functions vanish outside $K$, and $a_{i}$ belongs to the
domain of $t_{i}$. Since $\mathcal{E}_{t_{i}}$ comes from a partial
automorphism, we may view $a_{i}$ as an element of $\mathcal{E}_{t_{i}}$. It
belongs to $a\cdot\mathcal{E}_{t_{i}}$ because the support of $a_{i}$ is
contained in $V$. The product $a_{i}^{*}a_{j}$ is defined using the Fell
bundle structure. If $i\neq j$, then $a_{i}^{*}a_{j}=0$ because $(t_{i}\cdot
V_{i})\cap(t_{j}\cdot V_{j})=\emptyset$. Similarly, we get $a_{i}^{*}b=0$. And
$\sum_{i=1}^{n}a_{i}^{*}a_{i}=\sum_{i=1}^{n}(a-\varepsilon)_{+}\cdot
w_{i}=(a-\varepsilon)_{+}\approx_{\varepsilon}a.$
Hence $a$ is $\mathcal{E}$-infinite. ∎
###### Remark 5.24.
For strongly boundary group actions (see [Laca-Spielberg:Purely_infinite])
and, more generally, for filling actions (see [Jolissaint-
Robertson:Simple_purely_infinite]) condition (3) in Lemma 5.23 holds for every
nonempty open subset $V$. Thus if $\mathcal{E}$ comes from such an action,
then every element in $A^{+}\setminus\\{0\\}$ is $\mathcal{E}$-infinite. This
also holds when $A$ is noncommutative (see [Kwasniewski-
Szymanski:Pure_infinite]*Lemma 5.12).
###### Remark 5.25.
An étale, Hausdorff, locally compact groupoid $H$ is _locally contracting_ if
for each nonempty open set $U$ in the unit space $H^{0}$ of $H$ there is a
bisection $B\subseteq H$ with $\overline{r(B)}\subsetneq s(B)\subseteq U$ (see
[Anantharaman-Delaroche:Purely_infinite]). Given a wide inverse subsemigroup
$S\subseteq\operatorname{Bis}(H)$, we may strengthen this criterion by
requiring $B\subseteq t$ for some $t\in S$. Then we may rewrite
$\overline{r(B)}\subsetneq s(B)\subseteq U$ as follows: there is $t\in S$ and
$V\subseteq U$ contained in the domain of $t$ with $\overline{t\cdot
V}\subsetneq V$. This is the case $n=1$ of condition (3) in Lemma 5.23. As a
result, if the dual groupoid $\widehat{A}\rtimes S$ is locally contracting,
then for any $a\in A^{+}\setminus\\{0\\}$ there is $0\neq a_{2}\leq a$ that is
$\mathcal{E}$-infinite; namely, choose $U=\operatorname{supp}a$ and then
$a_{2}$ with $\operatorname{supp}a_{2}=V$ and $a_{2}\leq a$ for $V$ as above.
Condition (3) in Lemma 5.23 could be relaxed so that it still implies
$\mathcal{E}$-infiniteness, by using compact subsets of $V$. We formulate the
relevant condition implying $\mathcal{E}$-proper infiniteness:
###### Lemma 5.26.
Retain the assumptions of Lemma 5.23. In particular, let $a\in A^{+}$ and
$V\mathrel{\vcentcolon=}\\{x\in\widehat{A}\,{:}\,\mathopen{}a(x)\neq 0\\}$. If
for each compact subset $K\subseteq V$ there are $n,m\in\mathbb{N}$,
$t_{1},\dotsc,t_{n+m}\in S$, and open subsets $V_{1},\dotsc,V_{n+m}\subseteq
V$ such that $(t_{i}\cdot V_{i})\cap(t_{j}\cdot V_{j})=\emptyset$ if $1\leq
i<j\leq n+m$, $K\subseteq\bigcup_{i=1}^{n}V_{i}$ and
$K\subseteq\bigcup_{i=n+1}^{n+m}V_{i}$, then $a$ is $\mathcal{E}$-properly
infinite.
###### Proof.
Fix $\varepsilon>0$. Let
$K\mathrel{\vcentcolon=}\\{x\in\widehat{A}\,{:}\,\mathopen{}a(x)\geq\varepsilon\\}$.
Choose $n,m,t_{i},V_{i}$ as in the assumption of the lemma. Let
$w_{1},\dotsc,w_{n}\in A$ and $w_{n+1},\dotsc,w_{n+m}\in A$ be partitions of
unity subordinate to the open coverings $K\subseteq\bigcup_{i=1}^{n}V_{i}$ and
$K\subseteq\bigcup_{i=n+1}^{n+m}V_{i}$, respectively. Let
$a_{i}\mathrel{\vcentcolon=}(a-\varepsilon)_{+}^{1/2}\cdot w_{i}^{1/2}$ for
$i=1,\dotsc,n+m$. As in the proof of the implication (3)$\Rightarrow$(4) in
Lemma 5.23 one sees that treating $a_{i}$ as an element of
$\mathcal{E}_{t_{i}}$, the elements $a_{i}$ satisfy the relations in
Definition 5.18.(3). ∎
Now we assume, in addition, that the spectrum $\widehat{A}$ is totally
disconnected. This implies that the compact open bisections form a basis for
the topology and that $A$ is spanned by projections. We are going to see that
a projection is $\mathcal{E}$-paradoxical if and only if its support is
$(2,1)$-paradoxical as defined in [Boenicke-Li:Ideal]. Such open subsets give
purely infinite elements in the type semigroup considered in [Boenicke-
Li:Ideal, Rainone-Sims:Dichotomy, Ma:Purely_infinite_groupoids].
###### Definition 5.27 ([Boenicke-Li:Ideal]).
Let $G$ be an ample groupoid. We say that a compact open set $V\subseteq
G^{0}$ is _$(2,1)$ -paradoxical_ if there are $n,m\in\mathbb{N}$ and compact
open bisections $U_{i}\subseteq G$ for $1\leq i\leq n+m$ such that
$r(U_{i})\subseteq V$ for $1\leq i\leq n+m$ and
$V=\bigsqcup_{i=1}^{n}s(U_{i}),\qquad V=\bigsqcup_{i=n+1}^{n+m}s(U_{i}),\qquad
r(U_{i})\cap r(U_{j})=\emptyset\quad\text{for }i\neq j.$
###### Proposition 5.28.
Let $\mathcal{E}$ be an action of an inverse semigroup $S$ by Hilbert
bimodules on a commutative $\textup{C}^{*}$-algebra $A$ with totally
disconected spectrum $\widehat{A}$; equivalently, the dual groupoid
$G\mathrel{\vcentcolon=}\widehat{A}\rtimes S$ is ample. A projection $a\in
A^{+}$ is $\mathcal{E}$-paradoxical if and only if its support
$V\mathrel{\vcentcolon=}\\{x\in\widehat{A}\,{:}\,\mathopen{}a(x)\neq 0\\}$ is
$(2,1)$-paradoxical.
###### Proof.
Suppose first that $a\in A^{+}\setminus\\{0\\}$ is $\mathcal{E}$-paradoxical.
That is, there are $n,m\in\mathbb{N}$, $t_{1},\dotsc,t_{n+m}\in S$, and
$a_{i}\in a\mathcal{E}_{t_{i}}$ such that
$a=\sum_{i=1}^{n}a_{i}^{*}a_{i}=\sum_{i=n+1}^{n+m}a_{i}^{*}a_{i}$ and
$a_{i}^{*}a_{j}=0$ for $i\neq j$. Let $1\leq i\leq n+m$. Recall that we may
treat $\mathcal{E}_{t_{i}}$ as spaces of sections $A_{U_{i}}$ of a line bundle
over $G=\widehat{A}\rtimes S$ that are supported on open bisections
$U_{i}\in\operatorname{Bis}(G)$. Thus $U_{i}\mathrel{\vcentcolon=}\\{\gamma\in
G\,{:}\,\mathopen{}\lVert a_{i}(\gamma)\rVert>0\\}$ is an open bisection of
$G$ contained in $U_{i}$. Since $a_{i}\in aA_{U_{i}}$, we have
$r(U_{i})=\\{x\in\widehat{A}\,{:}\,\mathopen{}(a_{i}a_{i}^{*})(x)>0\\}\subseteq
V$. And $a_{i}^{*}a_{j}=0$ implies that $r(U_{i})\cap r(U_{j})=\emptyset$ for
all $i\neq j$. Since $\\{x\in
X\,{:}\,\mathopen{}(a_{i}^{*}a_{i})(x)>0\\}=s(U_{i})$, the equalities
$a=\sum_{i=1}^{n}a_{i}^{*}a_{i}$ and $a=\sum_{i=n+1}^{n+m}a_{i}^{*}a_{i}$
imply $V=\bigsqcup_{i=1}^{n}s(U_{i})$ and $V=\bigsqcup_{i=n+1}^{n+m}s(U_{i})$.
Hence the family $U_{i}\in\operatorname{Bis}(G)$ for $1\leq i\leq n+m$ has all
the desired properties, except that $U_{i}$ need not be compact. However,
since $G$ is ample, every $U_{i}$ is a union of some compact open bisections.
Since $V$ is compact and
$V=\bigsqcup_{i=1}^{n}s(U_{i})=\bigsqcup_{i=n+1}^{n+m}s(U_{i})$, we may, in
fact, replace each $U_{i}$ for $1\leq i\leq n+m$ by a finite union of compact
open bisections. This gives a compact open bisection.
Conversely, let $U_{i}\subseteq G$ for $1\leq i\leq n+m$ be a family of
bisection as in Definition 5.27. Let
$S^{\prime}\subseteq\operatorname{Bis}(G)$ be the family of open compact
bisections that trivialise the twist, that is, the restrictions of the
associated line bundle over $G$ to sets in $S^{\prime}$ are trivial. Note that
$S^{\prime}$ forms an inverse semigroup and a basis for the topology of $G$;
this holds for the family of all open bisections that trivialise the twist, by
the proof of [BussExel:Regular.Fell.Bundle]*Theorem 7.2, and for the family of
all compact open bisections because $G$ is ample. Since
$V=\bigsqcup_{i=1}^{n}s(U_{i})=\bigsqcup_{j=n+1}^{n+m}s(U_{j})$ is compact,
for each $i=1,\dotsc,n$ we may find a finite family of sets
$(U_{i,j})_{j=1}^{n_{i}}\subseteq S^{\prime}$ such that
$\bigcup_{j=1}^{n_{i}}U_{i,j}\subseteq U_{i}$ and
$V=\bigcup_{i=1}^{n}\bigcup_{j=1}^{n_{i}}s(U_{i,j})$. Since the bisections
$(U_{i,j})_{j=1}^{n_{i}}\subseteq U_{i}$ are closed and open, we may arrange
that the sets $(s(U_{i,j}))_{j=1}^{n_{i}}$ are pairwise disjoint. Then the
sets $(U_{i,j})_{i=1,j=1}^{n,n_{i}}$ are pairwise disjoint, and since
$\bigcup_{j=1}^{n_{i}}U_{i,j}\subseteq U_{i}$, for $i=1,\dotsc,n$, also
$(r(U_{i,j}))_{i=1,j=1}^{n,n_{i}}$ are pairwise disjoint. We put
$a_{i,j}\mathrel{\vcentcolon=}1_{U_{i,j}}$, for $i=1,\dots,n$,
$j=1,\dotsc,n_{i}$. By the choice of bisections in $S^{\prime}$, we may treat
$a_{i,j}$ as an element of the space $\textup{C}_{\textup{c}}(U_{ij})$ of
sections of the line bundle over $G$. By the construction of the Fell bundle
over $\widehat{A}\rtimes S$, by passing if necessary to smaller sets, we may
assume that each space $\textup{C}_{\textup{c}}(U_{i,j})$ is contained in
$\mathcal{E}_{t_{ij}}$ for some $t_{ij}\in S$. Hence
$a_{i,j}\in\mathcal{E}_{t_{ij}}$ for all $i,j$. Using the Fell bundle
structure, we get
$\sum_{i=1,j=1}^{n,n_{i}}a_{i,j}^{*}\cdot
a_{i,j}=\sum_{i=1,j=1}^{n,n_{i}}1_{s(U_{i,j})}=1_{V}=a.$
Similarly, we get $\sum_{i=n+1,j=1}^{n+m,n_{i}}a_{i,j}^{*}\cdot a_{i,j}=a$ and
$a_{i,j}^{*}a_{i^{\prime},j^{\prime}}=0$ for all
$(i,j)\neq(i^{\prime},j^{\prime})$. Hence $a$ is $\mathcal{E}$-paradoxical. ∎
###### Corollary 5.29.
Let $(G,\Sigma)$ be an essentially exact twisted groupoid where $G$ is ample
and residually topologically free with locally compact Hausdorff
$X\mathrel{\vcentcolon=}G^{0}$. If every compact open subset of $X$ is
$(2,1)$-paradoxical, then the essential $\textup{C}^{*}$-algebra
$\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)$ is purely infinite (and has the
ideal property).
###### Proof.
View $\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)$ as the essential crossed
product by an inverse semigroup action $\mathcal{E}$ on $\textup{C}_{0}(X)$ as
in Examples 3.21 and 3.22. The assertion follows from Proposition 5.28 and
Corollary 5.22. ∎
###### Remark 5.30.
When $G$ is Hausdorff, then
$\textup{C}^{*}_{\mathrm{ess}}(G,\Sigma)=\textup{C}^{*}_{\mathrm{r}}(G,\Sigma)$
and $(G,\Sigma)$ is essentially exact if and only if it is inner exact. Thus
Corollary 5.29 generalises the pure infiniteness criteria in [Boenicke-
Li:Ideal, Rainone-Sims:Dichotomy], where the authors considered Hausdorff
ample groupoids without a twist. They proved, in addition, that if the type
semigroup associated to $G$ is almost unperforated, then the implication in
Corollary 5.29 may be reversed. We will generalise this and some other results
of Ma [Ma:Purely_infinite_groupoids] to étale twisted groupoids in the
forthcoming paper [Kwasniewski-Meyer:Type_semigroups].
## References |
$S^{\prime}$-locally equivalent, i.e., the map $D_{n}^{S}F\to
D_{n}^{S^{\prime}}F$ is an $S^{\prime}$-local equivalence. For (2), since $F$
is reduced [We95, Corollary 8.3] implies that there is a commutative diagram
${{T_{n}^{S}F}}$${{T_{n-1}^{S}F}}$${{R_{n}^{S}F}}$${{T_{n}^{S^{\prime}}F}}$${{T_{n-1}^{S^{\prime}}F}}$${{R_{n}^{S^{\prime}}F}}$
in which both rows are homotopy fibre sequences. The map $R_{n}^{S}F\to
R_{n}^{S^{\prime}}F$ is an $S^{\prime}$-local equivalence by part (1), and the
map $T_{0}^{S}F\to T_{0}^{S^{\prime}}F$ is also an $S^{\prime}$-local
equivalence since $F$ is reduced. An induction argument on the degree of
polynomials yields the result. ∎
We obtain corollaries for both stable and unstable Bousfield classes.
###### Lemma 9.1.6.
Let $E$ and $E^{\prime}$ be spectra and $F$ an orthogonal functor. If $\langle
E\rangle\leq\langle E^{\prime}\rangle$, then
1. (1)
there is an $E^{\prime}$-local equivalence $D_{n}^{E}F\to
D_{n}^{D^{\prime}}F$; and,
2. (2)
if $F$ is reduced, then the $E$-local Weiss tower of $F$ is
$E^{\prime}$-locally equivalent to the $E^{\prime}$-local Weiss tower of $F$.
###### Lemma 9.1.7.
Let $W$ and $W^{\prime}$ be based spaces and $F$ an orthogonal functor. If
$\langle W\rangle\leq\langle W^{\prime}\rangle$, then
1. (1)
there is an $W^{\prime}$-local equivalence $D_{n}^{W}F\to
D_{n}^{W^{\prime}}F$; and,
2. (2)
if $F$ is reduced, then the $W$-local Weiss tower of $F$ is
$W^{\prime}$-locally equivalent to the $W^{\prime}$-local Weiss tower of $F$.
#### 9.2. The Telescope Conjecture
The height $n$ Telescope Conjecture relates the $T(n)$-localization and
$K(n)$-localization of spectra. There are numerous equivalent formalisations
of the conjecture see e.g., [BarthelChromaticConjectures, Proposition 3.6] and
we choose the following as it best suits any possible interaction with the
calculus.
###### Conjecture 9.2.1 (The height $n$ Telescope Conjecture).
Let $n\geq 0$. The Bousfield class of $T(n)$ agrees with the Bousfield class
of $K(n)$.
###### Lemma 9.2.2.
Let $n\geq 0$. The validity of the height $n$ Telescope Conjecture implies an
equality of model structures
${\sf{Fun}}(\mathcal{J}_{0},L_{K(n)}\operatorname{\sf{Top}_{\ast}})={\sf{Fun}}(\mathcal{J}_{0},L_{T(n)}\operatorname{\sf{Top}_{\ast}}),$
${\mathsf{Poly}^{\leq
n}}(\mathcal{J}_{0},L_{K(n)}\operatorname{\sf{Top}_{\ast}})={\mathsf{Poly}^{\leq
n}}(\mathcal{J}_{0},L_{T(n)}\operatorname{\sf{Top}_{\ast}}),$
${\mathsf{Homog}^{n}}(\mathcal{J}_{0},L_{K(n)}\operatorname{\sf{Top}_{\ast}})={\mathsf{Homog}^{n}}(\mathcal{J}_{0},L_{T(n)}\operatorname{\sf{Top}_{\ast}}).$
###### Proof.
The Telescope Conjecture implies that the Bousfield class of $T(n)$ and the
Bousfield class of $K(n)$, agree, hence the result follows by Theorem 9.1.4. ∎
The following is an immediate corollary to Theorem 9.1.1.
###### Theorem 9.2.3.
Let $n\geq 0$. The height $n$ Telescope Conjecture is equivalent to the
statement that for every orthogonal functor $F$ the $K(n)$-local Weiss tower
of $F$ and the $T(n)$-local Weiss tower of $F$ agree.
This provides new insight into the the height $n$ Telescope Conjecture. For
example, to find a counterexample it now suffices to find an orthogonal
functor such that one corresponding term in the $K(n)$-local and $T(n)$-local
Weiss towers disagree. This can also be seen through the spectral sequences
associated to the local Weiss towers. The $K(n)$-local and $T(n)$-local Weiss
towers of an orthogonal functor $F$ produce two spectral sequences,
$\pi_{t-s}D_{s}^{K(n)}F(V)\cong\pi_{t-s}((S^{\mathbb{R}^{s}\otimes
V}\wedge\partial_{s}^{K(n)}F)_{hO(n)})\Rightarrow\pi_{\ast}\underset{d}{\operatorname{\mathrm{holim}}}~{}T_{d}^{K(n)}F(V),$
and,
$\pi_{t-s}D_{s}^{T(n)}F(V)\cong\pi_{t-s}((S^{\mathbb{R}^{s}\otimes
V}\wedge\partial_{s}^{T(n)}F)_{hO(n)})\Rightarrow\pi_{\ast}\underset{d}{\operatorname{\mathrm{holim}}}~{}T_{d}^{T(n)}F(V),$
These are closely related to the telescope conjecture as follows.
###### Lemma 9.2.4.
Let $F$ be an orthogonal functor. If the height $n$ Telescope Conjecture
holds, then for all $r\geq 1$, the $E_{r}$-page of the $T(n)$-local Weiss
spectral sequence is isomorphic to the $E_{r}$-page of the $K(n)$-local Weiss
spectral sequence. In particular, the homotopy limit of the $T(n)$-local Weiss
tower is levelwise weakly equivalent to the homotopy limit of the $K(n)$-local
Weiss tower.
###### Proof.
It suffices to prove the claim for $r=1$. The validity of the height $n$
Telescope Conjecture implies that there is a natural transformation
$L_{K(n)}\to L_{T(n)}$. This natural transformation induces a map
$D_{d}^{K(n)}F\to D_{d}^{T(n)}F$, which by Theorem 9.2.3 is an levelwise weak
equivalence. It hence suffices to show that the natural map $D_{d}^{K(n)}F\to
D_{d}^{T(n)}F$ induces a map on the $E_{1}$-pages of the spectral sequences,
that is, we have to show that the induced diagram
${{\pi_{t-s}D_{s}^{K(n)}F(V)}}$${{\pi_{t-s+1}D_{s+1}^{K(n)}F(V)}}$${{\pi_{t-s}D_{s}^{T(n)}F(V)}}$${{\pi_{t-s+1}D_{s+1}^{T(n)}F(V)}}$$\scriptstyle{d_{1}^{K(n)}}$$\scriptstyle{d_{1}^{T(n)}}$
commutes for all $s$ and $t$. This follows from the commutativity of the
induced diagram of long exact sequences induced by the diagram of homotopy
fibre sequences,
${{D_{s}^{K(n)}F(V)}}$${{T_{s}^{K(n)}F(V)}}$${{T_{s-1}^{K(n)}F(V)}}$${{D_{s}^{T(n)}F(V)}}$${{T_{s}^{T(n)}F(V)}}$${{T_{s-1}^{T(n)}F(V)}}$
and the construction of the $d_{1}$-differential in the homotopy spectral
sequence associated to a tower of fibrations. ∎
### 10\. Postnikov sections
The classical theory of Postnikov sections of based spaces is obtained by the
nullification with respect to the spheres, that is, given a based space $A$,
the $k$-th Postnikov section of $A$ is the nullification of $A$ at $S^{k+1}$,
i.e., $P_{k}A=P_{S^{k+1}}A$. Given a diagram of (simplicial, left proper,
combinatorial) model categories, Barwick [BarwickLeftRight, Section 5
Application 1] and Bergner [BergnerHomotopyLimits] develop a general machinery
for producing a model structure which captures the homotopy theory of the
homotopy limit of the diagram of model categories. Gutiérrez and Roitzheim
[GR16, Section 4] applied this to the study of Postnikov sections for model
categories, which recovers the classical theory when $\mathcal{C}$ is the Kan-
Quillen model structure on simplicial sets. We consider the relationship
between Postnikov sections and orthogonal calculus via our local calculus.
#### 10.1. A combinatorial model for calculus
The current theory of homotopy limits of model categories requires that the
model categories in question be combinatorial, i.e., locally presentable and
cofibrantly generated. Since the category of based compactly generated weak
Hausdorff spaces is not locally presentable the Quillen model structure is not
combinatorial and hence none of our model categories for orthogonal functors
are either. We invite the reader to take for granted that all of our cellular
model categories may be replaced by combinatorial model categories by starting
with a combinatorial model for the Quillen model structure on based spaces,
and hence skip directly to Subsection 10.2.
We spell out the details of these combinatorial replacements here. We replace
compactly generated weak Hausdorff spaces with _$\Delta$ -generated spaces_; a
particular full subcategory of the category of topological spaces, which were
developed by Vogt [VogtConvenientHomotopy] and unpublished work of Smith,
which are surveyed by Dugger in [DuggerDeltaGeneratedSpaces]. The category of
$\Delta$-generated spaces may be equipped with a model structure analogous to
the Quillen model structure on compactly generated weak Hausdorff spaces.
###### Lemma 10.1.1.
There is a model category structure on the category of $\Delta$-generated
spaces with weak equivalences the weak homotopy equivalences and fibrations
the Serre fibrations. This model structure is combinatorial, proper and
topological.
###### Proof.
The existence of the model structure follows from [DuggerDeltaGeneratedSpaces,
Subsection 1.9]. The locally presentable (and hence combinatorial) property
follows from [FRDeltaGeneratedSpaces, Corollary 3.7] . ∎
The category of based $\Delta$-generated spaces is a convenient model category
for doing homotopy theory in the following sense, see
[DuggerDeltaGeneratedSpaces, Subsection 1.9].
###### Lemma 10.1.2.
The model category of based $\Delta$-generated spaces is Quillen equivalent to
the Quillen model structure on based compactly generated weak Hausdorff
spaces.
The combinatorial model for spaces transfers to categories of functors and we
obtain a projective model structure on the category of orthogonal functors
which is Quillen equivalent to our original projective model structure but is
now combinatorial.
A left or right Bousfield localization of a combinatorial model category is
again combinatorial, hence the $n$-polynomial, $n$-homogeneous and local
versions of these model categories are all combinatorial when we begin with
the combinatorial model for the projective model structure on orthogonal
fucntors.
###### Hypothesis 10.1.3.
For the remainder of this section, we will assume that all our model
structures are combinatorial, since they are all Quillen equivalent to
combinatorial model categories using the combinatorial model for based spaces.
#### 10.2. The model structure of $k$-types in orthogonal functors
Denote by $I$ the set of generating cofibrations of the projective model
structure of orthogonal functors, and denote by $W_{k}$ the set
$I\Box\\{S^{k+1}\to D^{k+2}\\}$, that is, the set of maps of the form
$B\wedge S^{k+1}\coprod_{A\wedge S^{k+1}}A\wedge D^{k+2}\longrightarrow
B\wedge D^{k+2},$
where $A\to B$ is a map in $I$. The model category of $k$-types in
$\sf{Fun}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$ is the left
Bousfield localization of the projective model structure at
$I\Box\\{S^{k+1}\to D^{k+2}\\}$ used by Gutiérrez and Roitzheim [GR16] to
model Postnikov sections.
###### Proposition 10.2.1.
Let $k\geq 0$. Under Hypothesis 10.1.3, the model structure of $k$-types in
orthogonal functors is identical to the $S^{k+1}$-local model structure, that
is, there is an equality of model structures,
$P_{k}{\sf{Fun}}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}}):=L_{W_{k}}{\sf{Fun}}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})={\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}).$
###### Proof.
It suffices to show that both model structures have the same fibrant objects
since the cofibrations in both model structures are identical. To see this,
note that by examining the pushout product we can rewrite the set $W_{k}$ as
$W_{k}=\\{\mathcal{J}_{0}(U,-)\wedge
S^{n+k+1}_{+}\longrightarrow\mathcal{J}_{0}(U,-)\wedge D^{n+k+2}_{+}\mid n\geq
0,U\in\mathcal{J}_{0}\\}.$
It follows by an adjunction argument that an orthogonal functor $Z$ is
$W_{k}$-local if and only if $\pi_{i}Z(U)$ is trivial for all $i\geq k+1$ and
all $U\in\mathcal{J}_{0}$. This last condition is equivalent to being
levelwise $S^{k+1}$-local. ∎
#### 10.3. The model structure of $k$-types in spectra
###### Proposition 10.3.1.
Let $k\geq 0$. Under Hypothesis 10.1.3, there is an equality of model
structures between the model category of $k$-types in spectra, and the
stablisation of $S^{k+1}$-local spaces, that is,
$P_{k}{\sf{Sp}}:=L_{W_{k}}{\sf{Sp}}={\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}).$
###### Proof.
Both model structures can be described as particular left Bousfield
localizations of the stable model structure on spectra, hence have the same
cofibrations. The proof reduces to the fact that the model structures have the
same fibrant objects. To see this, note that the fibrant objects of
$P_{k}{\sf{Sp}}$ are the $k$-truncated $\Omega$-spectra, and the fibrant
objects of ${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ are the
levelwise $k$-truncated $\Omega$-spectra. Since both fibrant objects are
$\Omega$-spectra a connectivity style argument yields that an
$\Omega$-spectrum is $k$-truncated if and only if it is levelwise
$k$-truncated, and hence both model structures have the same fibrant objects.
∎
#### 10.4. Postnikov reconstruction of orthogonal functors
The collection of $S^{k+1}$-local model structures on the category of
orthogonal functors assembles into a tower of model categories666A tower of
model categories is a special instance of a left Quillen presheaf, that is a
diagram of the form $F\colon\mathcal{J}^{\mathrm{op}}\to\sf{MCat}$ for some
small indexing category $\mathcal{J}$.
$\displaystyle{\sf{P}}_{\bullet}\colon\mathbb{N}^{\mathrm{op}}$
$\displaystyle\longrightarrow{\sf{MCat}},$ $\displaystyle k$
$\displaystyle\longmapsto{\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}),$
where $\sf{MCat}$ denotes the category of model categories and left Quillen
functors. The homotopy limit of this tower of model categories recovers the
projective model structure on orthogonal functors. The existence of a model
structure which captures the homotopy theory of the limit of these model
categories follows from [GR16, Proposition 2.2]. In particular, the homotopy
limit model structure is a model structure on the category of sections777A
section $X_{\bullet}$ of the tower ${\sf{P}}_{\bullet}$ is a sequence
$\cdots\longrightarrow X_{k}\longrightarrow
X_{k+1}\longrightarrow\cdots\longrightarrow X_{0},$ of orthogonal functors,
and a morphism of sections $f\colon X_{\bullet}\to Y_{\bullet}$ is given by
maps of orthogonal functors $f_{k}\colon X_{k}\to Y_{k}$ for all $k\geq 0$
subject to a commutative ladder condition. of the diagram ${\sf{P}}_{\bullet}$
formed by right Bousfield localizing the injective model structure in which a
map of sections is a weak equivalence or cofibration if it is a levelwise weak
equivalence or cofibration respectively.
###### Lemma 10.4.1 ([GR16, Theorem 1.3 $\&$ Proposition 2.2]).
There is a combinatorial model structure on the category of sections of
${\sf{P}}_{\bullet}$ where a map $f_{\bullet}\colon X_{\bullet}\to
Y_{\bullet}$ is a fibration if and only if $f_{0}$ is a fibration in
${\sf{Fun}}(\mathcal{J}_{0},L_{S^{1}}\operatorname{\sf{Top}_{\ast}})$ and for
every $k\geq 1$ the induced map
${{X_{k}}}$${{Y_{k}\times_{Y_{k-1}}X_{k-1}}}$${{X_{k-1}}}$${{Y_{k}}}$${{Y_{k-1}}}$$\scriptstyle{f_{k-1}}$$\scriptstyle{f_{k}}$
indicated by a dotted arrow in the above diagram is a fibration in
${\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$. A
section $X_{\bullet}$ is cofibrant if and only if $X_{n}$ is cofibrant in
${\sf{Fun}}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})$ and for every
$k\geq 0$, the map $X_{k+1}\to X_{k}$ is a weak equivalence in
${\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$. A map
of cofibrant sections is a weak equivalence if and only if the map is a weak
equivalence in
${\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for
each $k\geq 0$. We will refer to this model structure as the homotopy limit
model structure and denote it by
$\operatorname{\mathrm{holim}}{\sf{P}}_{\bullet}$.
###### Proposition 10.4.2.
Under Hypothesis 10.1.3 the adjoint pair
$\textstyle{\rm{const}:{\sf{Fun}}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}{\sf
P}_{\bullet}:\lim}$
is a Quillen equivalence.
###### Proof.
The adjoint pair exists, and is a Quillen adjunction by [GR16, Lemma 2.4].
To see that the adjoint pair is a Quillen equivalence let $X_{\bullet}$ be a
cofibrant and fibrant section in the homotopy limit model structure. Showing
that $\rm{const}\lim X_{\bullet}\to X_{\bullet}$ is a weak equivalence is
equivalent to showing that the map $\lim X_{\bullet}\to X_{k}$ is a weak
equivalence in
${\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for
all $k\geq 0$. This is in turn, equivalent to the map $(\lim
X_{\bullet})(U)\to X_{k}(U)$ being a weak equivalence in
$L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}$ for all $k\geq 0$. Since limits in
functor categories are computed levelwise, the fact that the unit is a weak
equivalence follows from [GR16, Theorem 2.5]. A similar argument, shows that
the counit is also a weak equivalence. ∎
#### 10.5. Postnikov reconstruction for spectra with an $O(n)$-action
The aim is to show that similar reconstruction theorems may be obtained for
the $n$-homogeneous functors. We first start by investigating analogous
theorems for spectra and show that such reconstructions are compatible with
the zigzag of Quillen equivalences between spectra with an $O(n)$-action and
the $n$-homogeneous model structure.
###### Lemma 10.5.1.
The functor
$\displaystyle{\sf{P}}_{\bullet}^{{\sf{Sp}}}\colon\mathbb{N}^{\mathrm{op}}$
$\displaystyle\longrightarrow{\sf{MCat}},$ $\displaystyle k$
$\displaystyle\longmapsto{\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}),$
defines a left Quillen presheaf.
###### Proof.
This follows from Proposition 10.3.1 and [GR16, Subsection 2.1] since the
stablisation of $S^{k+1}$-local spaces is precisely the model structure of
$k$-types in spectra. ∎
###### Remark 10.5.2.
Alternatively Lemma 10.5.1 may be proved by exhibiting that the adjoint pair
$\textstyle{\mathds{1}:{\sf{Sp}}(L_{S^{k+2}}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$
is a Quillen adjunction. This fact follows from the facts that both model
structures have the same cofibrations and a $S^{k+1}$-local space is
$S^{k+2}$-local as $\langle\Sigma W\rangle\leq\langle W\rangle$ for all based
spaces $W$, see e.g., [BousfieldPeriodicity, §9.9].
This left Quillen presheaf is ‘convergent’ in the following sense.
###### Proposition 10.5.3.
Under Hypothesis 10.1.3 the adjoint pair
$\textstyle{\rm{const}:{\sf{Sp}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}}:\lim}$
is a Quillen equivalence.
###### Proof.
The fact that the adjoint pair is a Quillen adjunction follows from [GR16,
Lemma 2.4].
The left adjoint reflects weak equivalences between cofibrant objects. Indeed,
if $X\to Y$ is a map between cofibrant spectra $X$ and $Y$, such that
$\mathrm{const}(X)\longrightarrow\mathrm{const}(Y),$
is a weak equivalence in
$\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}}$, then
$\mathrm{const}(X)\longrightarrow\mathrm{const}(Y),$
is a weak equivalence in
$\mathrm{Sect}(\mathbb{N},{\sf{P}}_{\bullet}^{{\sf{Sp}}})$ by the colocal
Whitehead’s theorem and the fact that the left adjoint is left Quillen and
thus preserves cofibrant objects. It follows that for each $k\in\mathbb{N}$,
the induced map
$\mathrm{const}(X)_{k}\longrightarrow\mathrm{const}(Y)_{k},$
is a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$, that is, $X\to Y$ is a
weak equivalence in ${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for
all $k$. Unpacking the definition of a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ and using the fact that
the right adjoint is a right Quillen functor and hence preserves weak
equivalences between fibrant objects, we see that the induced map
$\lim~{}P_{k}X\longrightarrow\lim~{}P_{k}Y,$
is a weak equivalence in ${\sf{Sp}}$, and hence, so is the map $X\to Y$.
It is left to show that the derived counit is an isomorphism. Let
$Y_{\bullet}$ be bifibrant in
$\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}}$. The
condition that the counit applied to $Y_{\bullet}$ is a weak equivalence is
equivalent to asking for the map
$\lim_{\geq k}~{}P_{k}Y_{\bullet}\longrightarrow Y_{k},$
to be a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for all
$k\in\mathbb{N}$. The structure maps of $Y_{\bullet}$ induce a map of towers
${\cdots}$${{Y_{j}}}$${\cdots}$${{Y_{k+3}}}$${{Y_{k+2}}}$${{Y_{k+1}}}$${\cdots}$${{Y_{k+1}}}$${\cdots}$${{Y_{k+1}}}$${{Y_{k+1}}}$${{Y_{k+1}}}$$\scriptstyle{=}$
in which each vertical arrow is a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$. This map of towers
induces a map
${0}$${{\lim^{1}_{\geq k}~{}\pi_{i+1}(Y_{\bullet})}}$${{\pi_{i}(\lim_{\geq
k}~{}Y_{\bullet})}}$${{\lim_{\geq
k}~{}\pi_{i}(Y_{\bullet})}}$${0}$${0}$${{\lim^{1}_{\geq
k}~{}\pi_{i+1}(Y_{k+1})}}$${{\pi_{i}(\lim_{\geq k}~{}Y_{k+1})}}$${{\lim_{\geq
k}~{}\pi_{i}(Y_{k+1})}}$${0}$
of short exact sequences. For $0\leq i<n$ the left and right hand side maps
are isomorphisms hence the map
$\lim_{\geq k}~{}Y_{\bullet}\longrightarrow Y_{k+1},$
is a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for all $k$, and it
follows that the required map
$\lim_{\geq k}~{}Y_{\bullet}\longrightarrow Y_{k+1}\longrightarrow Y_{k},$
is a weak equivalence in
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})$ for all $k$. ∎
A similar justification to Lemma 10.5.1 provides a left Quillen presheaf
$\displaystyle{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}\colon\mathbb{N}^{\mathrm{op}}$
$\displaystyle\longrightarrow{\sf{MCat}},$ $\displaystyle k$
$\displaystyle\longmapsto{\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})[O(n)],$
where ${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})[O(n)]$ is the
category of $O(n)$-objects in the category of $k$-types in spectra. This is
equivalent to the category of $k$-types in spectra with an $O(n)$-action.
###### Lemma 10.5.4.
Let $k,n\geq 0$. The model structure of the Borel stablisation of
$S^{k+1}$-local spaces with an $O(n)$-action is identical to the model
structure of $k$-types in the category of spectra with an $O(n)$-action, that
is, there is an equality of model structures
${\sf{Sp}}(L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}})[O(n)]=P_{k}({\sf{Sp}}[O(n)]).$
###### Proof.
Both model structures are identical to model structures transferred through
the same adjunction from identical model structures. ∎
As a corollary to Proposition 10.5.3, we obtain that the induced left Quillen
presheaf on spectra with an $O(n)$-action is also suitably convergent.
###### Corollary 10.5.5.
Under Hypothesis 10.1.3 the adjoint pair
$\textstyle{\rm{const}:{\sf{Sp}}[O(n)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}:\lim}$
is a Quillen equivalence.
#### 10.6. Postnikov reconstruction for the intermediate categories
Our attention now turns to the intermediate categories. We construct an
analogous left Quillen presheaf and show that it is also convergent in a
fashion which interacts well with the convergent left Quillen presheaf for
spectra with an $O(n)$-action.
###### Lemma 10.6.1.
The functor
$\displaystyle{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}\colon\mathbb{N}^{\mathrm{op}}$
$\displaystyle\longrightarrow{\sf{MCat}},$ $\displaystyle k$
$\displaystyle\longmapsto
L_{S^{k+1}}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}}),$
defines a left Quillen presheaf.
###### Proof.
As before, it suffices to show that there is an equality of model structures
between the $S^{k+1}$-local $n$-stable model structure and the model structure
of $k$-types in
${\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})$. The
proof of which is completely analogous to the case for spectra, see Lemma
10.5.1. ∎
###### Remark 10.6.2.
Since the $S^{k+1}$-local $n$-stable model structure agrees with the model
structure of $k$-types, we will denote both model structure by
$P_{k}{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})$.
The homotopy limit of the left Quillen presheaf of Lemma 10.6.1 agrees with
the homotopy limit of the left Quillen presheaf of Lemma 10.5.1, in the sense
that the homotopy limit model categories are Quillen equivalent. In detail,
the adjunction
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
59.48582pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}:{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
83.48582pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.48582pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\sf{Sp}}[O(n)]:(\alpha_{n})^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.48584pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$
of [BO13, §8] induces an adjunction
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
75.50989pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}^{\mathbb{N}}:\sf{Fun}(\mathbb{N},{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
99.50989pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
99.50989pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\sf{Fun}(\mathbb{N},{\sf{Sp}}[O(n)]):(\alpha_{n}^{*})^{\mathbb{N}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
75.5099pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$
where $(\alpha_{n}^{*})^{\mathbb{N}}=(\alpha_{n})^{*}\circ(-)$. This
adjunction in turn induces an adjunction
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
32.62474pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}^{\mathbb{N}}:\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
56.62474pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
56.62474pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}:(\alpha_{n}^{*})^{\mathbb{N}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.62476pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>
###### Proposition 10.6.3.
Under Hypothesis 10.1.3 the adjoint pair
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
32.62474pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}^{\mathbb{N}}:\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
56.62474pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
56.62474pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}:(\alpha_{n}^{*})^{\mathbb{N}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.62476pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$
is a Quillen equivalence.
###### Proof.
Fibrations of the homotopy limit model structure of
${\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}$ are precisely the fibrations of the
injective model structure on the category of sections of
${\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}$ since the homotopy limit model
structure is a right Bousfield localization of the injective model structure.
A similar characterisation holds for the left Quillen presheaf
${\sf{P}}_{\bullet}^{\mathcal{J}_{n}}$, hence to show that the right adjoint
preserves fibrations it suffices to show that the left adjoint preserves
acyclic cofibrations of the injective model structure on the categories of
sections. To see this, note that the adjunction
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
59.48582pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\alpha_{n})_{!}:{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
83.48582pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
83.48582pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\sf{Sp}}[O(n)]:(\alpha_{n})^{*}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.48584pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$
is a Quillen adjunction and hence, so to is the induced adjunction on the
injective model structures on the categories of sections.
To show that the left adjoint preserves cofibrations it suffices to show that
cofibrations between cofibrant objects are preserved. As the homotopy limit
model structures are right Bousfield localizations [Hi03, Proposition
3.3.16(2)] implies that cofibrations between cofibrant objects are
cofibrations of the injective model structures on the categories of sections
which by the analogous reasoning as above are preserved by the left adjoint.
This yields that the adjunction in question is a Quillen adjunction.
To show that the adjunction is a Quillen equivalence notice that the right
adjoint reflects weak equivalences between cofibrant objects by the colocal
Whitehead’s Theorem [Hi03, Theorem 3.2.13(2)], and the fact that the induced
adjunction on the injective model structures on the categories of sections is
a Quillen equivalence since for
$B_{\bullet}\in{\sf{Sect}}(\mathbb{N},{\sf{P}}_{\bullet}^{\mathcal{J}_{n}})$
and
$X_{\bullet}\in{\sf{Sect}}(\mathbb{N},{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]})$,
a map $B_{\bullet}\to(\alpha_{n}^{*})^{\mathbb{N}}X_{\bullet}$ is a weak
equivalence if and only if for each $k\in\mathbb{N}$, the map
$B_{k}\to(\alpha_{n}^{*})^{\mathbb{N}}X_{k}$ is a weak equivalence of spectra,
which in turn happens if and only if the adjoint map $(\alpha_{n})_{!}B_{k}\to
X_{k}$ is an $n$-stable equivalence, which is precisely the condition that the
adjoint map $(\alpha_{n})_{!}^{\mathbb{N}}B_{\bullet}\to X_{\bullet}$ is a
weak equivalence.
It is left to show that the derived counit is an isomorphism. Let
$Y_{\bullet}$ be bifibrant in the homotopy limit model structure of the left
Quillen presheaf ${\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}$. Then the derived
counit
$(\alpha_{n})_{!}^{\mathbb{N}}~{}\widehat{c}~{}((\alpha_{n}^{*})^{\mathbb{N}}Y_{\bullet})\longrightarrow
Y_{\bullet},$
is a map between cofibrant objects, hence a weak equivalence in the homotopy
limit model structure if and only if a weak equivalence in the injective model
structure on the category of sections i.e., if and only if for each
$k\in\mathbb{N}$, the induced map
$(\alpha_{n})_{!}(\alpha_{n})^{*}Y_{k}\longrightarrow Y_{k},$
is a weak equivalence. This last is always a weak equivalence by [BO13,
Proposition 8.3]. ∎
As a corollary, we see that the left Quillen presheaf
${\sf{P}}_{\bullet}^{\mathcal{J}_{n}}$ is convergent.
###### Corollary 10.6.4.
Under hypothesis 10.1.3 the adjoint pair
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
61.35667pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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85.35667pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
85.35667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}:\lim}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
61.35667pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$
is a Quillen equivalence.
###### Proof.
Consider the commutative diagram
${{{\sf{Fun}}_{O(n)}(\mathcal{J}_{n},O(n)\operatorname{\sf{Top}_{\ast}})}}$${{{\sf{Sp}}[O(n)]}}$${{\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}}}$${{\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{{\sf{Sp}}[O(n)]}}}$$\scriptstyle{(\alpha_{n})_{!}}$$\scriptstyle{(\alpha_{n})^{\ast}}$$\scriptstyle{\rm{const}}$$\scriptstyle{\lim}$$\scriptstyle{\rm{const}}$$\scriptstyle{\lim}$$\scriptstyle{(\alpha_{n})_{!}^{\mathbb{N}}}$$\scriptstyle{(\alpha_{n}^{\ast})^{\mathbb{N}}}$
of Quillen adjunctions in which three out of the four adjoint pairs are
Quillen equivalences by [BO13, Proposition 8.3], Corollary 10.5.5 and
Proposition 10.6.3. It follows since Quillen equivalences satisfy the $2$-out-
of-$3$ property, that the remaining Quillen adjunction is a Quillen
equivalence. ∎
#### 10.7. Postnikov reconstruction for homogeneous functors
Using the same approach as we have just employed from moving from spectra with
an $O(n)$-action to the intermediate categories we obtain similar results for
the homogeneous model structures. We choose to model $S^{k+1}$-local
$n$-homogeneous functors by the $S^{k+1}$-periodic $n$-homogeneous model
structures of Proposition 8.3.1.
###### Lemma 10.7.1.
The functor
$\displaystyle{\sf{P}}_{\bullet}^{\mathsf{Homog}^{n}}\colon\mathbb{N}^{\mathrm{op}}$
$\displaystyle\longrightarrow{\sf{MCat}},$ $\displaystyle k$
$\displaystyle\longrightarrow{\mathsf{Homog}^{n}}(\mathcal{J}_{0},P_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}),$
defines a left Quillen presheaf.
###### Proof.
It suffices to show that the adjoint pair
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{$\textstyle{\mathds{1}:\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{S^{k+2}}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
76.64969pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
76.64969pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Homog}^{n}(\mathcal{J}_{0},P_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
52.64969pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$
is a Quillen adjunction. The adjoint pair
$\textstyle{\mathds{1}:\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S^{k+2}}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$
is a Quillen adjunction since the composite of Quillen adjunctions is a
Quillen adjunction so the adjunction
$\textstyle{\mathds{1}:{\sf{Fun}}(\mathcal{J}_{0},L_{S^{k+2}}\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathsf{Poly}^{\leq
n}(\mathcal{J}_{0},L_{S^{k+1}}\operatorname{\sf{Top}_{\ast}}):\mathds{1}}$
is a Quillen adjunction, and by [Hi03, Proposition 3.3.18(1) $\&$ Theorem
3.1.6(1)], this composite Quillen adjunction extends to the $S^{k+2}$-local
$n$-polynomial model structure since $S^{k+1}$-local $n$-polynomial functors
are $S^{k+2}$-locally $n$-polynomial.
An application of [Hi03, Theorem 3.3.20(2)(a)] yields the desired result about
the $n$-homogeneous model structures. ∎
Similar proofs to Proposition 10.6.3 and Corollary 10.6.4 yeild the following
results relating the $n$-homogeneous model structure to the homotopy limit of
the tower of $S^{k+1}$-local $n$-homogeneous model structures.
###### Proposition 10.7.2.
Under Hypothesis 10.1.3 the adjunction
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
51.07452pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\operatorname{\mathsf{res}}_{0}^{n}/O(n))^{\mathbb{N}}:\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathsf{Homog}^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
75.07452pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
75.07452pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathcal{J}_{n}}:(\operatorname{\mathsf{ind}}_{0}^{n}\varepsilon^{*})^{\mathbb{N}}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
51.07452pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$
is a Quillen equivalence.
###### Corollary 10.7.3.
Under Hypothesis 10.1.3 the adjunction
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
52.02243pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{const}:\mathsf{Homog}^{n}(\mathcal{J}_{0},\operatorname{\sf{Top}_{\ast}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
76.02243pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
76.02243pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{holim}}~{}{\sf{P}}_{\bullet}^{\mathsf{Homog}^{n}}:\lim}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
52.02243pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$
is a Quillen equivalence.
### References |
# The ALMA REBELS Survey: The First Infrared Luminosity Function Measurement
at $\mathbf{z\sim 7}$
L. Barrufet1, P. A. Oesch1,2, R. Bouwens3, H. Inami4, L. Sommovigo5, H.
Algera4,6, E. da Cunha7<EMAIL_ADDRESS>M. Aravena8, P.
Dayal9, A. Ferrara5, Y. Fudamoto10,11, V. Gonzalez12,13 L. Graziani14,15 A.
Hygate3 I. de Looze16,17, T. Nanayakkara18, A. Pallottini5, R. Schneider14,15
M. Stefanon19,20 M. Topping21 and P. van Der Werf3
1Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290
Versoix, Switzerland
2Cosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen,
Jagtvej 128, København N, DK-2200, Denmark
3Leiden Observatory, Leiden University, PO Box 9500, 2300 RA Leiden, The
Netherlands
4 Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1,
Kagamiyama-Higashi, Hiroshima 739-8526, Japan
5 Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
6 National Astronomical Observatory of Japan, 2-21-1, Osawa, Mitaka, Tokyo,
Japan
7 International Centre for Radio Astronomy Research, University of Western
Australia, Stirling Hwy, Crawley, 26WA, 6009, Australia
8 Nucleo de Astronomia Facultad de Ingenieria y Ciencias, Universidad Diego
Portales, Av Ejercito 441, Santiago, Chile
9 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700,
AV Groningen, The Netherlands
10 Waseda Research Institute for Science and Engineering, Faculty of Science
and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555,
Japan
11National Astronomical Observatory of Japan, 2-21-1, Osawa, Mitaka, Tokyo,
Japan
12 Departmento de Astronomia, Universidad de Chile, Casilla 36-D, Santiago
7591245, Chile
13Centro de Astrofisica y Tecnologias Afines (CATA), Camino del Observatorio
1515, Las Condes, Santiago 7591245, Chile
14 Dipartimento di Fisica, Sapienza, Universita di Roma, Piazzale Aldo Moro 5,
I-00185 Roma, Italy
15 INAF/Osservatorio Astronomico di Roma, via Frascati 33, I-00078 Monte
Porzio Catone, Roma, Italy
16 Sterrenkundig Observatorium, Ghent University, Krijgslaan 281-59, 9000,
Gent, Belgium
17 Dept. of Physics & Astronomy, University College London, Gower Street,
London WC1E 6BT, United Kingdom
18 Centre for Astrophysics & Supercomputing, Swinburne University of
Technology, PO Box 218, Hawthorn, VIC 3112, Australia
19 Departament d’Astronomia i Astrofisica, Universitat de Valencia, C. Dr.
Moliner 50, E-46100 Burjassot, Valencia, Spain
cia, C. Dr. Moliner 50, E-46100 Burjassot, Valencia, Spain
20 Unidad Asociada CSIC "Grupo de Astrofisica Extragalactica y Cosmologia"
(Instituto de Fisica de Cantabria - Universitat de Valencia)
21 Steward Observatory University of Arizona, 933 N Cherry Ave, Tucson, AZ
85721, United States
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We present the first observational infrared luminosity function (IRLF)
measurement in the Epoch of Reionization (EoR) based on a UV-selected galaxy
sample with ALMA spectroscopic observations. Our analysis is based on the ALMA
large program Reionization Era Bright Emission Line Survey (REBELS), which
targets 42 galaxies at $\mathrm{z=6.4-7.7}$ with [CII] 158$\micron$ line
scans. 16 sources exhibit a dust detection, 15 of which are also
spectroscopically confirmed through the [CII] line. The IR luminosities of the
sample range from $\log L_{IR}/L_{\odot}=11.4$ to 12.2. Using the UVLF as a
proxy to derive the effective volume for each of our target sources, we derive
IRLF estimates, both for detections and for the full sample including IR
luminosity upper limits. The resulting IRLFs are well reproduced by a
Schechter function with the characteristic luminosity of $\log
L_{*}/L_{\odot}=11.6^{+0.2}_{-0.1}$. Our observational results are in broad
agreement with the average of predicted IRLFs from simulations at $z\sim 7$.
Conversely, our IRLFs lie significantly below lower redshift estimates,
suggesting a rapid evolution from $z\sim 4$ to $z\sim 7$, into the
reionization epoch. The inferred obscured contribution to the cosmic star-
formation rate density at $z\sim 7$ amounts to
$\mathrm{log(SFRD/M_{\odot}/yr/Mpc^{3})=-2.66^{+0.17}_{-0.14}}$ which is at
least $\sim$10% of UV-based estimates. We conclude that the presence of dust
is already abundant in the EoR and discuss the possibility of unveiling larger
samples of dusty galaxies with future ALMA and JWST observations.
###### keywords:
Galaxies: high-redshift, luminosity function. Infrared: galaxies
††pubyear: 2022††pagerange: The ALMA REBELS Survey: The First Infrared
Luminosity Function Measurement at $\mathbf{z\sim 7}$–The ALMA REBELS Survey:
The First Infrared Luminosity Function Measurement at $\mathbf{z\sim 7}$
## 1 Introduction
It is still a crucial open question in astrophysics when the first galaxies
formed and how they built up their mass. The continuous discovery of higher
redshift galaxies is pushing the boundaries of our knowledge of galaxy
evolution (e.g., Dunlop, 2013; Stark, 2016; Dayal & Ferrara, 2018; Schaerer et
al., 2022; Naidu et al., 2022b; Atek et al., 2022; Adams et al., 2022). In
particular, the discovery of a significant population of luminous and massive
galaxies at $\mathrm{z>9}$ has posed questions about the speed of early
stellar mass production (e.g. Oesch et al., 2016; Laporte et al., 2021; Naidu
et al., 2022a; Labbe et al., 2022).
Until recently, the knowledge of galaxies at $\mathrm{z>7}$ was mainly based
on rest-frame ultraviolet (UV) observations (Oesch et al., 2018a; Bouwens et
al., 2021a). These samples might not be complete, however, as they might miss
extremely dust obscured, but highly star-forming galaxies (e.g., Casey et al.,
2019).
From an observational point of view, the Atacama Large Millimeter Array (ALMA)
is the most powerful tool to study dust at high redshift (e.g., Capak et al.,
2015; Bouwens, 2016; Bowler et al., 2018; Béthermin et al., 2020). However,
the cost to obtain statistical samples of galaxies in the EoR results in the
fact that only a modest number of galaxies have been characterized in detail
so far (e.g., Watson et al., 2015; Smit et al., 2018; Laporte et al., 2019;
Faisst et al., 2020; Harikane et al., 2021; Schouws et al., 2022).
Furthermore, the study of dust at $\mathrm{2<z<6}$ was for a long time limited
to bright dusty galaxies such as submillimetre galaxies (SMGs; e.g., Gruppioni
et al., 2013; Wang et al., 2019b; Barrufet et al., 2020). However, ALMA is
bridging the gap between these extreme dusty massive galaxies and more
moderate star-forming galaxies (see Hodge & da Cunha 2020 for a review).
The recent observational improvements have allowed the discovery of the
emergence of high-z dusty galaxies at $\mathrm{z>6}$. In particular, Fudamoto
et al. (2021) has serendipitously detected two dusty galaxies at
$\mathrm{z_{spec}\sim 7}$ near massive neighbors at the same redshifts. This
shows that dusty galaxies in the EoR could be more common than previously
thought, which leads to the question of whether the number of dusty galaxies
at $\mathrm{z>6}$ is higher than expected (see also Barrufet et al., 2022;
Nelson et al., 2022; Rodighiero et al., 2022).
The possible underestimation of the number of dusty galaxies would have a
direct impact on the obscured Star Formation Rate Density (SFRD), which
remains uncertain at $\mathrm{z>3}$ (Casey et al., 2019). Several studies have
calculated the obscured SFRD at $\mathrm{z>5}$ based on serendipitous sources
resulting in largely differing conclusions (e.g Gruppioni et al., 2020;
Fudamoto et al., 2021; Talia et al., 2021; Casey et al., 2021; Viero et al.,
2022). While some studies find that 2mm selected, dusty galaxies contribute
$\mathrm{\sim 30\%}$ to the integrated star-formation rate density between
$\mathrm{3<z<6}$ (Casey et al., 2021), others report a significantly larger
obscured SFRD that remains constant over redshift (e.g., Gruppioni et al.,
2020; Talia et al., 2021). An approach to clarify the contribution of dust-
obscured star formation to the cosmic star formation history is to measure the
infrared luminosity function (IRLF) all the way into the EoR. The shape and
scale of the IRLF are crucial to understanding the abundance of dusty galaxies
and how rapidly dust is formed in the early universe. This directly affects
the fraction of star formation that is obscured in forming galaxies, and
thereby the formation (or rise) of metals.
Due to the wealth of rest-frame UV observations, the UV luminosity function
(UVLF) is well constrained up to $\mathrm{z\sim 9}$ (e.g., Bouwens et al.,
2007; Bouwens et al., 2015; Oesch et al., 2018b; Bowler et al., 2020; Bouwens
et al., 2021b), and we even have some information at $\mathrm{z\sim 9-10}$
(Oesch et al., 2018a; Harikane et al., 2022) and beyond now with JWST (e.g.
Naidu et al., 2022a; Donnan et al., 2022; Atek et al., 2022; Adams et al.,
2022; Finkelstein et al., 2022). In contrast, the IRLF is still quite
uncertain at high redshifts. Current measurements of the IRLF rely on small
numbers of dusty sources at $\mathrm{z>3.5}$ (e.g., Wang et al., 2019a;
Gruppioni et al., 2020). This leads to large uncertainties in the IRLF
parameters, including the faint-end slopes, and disagreements between
different survey results (e.g., Gruppioni et al., 2013; Koprowski et al.,
2017; Lim et al., 2020; Popping et al., 2020; Gruppioni et al., 2020).
The recent study of Zavala et al. (2021) compiled the results of several
surveys and combined those with semi-empirical modelling to constrain the
evolution of the IRLF out to $\mathrm{z>5}$, albeit with significant
uncertainties. However, an IRLF at $\mathrm{z\sim 7}$ has not been measured
directly using dust continuum observations yet. In this context, we use the
data from the Reionization Era Bright Emission Line Survey (REBELS), an ALMA
large program aimed at obtaining a statistical sample of normal star-forming
galaxies at $\mathrm{z>6.4}$ (see Bouwens et al. 2022 for details). REBELS has
increased the number of spectroscopically observed massive galaxies in the EoR
by a factor $\mathrm{\times\sim 4-5}$ compared to the previous literature
(Bouwens et al., 2021a). The same strategy of the REBELS selection was tested
in a pilot program presented in Schouws et al. (2022). This study showed the
potential of ALMA as a high redshift ‘machine’ and the six pilot galaxies are
also included in the main REBELS sample (Smit et al., 2018; Schouws et al.,
2021, 2022). While observations from the REBELS program were just recently
completed and analysis of the full data set now underway, its data have
already been used for a number of scientific analyses, including the discovery
of serendipitous dust-obscured sources at $z\sim 7$ (Fudamoto et al., 2021),
modelling the dust and ISM properties of $z>6$ galaxies (e.g., Sommovigo et
al., 2022; Dayal et al., 2022; Ferrara et al., 2022), measuring their detailed
specific SFRs (Topping et al., 2022), calculating their SFRD Algera et al.
(2022), estimating Ly$\alpha$ transmission around luminous sources in
overdense $z\sim 7$ environments (Endsley et al., 2022), and constraining the
neutral gas fraction out to the EoR (Heintz et al., 2022).
In this paper, we use this survey to calculate – for the first time – an IRLF
at $\mathrm{z\sim 7}$. In Section 2, we describe the ALMA observations and the
infrared luminosity calculations used in this work. The methodology for
calculating the IRLF and their values is described in Section 3. We present
the results on the obscured SFRD of REBELS galaxies in Section 4. We discuss
our results in Section 5 and present a summary and our conclusions in Section
6.
## 2 REBELS observations
### 2.1 ALMA observations and catalogue
In this work, we use data from REBELS (Bouwens et al., 2021a) which is a Cycle
7 ALMA large program of $\mathrm{\sim 40}$ UV bright galaxies at
$\mathrm{z>6.4}$. The selection was based on UV brightness
($\mathrm{-23<M_{UV}<-21.3}$) and photometric redshifts for galaxies
identified over a combined area of $\mathrm{\sim 7deg^{2}}$ in several fields
(see Bouwens et al. 2021a for details). This survey of spectral scan
observations identifies bright ISM cooling lines ([CII], [OIII]) while
simultaneously probing the dust-continuum in bands $\mathrm{158\ \mu m}$ and
$\mathrm{88\ \mu m}$, respectively, which is essential to derive the infrared
luminosity ($\mathrm{L_{IR}}$). Given its selection, the REBELS sample only
spans a limited range in redshift and UV luminosities. Even though it is UV
selected, the sample is representative of massive star-forming galaxies at
$\mathrm{z\sim 7}$, providing an extensive probe of ISM reservoirs in the EoR
(Bouwens et al., 2022; Ferrara et al., 2022).
In this work, we only focus on galaxies that were scanned for [CII], i.e.,
sources with $\mathrm{z_{phot}=6.4-7.7}$. The total sample used in this study
contains 42 galaxies with [CII] scanned, 16 of which with a dust continuum
detection at more than $3\sigma$. Notably, 15 of these 16 sources also do have
a significant [CII] emission line detection and thus a robust spectroscopic
redshift measurement (Inami et al., 2022).
### 2.2 Infrared luminosity from REBELS survey
In this section, we describe the infrared luminosity measurements from Inami
et al. (2022) and the average properties of the REBELS galaxies.
When deriving the infrared luminosities of our sample, we have to make an
assumption about the dust temperature. Estimating this based on a few
photometric detections in the far-infrared is very challenging. Sommovigo et
al. (2021) solve this difficulty using $\mathrm{L_{[CII]}}$ as a proxy for the
dust mass and the underlying continuum to constrain the dust temperature. This
is particularly useful for the REBELS survey, given that [CII] estimates (or
upper limits) are available for the full sample. Using these measurements,
Sommovigo et al. (2022) find an average dust temperature of
$\mathrm{T_{d}=46K}$ for the REBELS sample. Hence, Inami et al. (2022) assumed
a Spectral Energy Distribution (SED) with dust temperature and emissivity from
Sommovigo et al. (2022) ($\mathrm{T_{d}=46K}$ and $\mathrm{\beta=2}$
respectively) to calculate the infrared luminosity based on the ALMA dust
continuum flux. For the galaxies without dust continuum detection a 3$\sigma$
upper limit was derived both for the continuum flux and the corresponding
infrared luminosity. A cosmic microwave background correction was applied for
all galaxies, with and without dust detection. The correction depends on the
exact redshift, but lies in the range of $\mathrm{8-14\%}$ (see Inami et al.
2022 for details).
Figure 1: Infrared luminosity against UV absolute magnitude with the redshift
colour-coded for the REBELS (filled symbols) and ALPINE (empty symbols)
samples for both $\mathrm{3\sigma}$ detections (dots) and upper limits
(triangles). The REBELS sample does not show significant differences between
detections and upper limits. $\mathrm{L_{IR}}$ does not depend on
$\mathrm{M_{UV}}$ or redshift. The small $\mathrm{L_{IR}}$ dynamic range and
the flatness are comparable with the ALPINE sample at $\mathrm{4.5<z<6}$
although ALPINE extends to fainter UV galaxies (empty triangles and dots for
upper limits and detections respectively). The ALPINE relation presented in
Khusanova et al. (2021) is shown in the black dashed line.
Using the derived IR luminosity measurements, we plot in Figure 1 the relation
between UV and IR-luminosities. Given the selection of UV luminous sources,
the dynamic range both in UV and IR luminosities is limited. The REBELS sample
only probes the most massive, UV-luminous galaxies at these redshifts. It is
composed of luminous infrared galaxies (LIRGs;
$\mathrm{10^{11}<L_{IR}/L_{\odot}<10^{12}}$) except for REBELS-25, the
brightest galaxy in our sample with $\mathrm{log(L_{IR})\sim 12.2L_{\odot}}$
(see Hygate et al. 2022 for details). The fact that we found only one ultra
luminous infrared galaxy (ULIRG; $\mathrm{L_{IR}>10^{12}L_{\odot}}$) in the
REBELS sample could be due to the UV bright selection of REBELS galaxies with
$\mathrm{-23<M_{UV}<-21.3}$. We discuss this further in a later section.
We compare the IR luminosities from REBELS with the sample from the ALMA Large
Program to INvestigate [CII] at Early times (ALPINE, Le Fèvre et al., 2020)
which targets UV-selected sources at lower redshifts at $\mathrm{4.5<z<6}$.
The ALPINE sample spans a wider $\mathrm{M_{UV}}$ range
($\mathrm{-23.3<M_{UV}<-20}$) but is also mostly composed of LIRGs (see Figure
1) finding also in general dusty galaxies (Pozzi et al., 2021, Sommovigo et
al. 2022b in prep). Our REBELS sample shows that UV-selected galaxies at
$\mathrm{z\sim 7}$ have comparable infrared luminosities to UV-selected
galaxies at lower redshift ($\mathrm{4.5<z<6}$) (see Section 5 for
Discussion).
## 3 Infrared luminosity function at z $\sim$ 7
In this section, we explain the procedure to calculate the luminosity function
(LF). The main complication in computing a luminosity function using a
targeted survey such as REBELS is that it is not straightforward to derive a
selection volume for each source. This can be overcome by basing our volume
estimates on the UV luminosity function as a proxy, as was successfully
demonstrated in Yan et al. (2020) who used the ALPINE UV targeted sample to
derive the [CII] luminosity function. Here, we closely follow their approach.
### 3.1 Calculation of the luminosity function
Our derivation is based on the $\mathrm{z\sim 7}$ UVLF from Bouwens et al.
(2021a). This is used to derive a representative volume for the UV-selected
sources. In practice, we use the UVLF to compute the number of expected
galaxies in bins of UV luminosity assuming a volume-limited survey over the
full selection area of the REBELS sample of 7 deg2 and $\mathrm{z=6.4-7.7}$
(see Fig. 2). This is given by:
$\mathrm{N_{exp}=\phi_{UV}(M_{UV})\,\Delta M_{UV}\,V_{tot}}$ (1)
where $\mathrm{\phi_{UV}(M)}$ is the UVLF from Bouwens et al. (2021a) per
magnitude bin $\mathrm{\Delta M_{UV}}$, and $\mathrm{V_{tot}}$ is the total
survey volume over which REBELS sources were selected. REBELS only targets a
very small sub-sample of all galaxies expected in such a large survey. We can
compute a correction factor to account for this sampling incompleteness in
each UV luminosity bin as $\mathrm{f_{UV}=N_{exp}/N_{obs}}$, where
$\mathrm{N_{obs}}$ is the number of targeted REBELS galaxies in each
$\mathrm{M_{UV}}$ bin.
Figure 2: Number of $\mathrm{L_{IR}}$ detections against the UV absolute
magnitude. The histogram shows the detected sources in red and the non-
detections in grey with the fraction of detections/total indicated in the
lower numbers. Also shown is the UVLF from (Bouwens et al., 2021a) as a dashed
line. This is used to compute the representative volume for each of our
targets. The small numbers above the LF indicate how many galaxies are
expected per MUV bin in a volume-limited survey spanning the REBELS target
selection area of 7 deg2. Clearly, REBELS only targets a very small fraction
of the full galaxy population at faint UV luminosities, which we account for
in our analysis (see main text).
While the correction factor above is derived for a volume-limited survey, the
requirement of a dust continuum detection can further introduce a reduction in
the survey volume for each source. Namely, it can limit the maximum redshift
up to which a given source would remain detected. This is accounted for by
computing the so-called maximum comoving volume $\mathrm{V_{max,i}}$ for each
galaxy i (see Schmidt, 1968). Specifically,
$\mathrm{V_{max,i}}=\int_{z_{min}}^{z_{max,i}}\nicefrac{{d^{2}V}}{{dz\,d\Omega}}\,\Omega\,dz$,
where $\mathrm{z_{max,i}}$ is either the upper edge of the redshift bin of the
LF, or, if smaller, the maximum redshift up to which source i would remain
continuum detected at $>3\sigma$. $\Omega$ is the survey volume. In practice,
$\mathrm{z_{max,i}}=7.7$ for most galaxies, except for the faintest few
sources in the sample.
We now have all quantities to calculate the IR luminosity function
$\mathrm{\phi_{IR}}$ in bins of $\mathrm{L{IR}}$. This is given by:
$\mathrm{\phi_{IR}(\log L_{IR})=\frac{1}{\Delta logL_{IR}}\sum_{i\in
bin}\frac{f_{UV,i}}{V_{max,i}}}$ (2)
where i runs over all sources in a given IR luminosity bin $\mathrm{\log
L_{IR}\pm\Delta\log L_{IR}/2}$ (see Eq. 3 in Yan et al. 2020). The
uncertainties on the IRLF bins are computed as the Poisson errors in each
$\mathrm{L_{IR}}$ bin.
Note that this calculation is independent of the assumed survey area $\Omega$,
since both $\mathrm{{V_{max}}}$ and $\mathrm{{f_{UV}}}$ are directly
proportional to it.
We repeat the above calculation twice. In the first case, we only consider
continuum detected galaxies (16 sources); in the second case, we include the
full REBELS sample (42 sources), treating non-detections as upper-limits. The
completeness factors $\mathrm{f_{UV}}$ are computed separately for both cases.
The resulting IRLFs are in very good agreement, as discussed in the next
section.
### 3.2 The infrared luminosity function at $z\sim 7$
Figure 3: Infrared luminosity function at $\mathrm{z\sim 7}$ for the REBELS
sample (red dots and lines) compared with simulations (dashed lines) and
observations (solid lines). The IRLF was calculated both only using the
galaxies with dust continuum detections (16 galaxies, empty dots) as well as
using the full sample including upper limits (42 galaxies, filled red dots).
The red line shows the Schechter 1976 fit for the total sample. The shaded
area shows the uncertainty of the luminosity function Schechter function fit
with the total sample which is larger at the low luminosity end due to the
lack of data. The rest of the lines show both theoretical and observational
IRLF studies in several fields. Our study is in agreement with Li et al. in
prep (dark purple line) which predicts a similar number of dusty galaxies in a
broad range of luminosities. The dark grey line is the IRLF at $\mathrm{z\sim
7}$ from Zavala et al. (2021) and predicts a larger number of galaxies than
our study for the bright end with luminosities
($\mathrm{12.5<log(L_{IR}/L_{\odot}<13}$) whereas our luminosity function does
not predict a significant number of galaxies at $\mathrm{z\sim 7}$ with
$\mathrm{log(L_{IR}/L_{\odot}>12.5}$. TNG simulations at $\mathrm{z\sim 6}$
from Shen et al. (2021) show a systematic shift with respect to our fitting,
but consistent in shape (blue dashed line). Dayal et al. (2022) and Lagos et
al. (2020) simulations at $\mathrm{z\sim 7}$ (light blue and grey line
respectively) present a 1 dex difference in the lower luminosity with our
result in between them. The yellow line and dots indicate the IRLF at
$\mathrm{z\sim 5.25}$ predicted by the serendipitous galaxies found in the
ALPINE survey presented in Gruppioni et al. (2020), whereas the orange symbols
show Wang et al. (2019a) results at similar redshift.
#### 3.2.1 The Step-Wise IRLF
In this section, we first present the step-wise LF by using the methodology
described in the previous subsection, before we derive parametric Schechter
function fits. Figure 3 shows the resulting LFs in three equidistant
luminosity bins $\mathrm{log(L_{IR}/L_{\odot}}$: [11.3-11.6], [11.6-11.9] and
[11.9-12.2], both for our detections-only and our full sample. The derived
stepwise LFs are in excellent agreement, showing that the detection-only
sample is not biased significantly. In the rest of the paper, we use the total
sample as a baseline.
For the detection-only sample, we further test the possible impact of
uncertainties in the IR luminosity estimates. Specifically, we use a Monte
Carlo technique in which we perturb the initial $L_{IR}$ measurements by their
statistical (Gaussian) uncertainties 10,000 times and rederive the IRLF in
each case. We then use the median and 16th and 84th percentiles, respectively,
as the uncertainties. We do not find significant differences in the resulting
LF values, but the uncertainties are increased as can also be seen in Figure
3.
#### 3.2.2 Schechter Function Fits
We now derive a parametric estimate of the IRLF based on the classic Schechter
function from Schechter 1976, commonly used both in the local and the high-z
Universe (Johnston, 2011). The three parameters that define the Schechter
function are $\mathrm{\phi^{*}}$, $\mathrm{L_{*}}$ and $\mathrm{\alpha}$; the
normalization factor of the overall density of galaxies, the characteristic
luminosity, and the faint-end luminosity slope, respectively. Due to the lack
of data at low $\mathrm{L_{IR}}$, we have restricted $\mathrm{\alpha}$ taking
into account the faint-end slope values found in the literature (see Section 5
for details). We fix the slope to $\mathrm{\alpha=-1.3}$ in our fitting, which
is the value derived for the ALPINE high-z IRLF in Gruppioni et al. (2020).
We use a Bayesian Monte Carlo Markov Chain (MCMC) approach to derive the
posterior distribution of the Schechter function parameters. Hence, we compute
the $\mathrm{\phi_{IR}}$, $\mathrm{L_{*}}$, while keeping the slope fixed at
$\mathrm{\alpha=-1.3}$. We have set these initial parameters centered at the
values obtained by minimizing the error function first
($\mathrm{log(\phi_{IR})=-3.5}$, $\mathrm{log(L^{*})=11.7}$), and then use
non-informative Gaussian priors. We then perform 20,000 MCMC iterations and
ensure that these are converged. We find that posterior distribution of the
parameters is similar in both cases, either including the total sample
(considering upper limits) or only detections. Therefore, we only present the
Schechter function with uncertainties for total sample in Figure 3. The
$\mathrm{1\sigma}$ uncertainty of the fit function was also calculated from
the MCMC chains computing the 16th and 84th percentiles of the posterior
distributions. The $\mathrm{\phi_{IR}}$ uncertainties in the fainter end are
$\mathrm{\sim 0.5\ dex}$, while at the brighter end they are $\mathrm{<0.2\
dex}$. The IRLF is best constrained between
$\mathrm{11.5<log(L_{IR}/L_{\odot})<12}$, and shows that the density of
sources drops quickly ($\mathrm{log(\phi_{IR})<-6.5dex^{-1}Mpc^{-3}}$) at
luminosities above $\mathrm{log(L_{IR}/L_{\odot})>12.3}$.
The resulting Schechter function parameters are
$\mathrm{log(\phi_{IR})=-4.38^{+0.38}_{-0.35}dex^{-1}Mpc^{-3}}$ and
$\mathrm{log(L_{*}/L_{\odot})=11.60^{+0.23}_{-0.13}}$ with a fixed
$\mathrm{\alpha=-1.3}$ (see Table 1 for the summary of the main parameters).
Our analysis shows a $\mathrm{z\sim 7}$ IRLF with a considerable number of
LIRGs that drops in the ULIRG range suggesting a limit in luminosity at
$\mathrm{log(L_{IR}/L_{\odot})\sim 12.3}$. This is in general agreement with
some theoretical studies. The IRLF at $\mathrm{L_{IR}<11.5L_{\odot}}$ is
uncertain and a larger study with fainter galaxies should be carried out to
accurately measure the IRLF at the fainter luminosity end.
We compared our results to both theoretical and observational IRLF studies at
similar redshifts (see dashed and continuous lines respectively in Figure 3).
Generally, our results are in broad agreement with some simulated IRLFs at
similar redshift. When comparing to lower redshift observations at $z\sim
5-6$, however, we find that our IRLF is more than an order of magnitude lower.
Finally, our IRLF shows an interesting evolution with redshift, compared with
the literature, not only in number density (as was previously shown in
Koprowski et al. (2020); Fujimoto et al. (2023)), but also in
$\mathrm{L_{*}}$. This could be due to our UV-selected sample being biased to
bright sources and further study with a similar selection at different
redshift should be carried out to confirm the possibility of evolution with
$\mathrm{L_{*}}$.
We discuss the points above in more detail in Section 5. We also discuss in
subsection 5.3 the importance that our data is UV-bright selected which cannot
take into account extremely dust-obscured sources that are faint in the UV.
## 4 Obscured star formation rate density
In this section, we calculate the obscured SFRD directly through the IRLF
derived in the previous section. We calculate the SFRD in two different ways:
1) by simply summing up the step-wise infrared densities for the data in the
REBELS sample and 2) by integrating the Schechter IRLF over the luminosity
range $\mathrm{10.5<log(L_{IR}/L_{\odot})<13}$. These limits were selected in
the range over which we can define the Schechter function. Note that the
integration limits are narrow but, due to the luminosity bins, there is no
data to constrain a lower-limit integration. Further analysis is produced in
section 5. In both cases we use a conversation factor
$\mathrm{\kappa=10^{-10}M_{\odot}/yr/L_{\odot}}$.
For the step-wise estimates, we considered both the total sample and
detections. We find $\mathrm{log(SFRD/(M_{\odot}/yr/Mpc^{3}))=-3.21\pm 0.18}$
taking only into account the dust continuum detections, which is slightly
lower than for the total sample with
$\mathrm{log(SFRD/M_{\odot}/yr/Mpc^{3})=-2.93\pm 0.20}$. This SFRD estimate
needs to be considered as a lower limit, since it only takes into account the
three luminosity bins.
To extrapolate to fainter luminosities, we have calculated the SFRD for the
Schechter LFs. In particular, we use the MCMC chains to derive the median
posterior SFRD and the associated uncertainties. We find
$\mathrm{log(SFRD/M_{\odot}/yr/Mpc^{3})=-2.66^{+0.17}_{-0.14}}$ where the
uncertainties correspond to the 16-84th percentile (see Figure 4 ). As
expected, this SFRD is larger than the SFRD calculated from the observations,
since it is integrated over the full luminosity range (
$\mathrm{10.5<log(L_{IR}/L_{\odot})<13}$). Notice that REBELS is a UV-selected
sample and the obscured SFRD needs to be taken into account as a robust lower
limit (see caveats in Section 5.3). Finally, the SFRD was computed adding the
serendipitous sources from the REBELS sample presented in Fudamoto et al.
(2021). The sum of the two points, UV-selected galaxies and serendipitous
’dark’ systems, is
$\mathrm{log(SFRD/(M_{\odot}/yr/Mpc^{3}))=-2.53^{+0.17}_{-0.14}}$.
$\mathrm{\alpha}$ | $\mathrm{log(L^{*})}$ | $\mathrm{log(\phi_{IR})}$ | $\mathrm{log(SFRD)}$
---|---|---|---
| $\mathrm{[L_{\odot}]}$ | $\mathrm{[dex^{-1}Mpc^{-3}]}$ | $\mathrm{[M_{\odot}/yr/Mpc^{3}]}$
Schechter Function Fit
-1.3 (fix) | $\mathrm{11.60}^{+0.23}_{-0.13}$ | $\mathrm{-4.38^{+0.38}_{-0.35}}$ | $\mathrm{-2.66^{+0.17}_{-0.14}}$
Total sample | $\mathrm{11.15}$ | $\mathrm{-4.3^{+0.1}_{-0.1}}$ | $\mathrm{-2.93\pm 0.20}$
| $\mathrm{11.75}$ | $\mathrm{-4.6^{+0.2}_{-0.2}}$ |
| $\mathrm{12.05}$ | $\mathrm{-5.5^{+0.4}_{-0.5}}$ |
Detections | $\mathrm{11.15}$ | $\mathrm{-4.4^{+0.2}_{-0.2}}$ | $\mathrm{-3.21\pm 0.18}$
| $\mathrm{11.75}$ | $\mathrm{-4.6^{+0.3}_{-0.3}}$ |
| $\mathrm{12.05}$ | $\mathrm{-5.1^{+0.2}_{-0.5}}$ |
Table 1: Summary of the main parameters of this study. The first column shows
the faint luminosity slope ($\mathrm{\alpha}$), and the second column shows
the luminosity function at the determined luminosity bin (third column).
Finally, the fourth column shows the obscured star formation rate density
taking into account the three luminosity bins. The first row shows the best
fit Schechter function parameters for a fixed slope of $\mathrm{\alpha=-1.3}$,
while the subsequent rows show the total sample and only with detections.
We compare our results with previous studies in the literature for both
similar samples to REBELS and other dusty galaxies at high redshift. Our
derived obscured SFRD of the REBELS sample is $\mathrm{\sim 13\pm 1\%}$ of the
total CSFRD at $\mathrm{z\sim 7}$ from Madau & Dickinson (2014) and
$\mathrm{9\%}$ of the unobscured SFRD estimate from Bouwens et al. (2022).
This is in agreement with the range of obscured SFRD predictions of Zavala et
al. (2021), who use a compilation of several surveys to derive a model of the
IRLF evolution. Our resulting obscured SFRD lies in the upper part of their
inferred SFRD range being the first result at $\mathrm{z\sim 7}$ calculated
through [CII] spectroscopic scans. In an accompanying paper, Algera et al.
(2022) also derived the SFRD for the REBELS sample using the stellar mass as a
proxy to calculate the SFRD through a stacking analysis. While our best
estimates are a factor $\sim 2.5\times$ lower, the measurements are consistent
within the $\mathrm{1\sigma}$ uncertainties.
Figure 4: Star formation rate density against redshift for the REBELS sample
at $\mathrm{z\sim 7}$ and several works in the literature. The black line
shows the total SFRD from Madau & Dickinson (2014) whereas the orange shaded
region shows the obscured SFRD (Zavala et al., 2021). Our results show a
moderate SFRD calculated from the fitted IRLF (red triangle) which increases
if the two serendipitous normal dusty REBELS galaxies from Fudamoto et al.
(2021) are taken into account (orange dot). Similarly, Algera et al. (2022)
obtains a larger contribution to the obscured star formation but in agreement
within $\mathrm{1\sigma}$ error (dark orange dot). DSFGs from the ALMA 2 mm
photometric blind survey show a decrease in SFRD over redshift (purple
squares; Casey et al., 2021). The 1.3 mm ALMA blind survey presented in Dunlop
et al. (2017) shows a obscured SFRD at $\mathrm{1<z<4.5}$ that decreases at
$\mathrm{z>2}$ (purple diamonds). Khusanova et al. (2021) shows the SFRD from
the ALPINE survey at $\mathrm{z\sim 5}$ (brown area). Also from ALPINE,
Gruppioni et al., 2020 present a larger obscured SFRD which is decreasing at
$\mathrm{z>3}$ (pink area) with the last redshift bin at $\mathrm{z>4}$
containing only one source (dashed pink area). Similarly, Wang et al. (2019a)
shows a decreasing SFRD (light purple area) with large uncertainty in the last
bin at $\mathrm{z\sim 4}$ (dashed light purple area). Koprowski et al. (2020)
presented a constrained SFRD up to $\mathrm{z\sim 4}$ (purple area). REBELS
results shows the presence of dust at $\mathrm{z\sim 7}$ even in UV-selected
galaxies.
In Figure 4 we also present the obscured SFRD for several studies showing the
lack of consensus at $\mathrm{z>3}$ on the obscured SFRD. Our SFRD result is
comparable to DSFGs selected at 2mm from Casey et al. (2021), who reports a
decrease in the obscured SFRD over $\mathrm{4<z<6}$. In contrast to these
findings, the SFRD from serendipitous sources found in the ALPINE survey
present a non-evolving SFRD across the whole redshift range of the sample
($\mathrm{1<z<5.5}$). Their calculated SFRD is over two orders of magnitude
more than our results at $\mathrm{z\sim 7}$. Similarly, longer wavelength
studies support a flatter evolution of the SFRD at $\mathrm{3<z<6}$, albeit
with more moderate SFRD (Talia et al., 2021). In contrast, our results show
lower SFRD at $\mathrm{z\sim 7}$, which, when compared to literature at lower
redshifts, supports a non-flat SFRD across redshift (see section 5 for
discussion).
## 5 Discussion
In this section, we compare our IRLF results with observational and
theoretical studies. However, due to the underlying assumptions, IRLFs from
simulations are not directly comparable. As a result, our findings broadly
concur with theoretical research. On the observational side, the literature
shows a large range of IRLF suggesting SFRD discrepancies of $\mathrm{\sim 2}$
orders of magnitude. We also explore the causes for the different results in
the literature and compare to our IRLF and SFRD.
### 5.1 Comparison to Literature
Some theoretical IRLFs at $\mathrm{z\sim 6-7}$ agree quite well with our
findings. For example, Li et al. in prep. show a similar IRLF over the
luminosity range $\mathrm{10.5<log(L_{IR}/L_{\odot})<12.5}$, as do the TNG+300
simulations shown in Shen et al. (2021). But throughout the whole infrared
luminosity range, the latter exhibits larger number densities by $\mathrm{\sim
0.5dex}$. A plausible explanation for this shift is the difference in redshift
($\mathrm{\Delta z\sim 1}$) between our results and those of Shen et al.
(2021), as the IRLF is expected to decrease in number density at increasing
redshift (see e.g. (Koprowski et al., 2017; Fujimoto et al., 2023)).
Our results contrast with those from Lagos et al. (2020) which themselves
differ by $\mathrm{\sim 0.5dex}$ despite the fact that both utilise semi-
analytical models based on merger trees. Over the full range of our directly
observed luminosities ($\mathrm{log(L_{IR}/L_{\odot})>11.5}$), our results are
higher than both of these estimates.
Although the simulations described above are based on different assumptions,
the theoretical work does not contain a UV selected sample bias. This suggests
that, according to simulations, our IRLF estimate is not missing a significant
number of extremely luminous, UV-undetected galaxies at $\mathrm{z\sim 7}$
(for potential caveats, see Section 5.3).
We continue by contrasting with semi-empirical models from Zavala et al.
(2021) at $\mathrm{z\sim 7}$. Their IRLF changes very little at
$\mathrm{12<log(L_{IR}/L_{\odot})<12.5}$, whereas our IRLF sharply declines.
Our study shows an IRLF an order of magnitude higher for LIRGs and a
negligible number of galaxies with $\mathrm{log(L_{IR}/L_{\odot})>12.3}$.
Thus, we find a different distribution also for the bright luminosity end.
These differences in IRLF could be explained by the different methodology, due
to the lack of observational data at $\mathrm{z\sim 7}$, that leads to an
extrapolation of their IRLF at higher redshifts. To do that, it is necessary
to assume two different slopes for the LIRGs and the ULIRGs that might lead to
different outcomes between our study and Zavala et al. (2021).
Finally, we compare our results with IRLFs derived from observations. In
particular, we contrast with the ALPINE IRLF, since it is an analogous survey
to REBELS, but at lower redshift (see section 2 for details). Using the ALPINE
data, Gruppioni et al. (2020) provide the IRLF at $\mathrm{z\sim 5}$ for
serendipitous galaxies. Their IRLF agrees with ours for the lower luminosity
bin, but the overall normalisation is significantly higher. The reason of the
difference is the IRLF rely on several factors. Firstly, the redshift
difference ($\mathrm{\Delta z\sim 2}$) is an obvious reason for the density to
be lower. Furthermore, the REBELS sample was UV-selected, implying a selection
effect that is nonexistent in a blind survey (see section 5.3 for caveats).
Another cause for the disparity with Gruppioni et al. (2020) might the
difference it redshift calculation. Their redshifts were calculated with
multi-band photometry and with only three galaxies at $\mathrm{z\sim 5}$.
Finally, the differing dust temperature assumptions and the SED fitting may
lead to different infrared luminosities, but further analysis is required to
ensure that the differences are significant.
In order to continue the observational comparison, we contrast the IRLF
calculated with the maximum redshift observed to yet in Wang et al. (2019a).
This analysis presents an IRLF with bright infrared galaxies selected with
Herschel Space Observatory Pilbratt et al. (2010) at $\mathrm{z=5.5}$. At same
redshift, their results have a 2 dex greater luminosity function than ours at
the bright end, but a smaller overall luminosity function than the one stated
in Gruppioni et al. (2020). Again, the expected difference is caused by the
disparity in redshift, as does the bias to select massive galaxies with
Herschel.
### 5.2 What IRLF is needed to reproduce extreme SFRD?
This section discusses how changes in the IRLF impact the SFRD. Since there is
lack of consensus about obscured SFRDs at $\mathrm{z>5}$, we evaluate the key
variables that influence the SFRD computation: the IRLF faint end slope, the
$\mathrm{L_{IR}}$ integration limits, and the conversion factor between
$\mathrm{L_{IR}}$ and SFRD. To do that, we compute the SFRD derived for
extreme $\mathrm{\alpha}$ and integration limits to determine whether the most
extreme SFRD described in the literature could be reproduced. We also discuss
the likely causes of these variances.
First, we investigate changes in the IRLF slope. Lower redshift studies
frequently find a slope of $\mathrm{\alpha=-1.3}$, including more galaxies
with lower infrared luminosities (Hammer et al., 2012), but some high redshift
studies report shallower faint-end slopes of $\mathrm{\alpha=-0.4}$ (Koprowski
et al., 2017; Zavala et al., 2021). In Figure 5, we compute the IRLF for these
two extreme cases by using $\mathrm{\alpha=-2}$ and $\mathrm{\alpha=-0.4}$,
respectively. Additionally, we used a wider luminosity range for the
integration than in previous sections of this work, allowing for
$\mathrm{8<log(L_{IR}/L_{\odot})<13}$ as in (Gruppioni et al., 2020).
Nevertheless, we cannot recreate values close to their SFRD, even in the most
extreme scenario ($\mathrm{\alpha=-2}$), yielding a $\mathrm{SFRD\sim 6\cdot
10^{-3}M_{\odot}/Mpc^{3}/yr}$.
This SFRD is, however, consistent with the findings of Talia et al. (2021)
($\mathrm{SFRD\sim 5\cdot 10^{-3}M_{\odot}/yr/Mpc^{3}}$ at $\mathrm{z\sim
5}$). It should be noted that the analysis of Talia et al. (2021) was
conducted using radio galaxies with median $\mathrm{L_{IR}=2.3\pm 0.5\times
10^{12}L_{\odot}}$, and is thus based on a different set of assumptions than
our IR-based estimates.
Despite the fact that it is common to compute the obscured SFRD using the
IRLF, some studies directly calculate it by using the individual SFRs. For
instance, the MORA survey performed blind 2mm ALMA observations (Casey et al.,
2021), and identified a number of $\mathrm{z\sim 4-6}$ DSFGs. They find
$\mathrm{SFRD\sim 10^{-3}\ M_{\odot}/yr/Mpc^{3}}$ at $\mathrm{z\sim 6}$, which
is far lower than the previously mentioned studies such as Talia et al. (2021)
or Gruppioni et al. (2020). The key distinction is that their photometric
redshift estimates are based on submillimetre data, which can be degenerate
with dust temperature. Generally, however, the findings of Casey et al. (2021)
are in good agreement with ours, and their obscured SFRD is compatible with a
$\mathrm{z\sim 6}$ extension of our SFRD at $\mathrm{z\sim 7}$. This agreement
also extends to the 1.3 mm ALMA serendipitous sources at $\mathrm{z<4.5}$ from
Dunlop et al. (2017). Both Dunlop et al. (2017) and Casey et al. (2021)
present a decrease of obscured SFRD at $\mathrm{z>3}$ which likely continues
beyond $\mathrm{z>6}$, as suggested by our data.
Even if several obscured SFRD present large values at $\mathrm{z\sim 5}$ (i.e.
Wang et al. (2019a); Gruppioni et al. (2020); Khusanova et al. (2021)), we
also notice that the highest redshift bin in both Wang et al. (2019a) and
Gruppioni et al. (2020) have larger uncertainty than the rest to the low
number of sources (as shown the hatched areas in Figure 4). Given these larger
uncertainties, a declining SFRD cannot be excluded from these analyses. Hence,
although not in agreement, our results are not in contradiction with the
studies that show large SFRDs and the highest redshift surveys. Studies
including larger samples at $\mathrm{4<z<7}$ would be needed to corroborate
this hypothesis.
Figure 5: The SFRD depends on the IRLF shape and the luminosity range used in
the integration. The faint end slope $\alpha$ assumed in the low luminosity
end is key for the resulting SFRD. This plot shows the best fit IRLF for two
extreme slopes: $\mathrm{\alpha=-2}$ (red line) and $\mathrm{\alpha=-0.4}$
(orange line). The difference between slopes increases in IRLF being
$\mathrm{\sim 4}$ orders of magnitude higher at
$\mathrm{L_{IR}=10^{9}L_{\odot}}$. The inner plot shows the SFRD for these two
extreme cases which shows an order of magnitude difference depending on the
slope assumed with the same integration luminosity
($\mathrm{10^{8}<L_{IR}/L_{\odot}<10^{13}}$). The dark red dots show the total
REBELS sample for the three luminosity bins. The dark red line shows the
Schechter fit with $\mathrm{\alpha=-1.3}$ (dark red line) as presented
previously in Section 3.
### 5.3 Possible Caveats
In this section, we assess the importance of our data being based on a UV-
bright target selection. This directly implies that our study cannot account
for extremely dust-obscured sources, such as SMGs, that are faint in the UV.
However, given that there are several verified SMGs at $\mathrm{z>4}$, we know
that such galaxies are 100$\times$ less common than UV-based Lyman Break
Galaxies, given the SMG sky surface density of 0.01 arcmin-2 (e.g., Riechers
et al., 2013, 2017; Marrone et al., 2018). Furthermore, extremely dusty high
redshift galaxies have only been discovered up to a maximum $\mathrm{z=6.34}$
(Riechers et al., 2013). All of these findings are based on large surveys
conducted with the South Pole Telescope (SPT), SCUBA-2, or Herschel Space
Observatory.
The serendipitous detection of two dust-obscured galaxies in the REBELS
dataset with similar masses and SFRs as the main sample clearly shows that the
primary target sample of REBELS is not complete (Fudamoto et al., 2021).
While, the contribution of this class of galaxies to the SFRD is still very
uncertain, Fudamoto et al. (2021) estimate a value of $\mathrm{1.2\times
10^{-3}\,M_{\odot}/yr/Mpc^{3}}$, i.e. comparable to our estimate from the
IRLF. This would suggest that UV-undetected galaxies could contribute a
similar, but additional amount of obscured SFR as UV-bright galaxies.
Similar conclusions have been reached from recent JWST observations. The first
deep NIRCam observations revealed the existence of UV-undetected, dusty
galaxies at $\mathrm{z>6}$. Barrufet et al. (2022), in particular, present the
SFRD for high-z dusty galaxies, finding a
$\mathrm{log(SFRD/M_{\odot}/yr/Mpc^{3})\sim-3}$ at $\mathrm{z\sim 7}$ for
highly attenuated galaxies. We thus conclude that the galaxies we are missing
in UV selections might contribute the same order of magnitude as the REBELS
sample itself.
To compute a more complete IRLF it would be necessary to perform a deep, but
blind survey to probe galaxies at $z\sim 7$ at several wavelengths. For the
present, a good first step is to obtain results based on the UV-selected
REBELS galaxies. These results represent a firm lower limit on the total
obscured SFRD at $\mathrm{z\sim 7}$.
## 6 Summary and Conclusions
In this work, we have exploited the data from the REBELS survey, which
consists of ALMA spectroscopic data of UV-bright galaxies in the EoR. Our
sample consists of 42 galaxies at $\mathrm{6.4<z<7.7}$. 16 have revealed
significant dust continuum emission at rest-frame $\sim 158\micron$, and all
but one of these are spectroscopically confirmed through their [CII] emission
lines. This sample was used to:
* •
We have calculated the Infrared Luminosity Function (IRLF) at $\mathrm{z\sim
7}$ for the first time using a spectroscopically confirmed sample. We find a
$\mathrm{log(\phi_{IR})\sim-4.2\pm 0.2\ dex^{-1}Mpc^{-3}}$ in our faintest
luminosity bin of $\mathrm{log(L_{IR}/L_{\odot})\sim 11.5}$. At higher
luminosities, the IRLF decreases considerably.
* •
We have fit a Schechter (1976) function with a fix slope of
$\mathrm{\alpha=-1.3}$ for the low luminosity end finding the best fitting
values $\mathrm{log(\phi_{IR})\sim-4.38\ dex^{-1}Mpc^{-3}}$ and
$\mathrm{log(L_{IR}/L_{\odot})=11.6}$. Our results indicate that extremely
luminous galaxies with $\mathrm{log(L_{IR}/L_{\odot})>12.3}$ are extremely
rare at $z\sim 7$, with number densities
$\mathrm{log(\phi_{IR})<-6.5dex^{-1}Mpc^{-3}}$.
* •
We have derived the obscured Star Formation Rate Density through the IRLF.
From the observations we calculate a lower limit of
$\mathrm{log(SFRD/M_{\odot}/yr/Mpc^{3})=-2.93\pm 0.20}$ at $\mathrm{z\sim 7}$
which represents $\mathrm{\sim 13\%}$ of the total SFRD. When integrating over
the luminosity range $\mathrm{10.5<log(L_{IR}/L_{\odot})<13}$ we infer a
larger value of
$\mathrm{log(SFRD/M_{\odot}/yr/Mpc^{3})=-2.66^{+0.17}_{-0.14}}$.
* •
Our IRLF is broadly consistent with some simulations at $\mathrm{z\sim 7}$.
The inferred SFRD is a robust lower limit that shows a significant
contribution of obscured star formation at $\mathrm{z\sim 7}$.
We conclude that our results imply a significant amount of obscured SFR at
$\mathrm{z\sim 7}$ of at least
$\mathrm{log(SFRD/M_{\odot}/yr/Mpc^{3})\sim-3}$. Comparing with ALMA blind
surveys, our results suggest a steep evolution of the obscured SFRD over
redshift that continues to $\mathrm{z\sim 7}$, at least.
## Acknowledgements
We acknowledge the constructive feedback of the referee (MB) for his
constructive feedback that helped in the improvement of this paper. We
acknowledge support from: the Swiss National Science Foundation through the
SNSF Professorship grant 190079 (LB and PAO). The Cosmic Dawn Center (DAWN) is
funded by the Danish National Research Foundation under grant No. 140. PD
acknowledges support from the European Research Council’s starting grant ERC
StG-717001 (“DELPHI"), from the NWO grant 016.VIDI.189.162 (“ODIN") and the
European Commission’s and University of Groningen’s CO-FUND Rosalind Franklin
program. AF and AP acknowledge support from the ERC Advanced Grant
INTERSTELLAR H2020/740120. Generous support from the Carl Friedrich von
Siemens-Forschungspreis der Alexander von Humboldt-Stiftung Research Award is
kindly acknowledged. YF acknowledges support from NAOJ ALMA Scientific
Research Grant number 2020-16B. VG gratefully acknowledges support by the ANID
BASAL projects ACE210002 and FB210003.
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|
# Quantum $K$-theory of $G/P$ and $K$-homology of affine Grassmannian
Chi Hong Chow and Naichung Conan Leung
###### Abstract.
This paper is the $K$-theoretic analogue of a recent new proof, given by the
first named author, of Peterson-Lam-Shimozono’s theorem via Savelyev’s
generalization of Seidel representations. The outcome is a new proof of Lam-
Li-Mihalcea-Shimozono’s conjecture, including its extension to the parabolic
case, which was first verified by Kato.
## 1\. introduction
Let $G$ be a simple and simply connected complex Lie group. In Schubert
calculus, an unpublished result of Peterson [35], first proved by Lam-
Shimozono [30], states that the Pontryagin product of the homology of the
affine Grassmannian $Gr_{G}$ determines completely, via an explicit ring
homomorphism defined in terms of the (affine) Schubert bases, the quantum cup
product of the quantum cohomology of any flag variety $G/P$. Recently, Chow
[13] has given a new proof of this result by computing Savelyev’s parametrized
version [38] of Seidel representations [39].
In $K$-theory, Lam-Li-Mihalcea-Shimozono [28] conjectured a similar
homomorphism for the case of $G/B$ where the bases are replaced by the
structure sheaves of the (affine) Schubert varieties. Their conjecture was
first proved by Kato [20, 22]. Kato also extended the result to the general
parabolic case [21]. In this paper, we give an alternative proof of Kato’s
result by following the approach in [13].
###### Theorem 1.1.
The map
$\begin{array}[]{ccccc}\Phi&:&K^{T}(Gr_{G})&\rightarrow&QK_{T}(G/P)[\Lambda^{-1}]\\\\[5.0pt]
&&\mathcal{O}_{wt_{\lambda}}&\mapsto&q^{\lambda+Q^{\vee}_{P}}\mathcal{O}_{\widetilde{w}}\end{array}$
is an $R(T)$-algebra homomorphism, where $\widetilde{w}$ is the minimal length
coset representative of $wW_{P}$.
As already pointed out by Lam-Li-Mihalcea-Shimozono, Theorem 1.1 implies
immediately the finiteness property of the quantum $K$-product $\star$.
###### Corollary 1.2.
For any $v_{1},v_{2}\in W^{P}$,
$\mathcal{O}_{v_{1}}\star\mathcal{O}_{v_{2}}\in
K_{T}(G/P)\otimes\mathbb{Z}[\Lambda].$
Corollary 1.2 is not as obvious as the case of quantum cohomology because the
moduli spaces of stable maps of arbitrary dimension contribute. What’s more,
as proved by Givental [18], in order for $\star$ to be associative, it is
necessary to introduce a deformation of the Poincaré pairing
$\chi_{G/P}(-\otimes-)$ by two-pointed $K$-theoretic GW invariants which is in
general an infinite sum. Thus, the finiteness must follow from a non-trivial
cancellation of terms. This issue has already been settled by Kato [20, 21,
22] and Anderson-Chen-Tseng [4]. See also the earlier work [3] of the authors
of [4] and the work of Buch-Chaput-Mihalcea-Perrin [10, 11].
## Outline of the proof
Our approach is to define a map by Gromov-Witten theory and show that it is an
$R(T)$-algebra homomorphism and has the desired form.
We first recall the proof of Peterson’s original result given in [13]. Take
$Gr_{G}$ to be Pressley-Segal’s model [36]. In loc. cit., they constructed,
for any holomorphic map $f:\Gamma\rightarrow Gr_{G}$, a holomorphic principal
$G$-bundle $P_{f}$ over $\mathbb{P}^{1}\times\Gamma$ with a trivialization
over $(\mathbb{P}^{1}\setminus\\{|z|\leqslant 1\\})\times\Gamma$. In
particular, we obtain a $G/P$-fibration $P_{f}(G/P)$ over
$\mathbb{P}^{1}\times\Gamma$ by reduction of fibers. One should think of
$P_{f}(G/P)$ as a holomorphic family of $G/P$-fibrations over $\mathbb{P}^{1}$
parametrized by $\Gamma$. For any section class $\beta$, define
$\overline{\mathcal{M}}(f,\beta):=~{}$moduli stack of holomorphic sections in
$P_{f}(G/P)$ representing $\beta$
and
$\operatorname{ev}:\overline{\mathcal{M}}(f,\beta)\rightarrow G/P$
to be the evaluation map at $\infty\in\mathbb{P}^{1}$. Thanks to the above
trivialization, $\operatorname{ev}$ is well-defined.
For our purpose, we consider two classes of $f$: the $T$-fixed points of
$Gr_{G}$ and Bott-Samelson resolutions of the affine Schubert varieties. They
give rise to the localization basis $\\{\eta_{\mu}\\}_{\mu\in Q^{\vee}}$ and
the affine Schubert basis $\\{\xi_{wt_{\lambda}}\\}_{wt_{\lambda}\in
W_{af}^{-}}$ of $H^{T}(Gr_{G})$ respectively. Denote by
$\overline{\mathcal{M}}(\mu,\beta)$ and
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ the corresponding moduli stacks
defined above. Define the Savelyev-Seidel homomorphism [38, 39]
$\Phi_{QH}:H^{T}(Gr_{G})\rightarrow QH_{T}(G/P)[\Lambda^{-1}]$
either by
$\Phi_{QH}(\eta_{\mu}):=\sum_{\beta}q^{\beta}\operatorname{ev}_{*}[\overline{\mathcal{M}}(\mu,\beta)]^{vir}$
(1.1)
or
$\Phi_{QH}(\xi_{wt_{\lambda}}):=\sum_{\beta}q^{\beta}\operatorname{ev}_{*}[\overline{\mathcal{M}}(wt_{\lambda},\beta)]^{vir}.$
(1.2)
By the virtual localization formula [19], these two definitions are
equivalent. (1.1) is used when we show that $\Phi_{QH}$ is a ring homomorphism
and (1.2) is used for the computation. The former follows from a degeneration
argument and the latter relies on the following key observation:
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ is smooth and of expected
dimension.
Since $\operatorname{ev}$ is $B$-equivariant,
$\operatorname{ev}_{*}[\overline{\mathcal{M}}(wt_{\lambda},\beta)]$ is equal
to a multiple of a Schubert class or zero depending on whether the generic
fiber of $\operatorname{ev}$ has zero or positive dimension. This reduces our
computation to a purely combinatorial problem which can be solved in a
straightforward way.
Back to the situation in the present paper, we will prove Theorem 1.1 by
adapting the above approach to the $K$-theoretic settings. By recalling the
definition of the quantum $K$-product, one expects that the $K$-theoretic
Savelyev-Seidel homomorphism should be defined by incorporating a new feature
that the contribution of each $\overline{\mathcal{M}}(\mu,\beta)$ or
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ be corrected by the two-pointed
$K$-theoretic GW invariants (see the paragraph following Corollary 1.2).
Define
$A_{G/P}:=\sum_{\beta\neq
0}q^{\beta}(\operatorname{ev}^{\beta}_{2})_{*}(\operatorname{ev}^{\beta}_{1})^{*}$
where $\operatorname{ev}^{\beta}_{1},\operatorname{ev}^{\beta}_{2}$ are the
evaluation maps on $\overline{\mathcal{M}}_{0,2}(G/P,\beta)$. The
aforementioned correction is given by $(\operatorname{id}+A_{G/P})^{-1}$.
Therefore, we define
$\Phi_{QK}:K^{T}(Gr_{G})\rightarrow QK_{T}(G/P)[\Lambda^{-1}]$
by
$\Phi_{QK}(\mathcal{O}_{wt_{\lambda}}):=(\operatorname{id}+A_{G/P})^{-1}\left(\sum_{\beta}q^{\beta}\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}]\right).$
where $\mathcal{O}_{wt_{\lambda}}$ is the $K$-theoretic analogue of
$\xi_{wt_{\lambda}}$. That $\Phi_{QK}$ is a ring homomorphism follows from
similar localization and degeneration arguments as well as an argument used by
Givental [18] and Lee [31] in their proof of the $K$-theoretic WDVV equation.
The heart of the paper is the computation of
$\Phi_{QK}(\mathcal{O}_{wt_{\lambda}})$. Our strategy is to introduce a
$\mathbb{C}^{\times}$-action on $\overline{\mathcal{M}}(wt_{\lambda},\beta)$
by rescaling the domain of free loops in $G$, and apply Oprea’s stacky version
[34] of Białynicki-Birula’s theorem [8] to this action. We show that if
$\overline{\mathcal{M}}(wt_{\lambda},\beta)\neq\emptyset$, there exists a
unique component of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)^{\mathbb{C}^{\times}}$ whose
Białynicki-Birula cell is open, and hence
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ is either empty or irreducible.
A more in-depth analysis of this component gives
$\\{\beta|~{}\overline{\mathcal{M}}(wt_{\lambda},\beta)\neq\emptyset\\}=[\lambda]+\Lambda$
and
$\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}]=(\operatorname{ev}^{\beta-[\lambda]}_{2})_{*}(\operatorname{ev}^{\beta-[\lambda]}_{1})^{*}\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},[\lambda])}],\quad\beta\in[\lambda]+(\Lambda\setminus\\{0\\})$
(1.3)
where we put $[\lambda]:=\lambda+Q^{\vee}_{P}$ for simplicity, $\Lambda$ is
the semigroup of effective curve classes and
$\operatorname{ev}^{\beta-[\lambda]}_{1},\operatorname{ev}^{\beta-[\lambda]}_{2}:\overline{\mathcal{M}}_{0,2}(G/P,\beta-[\lambda])\rightarrow
G/P$ are the evaluation maps. Summing up (1.3) over all $\beta$, weighted by
$q^{\beta}$, we get
$\displaystyle\Phi_{QK}(\mathcal{O}_{wt_{\lambda}})$
$\displaystyle=(\operatorname{id}+A_{G/P})^{-1}\circ(\operatorname{id}+A_{G/P})\left(q^{[\lambda]}\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},[\lambda])}]\right)$
$\displaystyle=q^{[\lambda]}\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},[\lambda])}].$
The last term can easily be shown to be equal to
$q^{[\lambda]}\mathcal{O}_{\widetilde{w}}$ as stated in Theorem 1.1.
## 2\. Preliminaries
### 2.1. Some Lie-theoretic notations
Let $G$ be a simple and simply connected complex Lie group and $T\subset G$ a
maximal torus. Put $\mathfrak{g}:=\operatorname{Lie}(G)$ and
$\mathfrak{h}:=\operatorname{Lie}(T)$. We have the root space decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in R}\mathfrak{g}_{\alpha}$
where $R$ is the set of roots associated to the pair
$(\mathfrak{g},\mathfrak{h})$ and each $\mathfrak{g}_{\alpha}$ is a one-
dimensional eigenspace with respect to the adjoint action of $\mathfrak{h}$.
Denote by $W$ the Weyl group. Fix a set $\\{\alpha_{1},\ldots,\alpha_{r}\\}$
of simple roots of $R$. Denote by $\alpha_{i}^{\vee}$ the corresponding
coroots. Define $R^{+}\subset R$ to be the set of positive roots spanned by
$\alpha_{1},\ldots,\alpha_{r}$. Define $B^{-}$ (resp. $B^{+}$) to be the Borel
subgroup of $G$ containing $T$ with Lie algebra equal to
$\mathfrak{h}\oplus\bigoplus_{\alpha\in-R^{+}}\mathfrak{g}_{\alpha}$ (resp.
$\mathfrak{h}\oplus\bigoplus_{\alpha\in R^{+}}\mathfrak{g}_{\alpha}$).
Define the affine Weyl group $W_{af}:=W\ltimes Q^{\vee}$ where
$Q^{\vee}\subset\mathfrak{h}$ is the $\mathbb{Z}$-span of $\alpha^{\vee}_{i}$,
$i=1,\ldots,r$. Typical elements of $W_{af}$ are denoted by $wt_{\lambda}$.
Define the affine simple roots
$\widetilde{\alpha}_{0},\ldots,\widetilde{\alpha}_{r}$ by
$\widetilde{\alpha}_{0}:=-\alpha_{0}+1$ and
$\widetilde{\alpha}_{i}:=\alpha_{i}$ for $i=1,\ldots,r$, where $\alpha_{0}$ is
the highest positive root.
### 2.2. Algebraic K-theory
A good reference for the following materials is [14].
Let $Y$ be a (finite-dimensional) scheme over $\mathbb{C}$ with an action of a
complex torus $T$ (which will be the maximal torus fixed in Section 2.1
throughout). Define $K_{T}(Y)$ (resp. $K^{T}(Y)$) to be the Grothendieck group
of $T$-equivariant vector bundles (resp. $T$-equivariant coherent sheaves) on
$Y$. If $Y$ is smooth and quasi-projective, they are known to be isomorphic.
Pullback and tensor product of vector bundles give rise to the pullback
operator and a ring structure on $K_{T}(Y)$ respectively. Tensor product also
defines a $K_{T}(Y)$-module structure on $K^{T}(Y)$, since for any vector
bundle $E$, $E\otimes-$ is an exact functor on the abelian category of
coherent sheaves. If $f:Y\rightarrow Z$ is a $T$-equivariant proper morphism,
we define the pushforward operator
$f_{*}:K^{T}(Y)\rightarrow K^{T}(Z)$
by
$f_{*}([\mathcal{E}]):=\sum_{i\geqslant 0}(-1)^{i}[R^{i}f_{*}(\mathcal{E})].$
In particular, if $Y$ is proper and $Z$ is a point, the corresponding operator
is denoted by $\chi_{Y}$.
There is a natural $K_{T}(pt)$-module structure on $K_{T}(Y)$ and $K^{T}(Y)$,
defined via the pullback operator associated to the structure morphism
$Y\rightarrow pt$. It is well-known that $K_{T}(pt)$ is isomorphic to the
representation ring $R(T)$ of $T$. We will adopt the latter notation
throughout the paper.
###### Remark 2.1.
In this paper, we have to work with Deligne-Mumford stacks because moduli
spaces of stable maps are not schemes in general. While the above definitions
extend to Deligne-Mumford stacks, they are not strictly necessary for the
computational aspect of this paper. We will bypass them by following the
approach explained in [31, Remark 5].
Let $\mathcal{Y}$ be a Deligne-Mumford stack arising from the moduli of stable
maps to a smooth projective variety. Consider the canonical map
$p:\mathcal{Y}\rightarrow Y$
from $\mathcal{Y}$ to its coarse moduli $Y$. By the tameness property of
$\mathcal{Y}$ (see [1]), we have
$p_{*}[\mathcal{O}_{\mathcal{Y}}]=[\mathcal{O}_{Y}].$ (2.1)
The quantum $K$-invariants considered in this paper are of the form
$\chi_{\mathcal{Y}}(\mathcal{O}^{vir}_{\mathcal{Y}}\otimes\operatorname{ev}_{1}^{*}\alpha_{1}\otimes\cdots\otimes\operatorname{ev}_{k}^{*}\alpha_{k})$
where
* •
$\mathcal{O}^{vir}_{\mathcal{Y}}\in K^{T}(\mathcal{Y})$ is the virtual
structure sheaf constructed in [31];
* •
$\operatorname{ev}_{i}$ are the evaluation maps on $\mathcal{Y}$; and
* •
$\alpha_{i}$ are some $K$-theory classes on the target space.
The key observation is that each $\operatorname{ev}_{i}$ factors through $p$,
and hence the above invariant is equal to
$\chi_{Y}(p_{*}\mathcal{O}^{vir}_{\mathcal{Y}}\otimes\operatorname{ev}_{1}^{*}\alpha_{1}\otimes\cdots\otimes\operatorname{ev}_{k}^{*}\alpha_{k}),$
by the projection formula. For our computation, $\mathcal{Y}$ will be smooth
and of expected dimension. It follows that
$\mathcal{O}^{vir}_{\mathcal{Y}}=[\mathcal{O}_{\mathcal{Y}}]$. This allows us
to work only with the coarse moduli $Y$, by (2.1).
### 2.3. Flag varieties
A flag variety is a homogeneous space $G/P$ where $P$ is any parabolic
subgroup containing $B^{+}$. We have
$\operatorname{Lie}(P)=\operatorname{Lie}(B^{+})\oplus\bigoplus_{\alpha\in-
R_{P}^{+}}\mathfrak{g}_{\alpha}$
where $R_{P}^{+}:=R_{P}\cap R^{+}$ and $R_{P}$ is the set of roots of $P$.
Denote by $W_{P}$ the Weyl group of $P$ and by $W^{P}$ the set of minimal
length coset representatives in $W/W_{P}$. For any $v\in W^{P}$, put
$y_{v}:=vP\in G/P$. Then $\\{y_{v}\\}_{v\in W^{P}}$ is the set of $T$-fixed
points of $G/P$. Define
$\mathcal{O}_{v}:=[\mathcal{O}_{\overline{B^{-}\cdot y_{v}}}]\in K_{T}(G/P).$
###### Lemma 2.2.
$\\{\mathcal{O}_{v}\\}_{v\in W^{P}}$ is an $R(T)$-basis of $K_{T}(G/P)$.
###### Proof.
See the proof of Lemma 2.10. ∎
We recall the equivariant quantum $K$-theory of $G/P$ defined by Givental
[18]. See also the work of Lee [31] which deals with general smooth projective
varieties. Denote by $\Lambda\subset\pi_{2}(G/P)$ the semigroup of effective
curve classes. We identify $\Lambda$ with
$\bigoplus_{i=1}^{r}\mathbb{Z}_{\geqslant
0}\langle\alpha_{i}^{\vee}\rangle\subset Q^{\vee}$ via the dual of the
composition of three isomorphisms
$\left(Q^{\vee}/Q^{\vee}_{P}\right)^{*}\xrightarrow{\rho~{}\mapsto
L_{\rho}}\operatorname{Pic}(G/P)\xrightarrow{c_{1}}H^{2}(G/P)\simeq\pi_{2}(G/P)^{*}$
(2.2)
where
* •
$Q^{\vee}_{P}:=\operatorname{Span}_{\mathbb{Z}}\\{\alpha_{i}^{\vee}|~{}\alpha_{i}\in
R_{P}\\}$;
* •
$L_{\rho}:=G\times_{P}\mathbb{C}_{-\rho}$; and
* •
$\mathbb{C}_{-\rho}$ is the one-dimensional representation of weight $-\rho$
on which $P$ acts by forgetting the semi-simple and unipotent parts.
We have, as abelian groups,
$QK_{T}(G/P):=K_{T}(G/P)\otimes\mathbb{Z}[[\Lambda]]$
where $\mathbb{Z}[[\Lambda]]$ is the formal completion of the group ring
$\mathbb{Z}[\Lambda]$.
###### Remark 2.3.
The reason for enlarging the standard coefficient ring $\mathbb{Z}[\Lambda]$
for quantum cohomology is to ensure that the ring product we are going to
define is well-defined. It turns out that this is unnecessary by Corollary
1.2.
What is non-trivial is the definition of the quantum $K$-product $\star$ on
$QK_{T}(G/P)$. For any $\beta\in\Lambda$ and $\gamma_{1},\ldots,\gamma_{k}\in
K_{T}(G/P)$, define
$\operatorname{KGW}^{\beta}(\gamma_{1},\ldots,\gamma_{k}):=\chi_{\overline{\mathcal{M}}_{0,k}(G/P,\beta)}(\operatorname{ev}_{1}^{*}\gamma_{1}\otimes\cdots\otimes\operatorname{ev}_{k}^{*}\gamma_{k})\in
R(T)$
where $\overline{\mathcal{M}}_{0,k}(G/P,\beta)$ is the Deligne-Mumford moduli
stack of genus zero $k$-pointed stable maps to $G/P$ representing $\beta$.
Clearly, we can extend $\operatorname{KGW}$ to a linear map
$(QK_{T}(G/P))^{\otimes k}\rightarrow R(T)\otimes\mathbb{Z}[[\Lambda]]$ by
linearity. Take an $R(T)$-basis $\\{e_{i}\\}_{i\in I}$ of $K_{T}(G/P)$ (the
Schubert basis, for example) and denote by $\\{g^{ij}\\}_{i,j\in I}$ the
inverse of the matrix $\\{\chi_{G/P}(e_{i}\otimes e_{j})\\}_{i,j\in I}$. It is
well-known that the latter matrix is indeed invertible. Define a linear
operator
$A_{G/P}:QK_{T}(G/P)\rightarrow QK_{T}(G/P)$
by
$A_{G/P}(\gamma):=\sum_{i,j\in
I}\sum_{\beta\in\Lambda\setminus\\{0\\}}q^{\beta}g^{ij}\operatorname{KGW}^{\beta}(\gamma,e_{i})e_{j}.$
We have, for any $\gamma_{1},\gamma_{2}\in QK_{T}(G/P)$,
$\gamma_{1}\star\gamma_{2}:=\sum_{i,j\in
I}\sum_{\beta\in\Lambda}q^{\beta}g^{ij}\operatorname{KGW}^{\beta}(\gamma_{1},\gamma_{2},e_{i})(\operatorname{id}+A_{G/P})^{-1}(e_{j}).$
By [18] or [31], $\star$ defines a ring structure on $QK_{T}(G/P)$.
### 2.4. Affine Grassmannian
There are many models for the affine Grassmannian $Gr_{G}$. In this paper, we
work with Pressley-Segal’s version [36].
Define $H:=L^{2}(S^{1};\mathfrak{g})$ to be the Hilbert space of
$L^{2}$-functions on $S^{1}$ with values in $\mathfrak{g}$. We have an
orthogonal decomposition $H=H_{+}\oplus H_{-}$ where $H_{+}$ (resp. $H_{-}$)
consists of functions whose negative (resp. non-negative) Fourier coefficients
are zero. Let $\operatorname{pr}_{\pm}:H\rightarrow H_{\pm}$ denote the
orthogonal projections. Define
$Gr(H):=\\{W\subset H\text{ closed
subspaces}|~{}\operatorname{pr}_{+}|_{W}\text{ is Fredholm and
}\operatorname{pr}_{-}|_{W}\text{ is Hilbert-Schmidt}\\}.$
It is proved in [36] that $Gr(H)$ is a complex Hilbert manifold.
Fix a maximal compact subgroup $K$ of $G$. Define
$\displaystyle L_{sm}G$ $\displaystyle:=\\{\text{smooth free loops in }G\\}$
$\displaystyle L_{pol}G$ $\displaystyle:=\\{\text{polynomial free loops in
}G\\}$ $\displaystyle\Omega_{sm}K$ $\displaystyle:=\\{\text{smooth based loops
in }K\\}$ $\displaystyle\Omega_{pol}K$ $\displaystyle:=\\{\text{polynomial
based loops in }K\\}.$
Then $L_{sm}G$ (resp. $\Omega_{sm}K$) is an infinite-dimensional complex
(resp. real) Fréchet Lie group in the $C^{\infty}$-topology. Consider the
following action on $H$ by $L_{sm}G$:
$(\varphi\cdot f)(z):=\operatorname{Ad}(\varphi(z))f(z)\quad\varphi\in
L_{sm}G,~{}f\in H$
where $\operatorname{Ad}$ is the adjoint action. This action induces an
$L_{sm}G$-action on $Gr(H)$ with respect to which the stabilizer of $H_{+}\in
Gr(H)$ is equal to $L^{0}_{sm}G$, the subgroup of $L_{sm}G$ consisting of
$\varphi$ which extend to holomorphic functions defined on the unit disk.
###### Theorem 2.4.
[36, Theorem 8.6.2] There exists a diffeomorphism
$L_{sm}G\cdot H_{+}\simeq\Omega_{sm}K$
under which the sub-orbit $L_{pol}G\cdot H_{+}$ corresponds to
$\Omega_{pol}K$.
From now on, we identify the orbit $L_{sm}G\cdot H_{+}$ (resp. $L_{pol}G\cdot
H_{+}$) with $\Omega_{sm}K$ (resp. $\Omega_{pol}K$) via the above
diffeomorphism . Since $G$ is assumed to be simply connected, $\Omega_{sm}K$
is connected and so lies in the connected component $Gr(H)^{o}$ of $Gr(H)$
containing $H_{+}$. For any natural number $n$, define
$Gr^{(n)}(H):=\\{W\in Gr(H)^{o}|~{}z^{n}H_{+}\subseteq W\subseteq
z^{-n}H_{+}\\}$
and
$\Omega_{pol}^{(n)}K:=\Omega_{pol}K\cap Gr^{(n)}(H).$
Notice that $Gr^{(n)}(H)$ is a submanifold of $Gr(H)^{o}$ biholomorphic to the
type-A Grassmannian
$Gr(n\cdot\dim_{\mathbb{C}}\mathfrak{g},2n\cdot\dim_{\mathbb{C}}\mathfrak{g})$
and $\Omega_{pol}^{(n)}K$ is a possibly singular closed subvariety of
$Gr^{(n)}(H)$.
###### Theorem 2.5.
[36, Theorem 8.3.3] $\Omega_{pol}K=\bigcup_{n=0}^{\infty}\Omega_{pol}^{(n)}K$.
Consider the action on $\Omega_{pol}K$ by the maximal torus $T\subset G$. It
is easy to see that the fixed-point set $(\Omega_{pol}K)^{T}$ is equal to
$\\{x_{\mu}\\}_{\mu\in Q^{\vee}}$ where $x_{\mu}$ is the cocharacter of $T$
associated to any element $\mu\in Q^{\vee}$. Define
$\mathcal{B}^{0,-}_{sm}:=\\{\varphi\in L^{0}_{sm}G|~{}\varphi(0)\in B^{-}\\}.$
(By abuse of notation, the holomorphic extension of any $\varphi\in
L^{0}_{sm}G$ is denoted by the same symbol.)
###### Theorem 2.6.
[36, Theorem 8.6.3]
1. (1)
(Bruhat decomposition) We have
$\Omega_{pol}K=\bigcup_{\mu\in Q^{\vee}}\mathcal{B}_{sm}^{0,-}\cdot x_{\mu}.$
2. (2)
For any $\mu\in Q^{\vee}$, the orbit $\mathcal{B}_{sm}^{0,-}\cdot x_{\mu}$ is
biholomorphic to a complex affine space.
###### Definition 2.7.
Define the affine Grassmannian $Gr_{G}:=\Omega_{pol}K$.
Now we define, following [27], the $K$-homology $K^{T}(Gr_{G})$ of $Gr_{G}$.
Notice that the definition does not follow directly from Section 2.2 where we
only deal with finite-dimensional schemes. By Theorem 2.5, $Gr_{G}$ is the
union of the chain of projective varieties
$\Omega_{pol}^{(0)}K\subset\Omega_{pol}^{(1)}K\subset\Omega_{pol}^{(2)}K\subset\cdots.$
This chain induces a direct system of $R(T)$-modules
$K^{T}(\Omega_{pol}^{(0)}K)\rightarrow K^{T}(\Omega_{pol}^{(1)}K)\rightarrow
K^{T}(\Omega_{pol}^{(2)}K)\rightarrow\cdots.$
###### Definition 2.8.
Define
$K^{T}(Gr_{G}):=\varinjlim_{n}K^{T}(\Omega_{pol}^{(n)}K).$
Denote by $W_{af}^{-}$ the set of minimal length coset representatives in
$W_{af}/W$. Notice that the map $W_{af}^{-}\rightarrow Q^{\vee}$ sending
$wt_{\lambda}$ to $w(\lambda)$ is bijective.
###### Definition 2.9.
1. (1)
Let $\mu\in Q^{\vee}$. Define
$\mathcal{O}_{\mu}:=[\mathcal{O}_{x_{\mu}}]\in K^{T}(Gr_{G}).$
2. (2)
Let $wt_{\lambda}\in W_{af}^{-}$. Define
$\mathcal{O}_{wt_{\lambda}}:=[\mathcal{O}_{\overline{\mathcal{B}^{0,-}_{sm}\cdot
x_{w(\lambda)}}}]\in K^{T}(Gr_{G})$
where $\overline{\mathcal{B}^{0,-}_{sm}\cdot x_{w(\lambda)}}$ is the Zariski
closure of $\mathcal{B}^{0,-}_{sm}\cdot x_{w(\lambda)}$ taken in
$\Omega_{pol}^{(n)}K$ for some large $n$.
###### Lemma 2.10.
1. (1)
$\\{\mathcal{O}_{wt_{\lambda}}\\}_{wt_{\lambda}\in W_{af}^{-}}$ is an
$R(T)$-basis of $K^{T}(Gr_{G})$.
2. (2)
There exists a monomorphism
$K^{T}(Gr_{G})\hookrightarrow\bigoplus_{\mu\in
Q^{\vee}}\operatorname{Frac}(R(T))\langle\mathcal{O}_{\mu}\rangle$ (2.3)
which fixes every $\mathcal{O}_{\mu}$.
###### Proof.
(1) follows from an argument of Kumar [27] based on the following two standard
results:
1. (i)
(The excision sequence) If we have
$U\xhookrightarrow{i}X\xhookleftarrow{j}X\setminus U$ where $X$ is projective
and $U$ is open, then the sequence
$K^{T}(X\setminus
U)\xrightarrow{j_{*}}K^{T}(X)\xrightarrow{i^{*}}K^{T}(U)\rightarrow 0$ (2.4)
is exact.
2. (ii)
(The Thom isomorphism: a special case) If $T$ acts on a vector space
$\mathbb{C}^{r}$ linearly, then $K^{T}(\mathbb{C}^{r})$ is freely generated by
$[\mathcal{O}_{\mathbb{C}^{r}}]$.
Proofs of (i) and (ii) can be found in [14]. To prove (2), it suffices to show
that every $\mathcal{O}_{wt_{\lambda}}$ is an
$\operatorname{Frac}(R(T))$-linear combination of $\mathcal{O}_{w(\lambda)}$
and some other $\mathcal{O}_{w^{\prime}t_{\lambda^{\prime}}}$ with
$\ell(w^{\prime}t_{\lambda^{\prime}})<\ell(wt_{\lambda})$. This follows from
(2.4) and a local computation in $K^{T}(\mathbb{C}^{\ell(wt_{\lambda})})$. ∎
###### Definition 2.11.
Define an $R(T)$-algebra structure
$\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}$
on $\bigoplus_{\mu\in
Q^{\vee}}\operatorname{Frac}(R(T))\langle\mathcal{O}_{\mu}\rangle$ by
$\mathcal{O}_{\mu_{1}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathcal{O}_{\mu_{2}}:=\mathcal{O}_{\mu_{1}+\mu_{2}}\quad\mu_{1},\mu_{2}\in
Q^{\vee}.$ (2.5)
###### Lemma 2.12.
Via the monomorphism (2.3), $K^{T}(Gr_{G})$ is a sub-$R(T)$-algebra.
###### Proof.
This is proved by Lam-Schilling-Shimozono [29]. Notice that they first defined
an $R(T)$-algebra structure on $K^{T}(Gr_{G})$ and verified (2.5).
Alternatively, we show that it follows from our proof of Theorem 1.1, although
the statement of this theorem assumes the lemma we are proving. Take
$P=B^{+}$. In Section 4.1, we will construct an $R(T)$-algebra homomorphism
$\Phi:\bigoplus_{\mu\in
Q^{\vee}}\operatorname{Frac}(R(T))\langle\mathcal{O}_{\mu}\rangle\rightarrow
QK_{T}(G/B^{+})[\Lambda^{-1}]\otimes\operatorname{Frac}(R(T)).$
In Section 4.2, 4.3 and 4.4, we will show
$\Phi(\mathcal{O}_{wt_{\lambda}})=q^{\lambda}\mathcal{O}_{w}$
where $\mathcal{O}_{wt_{\lambda}}$ is regarded as an element of the domain of
$\Phi$ via (2.3). This implies that $\Phi$ sends an $R(T)$-basis of
$K^{T}(Gr_{G})$ injectively into an $R(T)$-basis of
$K_{T}(G/B^{+})\otimes\mathbb{Z}[\Lambda^{\pm}]$, and hence
$K^{T}(Gr_{G})=\Phi^{-1}(QK_{T}(G/B^{+})[\Lambda^{-1}])$
which is clearly a sub-$R(T)$-algebra of $\bigoplus_{\mu\in
Q^{\vee}}\operatorname{Frac}(R(T))\langle\mathcal{O}_{\mu}\rangle$. ∎
## 3\. The key moduli
### 3.1. The G/P-fibration
The following theorem is the starting point of everything.
###### Theorem 3.1.
[36, Theorem 8.10.2] For any complex manifold $\Gamma$, there exists a
bijection between
1. (1)
the set of holomorphic maps $\Gamma\rightarrow\Omega_{sm}K$; and
2. (2)
the set of isomorphism classes of holomorphic principal $G$-bundles over
$\mathbb{P}^{1}\times\Gamma$ with a trivialization over
$(\mathbb{P}^{1}\setminus\\{|z|\leqslant 1\\})\times\Gamma$.
We will need the $G/P$-bundle associated to the bundle in (2). To simplify the
exposition on how this bundle is constructed, we introduce some Banach Lie
groups as in [13]. Define
$\displaystyle D_{0}$ $\displaystyle:=\\{z\in\mathbb{C}|~{}|z|\leqslant 2\\}$
$\displaystyle D_{\infty}$
$\displaystyle:=\\{z\in\mathbb{C}\cup\\{\infty\\}|~{}1/2\leqslant|z|\\}$
$\displaystyle A$ $\displaystyle:=D_{0}\cap D_{\infty},$
and
$\displaystyle\mathcal{G}$ $\displaystyle:=\\{\varphi:A\rightarrow
G|~{}\varphi\text{ is continuous and }\varphi|_{\mathring{A}}\text{ is
holomorphic}\\}$ $\displaystyle\mathcal{G}^{0}$
$\displaystyle:=\\{\varphi:D_{0}\rightarrow G|~{}\varphi\text{ is continuous
and }\varphi|_{\mathring{D}_{0}}\text{ is holomorphic}\\}$
$\displaystyle\mathcal{G}^{\infty}$
$\displaystyle:=\\{\varphi:D_{\infty}\rightarrow G|~{}\varphi\text{ is
continuous and }\varphi|_{\mathring{D}_{\infty}}\text{ is holomorphic}\\}.$
These groups are complex Banach Lie groups in the $C^{0}$-topology. Moreover,
$\mathcal{G}^{0}$ and $\mathcal{G}^{\infty}$ naturally embed into
$\mathcal{G}$ as subgroups in the sense of [9].
Define $\widetilde{P(G/P)}$ to be the pushout of the diagram
$\mathring{D}_{0}\times\mathcal{G}\times
G/P$$\mathring{D}_{0}\times\mathcal{G}\times
G/P$$\mathring{A}\times\mathcal{G}\times G/P$inclusion
$~{}(z,\phi,y)\mapsto(z^{-1},\phi,\phi(z)\cdot y)$
.
We call the left (resp. right) copy $\mathring{D}_{0}\times\mathcal{G}\times
G/P$ the $0$-chart (resp. $\infty$-chart). We have a map
$\widetilde{\pi}:\widetilde{P(G/P)}\rightarrow\mathbb{P}^{1}\times\mathcal{G}$
defined by forgetting the factor $G/P$ in each of these charts.
Define a left $\mathcal{G}^{\infty}$-action and a right
$\mathcal{G}^{0}$-action on $\widetilde{P(G/P)}$ by
$\psi^{\infty}\cdot(z,\phi,y)\cdot\psi^{0}:=\left\\{\begin{array}[]{ll}(z,\psi^{\infty}\phi\psi^{0},\psi^{0}(z)^{-1}\cdot
y)&0\text{-chart}\\\ (z,\psi^{\infty}\phi\psi^{0},\psi^{\infty}(z^{-1})\cdot
y)&\infty\text{-chart}\\\ \end{array}\right.$
for any $\psi^{0}\in\mathcal{G}^{0}$ and
$\psi^{\infty}\in\mathcal{G}^{\infty}$. It is easy to see that these actions
are free and commute with each other. Moreover, $\widetilde{\pi}$ is
equivariant where the action on $\mathbb{P}^{1}$ is assumed to be trivial.
Define
$P(G/P):=\widetilde{P(G/P)}/\mathcal{G}^{0}$
and
$\pi:P(G/P)\rightarrow\mathbb{P}^{1}\times(\mathcal{G}/\mathcal{G}^{0})$
to be the map induced by $\widetilde{\pi}$. It is straightforward to verify
that
1. (1)
the left $\mathcal{G}^{\infty}$-action on $\widetilde{P(G/P)}$ induces a left
$\mathcal{G}^{\infty}$-action on $P(G/P)$;
2. (2)
$\pi$ is a $\mathcal{G}^{\infty}$-equivariant $G/P$-fibration; and
3. (3)
the $\infty$-chart of $\widetilde{P(G/P)}$ induces an $\infty$-chart of
$P(G/P)$ in the obvious sense.
Observe that the above construction works not only for $G/P$ but also any
$G$-spaces. In particular, if we take the $G$-equivariant line bundle
$L_{\rho}=G\times_{P}\mathbb{C}_{-\rho}$ associated to any
$\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$ (see Section 2.3), we obtain a line
bundle $\mathcal{L}_{\rho}$ on $P(G/P)$ which restricts to $L_{\rho}$ on each
fiber of $\pi$. Line bundles of this form will be useful in the next
subsection.
### 3.2. Definition of the moduli
Following [13], we introduce two moduli spaces
$\overline{\mathcal{M}}(\mu,\beta)$ and
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ which will be used for the
definition and computation of the $R(T)$-algebra homomorphism $\Phi$ stated in
Theorem 1.1 respectively. Recall the fibration $P(G/P)$ defined in the last
subsection.
Let $\mu\in Q^{\vee}$. We have the associated cocharacter $x_{\mu}$ of $T$
which is naturally an element of $\mathcal{G}$. By abuse of notation, the
corresponding point in $\mathcal{G}/\mathcal{G}^{0}$ is denoted by the same
symbol. Define
$P_{{\mu}}(G/P):=P(G/P)|_{\mathbb{P}^{1}\times\\{x_{\mu}\\}}$
and
$\pi_{\mu}:P_{{\mu}}(G/P)\rightarrow\mathbb{P}^{1}$
to be the map induced by $\pi$. By [13, Lemma 3.2], it is a smooth projective
variety. Define $D_{\mu}:=\pi_{\mu}^{-1}(\infty)$ and
$\iota_{\mu}:D_{\mu}\hookrightarrow P_{{\mu}}(G/P)$ to be the inclusion.
###### Definition 3.2.
Let $\beta\in\pi_{2}(G/P)$.
1. (1)
Define
$\overline{\mathcal{M}}(\mu,\beta):=\bigcup_{\widetilde{\beta}}\overline{\mathcal{M}}_{0,1}(P_{{\mu}}(G/P),\widetilde{\beta})\times_{(\operatorname{ev}_{1},\iota_{\mu})}D_{\mu}$
where $\widetilde{\beta}$ runs over all classes in $\pi_{2}(P_{{\mu}}(G/P))$
satisfying
1. (i)
$(\pi_{\mu})_{*}\widetilde{\beta}=[\mathbb{P}^{1}]\in\pi_{2}(\mathbb{P}^{1})$;
and
2. (ii)
$\langle\widetilde{\beta},c_{1}(\mathcal{L}_{\rho})\rangle=\langle\beta,c_{1}(L_{\rho})\rangle$
for any $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$.
(The line bundles $\mathcal{L}_{\rho}$ and $L_{\rho}$ are defined in Section
3.1 and Section 2.3 respectively.)
2. (2)
Define
$\operatorname{ev}:\overline{\mathcal{M}}(\mu,\beta)\rightarrow D_{\mu}\simeq
G/P$
to be the morphism induced by the evaluation map $\operatorname{ev}_{1}$ on
$\overline{\mathcal{M}}_{0,1}(P_{{\mu}}(G/P),\widetilde{\beta})$.
Next we define $\overline{\mathcal{M}}(wt_{\lambda},\beta)$. Define
$\mathcal{B}^{0,-}:=\\{\varphi\in\mathcal{G}^{0}|~{}\varphi(0)\in B^{-}\\}.$
For any affine simple root $\widetilde{\alpha}_{i}$, $i=0,\ldots,r$ (see
Section 2.1), there exists a unique connected subgroup
$\mathcal{P}_{\widetilde{\alpha}_{i}}$ of $\mathcal{G}$ with
$\mathcal{B}^{0,-}\subset\mathcal{P}_{\widetilde{\alpha}_{i}}$ such that
$\operatorname{Lie}(\mathcal{P}_{\widetilde{\alpha}_{i}})=\left\\{\begin{array}[]{ll}\operatorname{Lie}(\mathcal{B}^{0,-})\oplus
z^{-1}\mathfrak{g}_{-\alpha_{0}}&i=0\\\\[10.00002pt]
\operatorname{Lie}(\mathcal{B}^{0,-})\oplus\mathfrak{g}_{\alpha_{i}}&i=1,\ldots,r\\\
\end{array}\right..$
For any $wt_{\lambda}\in W_{af}^{-}$, choose a reduced word decomposition
$(i_{1},\ldots,i_{\ell(wt_{\lambda})})$ of it. Define the associated Bott-
Samelson variety
$\Gamma_{wt_{\lambda}}:=\mathcal{P}_{\widetilde{\alpha}_{i_{1}}}\times_{\mathcal{B}^{0,-}}\cdots\times_{\mathcal{B}^{0,-}}\mathcal{P}_{\widetilde{\alpha}_{i_{\ell(wt_{\lambda})}}}/\mathcal{B}^{0,-}.$
It is easy to see that $\Gamma_{wt_{\lambda}}$ is a smooth projective variety
with a structure of iterated $\mathbb{P}^{1}$-bundles. Define a holomorphic
map
$f_{wt_{\lambda}}:\Gamma_{wt_{\lambda}}\rightarrow\mathcal{G}/\mathcal{G}^{0}$
by
$f_{wt_{\lambda}}([\varphi_{1}:\cdots:\varphi_{\ell(wt_{\lambda})}]):=\varphi_{1}\cdots\varphi_{\ell(wt_{\lambda})}\mathcal{G}^{0}.$
Define
$P_{wt_{\lambda}}(G/P):=(\mathbb{P}^{1}\times\Gamma_{wt_{\lambda}})\times_{(\operatorname{id}\times
f_{wt_{\lambda}},\pi)}P(G/P)$
and
$\pi_{wt_{\lambda}}:P_{wt_{\lambda}}(G/P)\rightarrow\mathbb{P}^{1}\times\Gamma_{wt_{\lambda}}$
to be the map induced by $\pi$. By [13, Lemma 3.2], $P_{wt_{\lambda}}(G/P)$ is
a smooth projective variety. Define
$D_{wt_{\lambda}}:=\pi_{wt_{\lambda}}^{-1}(\\{\infty\\}\times\Gamma_{wt_{\lambda}})$
and $\iota_{wt_{\lambda}}:D_{wt_{\lambda}}\hookrightarrow
P_{wt_{\lambda}}(G/P)$ to be the inclusion. Then $D_{wt_{\lambda}}$ is a
smooth divisor of $P_{wt_{\lambda}}(G/P)$ and isomorphic to
$\Gamma_{wt_{\lambda}}\times G/P$ via the $\infty$-chart of
$P_{wt_{\lambda}}(G/P)$.
###### Definition 3.3.
Let $\beta\in\pi_{2}(G/P)$.
1. (1)
Define
$\overline{\mathcal{M}}(wt_{\lambda},\beta):=\bigcup_{\widetilde{\beta}}\overline{\mathcal{M}}_{0,1}(P_{wt_{\lambda}}(G/P),\widetilde{\beta})\times_{(\operatorname{ev}_{1},\iota_{wt_{\lambda}})}D_{wt_{\lambda}}$
where $\widetilde{\beta}$ runs over all classes in
$\pi_{2}(P_{wt_{\lambda}}(G/P))$ satisfying
1. (i)
$(\pi_{wt_{\lambda}})_{*}\widetilde{\beta}=[\mathbb{P}^{1}\times\\{pt\\}]\in\pi_{2}(\mathbb{P}^{1}\times\Gamma_{wt_{\lambda}})$;
and
2. (ii)
$\langle\widetilde{\beta},c_{1}(\mathcal{L}_{\rho})\rangle=\langle\beta,c_{1}(L_{\rho})\rangle$
for any $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$.
2. (2)
Define
$\operatorname{ev}:\overline{\mathcal{M}}(wt_{\lambda},\beta)\rightarrow G/P$
to be the composite
$\overline{\mathcal{M}}(wt_{\lambda},\beta)\rightarrow
D_{wt_{\lambda}}\simeq\Gamma_{wt_{\lambda}}\times G/P\rightarrow G/P$
where the first arrow is induced by the evaluation map $\operatorname{ev}_{1}$
on $\overline{\mathcal{M}}_{0,1}(P_{wt_{\lambda}}(G/P),\widetilde{\beta})$ and
the second arrow is the canonical projection.
In order to compute $\Phi$, we have to establish some geometric properties of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$. This will be done in Section
3.5. The intermediate subsections 3.3 and 3.4 will serve as preparations.
### 3.3. An extra torus action on the moduli
We define an algebraic $\mathbb{C}^{\times}$-action on
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$. To avoid confusion with other
actions, we will introduce the subscript “$t$” for the action.
The desired action is defined in several steps:
1. (1)
Define an $S^{1}_{t}$-action on $\mathcal{G}$ by
$(t\cdot\varphi)(z):=\varphi(tz)\quad t\in
S^{1}_{t},~{}\varphi\in\mathcal{G}.$
Observe that this action preserves the subgroups $\mathcal{G}^{0}$,
$\mathcal{G}^{\infty}$, $\mathcal{B}^{0,-}$ and
$\mathcal{P}_{\widetilde{\alpha}_{i}}$.
2. (2)
Define an $S^{1}_{t}$-action on $\widetilde{P(G/P)}$ by
$t\cdot(z,\phi,y)=\left\\{\begin{array}[]{cl}(t^{-1}z,t\cdot\phi,y)&0\text{-chart}\\\
(tz,t\cdot\phi,y)&\infty\text{-chart}\end{array}\right..$
It is straightforward to check that this action descends to an
$S^{1}_{t}$-action on $P(G/P)$ which satisfies the following properties:
1. (i)
it is compatible with the $\mathcal{G}^{\infty}$-action in the sense that
$t\cdot(\psi^{\infty}\cdot x)=(t\cdot\psi^{\infty})\cdot(t\cdot x)$
for any $t\in S^{1}_{t}$, $\psi^{\infty}\in\mathcal{G}^{\infty}$ and $x\in
P(G/P)$; and
2. (ii)
the map
$\pi:P(G/P)\rightarrow\mathbb{P}^{1}\times(\mathcal{G}/\mathcal{G}^{0})$ is
equivariant where $S^{1}_{t}$ acts on $\mathbb{P}^{1}$ by
$t\cdot z=\left\\{\begin{array}[]{cl}t^{-1}z&0\text{-chart}\\\
tz&\infty\text{-chart}\end{array}\right..$
3. (3)
Consider the $S^{1}_{t}$-action on $\Gamma_{wt_{\lambda}}$ induced by the one
defined in (1). Then $f_{wt_{\lambda}}$ is equivariant.
4. (4)
It follows that there is an induced $S^{1}_{t}$-action on
$P_{wt_{\lambda}}(G/P)$. It is not hard to show that this $S^{1}_{t}$-action
extends to a unique algebraic $\mathbb{C}^{\times}_{t}$-action. See Remark 3.4
below.
5. (5)
Clearly, $D_{wt_{\lambda}}$ is $\mathbb{C}^{\times}_{t}$-invariant, and hence
the $\mathbb{C}^{\times}_{t}$-action in (4) induces a
$\mathbb{C}^{\times}_{t}$-action on
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$.
###### Remark 3.4.
Every $S^{1}$-action on a smooth projective variety by biholomorphisms extends
to a unique holomorphic $\mathbb{C}^{\times}$-action. In this paper, we
require the latter action to be algebraic. One way of showing this is to
construct a lift of the given $S^{1}$-action on an ample line bundle. Observe
that the spaces in question, including $\Gamma_{wt_{\lambda}}$ and
$P_{wt_{\lambda}}(G/P)$, have a structure of iterated fibrations such that the
fibers at each step are Fano. Thus, it suffices to deal with the following
situation. Let $F\rightarrow E\xrightarrow{\pi}B$ be a fibration with $F$
being Fano. Suppose $S^{1}$ acts on $E$ and $B$ such that $\pi$ is equivariant
and $B$ admits an equivariant ample line bundle $\mathcal{L}_{B}$. For
sufficiently large $N$, the line bundle
$\mathcal{L}_{E}:=\omega_{E/B}^{\vee}\otimes\pi^{*}\left(\mathcal{L}_{B}^{\otimes
N}\right)$ is ample. Then $S^{1}$ acts on $\mathcal{L}_{E}$ naturally because
$\omega_{E/B}^{\vee}$ is formed out of the vertical tangent bundle of $\pi$
and $\pi$ is equivariant.
### 3.4. Constant sections
Let $\mu\in Q^{\vee}$. Recall the point
$x_{\mu}\in\mathcal{G}/\mathcal{G}^{0}$. Define
$P_{\mu}:=\\{g\in G|~{}g\cdot x_{\mu}=x_{\mu}\\}.$
###### Lemma 3.5.
$P_{\mu}$ is a parabolic subgroup with Lie algebra
$\operatorname{Lie}(P_{\mu})=\mathfrak{h}\oplus\bigoplus_{\alpha(\mu)\leqslant
0}\mathfrak{g}_{\alpha}.$ (3.1)
###### Proof.
First notice that $P_{\mu}$ is an algebraic subgroup of $G$. It is clear that
the RHS of (3.1) is a parabolic subalgebra and so defines a parabolic subgroup
$P^{\prime}$. Then $P^{\prime}\subseteq P_{\mu}$, and hence $P_{\mu}$ is
connected. Suppose $P^{\prime}\subsetneq P_{\mu}$. Then there exists
$v\in\mathfrak{g}_{\alpha}\setminus\\{0\\}$ for some $\alpha\in R$ with
$\alpha(\mu)>0$ such that $g:=\exp(v)\in P_{\mu}$. This implies that the
holomorphic function $z\mapsto x_{\mu}(z^{-1})gx_{\mu}(z)$ extends to a
holomorphic function on $\mathbb{P}^{1}$. Since $G$ is affine, this function
is constant but this is impossible. ∎
Recall $P_{{\mu}}(G/P)=P(G/P)|_{\mathbb{P}^{1}\times\\{x_{\mu}\\}}$. Observe
it can also be defined as the pushout of the diagram
$\mathbb{C}\times G/P$$\mathbb{C}\times G/P$$\mathbb{C}^{\times}\times
G/P$inclusion $~{}(z,y)\mapsto(z^{-1},x_{\mu}(z)\cdot y)$
.
Notice $x_{\mu}\in(\mathcal{G}/\mathcal{G}^{0})^{S^{1}_{t}}$, and hence the
$S^{1}_{t}$-action preserves $P_{{\mu}}(G/P)$. One checks easily that the
induced action on $P_{{\mu}}(G/P)$, which actually extends to a
$\mathbb{C}^{\times}_{t}$-action, reads
$t\cdot(z,y)=\left\\{\begin{array}[]{cl}(t^{-1}z,x_{\mu}(t)\cdot
y)&0\text{-chart}\\\ (tz,y)&\infty\text{-chart}\end{array}\right..$
Consider the $\mathbb{C}^{\times}$-action on $G/P$ induced by the cocharacter
$x_{\mu}:\mathbb{C}^{\times}\rightarrow T$. We will write
$\mathbb{C}^{\times}_{\mu}$ in place of $\mathbb{C}^{\times}$ in this context.
Let $y\in G/P$. Since $G/P$ is complete, the morphism
$\mathbb{C}^{\times}\rightarrow G/P:z\mapsto x_{\mu}(z^{-1})\cdot y$ extends
to a morphism defined on $\mathbb{C}$.
###### Definition 3.6.
Let $y\in G/P$. Define a section $u_{y}$ of $P_{\mu}(G/P)$ by
$u_{y}(z):=\left\\{\begin{array}[]{cl}(z,x_{\mu}(z^{-1})\cdot
y)&0\text{-chart}\\\ (z,y)&\infty\text{-chart}\end{array}\right..$
Any sections of the form $u_{y}$ are called constant sections (meaning
constant in the $\infty$-chart).
###### Lemma 3.7.
Let $u:\mathbb{P}^{1}\rightarrow P_{{\mu}}(G/P)$ be a holomorphic section.
Suppose for any $t\in\mathbb{C}^{\times}_{t}$, there exists
$\phi\in\operatorname{Aut}(\mathbb{P}^{1})$ such that for any
$z\in\mathbb{P}^{1}$,
$t\cdot u(z)=u(\phi(z)).$
Then $u$ is a constant section.
###### Proof.
Restricting $u$ to the $0$-chart and $\infty$-chart, we get two maps
$u_{0},u_{\infty}:\mathbb{C}\rightarrow G/P$ satisfying
$u_{\infty}(z^{-1})=x_{\mu}(z)\cdot u_{0}(z)\quad\text{for any
}z\in\mathbb{C}^{\times}.$
The given condition implies $u_{\infty}$ is constantly equal to a point $y\in
G/P$. This forces $u\equiv u_{y}$. ∎
Denote by $F^{attr}_{\mathbb{C}^{\times}_{\mu}}(G/P)$ the unique component of
$(G/P)^{\mathbb{C}^{\times}_{\mu}}$ whose normal bundle has only positive
weights. The superscript “attr” will be explained in Definition 3.11.
###### Lemma 3.8.
$P_{\mu}$ preserves and acts transitively on
$F^{attr}_{\mathbb{C}^{\times}_{\mu}}(G/P)$.
###### Proof.
It suffices to look at the infinitesimal action. Let $y\in
F^{attr}_{\mathbb{C}^{\times}_{\mu}}(G/P)$. Notice that $T_{y}(G/P)$ is a
direct sum of weight spaces (with respect to the
$\mathbb{C}^{\times}_{\mu}$-action) of non-negative weights and the set of
these weights (counted with multiplicities) is a subset of the set of weights
of the $\mathbb{C}^{\times}_{\mu}$-module $\mathfrak{g}$. By definition,
$\operatorname{Lie}(P_{\mu})$ contains all non-positive weights of
$\mathfrak{g}$. It follows that the weights contributed by the infinitesimal
action of $P_{\mu}$ are precisely all the zero weights. ∎
###### Definition 3.9.
Define $\beta_{\mu}\in\pi_{2}(G/P)$ to be the unique element such that
$\deg(u^{*}\mathcal{L}_{\rho})=\langle\beta_{\mu},c_{1}(L_{\rho})\rangle$
for any $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$ where $u$ is the constant section
of $P_{{\mu}}(G/P)$ corresponding to a point in
$F^{attr}_{\mathbb{C}^{\times}_{\mu}}(G/P)$. (The line bundles
$\mathcal{L}_{\rho}$ and $L_{\rho}$ are defined in Section 3.1 and Section 2.3
respectively.)
### 3.5. Some properties of the moduli
###### Proposition 3.10.
[13, Proposition 4.5] The stack $\overline{\mathcal{M}}(wt_{\lambda},\beta)$
is smooth and of expected dimension.
In what follows, we prove some further properties of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$. More precisely, we determine the
set of $\beta$ for which
$\overline{\mathcal{M}}(wt_{\lambda},\beta)\neq\emptyset$ and show that
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ is irreducible for these $\beta$.
Our approach is to apply Oprea’s stacky version [34] of a theorem of
Białynicki-Birula [8] to the $\mathbb{C}^{\times}_{t}$-action on
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ defined in Section 3.3 and study
the geometry of a particular fixed-point component. It turns out that this
component is determined by $\overline{\mathcal{M}}_{0,2}(G/P,\beta^{\prime})$
for some other $\beta^{\prime}$ and
$F^{attr}_{\mathbb{C}^{\times}_{\mu}}(G/P)$ defined in Section 3.4.
###### Definition 3.11.
Let $\mathbb{C}^{\times}$ act on a smooth Deligne-Mumford stack $\mathcal{X}$.
A component $\mathcal{F}$ of $\mathcal{X}^{\mathbb{C}^{\times}}$ is said to be
attractive if for a (and hence any) geometric point $x\in\mathcal{F}$, the
weights (more precisely, the orbi-weights) of the tangent space
$T_{x}\mathcal{X}$ with respect to the $\mathbb{C}^{\times}$-action are all
non-negative.
###### Theorem 3.12.
[8] Let $X$ be a smooth quasi-projective variety with a
$\mathbb{C}^{\times}$-action and $F$ an attractive component of
$X^{\mathbb{C}^{\times}}$. There exists a unique
$\mathbb{C}^{\times}$-invariant open subscheme $U$ of $X$ containing $F$ which
is isomorphic to a $\mathbb{C}^{\times}$-equivariant affine fibration over
$F$.
For other components of $X^{\mathbb{C}^{\times}}$, there are similar affine
fibrations which are in general locally closed subschemes. Białynicki-Birula
also showed that if $X$ is proper, these subschemes form a decomposition of
$X$. His result has been generalized by Oprea [34] to Deligne-Mumford stacks.
For our purpose, we only need the following application.
###### Theorem 3.13.
Let $\mathcal{X}$ be a non-empty proper smooth Deligne-Mumford stack with a
$\mathbb{C}^{\times}$-action. Suppose $\mathcal{X}$ admits a
$\mathbb{C}^{\times}$-equivariant étale atlas. Then
$\mathcal{X}^{\mathbb{C}^{\times}}$ has an attractive component. It is unique
if and only if $\mathcal{X}$ is irreducible.
Denote by $F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$ the
unique attractive component of
$\Gamma_{wt_{\lambda}}^{\mathbb{C}^{\times}_{t}}$.
###### Lemma 3.14.
$f_{wt_{\lambda}}(F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}}))\subseteq
G\cdot x_{w(\lambda)}$.
###### Proof.
Let $\gamma\in\Gamma_{wt_{\lambda}}$ be the unique point such that
$f_{wt_{\lambda}}(\gamma)=x_{w(\lambda)}$. The result follows from the
observations that $\mathcal{B}^{0,-}\cdot\gamma$ is open and the weights of
$\operatorname{Lie}(B^{-})$ (resp.
$\operatorname{Lie}(\mathcal{B}^{0,-})/\operatorname{Lie}(B^{-})$) are all
zero (resp. positive). ∎
By Lemma 3.8,
$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$ is
well-defined. Consider the diagram
$F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$$G\cdot
x_{w(\lambda)}\simeq
G/P_{w(\lambda)}$$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$$G/P$$f$$\pi$$j$
(3.2)
where
* •
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$ and $P_{w(\lambda)}$ are
defined in Section 3.4;
* •
$f$ is the restriction of $f_{wt_{\lambda}}$ to
$F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$ (it does land in
$G\cdot x_{w(\lambda)}$ by Lemma 3.14);
* •
$\pi$ is the canonical projection; and
* •
$j$ is the unique $G$-equivariant map extending the inclusion
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)\hookrightarrow G/P$.
###### Definition 3.15.
Define a smooth variety
$F_{wt_{\lambda}}:=F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})\times_{(f,\pi)}(G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P))$
and $h_{wt_{\lambda}}:F_{wt_{\lambda}}\rightarrow G/P$ to be the composite
$F_{wt_{\lambda}}\xrightarrow{f^{\prime}}G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)\xrightarrow{j}G/P$
where $f^{\prime}$ is induced by $f$ in the fiber product.
The role of $F_{wt_{\lambda}}$ is to parametrize a class of holomorphic
sections of $P_{wt_{\lambda}}(G/P)|_{\mathbb{P}^{1}\times
F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})}$. Notice
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$ parametrizes a component of
the space of constant sections of $P_{{w(\lambda)}}(G/P)$. Indeed,
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$ is attractive so we have
$\displaystyle\lim_{z\to 0}x_{\mu}(z^{-1})\cdot y\in
F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)~{}\Longrightarrow~{}y\in
F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P).$ (3.3)
Since $P(G/P)|_{\mathbb{P}^{1}\times(G\cdot x_{w(\lambda)})}\simeq
G\times_{P_{w(\lambda)}}P_{{w(\lambda)}}(G/P)$, we see that $F_{wt_{\lambda}}$
parametrizes the pullbacks of the $G$-translates of these constant sections.
###### Definition 3.16.
Let $\beta\in\pi_{2}(G/P)$. Define a morphism
$F_{wt_{\lambda}}\times_{(h_{wt_{\lambda}},\operatorname{ev}_{1})}\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})\rightarrow\overline{\mathcal{M}}(wt_{\lambda},\beta)$
(3.4)
as follows.
* •
Every point of the domain of (3.4) is of the form $(\gamma,[g:y],u)$ where
1. (a)
$\gamma\in F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$;
2. (b)
$[g:y]\in
G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$; and
3. (c)
$u\in\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})$
such that $f_{wt_{\lambda}}(\gamma)=g\cdot x_{w(\lambda)}$ and
$u(z_{1})=g\cdot y$, where $z_{1}$ is the first marked point on the domain of
$u$.
* •
We send this point to $u_{1}\\#u_{2}$ where
1. (i)
$u_{1}$ is the section of
$P_{wt_{\lambda}}(G/P)|_{\mathbb{P}^{1}\times\\{\gamma\\}}\simeq
P(G/P)|_{\mathbb{P}^{1}\times\\{g\cdot x_{w(\lambda)}\\}}$ which is the
$g$-translate of the constant section of $P_{{w(\lambda)}}(G/P)$ corresponding
to $y$; and
2. (ii)
$u_{2}$ is just $u$ but regarded as a stable map to the fiber of
$P_{wt_{\lambda}}(G/P)$ over $(\infty,\gamma)$.
* •
If $\beta=\beta_{w(\lambda)}$, the domain of (3.4) is understood to be
$F_{wt_{\lambda}}$. In this case, the morphism is defined in a similar way.
It is easy to see that morphism (3.4) is injective. One can show that it is
even a closed immersion. But we will not use this fact.
###### Proposition 3.17.
Suppose $\overline{\mathcal{M}}(wt_{\lambda},\beta)\neq\emptyset$. Then
$\overline{\mathcal{M}}(wt_{\lambda},\beta)^{\mathbb{C}^{\times}_{t}}$ has a
unique attractive component. Set-theoretically, it is equal to the image of
morphism (3.4).
Recall $\Lambda\subset\pi_{2}(G/P)$ is the semigroup of effective curve
classes in $G/P$ and $\beta_{w(\lambda)}\in\pi_{2}(G/P)$ is defined in
Definition 3.9.
###### Corollary 3.18.
The set
$\\{\beta\in\pi_{2}(G/P)|~{}\overline{\mathcal{M}}(wt_{\lambda},\beta)\neq\emptyset\\}$
is equal to $\beta_{w(\lambda)}+\Lambda$.
###### Proof.
If $\beta=\beta_{w(\lambda)}$, then the domain of (3.4) is $F_{wt_{\lambda}}$
which is clearly non-empty. If $\beta\neq\beta_{w(\lambda)}$, then the stack
$\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})$ is non-empty if
and only if $\beta\in\beta_{w(\lambda)}+(\Lambda\setminus\\{0\\})$. ∎
###### Corollary 3.19.
For any $\beta\in\beta_{w(\lambda)}+\Lambda$,
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ is irreducible.
###### Proof.
This follows from Proposition 3.17 and Theorem 3.13. Notice that
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ does admit a
$\mathbb{C}^{\times}_{t}$-equivariant étale atlas, provided we reparametrize
the torus $\mathbb{C}^{\times}_{t}$. (So we actually apply Theorem 3.13 to
this reparametrized action, but this will not affect our arguments.) See
Remark 4.7. ∎
#### Proof of Proposition 3.17.
First observe that the domain of (3.4) is irreducible, by a result of Kim-
Pandharipande [24], and that this morphism sends every point into
$\overline{\mathcal{M}}(wt_{\lambda},\beta)^{\mathbb{C}^{\times}_{t}}$. It
follows that the image of (3.4) is contained in a unique component of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)^{\mathbb{C}^{\times}_{t}}$. Let
$\mathcal{F}$ be an attractive component of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)^{\mathbb{C}^{\times}_{t}}$ which
exists by Theorem 3.13. Let $u\in\mathcal{F}$. Observe that $u$ lies over a
point $p$ in a component $F$ of
$\Gamma_{wt_{\lambda}}^{\mathbb{C}^{\times}_{t}}$.
We claim $F=F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$.
Suppose the contrary. Since the $T$-action commutes with the
$\mathbb{C}^{\times}_{t}$-action (see Section 3.3(2)(i)), $T$ preserves
$\mathcal{F}$, and hence we can replace $u$ with another
$\overline{u}\in\mathcal{F}$ which is also a $T$-fixed point of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$. Then $\overline{u}$ lies over a
$T$-fixed point $\overline{p}$ of $F$. Since $F\neq
F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$, the tangent space
$T_{\overline{p}}\Gamma_{wt_{\lambda}}$ contains a weight vector $v$ of
negative weight. Consider $\mathcal{B}^{0,-}\cdot\overline{p}$, the
$\mathcal{B}^{0,-}$-orbit passing through $\overline{p}$. Then $v\not\in
T_{\overline{p}}(\mathcal{B}^{0,-}\cdot\overline{p})$. By [13, Proposition
4.5], $\overline{u}$ is still unobstructed when it is regarded as a stable map
to
$P_{wt_{\lambda}}(G/P)|_{\mathbb{P}^{1}\times(\mathcal{B}^{0,-}\cdot\overline{p})}$
(a smooth $\mathcal{B}^{0,-}$-equivariant compactification of
$\mathcal{B}^{0,-}\cdot\overline{p}$ is not required since $\overline{u}$ is
$T$-invariant). Therefore, $v$ lifts to a weight vector in the tangent space
$T_{\overline{u}}\overline{\mathcal{M}}(wt_{\lambda},\beta)$ which has the
same weight as $v$. By assumption, the weight is negative, a contradiction.
By Lemma 3.14,
$f_{wt_{\lambda}}(F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}}))\subseteq
G\cdot x_{w(\lambda)}$ and
$P_{wt_{\lambda}}(G/P)|_{\mathbb{P}^{1}\times\\{p\\}}\simeq
P(G/P)|_{\mathbb{P}^{1}\times{\\{f_{wt_{\lambda}}(p)}\\}}\simeq
P_{{w(\lambda)}}(G/P)$
as $\mathbb{C}^{\times}_{t}$-varieties. Since the rest of the proof relies
only on constructing some deformation vector fields of $u$ in
$P_{wt_{\lambda}}(G/P)|_{\mathbb{P}^{1}\times\\{p\\}}$, we may assume
$f_{wt_{\lambda}}(p)=x_{w(\lambda)}$ so that $u$ is a stable map to
$P_{{w(\lambda)}}(G/P)$ which represents a section class and is a
$\mathbb{C}^{\times}_{t}$-fixed point in the moduli. Write
$u=u_{0}\\#u_{s}\\#u_{\infty}$ where $u_{s}$ is a section and $u_{0}$ (resp.
$u_{\infty}$) is a stable map to the fiber of $P_{{w(\lambda)}}(G/P)$ over $0$
(resp. $\infty$). By Lemma 3.7, $u_{s}$ is the constant section corresponding
to a point $y\in G/P$.
We first reduce the situation to the case
$y\in(G/P)^{\mathbb{C}^{\times}_{w(\lambda)}}$. More precisely, we show that
there exists another stable map $u^{\prime}\in\mathcal{F}$ such that if we
write $u^{\prime}=u^{\prime}_{0}\\#u^{\prime}_{s}\\#u^{\prime}_{\infty}$ as
before, then $u^{\prime}_{s}$ is the constant section corresponding to a point
in $(G/P)^{\mathbb{C}^{\times}_{w(\lambda)}}$. For any
$\eta\in\mathbb{C}^{\times}$, define $y_{\eta}:=x_{\mu}(\eta^{-1})\cdot y$.
Let $u_{y_{\eta}}$ be the constant section of $P_{w(\lambda)}(G/P)$
corresponding to $y_{\eta}$. Define a morphism
$\begin{array}[]{ccc}\mathbb{C}^{\times}&\rightarrow&\overline{\mathcal{M}}(wt_{\lambda},\beta)\\\\[5.0pt]
\eta&\mapsto&u_{0}\\#u_{y_{\eta}}\\#(x_{\mu}(\eta^{-1})\cdot
u_{\infty})\end{array}$ (3.5)
By [17, Proposition 6], after a base change
$\mathbb{C}^{\times}\rightarrow\mathbb{C}^{\times}$, the above morphism
extends to a morphism defined on $\mathbb{C}$. The stable map at $\eta=0$ will
be our $u^{\prime}$.
From now on, we assume $y\in(G/P)^{\mathbb{C}^{\times}_{w(\lambda)}}$. We show
1. (1)
$u_{0}$ does not exist; and
2. (2)
$y\in F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$.
For (1), recall $\mathbb{C}^{\times}_{t}$ acts on $\mathbb{P}^{1}$, the base
of $P_{{w(\lambda)}}(G/P)$, in the following way:
$t\cdot z=\left\\{\begin{array}[]{cl}t^{-1}z&0\text{-chart}\\\
tz&\infty\text{-chart}\end{array}\right..$
Take a weight vector $\zeta\in H^{0}(\mathbb{P}^{1};T\mathbb{P}^{1})$ such
that $\zeta(0)\neq 0$. Then it has weight $-1$. It is easy to show that there
exists a weight vector $\zeta^{\prime}$ in
$H^{0}(u^{*}TP_{{w(\lambda)}}(G/P))$ which is non-tangential to $u$ such that
$\zeta^{\prime}|_{u_{s}}$ projects to $\zeta$. It follows that
$\zeta^{\prime}$ defines a weight vector in
$T_{u}\overline{\mathcal{M}}(wt_{\lambda},\beta)$ of weight $-1$, a
contradiction.
For (2), notice that $T_{y}(G/P)$ is isomorphic, as
$\mathbb{C}^{\times}_{w(\lambda)}$-modules, to a direct sum of weight spaces
of the form $\mathbb{C}_{\alpha(w(\lambda))}$ where $\alpha\in R$. If
$y\not\in F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$, then $T_{y}(G/P)$
contains a weight vector $v\in\mathbb{C}_{\alpha(w(\lambda))}$ for some
$\alpha$ with $\alpha(w(\lambda))<0$. Define a vector field $\xi\in
H^{0}(\mathbb{P}^{1};u_{s}^{*}TP_{{w(\lambda)}}(G/P))$ by
$\xi(z)=\left\\{\begin{array}[]{cl}v&0\text{-chart}\\\
z^{-\alpha(w(\lambda))}v&\infty\text{-chart}\end{array}\right..$
Since $\xi(\infty)=0$, we can extend $\xi$ trivially to a deformation vector
field $\xi^{\prime}$ of $u$. (Recall we have proved that $u_{0}$ does not
exist.) It is clear that $\xi^{\prime}$ is non-tangential to $u$, and hence it
defines a weight vector in $T_{u}\overline{\mathcal{M}}(wt_{\lambda},\beta)$
of weight $\alpha(w(\lambda))<0$, a contradiction.
Thus, every $u\in\mathcal{F}$ is contained in the set-theoretic image of
morphism (3.4) after passing to the limit of morphism (3.5). By (3.3), $u$
actually lies in the image set before passing to the limit. The proof of
Proposition 3.17 is complete. $\square$
## 4\. Proof of the main theorem
### 4.1. Construction of the homomorphism
###### Definition 4.1.
Define an $R(T)$-linear map
$\Phi:\bigoplus_{\mu\in
Q^{\vee}}\operatorname{Frac}(R(T))\langle\mathcal{O}_{\mu}\rangle\rightarrow
QK_{T}(G/P)[\Lambda^{-1}]\otimes\operatorname{Frac}(R(T))$
by
$\Phi(\mathcal{O}_{\mu}):=\sum_{i,j\in
I}\sum_{\beta\in\pi_{2}(G/P)}q^{\beta}g^{ij}\chi_{\overline{\mathcal{M}}(\mu,\beta)}(\mathcal{O}_{\overline{\mathcal{M}}(\mu,\beta)}^{vir}\otimes\operatorname{ev}^{*}e_{i})(\operatorname{id}+A_{G/P})^{-1}(e_{j}),$
where $\mathcal{O}_{\overline{\mathcal{M}}(\mu,\beta)}^{vir}\in
K^{T}(\overline{\mathcal{M}}(\mu,\beta))$ is the virtual structure sheaf
constructed in [31]. See Section 2.3 for the definition of $\\{e_{i}\\}_{i\in
I}$, $\\{g^{ij}\\}_{i,j\in I}$ and $A_{G/P}$, and Section 3.2 for the
definition of $\overline{\mathcal{M}}(\mu,\beta)$.
In order for $\Phi$ to be well-defined, we must verify
###### Lemma 4.2.
$\Phi(\mathcal{O}_{\mu})$ lands in $QK_{T}(G/P)[\Lambda^{-1}]$.
###### Proof.
This follows from Corollary 3.18 since
$\overline{\mathcal{M}}(\mu,\beta)\subseteq\overline{\mathcal{M}}(wt_{\lambda},\beta)$
for some $wt_{\lambda}\in W_{af}^{-}$. ∎
###### Proposition 4.3.
$\Phi$ is an $R(T)$-algebra homomorphism.
###### Proof.
The proof relies heavily on Appendix A. Let $\mu_{1},\mu_{2}\in Q^{\vee}$. We
have
$\Phi(\mathcal{O}_{\mu_{1}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathcal{O}_{\mu_{2}})=\Phi(\mathcal{O}_{\mu_{1}+\mu_{2}})=\sum_{i,j\in
I}g^{ij}\operatorname{PKGW}_{\mu_{1}+\mu_{2}}(e_{i})(\operatorname{id}+A_{G/P})^{-1}(e_{j})$
where $\operatorname{PKGW}$ is defined in Definition A.2. By Proposition A.3,
the last expression is equal to
$\displaystyle\sum_{i,j\in I}\sum_{i^{\prime},j^{\prime}\in
I}g^{ij}g^{i^{\prime}j^{\prime}}\operatorname{PKGW}_{\mu_{1}}\left(e_{i},(\operatorname{id}+A_{G/P})^{-1}(e_{i^{\prime}})\right)\operatorname{PKGW}_{\mu_{2}}(e_{j^{\prime}})(\operatorname{id}+A_{G/P})^{-1}(e_{j})$
$\displaystyle=~{}$ $\displaystyle\sum_{i,j\in
I}g^{ij}\operatorname{PKGW}_{\mu_{1}}\left(e_{i},\Phi(\mathcal{O}_{\mu_{2}})\right)(\operatorname{id}+A_{G/P})^{-1}(e_{j}).$
Applying Proposition A.3 again, to the splitting $\mu_{1}=\mu_{1}+0$, the last
expression is equal to
$\displaystyle\sum_{i,j\in I}\sum_{i^{\prime},j^{\prime}\in
I}g^{ij}g^{i^{\prime}j^{\prime}}\operatorname{PKGW}_{\mu=0}\left(e_{i},\Phi(\mathcal{O}_{\mu_{2}}),(\operatorname{id}+A_{G/P})^{-1}(e_{i^{\prime}})\right)\operatorname{PKGW}_{\mu_{1}}(e_{j^{\prime}})(\operatorname{id}+A_{G/P})^{-1}(e_{j})$
$\displaystyle=~{}$ $\displaystyle\sum_{i,j\in
I}g^{ij}\operatorname{PKGW}_{\mu=0}\left(e_{i},\Phi(\mathcal{O}_{\mu_{2}}),\Phi(\mathcal{O}_{\mu_{1}})\right)(\operatorname{id}+A_{G/P})^{-1}(e_{j}).$
But we have (cf. [13, Lemma 3.7])
$\operatorname{PKGW}_{\mu=0}\left(e_{i},\Phi(\mathcal{O}_{\mu_{2}}),\Phi(\mathcal{O}_{\mu_{1}})\right)=\sum_{\beta\in\Lambda}q^{\beta}\operatorname{KGW}^{\beta}(\Phi(\mathcal{O}_{\mu_{1}}),\Phi(\mathcal{O}_{\mu_{2}}),e_{i}).$
Therefore,
$\Phi(\mathcal{O}_{\mu_{1}}\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}\mathcal{O}_{\mu_{2}})=\Phi(\mathcal{O}_{\mu_{1}})\star\Phi(\mathcal{O}_{\mu_{2}})$
as desired. ∎
### 4.2. Step 1 of the computation: reducing to the initial term
The main result of this subsection is Proposition 4.8. Let $wt_{\lambda}\in
W_{af}^{-}$. Recall the map
$f_{wt_{\lambda}}:\Gamma_{wt_{\lambda}}\rightarrow\mathcal{G}/\mathcal{G}^{0}$
defined in Section 3.2. By abuse of notation, we also denote by
$f_{wt_{\lambda}}$ the composite
$\Gamma_{wt_{\lambda}}\xrightarrow{f_{wt_{\lambda}}}\mathcal{G}/\mathcal{G}^{0}\rightarrow
L_{sm}G/L_{sm}^{0}G\xrightarrow{\sim}\Omega_{sm}K$
where the second arrow is the obvious map and the third is the diffeomorphism
in Theorem 2.4.
###### Lemma 4.4.
$f_{wt_{\lambda}}$ is a $\mathcal{B}^{0,-}$-equivariant resolution of
$\overline{\mathcal{B}^{0,-}_{sm}\cdot x_{w(\lambda)}}$. In particular, the
image of $f_{wt_{\lambda}}$ lies in $Gr_{G}=\Omega_{pol}K$.
###### Proof.
First notice that for any $n\in\mathbb{N}$ and $i=0,\ldots,r$, we have
$\mathcal{P}_{\widetilde{\alpha}_{i}}\cdot Gr^{(n)}(H)\subseteq
Gr^{(n+1)}(H).$
It follows that $f_{wt_{\lambda}}$ lands in $Gr^{(N)}(H)$ for sufficiently
large $N$. In particular, $f_{wt_{\lambda}}$ is algebraic. Since
$wt_{\lambda}\in W_{af}^{-}$ and the word defining $\Gamma_{wt_{\lambda}}$ is
reduced, there exists a unique point $\gamma\in\Gamma_{wt_{\lambda}}$ such
that $f_{wt_{\lambda}}(\gamma)=x_{w(\lambda)}$. Moreover, the orbit
$\mathcal{B}^{0,-}\cdot\gamma$ is open and
$f_{wt_{\lambda}}|_{\mathcal{B}^{0,-}\cdot\gamma}$ is bijective onto
$\mathcal{B}^{0,-}_{sm}\cdot x_{w(\lambda)}$. The rest of the proof is clear.
∎
###### Lemma 4.5.
$(f_{wt_{\lambda}})_{*}[\mathcal{O}_{\Gamma_{wt_{\lambda}}}]=[\mathcal{O}_{\overline{\mathcal{B}^{0,-}_{sm}\cdot
x_{w(\lambda)}}}]=\mathcal{O}_{wt_{\lambda}}\in K^{T}(Gr_{G})$.
###### Proof.
This follows from the fact that affine Schubert varieties have rational
singularities. A proof can be found in [26, Theorem 8.2.2]. Although our
definition of these varieties is a priori different from the one in loc. cit.,
the arguments there apply well to our case. ∎
Recall $\mathcal{O}_{wt_{\lambda}}$ is regarded as an element of the domain of
$\Phi$ via (2.3).
###### Lemma 4.6.
We have
$\Phi(\mathcal{O}_{wt_{\lambda}})=\sum_{i,j\in
I}\sum_{\beta\in\beta_{w(\lambda)}+\Lambda}q^{\beta}g^{ij}\chi_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}(\operatorname{ev}^{*}e_{i})(\operatorname{id}+A_{G/P})^{-1}(e_{j})$
where $\beta_{w(\lambda)}$ is defined in Definition 3.9.
###### Proof.
By the classical localization formula,
$[\mathcal{O}_{\Gamma_{wt_{\lambda}}}]=\sum_{\gamma\in\Gamma_{wt_{\lambda}}^{T}}\frac{1}{\Lambda_{-1}((T_{\gamma}\Gamma_{wt_{\lambda}})^{\vee})}[\mathcal{O}_{\gamma}]\in
K^{T}(\Gamma_{wt_{\lambda}})$
where $\Lambda_{-1}(V):=\sum_{i\geqslant 0}(-1)^{i}[\Lambda^{i}V]\in R(T)$ for
any $T$-module $V$. Applying $(f_{wt_{\lambda}})_{*}$ to both sides of the
last equation and using Lemma 4.5, we get
$\mathcal{O}_{wt_{\lambda}}=\sum_{\mu\in
Q^{\vee}}\left(\sum_{\gamma\in\Gamma_{wt_{\lambda}}^{T}\cap
f_{wt_{\lambda}}^{-1}(\mu)}\frac{1}{\Lambda_{-1}((T_{\gamma}\Gamma_{wt_{\lambda}})^{\vee})}\right)\mathcal{O}_{\mu}.$
The rest follows from a parallel argument used in the proof of [13, Lemma 3.9]
which deals with the case of quantum cohomology. In our case, we need the
$K$-theoretic version of virtual localization formula in [19]. See [37] for
the explicit formula and its proof. ∎
###### Remark 4.7.
Some care needs to be taken when we apply the virtual localization formula:
The proof of this formula given in [37] assumes an extra condition which, by
[23, Proposition 5.13], is satisfied if our stack admits a $T$-equivariant
étale atlas of finite type. According to the remark following that
proposition, which cites [2, Theorem 4.3], every separated Deligne-Mumford
stack of finite type with a $T$-action admits such an atlas, after possibly
reparametrizing $T$. Notice that such reparametrization will not affect the
argument in the proof of Lemma 4.6. Alternatively, the existence of the
required atlas for our particular stack follows from [34, Corollary 4].
Define
$\varphi:=\sum_{i,j\in
I}\sum_{\beta\in\beta_{w(\lambda)}+\Lambda}q^{\beta}g^{ij}\chi_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}(\operatorname{ev}^{*}e_{i})e_{j}.$
Notice the absence of $(\operatorname{id}+A_{G/P})^{-1}$. Write
$\varphi=\varphi_{0}+\varphi_{+}$ where $\varphi_{0}$ (resp. $\varphi_{+}$) is
the expression contributed by $\beta=\beta_{w(\lambda)}$ (resp.
$\beta\neq\beta_{w(\lambda)}$).
###### Proposition 4.8.
$\Phi(\mathcal{O}_{wt_{\lambda}})=\varphi_{0}$.
###### Proof.
By Lemma 4.10 below, we have $\varphi_{+}=A_{G/P}(\varphi_{0})$, and hence
$\Phi(\mathcal{O}_{wt_{\lambda}})=(\operatorname{id}+A_{G/P})^{-1}(\varphi_{0}+\varphi_{+})=(\operatorname{id}+A_{G/P})^{-1}\circ(\operatorname{id}+A_{G/P})(\varphi_{0})=\varphi_{0}.$
∎
Before proving Lemma 4.10 which is used in the proof of Proposition 4.8, we
first prove another lemma. Recall morphism (3.4). Its domain is
$F_{wt_{\lambda}}\times_{(h_{wt_{\lambda}},\operatorname{ev}_{1})}\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})$
if $\beta\neq\beta_{w(\lambda)}$ and $F_{wt_{\lambda}}$ if
$\beta=\beta_{w(\lambda)}$.
###### Lemma 4.9.
We have
$\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}]=\left\\{\begin{array}[]{cc}(\operatorname{ev}^{\prime}_{2})_{*}[\mathcal{O}_{F_{wt_{\lambda}}\times_{(h_{wt_{\lambda}},\operatorname{ev}_{1})}\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})}]&\beta\neq\beta_{w(\lambda)}\\\\[10.00002pt]
(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}]&\beta=\beta_{w(\lambda)}\end{array}\right.$
where $\operatorname{ev}^{\prime}_{2}$ is induced by the evaluation map on
$\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})$ at the second
marked point.
###### Proof.
By Remark 2.1, it suffices to verify the corresponding equality for their
coarse moduli. Let $M$ and $F$ be the coarse moduli of
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ and the domain of (3.4). They are
projective varieties with only finite quotient singularities. Since
$\overline{\mathcal{M}}(wt_{\lambda},\beta)$ is irreducible by Corollary 3.19
and the canonical morphism
$\overline{\mathcal{M}}(wt_{\lambda},\beta)\rightarrow M$ is surjective, $M$
is also irreducible. Similarly, $F$ is irreducible. By the definition of
coarse moduli, morphism (3.4) induces a unique morphism $\iota:F\rightarrow M$
which is equal to (3.4) set-theoretically. The uniqueness implies that $\iota$
is $T$-equivariant.
Denote by $M^{sm}$ the smooth locus of $M$. It is easy to see that
$\iota^{-1}(M^{sm})\neq\emptyset$ (look at chains of embedded spheres) and
$\iota$ maps $\iota^{-1}(M^{sm})$ bijectively onto an attractive component
$F^{\prime}$ of $(M^{sm})^{\mathbb{C}^{\times}_{t}}$. Since
$\iota^{-1}(M^{sm})$ is reduced, $\iota|_{\iota^{-1}(M^{sm})}$ factors through
the inclusion $F^{\prime}\hookrightarrow M^{sm}$. There exists a non-empty
open subscheme $V\subseteq\iota^{-1}(M^{sm})$ such that $\iota|_{V}$ is smooth
over $F^{\prime}$. Since $\iota$ is injective, $\iota|_{V}$ is étale over
$F^{\prime}$ and hence an isomorphism onto its image.
By Theorem 3.12, there exists a $\mathbb{C}^{\times}_{t}$-invariant open
subscheme $U\subseteq M^{sm}$ containing $F^{\prime}$ and an affine fibration
$U\rightarrow F^{\prime}$. The latter morphism induces, via the morphism
$\iota|_{V}:V\xrightarrow{\sim}\iota(V)\subseteq F^{\prime}$, a rational map
$\phi:M\dashrightarrow F$. Since $\iota$ is $T$-equivariant, $U$ is unique and
the $T$-action commutes with the $\mathbb{C}^{\times}_{t}$-action (see Section
3.3(2)(i)), it follows that everything is $T$-equivariant.
By resolving the indeterminacy locus of $\phi$, we obtain a smooth irreducible
projective variety $Z$ and morphisms $\nu_{M}:Z\rightarrow M$ and
$\nu_{F}:Z\rightarrow F$ such that $\nu_{M}$ is birational and
$\phi\circ\nu_{M}=\nu_{F}$. Since equivariant resolutions of singularities
exist (see e.g. [25]), we may assume $Z$ has a $T$-action and $\nu_{M}$,
$\nu_{F}$ are $T$-equivariant. Define
$\operatorname{ev}_{F}:=\operatorname{ev}_{2}^{\prime}$ if
$\beta\neq\beta_{w(\lambda)}$ and $\operatorname{ev}_{F}:=h_{wt_{\lambda}}$
otherwise. By the fact that $\operatorname{ev}$ is
$\mathbb{C}^{\times}_{t}$-invariant, we have
$\operatorname{ev}=\operatorname{ev}_{F}\circ\phi$, and hence
$\operatorname{ev}\circ\nu_{M}=\operatorname{ev}_{F}\circ\nu_{F}$, giving
$\operatorname{ev}_{*}(\nu_{M})_{*}[\mathcal{O}_{Z}]=(\operatorname{ev}_{F})_{*}(\nu_{F})_{*}[\mathcal{O}_{Z}]\in
K_{T}(G/P).$
Since $Z$ has only finite quotient singularities and singularities of this
kind are rational, by [40], we have
$(\nu_{M})_{*}[\mathcal{O}_{Z}]=[\mathcal{O}_{M}]$. To conclude the proof, it
suffices to show $(\nu_{F})_{*}[\mathcal{O}_{Z}]=[\mathcal{O}_{F}]$. This
follows from [12, Theorem 3.1], given the following conditions:
1. (1)
$\nu_{F}$ is surjective and $T$-equivariant;
2. (2)
$Z$ and $F$ are projective with rational singularities; and
3. (3)
the general fiber of $\nu_{F}$ is rational.
Condition (1) is obvious. To verify (2), we use the above cited result [40].
For (3), take a non-empty open subscheme $W\subseteq V$ such that
$\nu_{F}|_{\nu_{F}^{-1}(W)}$ is smooth. Since $Z$ contains an open dense
subscheme $U^{\prime}$ such that $\nu_{F}|_{U^{\prime}}$ is an affine
fibration over $W$, it follows that every geometric fiber of
$\nu_{F}|_{\nu_{F}^{-1}(W)}$ is connected and contains the affine space as an
open subscheme, i.e. it is rational. ∎
###### Lemma 4.10.
$\varphi_{+}=A_{G/P}(\varphi_{0})$.
###### Proof.
By the projection formula,
$\varphi_{+}=\sum_{\beta\in\beta_{w(\lambda)}+(\Lambda\setminus\\{0\\})}q^{\beta}\operatorname{ev}_{*}[\mathcal{O}_{\overline{\mathcal{M}}(wt_{\lambda},\beta)}]$
which is equal to
$\sum_{\beta\in\beta_{w(\lambda)}+(\Lambda\setminus\\{0\\})}q^{\beta}(\operatorname{ev}^{\prime}_{2})_{*}[\mathcal{O}_{F_{wt_{\lambda}}\times_{(h_{wt_{\lambda}},\operatorname{ev}_{1})}\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})}]$
by Lemma 4.9.
Denote by $\operatorname{ev}_{1}$ and $\operatorname{ev}_{2}$ the evaluation
maps on $\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})$. Since
$\operatorname{ev}_{1}$ is flat, we have
$(\operatorname{ev}^{\prime}_{2})_{*}[\mathcal{O}_{F_{wt_{\lambda}}\times_{(h_{wt_{\lambda}},\operatorname{ev}_{1})}\overline{\mathcal{M}}_{0,2}(G/P,\beta-\beta_{w(\lambda)})}]=(\operatorname{ev}_{2})_{*}(\operatorname{ev}_{1})^{*}(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}],$
(4.1)
by the base change formula. Summing up (4.1) over all
$\beta\in\beta_{w(\lambda)}+(\Lambda\setminus\\{0\\})$, weighted by
$q^{\beta}$, we get
$\varphi_{+}=A_{G/P}(q^{\beta_{w(\lambda)}}(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}])$.
By Lemma 4.9 applied to $\beta=\beta_{w(\lambda)}$, we get
$\varphi_{0}=q^{\beta_{w(\lambda)}}(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}]$.
The result follows. ∎
### 4.3. Step 2 of the computation: determining the initial term
By Proposition 4.8, it suffices to determine $\varphi_{0}$. By Lemma 4.9
applied to $\beta=\beta_{w(\lambda)}$, we have
$\varphi_{0}=q^{\beta_{w(\lambda)}}(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}]$
(4.2)
where $F_{wt_{\lambda}}$ and $h_{wt_{\lambda}}$ are defined in Definition
3.15. Since $F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})$ is
$B^{-}$-invariant and
$f=f_{wt_{\lambda}}|_{F^{attr}_{\mathbb{C}^{\times}_{t}}(\Gamma_{wt_{\lambda}})}$
is $B^{-}$-equivariant, $f$ is birational onto the Schubert variety
$\overline{B^{-}\cdot x_{w(\lambda)}}\subseteq G\cdot x_{w(\lambda)}\simeq
G/P_{w(\lambda)}$. Since Schubert varieties have rational singularities (see
Lemma 4.5), we have, by the base change formula,
$(h_{wt_{\lambda}})_{*}[\mathcal{O}_{F_{wt_{\lambda}}}]=j_{*}[\mathcal{O}_{\pi^{-1}(\overline{B^{-}\cdot
x_{w(\lambda)}})}].$ (4.3)
See (3.2) for the definition of $j$ and $\pi$.
Let us deal with the case $P=B^{+}$ first. Define $y_{w}^{\prime}:=wB^{+}\in
G/B^{+}$ and $B_{w}:=wBw^{-1}$.
###### Lemma 4.11.
1. (1)
We have $y_{w}^{\prime}\in
F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})$ and $B_{w}\subseteq
P_{w(\lambda)}$.
2. (2)
There exists a $G$-equivariant isomorphism
$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})\simeq
G/B_{w}$
under which $\pi$ and $j$ (see diagram (3.2)) are identified with the
projection $G/B_{w}\rightarrow G/P_{w(\lambda)}$ and the isomorphism
$G/B_{w}\xrightarrow{\sim}G/B^{+}:gB_{w}\mapsto gwB^{+}$ respectively.
3. (3)
$j$ maps $\pi^{-1}(\overline{B^{-}\cdot x_{w(\lambda)}})$ isomorphically onto
$\overline{B^{-}\cdot y_{w}^{\prime}}$.
###### Proof.
(1) is proved by looking at the weight spaces:
$T_{y_{w}^{\prime}}(G/B^{+})\simeq\bigoplus_{\alpha\in-
wR^{+}}\mathfrak{g}_{\alpha}\quad\text{ and
}\quad\operatorname{Lie}(B_{w})=\mathfrak{h}\oplus\bigoplus_{\alpha\in
wR^{+}}\mathfrak{g}_{\alpha}.$
Since $wt_{\lambda}\in W_{af}^{-}$, $\lambda$ is anti-dominant, and hence
$\pm\alpha(w(\lambda))\geqslant 0$ for any $\alpha\in\mp wR^{+}$.
To prove (2), recall (Lemma 3.8) $P_{w(\lambda)}$ acts transitively on
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})$. Hence, by (1), we have
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})\simeq
P_{w(\lambda)}/B_{w}$ so that
$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})\simeq
G\times_{P_{w(\lambda)}}(P_{w(\lambda)}/B_{w})\simeq G/B_{w}.$
The rest of the proof is clear.
For (3), we use the identifications in (2). Denote by $C$ the dominant Weyl
chamber. It suffices to show that $wC$ has the smallest length (with respect
to $C$) among other chambers which contain $-w(\lambda)$. It amounts to
showing
$\alpha(w(\lambda))=0\text{ and }\alpha\in R^{+}\Longrightarrow
w^{-1}\alpha\in R^{+}.$ (4.4)
This requires the assumption $wt_{\lambda}\in W_{af}^{-}$. Denote by $\Delta$
the dominant alcove. By definition, the alcove
$wt_{\lambda}(\Delta)=w(\lambda)+w(\Delta)$ has the smallest length (with
respect to $\Delta$) among other alcoves which contain $w(\lambda)$. If
$\alpha(w(\lambda))=0$, then $w(\lambda)+w(\Delta)$ and $\Delta$ lie in the
same side with respect to the wall $\\{\alpha=0\\}$. This proves (4.4). ∎
For general $P$, we have the commutative diagram
$G\cdot
x_{w(\lambda)}$$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})$$G\times_{P_{w(\lambda)}}F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/P)$$G/B^{+}$$G/P$$\simeq$$\pi_{G/B^{+}}$$\pi_{G/P}$$p$$j_{G/B^{+}}$$j_{G/P}$
where the horizontal arrows are some canonical projections. By Lemma 4.11 and
the fact that the upper horizontal arrow has rational fibers, we have
$(j_{G/P})_{*}[\mathcal{O}_{(\pi_{G/P})^{-1}(\overline{B^{-}\cdot
x_{w(\lambda)}})}]=p_{*}[\mathcal{O}_{\overline{B^{-}\cdot y_{w}^{\prime}}}].$
(4.5)
Denote by $\widetilde{w}\in W/W_{P}$ the minimal length coset representative
of $wW_{P}$. Notice $p(\overline{B^{-}\cdot
y_{w}^{\prime}})=\overline{B^{-}\cdot y_{\widetilde{w}}}$ but the dimension of
some fibers of $p|_{\overline{B^{-}\cdot y_{w}^{\prime}}}$ may be positive.
Choose Bott-Samelson resolutions
$\Gamma^{\prime}\rightarrow\overline{B^{-}\cdot y_{w}^{\prime}}$ and
$\Gamma\rightarrow\overline{B^{-}\cdot y_{\widetilde{w}}}$ such that there is
a map $p^{\prime}:\Gamma^{\prime}\rightarrow\Gamma$ defined by forgetting last
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By the facts that $p^{\prime}$ has rational fibers and Schubert varieties have
rational singularities, we have
$p_{*}[\mathcal{O}_{\overline{B^{-}\cdot
y_{w}^{\prime}}}]=[\mathcal{O}_{\overline{B^{-}\cdot y_{\widetilde{w}}}}].$
(4.6)
Alternatively, (4.6) follows from [12, Theorem 3.1].
### 4.4. Proof of Theorem 1.1
We start with summarizing what we have done in the previous subsections. We
defined an $R(T)$-linear map $\Phi$ in Definition 4.1 and proved in
Proposition 4.3 that it is an $R(T)$-algebra homomorphism. A priori, $\Phi$
was defined in terms of the localization basis
$\\{\mathcal{O}_{\mu}\\}_{\mu\in Q^{\vee}}$. But by Lemma 4.6, it can also be
defined in the same style in terms of the affine Schubert basis
$\\{\mathcal{O}_{wt_{\lambda}}\\}_{wt_{\lambda}\in W_{af}^{-}}$. By
Proposition 4.8, (4.2), (4.3), (4.5) and (4.6), we have
$\Phi(\mathcal{O}_{wt_{\lambda}})=q^{\beta_{w(\lambda)}}[\mathcal{O}_{\overline{B^{-}\cdot
y_{\widetilde{w}}}}].$
It remains to show that $\beta_{w(\lambda)}$ corresponds to
$\lambda+Q^{\vee}_{P}$ via the dual of isomorphism (2.2). By making use of the
canonical projection $G/B^{+}\rightarrow G/P$, we may assume $P=B^{+}$. Let
$\rho\in(Q^{\vee})^{*}$. We have to show
$\deg(u^{*}\mathcal{L}_{\rho})=\rho(\lambda)$ for the constant section $u$ of
$P_{w(\lambda)}(G/B^{+})$ corresponding to a point of
$F^{attr}_{\mathbb{C}^{\times}_{w(\lambda)}}(G/B^{+})$. By Lemma 4.11, we can
take that point to be $y^{\prime}_{w}:=wB^{+}$. Recall
$L_{\rho}=G\times_{B^{+}}\mathbb{C}_{-\rho}$ so that
$(L_{\rho})_{y^{\prime}_{w}}\simeq\mathbb{C}_{-w\rho}$ as $T$-modules, and
hence $(L_{\rho})_{y^{\prime}_{w}}\simeq\mathbb{C}_{-\rho(\lambda)}$ as
$\mathbb{C}^{\times}_{w(\lambda)}$-modules. Therefore,
$u^{*}\mathcal{L}_{\rho}\simeq\mathcal{O}_{\mathbb{P}^{1}}(\rho(\lambda))$ as
desired. The proof of Theorem 1.1 is complete.
## Appendix A A K-theoretic degeneration formula
Let $\mu\in Q^{\vee}$. Recall the $G/P$-fibration $P_{\mu}(G/P)$ over
$\mathbb{P}^{1}$ and the projection
$\pi_{\mu}:P_{\mu}(G/P)\rightarrow\mathbb{P}^{1}$ defined in Section 3.2. Let
$k\in\mathbb{N}$. Fix some points $z_{1},\ldots,z_{k}\in\mathbb{P}^{1}$. For
each $i=1,\ldots,k$, define $D_{\mu,i}:=\pi_{\mu}^{-1}(z_{i})$ and
$\iota_{\mu,i}:D_{\mu,i}\hookrightarrow P_{\mu}(G/P)$ to be the inclusion.
###### Definition A.1.
Let $\beta\in\pi_{2}(G/P)$.
1. (1)
Define
$\overline{\mathcal{M}}_{k}(\mu,\beta):=\bigcup_{\widetilde{\beta}}\overline{\mathcal{M}}_{0,k}(P_{{\mu}}(G/P),\widetilde{\beta})\times_{(\vec{\operatorname{ev}},\iota_{\mu,1}\times\cdots\times\iota_{\mu,k})}\left(D_{\mu,1}\times\cdots\times
D_{\mu,k}\right)$
where $\widetilde{\beta}$ runs over all classes in $\pi_{2}(P_{{\mu}}(G/P))$
satisfying
1. (i)
$(\pi_{\mu})_{*}\widetilde{\beta}=[\mathbb{P}^{1}]\in\pi_{2}(\mathbb{P}^{1})$;
and
2. (ii)
$\langle\widetilde{\beta},c_{1}(\mathcal{L}_{\rho})\rangle=\langle\beta,c_{1}(L_{\rho})\rangle$
for any $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$.
(The line bundles $\mathcal{L}_{\rho}$ and $L_{\rho}$ are defined in Section
3.1 and Section 2.3 respectively.)
2. (2)
By abuse of notation, define
$\operatorname{ev}_{i}:\overline{\mathcal{M}}_{k}(\mu,\beta)\rightarrow
D_{\mu,i}\simeq G/P$
to be the morphism induced by the evaluation map $\operatorname{ev}_{i}$ on
$\overline{\mathcal{M}}_{0,k}(P_{{\mu}}(G/P),\widetilde{\beta})$.
###### Definition A.2.
1. (1)
Let $\gamma_{1},\ldots,\gamma_{k}\in K_{T}(G/P)$ and $\beta\in\pi_{2}(G/P)$.
Define
$\operatorname{PKGW}^{\beta}_{\mu}(\gamma_{1},\ldots,\gamma_{k}):=\chi_{\overline{\mathcal{M}}_{k}(\mu,\beta)}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}_{k}(\mu,\beta)}\otimes\operatorname{ev}_{1}^{*}\gamma_{1}\otimes\cdots\otimes\operatorname{ev}_{k}^{*}\gamma_{k}\right)\in
R(T)$
where $\mathcal{O}^{vir}_{\overline{\mathcal{M}}_{k}(\mu,\beta)}$ is the
virtual structure sheaf constructed in [31].
2. (2)
Let $\gamma_{1},\ldots,\gamma_{k}\in K_{T}(G/P)$. Define
$\operatorname{PKGW}_{\mu}(\gamma_{1},\ldots,\gamma_{k}):=\sum_{\beta\in\pi_{2}(G/P)}q^{\beta}\operatorname{PKGW}^{\beta}_{\mu}(\gamma_{1},\ldots,\gamma_{k})\in
R(T)\otimes\mathbb{Z}[[\Lambda,\Lambda^{-1}].$
(By Corollary 3.18, the ring $\mathbb{Z}[[\Lambda,\Lambda^{-1}]$ is large
enough in order for $\operatorname{PKGW}_{\mu}$ to be well-defined.)
Recall the notations $\\{e_{i}\\}_{i\in I}$, $\\{g^{ij}\\}_{i,j\in I}$ and
$A_{G/P}$ defined in Section 2.4. The following proposition is a $K$-theoretic
degeneration formula which we need in order to prove Proposition 4.3.
###### Proposition A.3.
For any $\mu_{1},\mu_{2}\in Q^{\vee}$ and tuples
$\vec{\gamma}^{(1)}=(\gamma^{(1)}_{1},\ldots,\gamma^{(1)}_{k_{1}})$,
$\vec{\gamma}^{(2)}=(\gamma^{(2)}_{1},\ldots,\gamma^{(2)}_{k_{2}})$ of
elements of $K_{T}(G/P)$,
$\operatorname{PKGW}_{\mu_{1}+\mu_{2}}(\vec{\gamma}^{(1)},\vec{\gamma}^{(2)})=\sum_{i,j\in
I}g^{ij}\operatorname{PKGW}_{\mu_{1}}\left(\vec{\gamma}^{(1)},(\operatorname{id}+A_{G/P})^{-1}(e_{i})\right)\operatorname{PKGW}_{\mu_{2}}(\vec{\gamma}^{(2)},e_{j}).$
The rest of the appendix is devoted to the proof of Proposition A.3. While it
should not be difficult to prove the result using the machineries developed by
Li [32, 33], we take a more direct approach, namely we work with the moduli of
stable maps to the degeneration family instead of working with the stack of
expanded degenerations and its associated moduli of relative stable maps.
Define
$C:=\\{(t,[x:y:z])\in\mathbb{A}^{1}\times\mathbb{P}^{2}|~{}xy=tz^{2}\\}.$
Then the projection onto $\mathbb{A}^{1}$ defines a flat family
$C\rightarrow\mathbb{A}^{1}$. Denote by $C_{t}$ the fiber over
$t\in\mathbb{A}^{1}$. If $t\neq 0$, $C_{t}$ is isomorphic to $\mathbb{P}^{1}$,
and if $t=0$, it is isomorphic to the nodal rational curve $C_{1}\cup C_{2}$
where $C_{1},C_{2}\simeq\mathbb{P}^{1}$. In [13], we constructed a locally
trivial $G/P$-fibration
$p:P_{\mu_{1},\mu_{2}}\rightarrow C$
such that
$P_{\mu_{1},\mu_{2}}|_{C_{t\neq 0}}\simeq P_{\mu_{1}+\mu_{2}}(G/P)~{}\text{
and }~{}P_{\mu_{1},\mu_{2}}|_{C_{i}}\simeq P_{\mu_{i}}(G/P).$
By the construction, $P_{\mu_{1},\mu_{2}}$ admits a $T$-action such that $p$
is $T$-equivariant where $T$ acts $C$ trivially. Moreover, the above
isomorphisms are $T$-equivariant. Let $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$.
Like $P_{\mu}(G/P)$, there is a line bundle on $P_{\mu_{1},\mu_{2}}$ which
restricts to $L_{\rho}=G\times_{P}\mathbb{C}_{-\rho}$ on each fiber of $p$. By
abuse of notation, we denote this line bundle by $\mathcal{L}_{\rho}$.
Fix some sections
$s_{1}^{(1)},\ldots,s_{k_{1}}^{(1)},s_{1}^{(2)},\ldots,s_{k_{2}}^{(2)}$ of the
family $C\rightarrow\mathbb{A}^{1}$ such that
$s_{1}^{(i)}(0),\ldots,s_{k_{i}}^{(i)}(0)$ lie in $C_{i}\subset C_{t=0}$ away
from the intersection $C_{1}\cap C_{2}$. For each $s=1,\ldots,k_{1}$, the
divisor
$D_{s}^{(1)}:=P_{\mu_{1},\mu_{2}}\times_{(p,s_{s}^{(1)})}\mathbb{A}^{1}$ of
$P_{\mu_{1},\mu_{2}}$ is canonically isomorphic to $\mathbb{A}^{1}\times G/P$.
Denote by $\iota_{s}^{(1)}:D_{s}^{(1)}\hookrightarrow P_{\mu_{1},\mu_{2}}$ the
inclusion and $j_{s}^{(1)}:D_{s}^{(1)}\rightarrow G/P$ the projection. For
each $t=1,\ldots,k_{2}$, we define $D_{t}^{(2)}$, $\iota_{t}^{(2)}$ and
$j_{t}^{(2)}$ in a similar way.
From now on, fix $\beta\in\pi_{2}(G/P)$. We have to show
$\operatorname{PKGW}^{\beta}_{\mu_{1}+\mu_{2}}(\vec{\gamma}^{(1)},\vec{\gamma}^{(2)})=\sum_{r=0}^{\infty}\sum_{i,j\in
I}\sum_{\vec{\beta}}(-1)^{r}g^{ij}\operatorname{PKGW}^{\beta_{1}}_{\mu_{1}}(\vec{\gamma}^{(1)},I^{\beta_{11},\ldots,\beta_{1r}}(e_{i}))\operatorname{PKGW}^{\beta_{2}}_{\mu_{2}}(\vec{\gamma}^{(2)},e_{j}).$
(A.1)
Here,
* •
the third sum runs over all tuples
$\vec{\beta}=(\beta_{1};\beta_{11},\ldots,\beta_{1r};\beta_{2})$ of elements
of $\pi_{2}(G/P)$ such that $\beta_{1i}\neq 0$ for all $i=1,\ldots,r$ and
$\beta_{1}+\sum_{i=1}^{r}\beta_{1i}+\beta_{2}=\beta$; and
* •
$I^{\beta_{11},\ldots,\beta_{1r}}$ is a linear operator on $K_{T}(G/P)$
defined by
$I^{\beta_{11},\ldots,\beta_{1r}}:=(\operatorname{ev}_{2}^{\beta_{11}})_{*}\circ(\operatorname{ev}_{1}^{\beta_{11}})^{*}\circ\cdots\circ(\operatorname{ev}_{2}^{\beta_{1r}})_{*}\circ(\operatorname{ev}_{1}^{\beta_{1r}})^{*}$
where
$\operatorname{ev}_{1}^{\beta^{\prime}},\operatorname{ev}_{2}^{\beta^{\prime}}:\overline{\mathcal{M}}_{0,2}(G/P,\beta^{\prime})\rightarrow
G/P$ are the evaluation maps.
Denote by $A$ the semigroup of effective curve classes in
$P_{\mu_{1},\mu_{2}}$. Let $a\in A$ and $k\in\mathbb{N}$. Consider the Artin
stack $\mathfrak{M}_{0,k,A,a}$ defined in [15]. Roughly speaking, it
parametrizes genus zero $k$-pointed prestable curves each of whose irreducible
components is assigned an element of $A$ such that the sum of these elements
is equal to $a$ and this assignment satisfies a stability condition. There is
a natural morphism
$\overline{\mathcal{M}}_{0,k}(P_{\mu_{1},\mu_{2}},a)\rightarrow\mathfrak{M}_{0,k,A,a}$
defined by forgetting the target but remembering the degrees of the
restrictions of any stable maps to the irreducible components of the domain
curves. It is shown in loc. cit. that $\mathfrak{M}_{0,k,A,a}$ is étale over
the usual Artin stack $\mathfrak{M}_{0,k}$ of prestable curves. It follows
that $\mathfrak{M}_{0,k,A,a}$ is smooth and the standard virtual structure
sheaf
$\mathcal{O}^{vir}_{\overline{\mathcal{M}}_{0,k}(P_{\mu_{1},\mu_{2}},a)}$ is
equal to the one constructed using the relative perfect obstruction theory
associated to the above morphism.
###### Definition A.4.
1. (1)
Define a Deligne-Mumford stack
$\overline{\mathcal{M}}:=\bigcup_{\widetilde{\beta}}\overline{\mathcal{M}}_{0,k_{1}+k_{2}}(P_{\mu_{1},\mu_{2}},\widetilde{\beta})$
where $\widetilde{\beta}$ runs over all classes in
$\pi_{2}(P_{\mu_{1},\mu_{2}})$ satisfying
1. (i)
$p_{*}\widetilde{\beta}=[C_{1}]+[C_{2}]$, the fiber class of the flat family
$C\rightarrow\mathbb{A}^{1}$; and
2. (ii)
$\langle\widetilde{\beta},c_{1}(\mathcal{L}_{\rho})\rangle=\langle\beta,c_{1}(L_{\rho})\rangle$
for any $\rho\in(Q^{\vee}/Q^{\vee}_{P})^{*}$.
2. (2)
Define an Artin stack
$\mathfrak{M}_{A}:=\bigcup_{\widetilde{\beta}}\mathfrak{M}_{0,k_{1}+k_{2},A,\widetilde{\beta}}$
where $\widetilde{\beta}$ runs over the same set in (1) above.
There is a proper morphism
$\pi:\overline{\mathcal{M}}\rightarrow\mathbb{A}^{1}$ sending each stable map
$u$ to the unique $t\in\mathbb{A}^{1}$ such that $p\circ u$ lands in $C_{t}$.
There is also a natural morphism
$\nu:\overline{\mathcal{M}}\rightarrow\mathfrak{M}_{A}$ which is the union of
the morphisms
$\overline{\mathcal{M}}_{0,k_{1}+k_{2}}(P_{\mu_{1},\mu_{2}},\widetilde{\beta})\rightarrow\mathfrak{M}_{0,k_{1}+k_{2},A,\widetilde{\beta}}$
mentioned above.
###### Definition A.5.
Let $r$ be a non-negative integer.
1. (1)
Define $\Omega^{r}$ to be the set of modular graphs with $A$-structure [7]
satisfying
1. (a)
the number of vertices is $r+2$;
2. (b)
the genus associated to every vertex is zero;
3. (c)
the underlying graph without tails is a linear graph;
4. (d)
the degrees associated to the two end vertices of the graph in (c) are classes
which are projected via $p$ to $[C_{1}]$ and $[C_{2}]$ respectively;
5. (e)
the sum of degrees satisfies conditions (i) and (ii) in the definition of
$\overline{\mathcal{M}}$ above; and
6. (f)
the number of tails is $k_{1}+k_{2}$.
2. (2)
Define $\widetilde{\Omega}^{r}$ to be the set of modular graphs with
$A$-structure satisfying (a) to (e) above and that the number of tails is
zero.
It follows from (d) and (e) that for any $\sigma\in\Omega^{r}$ or
$\widetilde{\Omega}^{r}$, the degrees associated to the intermediate vertices
are fiber classes with respect to the fibration
$p:P_{\mu_{1},\mu_{2}}\rightarrow C$.
Define an injective map
$\widetilde{\Omega}^{r}\hookrightarrow\Omega^{r}:\sigma\mapsto\overline{\sigma}$
as follows. For each $\sigma\in\widetilde{\Omega}^{r}$, define
$\overline{\sigma}$ to be the modular graph obtained from $\sigma$ by
attaching the first $k_{1}$ tails to the end vertex corresponding to $C_{1}$
and the rest to the other end vertex. See (d) in the definition of
$\Omega^{r}$. We will identify $\widetilde{\Omega}^{r}$ with the image of this
injective map.
For any $\sigma\in\Omega^{r}$, we have a similarly-defined Deligne-Mumford
stack $\overline{\mathcal{M}}(\sigma)$ (resp. Artin stack
$\mathfrak{M}_{A}(\sigma)$) parametrizing stable maps (resp. prestable curves)
which are at least singular as described by the graph $\sigma$. If
$\sigma\in\Omega^{0}$, then $\mathfrak{M}_{A}(\sigma)$ is a Weil divisor of
$\mathfrak{M}_{A}$. Since $\mathfrak{M}_{A}$ is smooth, we have the associated
Cartier divisor $\mathcal{O}_{\mathfrak{M}_{A}}(\mathfrak{M}_{A}(\sigma))$
(for simplicity, the section is hidden from the notation). For each
$\ell\in\mathbb{N}$, define $\mathfrak{M}^{\leqslant\ell}_{0,0}$ to be the
moduli stack of genus zero prestable curves with at most $\ell$ nodes. We have
a smooth divisor $\mathcal{D}$ of $\mathfrak{M}^{\leqslant 1}_{0,0}$ which is
the complement of the open substack $\mathfrak{M}^{\leqslant 0}_{0,0}$.
###### Lemma A.6.
There exists a commutative diagram of stacks
${\overline{\mathcal{M}}}$${\mathfrak{M}_{A}}$${\mathbb{A}^{1}}$${\mathfrak{M}^{\leqslant
1}_{0,0}}$$\scriptstyle{\pi}$$\scriptstyle{\nu}$$\scriptstyle{g}$$\scriptstyle{f}$
(A.2)
such that $f^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\mathcal{O}_{\mathbb{A}^{1}}(\mathbf{0})$ and
$g^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\bigotimes_{\sigma\in\Omega^{0}}\mathcal{O}_{\mathfrak{M}_{A}}(\mathfrak{M}_{A}(\sigma))$
as Cartier divisors.
###### Proof.
We postpone the proof until the end. ∎
Observe that
$\bigcup_{\sigma\in\Omega^{0}}\mathfrak{M}_{A}(\sigma)\subset\mathfrak{M}_{A}$
is a normal crossing divisor and the intersection of these divisors
$\mathfrak{M}_{A}(\sigma)$ over any finite subset of $\Omega^{0}$ is of the
form $\mathfrak{M}_{A}(\sigma^{\prime})$ for some modular graph
$\sigma^{\prime}$ with $A$-structure. It is easy to see that in our case these
modular graphs are precisely the elements of $\Omega^{r}$, $r\in\mathbb{N}$.
Therefore, by the inclusion-exclusion principle [31],
$\left[\mathcal{O}_{\bigcup_{\sigma\in\Omega^{0}}\mathfrak{M}_{A}(\sigma)}\right]=\sum_{r=0}^{\infty}(-1)^{r}\sum_{\sigma\in\Omega^{r}}\left[\mathcal{O}_{\mathfrak{M}_{A}(\sigma)}\right].$
(A.3)
By Lemma A.6, (A.3) and some standard arguments in virtual pullbacks [37], we
have
$\text{LHS of
}\eqref{appeq1}=\sum_{r=0}^{\infty}(-1)^{r}\sum_{\sigma\in\Omega^{r}}\chi_{\overline{\mathcal{M}}(\sigma)}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}(\sigma)}\otimes
e_{1}\otimes e_{2}\right)$ (A.4)
where
$e_{1}:=\bigotimes_{s=1}^{k_{1}}\operatorname{ev}_{s}^{*}(\iota_{s}^{(1)})_{*}(j_{s}^{(1)})^{*}\gamma_{s}^{(1)}$
and
$e_{2}:=\bigotimes_{t=1}^{k_{2}}\operatorname{ev}_{k_{1}+t}^{*}(\iota_{t}^{(2)})_{*}(j_{t}^{(2)})^{*}\gamma_{t}^{(2)}$.
Let $\sigma\in\Omega^{r}$. Recall the subset
$\widetilde{\Omega}^{r}\subseteq\Omega^{r}$ defined above. Suppose
$\sigma\not\in\widetilde{\Omega}^{r}$. It is easy to see that one of the
evaluation maps on $\overline{\mathcal{M}}(\sigma)$ factors through the open
immersion $P_{\mu_{1},\mu_{2}}\setminus D_{s}^{(1)}\hookrightarrow
P_{\mu_{1},\mu_{2}}$ or $P_{\mu_{1},\mu_{2}}\setminus
D_{t}^{(2)}\hookrightarrow P_{\mu_{1},\mu_{2}}$ for some $s=1,\ldots,k_{1}$ or
$t=1,\ldots,k_{2}$. It follows that the summand in (A.4) is zero for $\sigma$.
Therefore, we have
$\text{RHS of
}\eqref{appeq4}=\sum_{r=0}^{\infty}(-1)^{r}\sum_{\sigma\in\widetilde{\Omega}^{r}}\chi_{\overline{\mathcal{M}}(\overline{\sigma})}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}(\overline{\sigma})}\otimes
e_{1}\otimes e_{2}\right).$ (A.5)
Now let $\sigma\in\widetilde{\Omega}^{r}$. Denote the degrees of $\sigma$ by
$a_{1}^{\sigma},a_{11}^{\sigma},\ldots,a_{1r}^{\sigma},a_{2}^{\sigma}$, in the
order compatible with the linear graph, starting with the end vertex
corresponding to $C_{1}$. Define $P_{\mu_{i}}:=P_{\mu_{1},\mu_{2}}|_{C_{i}}$,
$F:=P_{\mu_{1},\mu_{2}}|_{C_{1}\cap C_{2}}$ and
$\iota_{F}^{(i)}:F\hookrightarrow P_{\mu_{i}}$ to be the inclusion. For each
$i=1,2$, define
$\overline{\mathcal{M}}_{i}:=\overline{\mathcal{M}}_{0,k_{i}+1}(P_{\mu_{i}},a_{i}^{\sigma})\times_{(\operatorname{ev}_{k_{i}+1},\iota_{F}^{(i)})}F$
and $\operatorname{ev}_{(i)}:\overline{\mathcal{M}}_{i}\rightarrow F$ to be
the morphism induced by $\operatorname{ev}_{k_{i}+1}$. Then Proposition A.3
follows if we can show
$\displaystyle\chi_{\overline{\mathcal{M}}(\overline{\sigma})}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}(\overline{\sigma})}\otimes
e_{1}\otimes e_{2}\right)$ $\displaystyle=~{}$ $\displaystyle\sum_{i,j\in
I}g^{ij}\chi_{\overline{\mathcal{M}}_{1}}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}_{1}}\otimes
e_{1}\otimes\operatorname{ev}_{(1)}^{*}\left(I^{a_{11}^{\sigma},\ldots,a_{1r}^{\sigma}}(e_{i})\right)\right)\chi_{\overline{\mathcal{M}}_{2}}\left(\mathcal{O}^{vir}_{\overline{\mathcal{M}}_{2}}\otimes
e_{2}\otimes\operatorname{ev}_{(2)}^{*}e_{j}\right).$ (A.6)
The last equation follows from a similar argument which is used to prove the
“cutting edges” axiom in [6]. More precisely, we prove
###### Lemma A.7.
There exists a Cartesian diagram
${\overline{\mathcal{M}}(\overline{\sigma})}$${\overline{\mathcal{M}}_{1}\times\displaystyle\left(\prod_{i=1}^{r}\overline{\mathcal{M}}_{0,2}(F,a^{\sigma}_{1i})\right)\times\overline{\mathcal{M}}_{2}}$${F^{r+1}}$${F^{2r+2}}$$\scriptstyle{\Delta_{F}^{r+1}}$
(A.7)
such that the obstruction theories on the two stacks in the upper row are
compatible over $\Delta_{F}^{r+1}$.
###### Proof.
Denote by
$\mathfrak{C}_{1},\mathfrak{C}_{11},\ldots,\mathfrak{C}_{1r},\mathfrak{C}_{2}$
the universal curves of $\overline{\mathcal{M}}(\overline{\sigma})$
corresponding to different vertices of $\overline{\sigma}$, and by
$u_{1},u_{11},\ldots,u_{1r},u_{2}$ the universal stable maps. Observe that
over any $\mathbb{C}$-point of $\overline{\mathcal{M}}(\overline{\sigma})$,
$u_{1}$, $u_{1i}$ and $u_{2}$ factor through scheme-theoretically the
inclusions of $P_{\mu_{1}}$, $F$ and $P_{\mu_{2}}$ into $P_{\mu_{1},\mu_{2}}$
respectively. Using the fact that $C_{1}$ and $C_{2}$ are $(-1)$-curves in $C$
and the theorem of cohomology and base change, these stable maps (over
$\overline{\mathcal{M}}(\overline{\sigma})$) in fact land in these subschemes.
This defines the horizontal arrow in (A.7). The vertical arrows are defined to
be the evaluation maps at those marked points which do not correspond to any
of the insertions $\vec{\gamma}^{(1)}$ and $\vec{\gamma}^{(2)}$. It is
straightforward to verify that (A.7) is Cartesian.
It remains to verify the compatibility of obstruction theories. Let
$\mathfrak{C}$ denote the universal curve of
$\overline{\mathcal{M}}(\overline{\sigma})$ obtained by gluing
$\mathfrak{C}_{1},\mathfrak{C}_{11},\ldots,\mathfrak{C}_{1r},\mathfrak{C}_{2}$
according to the combinatorics of $\overline{\sigma}$. Let
$u:\mathfrak{C}\rightarrow P_{\mu_{1},\mu_{2}}$ be the universal stable map,
$\eta:\mathfrak{C}_{1}\amalg\cdots\amalg\mathfrak{C}_{2}\rightarrow\mathfrak{C}$
the gluing map, and
$z_{0},\ldots,z_{r}:\overline{\mathcal{M}}(\overline{\sigma})\rightarrow\mathfrak{C}$
the sections “lying between” $\mathfrak{C}_{1}$ and $\mathfrak{C}_{11}$,
$\mathfrak{C}_{11}$ and $\mathfrak{C}_{12}$, etc. For each $i=1,2$, we remove
the component $\mathfrak{C}_{3-i}$ from $\mathfrak{C}$, that is, we glue all
$\mathfrak{C}_{1},\mathfrak{C}_{11},\ldots,\mathfrak{C}_{1r},\mathfrak{C}_{2}$
except $\mathfrak{C}_{3-i}$. Denote by $\widetilde{\mathfrak{C}}_{i}$ the
resulting curve and by
$\eta_{i}:\widetilde{\mathfrak{C}}_{i}\hookrightarrow\mathfrak{C}$ the
inclusion.
Define $\mathcal{F}_{i}$ to be the kernel of the natural epimorphism
$u_{i}^{*}T_{P_{\mu_{i}}}\rightarrow(x_{i})_{*}(u_{i}\circ
x_{i})^{*}N_{F/P_{\mu_{i}}}\rightarrow 0$
where
$x_{1}:\overline{\mathcal{M}}(\overline{\sigma})\rightarrow\mathfrak{C}_{1}$
(resp.
$x_{2}:\overline{\mathcal{M}}(\overline{\sigma})\rightarrow\mathfrak{C}_{2}$)
is the section which is used to glue $\mathfrak{C}_{1}$ and
$\mathfrak{C}_{11}$ (resp. $\mathfrak{C}_{1r}$ and $\mathfrak{C}_{2}$).
(Recall $u_{i}$ lands in $P_{\mu_{i}}$.) Notice that $\mathcal{F}_{i}$ is the
sheaf giving rise to the deformation and obstruction spaces for
$\overline{\mathcal{M}}_{i}$ relative to
$\mathfrak{M}_{A}(\overline{\sigma})$. It is straightforward to verify that
there exists a coherent sheaf $\mathcal{E}$ on $\mathfrak{C}$ which fits into
the following two short exact sequences simultaneously:
$0\rightarrow\mathcal{E}\rightarrow\eta_{*}\left(\mathcal{F}_{1}\oplus\bigoplus_{i=1}^{r}u_{1i}^{*}T_{F}\oplus\mathcal{F}_{2}\right)\rightarrow\bigoplus_{j=0}^{r}(z_{j})_{*}(u\circ
z_{j})^{*}T_{F}\rightarrow 0$ (A.8) $0\rightarrow\mathcal{E}\rightarrow
u^{*}T_{P_{\mu_{1},\mu_{2}}}\rightarrow\bigoplus_{i=1}^{2}(\eta_{i})_{*}(u\circ\eta_{i})^{*}N_{P_{\mu_{i}}/P_{\mu_{1},\mu_{2}}}\rightarrow
0.$ (A.9)
Since $(u\circ\eta_{i})^{*}N_{P_{\mu_{i}}/P_{\mu_{1},\mu_{2}}}\simeq(p\circ
u\circ\eta_{i})^{*}\mathcal{O}_{C_{i}}(-1)$, the derived pushforward functor
sends the cokernel in (A.9) to the zero object in the derived category
$D(\overline{\mathcal{M}}(\overline{\sigma}))$ and so defines an isomorphism
between the objects corresponding to the other two coherent sheaves in the
same short exact sequence. Using this isomorphism and applying the argument in
[6] to (A.8), we obtain a homomorphism of distinguished triangles
${h^{*}E_{\overline{\mathcal{M}}_{right}}}$${E_{\overline{\mathcal{M}}_{left}}}$${v^{*}L_{\Delta^{r+1}_{F}}}$${h^{*}E_{\overline{\mathcal{M}}_{right}}[1]}$${h^{*}L_{\overline{\mathcal{M}}_{right}/\mathfrak{M}_{A}(\overline{\sigma})}}$${L_{\overline{\mathcal{M}}_{left}/\mathfrak{M}_{A}(\overline{\sigma})}}$${L_{\overline{\mathcal{M}}_{left}/\overline{\mathcal{M}}_{right}}}$${h^{*}L_{\overline{\mathcal{M}}_{right}/\mathfrak{M}_{A}(\overline{\sigma})}[1]}$
where $\overline{\mathcal{M}}_{left}$ (resp. $\overline{\mathcal{M}}_{right}$)
is the stack at the top left (resp. right) corner in diagram (A.7), and
$h:\overline{\mathcal{M}}_{left}\rightarrow\overline{\mathcal{M}}_{right}$ and
$v:\overline{\mathcal{M}}_{left}\rightarrow F^{r+1}$ are the morphisms in the
same diagram. This is the compatibility we need to prove. ∎
Equation (A.6) now follows from the functoriality of virtual structure sheaves
[31]. The proof is Proposition A.3 is complete.
#### Proof of Lemma A.6.
By making use of the “forgetting tails” morphisms for $\overline{\mathcal{M}}$
and $\mathfrak{M}_{A}$, we may assume $k_{1}=k_{2}=0$. Notice that the latter
morphism exists by [15, Proposition 2.1.1]. Define $f$ to be the morphism
representing the flat family $C\rightarrow\mathbb{A}^{1}$ defined at the
beginning.
We define $g$ as follows. Denote by $B$ the semigroup of effective curve
classes in $C$. The projection $p:P_{\mu_{1},\mu_{2}}\rightarrow C$ induces a
homomorphism $\phi:A\rightarrow B$ of semigroups. There is a morphism
$\mathfrak{M}_{A}\rightarrow\mathfrak{M}_{0,0,B,[C_{1}]+[C_{2}]}$ sending each
curve $S$ over $\operatorname{Spec}\mathbb{C}$ to the curve obtained by
replacing the degree $a\in A$ assigned to each irreducible component with
$\phi(a)\in B$ and then stabilizing the resulting curve in the sense of [15],
i.e. contracting those irreducible components labelled by $0\in B$ and having
less than three special points. Although the construction of this morphism is
standard, we provide the details in the next paragraph.
Let $T$ be a scheme and $S\rightarrow T$ a family of prestable curves
representing a $T$-point of $\mathfrak{M}_{0,0,A,a}$. By definition, we have a
constructible function $\deg:S\rightarrow A$ which records the degree assigned
to each irreducible component of the fiber over any $\mathbb{C}$-point of $T$.
We have to stabilize the family $S\rightarrow T$ with respect to the degree
function $\phi\circ\deg:S\rightarrow B$. As in the usual case, this type of
stabilization is universal so that we can construct it étale locally. Étale
locally, we can make $S$ stable in the usual sense by inserting sufficiently
many sections. Then we remove these sections one by one. Each time we remove a
section, we stabilize the family by contracting those components labelled by
$0\in B$ and having two special points. Such a stabilization does exist, by
[15, Proposition 2.1.1] which says that
$\mathfrak{C}_{0,N,B,b}\simeq\mathfrak{M}_{0,N+1,B,b}$ where
$\mathfrak{C}_{0,N,B,b}$ is the universal curve of $\mathfrak{M}_{0,N,B,b}$.
The family obtained after removing all these sections will be the desired
stabilization.
Our $g$ is then defined to be the composite
$\mathfrak{M}_{A}\rightarrow\mathfrak{M}_{0,0,B,[C_{1}]+[C_{2}]}\rightarrow\mathfrak{M}^{\leqslant
1}_{0,0}$
where the first morphism is the above morphism and the second is defined by
forgetting all the degrees without stabilizing. It is easy to see that $g$
does land in $\mathfrak{M}^{\leqslant 1}_{0,0}$.
It remains to verify the stated properties.
The commutativity of diagram (A.2).
Let $T$ be a scheme and $u:S\rightarrow P_{\mu_{1},\mu_{2}}$ a stable map over
$T$ representing a $T$-point of $\overline{\mathcal{M}}$. By [7, Lemma 2.2],
$u$ lies over a unique morphism $v:T\rightarrow\mathbb{A}^{1}$. By definition,
$f\circ\pi$ sends this $T$-point to the $T$-point of $\mathfrak{M}^{\leqslant
1}_{0,0}$ represented by the flat family $C\times_{\mathbb{A}^{1}}T$. Consider
the morphism $w:S\rightarrow C\times_{\mathbb{A}^{1}}T$ induced by $p\circ u$.
Observe that, over every $\mathbb{C}$-point of $T$, $w$ contracts precisely
those irreducible components which are mapped into the fibers of $p$. It
follows that $C\times_{\mathbb{A}^{1}}T$ is the stabilization of $p\circ u$
which is equal to the $T$-point with respect to the composite $g\circ\nu$.
The equality $f^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\mathcal{O}_{\mathbb{A}^{1}}(\mathbf{0})$.
This is straightforward. For example, one can make use of the quotient stack
presentation of $\mathfrak{M}^{\leqslant 1}_{0,0}$ given in [16]. More
precisely, $\mathfrak{M}^{\leqslant 1}_{0,0}$ is isomorphic to $[V/GL_{3}]$
where $V\subset\operatorname{Sym}^{2}\mathbb{C}^{3}$ is the space of symmetric
2-tensors on $\mathbb{C}^{3}$ with rank at least two, and the divisor
$\mathcal{D}$ corresponds to the space of those tensors whose rank is
precisely two.
The equality $g^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\bigotimes_{\sigma\in\Omega^{0}}\mathcal{O}_{\mathfrak{M}_{A}}(\mathfrak{M}_{A}(\sigma))$.
Observe that, in a smooth atlas, the equality
$g^{-1}(\mathcal{D})=\bigcup_{\sigma\in\Omega^{0}}\mathfrak{M}_{A}(\sigma)$
holds set-theoretically. It follows that
$g^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\bigotimes_{\sigma\in\Omega^{0}}\mathcal{O}_{\mathfrak{M}_{A}}(\mathfrak{M}_{A}(\sigma))^{\otimes
m_{\sigma}}$ as Cartier divisors for some $m_{\sigma}\in\mathbb{Z}_{\geqslant
1}$. To determine $m_{\sigma}$, consider the morphism
$h_{\sigma}:\mathbb{A}^{1}\rightarrow\mathfrak{M}_{A}$ which represents the
flat family $C\rightarrow\mathbb{A}^{1}$ where $C_{1}$ and $C_{2}$ are
labelled by the degrees given by $\sigma$. By the previous paragraph, we have
$h_{\sigma}^{*}g^{*}\mathcal{O}_{\mathfrak{M}^{\leqslant
1}_{0,0}}(\mathcal{D})=\mathcal{O}_{\mathbb{A}^{1}}(\mathbf{0})$. On the other
hand, it is easy to see that for any $\sigma^{\prime}\in\Omega^{0}$ not equal
to $\sigma$,
$h_{\sigma}^{*}\mathcal{O}_{\mathfrak{M}_{A}}(\mathfrak{M}_{A}(\sigma^{\prime}))$
is the trivial Cartier divisor. This gives $m_{\sigma}=1$. $\square$
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|
11institutetext: National Radio Astronomy Observatory, 520 Edgemont Road,
Charlottesville, VA, USA 22903 11email<EMAIL_ADDRESS>22institutetext:
LERMA, Observatoire de Paris, PSL Research University CNRS Sorbonne Université
22email<EMAIL_ADDRESS>
# The Dark Neutral Medium is (Mostly) Molecular Hydrogen
H. Liszt and M. Gerin 1122<EMAIL_ADDRESS>
(received )
###### Abstract
Context. More gas is sometimes inferred in molecular cloud complexes than is
represented in HI or CO emission, and this is called dark neutral medium
(DNM).
Aims. Our aim is to extend a study of DNM along 13 directions in the outskirts
of Chamaeleon by determining the atomic or molecular character of the DNM
along a larger sample of sightlines.
Methods. We acquired ALMA ground rotational state absorption profiles of
$\mathrm{HCO^{+}}$ and other molecules toward 33 compact extragalactic
continuum background sources seen toward the Galactic anticenter, deriving
N(H2) = N($\mathrm{HCO^{+}}$)/$3\times 10^{-9}$ as before. We observed J=1-0
CO emission with the IRAM 30m telescope in directions where $\mathrm{HCO^{+}}$
was newly detected.
Results. $\mathrm{HCO^{+}}$ absorption was detected in 28 of 33 new directions
and CO emission along 19 of those 28. The five sightlines lacking detectable
$\mathrm{HCO^{+}}$ have three times lower $<$EB-V$>$ and $<$N(DNM)$>$. Binned
in EB-V, N(H2) and N(DNM) are strongly correlated and vary by factors of
50-100 over the observed range EB-V $\approx$ 0.05 - 1 mag, while N(HI) varies
by factors of only 2-3. On average N(DNM) and N(H2) are well matched, and
detecting $\mathrm{HCO^{+}}$ absorption adds little to no H2 in excess of the
previously inferred DNM. There are five cases where 2N(H2) $<$ N(DNM)/2
indicates saturation of the HI emission profile. For sightlines with ${\rm
W}_{\mathrm{CO}}$ $\geq$ 1 K-km s-1, the CO-H2 conversion factor N(H2)/${\rm
W}_{\mathrm{CO}}$ $=2-3\times 10^{20}\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm cm}^{-2}$/(1 K-km s-1) is higher than is derived from studies of resolved
clouds in $\gamma$-rays.
Conclusions. Our work sampled primarily atomic gas with a mean H2 fraction
$\approx$ 1/3, but the DNM is almost entirely molecular. CO fulfills its role
as an H2 tracer when its emission is strong, but large-scale CO surveys are
not sensitive to H2 columns associated with typical values N(DNM) = $2-6\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$. Lower $X_{\mathrm{CO}}$ values
from $\gamma$-ray studies arise in part from different definitions and usage.
Sightlines with ${\rm W}_{\mathrm{CO}}$ $\geq$ 1 K-km s-1 represent 2/3 of the
H2 detected in $\mathrm{HCO^{+}}$ and detecting 90% of the H2 would require
detecting CO at levels ${\rm W}_{\mathrm{CO}}$ $\approx$ 0.2-0.3 K-km s-1.
###### Key Words.:
interstellar medium – abundances
††offprints: H. S. Liszt
## 1 Introduction
Between the atomic and molecular interstellar gases there is a transition that
is imperfectly traced in $\lambda 21$cm HI and/or $\lambda 2.6$mm CO emission.
Gas that is not well represented in one or both tracers is described as dark
neutral medium (DNM) (Grenier et al., 2005; Planck Collaboration et al., 2015;
Remy et al., 2017, 2018), and the atomic or molecular character of the DNM can
only be decided by appealing to other information.
In the outskirts of Chamaeleon, we used millimeter-wave $\mathrm{HCO^{+}}$
absorption to show that the DNM was almost exclusively molecular, even while
the gas as a whole was mostly atomic (Liszt et al., 2018). $\mathrm{HCO^{+}}$
was detected along all 13 sampled sightlines, of which 12 lacked detectable CO
emission at a level of 1.5 K-km s-1. The amount of inferred H2 was comparable
to that of the DNM (confirming the amount of the detected DNM), and only for
three sightlines did it seem likely that saturation of the HI emission profile
could account for much of the DNM. Subsequent detections of CO absorption
along six directions by Liszt et al. (2019) showed that the lack of CO
emission was due to low CO column densities. The linewidth of CO absorption
was shown to be smaller than that of its parent molecule $\mathrm{HCO^{+}}$,
illustrating the complex chemical nature of the CO formation process in
diffuse gas.
In this work we extend the analysis of the composition of DNM to 33 sightlines
in directions toward the Galactic anticenter (Remy et al., 2017, 2018) and
consider the larger sample of 46 sightlines. The present study employs a
relatively small number of diffuse (AV $\leq 1$ mag) and translucent (AV $\leq
3$ mag) sightlines where background sources are serendipitously available, and
is not a revision of the earlier study, whose derived DNM column densities and
other results are assumed here. Rather, we use the newly derived H2
abundances, independent of the presence of CO emission, to consider such
topics as the atomic or molecular character of the DNM, the degree to which
the $\lambda$21cm HI profile represents N(HI) and the influence of the CO-H2
conversion factor on the DNM determination.
Section 2 summarizes the observational material considered here and Section 3
compares the prior results for DNM with atomic hydrogen measured in the
$\lambda$21cm emission and H2 as sampled by its proxy 89 GHz absorption of
$\mathrm{HCO^{+}}$. The role of CO emission data is explored in Section 4
where the CO-H2 conversion factor is derived. Section 5 presents a summary,
with conclusions and a discussion of later work.
## 2 Observations and data reduction
The methods of work were described in Liszt et al. (2018) and are not repeated
in full detail here. From the existing DNM analyses of Planck Collaboration et
al. (2015) and Remy et al. (2017, 2018) we take N(DNM) and $\lambda$21cm HI
column densities N(HI) and N(HI)cl representing respectively the line profile
integral integrated over all velocities and the column density that is
associated with the cloud and/or kinematic features hosting the neutral gas
and DNM, as derived by a decomposition that accounts for the compound nature
of HI emission profiles from individual clouds.
To the existing analyses we add ALMA observations of absorption from
$\mathrm{HCO^{+}}$ and profiles of $\lambda$2.6mm CO emission from the IRAM
30m telescope, as described below.
### 2.1 Millimeter-wave absorption
The new absorption data discussed here are spectra of $\mathrm{HCO^{+}}$ in 33
directions along Galactic anticenter sightlines toward ALMA phase calibrators
as given in the Tables here and projected on sky maps in Figure 1. As before,
the $\mathrm{HCO^{+}}$ spectra were acquired with a spectral resolution of
0.205 km s-1 sampled at 0.102 km s-1 intervals without the so-called spectral
averaging that bins data. We also acquired spectra of HCN, C2H, HCO, and
several isotopologs. Counts of detections are shown in Figure 2, but results
for species other than $\mathrm{HCO^{+}}$ will be discussed in detail in later
work. Statistics of the $\mathrm{HCO^{+}}$ detections are shown in Figure 9.
Spectra in the 28 directions with detectable $\mathrm{HCO^{+}}$ absorption are
shown in Figure 10.
### 2.2 IRAM 30m CO emission
We took frequency-switched CO emission profiles at the IRAM 30m telescope in
August 2019 toward the 28 anticenter sightlines with detected
$\mathrm{HCO^{+}}$ absorption. The CO spectra are shown in Figure 10. These CO
data were taken as five-point maps toward the target and displaced by 2 HPBW
(2x22′′) in the four cardinal directions, using a pointing pattern that was
designed to detect $\mathrm{HCO^{+}}$ and HCN emission in the presence of
spectral line absorption in the target direction. For the $\mathrm{HCO^{+}}$
emission that will be discussed in subsequent work, the $\mathrm{HCO^{+}}$
emission profile is formed by averaging spectra along the four outlying
directions. For CO the present results use the average of all five pointings
because contamination by absorption is not measurable given the strength of
the emission and/or the continuum. The results are presented on the main beam
antenna temperature scale that is native to the 30m telescope.
### 2.3 Conversion from integrated HCO+ absorption to N(H2)
The suitability of $\mathrm{HCO^{+}}$ as a proxy for H2 in diffuse molecular
gas was explored extensively in Liszt & Gerin (2023) (hereafter Paper 1). As
in our earlier work, we use N(H2) = N($\mathrm{HCO^{+}}$)/$3\times 10^{-9}$
and N($\mathrm{HCO^{+}}$) = $1.10\times 10^{12}$$\Upsilon_{\mathrm{HCO^{+}}}$
where $\Upsilon_{\mathrm{HCO^{+}}}$ is the integrated $\mathrm{HCO^{+}}$
optical depth expressed in km s-1.
Figure 1: Sky maps of the observed sightlines. Left: Positions of the
background sources observed here are projected against a map of EB-V from the
work of Schlegel et al. (1998). The sightlines lacking detected
$\mathrm{HCO^{+}}$ absorption (largely due to weak continuum) are indicated
with smaller symbols. Right: Same as left, but shown against a map of N(DNM)
from Remy et al. (2017, 2018).
### 2.4 Reddening and dust optical depth
The 6′ resolution dust-emission maps scaled to optical reddening EB-V by
Schlegel et al. (1998) are cited here. Those reddening values can be converted
to Planck 353 GHz dust optical depth using the relationship established by
Planck Collaboration et al. (2014) between the 353 GHz dust optical depth and
the EB-V values of Schlegel et al. (1998), EB-V/$\tau_{353}=(1.49\pm
0.03)\times 10^{4}$ mag.
### 2.5 Conventions
Velocities presented with the spectra are taken with respect to the kinematic
definition of the Local Standard of Rest. N(H) is the column density of H
nuclei detected in neutral atomic and molecular form, N(H) = N(HI)+2N(H2), and
the molecular hydrogen and DNM fractions 2N(H2)/N(H) and N(DNM)/N(H) are
respectively represented by f${}_{{\rm H}_{2}}$ and fDNM. The integrated
absorption of the $\mathrm{HCO^{+}}$ profile in units of km s-1 is denoted by
$\Upsilon_{{\rm HCO}\mbox{${}^{+}$}}$ and similarly for other species. The
integrated emission of the J=1-0 CO line is denoted by ${\rm W}_{\mathrm{CO}}$
with units of K-km s-1 and the CO-H2 conversion factor N(H2)/${\rm
W}_{\mathrm{CO}}$ is denoted by $X_{\mathrm{CO}}$. Where reference is made to
a typical Galactic or standard CO-H2 conversion factor, the value N(H2)/${\rm
W}_{\mathrm{CO}}$ $=2\times 10^{20}\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm cm}^{-2}$/(1 K-km s-1) should be understood, as summarized in Table E.1
of Remy et al. (2017).
### 2.6 Overview
Observational results of importance to this work are given in Table 1 and
Table 2, and summaries of mean properties of various subsamples are presented
in Table 3. Some statistics of the continuum targets and noise in the
detections of $\mathrm{HCO^{+}}$ absorption are discussed in Appendix A and
the CO emission and $\mathrm{HCO^{+}}$ absorption line profiles are shown in
Figure 10 and discussed in Appendix B.
## 3 HI, DNM, and H2 sampled in HCO+ absorption
Figure 1 shows the new Galactic anticenter sightlines projected on maps of
EB-V and N(DNM). The region at $b<-30$o is described by Remy et al. (2017) as
Cetus, and that at $b>-22$o as Taurus-Main. California lies to the north
around $b=-10$o and Perseus is near (l,b) = 160o,-20o. Taurus-North overlays
most of the map region. The individual subregions are not discussed here owing
to the sparse sampling.
Counts of sources and detected molecular tracers binned in 0.05 mag intervals
of reddening are shown in Figure 2. $\mathrm{HCO^{+}}$ is the molecular tracer
detected most commonly (28 of 33 anticenter sightlines and 41 of 46 overall),
followed by C2H (26 total) and HCN (21 total) . The properties of molecular
absorption line tracers other than $\mathrm{HCO^{+}}$ will be discussed in
later work.
The Chamaeleon and Galactic anticenter subsamples have the same mode at EB-V =
0.25 - 0.3 mag in Figure 2, but the anticenter subsample has a high-reddening
tail at EB-V $>0.5$ mag that is absent in Chamaeleon. The Galactic anticenter
sample has higher mean extinction and molecular fraction. Overall (see Table
3) the anticenter sources have the following characteristics:
* •
33% higher $<$EB-V$>$ = 0.36 vs 0.27 mag;
* •
Same $<$N(HI)$>$ = $1.3\times 10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ and
fDNM = 0.12;
* •
80% higher $<$N(HI)cl$>$ = 1.12 vs 0.62 $\times 10^{21}\leavevmode\nobreak\
{\rm cm}^{-2}$;
* •
2 times higher $<$2N(H2)$>$ = 0.80 vs 0.39 $\times 10^{20}\leavevmode\nobreak\
{\rm cm}^{-2}$;
* •
65% higher f${}_{{\rm H}_{2}}$ = 0.38 vs 0.23;
* •
Higher fraction of detected CO, 9/33 (18/33 in the new work) vs 1/13;
* •
Higher incidence of nondetection of $\mathrm{HCO^{+}}$, 5/33 vs 0/13, in some
cases due to low flux, see Figure 9.
The DNM fraction is nearly the same in the two samples (0.12-0.13) and much
smaller (0.06) in both subsamples when CO emission was detected at the level
of 1 K-km s-1. The DNM fraction is also noticeably smaller when molecular gas
sampled in $\mathrm{HCO^{+}}$ absorption is absent, and similarly at EB-V $<$
0.2 mag. The strongest variations in fDNM in Table 3 arise in selections based
on the strength of CO emission and are discussed in Section 4.
### 3.1 Quantitative summary results
Figure 3 shows an overview of trends in means of N(HI), N(DNM), and N(H2)
using data binned in 0.05 mag intervals of reddening as in Figure 2, plotted
horizontally at the mean EB-V in every bin. The counts in each bin are shown
and the bins at EB-V $>$ 0.5 mag are occupied by only one or two sightlines.
The association of DNM with molecular gas is unmistakable.
The mean HI column density varies by only a factor of 3 over a wider range
EB-V = 0.09 - 1 mag. By contrast, means of N(NDM) and N(H2) vary by factors of
50-100. The values of $<$N(DNM)$>$ and $<$N(H2)$>$ are comparable and increase
steadily for EB-V $\la$ 0.7 mag, beyond which $<$N(DNM)$>$ either fails to
increase or falls: the CO emission tracer used to derive N(DNM) was used
effectively to account for H nuclei in H2. The cloud-associated mean N(HI)cl
declines at EB-V $\geq 0.7$ mag as H2 sequesters H nuclei. The value of
$<$N(H2)$>$ increases up to the bins at highest EB-V where $<$N(H2)$>$
$\approx$ $<$N(HI$>$) $\approx 2-3$ $<$N(HI)cl$>$ and whole sightlines are
dominated by molecular gas.
Figure 2: Counts of sightlines and detected tracers binned in 0.05 intervals
of reddening, EB-V. The histogram of all sources is overlaid (dashed red) in
the other panels.
The correlation between DNM and H2 is remarkable considering the vastly
different scales on which EB-V, N(HI), and N(NDM) are measured (6-20′), as
contrasted with N(H2) $\propto$ $\Upsilon_{\mathrm{HCO^{+}}}$ that is derived
on submilliarcsecond scales from absorption against scatter-broadened compact
millimeter-wave ALMA calibrator continuum sources. The same kind of
correlation of tracers observed on different angular scales occurs in the
correlation of integrated $\lambda$21cm HI optical depth with EB-V (Liszt,
2021).
When $\mathrm{HCO^{+}}$ is not detected (5/46 directions) the following
characteristics are found compared to sightlines with detected
$\mathrm{HCO^{+}}$:
* •
3 times lower $<$EB-V$>$ = 0.13 mag;
* •
1.3 times higher $<$N(H)$>$/$<$EB-V$>$ = $8\times 10^{21}\leavevmode\nobreak\
{\rm cm}^{-2}$ mag-1;
* •
3.5 times lower $<$N(DNM)$>$ $=8\times 10^{19}\leavevmode\nobreak\ {\rm
cm}^{-2}$;
* •
2 times lower $<$fDNM$>$ = 0.07;
* •
no DNM in 3/5 cases vs. 15/46 overall.
Instead, using 3$\sigma$ upper limits for N(H2) we find:
* •
$>$5 times lower $<$2N(H2)$>\leq 1.5\times 10^{20}\leavevmode\nobreak\ {\rm
cm}^{-2}$ and
* •
$>$2.5 times lower $<$f${}_{{\rm H}_{2}}$$>\leq 0.14$
The ensemble of sightlines with N(DNM) $<5\times 10^{19}\leavevmode\nobreak\
{\rm cm}^{-2}$ does not differ by more than 10-20% from other sightlines in
any mean property not involving N(DNM). The variations in N(DNM) and N(H2)
stand in great contrast with N(HI) that increases only by a factor of 3 across
the figure. The rise in N(H2) accounts for the failure of N(HI) to increase at
high EB-V, but this could in principle also be explained by increasing
saturation of the HI emission. As noted in Planck Collaboration et al. (2015)
and further refined by Remy et al. (2018), the global DNM analysis finds the
best fit with the optically thin estimate of N(HI), implicitly leaving a
bigger overall role for molecular gas than for saturation of the HI profile.
### 3.2 DNM and HI
Figure 3: Trends in mean N(HI), N(H2), and N(DNM). The results are binned in
0.05 mag intervals of EB-V. Left: N(HI) is calculated over the whole line
profile. Right: N(HI) = N(HI)cl the HI column density associated with the
clouds studied here. The number of sightlines in each bin is shown in both
panels.
Shown in Figure 4 is the variation of $<$N(DNM)$>$ with $<$N(HI)$>$ and
$<$N(HI)cl$>$. Chamaeleon differs from the Galactic anticenter in having more
extraneous atomic gas along the line of sight and smaller values of cloud-
associated gas. The cloud-associated HI in Chamaeleon is clustered at the low
end of the range, even though the samples are evenly matched in the span of
their total HI column density and have the same mean N(HI) in Table 3.
Sightlines lacking DNM are present at all N(HI), but N(DNM) is uniformly small
along sightlines at the highest column densities N(HI) $\ga 2.3\times
10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ where CO emission is more commonly
observed. The sources with N(DNM) $>6\times 10^{20}\leavevmode\nobreak\ {\rm
cm}^{-2}$ and high N(DNM)/N(HI) ratios are perhaps the most obvious candidates
for hosting significant quantities of DNM in atomic form as the result of
saturation of the HI profile, but only two of these actually have small
contributions from N(H2). Assessing the DNM contribution arising from possible
saturation of the HI profile is discussed in Section 3.3 where it is seen that
the strongest cases of atomic DNM actually have more modest N(DNM).
### 3.3 DNM, EB-V, and N(H2)
As with HI, sightlines lacking DNM are present over the full range of EB-V in
Figure 5, top, but they are predominant at smaller N(H2). Sightlines with
2N(H2) $\la 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ lack DNM. Along
sightlines with column densities too low to host molecular gas, the atomic gas
is well represented by the optically thin estimates of N(HI) used in the DNM
analysis.
Care must be taken in interpreting the relationship of DNM and N(H2) in Figure
5 because the DNM in part represents material that is missing in the CO
emission tracer while some of the H2 traced by $\mathrm{HCO^{+}}$ is actually
visible in CO emission. To minimize this cross-contamination, Figure 6 shows a
plot of N(DNM) vs. N(H2) for sightlines lacking a detection of CO, using the
more sensitive IRAM data for the Galactic anticenter subsample. Included are
all but one (12/13) of the Chamaeleon sightlines and 60% of those toward the
anticenter.
Sightlines lacking DNM in the absence of CO emission in Figure 6 are almost
entirely confined at 2N(H2) $\la 2\times 10^{20}\leavevmode\nobreak\ {\rm
cm}^{-2}$, so the detections of $\mathrm{HCO^{+}}$ absorption do not imply
much additional gas along sightlines where DNM was not found. Missing from
Figure 6 are sightlines with 2N(H2) $\ga 10^{21}\leavevmode\nobreak\ {\rm
cm}^{-2}$ corresponding to ${\rm W}_{\mathrm{CO}}$= 2.5 K-km s-1 for the
Galactic CO-H2 conversion factor.
There are two or three cases in each subsample where N(DNM)/2N(H2) $\ga 2$ and
saturation of the $\lambda$21cm emission profile may be important. Most of
these sightlines have modest values N(DNM) $\approx 4-5\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ and are not the sightlines with
high N(DNM) that were singled out for discussion in Section 3.2 when comparing
N(DNM) and N(HI).
There are also two or three sightlines at N(H2) $>6\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ where N(DNM)/2N(H2) $\la 2$.
Overall $<$2N(H2)$>$ = 1.3$<$N(DNM)$>$ for sources lacking CO emission, so H2
accounts for DNM without adding extra “dark” gas, and the DNM is mostly
molecular. As before in Chamaeleon, the DNM is largely molecular, even if the
medium overall is predominantly atomic.
## 4 The role of CO emission
The CO emission that was detected in previous studies used to determine N(DNM)
along our sightlines was well above 1 K-km s-1 in every case (Table 2), with
$<$${\rm W}_{\mathrm{CO}}$$>$ = 4.9K-km s-1 (Table 3). This is very nearly the
same mean as for the sample of sightlines with ${\rm W}_{\mathrm{CO}}$ $\geq
1$K-km s-1 observed at the IRAM 30m, 5.5 K-km s-1 (see Table 2). Compared to
the overall average, the sightlines with $<$${\rm W}_{\mathrm{CO}}$$>$ $\geq$
1K-km s-1 have the following characteristics:
* •
50-100% higher $<$EB-V$>$ = $0.5-0.7$ mag;
* •
3-4 times higher $<$2N(H2)$>$ = $2-3\times 10^{21}\leavevmode\nobreak\ {\rm
cm}^{-2}$;
* •
2 times higher $<{\rm f}_{\mbox{H${}_{2}$}}>$ = 0.6;
* •
2 times lower $<$fDNM$>$ = 0.06.
Figure 4: N(DNM) plotted against total sightline N(HI) (bottom) and cloud-
associated atomic hydrogen N($\mathrm{H\textsc{i}}$)cl (top). Figure 5: N(DNM)
plotted against EB-V (top) and 2N(H2) (bottom). Larger symbols represent
sightlines where ${\rm W}_{\mathrm{CO}}$$\ga$ 1 K-km s-1.
The 50% smaller DNM fractions $<$fDNM$>$ = 0.06 along samples of sightlines
with $<$${\rm W}_{\mathrm{CO}}$$>$ $\geq$ 1K-km s-1 indicate that CO emission
is doing a good job of tracing H2 in gas where the molecular fraction is high
(Liszt, 2017), CO emission is strong, and the cloud-associated HI fraction
declines (Figure 2). However, CO emission in general represents a small
portion of the total molecular gas present along the sightlines in this work.
For instance, $<$2N(H2)$>$ = $7\times 10^{20}\leavevmode\nobreak\ {\rm
cm}^{-2}$ along 46 sightlines, while $<$${\rm W}_{\mathrm{CO}}$$>$ $\approx 5$
K-km s-1 along the 6-10 sightlines where $<$${\rm W}_{\mathrm{CO}}$$>$ $\geq$
1K-km s-1 (Table 3). For a typical Galactic CO-H2 conversion factor, the
summed molecular gas column represented in $\mathrm{HCO^{+}}$ is three to four
times that inferred from the summed CO emission. Most of the molecular gas in
our sample was hidden in the DNM prior to our work and is still only revealed
by the $\mathrm{HCO^{+}}$ absorption, even with more sensitve CO observations.
The IRAM 30m detections with $<$${\rm W}_{\mathrm{CO}}$$>$ below 1K-km s-1
comprise less than 20% of the total amount of CO emission.
### 4.1 The CO-H2 conversion factor
Comparison of the scaled $\mathrm{HCO^{+}}$ absorption and IRAM 30 emission
measurements provides the most direct determination of the actual CO-H2
conversion factor along the lines of sight studied in this work. Figure 7
shows the trends in $<$N(H2)$>$, $<$${\rm W}_{\mathrm{CO}}$$>$, and
$<$N(H2)$>$/$<$${\rm W}_{\mathrm{CO}}$$>$ binned in 0.05 mag increments of
EB-V as in Figure 3. Included are the 28 Galactic anticenter sightlines where
$\mathrm{HCO^{+}}$ was detected, with ${\rm W}_{\mathrm{CO}}$ taken at the
3$\sigma$ upper limit for sightlines where CO emission was not detected with
greater significance. Also included is the sightline toward J1733 in
Chamaeleon where CO emission was detected. The $3\sigma$ upper limits ${\rm
W}_{\mathrm{CO}}$ $\leq$ 1.5 K-km s-1 in Chamaeleon are not useful.
Figure 7 illustrates the behavior of the CO-H2 conversion factor that is
tabulated for different samples in Table 3. The values of $<$N(H2)$>$ and
$<$${\rm W}_{\mathrm{CO}}$$>$ both increase steadily with EB-V in Figure 7,
but at different rates so that their ratio declines by a factor $\approx 7$ to
$<$N(H2)$>$/$<$${\rm W}_{\mathrm{CO}}$$>$ $=2.5-3\times
10^{20}$H2$\leavevmode\nobreak\ {\rm cm}^{-2}$(K-km s-1)-1 for EB-V $\ga$ 0.5
mag. Variations of similar magnitude in individual diffuse and/or translucent
MBM clouds were reported by Magnani et al. (1998) and Cotten & Magnani (2013).
Mean values of N(H2)/${\rm W}_{\mathrm{CO}}$ are $2-2.5\times
10^{20}$H2$\leavevmode\nobreak\ {\rm cm}^{-2}$(K-km s-1)-1 for the old and new
samples with $<$${\rm W}_{\mathrm{CO}}$$>$ $\geq 1$ K-km s-1, increasing by
factors of 2-3 for the samples with IRAM 30m detections below 1K-km s-1 and
upper limits. Also see the inset in Figure 7 on this point..
Figure 6: N(DNM) plotted against N(H2) for sightlines lacking detected CO
emission in the analysis of Remy et al. (2018). Shown are loci at which the
number of H nuclei in H2 is 50, 100, and 200% of that in DNM. Figure 7:
Trends in mean N(H2) (black squares), ${\rm W}_{\mathrm{CO}}$ (pink diamonds),
and N(H2)/${\rm W}_{\mathrm{CO}}$ (blue triangles). The results are binned in
0.05 mag intervals of EB-V, as in Figure 3. Bins in which all sightlines have
only upper limits on ${\rm W}_{\mathrm{CO}}$ are indicated by upper and lower
limits. The scales for N(H2) $\leavevmode\nobreak\ {\rm cm}^{-2}$ and
N(H2)/${\rm W}_{\mathrm{CO}}$ ($\leavevmode\nobreak\ {\rm cm}^{-2}$ (K-km
s-1)${}^{-}1$) are read on the left vertical axis and that for ${\rm
W}_{\mathrm{CO}}$ (K-km s-1) on the right. The number of sightlines in each
bin is shown as in Figure 3 and the count is carried separately for ${\rm
W}_{\mathrm{CO}}$ at low EB-V. The light gray dashed line is a power-law fit
N(H2) $=10^{21.0973}$EB-V1.335.
The values of the CO-H2 conversion factor derived are comparable to those
derived in extant Galactic-scale $\gamma$-ray analyses (Table E.1 in Remy et
al. 2017) when ${\rm W}_{\mathrm{CO}}$ $\ga$ 1 K-km s-1, but are uniformly
larger than those derived in cloud-level studies like the DNM analysis whose
results are summarized in Table 2 of Remy et al. (2017). Those results ranged
from $1-1.6\times 10^{20}$H2$\leavevmode\nobreak\ {\rm cm}^{-2}$(K-km s-1)-1
for determinations based on dust and from $0.44-1.00\times
10^{20}$H2$\leavevmode\nobreak\ {\rm cm}^{-2}$ (K-km s-1)-1 for determinations
based on gamma rays.
There is no contradiction with the larger values found in this work whose
method of studying widely separated, semi-randomly placed sightlines is
similar to the larger scale studies. The strongly CO-emitting (${\rm
W}_{\mathrm{CO}}$ $>$ 10 K-km s-1) regions encountered in the cloud-level
studies that have generally small values of $X_{\mathrm{CO}}$ (see Figure 13
in Remy et al. 2018) were not sampled here.
There may also be a difference arising from the operational definition of the
conversion factor. The conversion factors in Table 2 in Remy et al. (2017) are
those that optimize the fit of the $\gamma$-ray or dust emissivity to a total
hydrogen column density that is represented schematically as N(H) =
N(HI)+N(DNM)+2N(H2) = N(HI)+N(DNM)+2$X_{\mathrm{CO}}$${\rm W}_{\mathrm{CO}}$.
After the analysis, there remains a DNM constituent whose molecular fraction
is undetermined and not explicitly considered in the definition of the
multiplier $X_{\mathrm{CO}}$; we showed that the DNM is largely molecular. By
contrast, the present analysis determines N(H2) independent of CO emission and
defines $X_{\mathrm{CO}}$ = N(H2)/${\rm W}_{\mathrm{CO}}$. This N(H2) includes
the molecular component of the DNM that we took pains to consider separately
in the discussion of Figure 6.
### 4.2 Achievable limits and detection thresholds for CO, DNM, and H2
Sightlines in our study often had rms CO emission noise $\Delta$${\rm
W}_{\mathrm{CO}}$ $\ga 1/3-1/2$ K-km s-1 in the prior DNM analysis (Table 2).
For a typical Galactic conversion 2N(H2)/${\rm W}_{\mathrm{CO}}$ $=4\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ (K-kms)-1, the equivalent $3\sigma$
threshold detection limits on the hydrogen column are 2N(H2) $=4-6\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$, above the actual values of N(DNM)
along most of the sightlines we observed. By contrast, the median $3\sigma$
detection threshold from $\mathrm{HCO^{+}}$ in our work is 2N(H2) $>1.1\times
10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ and typical N(DNM) values are
N(DNM) $\ga 2\times 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ (Figure 6).
The effectiveness of reducing the detection threshold of CO emission below 1
K-km s-1 can be assessed by examining the distribution of CO detections and
upper limits in the IRAM 30m CO observations. The subsamples of IRAM CO
observations summarized in Table 3 show 6 detections with ${\rm
W}_{\mathrm{CO}}$ $>$ 1 K-km s-1, 12 detections with ${\rm W}_{\mathrm{CO}}$
$<$ 1 K-km s-1, and ten upper limits at levels ${\rm W}_{\mathrm{CO}}$
$<0.1-0.2$ K-km s-1 (see Table 1 for values of upper limits and Figure 3 for a
graphical representation of the data). $\mathrm{HCO^{+}}$ was detected along
all of these sightlines, and the three subsamples respectively represent
fractions 0.66, 0.26, and 0.08 of the total amount of H2.
Thus, a survey with a detection limit of 1 K-km s-1 would detect approximately
two-thirds of the molecular gas along these diffuse and/or translucent lines
of sight, and a much increased effort to reduce the detection limit to 0.2
K-km s-1 might find another one-fourth of the H2. Comparable reductions in the
fraction of undetected H2 with increasing CO sensitivity down to rms levels
$\Delta$${\rm W}_{\mathrm{CO}}$ $\approx 0.1$ K-km s-1 were achieved by Donate
& Magnani (2017). Missing one-third of the H2 at the 1 K-km s-1 threshold is
consistent with the dark gas fraction derived by Wolfire et al. (2010) and
Gong et al. (2018).
### 4.3 The wider view
In Section 4.2 we note that two-thirds of the H2 was found along the
sightlines with ${\rm W}_{\mathrm{CO}}$ $>$ 1 K-km s-1 and that a deeper
survey with a detection limit of 0.2 K-km s-1 would have found another 25% of
the H2. At this point we can ask how the results of this sparse sampling are
reflected in the region as a whole. We drew a hull around the observed
anticenter sightlines, as shown in Figure 11 and derived the pixel-by-pixel
statistics for the contained area, the amount of material (taken proportional
to EB-V), and the amount of H2. For H2 we used the N(H2)-EB-V relationship
N(H2) $=10^{21.0973}$EB-V1.335 111f${}_{{\rm H}_{2}}$ = 2N(H2)/N(H) = 0.4-0.5
at EB-V = 2 mag if N(H)/EB-V $=6-8\times 10^{21}{\rm mag}^{-1}$ and the fact
that ${\rm W}_{\mathrm{CO}}$ $\ga 1$ K-km s-1 at EB-V $\ga$ 0.4 mag or N(H2)
$\ga 4\times 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ in Figure 7 (see also
Figure 5 of Paper 1).
Figure 8 shows the probability densities (at left) and cumulative
distributions for the derived quantities. Reading values off the cumulative
probability distributions at right in Figure 8, conditions with EB-V $\geq$
0.4 mag, N(H2) $\ga 4\times 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ occur
over one-fourth of the total area containing one-half of the total projected
gas column and two-thirds of the H2. Apparently, the sampled sightlines were
representative of the region as a whole regarding the H2 distribution.
Sampling 90% of the H2 would require detecting CO emission at EB-V $\ga 0.2$
mag where N(H2) $\approx 2\times 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$.
The sampling here is too sparse to derive an equivalent value of ${\rm
W}_{\mathrm{CO}}$ in Figure 7, but the broader sample in Paper 1 suggests
${\rm W}_{\mathrm{CO}}$ $\ga$ 0.3 K-km s-1 would be appropriate. Sightlines
with EB-V $<0.32$ mag or AV $<1$ mag comprise two-thirds of the area, one-
third of the mass, and one-fourth of the H2.
Figure 8: Probability distributions of area, amount of material, and
molecular hydrogen with respect to EB-V. Left: Counts (probability density)
scaled to the same area. Right: Cumulative distribution function. N(H2) is
calculated using the mean power-law fit N(H2) $=10^{21.0973}$EB-V1.335 derived
as shown in Figure 7.
### 4.4 Comparison with predictions and models
For models, the critical factors in predicting the amount of dark gas are the
threshold where ${\rm W}_{\mathrm{CO}}$ reaches the 1 K-km s-1 brightness
level and the amount of material above the threshold. In the regime of diffuse
molecular gas the integrated brightness of the J=1-0 line is determined almost
exclusively by the CO column density, with ${\rm W}_{\mathrm{CO}}$ (K-km s-1)
= N(CO)$/10^{15}\leavevmode\nobreak\ {\rm cm}^{-2}$ for hydrogen number
densities n(H) $=64-500\leavevmode\nobreak\ {\rm cm}^{-3}$ (Liszt, 2007, 2017;
Hu et al., 2021) and kinetic temperatures appropriate to the local thermal
pressure. This simple proportionality between column density and brightness
persists well beyond the CO column density where the J=1-0 transition becomes
optically thick, owing to the strongly subthermal excitation (Goldreich &
Kwan, 1974).
Thus, the CO chemistry dominates the CO brightness, and the sky map of CO
emission is a map of the chemistry. The observations can be reproduced by an
ad hoc CO formation chemistry in which a fixed relative abundance of
$\mathrm{HCO^{+}}$ N($\mathrm{HCO^{+}}$)/N(H2) $=3\times 10^{-9}$ thermally
recombines to CO if the H2 formation and shielding of H2 and CO are treated
self-consistently along with the heating and cooling Liszt (2017). However,
networks of chemical reactions in quiescent gas may fail to form the observed
quantity of $\mathrm{HCO^{+}}$ and produce too little CO. This causes the CO
brightness to reach 1 K-km s-1 at overly large values of EB-V and N(H2) where
shielding of CO by dust and H2 make up for the deficit in the CO formation
rate (e.g., Gong et al. 2018 Figure 7; Hu et al. 2021 Figure 9).
Models with a weak CO formation chemistry have an innate tendency to
overestimate the amount of CO-dark gas, but the actual amount of CO-dark gas
depends on the distribution of material. In practice the DNM fraction seen by
Remy et al. (2018) varied from 0 to 0.3 (their Figure 8) and the sightlines
sampled here had $<$fDNM$>$ = 0.13.
Table 1: Line-of-sight properties and new results for $\mathrm{HCO^{+}}$ and
CO
Source | RA(J2000) | Dec(J2000) | $l$ | $b$ | EB-V1 | 89 GHz flux | $\Upsilon_{{\rm HCO}\mbox{${}^{+}$}}$2 | $\sigma$$\Upsilon_{{\rm HCO}\mbox{${}^{+}$}}$3 | ${\rm W}_{\mathrm{CO}}$4 | $\sigma$${\rm W}_{\mathrm{CO}}$5
---|---|---|---|---|---|---|---|---|---|---
| hh.mmsss | dd.mmsss | degrees | degrees | mag | Jy | km s-1 | km s-1 | K-km s-1 | K-km s-1
J0203+1134 | 2.03464 | 11.34492 | 149.6826 | -47.4992 | 0.144 | 0.126 | $\leq$0.263 | 0.088 | |
J0209+1352 | 2.09357 | 13.52045 | 150.1800 | -44.8290 | 0.094 | 0.223 | 0.20 | 0.050 | $\leq$0.071 | 0.024
J0211+1051 | 2.11132 | 10.51348 | 152.5781 | -47.3674 | 0.145 | 0.462 | 0.76 | 0.029 | 0.36 | 0.025
J0213+1820 | 2.13105 | 18.20255 | 148.7289 | -40.4014 | 0.130 | 0.093 | $\leq$0.345 | 0.115 | |
J0231+1322 | 2.31459 | 13.22547 | 157.0917 | -42.7380 | 0.121 | 0.430 | 0.14 | 0.025 | $\leq$0.064 | 0.021
J0242+1742 | 2.42243 | 17.42589 | 157.0180 | -37.7033 | 0.077 | 0.227 | $\leq$0.151 | 0.050 | |
J0252+1718 | 2.52077 | 17.18427 | 159.7420 | -36.7885 | 0.220 | 0.172 | 0.25 | 0.077 | $\leq$0.096 | 0.032
J0325+2224 | 3.25368 | 22.24004 | 163.6700 | -28.0213 | 0.213 | 1.162 | 1.01 | 0.017 | 0.94 | 0.051
J0329+3510 | 3.29154 | 35.10060 | 155.9152 | -17.4042 | 0.267 | 0.570 | 0.51 | 0.032 | $\leq$0.143 | 0.048
J0329+2756 | 3.29577 | 27.56155 | 160.7030 | -23.0743 | 0.198 | 0.158 | $\leq$0.193 | 0.064 | |
J0334+0800 | 3.34533 | 8.00144 | 177.2396 | -37.0871 | 0.391 | 0.150 | 0.44 | 0.088 | $\leq$0.162 | 0.054
J0336+3218 | 3.36301 | 32.18293 | 158.9998 | -18.7650 | 0.733 | 1.689 | 0.16 | 0.009 | $\leq$0.219 | 0.073
J0356+2903 | 3.56085 | 29.03423 | 164.6120 | -18.4927 | 0.212 | 0.139 | 1.50 | 0.090 | 1.62 | 0.042
J0357+2319 | 3.57216 | 23.19538 | 169.0302 | -22.4661 | 0.185 | 0.224 | 0.28 | 0.027 | $\leq$0.192 | 0.064
J0400+0550 | 4.00117 | 5.50431 | 184.2710 | -33.7266 | 0.266 | 0.159 | 0.25 | 0.063 | $\leq$0.172 | 0.057
J0401+0413 | 4.01199 | 4.13344 | 186.0261 | -34.4947 | 0.341 | 0.405 | 0.49 | 0.021 | 0.19 | 0.048
J0403+2600 | 4.03056 | 26.00015 | 168.0260 | -19.6483 | 0.201 | 0.331 | 0.60 | 0.029 | 0.62 | 0.067
J0406+0637 | 4.06343 | 6.37150 | 184.7075 | -32.0009 | 0.283 | 0.264 | 0.54 | 0.051 | 0.63 | 0.040
J0407+0742 | 4.07291 | 7.42075 | 183.8723 | -31.1558 | 0.265 | 0.387 | 0.53 | 0.031 | 0.19 | 0.042
J0426+0518 | 4.26366 | 5.18199 | 189.3631 | -28.7705 | 0.291 | 0.516 | 0.12 | 0.020 | $\leq$0.170 | 0.057
J0426+2327 | 4.26557 | 23.27396 | 173.8881 | -17.4457 | 0.539 | 0.304 | 2.57 | 0.057 | 4.84 | 0.045
J0427+0457 | 4.27476 | 4.57083 | 189.8857 | -28.7306 | 0.335 | 0.414 | 0.62 | 0.024 | 0.39 | 0.045
J0437+2037 | 4.31038 | 20.37343 | 176.8096 | -18.5565 | 0.532 | 0.217 | 1.54 | 0.073 | 0.67 | 0.026
J0431+1731 | 4.31574 | 17.31358 | 179.4942 | -20.3579 | 0.464 | 0.104 | 1.01 | 0.110 | 0.70 | 0.049
J0433+0521 | 4.33111 | 5.21156 | 190.3730 | -27.3967 | 0.298 | 4.911 | 0.35 | 0.003 | $\leq$0.109 | 0.036
J0437+2940 | 4.37044 | 29.40138 | 170.5818 | -11.6609 | 0.979 | 0.059 | 5.92 | 1.138 | 10.42 | 0.027
J0438+3004 | 4.38049 | 30.04455 | 170.4116 | -11.2283 | 0.952 | 0.689 | 6.25 | 0.038 | 7.11 | 0.026
J0439+3045 | 4.39178 | 30.45076 | 170.0655 | -10.5913 | 0.867 | 0.195 | 5.05 | 0.082 | 6.69 | 0.027
J0440+1437 | 4.40211 | 14.37570 | 183.2538 | -20.5438 | 0.681 | 0.337 | 1.21 | 0.031 | 0.83 | 0.029
J0445+0715 | 4.45014 | 7.15539 | 190.4535 | -23.8898 | 0.121 | 0.275 | $\leq$0.083 | 0.028 | |
J0449+1121 | 4.49077 | 11.21286 | 187.4274 | -20.7365 | 0.504 | 0.521 | 0.65 | 0.022 | 0.23 | 0.033
J0502+1338 | 5.02332 | 13.38110 | 187.4143 | -16.7456 | 0.564 | 0.271 | 1.81 | 0.059 | 2.46 | 0.031
J0510+1800 | 5.10024 | 18.00416 | 184.7304 | -12.7895 | 0.328 | 2.411 | 0.13 | 0.005 | 0.17 | 0.036
1From Schlegel et al. (1998)
2Integrated optical depth or $3\sigma$ upper limit over a 3 km s-1 interval
3Integrated emission from our results or $3\sigma$ upper limit over a 3 km s-1
interval
4 rms of detected emission or rms over a 3km s-1 interval
Table 2: Line-of-sight properties and derived quantities
Source | E(B-V) | N(HI)1 | N(HI)cl1 | 2N(H2) 2 | ${\rm W}_{\mathrm{CO}}$3 | N(DNM) 1 | 2N(H2)/N(HI) | N(DNM)/N(HI) | $X_{\mathrm{CO}}$4
---|---|---|---|---|---|---|---|---|---
| mag | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | K-km s-1 | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | | |
J0203+1134 | 0.144 | 0.664 | 0.646 | $\leq$0.193 | | 0.201 | $\leq$0.290 | 0.303 |
J0209+1352 | 0.094 | 0.696 | 0.686 | 0.144 | | 0.00 | 0.206 | 0.00 |
J0211+1051 | 0.145 | 0.618 | 0.617 | 0.561 | | 0.240 | 0.908 | 0.388 |
J0213+1820 | 0.130 | 0.933 | 0.911 | $\leq$0.253 | | 0.177 | $\leq$0.271 | 0.190 |
J0231+1322 | 0.121 | 0.926 | 0.852 | 0.103 | | 0.00 | 0.111 | 0.00 |
J0242+1742 | 0.077 | 0.820 | 0.819 | $\leq$0.111 | | 0.00 | $\leq$0.135 | 0.00 |
J0252+1718 | 0.220 | 1.142 | 1.139 | 0.184 | | 0.299 | 0.161 | 0.262 |
J0325+2224 | 0.213 | 0.999 | 0.926 | 0.742 | 2.82 | 0.180 | 0.743 | 0.180 | 1.314
J0329+3510 | 0.267 | 1.418 | 1.131 | 0.372 | | 0.150 | 0.263 | 0.106 |
J0329+2756 | 0.198 | 1.129 | 1.002 | $\leq$0.142 | $\leq 1.97$ | 0.00 | $\leq$0.126 | 0.00 |
J0334+0800 | 0.391 | 1.893 | 1.855 | 0.322 | | 0.223 | 0.170 | 0.118 |
J0336+3218 | 0.733 | 1.435 | 1.165 | 0.119 | $\leq$2.60 | 0.606 | 0.082 | 0.423 |
J0356+2903 | 0.212 | 0.666 | 0.520 | 1.099 | 1.40 | 0.178 | 1.650 | 0.267 | 3.931
J0357+2319 | 0.185 | 1.022 | 0.944 | 0.205 | $\leq$0.79 | 0.00 | 0.200 | 0.00 |
J0400+0550 | 0.266 | 1.266 | 1.210 | 0.182 | | 0.238 | 0.144 | 0.188 |
J0401+0413 | 0.341 | 1.209 | 1.202 | 0.357 | | 0.426 | 0.296 | 0.352 |
J0403+2600 | 0.201 | 0.830 | 0.667 | 0.438 | 1.99 | 0.00 | 0.528 | 0.00 | 1.101
J0406+0637 | 0.283 | 1.182 | 1.143 | 0.397 | | 0.266 | 0.335 | 0.225 |
J0407+0742 | 0.265 | 1.033 | 1.006 | 0.387 | | 0.258 | 0.374 | 0.250 |
J0426+0518 | 0.291 | 1.305 | 1.141 | 0.091 | $\leq$1.40 | 0.414 | 0.070 | 0.317 |
J0426+2327 | 0.539 | 1.586 | 1.331 | 1.887 | 5.78 | 0.00 | 1.190 | 0.00 | 1.632
J0427+0457 | 0.335 | 1.361 | 1.192 | 0.457 | 2.49 | 0.590 | 0.336 | 0.433 | 0.918
J0437+2037 | 0.532 | 2.324 | 1.992 | 1.130 | $\leq$1.61 | 0.00 | 0.486 | 0.00 |
J0431+1731 | 0.464 | 1.714 | 1.457 | 0.739 | 3.09 | 0.00 | 0.431 | 0.00 | 1.198
J0433+0521 | 0.298 | 1.159 | 0.950 | 0.255 | $\leq$1.48 | 0.795 | 0.220 | 0.686 |
J0437+2940 | 0.979 | 1.320 | 0.936 | 4.345 | 11.98 | 0.126 | 3.292 | 0.095 | 1.814
J0438+3004 | 0.952 | 2.019 | 1.002 | 4.586 | 7.70 | 0.771 | 2.271 | 0.382 | 2.979
J0439+3045 | 0.867 | 2.361 | 0.745 | 3.701 | 9.42 | 0.00 | 1.567 | 0.00 | 1.964
J0440+1437 | 0.681 | 1.659 | 1.524 | 0.888 | $\leq$1.97 | 1.230 | 0.535 | 0.741 |
J0445+0715 | 0.121 | 1.044 | 0.915 | $\leq$0.061 | | 0.00 | $\leq$0.058 | 0.00 |
J0449+1121 | 0.504 | 1.416 | 1.347 | 0.476 | | 0.791 | 0.336 | 0.558 |
J0502+1338 | 0.564 | 2.188 | 1.957 | 1.324 | $\leq$1.90 | 0.672 | 0.605 | 0.307 |
J0510+1800 | 0.328 | 2.347 | 1.986 | 0.092 | $\leq$2.00 | 0.00 | 0.039 | 0.00 |
J0942-7731 | 0.336 | 1.000 | 0.617 | 0.837 | $\leq$1.50 | 0.382 | 0.838 | 0.382 |
J1058-8003 | 0.152 | 0.651 | 0.470 | 0.147 | “ | 0.371 | 0.226 | 0.570 |
J1136-6827 | 0.460 | 2.407 | 0.915 | 0.910 | “ | 0.172 | 0.378 | 0.071 |
J1145-6954 | 0.387 | 1.925 | 0.998 | 0.638 | “ | 0.223 | 0.331 | 0.116 |
J1147-7935 | 0.300 | 2.506 | 0.773 | 0.040 | “ | 0.00 | 0.016 | 0.00 |
J1152-8344 | 0.283 | 1.088 | 0.505 | 0.176 | “ | 0.515 | 0.162 | 0.473 |
J1224-8313 | 0.257 | 0.962 | 0.563 | 0.693 | “ | 0.335 | 0.720 | 0.348 |
J1254-7138 | 0.282 | 1.902 | 0.660 | 0.102 | “ | 0.00 | 0.053 | 0.00 |
J1312-7724 | 0.476 | 1.257 | 0.671 | 0.199 | “ | 0.396 | 0.158 | 0.315 |
J1550-8258 | 0.107 | 0.679 | 0.416 | 0.177 | “ | 0.027 | 0.261 | 0.039 |
J1617-7717 | 0.091 | 0.659 | 0.431 | 0.043 | “ | 0.045 | 0.065 | 0.068 |
J1723-7713 | 0.255 | 0.834 | 0.500 | 1.105 | 2.40 | 0.195 | 1.325 | 0.234 | 2.302
J1733-7935 | 0.139 | 0.792 | 0.525 | 0.041 | $\leq$ 1.50 | 0.00 | 0.052 | 0.00 |
1 HI and DNM column densities are from Planck Collaboration et al. (2015) and
Remy et al. (2017, 2018)
2 N(H2) = N($\mathrm{HCO^{+}}$)/$3\times 10^{-9}$
3 Integrated J=1-0 CO emission used by Planck Collaboration et al. (2015) and
Remy et al. (2017, 2018)
4 $X_{\mathrm{CO}}$ = N(H2)/${\rm W}_{\mathrm{CO}}$
Table 3: Mean line-of-sight properties and derived quantities
Sample | E(B-V) | N(HI) | N(HI)cl | 2N(H2) | ${\rm W}_{\mathrm{CO}}$ | N(DNM) | f${}_{{\rm H}_{2}}$ | fDNM | $X_{\mathrm{CO}}$
---|---|---|---|---|---|---|---|---|---
| mag | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | K-km s-1 | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | | |
Galactic anticenter | 0.362 | 1.324 | 1.119 | 0.798 | 5.185 | 0.268 | 0.38 | 0.13 | 1.93
# | 33 | 33 | 33 | 33 | 9 | 33 | 33 | 33 | 9
Chamaeleon | 0.271 | 1.282 | 0.619 | 0.393 | 2.400 | 0.205 | 0.23 | 0.12 | 2.30
# | 13 | 13 | 13 | 13 | 1 | 13 | 13 | 13 | 1
All LOS | 0.336 | 1.312 | 0.977 | 0.684 | 4.906 | 0.250 | 0.34 | 0.13 | 1.95
# | 46 | 46 | 46 | 46 | 10 | 46 | 46 | 46 | 10
EB-V $\leq 0.2$mag | 0.131 | 0.818 | 0.710 | 0.168 | | 0.082 | 0.18 | 0.08 |
# | 13 | 13 | 13 | 13 | | 13 | 13 | 13 |
Lacking $\mathrm{HCO^{+}}$1 | 0.134 | 0.918 | 0.859 | $\leq$0.152 | | 0.076 | $\leq$0.14 | $\geq$0.08 |
# | 5 | 5 | 5 | 5 | | 5 | 5 | 5 |
DNM $\leq 10^{20}\leavevmode\nobreak\ {\rm cm}^{-2}$ | 0.273 | 1.373 | 0.953 | 0.539 | 5.070 | 0.004 | 0.28 | 0.00 | 1.67
# | 17 | 17 | 17 | 17 | 4 | 17 | 17 | 17 | 4
CO detected earlier2 | 0.502 | 1.369 | 0.928 | 1.910 | 4.906 | 0.204 | 0.58 | 0.06 | 1.95
# | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10
30m CO detected | 0.467 | 1.491 | 1.197 | 1.312 | 2.170 | 0.318 | 0.47 | 0.11 | 3.02
# | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18
30m ${\rm W}_{\mathrm{CO}}$ $>$1K-km s-1 | 0.685 | 1.690 | 1.082 | 2.824 | 5.522 | 0.291 | 0.63 | 0.06 | 2.56
# | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6
30m ${\rm W}_{\mathrm{CO}}$ $\leq$1K-km s-1 | 0.358 | 1.391 | 1.255 | 0.555 | 0.494 | 0.332 | 0.29 | 0.17 | 5.62
# | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12
30m UL $\leq$1K-km s-1 | 0.287 | 1.226 | 1.107 | 0.198 | $<$0.140 | 0.273 | 0.14 | 0.19 | $>$7.07
# | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10
Sample | E(B-V) | N(HI) | N(HI)cl | 2N(H2) | ${\rm W}_{\mathrm{CO}}$ | N(DNM) | f${}_{{\rm H}_{2}}$ | fDNM | XCO
| mag | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | | $10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ | | |
1 Limits in this table use $3\sigma$ upper limits on undetected quantities
2 All sightlines detected earlier in CO had ${\rm W}_{\mathrm{CO}}$ $>$1K-km
s-1
## 5 Summary
We took 89.2 GHz ALMA $\mathrm{HCO^{+}}$ absorption spectra toward 33 compact
millimeter-wave extragalactic continuum sources seen against the Galactic
anticenter (Figure 1 and the tables above). We also observed J=1-0 CO emission
at the IRAM 30m telescope in the 28 anticenter directions where
$\mathrm{HCO^{+}}$ was detected. Inferring N(H2) from N($\mathrm{HCO^{+}}$)
using the ratio N($\mathrm{HCO^{+}}$)/N(H2) $=3\times 10^{-9}$, we combined
these results with those from our earlier study of 13 directions where
$\mathrm{HCO^{+}}$ absorption was detected in the outskirts of Chamaeleon. We
compared the inferred N(H2) with prior determinations of the column densities
of the dark neutral medium, the neutral gas of uncertain (atomic or molecular)
character that had been found to be missing in HI and/or CO emission when
compared with the submillimeter dust and $\gamma$-ray emissivities of large-
scale molecular cloud complexes.
Binning the HI, H2, and DNM column densities in reddening, we showed in Figure
3 that the mean DNM and molecular gas column densities were comparable and
varied compatibly by factors of 50-100 over the observed range EB-V = 0.09 - 1
mag, while N(H(I) varied by only factors of 2-3. The means of N(H2) and N(DNM)
are small at low mean reddening, and increase in similar fashion up to EB-V=
0.5 mag where molecular gas begins to dominate and CO emission is strong.
N(H2) continues to increase with higher reddening, but N(DNM) and the column
density of cloud-associated HI fall where H2 dominates (sequestering hydrogen)
and CO emission is stronger and more closely representative of N(H2).
We made detailed individual sightline-level comparisons of N(DNM) with EB-V,
N(HI), and N(H2) in Figures 4-6. Sightlines with appreciable DNM appear at all
N(HI) (Figure 4); the overall mean DNM fraction fDNM = N(DNM)/(N(HI)+2N(H2)) =
0.12 is modest. Sightlines with appreciable DNM are lacking when EB-V $\la$
0.15 mag (Figure 5) and when 2N(H2) $\la 10^{20}\leavevmode\nobreak\ {\rm
cm}^{-2}$ or 2N(H2) $\ga 10^{21}\leavevmode\nobreak\ {\rm cm}^{-2}$ (Figure
6). In Figure 6 we compared N(DNM) and N(H2) in directions lacking detected CO
emission in order to eliminate the case that H2 was already represented by CO
emission in the DNM analysis. This figure showed that there were 2-3
sightlines in each subsample (Chamaeleon and anticenter) or 5-6/46 overall
where 2N(H2) $<\leq$ N(DNM)/2 and H2 accounted for the minority of the DNM.
There are also a few directions at 2N(H2) $\approx 10^{21}\leavevmode\nobreak\
{\rm cm}^{-2}$ where 2N(H2) $>$ 2N(DNM). Overall, the amounts of DNM and H2
are similar $<$2N(H2)$>$ = 1.3 $<$N(DNM)$>$ for the unambiguous cases lacking
CO emission. The form of the DNM is overwhelmingly molecular hydrogen.
Directions with $<$${\rm W}_{\mathrm{CO}}$$>$ $>$1 K-km s-1 have two times
smaller mean DNM fractions $<$fDNM$>$ = 0.06, while sightlines with $<$${\rm
W}_{\mathrm{CO}}$$>$ $<$ 1 K-km s-1 have three times higher fDNM $\ga$
0.17-0.19 (Table 3). The relatively few sightlines with ${\rm
W}_{\mathrm{CO}}$ $\geq$ 1 K-km s-1 contain two-thirds of the H2 detected in
$\mathrm{HCO^{+}}$, and detecting 90% of the H2 would require detecting CO at
levels ${\rm W}_{\mathrm{CO}}$ $\approx$ 0.2-0.3 K-km s-1.
The CO-H2 conversion factor falls steadily with increasing EB-V or ${\rm
W}_{\mathrm{CO}}$ in Figure 7. Because the H2 is concentrated in the
sightlines with ${\rm W}_{\mathrm{CO}}$ $\geq$ 1 K-km s-1, the overall mean
CO-H2 conversion factors in our work are $<$N(H2)$>$/$<$${\rm
W}_{\mathrm{CO}}$$>$ = $2-3\times 10^{20}$H2$\leavevmode\nobreak\ {\rm
cm}^{-2}$ for samples of sightlines with detectable CO emission. These values
are comparable to previously determined large-scale Galactic averages, and are
substantially higher than the global values determined by the cloud-level
analyses quoted here to derive N(DNM). We ascribed these differences in part
to the present sampling of widely scattered sightlines (i.e., that we did not
do a cloud-level analysis) and perhaps to differences in the operational
definition of the conversion factor, as discussed in Section 4.1.
Subsequent papers derived from the observations discussed here will focus on
the physics and chemistry of the molecules observed in the course of this
work, chiefly $\mathrm{HCO^{+}}$, C2H, and HCN.
###### Acknowledgements.
The National Radio Astronomy Observatory (NRAO) is a facility of the National
Science Foundation, operated by Associated Universities, Inc. This work was
supported by the French program “Physique et Chimie du Milieu Interstellaire”
(PCMI) funded by the Conseil National de la Recherche Scientifique (CNRS) and
Centre National d’Etudes Spatiales (CNES). This work is based in part on
observations carried out under project number 003-19 with the IRAM 30m
telescope]. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN
(Spain). This paper makes use of the following ALMA data –
ADS/JAO.ALMA#2016.1.00714.S
– ADS/JAO.ALMA#2018.1.00115.S ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. We thank Isabelle Grenier for providing results of the DNM analysis and
we thank the anonymous referee for many helpful remarks. HSL is grateful for
the hospitality of the ITU-R and the Hotel de la Cigogne in Geneva during the
completion of this manuscript.
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## Appendix A Statistics of HCO+ detections
Figure 9: Integrated $\mathrm{HCO^{+}}$ optical depth
$\Upsilon_{\mathrm{HCO^{+}}}$ plotted against its rms error. The upper limits
along five sightlines not detected in $\mathrm{HCO^{+}}$ are $3\sigma$.
$\mathrm{HCO^{+}}$ absorption was not detected in five directions, all in the
Galactic anticenter sample. Figure 9 shows $\Upsilon_{\mathrm{HCO^{+}}}$
plotted against its rms error $\Delta$$\Upsilon_{\mathrm{HCO^{+}}}$, in
essence against the strength of the continuum background since all sightlines
were observed for the same length of time. The plot shows that even the
noisiest sightlines miss relatively small amounts of molecular gas. The bias
in the plot, whereby $<$$\Upsilon_{\mathrm{HCO^{+}}}$$>$ increases with
$\Delta$$\Upsilon_{\mathrm{HCO^{+}}}$, was to some extent built into the
observing to increase the yield. That is, the source selection began with
flux-limited samples that would achieve a minimum signal-to-noise ratio on the
weakest source, and added 50% additional targets at lower flux Sν and higher
EB-V/Sν that could be serendipitously observed without a proportionate
increase in the required observing time.
## Appendix B Spectra of HCO+ and CO
Figure 10: Spectra of $\mathrm{HCO^{+}}$ absorption (red) and CO emission
(black) from the Galactic anticenter sample. For an explanation of the
appearance of the frequency-switched IRAM 30m emission spectra, see Appendix
B.
Spectra of $\mathrm{HCO^{+}}$ absorption and CO emission in the 28 directions
with detected $\mathrm{HCO^{+}}$ absorption are shown in Figure 10. The
$\mathrm{HCO^{+}}$ absorption is the negative-going signal shown in red. The
black histogram shows the CO emission. The frequency-switched IRAM 30m CO
spectra were delivered only after folding in frequency, preventing a
separation of the phases of the frequency-switching cycle using the methods of
Liszt (1997). Negative-going features in the CO spectrum are artifacts; only
the positive-going CO signal coincident with $\mathrm{HCO^{+}}$ absorption is
interstellar.
## Appendix C Area used to derive large-scale statistics
Figure 11: As in Figure 1, left, but showing the outlines of an area
containing the observed sightlines used to derive large-scale properties of
the anticenter region (see Section 4.3).
To derive the statistical distributions shown in Figure 8 and discussed in
Section 4.4, we drew a hull around the observed sightlines as illustrated in
Figure 11 and summed over the interior pixels.
|
# Functional Generalized Canonical Correlation Analysis for studying multiple
longitudinal variables
Lucas Sort Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des
Signaux et Systèmes, Gif-sur-Yvette, France Laurent Le Brusquet Université
Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, Gif-
sur-Yvette, France Arthur Tenenhaus Université Paris-Saclay, CNRS,
CentraleSupélec, Laboratoire des Signaux et Systèmes, Gif-sur-Yvette, France
###### Abstract
In this paper, we introduce Functional Generalized Canonical Correlation
Analysis (FGCCA), a new framework for exploring associations between multiple
random processes observed jointly. The framework is based on the multiblock
Regularized Generalized Canonical Correlation Analysis (RGCCA) framework. It
is robust to sparsely and irregularly observed data, making it applicable in
many settings. We establish the monotonic property of the solving procedure
and introduce a Bayesian approach for estimating canonical components. We
propose an extension of the framework that allows the integration of a
univariate or multivariate response into the analysis, paving the way for
predictive applications. We evaluate the method’s efficiency in simulation
studies and present a use case on a longitudinal dataset.
Keywords: Longitudinal data, Functional Data, Generalized Canonical
Correlation Analysis
## 1 Introduction
Measuring multiple biomarkers jointly over time is common in observational
studies and clinical trials. As they characterize various biological processes
which are often interdependent, those biomarkers are usually correlated.
Hence, separately analyzing those longitudinal variables may hide parts of the
biological mechanisms at stake and give redundant information. Furthermore, as
subjects often miss one or more visits, the biomarkers may be observed
sparsely and irregularly. Therefore, along with the complex time-dependent
continuous structure of the data, the statistical analysis of multiple
longitudinal variables requires using specific methodologies for efficiently
harvesting information and integrating the interaction between the variables.
In the multivariate setting, the analysis of data coming from multiple
sources, usually represented by multiple sets of variables, is often referred
to as ”multiblock” or ”multi-set” analysis. Canonical Correlation Analysis
(CCA) (Hotelling (1936)) is one of the most notorious approaches, but it is
limited to exploring associations between only two sets of variables.
Therefore, various methods were proposed to generalize the CCA problem to more
than two sets of variables (Horst (1961); Carroll (1968); Kettenring (1971)).
More recently, Tenenhaus and Tenenhaus (2011) introduced Regularized
Generalized Canonical Correlation Analysis (RGCCA), a regularized and more
flexible framework for studying multiple sets of variables, giving birth to a
new generation of methods (Tenenhaus et al. (2014, 2015); Singh et al.
(2019)). Various extensions of RGCCA were proposed to handle emerging data
types, such as tensor data (Girka et al. (2023)). However, most are still
limited to finite-dimensional Euclidian spaces and are not designed to handle
large amounts of missing observations.
In the longitudinal literature, approaches based on linear mixed-effects
models have been widely employed over the past decades for studying
longitudinal biomarkers (Rizopoulos (2011); German et al. (2021)). However,
for adapting multivariate data analysis methods to the longitudinal setting,
functional approaches have been preferred as functional spaces are easy to
handle and allow to describe the underlying smooth structure of the time-
dependent variables. In this context, adaptations of Principal Component
Analysis (PCA) to the longitudinal setting have flourished (Rice and Silverman
(1991)). Most notably, Yao et al. (2005) proposed an adaptation using a
covariance-based procedure and a Bayesian approach to estimate the principal
components. The method is thus robust to sparse and irregular data, making it
applicable to numerous problems.
Multiple extensions of CCA were proposed to explore associations between two
longitudinal variables (Leurgans et al. (1993); He et al. (2003); Zhou et al.
(2008); Shin and Lee (2015)). Regularization is crucial in this infinite-
dimensional context as CCA requires inverting covariance matrices. Inspired by
Yao et al. (2005) approach, Yang et al. (2011) introduced the Functional
Singular Value Decomposition (FSVD), which moves the CCA criterion to a
covariance criterion and also uses a Bayesian approach to estimate the
canonical components. However, as in the multivariate setting, those
adaptations are limited to a pair of longitudinal variables. Few methods can
go beyond this limitation. To our knowledge, Hwang et al. (2011) proposed the
first approach to find associations between any number of longitudinal
variables using a homogenous components criterion. More recently, Górecki et
al. (2020) proposed to adapt Horst (1961) approach to functional spaces using
basis decomposition. Although it is not designed to explore association among
several longitudinal variables, it is worth mentioning Multivariate Functional
Principal Component Analysis (MFPCA) (Happ and Greven (2018)), designed to
retrieve the principal modes of variation on multivariate longitudinal data.
The method can also handle sparse and irregular data.
In this context, we propose Functional Generalized Canonical Correlation
Analysis (FGCCA), a framework based on RGCCA that allows exploring and
studying associations between several longitudinal markers in a flexible way.
The method proposed is robust to sparse and irregular longitudinal data.
Furthermore, it is designed so it can integrate a multivariate block in the
analysis to perform, for instance, supervised learning. As RGCCA, the
framework provided by FGCCA is so vast that it encompasses many existing
methods, notably Yao et al. (2005) FPCA, FSVD, or Functional Partial Least
Squares (FPLS) as presented by Preda et al. (2007).
The paper is organized as follows. First, in Section 2.1, we recall the
Regularized Generalized Canonical Correlation Analysis (RGCCA) framework.
Then, in Section 2.2, we introduce the Functional Generalized Canonical
Correlation Analysis (FGCCA) context and optimization problem. We present the
solving procedure of FGCCA in Section 2.3 and introduce a scheme for
retrieving higher-order functions and estimating components in Section 2.4 and
2.5 respectively. We validate our method on simulation studies in Section 3.1
and propose an application to a real dataset in Section 3.2. Finally, our
approach’s limitations and possible extensions are discussed in Section 4.
Proofs of propositions are given in Supplementary Materials. The code used to
run the experiments and the R implementation of FGCCA are freely available on
github: https://github.com/Sort-L/FGCCA-Code.
## 2 Method
### 2.1 Regularized Generalized Canonical Correlation Analysis (RGCCA)
Regularized Generalized Canonical Correlation Analysis (RGCCA) (Tenenhaus et
al. (2017)) is an optimization and statistical framework for studying
associations between multiple sets of random variables. Denoting
$\operatorname{x}_{1},\dots,\operatorname{x}_{J}$ the sets of respectively
$p_{1},\dots,p_{J}$ random variables, and $\boldsymbol{\Sigma}_{jj^{\prime}}$
the $p_{j}\times p_{j^{\prime}}$ matrix of (cross-)covariance between
$\operatorname{x}_{j}$ and $\operatorname{x}_{j^{\prime}}$, the RGCCA
optimization problem can be expressed as:
$\operatorname*{argmax}_{\operatorname{a}_{1},\dots,\operatorname{a}_{J}\in\Omega_{1}\times\dots\times\Omega_{J}}\sum_{j\neq
j^{\prime}}c_{j,j^{\prime}}g(\operatorname{a}_{j}^{\top}\boldsymbol{\Sigma}_{jj^{\prime}}\operatorname{a}_{j^{\prime}})\\\
$ (1)
where $\Omega_{j}$ is defined as
$\Omega_{j}=\left\\{\operatorname{a}_{j}\in\mathbb{R}^{p_{j}}\mid\operatorname{a}_{j}^{\top}\boldsymbol{M}_{j}\operatorname{a}_{j}=1\right\\}$,
with $\boldsymbol{M}_{j}$ being a symmetric-positive matrix, $g$ is a convex
differentiable function and the matrix $C=(c_{j,j^{\prime}})$ is a $J\times J$
symmetric matrix with positive elements specifying the desired connection
design to study associations between the blocks. Classically we set
$c_{j,j^{\prime}}=1$ if we want to consider the interaction between the blocks
$j$ and $j^{\prime}$ and $c_{j,j^{\prime}}=0$ otherwise. Additionally, we
often consider
$\boldsymbol{M}_{j}=\tau_{j}\boldsymbol{I}_{p_{j}}+(1-\tau_{j})\boldsymbol{\Sigma}_{jj}$
with $\tau_{j}\in[0,1]$ to interpolate smoothly the criterion between a
correlation criterion, when $\tau_{j}=0$, and a covariance criterion, when
$\tau_{j}=1$. In this context, the goal of RGCCA is to retrieve block weight
vectors $\operatorname{a}_{j}$ giving block components
$\operatorname{y}_{j}=\operatorname{x}_{j}^{\top}\operatorname{a}_{j}$, which
are a compromise of the information from each set of variables and the
information shared with the other sets of variables.
Finally, two strategies are often used to compute higher-level weight vectors
for each set of variables. The first strategy leads to new weight vectors
associated with components uncorrelated to the previous ones. The second
strategy yields new weight vectors orthogonal to the previous ones. Both
strategies require transforming the original sets of variables
$\operatorname{x}_{j}$ into new sets of variables
$\operatorname{x}_{j}^{\prime}$ called ”deflated” vectors. The first
transformation, more often used, consists in regressing out from each set
$\operatorname{x}_{j}$ its associated component
$\operatorname{y}_{j}=\operatorname{x}_{j}^{\top}a_{j}$: the transformation
can be written
$\operatorname{x}_{j}^{\prime}=\operatorname{x}_{j}-(a_{j}^{\top}\boldsymbol{\Sigma}_{jj}a_{j})^{-1}\boldsymbol{\Sigma}_{jj}a_{j}a_{j}^{\top}\operatorname{x}_{j}$.
The second transformation consists in projecting each set
$\operatorname{x}_{j}$ onto the orthogonal of the space spanned by the
previous weight vectors: it is defined as
$\operatorname{x}_{j}^{\prime}=\operatorname{x}_{j}-\operatorname{a}_{j}\operatorname{a}_{j}^{\top}\operatorname{x}_{j}$.
To retrieve new weight vectors and new components with the desired properties,
the solving procedure is rerun, replacing original vectors
$\operatorname{x}_{j}$ with deflated vectors $\operatorname{x}_{j}^{\prime}$.
The transformation equations, often called ”deflation” equations, can be used
repeatedly to retrieve multiple weight vectors and components.
As demonstrated in Tenenhaus et al. (2017), the framework of RGCCA is very
general and subsumes many notorious data analysis methods such as Principal
Component Analysis (PCA), Canonical Correlation Analysis (CCA) (Hotelling
(1936)), Partial Least Squares (PLS) regression (Wold et al. (2001)), and
Generalized Canonical Correlation Analysis (GCCA) (Carroll (1968), Horst
(1961), Kettenring (1971)), to name a few. Many extensions and adaptations
have been proposed to tackle a wide variety of problems, but, to our
knowledge, none exists to integrate the time continuous structure of
longitudinal data and, especially, to handle highly sparse and irregular
observations.
### 2.2 Functional Generalized Canonical Correlation Analysis (FGCCA)
We now consider multiple time-dependent variables, such as longitudinal
biomarkers. We propose to adapt the previous framework to functional spaces,
and more precisely square-integrable random processes, since random processes
can represent the time continuous structure of the data. Our approach, named
Functional Generalized Canonical Correlation Analysis (FGCCA) is now
introduced.
#### 2.2.1 Notations and definitions
From now on, we consider $\operatorname{X}_{1},\dots,\operatorname{X}_{J}$,
$J$ square-integrable random processes defined on compact intervals of
$\mathbb{R}$, $I_{1},\dots,I_{J}$ respectively. Note that the random objects
are thus part of infinite-dimensional Hilbert spaces
$L^{2}(I_{1}),\dots,L^{2}(I_{J})$. In this context, we define for the process
$j$ the mean function $\mu_{j}$ as
$\mu_{j}(t)=\mathbb{E}(\operatorname{X}_{j}(t))$ for $t\in I_{j}$.
Additionally, the (cross-)covariance function (or ”surface”)
$\Sigma_{jj^{\prime}}$ between the processes $j$ and $j^{\prime}$ is defined
as
$\Sigma_{jj^{\prime}}(s,t)=\mathbb{E}((\operatorname{X}_{j}(s)-\mu_{j}(s))(\operatorname{X}_{j^{\prime}}(t)-\mu_{j^{\prime}}(t)))$
with $s\in I_{j}$ and $t\in I_{j^{\prime}}$. Finally, the (cross-)covariance
operator between the processes $j$ and $j^{\prime}$,
$\boldsymbol{\Sigma}_{jj^{\prime}}$, is defined as :
$\displaystyle\boldsymbol{\Sigma}_{jj^{\prime}}$
$\displaystyle:L^{2}(I_{j^{\prime}})\rightarrow L^{2}(I_{j}),\;f\mapsto
g,\;g(s)=\int_{I_{j^{\prime}}}\Sigma_{jj^{\prime}}(s,t)f(t)\text{dt}$
#### 2.2.2 Model
Following the optimization problem (1), defining RGCCA, we define Functional
Generalized Canonical Correlation Analysis (FGCCA) optimization problem by
moving from the multivariate setting of $\mathbb{R}^{p_{j}}$ spaces to the
functional setting of $L^{2}(I_{j})$ spaces, replacing the sets of variables
$\operatorname{x}_{j}$ by the random processes $\operatorname{X}_{j}$ and,
therefore, the euclidean dot product $a^{\top}b$ by the functional scalar
product defined by $\langle f,g\rangle_{L^{2}}=\int fg$. The FGCCA
optimization problem can therefore be written:
$\operatorname*{argmax}_{f_{1},\dots,f_{J}\in\Omega_{1}\times\dots\times\Omega_{J}}\sum_{j\neq
j^{\prime}}c_{j,j^{\prime}}g(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}})$ (2)
where $\Omega_{j}$ is defined as $\Omega_{j}=\left\\{f_{j}\in
L^{2}(I_{j})\mid\langle
f_{j},\boldsymbol{M}_{j}f_{j}\rangle_{L^{2}}=1\right\\}$ with
$\boldsymbol{M}_{j}$ being a symmetric positive-definite operator, and where
$g$ and $\mathbb{C}=(c_{j,j^{\prime}})$ are defined similarly as before.
Similarly to RGCCA, we suggest setting
$\boldsymbol{M}_{j}=\tau_{j}\boldsymbol{I}_{I_{j}}+(1-\tau_{j})\boldsymbol{\Sigma}_{jj}$.
However, to ensure the positive definiteness of the operator
$\boldsymbol{M}_{j}$, regularization parameters $\tau_{j}$ must be strictly
superior to $0$ (and thus, lie in $]0,1]$) as covariance operators
$\boldsymbol{\Sigma}_{jj}$ are not necessarily definite in the infinite-
dimensional setting. Functions $f_{j}$ and components
$\operatorname{y}_{j}=\langle\operatorname{X}_{j},f_{j},\rangle_{L^{2}}$,
allow capturing information for each process which, depending on the model
parameters, is a summary of both the information from each process and the
information shared with the others.
#### 2.2.3 (Cross-)Covariance estimation
In the multivariate setting, the most straightforward estimation for
(cross-)covariance matrices $\boldsymbol{\Sigma}_{jj^{\prime}}$ is the sample
covariance matrix, which is fast and easy to compute. However, in the
functional setting, it is usually preferable to use alternative strategies
that integrate the data’s time-continuous structure.
Various methods have been proposed over the past decades to integrate this
structure. In the functional data analysis literature, (cross-)covariance
operators $\boldsymbol{\Sigma}_{jj^{\prime}}$ are often discretized and
estimated on dense & regular grids using kernel smoothing methods (Yao et al.
(2005), Yang et al. (2011)) or Generative Additive Models (GAM), which are
easy to implement. In this context, estimation methods often have several
hyperparameters that must be set. Usually, those hyperparameters are manually
specified prior to the analysis. However, cross-validation procedures, such as
leave-one-out cross-validation or criterion-based procedures, can be used for
selecting them (Leurgans et al. (1993)). Additionally, due to approximation
errors, estimated operators are rarely positive in practice, making the choice
of the regularization parameters $\tau_{j}$ in FGCCA crucial, as the interval
they are defined on may not be clearly identified. Consequently, we advise
setting regularization parameters to $1$ as it will always prevent the
optimization problem from being ill-posed.
Finally, we recommend normalizing the different processes before estimating
(cross-)covariance operators. Like in the multivariate setting, considerable
differences in the variance of the processes may lead to biased results. For
this purpose, we suggest using the normalization presented in a similar
context by Happ and Greven (2018), enforcing the integrated variance to be the
same for each process. To achieve this, each process $j$ is multiplied by the
following normalization quantity:
$w_{j}=\left(\int_{{I_{j}}}\operatorname*{var}(\operatorname{X}_{j}(t))\text{dt}\right)^{-1/2}$
### 2.3 Resolution
We now introduce a procedure to retrieve solutions to the FGCCA optimization
problem. Convergence properties are given, ensuring the stability of the
solving procedure.
#### 2.3.1 Procedure
Let $\Psi$ be the objective function:
$\Psi(f_{1},\dots,f_{J})=\Psi(\boldsymbol{f})=\sum_{j\neq
j^{\prime}}c_{j,j^{\prime}}g(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}})$
As $\Psi$ is a function of multiple arguments, we suggest using a block
coordinate ascent (BCA) strategy (de Leeuw (1994)) for finding solutions to
the maximization problem. This strategy consists in maximizing $\Psi$ argument
by argument until convergence is reached. The properties of $g$ implies that
the objective function is differentiable and multi-convex, meaning that it is
convex with respect to each argument $f_{j}$ when all the others are fixed.
Note also that we consider here the ”functional differentiability” since
$\Psi$ has functional arguments. A proof of the definition of the gradient is
given in Supplementary Materials. From these properties we can derive the
following inequality for $\tilde{f}_{j}\in\Omega_{j}$:
$\displaystyle\Psi(f_{1},\dots,\tilde{f}_{j},$
$\displaystyle\dots,f_{J})\geq\Psi(\boldsymbol{f})+\langle\nabla_{j}\Psi(\boldsymbol{f}),\tilde{f}_{j}-f_{j}\rangle=m_{j}(\boldsymbol{f},\tilde{f}_{j})$
(3)
where $\nabla_{j}\Psi(\boldsymbol{f})$ is the functional partial derivative of
$\Psi$ with respect to the $j$th function. With this expression, we notice
that maximizing $\Psi$ for the $j$th argument can be achieved by maximizing
the minorizing function $m_{j}$. In this minorizing function, only the term
$\langle\nabla_{j}\Psi(\boldsymbol{f}),\tilde{f}_{j}\rangle$ is relevant since
all the others are fixed. Therefore, the maximum of $m_{j}$ under the
constraint that $\tilde{f}_{j}\in\Omega_{j}$ is reached for:
$\hat{f}_{j}=\operatorname*{argmax}_{\tilde{f}_{j}\in\Omega_{j}}\langle\nabla_{j}\Psi(\boldsymbol{f}),\tilde{f}_{j}\rangle=\frac{\boldsymbol{M}_{j}^{-1}\nabla_{j}\Psi(\boldsymbol{f})}{||\boldsymbol{M}_{j}^{-1/2}\nabla_{j}\Psi(\boldsymbol{f})||}:=r_{j}(\boldsymbol{f})$
(4)
where the partial derivative can be expressed as:
$\nabla_{j}\Psi(\boldsymbol{f})=2\sum_{\begin{subarray}{c}j^{\prime}=1\\\
j^{\prime}\neq j\end{subarray}}^{J}c_{j,j^{\prime}}g^{\prime}(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle)\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}$
(5)
From this, we propose the Algorithm (1) for retrieving solutions to
optimization problem (13).
Algorithm 1 FGCCA algorithm
Data: $(\boldsymbol{\Sigma}_{jj^{\prime}})_{1\leq j,j^{\prime}\leq
J},\,g,C,\epsilon,\boldsymbol{f}^{0}$
Result: $f_{1}^{s+1},\dots,f_{J}^{s+1}$
repeat
for _$j=1$ to $J$_ do
$\tilde{f}_{j}^{s+1}=\frac{\boldsymbol{M}_{j}^{-1}\nabla_{l}\Psi(f_{1}^{s+1},\dots,f_{j-1}^{s+1},f_{j}^{s},f_{j+1}^{s},\dots,f_{J}^{s})}{||\boldsymbol{M}_{j}^{-1/2}\nabla_{l}\Psi(f_{1}^{s+1},\dots,f_{j-1}^{s+1},f_{j}^{s},f_{j+1}^{s},\dots,f_{J}^{s})||}$
end for
$s=s+1$
until _$\Psi(f_{1}^{s+1},\dots,f_{J}^{s+1})-\Psi(f_{1}^{s},\dots,f_{J}^{s})
<\epsilon$_;
#### 2.3.2 Monotone convergence
Denoting $\Omega=\Omega_{1}\times\dots\times\Omega_{J}$ we define
$c_{j}:\Omega\rightarrow\Omega$ as the operator
$c_{j}(\boldsymbol{f})=(f_{1},\dots;f_{j-1},r_{j}(\boldsymbol{f}),f_{j+1},\dots,f_{J})$
with $r_{j}(\boldsymbol{f})$ being the update function for the $j$th function
of the solving procedure in Section 2.3.1. We also define
$c:\Omega\rightarrow\Omega$ as the operator $c=c_{J}\circ\dots\circ c_{1}$
We consider the sequence
$\\{\boldsymbol{f}^{s}=(f_{1}^{s},\dots,f_{J}^{s})\\}$ generated by
$\boldsymbol{f}^{s+1}=c(\boldsymbol{f}^{s})$. The following proposition states
the monotone convergence of the generated sequence
$\\{\boldsymbol{f}^{s}\\}^{\infty}_{s=0}$ and holds as long as the update
$r_{j}(\boldsymbol{f})$ exists, is unique and $\Omega$ is bounded:
###### Proposition 1.
Considering any sequence $\\{\boldsymbol{f}^{s}\\}_{s=0}^{\infty}$ generated
recursively by the relation $\boldsymbol{f}^{s+1}=c(\boldsymbol{f}^{s})$ with
$\boldsymbol{f}^{0}\in\Omega$. The sequence $\\{\Psi(\boldsymbol{f}^{s})\\}$
is monotonically increasing and therefore convergent as $\Psi$ is bounded on
$\Omega$, implying the convergence of the FGCCA algorithm.
Using this proposition, since $\Omega$ is bounded and $r_{j}(\boldsymbol{f})$
properly and uniquely defined for all $j$ with the functional gradient, we can
conclude that the Algorithm (1) is monotone and convergent.
### 2.4 Retrieving higher-order orthogonal functions
Solving the optimization problem as described previously only yields one
function per block. However, it is often preferable to retrieve multiple
functions leading to multiple components. For this purpose, we suggest, as in
the RGCCA framework, using a deflation strategy.
First, we propose considering a deflation strategy for retrieving orthogonal
vectors. As detailed in Section 2.1, the deflation equation associated with
this strategy is
$\operatorname{x}_{j}^{\prime}=\operatorname{x}_{j}-\operatorname{a}_{j}\operatorname{a}_{j}^{\top}\operatorname{x}_{j}$.
This expression may be unusable in the functional setting, with sparse and
irregular observations, as processes $\operatorname{X}_{j}$ are not fully
observed. Moreover, only the (cross-)covariance operators are involved in the
solving procedure, making the deflation of the blocks appear as an unnecessary
step in the procedure. Therefore, we establish the following proposition,
allowing us to deflate the (cross-)covariance operators directly without
involving the possibly ill-defined blocks:
###### Proposition 2.
Denoting $\boldsymbol{\Sigma}_{jj^{\prime}}^{\prime}$ the deflated
(cross-)covariance operator of $\boldsymbol{\Sigma}_{jj^{\prime}}$, obtained
after projecting the processes onto the orthogonal of the space spanned by
their associated vectors. The following equality holds:
$\displaystyle\boldsymbol{\Sigma}_{jj^{\prime}}^{\prime}$
$\displaystyle=(\boldsymbol{I}_{I_{j}}-\boldsymbol{\Phi}_{j})\boldsymbol{\Sigma}_{jj^{\prime}}(\boldsymbol{I}_{I_{j^{\prime}}}-\boldsymbol{\Phi}_{j^{\prime}})$
(6)
where $\boldsymbol{\Phi}_{j}:L^{2}(I_{j})\rightarrow L^{2}(I_{j})$ is the
operator defined by :
$(\boldsymbol{\Phi}_{j})(f)=(f_{j}\otimes f_{j})(f)=\langle f_{j},f\rangle
f_{j}$
To retrieve new orthogonal functions, deflated operators are plugged into the
optimization problem and Algorithm 1 is run again. The deflation procedure can
be repeated multiple times, allowing to retrieve a set of canonical functions
$\\{f_{j}^{m}\\}_{1\leq m\leq M}$ for each random process. The number of
canonical functions to retrieve is often manually set but it can be chosen
using cross-validation or criterion-based approaches. Finally, in this
context, the set of components $\\{\operatorname{y}_{j}^{m}\\}_{1\leq m\leq
M}$ obtained for each process can be directly estimated from the original
processes $\operatorname{X}_{j}$ without using the deflated processes
$\operatorname{X}_{j}^{m}$, a desirable property in our setting as it may be
difficult to compute and manipulate the deflated processes. Indeed, we can
demonstrate easily that
$\langle\operatorname{X}_{j},f_{j}^{m}\rangle_{L^{2}}=\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle_{L^{2}}=\operatorname{y}_{j}^{m}$,
where $\operatorname{X}_{j}^{m}$ stands for the $m$th deflation of process
$j$.
### 2.5 Estimating components
Computing the components
$\operatorname{y}_{j}^{m}=\langle\operatorname{X}_{j},f_{j}^{m}\rangle_{L^{2}}$
may be difficult in the sparse and irregular setting, as the numerical
estimation of the $L^{2}$ scalar product can be unstable and untracktable,
particularly if the number of observations is small. In this context, inspired
from Yao et al. (2005) and Yang et al. (2011), we propose to estimate the
components using a Bayesian approach.
#### 2.5.1 Notations
In the following, subscripts $i$, $j$, $k$ denote respectively the subject
number, the process number, and the observation number. We denote $n_{ij}$ the
number of observations,
$\mathbb{U}_{ij}=(U_{ij1},\dots,U_{ijn_{ij}})^{\top}\in\mathbb{R}^{n_{ij}\times
1}$ the observations, and $t_{ij}=(t_{ij1},\dots,t_{ijn_{ij}})$ the
observation time points. Finally the observations are modeled as:
$U_{ijk}=X_{ij}(t_{ijk})+\varepsilon_{ijk}$ (7)
where $X_{ij}$ is the realization for the subject $i$ of the random process
$\operatorname{X}_{j}$ and $\varepsilon_{ijk}$ is a measurement error. The
measurement errors are supposed i.i.d and following a normal distribution
$\mathcal{N}(0,\sigma_{j}^{2})$.
#### 2.5.2 Process modeling
As previously stated, considering the set of orthonormal canonical functions
$\\{f_{j}^{m}\\}_{1\leq m\leq M}$, each process $j$ from any subject $i$ can
be decomposed as:
$X_{ij}(t)=\mu_{j}(t)+\sum_{m=1}^{M}\xi_{ij}^{m}f^{m}_{j}(t)$ (8)
where the coefficients $\xi_{ij}^{m}$ are the basis coefficients associated
with the basis $\\{f_{j}^{m}\\}_{1\leq m\leq M}$. Therefore, at the sample
level we have:
$U_{ijk}=\mu_{j}(t_{ijk})+\sum_{m=1}^{M}\xi_{ij}^{m}f^{m}_{j}(t_{ijk})+\varepsilon_{ijk}$
(9)
This formulation allows to see the basis decomposition as a linear mixed-
effects model with the fixed-effects part being the mean term and the random-
effects part being the decomposition term. Moreover, as $\xi_{ij}^{m}=\langle
X_{ij},f_{j}^{m}\rangle_{L^{2}}$ and since the deflation strategy introduced
in Section 2.4 leads as previously stated to
$\langle\operatorname{X}_{j},f_{j}^{m}\rangle_{L^{2}}=\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle_{L^{2}}=\operatorname{y}_{j}^{m}$
we have that $\xi_{ij}^{m}=\operatorname{y}_{ij}^{m}$. Therefore, estimating
basis coefficients is equivalent to estimating components.
#### 2.5.3 Estimation
For simplifying expressions, denoting $N_{i}=\sum_{j}n_{ij}$, we write the
vector of observations
$\mathbb{U}_{i}=(\mathbb{U}_{i1}^{\top},\dots,\mathbb{U}_{iJ}^{\top})^{\top}\in\mathbb{R}^{N_{i}\times
1}$, and the mean function vector (at the observation time points)
$\boldsymbol{\mu}_{i}=(\boldsymbol{\mu}_{i,1}^{\top},\dots,\boldsymbol{\mu}_{i,J}^{\top})^{\top}\in\mathbb{R}^{N_{i}\times
1}$ with
$\boldsymbol{\mu}_{i,j}=(\mu_{i}(t_{ij1}),\dots,\mu_{i}(t_{ijn_{ij}}))^{\top}\in\mathbb{R}^{n_{ij}\times
1}$. We also write
$\mathbb{F}_{ij}^{m}=(f_{j}^{m}(t_{ij1}),\dots,f_{j}^{m}(t_{ijn_{ij}}))^{\top}\in\mathbb{R}^{n_{ij}\times
1}$ and
$\mathbb{F}_{ij}=(\mathbb{F}_{ij}^{1},\dots,\mathbb{F}_{ij}^{M})^{\top}\in\mathbb{R}^{n_{ij}\times
M}$, the matrix of the $M$ canonical functions at the observation time points
for subject $i$, process $j$, and finally
$\boldsymbol{\xi}_{j}=(\xi_{j}^{1},\dots,\xi_{j}^{M})^{\top}\in\mathbb{R}^{M\times
1}$,
$\boldsymbol{\xi}=(\boldsymbol{\xi}_{1}^{\top},\dots,\boldsymbol{\xi}_{J}^{\top})^{\top}\in\mathbb{R}^{MJ\times
1}$, the vector of basis coefficients. Considering that the basis coefficients
and the measurement errors are centered and jointly Gaussian, we establish the
following proposition, allowing estimating coefficients:
###### Proposition 3.
Denoting
$\mathbb{F}_{i}=\operatorname*{diag}(\mathbb{F}_{i1},\dots,\mathbb{F}_{iJ})$,
$\mathbb{\Sigma}=\mathbb{E}[\boldsymbol{\xi}\boldsymbol{\xi}^{\top}]$ and
$\boldsymbol{\sigma}_{i}=\operatorname*{diag}(\sigma_{1}^{2}\mathbb{I}_{n_{i1}},\dots,\sigma_{J}^{2}\mathbb{I}_{n_{iJ}})$,
the best linear unbiaised predictor (BLUP) for $\boldsymbol{\xi}_{i}$ is given
by
$\mathbb{E}(\boldsymbol{\xi}_{i}|\boldsymbol{U}_{i})=\mathbb{\Sigma}\mathbb{F}_{i}^{\top}(\mathbb{F}_{i}\mathbb{\Sigma}\mathbb{F}_{i}^{\top}+\boldsymbol{\sigma}_{i})^{-1}(\boldsymbol{U}_{i}-\boldsymbol{\mu}_{i})$
(10)
In this expression, all the terms can be estimated. Notably, the canonical
functions can be interpolated at the observation time points of each subject
allowing estimating matrices $\mathbb{F}_{i}$. The noise standard deviation
$\sigma_{j}$ can be approximated for each process using the estimated
covariance surface of each process (a procedure is presented in Yao et al.
(2005)). Finally, the mean functions $\boldsymbol{\mu}_{i}$ are usually
estimated when estimating (cross-)covariance surfaces, often using smoothing
techniques or GAMs.
### 2.6 Retrieving higher-order uncorrelated components
The deflation strategy introduced in Section 2.4 allows recovering multiple
orthogonal functions for each process. However, as introduced in section 2.1,
retrieving uncorrelated components is often preferable. In the multivariate
setting, this deflation strategy is carried out using the following deflation
equation
$\operatorname{x}_{j}^{\prime}=\operatorname{x}_{j}-(a_{j}^{\top}\boldsymbol{\Sigma}_{jj}a_{j})^{-1}\boldsymbol{\Sigma}_{jj}a_{j}a_{j}^{\top}\operatorname{x}_{j}$.
As previously, we propose to adapt this strategy in the functional setting to
deflate the (cross-)covariance operators directly. For this purpose, we
establish the following proposition:
###### Proposition 4.
Denoting $\boldsymbol{\Sigma}_{jj^{\prime}}^{\prime}$ the deflated
(cross-)covariance operator of $\boldsymbol{\Sigma}_{jj^{\prime}}$, obtained
from regressing out the components from their associated block. The following
equality holds :
$\displaystyle\boldsymbol{\Sigma}_{jj^{\prime}}^{\prime}$
$\displaystyle=(\boldsymbol{I}_{I_{j}}-d_{j}\boldsymbol{\Sigma}_{jj}\boldsymbol{\Phi}_{j})\boldsymbol{\Sigma}_{jj^{\prime}}(\boldsymbol{I}_{I_{j^{\prime}}}-d_{j^{\prime}}\boldsymbol{\Phi}_{j^{\prime}}\boldsymbol{\Sigma}_{j^{\prime}j^{\prime}})$
(11)
where $d_{j}=(\operatorname{y}_{j}^{\top}\operatorname{y}_{j})^{-1}$ and, as
previously, $\boldsymbol{\Phi}_{j}:L^{2}(I_{j})\rightarrow L^{2}(I_{i})$ is
the operator defined by :
$(\boldsymbol{\Phi}_{j})(f)=(f_{j}\otimes f_{j})(f)=\langle f_{j},f\rangle
f_{j}$
As previously, new functions associated with uncorrelated components can be
obtained by replacing (cross-)covariance operators in the optimization problem
by deflated ones and running the solving procedure. However, to obtain
uncorrelated estimates of $\operatorname{y}_{j}^{m}$, additional steps are
required. Indeed, the equality
$\langle\operatorname{X}_{j},f_{j}^{m}\rangle_{L^{2}}=\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle_{L^{2}}=\operatorname{y}_{j}^{m}$,
which was previously used to estimate the components in the mixed-effects
model, no longer holds in this setting. Nevertheless, the orthogonal property
of the retrieved functions is still holding by construction. Therefore,
canonical functions $f_{j}^{m}$ can still be used as a decomposition basis in
the mixed-effects framework presented previously. Furthermore, using deflation
equations, basis coefficients and components can be linked with the following
recursive equation:
$\operatorname{y}_{j}^{m+1}=\xi_{j}^{m+1}-\sum_{k=1}^{m}P_{j}^{k}\xi_{j}^{m+1}$
(12)
With
$P_{j}^{k}=({\operatorname{y}_{j}^{k}}^{\top}\operatorname{y}_{j}^{k})^{-1}\operatorname{y}_{j}^{k}{\operatorname{y}_{j}^{k}}^{\top}$
being the projection matrix from the regression of $\xi_{j}^{k}$ on
$\operatorname{y}_{j}^{k}$, starting with
$\xi_{j}^{1}=\operatorname{y}_{j}^{1}$. This equation can be seen as a
decorrelation procedure: for each process, each new basis coefficient estimate
is decorrelated from the previous components.
Finally, the choice of the deflation type depends on the desired usage of the
components and the canonical functions. For reconstructing trajectories,
orthogonal functions deflation may be more adapted as the orthonormal
functions retrieved are preferable for decomposition purposes. For doing
clustering or dimension reduction for further analysis, such as regression,
using uncorrelated components seem more suitable.
### 2.7 Integrating a multivariate response
Inspired from the PLS-framework, we propose to modify slightly the FGCCA
optimization problem to include a multivariate response
$\operatorname{Y}\in\mathbb{R}^{p}$ with $p\in\mathbb{N}^{*}$ for expanding
the possibilities given by the framework:
$\operatorname*{argmax}_{\begin{subarray}{c}f_{1},\dots,f_{J},\in\Omega_{1}\times\dots\times\Omega_{J}\\\
||a||_{2}=1\end{subarray}}\sum_{j\neq j^{\prime}}c_{j,j^{\prime}}g(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}})+2\sum_{j}g(\langle
f_{j},\boldsymbol{\Sigma}_{j\operatorname{Y}}a\rangle_{L^{2}})$ (13)
where $\boldsymbol{\Sigma}_{j\operatorname{Y}}$ is the cross-covariance
operator between the process $j$ and the response $\operatorname{Y}$, defined
as:
$\displaystyle\boldsymbol{\Sigma}_{j\operatorname{Y}}$
$\displaystyle:\mathbb{R}^{p}\rightarrow L^{2}(I_{j}),\;a\mapsto
g,\;g(s)=\sum_{i=1}^{p}\mathbb{E}[\operatorname{X}_{j}(t)\operatorname{Y}_{i}]a_{i}$
As previously, this operator can be estimated with kernel smoothing methods or
GAMs. This additional interaction allows recovering canonical functions for
each process that explains best the process, interaction with the others
(depending on the design matrix $C$) and interaction with the response. This
design is particularly relevant in a predictive framework as it could be
interesting to use the components retrieved to predict the response
$\operatorname{Y}$.
Finally, note that the various procedures presented previously are not
significantly affected by this change and can be easily rewritten to integrate
the multivariate vector using the multivariate-functional cross-covariance
operator presented above.
## 3 Results
### 3.1 Simulations studies
#### 3.1.1 Simulation 1: validating the Bayesian approach.
For $J=3$ processes, we propose to compare the components retrieved using the
Bayesian approach previously described in Section (2.5) to the components
obtained by computing the scalar product, as usually done. For this purpose,
we generate data according to Equation (8) so that true component values are
known. We choose the first $M=6$ Fourier basis functions in the $[0,1]$
interval as our orthonormal basis. The components $\xi_{ij}^{m}$ for each
subject are generated jointly with a centered Gaussian distribution of
covariance $\Sigma$ having a decreasing variance structure. For each subject
and each process, the time points are generated by sparsifying a grid of size
$50$ in the $[0,1]$ interval. Various sparsity levels are compared: Dense (100
% of observations retained), Low Sparsity (100 % to 80 % of observations
retained), Medium Sparsity (80 % to 40 % of observations retained), and High
Sparsity (40 % to 10 % of observations retained). We compare the two
approaches by computing the mean squared error on the canonical components for
100 simulations. Results are reported in Figure 1.
We observe a clear advantage of the Bayesian approach in the Medium and in the
High sparsity settings. The gap between the two approaches increases for
higher-order components. Furthermore, we notice a slight advantage of the
Bayesian approach in the Dense and in the Low sparsity cases, again,
especially for the higher-order components. Therefore, we can conclude that
the estimation error when using the Bayesian approach is smaller than the
estimation error due to the numerical approximation of the integral when using
the standard scalar product approach. Thus, we advise using the Bayesian
approach in all cases as it is also computationally inexpensive.
Figure 1: (top) Mean squared error (MSE) boxplots of the components
$\operatorname{y}_{j}^{m}$ estimated with the Bayesian approach or the scalar
product (integration) for $m=1,2,3,4,5,6$ obtained from $100$ simulations with
$N=100$ and averaged over the $J=3$ processes, $\sigma^{2}=1$ and various
sparsity settings.
#### 3.1.2 Simulation 2: comparing results
To validate FGCCA further, we suggest to compare it to some of its subsumed
methods. We propose to consider two other approaches for $J=2$ processes:
PACE-based Functional Principal Component Analysis (FPCA) (Yao et al. (2005))
and Functional Singular Value Decomposition (FSVD) (Yang et al. (2011)). Both
approaches can handle sparse and irregular data. The first gives similar
results to FGCCA for specific component’s covariance designs. The latter is
based on an optimization problem equivalent to FGCCA in the 2 processes
setting when regularization parameters $\tau_{j}$ are set to 1. The simulation
setting previously described is used again to generate data. The covariance
matrix $\Sigma$ is designed so FPCA, FSVD and FGCCA recover components and
functions in the same order. Results are reported on Figure 2.
As FGCCA and FSVD are supposed to retrieve similar canonical functions, we
observe similar distributions over the mean squared errors for functions.
Components estimation rely on a slightly different formula for FSVD. Indeed,
the singular values retrieved from the analysis are used, which should, in
theory, increase estimation accuracy. However, it seems that in our case, the
estimations obtained from a FSVD are significantly worse than with FGCCA,
especially for high and medium sparsity settings and for high-order functions.
On another hand, FPCA gives slightly better function estimations both for
functions and components, especially in the sparse setting and for the first
functions and components. For high order components, FGCCA sometimes
outperforms FPCA. Additional results along with simulation details are
presented in Supplementary Materials.
Figure 2: Mean squared errors (MSE) of functions $f^{m}$ (top) and components
$\xi^{m}$ (bottom) for $m=1,2,3,4,5,6$ obtained from $100$ simulations with
$N=100$, $\sigma^{2}=1$ and various sparsity settings. Comparison between
FPCA, FSVD and FGCCA.
#### 3.1.3 Simulation 3 : comparing reconstruction
Alternatively, we propose to compare the estimation quality of the
reconstructed trajectories between FGCCA with a fully-connected design and an
orthogonal deflation, and Multivariate Functional Principal Component Analysis
(MFPCA) (Happ and Greven (2018)). The reconstructed trajectories are obtained
for FGCCA using the decomposition equation (8) with the estimated canonical
functions and components, and for MFPCA using the multi-dimensional Karhunen-
Loeve decomposition, presented in the aforementioned paper. This time, using
the package funData (Happ-Kurz (2020)), 3 processes are generated based on the
first $M=6$ Fourier basis functions and with a linear decreasing variance over
the components.
The mean squared relative error is compared over 100 simulations in various
configurations. Results are presented on Figure 3. We can see that our
approach improves slightly the reconstruction of the processes, especially
when the number of subject $N$ is not too small. The gap between the two
methods appears to be stable among all sparsity settings. Further results,
available in the Supplementary Materials, show that the difference is bigger
as the noise $\sigma^{2}$ is smaller.
Figure 3: Mean relative squared errors (MRSE) of reconstructed trajectories,
using estimated canonical functions and components. Comparing FGCCA and MFPCA
with $M=6$, $\sigma^{2}=1$, for various number of subjects $N$ and various
sparsity settings. Statistical significance displayed : (***) $p<0.001$ (****)
$p<0.0001$
### 3.2 Application to Primary Biliary Cirrhosis dataset
The Primary Biliary Cirrhosis dataset (Murtaugh et al. (1994)) is a dataset
from Mayo Clinic containing the follow-up of various biomarkers extracted from
blood analyses of 312 patients who have been diagnosed with primary biliary
cirrhosis of the liver, a rare autoimmune disease. We propose to use this
multi-biomarker dataset to show various usages of FGCCA. For this purpose,
three biomarkers were considered: albumin, bilirubin, which is log
transformed, and prothrombin time, observed up to 10 years after the first
visit. Those biomarkers were chosen as they have been proven to be good
predictors of patient outcomes. Figure 4 represents an aggregated view of the
data.
Figure 4: Longitudinal trajectories for albumin, bilirubin and prothrombin
time in the pbc2 dataset for all individuals. As usually done, the bilirubin
marker is log transformed.
#### 3.2.1 Exploratory analysis
We first propose visualizing the canonical functions and components obtained
with FGCCA when using a fully connected design and deflation leading to
uncorrelated components. We compare the results to the principal functions and
components obtained using PACE-based FPCA (Yao et al. (2005)). For both
methods, the bandwidths are manually set to 1 for interpretability. The
results are displayed in Figure 5.
The first principal and canonical functions have a similar flat shape,
implying that the difference in the trajectories between subjects for all
biomarkers comes primarily from an overall shift around the mean. On the other
hand, the second canonical and principal functions are either decreasing or
increasing during the 10 years interval, indicating that the next source of
variation between subjects thus comes from the monotonicity of the
trajectories: for the different biomarkers, subjects have either an increasing
or decreasing trend. Additionally, we notice a more significant difference
between principal and canonical functions for prothrombin, suggesting that
this biomarker is particularly correlated to the others. The differences are,
however, difficult to interpret. Additionally, note that the functions
retrieved with FGCCA are slightly smoother, notably at the end of the
interval. We can explain this by the border effects when estimating the
(cross-)covariance operators. Indeed, for FGCCA, the functions are estimated
using information from multiple processes and not just one (as done in FPCA),
leading to more stable and reliable results.
Component plots allow us to see the differences between the two approaches
more clearly. First, we notice that the components are spread more evenly for
FGCCA, especially for prothrombin, thanks to the normalization. For the three
biomarkers, the components given by FGCCA seem to separate better the two
outcomes. This property will be confirmed in the predictive analysis.
Figure 5: (top) First 3 functional modes retrieved by FGCCA (canonical
functions) and FPCA (principal functions), for each biomarker. Functions were
flipped to minimize the differences between the two methods. (bottom) Biplots,
for each biomarker, of the first 2 components obtained with FGCCA and FPCA
coloured by final status. Ellipses represent the estimated Gaussian
distributions of the components for each outcome.
#### 3.2.2 Prediction
Inspired by Singh et al. (2019), we propose using a multiblock functional PLS
framework to predict each patient’s outcome and demonstrate the ability of
FGCCA to integrate a multivariate or univariate response. The multiblock
functional PLS design is defined as a FGCCA design where only the associations
with the response are considered in the problem. It allows to recover
biomarker information correlated to the response, which is particularly useful
for predictive purposes. A simple logistic regression model is fitted per
biomarker to predict the response using the first component retrieved with
FGCCA. For a new subject, the components are predicted using the observed
trajectories. The final prediction is obtained by computing a weighted average
of the predicted outcomes, where each biomarker prediction is weighted by the
correlation of its component with the response. The predictive performances
obtained are compared to a similar model where the principal components from
FPCA are used instead.
The results, summarized in Figure 6, show that the FGCCA-based components
provide a significantly better outcome estimation. The apparent difference in
the canonical and principal functions implies that FGCCA has indeed retrieved
components highlighting the association with the outcome for each biomarker.
As the canonical functions have a monotonic trend, we can argue that survival
is mainly associated with a decreasing or increasing behavior of the various
biomarkers. More precisely, mortality seems to be associated with an increase
in bilirubin, prothrombin and a decrease in albumin. It is consistent with the
fact that a decreasing albumin or increasing bilirubin and prothrombin are
usually associated with a bad prognosis of the liver.
Figure 6: (left) First principal/canonical function retrieved with FPCA/FGCCA.
For FGCCA a multiblock-FPLS design is used, integrating only the interaction
between each biomarker and the response. Functions are flipped to ensure that
the first component is positively correlated to the outcome, to improve
interpretability. (right) Boxplot of the balanced accuracy computed on the
test set for 100 monte-carlo runs. p-value and significance level of the
difference between the two distributions (t-test) are given.
#### 3.2.3 Reconstruction
Finally, we propose to evaluate the ability of FGCCA to reconstruct
trajectories and predict biomarker values at unobserved times. To this end, we
propose dividing the data into training and test datasets. The training
dataset is used to estimate (cross-)covariance operators and to run FGCCA. The
test dataset contains trajectories on which we seek to predict the last
observation, which has been removed. Each subject’s prediction is made from
reconstructed trajectories computed from previously obtained canonical
functions (from the training dataset) and components estimated using data
before the last observation. We propose to compare the results to an FPCA-
based approach, where the components and functions are estimated from an FPCA.
Furthermore, to evaluate the robustness of the two approaches, we sparsify the
test trajectories at various levels. The results are reported in Figure 7.
We observe a significant advantage of the FGCCA-based approach over the FPCA-
based approach except for bilirubin in the (1M) scenario, asserting that FGCCA
has integrated additional or more stable knowledge. We note that the gap
widens as the sparsity increases. These results pave the way to joint
modeling, as the components could be used both to predict the future
trajectories of biomarkers, as it is done here, and the survival of subjects.
This promising application is currently investigated.
Figure 7: (left) Reconstruction obtained with FGCCA and FPCA in 3 scenarios :
(1M) last observation removed, (2M) two last observations removed (3M) three
last observations removed. Crosses correspond to observations used to estimate
the components, circles correspond to observations we aim to predict. (right)
Boxplots of last observation prediction mean squared error obtained on 100
runs. Statistical significance displayed : (*) $p<0.05$ (****) $p<0.0001$
## 4 Discussion
We introduced Functional Generalized Canonical Correlation Analysis (FGCCA), a
flexible framework for exploring associations among multiple longitudinal
variables, by finding the main joint modes of variation. The method relies on
a monotone and globally convergent algorithm, which only requires
(cross-)covariance operators. We proposed a Bayesian approach for estimating
the components. Consequently, the method is robust to irregular and sparse
data, making it applicable to numerous settings. In addition, we allow
integrating a multivariate response in the analysis by slightly modifying the
optimization problem, paving the way to mixed data uses. Simulation studies
assess the validity of our approach and its underlying design. A wide variety
of usages are presented in the application.
As previously mentioned, the method relies significantly on the estimations of
(cross-)covariance operators. Therefore, studying more in-depth those
estimation procedures could be interesting as new methods have been proposed
recently (Xiao et al. (2018)). Computing confidence bands for the estimated
scores as it is done in sparse and irregular FPCA (Yao et al. (2005)) is
investigated. However, difficulties arise since the FGCCA algorithm does not
have a closed-form solution. Finally, an implementation allowing the user to
change the regularization parameter was developed. In this context, analyzing
the impact of the parameter on the algorithm and the results obtained could be
further investigated.
In numerous studies, longitudinal variables can be grouped in blocks
representing different modalities. For instance, in imaging genetics, multiple
longitudinal variables representing the evolution of several neuroimaging
features can be observed along multiple genetic features. In this context,
considering a block for each variable, as done with FGCCA, may be inefficient
as it would require a complex design and intensive computational resources.
Another approach, which is currently being investigated, would be to integrate
the multiple longitudinal variables in blocks. This approach was used by Happ
and Greven (2018) and can be referred to as multivariate functional data
modeling. Inspired by the multi-way/tensor literature, a reduced rank model
could also significantly help reduce the problem’s complexity. In this
context, numerous works have been proposed, notably for tensor regression
(Zhou et al. (2008)) and, as evoked in the introduction, for RGCCA (Girka et
al. (2023)).
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## Appendix A Proofs
### A.1 Definition of the functional gradient
Let $\Psi$ be the objective function:
$\Psi(f_{1},\dots,f_{J})=\Psi(\boldsymbol{f})=\sum_{j\neq
j^{\prime}}c_{j,j^{\prime}}g(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}})$
Let $j\in\\{1,\ldots,J\\}$, $f_{j^{\prime}}\,\forall j^{\prime}\neq j$. Let
$\begin{array}[]{lcl}\phi_{j}:L^{2}(I_{j})&\to&\mathbb{R}\\\
f_{j}&\mapsto&\Psi(f_{1},\ldots,f_{j},\ldots,f_{J})\\\ \end{array}$
Proposition. $\phi_{j}$ is Gâteaux-differentiable and its gradient is:
$\nabla_{j}\phi_{j}(f_{j})=2\sum_{j^{\prime}\neq
j}c_{j,j^{\prime}}g^{\prime}(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle)\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}$
Proof. Let $f_{j}\in L^{2}(I_{j})$. Let’s show that it exists a continuous
(bounded) linear operator $\phi_{j}^{\prime}(f_{j})$ such that
$\forall h\in
L^{2}(I_{j}),\,\phi_{j}^{\prime}(f_{j})h=\lim_{\begin{subarray}{c}\alpha\to
0\\\ \alpha>0\end{subarray}}\dfrac{\phi_{j}(f_{j}+\alpha
h)-\phi_{j}(f_{j})}{\alpha}$ (14)
First notice that according to Fubini theorem, expression of $\phi_{j}$ may be
simplified:
$\begin{array}[]{lcl}\phi_{j}(f_{j})&=&\displaystyle\sum_{j^{\prime}\neq
j}c_{j,j^{\prime}}g(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}})+\sum_{j^{\prime}\neq
j}c_{j^{\prime},j}g(\langle
f_{j}^{\prime},\boldsymbol{\Sigma}_{j^{\prime}j}f_{j}\rangle_{L^{2}})\\\
&=&\displaystyle 2\sum_{j^{\prime}\neq j}c_{j,j^{\prime}}g(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}})\\\
\end{array}$
Then, let’s show that the Gâteaux-differential of $\phi_{j}$ at $f_{j}$ in the
direction $h$ is defined by studying the limit in (14):
$\begin{array}[]{lcl}\dfrac{\phi_{j}(f_{j}+\alpha
h)-\phi_{j}(f_{j})}{\alpha}&=&2\displaystyle\sum_{j^{\prime}\neq
j}c_{j,j^{\prime}}\dfrac{g(\langle f_{j}+\alpha
h,\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}})-g(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}})}{\alpha}\\\
&=&2\displaystyle\sum_{j^{\prime}\neq j}c_{j,j^{\prime}}g^{\prime}(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}})\langle
h,\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}}+o(\alpha)\\\
\end{array}$
Thus, limit in (14) exists and leads to a linear operator:
$\phi_{j}^{\prime}(f_{j})h=\left\langle h,2\displaystyle\sum_{j^{\prime}\neq
j}c_{j,j^{\prime}}g^{\prime}(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}})\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\right\rangle_{L^{2}}$
Applying Cauchy-Schwarz and the monotonicity of $g^{\prime}$, it comes that
$\phi_{j}^{\prime}(f_{j})$ is bounded (and thus continuous) and so that
$\phi_{j}(f_{j})$ is Gâteaux-differentiable at $f_{j}$. By definition of the
gradient:
$\nabla_{j}\phi_{j}(f_{j})=2\sum_{j^{\prime}\neq
j}c_{j,j^{\prime}}g^{\prime}\left(\langle
f_{j},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}\rangle_{L^{2}}\right)\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}$
### A.2 Monotone convergence of the solving procedure (Proposition 1)
The proof of Proposition 1 is based on the proof given in Tenenhaus et al.
[2017]. In the following, we use the notations of the paper. We consider the
following lemma:
###### Lemma 1.
Consider the set $\Omega=\Omega_{1}\times\dots\times\Omega_{J}$, the function
$\Psi:\Omega\rightarrow\mathbb{R}$ and the operator
$c:\Omega\rightarrow\Omega$ defined in the paper. The following properties
hold:
1. (a)
$\Omega$ is a bounded set
2. (b)
$c$ is a continuous operator
3. (c)
$\Psi(\boldsymbol{f})\leq\Psi(c(\boldsymbol{f}))$ for any
$\boldsymbol{f}\in\Omega$
Proof of lemma 1 :
(a) $\Omega$ is bounded as it is the Cartesian product of $J$ bounded sets
(b) As $\Psi$ is a continuous differentiable operator, $r_{j}$ is continuous.
Thus, $c_{j}$ is continuous and $c=c_{1}\circ\dots c_{J}$ being the
composition of $J$ continuous operators is also continuous
(c) Let
$\operatorname*{\boldsymbol{f}}=({\operatorname*{\boldsymbol{f}}}_{1},\dots,{\operatorname*{\boldsymbol{f}}}_{J})\in\Omega$.
We want to find an update
$\hat{\operatorname*{\boldsymbol{f}}}_{j}\in\Omega_{j}$ of
${\operatorname*{\boldsymbol{f}}}_{j}$ such that
$\Psi(\operatorname*{\boldsymbol{f}})\leq\Psi({\operatorname*{\boldsymbol{f}}}_{1},\dots,{\operatorname*{\boldsymbol{f}}}_{j-1},\hat{\operatorname*{\boldsymbol{f}}}_{j},{\operatorname*{\boldsymbol{f}}}_{j+1},\dots,{\operatorname*{\boldsymbol{f}}}_{J})$.
For that purpose we use the fact that a convex function lies above its linear
approximation. Thus at ${\operatorname*{\boldsymbol{f}}}_{j}$ and for any
$\tilde{\operatorname*{\boldsymbol{f}}}_{j}\in\Omega_{j}$ we have
$\Psi({\operatorname*{\boldsymbol{f}}}_{1},\dots,{\operatorname*{\boldsymbol{f}}}_{j-1},\tilde{\operatorname*{\boldsymbol{f}}}_{j},{\operatorname*{\boldsymbol{f}}}_{j+1},\dots,{\operatorname*{\boldsymbol{f}}}_{J})\geq\Psi(\operatorname*{\boldsymbol{f}})+\langle\nabla_{j}\Psi(\operatorname*{\boldsymbol{f}}),\tilde{\operatorname*{\boldsymbol{f}}}_{j}-{\operatorname*{\boldsymbol{f}}}_{j}\rangle=l_{j}(\tilde{\operatorname*{\boldsymbol{f}}}_{j},\operatorname*{\boldsymbol{f}})$
Using the Cauchy-Schwartz inequality we obtain the unique maximizer
$\hat{\operatorname*{\boldsymbol{f}}}_{j}\in\Omega_{j}$ of
$l_{j}(\tilde{\operatorname*{\boldsymbol{f}}}_{j},\operatorname*{\boldsymbol{f}})$
w.r.t. $\tilde{\operatorname*{\boldsymbol{f}}}_{j}\in\Omega_{j}$
$\hat{\operatorname*{\boldsymbol{f}}}_{j}=\operatorname*{argmax}_{\tilde{\operatorname*{\boldsymbol{f}}}_{j}\in\Omega_{j}}l_{j}(\tilde{\operatorname*{\boldsymbol{f}}}_{j},\operatorname*{\boldsymbol{f}})=\frac{\nabla_{j}\Psi(\operatorname*{\boldsymbol{f}})}{||\nabla_{j}\Psi(\operatorname*{\boldsymbol{f}})||}=r_{j}(\operatorname*{\boldsymbol{f}})$
Thus we have,
$\Psi(\operatorname*{\boldsymbol{f}})=l_{j}({\operatorname*{\boldsymbol{f}}}_{j},\operatorname*{\boldsymbol{f}})\leq
l_{j}(r_{j}(\operatorname*{\boldsymbol{f}}),\operatorname*{\boldsymbol{f}})\leq\Psi({\operatorname*{\boldsymbol{f}}}_{1},\dots,{\operatorname*{\boldsymbol{f}}}_{j-1},r_{j}(\operatorname*{\boldsymbol{f}}),{\operatorname*{\boldsymbol{f}}}_{j+1},\dots,{\operatorname*{\boldsymbol{f}}}_{J})=\Psi(c_{j}(\operatorname*{\boldsymbol{f}}))$
(15)
And also,
$\Psi(c_{j-1}\circ\dots\circ
c_{1}(\operatorname*{\boldsymbol{f}}))\leq\Psi(c_{j}\circ\dots\circ
c_{1}(\operatorname*{\boldsymbol{f}}))$
Leading to the desired inequality for any
$\operatorname*{\boldsymbol{f}}\in\Omega$:
$\Psi(\operatorname*{\boldsymbol{f}})\leq\Psi(c_{1}(\operatorname*{\boldsymbol{f}}))\leq\Psi(c_{2}\circ
c_{1}(\operatorname*{\boldsymbol{f}}))\leq\dots\leq\Psi(c_{J}\circ\dots\circ
c_{1}(\operatorname*{\boldsymbol{f}}))=\Psi(c(\operatorname*{\boldsymbol{f}}))$
Now, using Lemma 1, we can prove Proposition 1. Indeed, Point (c) of Lemma 1
implies that the sequence $\\{\Psi(\operatorname*{\boldsymbol{f}}^{s})\\}$ is
monotonically increasing and, since $\Psi$ is bounded on $\Omega$ (Cauchy-
Schwartz), it is convergent.
### A.3 Deflation equations (Proposition 2 and 4)
To retrieve orthogonal functions, we project each process
$\operatorname{X}_{j}$ onto the orthogonal of the space spanned by its
associated normalized canonical function $f_{j}$. The deflation equation
associated to this transformation, giving the transformed process
$\operatorname{X}_{j}^{\prime}$, is:
$\operatorname{X}_{j}^{\prime}=\operatorname{X}_{j}-\langle
f_{j},\operatorname{X}_{j}\rangle f_{j}$
Denoting $\boldsymbol{F}_{j}=f_{j}\otimes f_{j}$ where $\otimes$ denotes the
functional tensor product, defined as $(f\otimes g)(h)=\langle g,h\rangle f$,
we can rewrite the previous expression:
$\operatorname{X}_{j}^{\prime}=(\boldsymbol{I}_{I_{j}}-\boldsymbol{F}_{j})(\operatorname{X}_{j})$
Now, we consider the deflated cross-covariance between deflated processes
$\operatorname{X}_{j}^{\prime}$ and $\operatorname{X}_{j^{\prime}}^{\prime}$:
$\displaystyle(\boldsymbol{\Sigma}_{jj^{\prime}}^{\prime}f)(s)$
$\displaystyle=\int_{I_{j^{\prime}}}\Sigma_{jj^{\prime}}^{\prime}(s,t)f(t)$
$\displaystyle=\int_{I_{j^{\prime}}}\mathbb{E}[\operatorname{X}_{j}^{\prime}(s)\operatorname{X}_{j^{\prime}}^{\prime}(t)]f(t)$
$\displaystyle=\mathbb{E}[\int_{I_{j^{\prime}}}\operatorname{X}_{j}^{\prime}(s)\operatorname{X}_{j^{\prime}}^{\prime}(t)f(t)]$
$\displaystyle=\mathbb{E}[\langle\operatorname{X}_{j^{\prime}}^{\prime},f\rangle\operatorname{X}_{j}^{\prime}(s)]$
Allowing us to write :
$\displaystyle\boldsymbol{\Sigma}_{jj^{\prime}}^{\prime}f$
$\displaystyle=\mathbb{E}[\langle\operatorname{X}_{j^{\prime}}^{\prime},f\rangle\operatorname{X}_{j}^{\prime}]$
$\displaystyle=\mathbb{E}[\langle(\boldsymbol{I}_{I_{j^{\prime}}}-\boldsymbol{F}_{j^{\prime}})(\operatorname{X}_{j^{\prime}}),f\rangle(\boldsymbol{I}_{I_{j}}-\boldsymbol{F}_{j})(\operatorname{X}_{j})]$
$\displaystyle=(\boldsymbol{I}_{I_{j}}-\boldsymbol{F}_{j})\mathbb{E}[\langle\operatorname{X}_{j^{\prime}},(\boldsymbol{I}_{I_{j^{\prime}}}-\boldsymbol{F}_{j^{\prime}})(f)\rangle\operatorname{X}_{j}]$
$\displaystyle=(\boldsymbol{I}_{I_{j}}-\boldsymbol{F}_{j})\boldsymbol{\Sigma}_{jj^{\prime}}(\boldsymbol{I}_{I_{j^{\prime}}}-\boldsymbol{F}_{j^{\prime}})(f)$
Leading to desired expression of the cross covariance operators:
$\boldsymbol{\Sigma}_{jj^{\prime}}^{\prime}=(\boldsymbol{I}_{I_{j}}-\boldsymbol{F}_{j})\boldsymbol{\Sigma}_{jj^{\prime}}(\boldsymbol{I}_{I_{j^{\prime}}}-\boldsymbol{F}_{j^{\prime}})$
For retrieving uncorrelated components, $\operatorname{X}_{j}^{\prime}$ is
defined by regressing out from the process the previously retrieved components
$\operatorname{y}_{j}$. The deflation process can be written as:
$\operatorname{X}_{j}^{\prime}=(\boldsymbol{I}_{I_{j}}-d_{j}\boldsymbol{\Sigma}_{jj}\boldsymbol{F}_{j})(\operatorname{X}_{j})$
(16)
where $d_{j}=\langle f_{j},\boldsymbol{\Sigma}_{jj}f_{j}\rangle^{-1}$.
Following the same steps as before we obtain
$\boldsymbol{\Sigma}_{jj^{\prime}}^{\prime}=(\boldsymbol{I}_{I_{j}}-d_{j}\boldsymbol{\Sigma}_{jj}\boldsymbol{F}_{j})\boldsymbol{\Sigma}_{jj^{\prime}}(\boldsymbol{I}_{I_{j^{\prime}}}-d_{j^{\prime}}\boldsymbol{F}_{j^{\prime}}\boldsymbol{\Sigma}_{j^{\prime}j^{\prime}})$
### A.4 Components conditional expectation (Proposition 3)
In the following we denote
$\boldsymbol{\xi}_{ij}=(\xi_{ij}^{1},\dots,\xi_{ij}^{M})^{\top}$,
$\boldsymbol{\epsilon}_{ij}=(\epsilon_{ij1},\dots,\epsilon_{ijn_{ij}})^{\top}$,
$\boldsymbol{\xi}_{i}=(\boldsymbol{\xi}_{i1}^{\top},\dots,\boldsymbol{\xi}_{iJ}^{\top})^{\top}$
and
$\boldsymbol{\epsilon}_{i}=(\boldsymbol{\epsilon}_{i1}^{\top},\dots,\boldsymbol{\epsilon}_{iJ}^{\top})^{\top}$.
We propose to write the vector of observations for the process $j$ and subject
$i$ with matrix notations:
$\boldsymbol{U}_{ij}=\boldsymbol{\mu}_{ij}+\mathbb{F}_{ij}\boldsymbol{\xi}_{ij}+\boldsymbol{\epsilon}_{ij}$
With $\mathbb{F}_{ij}$ corresponding to a matrix whose columns are the
canonical functions at the observed time points. Denoting
$\mathbb{F}_{i}=\operatorname*{diag}(\mathbb{F}_{i1},\dots,\mathbb{F}_{iJ})$,
we also suggest writing the vector of all the observations for the subject $i$
using matrix notations:
$\boldsymbol{U}_{i}=\boldsymbol{\mu}_{i}+\mathbb{F}_{ij}\boldsymbol{\xi}_{i}+\boldsymbol{\epsilon}_{i}$
From this, we propose to rewrite jointly the observations and the scores, as:
$\begin{bmatrix}\boldsymbol{U}_{i}\\\ \boldsymbol{\xi}_{i}\\\
\end{bmatrix}=\begin{bmatrix}\boldsymbol{\mu}_{i}\\\ 0\\\
\end{bmatrix}+\begin{bmatrix}\mathbb{F}_{i}&\mathbb{I}\\\
\mathbb{I}&0\end{bmatrix}\begin{bmatrix}\boldsymbol{\xi}_{i}\\\
\boldsymbol{\epsilon}_{i}\end{bmatrix}$
Using this expression, and the jointly Gaussian assumptions on $\xi_{i}$ and
$\epsilon_{i}$ we clearly see that $\xi_{i}$ and $\boldsymbol{U}_{i}$ are also
jointly Gaussian with joint law:
$\displaystyle\begin{bmatrix}\boldsymbol{U}_{i}\\\ \boldsymbol{\xi}_{i}\\\
\end{bmatrix}$ $\displaystyle\sim
N\left(\begin{bmatrix}\boldsymbol{\mu}_{i}\\\ 0\\\
\end{bmatrix},\begin{bmatrix}\mathbb{F}_{i}&\mathbb{I}\\\
\mathbb{I}&0\end{bmatrix}\begin{bmatrix}\mathbb{\Sigma}&0\\\
0&\sigma^{2}\mathbb{I}\end{bmatrix}\begin{bmatrix}\mathbb{F}_{i}^{\top}&\mathbb{I}\\\
\mathbb{I}&0\end{bmatrix}\right)$ $\displaystyle\sim
N\left(\begin{bmatrix}\boldsymbol{\mu}_{i}\\\ 0\\\
\end{bmatrix},\begin{bmatrix}\mathbb{F}_{i}\mathbb{\Sigma}\mathbb{F}_{i}^{\top}+\sigma^{2}\mathbb{I}&\mathbb{F}_{i}\mathbb{\Sigma}\\\
\mathbb{\Sigma}\mathbb{F}_{i}^{\top}&\mathbb{\Sigma}\end{bmatrix}\right)$
To obtain the best predictions for the scores, we now consider the conditional
distribution of $\xi_{i}$ and suggest using the conditional expectation as the
best predictor. Using the standard formulation of the Gaussian conditional
distribution (Rasmussen and Williams [2006]) we obtain :
$\mathbb{E}(\boldsymbol{\xi}_{i}|\boldsymbol{U}_{i})=\mathbb{\Sigma}\mathbb{F}_{i}^{\top}(\mathbb{F}_{i}\mathbb{\Sigma}\mathbb{F}_{i}^{\top}+\sigma^{2}\mathbb{I})^{-1}(\boldsymbol{U}_{i}-\boldsymbol{\mu}_{i})$
### A.5 Properties of deflated processes and components
#### A.5.1 Orthogonal deflation: correspondence between $\xi_{j}^{m}$ and
$\operatorname{y}_{j}^{m}$
In the decomposition model (8), basis coefficients are defined as
$\langle\operatorname{X}_{j},f_{j}^{m}\rangle=\xi_{j}^{m}$, since $f_{j}^{m}$
are orthonormal functions. As in RGCCA, the components are defined in the
FGCCA framework as
$\operatorname{y}_{j}^{m}=\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle$
where $\operatorname{X}_{j}^{m}$ stands for the $m$th deflated processes $j$,
with $\operatorname{X}_{j}^{0}=\operatorname{X}_{j}$, and $f_{j}^{0}$ and
$\operatorname{y}_{j}^{0}$ being respectively the first canonical function and
the first component retrieved. We can easily demonstrate recursively that we
have for $1\leq m\leq M$:
$\operatorname{X}_{j}^{m}=\operatorname{X}_{j}-\sum_{k=0}^{m-1}\langle\operatorname{X}_{j},f_{j}^{k}\rangle
f_{j}^{k}$
from this we can write
$\displaystyle\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle$
$\displaystyle=\langle\operatorname{X}_{j},f_{j}^{m}\rangle-\sum_{k=0}^{m-1}\langle\operatorname{X}_{j},f_{j}^{k}\rangle\langle
f_{j}^{k},f_{j}^{m}\rangle$
$\displaystyle=\langle\operatorname{X}_{j},f_{j}^{m}\rangle$
since, again, the set of canonical functions $\\{f_{j}^{m}\\}_{0\leq m\leq M}$
is a set of orthonormal functions. Therefore, we have proven that
$\operatorname{y}_{j}^{m}=\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle=\langle\operatorname{X}_{j},f_{j}^{m}\rangle=\xi_{j}^{m}$
#### A.5.2 Uncorrelated deflation: orthogonal property of $f_{j}^{m}$
First, we demonstrate that the functions retrieved using an uncorrelated
components deflation strategy are orthogonal. The $(m+1)$th deflation equation
for the process $j$ is:
$\operatorname{X}_{j}^{m+1}=\operatorname{X}_{j}^{m}-d_{j}^{m}\boldsymbol{\Sigma}_{jj}^{m}\boldsymbol{F}_{j}^{m}\operatorname{X}_{j}^{m}$
(17)
where $d_{j}^{m}=\langle
f_{j}^{m},\boldsymbol{\Sigma}_{jj}^{m}f_{j}^{m}\rangle^{-1}$. We recall that
the gradient operator, giving at each iteration the updated block weight
function is:
$\langle
f_{j}^{m},\hat{f}_{j}^{m+1}\rangle=\nabla_{j}\Psi(\boldsymbol{f})=2\sum_{\begin{subarray}{c}j^{\prime}=1\\\
j^{\prime}\neq j\end{subarray}}^{J}c_{j,j^{\prime}}g^{\prime}(\langle
f_{j}^{m+1},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}^{m+1}\rangle)\langle
fm_{j}^{m},\boldsymbol{\Sigma}_{jj^{\prime}}^{m+1}f_{j^{\prime}}^{m+1}\rangle$
(18)
Focusing on the elements in the sum we have that
$\displaystyle\langle
f_{j}^{m},\boldsymbol{\Sigma}_{jj^{\prime}}^{m+1}f_{j^{\prime}}^{m+1}\rangle$
$\displaystyle=\langle\mathbb{E}[\langle\operatorname{X}_{j^{\prime}}^{m+1},f_{j^{\prime}}^{m+1}\rangle\operatorname{X}_{j}^{m+1}],f_{j}^{m}\rangle$
$\displaystyle=\mathbb{E}[\langle\operatorname{X}_{j^{\prime}}^{m+1},f_{j^{\prime}}^{m+1}\rangle\langle\operatorname{X}_{j}^{m+1},f_{j}^{m}\rangle]$
Additionally,
$\displaystyle\langle\operatorname{X}_{j}^{m+1},f_{j}^{m}\rangle$
$\displaystyle=\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle-
d_{j}^{m}\langle(\boldsymbol{\Sigma}_{jj}^{m}\mathbb{F}_{j}^{m})(\operatorname{X}_{j}^{m}),f_{j}^{m}\rangle$
$\displaystyle=\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle-
d_{j}^{m}\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle\langle\boldsymbol{\Sigma}_{jj}^{m}f_{j}^{m},f_{j}^{m}\rangle$
$\displaystyle=\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle-
d_{j}^{m}\langle\operatorname{X}_{j}^{m},f_{j}^{m}\rangle(d_{j}^{m})^{-1}$
$\displaystyle=0$
Thus we have $\langle
f_{j}^{m},\boldsymbol{\Sigma}_{jj^{\prime}}f_{j^{\prime}}^{m+1}\rangle=0$ and
finally $\langle f_{j}^{m},f_{j}^{m+1}\rangle=0$, since all the terms in the
sum of the update function are null.
## Appendix B Additional experiments
### B.1 Simulation 2: additional results
In this additional section, we investigate the role of $\sigma$ on the
estimation accuracy. Results are displayed in figure 8 and 9.
### B.2 Simulation 3: additional results
As previously, we propose additional results obtained with different values of
noise $\sigma$. Results are displayed in figure 10.
Figure 8: Mean squared errors (MSE) of functions $f_{j}^{m}$ (top) and
components $\xi_{j}^{m}$ (bottom) for $m=1,2,3,4,5,6$ obtained from $100$
simulations with $N=100$, $\sigma^{2}=0$ and various sparsity settings.
Comparison between FPCA (Yao et al. [2005]), FSVD (Yang et al. [2011]) and
FGCCA.
Figure 9: Mean squared errors (MSE) of functions $f_{j}^{m}$ (top) and
components $\xi_{j}^{m}$ (bottom) for $m=1,2,3,4,5,6$ obtained from $100$
simulations with $N=100$, $\sigma^{2}=0.1$ and various sparsity settings.
Comparison between FPCA (Yao et al. [2005]), FSVD (Yang et al. [2011]) and
FGCCA.
Figure 10: Mean relative squared errors (MRSE) of reconstructed trajectories,
using estimated canonical functions and components. Comparing FGCCA and MFPCA
with $M=6$, for various number of subjects $N$, various sparsity settings and
various noise levels: (top) $\sigma_{2}=0$ and (bottom) $\sigma^{2}=0.1$.
|
# TensorCodec: Compact Lossy Compression of Tensors without Strong Data
Assumptions
Taehyung Kwon1, Jihoon Ko1, Jinhong Jung2, and Kijung Shin1
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Kim Jaechul Graduate School of AI, KAIST, 2School of
Software, Soongsil University
###### Abstract
Many real-world datasets are represented as tensors, i.e., multi-dimensional
arrays of numerical values. Storing them without compression often requires
substantial space, which grows exponentially with the order. While many tensor
compression algorithms are available, many of them rely on strong data
assumptions regarding its order, sparsity, rank, and smoothness.
In this work, we propose TensorCodec, a lossy compression algorithm for
general tensors that do not necessarily adhere to strong input data
assumptions. TensorCodec incorporates three key ideas. The first idea is
Neural Tensor-Train Decomposition (NTTD) where we integrate a recurrent neural
network into Tensor-Train Decomposition to enhance its expressive power and
alleviate the limitations imposed by the low-rank assumption. Another idea is
to fold the input tensor into a higher-order tensor to reduce the space
required by NTTD. Finally, the mode indices of the input tensor are reordered
to reveal patterns that can be exploited by NTTD for improved approximation.
Our analysis and experiments on $8$ real-world datasets demonstrate that
TensorCodec is (a) Concise: it gives up to $7.38\times$ more compact
compression than the best competitor with similar reconstruction error, (b)
Accurate: given the same budget for compressed size, it yields up to
$3.33\times$ more accurate reconstruction than the best competitor, (c)
Scalable: its empirical compression time is linear in the number of tensor
entries, and it reconstructs each entry in logarithmic time. Our code and
datasets are available at https://github.com/kbrother/TensorCodec.
###### Index Terms:
Tensor, Decomposition, Data Compression
## I Introduction
A tensor is a multi-dimensional array of numerical values [1] and can be
considered a higher-order generalization of a matrix. Various real-world
datasets, including sensory data [2], stock market history [3], and extracted
features from motion videos [4], are represented as tensors.
Many real-world tensors are large, making compression crucial for efficient
storage. Storing a tensor in its uncompressed form requires space proportional
to the total number of entries, which grows exponentially with the tensor’s
order. For instance, storing a tensor of size $704\times 2049\times 7997$ from
the music dataset [5] as a double-precision floating-point value per entry
consumes approximately $92$GB. This storage approach places a significant
burden on memory-limited devices such as mobile and IoT devices. Moreover,
transmitting such large datasets online can lead to substantial communication
costs.
Consequently, a variety of tensor compression methods have been devised, but
many of them rely on specific assumptions regarding input data, particularly
in terms of order, sparsity, rank, and smoothness. For instance, numerous
compression methods [6, 7] assume that the input tensor is of order two, i.e.,
a matrix. Additionally, many other compression methods [8, 9] are tailored for
compressing sparse tensors, which are tensors with a majority of zero entries.
Another category of compression methods based on tensor decomposition [10, 11,
12, 13] assumes that the input tensor has (approximately) a low-rank
structure. Furthermore, some compression methods [14, 15, 16, 17] presume that
tensor entries exhibit smooth variations, meaning that adjacent entries tend
to be similar, as in images and videos. However, many real-world tensors do
not necessarily conform to these assumptions, as shown in Section V-B.
How can we compactly compress tensors with minimal reconstruction error,
without imposing strong assumptions on data properties? To address this
question, we present TensorCodec, a lossy compression algorithm for general
tensors that do not depend on strict input data assumptions. To achieve this,
we first introduce Neural Tensor-Train Decomposition (NTTD). In contrast to
the original Tensor-Train Decomposition, where factor matrices are fixed for
all entries of a mode index, we use a recurrent neural network to obtain these
matrices, making them dependent on the other mode indices of the entries. This
modification allows NTTD to effectively approximate input tensors with a
limited number of parameters, even when they exhibit high rank. Secondly, we
fold the input tensor into a higher-order tensor, further reducing the number
of parameters needed for NTTD. Lastly, we introduce a reordering algorithm for
the mode indices of the input tensor, which uncovers patterns that NTTD can
exploit for accurate approximation of the tensor entries. The output of
TensorCodec consists of (a) the parameters of the recurrent neural network in
NTTD and (b) the mapping between the original and reordered mode indices,
which are used to approximate the entries of the input tensor.
We demonstrate the advantages of TensorCodec through complexity analysis and
comprehensive experiments on 8 real-world datasets, which are summarized as
follows:
* •
Concise: It gives up to 7.38$\times$ smaller output than the best-performing
competitor with similar approximation error.
* •
Accurate: It outperforms the most accurate competitor in terms of
approximation error by up to 3.33$\times$, while achieving comparable
compression sizes.
* •
Scalable: Its empirical compression time scales linearly with the number of
entries, while the reconstruction time scales logarithmically with the largest
mode size.
We present related works in Section II, give preliminaries in Section III,
propose TensorCodec in Section V, review experiments in Section IV, and make
conclusions in Section VI.
## II Related work
Compression methods for low-rank tensors: Tensor decomposition compresses
tensors lossily into smaller matrices and tensors. CP Decomposition (CPD) [18]
approximates the input tensor by the weighted sum of the outer products of
columns of the same order in the matrices. That is, the outer product of the
$i$-th columns of the matrices is computed for each $i$. Tucker Decomposition
(TKD) [11] generalizes CPD by allowing outer products of columns of different
orders, and TTHRESH [14] further compresses the outputs of TKD using run-
length and arithmetic coding. Tensor Train Decomppsition (TTD) [12]
approximates each entry by the product of matrices that vary depending on mode
indices, while Tensor Ring Decomposition (TRD) [13] uses the trace of the
product for approximation. That is, if two entries share $k$ mode indices, $k$
matrices are shared for their approximation. The approximation power of these
methods is limited by their reliance on low-rank assumptions and linear
operations, especially when the input tensor does not have a low rank and the
parameter size is restricted, as shown in Section V-B.
Compression methods for smooth tensors: Numerous tensor compression techniques
assume smoothness in the input data, meaning that adjacent entries tend to
have similar values. This assumption is especially prominent for image and
video data. Compression of images and videos constitutes a distinct and
advanced field of study [17, 16], and consequently, this paper focuses on
compression methods for tensors that are neither images nor videos. Smoothness
is also assumed when compressing scientific simulation data [15]. A notable
example, SZ3 [15] performs interpolation for each entry, based on smoothness,
and compactly but lossily encodes the errors using Huffman coding. Naturally,
the compression power of these methods diminishes when the smoothness
assumption is not met, as shown empirically in Section V-B.
Compression methods for sparse tensors: Many real-world tensors are sparse,
i.e. the majority of their entries are zero. This characteristic is commonly
leveraged in the aforementioned tensor decomposition methods for speed and
memory efficiency [19, 20, 21] without affecting the compressed outputs.
NeuKron [9] optimizes the entries of a small tensor, referred to as a seed
tensor, so that its generalized Kronecker power closely approximates the input
tensor. To improve the fit, NeuKron first reorders the mode indices in the
input tensor. NeuKron leverages the sparsity of the input tensor in three
ways: (a) it builds on the observation that real-world sparse matrices exhibit
self-similarity and can thus be approximated by Kronecker powers [22], (b) it
reorders mode indices based on the sparsity patterns of the sub-tensors formed
by the entries with each mode index, and (c) it enables rapid computation of
the objective function by taking advantage of the sparsity. Therefore, NeuKron
has been applied only to extremely sparse tensors where the ratio of non-zero
entries is less than 0.00354 [9]. NeuKron and our proposed TensorCodec share
similarities in that they both employ reordering and generalization using
auto-regressive models. However, TensorCodec generalizes TTD, instead of
Kronecker powers, and reorders mode indices based on the entry values, not on
sparsity patterns. It should be noticed that, unlike Kronecker powers, TTD has
been used to model various real-world tensors without being limited to sparse
ones. Due to these differences, TensorCodec significantly outperforms NeuKron
in terms of both compressed size and approximation accuracy for tensors that
are not extremely sparse, as demonstrated in Section V-B.
Usage of TTD for neural networks: TTD [12] has been utilized to compress
neural-network parameters in the form of tensors, including (a) weights of
fully connected layers [23] and Recurrent Neural Networks [24], and (b)
embeddings in recommender systems [25] and Graph Neural Networks [26]. These
works employ TTD for compressing neural networks, which is distinctly
different from our approach of employing a neural network to enhance the
expressiveness of TTD.
## III preliminaries
In this section, we introduce preliminaries, followed by a formal definition
of the considered problem. Frequently-used notations are listed in Table I.
For any positive integer $n$, $[n]:=\\{0,\cdots,n-1\\}$ denotes the set of
integers from 0 to $n-1$.
### III-A Basic Concepts and Notations
Matrix and tensor: We denote matrices in boldface capital letters. Given a
real-valued matrix $\mathbf{M}$ of size $N_{1}\times N_{2}$, the entry located
in the $i$-th row and $j$-th column is denoted by $\mathbf{M}(i,j)$. Tensors
are denoted by boldface Euler script letters. The order of a tensor refers to
the number of modes. Let $\mathbf{\mathcal{X}}$ be a real-valued $d$-order
tensor of size $N_{1}\times\cdots\times N_{d}$. The entry at the
$(i_{1},\cdots,i_{d})$-th position of $\mathbf{\mathcal{X}}$ is denoted by
$\mathbf{\mathcal{X}}(i_{1},\cdots,i_{d})$.
Slicing and reordering a tensor: For a mode-$j$ index $i\in[N_{j}]$,
$\mathbf{\mathcal{X}}^{(j)}(i)\in\mathbb{R}^{N_{1}\cdots N_{j-1}\times
N_{j+1}\cdots N_{d}}$ denotes the $i$-th slice of $\mathbf{\mathcal{X}}$ along
the $j$-th mode, i.e.,
$\mathbf{\mathcal{X}}^{(j)}(i)\coloneqq\mathbf{\mathcal{X}}(:_{1},\cdots,:_{j-1},i,:_{j+1},\cdots,:_{d})$,
where $:_{k}$ indicates all possible mode-$k$ indices (i.e., those in
$[N_{k}]$). We consider reordering of mode indices. Let
${\mathbf{\mathcal{X}}}_{\bm{\pi}}$ denote the tensor reordered from
$\mathbf{\mathcal{X}}$ by a set $\bm{\pi}=\\{\pi_{1},\cdots,\pi_{d}\\}$ of
reordering functions, where each $\pi_{i}:[N_{i}]\rightarrow[N_{i}]$ is a
bijective function from the set of the mode-$i$ indices to themselves. In
${\mathbf{\mathcal{X}}}_{\bm{\pi}}$, the $(i_{1},\cdots,i_{d})$-th entry
corresponds to the $(\pi_{1}(i_{1}),\pi_{2}(i_{2}),...,\pi_{d}(i_{d}))$-th
entry of $\mathbf{\mathcal{X}}$.
Frobenius norm: The Frobenius norm $\lVert\mathbf{\mathcal{X}}\rVert_{F}$ of
$\mathbf{\mathcal{X}}$ is defined as the squared root of the squared sum of
all its entries, i.e.,
$\lVert\mathbf{\mathcal{X}}\rVert_{F}=\sqrt{\sum_{(i_{1},\cdots,i_{d})\in[N_{1}]\times\cdots\times[N_{d}]}\big{(}\mathbf{\mathcal{X}}(i_{1},\cdots,i_{d})\big{)}^{2}}.$
(1)
### III-B Tensor-Train Decomposition (TTD)
Tensor-Train Decomposition (TTD) [12] decomposes a given $d$-order tensor
$\mathbf{\mathcal{X}}$ into $d$ tensors $\mathbf{\mathcal{G}}_{1}$, $\cdots$,
$\mathbf{\mathcal{G}}_{d}$, called TT cores, so that each entry of
$\mathbf{\mathcal{X}}$ is approximated as follows:
$\mathcal{X}(i_{1},\cdots,i_{d})\approx\mathbf{\mathcal{G}}_{1}^{(2)}(i_{1})\mathbf{\mathcal{G}}_{2}^{(2)}(i_{2})\cdots\mathbf{\mathcal{G}}_{d}^{(2)}(i_{d}),$
(2)
where, for all $k$, $\mathbf{\mathcal{G}}_{k}\in\mathbb{R}^{r_{k-1}\times
N_{k}\times r_{k}}$, and
$\mathbf{\mathcal{G}}_{k}^{(2)}(i)\in\mathbb{R}^{r_{k-1}\times r_{k}}$ is the
$i$-th slice of $\mathbf{\mathcal{G}}_{k}$ along the second mode. Note that
$r_{0}$ and $r_{d}$ are always set to $1$. For simplicity, in this paper, we
unify all other TT ranks (i.e., $r_{1},\cdots,r_{d-1}$) to a single value
denoted by $R$. The representative optimization algorithm for TTD is TT-SVD
[12], which aims to obtain
$\mathbf{\mathcal{G}}_{1},\cdots,\mathbf{\mathcal{G}}_{d}$ satisfying
$\lVert\mathbf{\mathcal{X}}-\mathbf{\mathcal{\tilde{X}}}_{TT}\rVert_{F}\leq\epsilon\lVert\mathbf{\mathcal{X}}\rVert_{F}$
for a prescribed accuracy $\epsilon$, where
$\mathbf{\mathcal{\tilde{X}}}_{TT}$ is the approximated tensor by TTD. In TT-
SVD, Truncated SVD is applied after reshaping a tensor to a matrix. TTD is
naturally used as a lossy tensor compression algorithm, with the compressed
results being the entries of the TT-core tensors. The number of these entries
is $R^{2}\sum_{k=1}^{d}N_{k}=O(dNR^{2})$, where $N$ represents the maximum
mode length.
TABLE I: Symbol description Symbol | Description
---|---
$\mathbf{A}\in\mathbb{R}^{N_{1}\times N_{2}}$ | $N_{1}$-by-$N_{2}$ matrix
$\mathbf{A}(i,j)$ | $(i,j)$-th entry of $\mathbf{A}$
$\mathbf{\mathcal{X}}\in\mathbb{R}^{N_{1}\times\cdots\times N_{d}}$ | tensor
$d$ | order of $\mathbf{\mathcal{X}}$
$\mathbf{\mathcal{X}}(i_{1},\cdots,i_{d})$ | $(i_{1},\cdots,i_{d})$-th entry of $\mathbf{\mathcal{X}}$
$N_{\text{max}}$ | maximum length of modes in $\mathbf{\mathcal{X}}$
${\mathbf{\mathcal{X}}}^{(j)}(i)$ | $i$-th slice of $\mathbf{\mathcal{X}}$ along the $j$-th mode
$[n]$ | set of consecutive integers from $0$ to $n-1$
$\bm{\pi}=(\pi_{1},\cdots,\pi_{d})$ | set of reordering functions for a tensor
$\pi_{i}$ | reordering function for the $i$-th mode indices
${\mathbf{\mathcal{X}}}_{\bm{\pi}}$ | reordered tensor of $\mathbf{\mathcal{X}}$ by $\bm{\pi}$
$\lVert\mathbf{\mathcal{X}}\rVert_{F}$ | Frobenius norm of $\mathbf{\mathcal{X}}$
$R$ | ranks of TT cores
$h$ | hidden dimension of LSTM
$\theta$ | set of the parameters of NTTD
$\mathbf{\mathcal{\tilde{X}}}$ | approximated tensor of $\mathbf{\mathcal{X}}$ by TensorCodec
$\mathbf{\mathcal{X}}^{\texttt{folded}}$ | folded tensor of $\mathbf{\mathcal{X}}$
$d^{\prime}$ | order of $\mathbf{\mathcal{X}}^{\texttt{folded}}$
$\mathbf{T}_{i}$ | $i$-th TT core generated by $\theta$
### III-C Problem Definition
We provide the formal definition of the lossy tensor compression problem
addressed in this paper as follows:
###### Problem 1.
(Lossy Compression of a Tensor) • Given: a tensor
$\mathcal{X}\in\mathbb{R}^{N_{1}\times\cdots\times N_{d}}$, • Find: the
compressed data $D$ • to Minimize: (1) the size of $D$
(2) the approximation error (e.g.,
$\lVert\mathbf{\mathcal{X}}-\mathbf{\mathcal{Y}}\rVert_{F}^{2}$,
where $\mathbf{\mathcal{Y}}$ is the tensor reconstructed from $D$)
In TensorCodec, the compressed data $D$ is composed of (a) the set $\theta$ of
parameters in Neural Tensor-Train Decomposition (NTTD), and (b) the set
$\bm{\pi}$ of reordering functions.
## IV Proposed Method
In this section, we propose TensorCodec, a precise and concise tensor-
compression algorithm.
### IV-A Overview
For our method, we focus on the following challenges in devising a compression
algorithm based on TTD:
1. Q1.
Expressiveness. Since TTD (see Section III-B) relies only on linear operations
of small-sized matrices, its expressiveness is limited.111For a more in-depth
discussion of expressiveness, refer to [12]. How can we enhance its
expressiveness to closely approximate a wider variety of tensors?
2. Q2.
Conciseness. The number of parameters of TTD increases with the order, mode
length, and TT-rank. Can we further reduce the parameter number for additional
compression?
3. Q3.
Data arrangement. The approximation by TTD depends on the arrangement of
tensor entries since it shares TT cores based on mode indices. How can we re-
arrange tensor entries to achieve a more accurate approximation?
TensorCodec is based on the following ideas for addressing each of the
aforementioned challenges:
1. A1.
Neural Tensor-Train Decomposition (NTTD). We incorporate an auto-regressive
neural network into TTD to improve its expressiveness while maintaining the
number of parameters small (Section IV-B).
2. A2.
Folding. We fold the input tensor into a higher-order tensor to reduce the
number of parameters and thus the size of the compressed output (Section
IV-C).
3. A3.
Reordering. We aid the model in better fitting the folded tensor by reordering
the mode indices of the input tensor (Section IV-D).
We summarize the overall process of TensorCodec in Algorithm 1. First,
TensorCodec initializes the NTTD model $\theta$ and the reordering functions
$\bm{\pi}$. Next, it reorders and folds $\mathbf{\mathcal{X}}$, creating a
higher-order tensor ${\mathbf{\mathcal{X}}}_{\bm{\pi}}^{\texttt{folded}}$.
Subsequently, the model parameters and reordering functions are updated to
minimize the approximation error. TensorCodec repeats this process until
convergence is reached, meaning that the approximation error no longer
exhibits significant changes. The outputs of the compression process are
optimized $\theta$ and $\bm{\pi}$ based on which each tensor entry is
approximated in logarithmic time, as proven in Section IV-E.
To simplify the explanation, we assume that the input tensor
${\mathbf{\mathcal{X}}}$ is already properly ordered in Sections IV-B and
IV-C. Afterward, we elaborate on how to initialize and update reordering
functions $\bm{\pi}$ in Section IV-D.
Figure 1: The overall process of Neural Tensor-Train Decomposition (NTTD),
which is the key component of TensorCodec. In order to generate TT cores for
approximation, the mode indices of a target entry are encoded through an
embedding layer and fed into an LSTM layer. Each TT core is obtained through a
linear layer from the LSTM output. An approximation is computed by the matrix
product of the TT cores. Note that the parameters of NTTD are those of the
neural networks, not TT cores themselves.
Input: a tensor $\mathbf{\mathcal{X}}\in\mathbb{R}^{N_{1}\times\cdots\times
N_{d}}$
Output: orders $\bm{\pi}$ and an NTTD model $\theta$
1 intialize $\theta$ and $\bm{\pi}$ $\triangleright$ Section IV-D and Alg. 4
of [27]
2 while _${fitness}$ does not converge_ do
3 ${\mathbf{\mathcal{X}}}_{\bm{\pi}}^{\texttt{folded}}\leftarrow$ reorder and
fold $\mathbf{\mathcal{X}}$ $\triangleright$ Section IV-C
4 Update $\theta$ $\triangleright$ Section IV-B
5 Update $\bm{\pi}$ $\triangleright$ Section IV-D and Alg. 3
6 $\mathbf{\mathcal{\tilde{X}}}\leftarrow$ approximation using $\theta$ and
$\bm{\pi}$
7 ${fitness}\leftarrow
1-\lVert\mathbf{\mathcal{X}}-\mathbf{\mathcal{\tilde{X}}}\rVert_{F}/\lVert\mathbf{\mathcal{X}}\rVert_{F}$
8
return $\theta$ and $\bm{\pi}$
Algorithm 1 Overview of TensorCodec
### IV-B Neural TT Decomposition (NTTD) for Fitting a Tensor
TensorCodec is a lossy compression algorithm, and its primary goal is to
accurately approximate tensor entries with a small number of parameters. As a
key component of TensorCodec, we propose Neural Tensor-Train Decomposition
(NTTD), a generalization of Tensor-Train Decomposition (TTD) that incorporates
an auto-regressive neural network.
Input: (a) an index $(i_{1},\cdots,i_{d})$
(b) parameters of embedding layers ($E_{1},\cdots,E_{d}$)
(c) parameters of an LSTM layer
(d) parameters of linear layers ($\mathbf{W}_{\\{1,d\\}}$, $\mathbf{W}$,
$\mathbf{b}_{\\{1,d\\}}$, $\mathbf{b}$)
Output: an approximated entry $\theta(i_{1},\cdots,i_{d})$
1 for _$k\leftarrow 1$ to $d$_ do
2 $\mathbf{e}_{k}\leftarrow E_{k}(i_{k})$
3
4$\mathbf{h}_{1},\cdots,\mathbf{h}_{d}\leftarrow\texttt{LSTM}(\mathbf{e}_{1},\cdots,\mathbf{e}_{d})$
$\triangleright$ $\mathbf{h}_{k}\in\mathbb{R}^{h}$ for $1\leq k\leq d$
5 $\mathbf{T}_{1}\leftarrow\mathbf{W}_{1}\mathbf{h}_{1}+\mathbf{b}_{1}$
$\triangleright$ $\mathbf{W}_{1}\in\mathbb{R}^{1\times R\times h}$ and
$\mathbf{b}_{1}\in\mathbb{R}^{1\times R}$
6 for _$k\leftarrow 2$ to $d-1$_ do
7 $\mathbf{T}_{k}\leftarrow\mathbf{W}\mathbf{h}_{k}+\mathbf{b}$
$\triangleright$ $\mathbf{W}\in\mathbb{R}^{R\times R\times h}$ and
$\mathbf{b}\in\mathbb{R}^{R\times R}$
8
9$\mathbf{T}_{d}\leftarrow\mathbf{W}_{d}\mathbf{h}_{d}+\mathbf{b}_{d}$
$\triangleright$ $\mathbf{W}_{d}\in\mathbb{R}^{R\times 1\times h}$ and
$\mathbf{b}_{d}\in\mathbb{R}^{R\times 1}$
return $\mathbf{T}_{1}\mathbf{T}_{2}\cdots\mathbf{T}_{d}$
Algorithm 2 Approximation of an entry by NTTD ($\theta$)
Instead of directly learning TT cores as free variables, for each
$(i_{1},\cdots,i_{d})$-th entry of the tensor, TT cores are obtained as the
output of a neural network that takes the mode indices of the entry as an
input. The neural network, which we denote by $\theta$, is trained to
approximate the entry as follows:
$\mathbf{\mathcal{X}}(i_{1},\cdots,i_{d})\approx\theta(i_{1},\cdots,i_{d})=\mathbf{T}_{1}\mathbf{T}_{2}\cdots\mathbf{T}_{d},$
(3)
where $\mathbf{T}_{1}\in\mathbb{R}^{1\times R}$,
$\mathbf{T}_{2}\in\mathbb{R}^{R\times R}$, $\cdots$,
$\mathbf{T}_{d-1}\in\mathbb{R}^{R\times R}$, and
$\mathbf{T}_{d}\in\mathbb{R}^{R\times 1}$ are the TT cores generated by
$\theta$.
Detailed procedures of NTTD are given in Algorithm 2 and illustrated in Figure
1. Note that NTTD consists of embedding, LSTM [28], and linear layers. To
encode each mode index $i_{k}$, our model first looks up the embedding
$\mathbf{e}_{k}$ from the embedding layer $E_{k}$ (lines 2 and 2). Then, it
feeds the embeddings $\mathbf{e}_{k}$ into the LSTM layer and obtains the
hidden embeddings $\mathbf{h}_{k}$ for $1\leq k\leq d$ (line 2). After
generating TT cores $\mathbf{T}_{k}$ from $\mathbf{h}_{k}$ using the linear
layers (lines 2-2), the product of the TT cores is returned as the
approximated entry value (line 2)222To further reduce the model size, we share
the same embedding parameters for modes of the same length (i.e.,
$E_{k}=E_{j}$ if $N_{k}=N_{j}$). The effect of this approach is explored in
Section V of [27]..
Note that we employ an auto-regressive model to allow for the dependency
between TT cores and mode indices. However, in NTTD, each $\mathbf{T}_{k}$
depends not only on the mode-$k$ index of the target entry (as in TTD) but
also on its mode-$j$ indices for all $j\leq k$. While we employ LSTM [28], any
alternatives, such as GRU [29] and Scaled Dot-product Attention [30], can be
used instead. In Section 8 of [27], we examine the performance of TensorCodec
equipped with alternatives.
Benefits over TTD: The NTTD model $\theta$ has the following advantages over
traditional TTD:
* •
Contextual: In NTTD, as mentioned earlier, each TT core varies depending on
not only the current mode index but also all preceding mode indices. For
example, consider approximating $\mathbf{\mathcal{X}}(2,1,2)$ and
$\mathbf{\mathcal{X}}(1,2,2)$. The TT cores used for the third mode in NTTD
are different in these two cases. However, in TTD, the same TT core is used
for both cases since the third mode indices are identical. By being contextual
and non-linear (described below), NTTD is able to model tensors that cannot be
easily approximated by TTD, even with more parameters, as shown experimentally
in Section V-E. This improved expressiveness reduces the reliance on
structural assumptions about input tensors.
* •
Non-linear: NTTD incorporates non-linear operations introduced by the LSTM
layer, while TTD does not. This contributes to enhancing the expressiveness of
NTTD, enabling TensorCodec to better approximate tensor entries.
* •
Concise: NTTD shares parameters (specifically, $\mathbf{W}$ and $\mathbf{b}$
in line 2 of Algorithm 2) for different modes, which enables the model to be
concise with fewer parameters than TTD. In contrast, TTD requires a unique TT
core for each mode.
We empirically examine the contribution of each component of NTTD to
approximation accuracy in Section V-C.
Space complexity analysis: We present the compressed output size of NTTD in
Theorem 1. The hidden dimension of LSTM and the rank of TT cores are denoted
by $h$ and $R$.
###### Theorem 1 (Compressed Output Size of NTTD).
The size of the compressed output of NTTD (i.e., the number of its parameters)
is $O(h(h+R^{2}+\sum_{i=1}^{d}N_{i}))$, which becomes $O(\sum_{i=1}^{d}N_{i})$
if we treat $h$ and $R$ as constants.
###### Proof.
The embedding layers have $O(\sum_{i=1}^{d}N_{i}h)$ parameters. The LSTM and
fully connected layers have $O(h^{2}+hR^{2})$ parameters. Hence, the total
size is $O(h(h+R^{2}+\sum_{i=1}^{d}N_{i}))$. ∎
Optimization method for $\theta$: We update the parameters $\theta$ of NTTD
using mini-batch gradient descent to minimize the loss function in Problem 1,
i.e., $\lVert\mathbf{\mathcal{X}}-\mathbf{\mathcal{Y}}\rVert_{F}^{2}$, where
$\mathbf{\mathcal{Y}}$ is the tensor approximated by $\theta$. As outlined in
Section IV-A, we alternately update $\theta$ and the reordering functions
$\pi$. After updating $\pi$, we reinitialize the optimizer (spec., Adam [31])
since the loss surface changes after reordering, as detailed in Section IV-D.
### IV-C Folding Technique for Lightweight NTTD
Figure 2: An example of folding an 8$\times$8 matrix into a
4$\times$4$\times$4 tensor, where entries with similar values have similar
colors. Note that when the rows of the matrix are reordered (R) such that
adjacent rows are similar, the similar entries are located close to each other
when the tensor is folded (F). In particular, similar entries share mode-$1$
and $2$ indices in the folded tensor.
We discuss the folding technique, which serves as the second component of
TensorCodec. It aims to further reduce the size of the compressed output of
NTTD, which is proportional to $\sum_{i=1}^{d}N_{i}$ according to Theorem 1.
The main idea is to fold the input tensor into a higher-order tensor,
maintaining the same number of entries but with smaller mode lengths. In
TensorCodec, the NTTD model $\theta$ aims to fit the folded tensor rather than
the input tensor after the arrangement process discussed in Section IV-D.
TT-matrix format: Our folding technique is inspired by the TT-matrix format
[12], which aims to fold a matrix into a tensor for reducing the number of
parameters in TTD. Given a matrix $\mathbf{A}$ of size $N\times M$ where
$N=\prod_{k=1}^{d}n_{k}$ and $M=\prod_{k=1}^{d}m_{k}$, the format folds
$\mathbf{A}$ into a $d$-order tensor $\mathbf{\mathcal{A}}$ of size
$n_{1}m_{1}\times\cdots\times n_{d}m_{d}$ (see the example with $d=3$ in
Figure 2) so that each entry of $\mathbf{A}$ is mapped to an entry of
$\mathbf{\mathcal{A}}$ as follows:
$\mathbf{A}(i,j)=\mathbf{\mathcal{A}}(i_{1}m_{1}+j_{1},\cdots,i_{d}m_{d}+j_{d}),$
where $i_{k}\in[n_{k}]$ and $j_{k}\in[m_{k}]$ for each $1\leq k\leq d$ are
those satisfying $i=\sum_{k=1}^{d}i_{k}\prod_{l=k+1}^{d}n_{l}$ and
$j=\sum_{k=1}^{d}j_{k}\prod_{l=k+1}^{d}m_{l}$. The impact of this folding
technique on NTTD is discussed below in the context of more general cases.
TT-tensor format: While the TT-matrix format can be naturally extended to
tensors, to the best of our knowledge, there has been no attempt to do so. In
this work, we extend the TT-matrix format to tensors by folding the input
tensor into a higher-order tensor with smaller mode lengths. We call this
process the TT-tensor format. Given a tensor $\mathbf{\mathcal{X}}$ of size
$N_{1}\times\cdots\times N_{d}$ where $N_{k}=\prod_{l=1}^{d^{\prime}}n_{k,l}$,
we fold $\mathcal{X}$ into a $d^{\prime}$-order tensor
$\mathbf{\mathcal{X}}^{\texttt{folded}}$ of size
$\prod_{k=1}^{d}n_{k,1}\times\cdots\times\prod_{k=1}^{d}n_{k,d^{\prime}}$.
Then, the mapping between the entries of $\mathbf{\mathcal{X}}$ and
$\mathbf{\mathcal{X}}^{\texttt{folded}}$ is:
$\displaystyle\mathbf{\mathcal{X}}\Big{(}\sum_{k=1}^{d^{\prime}}(i_{1,k}\prod_{l=k+1}^{d^{\prime}}n_{1,l}),\cdots,\sum_{k=1}^{d^{\prime}}(i_{d,k}\prod_{l=k+1}^{d^{\prime}}n_{d,l})\Big{)}$
(4)
$\displaystyle\rightarrow\mathbf{\mathcal{X}}^{\texttt{folded}}\Big{(}\sum_{k=1}^{d}(i_{k,1}\prod_{l=k+1}^{d}n_{l,1}),\cdots,\sum_{k=1}^{d}(i_{k,d^{\prime}}\prod_{l=k+1}^{d}n_{l,d^{\prime}})\Big{)},$
where $i_{k,l}\in[n_{k,l}]$ for all $k\in\\{1,\cdots,d\\}$ and
$l\in\\{1,\cdots,d^{\prime}\\}$.
In TensorCodec, we select the new order $d^{\prime}$ such that the folded
tensor has a higher order than the input tensor (i.e., $d^{\prime}>d$), while
maintaining $d^{\prime}=O(\log N_{\text{max}})$, where $N_{\text{max}}$
represents the maximum mode length in the input tensor $\mathbf{\mathcal{X}}$.
In real-world tensors, this is usually feasible since their mode lengths are
usually much larger than their orders. For instance, a 4-order tensor of size
$256\times 256\times 256\times 256$ can be folded into an 8-order tensor with
each mode having a length of 16. It may not always be feasible to construct a
folded tensor that meets the above criteria while having the same number of
entries as the input tensor. In such cases, the folded tensor may contain
extra entries, the values of which are disregarded. For real-world tensors
(see Section V), we initially assign $2$ to $n_{k,l}$ for all
$k\in\\{1,\cdots,d\\}$ and $l\in\\{1,\cdots,d^{\prime}\\}$ and modify some of
them using integers at most $5$ to ensure that the input and folded tensors
have similar numbers of entries. For example, for the PEMS-SF dataset (a
3-order tensor of size $963\times 144\times 440$), the assigned values in the
form of a $d$ by $d^{\prime}$ matrix are
$\left[\begin{array}[]{*{20}c}2&2&2&2&2&2&2&2&2&2\\\ 2&2&2&2&2&5&1&1&1&1\\\
2&2&2&2&2&2&2&2&2&1\end{array}\right],$
which result in a $10$-order tensor of size $8\times 8\times 8\times 8\times
8\times 20\times 4\times 4\times 4\times 2$. Note that
$\prod_{l=1}^{d^{\prime}}n_{1,l}=1024$, $\prod_{l=1}^{d^{\prime}}n_{2,l}=160$,
$\prod_{l=1}^{d^{\prime}}n_{3,l}=512$ are close to $963$, $144$, and $440$,
respectively. See Section IV of [27] for the values used for other datasets.
Space complexity analysis: We analyze the effect of folding on the number of
parameters (i.e., size of compressed output) in $\theta$. For simplicity, we
assume $n_{k,l}=\sqrt[d^{\prime}]{N_{k}}$ for all $k\in[d]$ and
$l\in[d^{\prime}]$. According to Theorem 1, the number of parameters of NTTD
of the original $\mathbf{\mathcal{X}}$ is
$O\Big{(}\sum_{k=1}^{d}\prod_{l=1}^{d^{\prime}}n_{k,l}\Big{)}=O(N_{1}+\cdots+N_{d}),$
if we treat $h$ and $R$ as constants.
The number of parameters of NTTD of $\mathbf{\mathcal{X}}^{\texttt{folded}}$
is
$O\Big{(}\sum_{l=1}^{d^{\prime}}\prod_{k=1}^{d}n_{k,l}\Big{)}=O\Big{(}d^{\prime}\\!\\!\sqrt[d^{\prime}]{N_{1}\cdots
N_{d}}\Big{)},$
which is significantly smaller than $O(N_{1}+\cdots+N_{d})$ in NTTD of the
original tensor since
$O\left(d^{\prime}\cdot\\!\\!\sqrt[d^{\prime}]{\prod_{k=1}^{d}{N_{k}}}\right)\in
o(N_{\text{max}})$, where $N_{\text{max}}$ is the maximum mode length in
$\mathbf{\mathcal{X}}$. This is because $d^{\prime}=O(\log N_{\text{max}})$,
$d^{\prime}>d$, and thus
$O\left(d^{\prime}\cdot\\!\\!\sqrt[d^{\prime}]{\prod_{k=1}^{d}{N_{k}}}\right)=O(N_{\text{max}}^{e}\log
N_{\text{max}})$ for some $e<1$. If we consider $R$ and $h$, according to
Theorem 1, the space complexity becomes
$O\Big{(}h\Big{(}h+R^{2}+d^{\prime}\\!\\!\sqrt[d^{\prime}]{N_{1}\cdots
N_{d}}\Big{)}\Big{)}.$ (5)
### IV-D Reordering Technique for Better Fitting a Folded Tensor
We present the reordering technique, the last component of TensorCodec.
Essentially, we reorder the mode indices in the input tensor before folding so
that entries with similar values are placed close to each other by sharing
mode indices in the folded tensor. This arrangement improves the ability of
our NTTD model $\theta$, to fit the folded tensor more effectively, as
$\theta$ generates TT cores based on mode indices of target entries that serve
as input for the model.
In the example in Figure 2, the closer two entries are located in the original
tensor, the more indices they tend to share in the folded tensor.
Specifically, the entries in the black region share only the first mode index
in the folded tensor. The adjacent entries in the blue region share both the
first and second mode indices in the folded tensor. It is important to note
that in our model $\theta$, the $k$-th TT cores $\mathbf{T}_{k}$ in Eq. (3)
for approximating two entries are the same if their first $k$ indices are the
same. Consequently, two TT cores are shared for the entries in the blue
region. Therefore, the closer two entries are located in the original tensor,
the more inputs and TT cores of $\theta$ are likely to share for these
entries. Due to this property, positioning similar entries close to each other
helps $\theta$ easily approximate entries more accurately. We achieve such re-
locations by reordering mode indices in the input tensor, as demonstrated in
the example in Figure 2, where the blue region comprises more similar entries
after reordering.
In TensorCodec, the mode-index reordering is accomplished by learning
reordering functions $\bm{\pi}$. As outlined in Section IV-A, we alternately
update the model $\theta$ and $\bm{\pi}$. In the following, we describe the
initialization and update procedures for $\bm{\pi}$. It is worth noting that
mode-index ordering by TensorCodec is associated with increasing smoothness,
which is discussed in Section I, while many other compression methods presume
that the input tensor is already highly smooth.
Initializing the orders: We initialize the set $\bm{\pi}$ of reordering
functions using a surrogate loss function. For all $k$, reordering the
mode-$k$ indices (i.e., optimizing $\pi_{k}$) is formulated as:
$\vspace{-2mm}\min_{\pi_{k}}\sum_{i=1}^{N_{k}-1}\left(\lVert{\mathbf{\mathcal{X}}}^{(k)}(\pi_{k}(i))-{\mathbf{\mathcal{X}}}^{(k)}(\pi_{k}(i+1))\rVert_{F}\right),$
(6)
where ${\mathbf{\mathcal{X}}}^{(k)}(i)$ is the $i$-th slice of
$\mathbf{\mathcal{X}}$ along the $k$-th mode. Note that minimizing Eq. (6)
makes adjacent slices similar.
The problem in Eq. (6) can be reduced to Metric Travelling Salesman Problem
[32]. Suppose each node represents a slice of the tensor, and each pair of
nodes forms an edge with a weight equal to the Frobenius norm of the
difference between their slices. Then, the optimal solution of the TSP problem
on the resulting complete graph can be used to minimize Eq. (6). However,
since computing it is NP-hard, we instead obtain a 2-approximation solution
based on the fact that the Frobenius norm satisfies the triangle inequality.
Then, we delete the edge with the largest weight from the obtained solution,
which is a TSP cycle, and set each $i$-th node in the resulting path to
$\pi_{k}(i)$. Refer to Section II of [27] for details on the TSP problem and
the 2-approximation algorithm, including pseudocode.
Updating the orders based on $\theta$ (Algorithm 3): As outlined in Section
IV-A, after updating the NTTD model $\theta$, we update the set $\bm{\pi}$ of
reordering functions based on updated $\theta$ and the loss function in
Problem 1. This update step is described in Algorithm 3. As defined in Section
III-A, we use ${\mathbf{\mathcal{X}}}_{\bm{\pi}}$ to denote the tensor
reordered from $\mathbf{\mathcal{X}}$ by $\bm{\pi}$. For each $k$-th mode, we
consider $\left\lfloor N_{k}/2\right\rfloor$ disjoint candidate pairs of
mode-$k$ indices (lines 17-18). The process of obtaining candidate pairs is
described in the following paragraph. For each pair $(i,i^{\prime})$ of mode
indices, we consider the corresponding slices
${\mathbf{\mathcal{X}}}_{\bm{\pi}}^{(k)}(i)$ and
${\mathbf{\mathcal{X}}}_{\bm{\pi}}^{(k)}(i^{\prime})$. If swapping them
reduces the loss function in Problem 1, we swap the values of $\pi_{k}(i)$ and
$\pi_{k}(i^{\prime})$ (lines 22-24). Note that, since pairs are disjoint, we
can compute the changes in the loss and update $\pi_{k}$ in parallel using
GPUs.
In the above process, each pair is composed so that swapping them tends to
make similar slices located nearby in ${\mathbf{\mathcal{X}}}_{\bm{\pi}}$. We
find such pairs using locality-sensitivity hashing (LSH) for Euclidean
distance [33]. We sample half of the indices in each mode and vectorize the
corresponding slices as points in a high-dimensional space. We project the
vectorized slices onto a random vector (lines 6-10) and evenly divide the
projected points to create buckets. We then repeatedly select two points in
the same bucket. Assuming that corresponding mode indices are $i_{1}$ and
$i_{2}$, two pairs $(i_{1},i_{2}\oplus 1)$ and $(i_{1}\oplus 1,i_{2})$ are
added as candidate pairs where $\oplus$ denotes the XOR operation (lines
17-18). This approach aims to locate indices corresponding to similar slices
nearby. The remaining mode indices are paired randomly (lines 19-21). The
effectiveness of the reordering process is experimentally validated in Section
V-C.
Input: (a) a tensor
$\mathbf{\mathcal{X}}\in\mathbb{R}^{N_{1}\times\cdots\times N_{d}}$, (b) a
NTTD model $\theta$,
(c) orders $\pi_{1},\cdots,\pi_{d}$
Output: updated orders $\pi_{1},\cdots,\pi_{d}$
1
2for _$k$ $\leftarrow$ $1$ to $d$_ do
// Project each slice
3 $I\leftarrow\emptyset$
4 for _$j\leftarrow 0$ to $N_{k}-1$ by $2$_ do
5 $u\sim U(0,1)$
6 if $u$ $<$ $1/2$ then $I$ $\leftarrow$ $I\cup\\{j\\}$ else $I$ $\leftarrow$
$I\cup\\{j+1\\}$
7
8 $s\leftarrow(\prod_{j=1}^{d}N_{j})/N_{k}$; $P\leftarrow\emptyset$
9 $r\leftarrow$ a random point from $\mathbb{R}^{s}$
10 foreach _$j\in I$_ do
11 $v\leftarrow$ vec$({\mathbf{\mathcal{X}}}^{(k)}(j))$
12 $P[j]\leftarrow r\cdot v/(\lVert r\rVert_{F}\lVert v\rVert_{F})$
// Hash points to buckets
13 $num\\_buckets\leftarrow\left\lfloor N_{k}/8\right\rfloor$
14 $bs\leftarrow\left\lfloor(\max(P)-\min(P))/num\\_buckets\right\rfloor$;
$B\leftarrow\emptyset$
15 foreach _$j\in I$_ do
16 $bi\leftarrow\left\lfloor(P[j]-\min(P))/bs\right\rfloor$
17 $B[bi]\leftarrow(B[bi]\cup\\{j\\}$)
// Build pairs (run in parallel)
18 $S\leftarrow\emptyset$; $S^{\prime}\leftarrow\emptyset$
19 foreach _$b\in B$_ do
20 $\textnormal{{AddPairs}}(b,S,True)$ $\triangleright$ Defined below
21 $S^{\prime}\leftarrow S^{\prime}\cup b$
22 $S^{\prime}\leftarrow S^{\prime}\cup\\{j\oplus 1:j\in S^{\prime}\\}$
23 $\textnormal{{AddPairs}}(S^{\prime},S,False)$
// Update orders (run in parallel)
24 foreach _$(i,i^{\prime})\in S$_ do
25 $\Delta\leftarrow$ change in the loss when $\pi_{k}(i)$,
$\pi_{k}(i^{\prime})$ are exchanged
26 if _$\Delta <0$_ then
$\pi_{k}(i),\pi_{k}(i^{\prime})\leftarrow\pi_{k}(i^{\prime}),\pi_{k}(i)$
27
28return $\pi_{1},\cdots,\pi_{d}$
29 Function _AddPairs( C, S, xor)_
30 while _$|C| >1$_ do
31 Randomly sample $(i_{1},i_{2})$ from $C$
32 if _$xor$_ then $S\leftarrow S\cup\\{(i_{1},i_{2}\oplus
1)\\}\cup\\{(i_{1}\oplus 1,i_{2})\\}$
33 else $S\leftarrow S\cup\\{(i_{1},i_{2})\\}$
34 $C\leftarrow C\setminus\\{i_{1},i_{2}\\}$
Algorithm 3 Update of the reordering functions $\bm{\pi}$
### IV-E Theoretical Analysis
We theoretically analyze TensorCodec’s compressed-output size, entry-
reconstruction speed, and compression speed. For simplicity, we assume that
all mode sizes of the input tensor
$\mathbf{\mathcal{X}}\in\mathbb{R}^{N_{1}\times\cdots\times N_{d}}$ are powers
of 2 (i.e., $n_{l,k}\in\\{1,2\\}$ for all $l\in\\{1,\cdots,d\\}$ and
$k\in\\{1,\cdots,d^{\prime}\\}$ in Section IV-C). We use $N_{\text{max}}$ to
denote the maximum size of modes in $\mathbf{\mathcal{X}}$, and use $h$ and
$R$ to denote the hidden dimension and TT rank of our model.
Size of compressed outputs: We present the space complexity of the outputs
produced by TensorCodec in Theorem 2. It is important to note that the
complexity is much lower than $O(\prod_{i=1}^{d}N_{i})$ of the original tensor
and can also be lower than $O(R^{2}\sum_{i=1}^{d}N_{i})$ of TTD and
$O(R\sum_{i=1}^{d}N_{i})$ of CP Decomposition (CPD), especially for large
values of $R$.
###### Theorem 2 (Size of Compressed Outputs).
The size of the compressed output $D=(\theta,\bm{\pi})$ produced by
TensorCodec (i.e., Algorithm 1) is $O(h(2^{d}+h+R^{2})+\sum_{i=1}^{d}N_{i}\log
N_{i})$.
###### Proof.
The embedding layer of $\theta$ has $O(h2^{d})$ parameters because our model
shares the embedding layers across different modes in
${X}_{\bm{\pi}}^{\texttt{folded}}$ and the largest mode size of
${X}_{\bm{\pi}}^{\texttt{folded}}$ is $2^{d}$, as detailed in Section IV-C.
The numbers of parameters of LSTM and fully connected layers are
$O(h^{2}+hR^{2})$ since the number of parameters of each linear layer is
proportional to the product of the input dimension and the output dimension.
For each mode $i$, the number of all possible orderings of $\pi_{i}$ is
$N_{i}!$. Thus, we need $O(\log N_{i}!)\in O(N_{i}\log N_{i})$ bits to save
one among them. Hence, the total size of the compressed output is
$O(h(2^{d}+h+R^{2})+\sum_{i=1}^{d}N_{i}\log N_{i})$. ∎
Speed of reconstruction: Another important aspect of a compression algorithm
is the speed of reconstruction. Theorem 3 formalizes the reconstruction speed
for the output of TensorCodec. While the complexity is higher than $O(dR^{2})$
of TTD or $O(dR)$ of CPD, it is only logarithmic in mode lengths.
###### Theorem 3 (Reconstruction Speed).
Given the output $D=(\theta,\bm{\pi})$ of TensorCodec (i.e., Algorithm 1),
approximating the value of an input tensor entry (i.e., Algorithm 2 on
${\mathbf{\mathcal{X}}}_{\bm{\pi}}$) takes $O((d+h^{2}+hR^{2})\log
N_{\textnormal{max}})$ time.
###### Proof.
For each entry of $\mathbf{\mathcal{X}}$, earning the mode indices in
${\mathbf{\mathcal{X}}}_{\bm{\pi}}$ requires $O(d)$ time. Computing the mode
indices in ${\mathbf{\mathcal{X}}}_{\bm{\pi}}^{\texttt{folded}}$ requires
$O(d\log N_{\text{max}})$ time because all $i_{kl}$ in Eq. (4) must be
computed where $k$ ranges from 1 to $d$ and $l$ ranges from 1 to
$d^{\prime}=O(\log N_{\text{max}})$. Processing the inputs through the
embedding layer and LSTM in $\theta$ requires $O(h^{2}\log N_{\text{max}})$
time. The time complexity of computing TT cores with fully connected layers is
$O(hR^{2}\log N_{\text{max}})$, and that of computing products of TT cores is
$O(R^{2}\log N_{\text{max}})$, provided that the order of computations is
optimized. Therefore, the total time complexity of approximating each entry is
$O((d+h^{2}+hR^{2})\log N_{\text{max}})$. ∎
Speed of compression: We analyze the speed of the compression process of
TensorCodec.
###### Theorem 4 (Compression Speed).
The time complexity of TensorCodec (i.e., Algorithm 1) with $T$ update steps
is $O((Td(d+h^{2}+hR^{2})\log
N_{\textnormal{max}}+\sum_{i=1}^{d}N_{i})\prod_{i=1}^{d}N_{i})$, where
$\prod_{i=1}^{d}N_{i}$ is the number of tensor entries.
Proof Sketch. Initializing all reordering functions $\bm{\pi}$ takes
$O((\sum_{i=1}^{d}N_{i})\cdot(\prod_{i=1}^{d}N_{i}))$ time. Updating $\theta$
and $\pi$ once (i.e., lines 1-1) takes
$O(d(d+h^{2}+hR^{2})\prod_{i=1}^{d}N_{i}\log N_{\textnormal{max}})$ time. For
a full proof, see Section I of [27]. ∎
Connection to actual running time: In Sections V-D, we measure the actual
running time for compression and reconstruction to confirm the time
complexity. In practice, the term $\prod_{i=1}^{d}N_{i}$, which corresponds to
the number of entries, is much larger and also increases much faster than all
other terms, and as thus the compression time increases near linearly with it.
Memory requirements: The complexity of memory space required for compression
by TensorCodec does not exceed the combined memory requirements for a mini-
batch, the compressed output, and the reordering functions.
###### Theorem 5 (Memory Requirements for Compression).
TensorCodec (Algorithm 1) requires $O(Bd+h(2^{d}+B(h+R^{2})\log
N_{\textnormal{max}})+\sum_{i=1}^{d}N_{i})$ memory space, where $B$ is the
number of tensor entries in a mini-batch.
###### Proof.
Refer to Section I of [27]. ∎
## V Experiments
We review our experiments for the following questions:
1. Q1.
Compression Performance: How accurately and compactly does TensorCodec
compress real-world tensors?
2. Q2.
Ablation Study: How effective is each component of TensorCodec for
reconstruction accuracy?
3. Q3.
Scalability: How does TensorCodec’s compression and reconstruction time scale
with input tensor size?
4. Q4.
Further Inspection: Can our method accurately compress high-rank tensors? How
does it reorder mode indices?
5. Q5.
Compression Time: How does the speed of compression by TensorCodec compare to
that of its competitors?
6. Q6.
Effect of Hyperparameters (Section VII of [27]): How sensitive is TensorCodec
to hyperparameter settings?
TABLE II: Statistics of public real-world datasets used in the paper.
Name | Size | Order | Order | Density | Smoothness
---|---|---|---|---|---
(Original) | (Folded)
Uber | $183\times 24\times 1140$ | 3 | 9 | 0.138 | 0.861
Air Quality | $5600\times 362\times 6$ | 12 | 0.917 | 0.513
Action | $100\times 570\times 567$ | 8 | 0.393 | 0.484
PEMS-SF | $963\times 144\times 440$ | 10 | 0.999 | 0.461
Activity | $337\times 570\times 320$ | 8 | 0.569 | 0.553
Stock | $1,317\times 88\times 916$ | 10 | 0.816 | 0.976
NYC | $265\times 265\times 28\times 35$ | 4 | 7 | 0.118 | 0.788
Absorb | $192\times 288\times 30\times 120$ | 7 | 1.000 | 0.935
(a) Stock (b) Activity (c) Action (d) Air Quality (e) PEMS-SF (f) Uber (g)
Absorb (h) NYC
Figure 3: TensorCodec provides compact and accurate tensor compression. Its
compressed size is up to 7.38$\times$ smaller than that of the best competitor
showing similar fitness. Given a budget for compressed size, TensorCodec shows
up to 3.33$\times$ higher fitness than the best baseline.
### V-A Experimental Settings
Machine: For TensorCodec and NeuKron, which require GPUs, we used a machine
with 4 RTX 2080Ti GPUs and 128GB RAM. For all other algorithms, we used a
machine with an i5-9600K (6 cores) and 64GB RAM. Note that the outputs are
independent of machine specifications.
Datasets: We used 8 real-valued tensors from real-world datasets to verify the
effectiveness of our method. We provide basic statistics of the tensors in
Table II. Smoothness is defined as
$1-\frac{\mathbb{E}_{i}[\sigma_{3}(i)]}{\sigma}$, where $\sigma_{3}(i)$
denotes the standard deviation of values in the sub-tensor of size $3^{d}$
centered at the position $i$, and $\sigma$ denotes the standard deviation of
the values of the entire tensor. Note that we used tensors with diverse
properties, including varying density and smoothness, from different domains,
without any assumption on data property. See Section III of [27] for data
semantics and pre-processing steps.
Competitors: We compared TensorCodec with 7 state-of-the-art methods (refer to
Section II) to evaluate their performance in terms of tensor compression. We
employed four well-known tensor decomposition methods: CPD [18], TKD [11], TTD
[12], and TRD [13] (Section II). These methods are implemented in MATLAB
R2020a, which offers a highly efficient linear algebra library.333Tensor
Toolbox. https://tensortoolbox.org/. 444TT-Toolbox.
https://github.com/oseledets/TT-Toolbox.
555https://github.com/oscarmickelin/tensor-ring-decomposition We used the
official C++ implementation of TTHRESH [14] and SZ3 [15], which are for
compressing tensors of visual data. Since the compression results of TTHRESH
and SZ3 depend on mode-index orders, we applied our order-initialization
technique to these methods only when it was helpful. We also used the official
PyTorch implementation of NeuKron [9], the state-of-the-art deep-learning-
based method for compressing sparse tensors. We implemented TensorCodec using
PyTorch.
Experimental Setup: To measure the accuracy of the compression methods, we
employed fitness, which is defined as:
$\vspace{-1mm}{fitness}=1-({\lVert\mathbf{\mathcal{X}}-\hat{\mathbf{\mathcal{X}}}\rVert_{F}}/{\lVert\mathbf{\mathcal{X}}\rVert_{F}})$
where $\hat{\mathbf{\mathcal{X}}}$ is an approximation of
$\mathbf{\mathcal{X}}$ obtained by the evaluated method. This metric has been
widely used to assess the approximation accuracy of tensor decomposition
methods [34, 35]. The fitness value is smaller than 1, with higher fitness
indicating more accurate approximation. For comparing compressed sizes, we
used the double-precision floating-point format for all methods. We encoded
the order of $N_{k}$ indices in each $k$-th mode using $N_{k}\log_{2}N_{k}$
bits, by representing each integer from 0 to $N_{k}-1$ in $\log_{2}N_{k}$
bits. Orders of all mode indices were stored separately in TensorCodec and
NeuKron (and also in SZ3 and TTHRESH when our reordering scheme was applied to
them). We conducted each experiment 5 times to obtain the average and standard
deviation of running time and fitness. Refer to Section IV of [27] for more
details.
### V-B Q1. Compression Performance
We evaluated the compression performance of TensorCodec and its competitors,
focusing on the trade-off between compressed size and fitness. The
hyperparameters of the compared methods were configured to yield similar
compressed sizes, as detailed in Section IV of [27]. A time limit of 24 hours
was set for running each method.
TensorCodec outperforms all competitors by offering a superior trade-off
between compressed size and fitness, as shown in Figure 3. Particularly, on
the Stock dataset, TensorCodec achieves a compressed size 7.38$\times$ smaller
than that of the second-best method, while offering similar fitness. Moreover,
when the compressed size is almost the same, the fitness of TensorCodec is
3.33$\times$ greater than that of the competitor with the best fitness. While
NeuKron is specifically designed for sparse tensors and performs relatively
better in such datasets, including the Uber dataset, TensorCodec still
outperforms it on all datasets.
(a) Action (b) Air Quality (c) PEMS-SF (d) Uber
Figure 4: Every component of TensorCodec is effective for improved
compression. Fitness significantly increases as more components are added.
### V-C Q2. Ablation Study
To demonstrate the effectiveness of each component of TensorCodec, we
considered the following variants of it:
1. 1.
TensorCodec-R: a variant of TensorCodec that does not include the repeated
reordering of mode indices.
2. 2.
TensorCodec-T: a variant of TensorCodec-R that does not initialize the order
of mode indices based on the $2$-approximate solution of Metric TSP.
3. 3.
TensorCodec-N: a variant of TensorCodec-T without an auto-regressive neural
network for generating TT cores. Instead, it simply applies TTD to the folded
tensor.
We compared the reconstruction accuracy in terms of fitness for the four small
datasets, to evaluate the performance of the TensorCodec variants.
TensorCodec-N is optimized by TT-SVD, while the other variants are optimized
by gradient descents. We set the TT ranks of TensorCodec-N so that its number
of parameters is closest to that of the other variants, which have exactly the
same number of parameters.
Each component of TensorCodec contributes to improved compression. As shown in
Figure 4, the fitness of TensorCodec increases as more components are added.
An exception occurs in the Uber dataset, where the fitness shows a slight
increase from TensorCodec-T to TensorCodec-N. This is likely to be influenced
by the difference in the number of parameters (i.e., TensorCodec-N uses 10,776
bytes while TensorCodec-T uses 10,104 bytes for parameters).
Figure 5: The compression time of TensorCodec scales near linearly with the
number of entries in the input $4$-order tensor. See Section VI of [27] for
the near-linear scalability on $3$-order tensors.
### V-D Q3. Scalability
We investigate the scalability of TensorCodec with respect to the tensor size
while fixing $R$ and $h$ values to 8.
Compression speed: We examined the increase in compression time for
TensorCodec as the number of entries in the input tensors grows. Specifically,
we measured the time taken for order initialization and a single iteration of
the model and order optimization on five synthetic full tensors with varying
sizes. Their sizes can be found in Table V of [27]. Each entry of them is
sampled uniformly between 0 and 1.
The compression time of TensorCodec, specifically, all its three steps.
increases near linearly with the number of entries in the input tensor, as
illustrated in Figure 5. This is because the number of tensor entries is much
larger and also increases much faster than all other terms in the time
complexity in Theorem 4 in Section IV-E.
Reconstruction speed: We also examined the scalability of reconstruction from
the compressed output of TensorCodec. We generated synthetic tensors of orders
3 and 4, with mode sizes increasing from $2^{6}$ to $2^{18}$ by a factor of 2.
Then, we measured the total elapsed time for reconstructing $2^{18}$ entries
sampled uniformly at random from each tensor.
The reconstruction time of TensorCodec is sublinear as shown in Figure 6.
Specifically, the time increases logarithmically with the size of the largest
mode in the input tensor. This result is in line with Theorem 3 presented in
Section IV-E.
(a) 3-order tensor (b) 4-order tensor
Figure 6: The reconstruction time of TensorCodec is sub-linear. When
reconstructing the input tensor entries from the outputs of TensorCodec, the
required time is logarithmic with respect to the largest mode size.
### V-E Q4. Further Investigation
Order of mode indices: We compare the mode index orders obtained by
TensorCodec and NeuKron in the NYC dataset.666We set $R$ to 10 and $h$ to 9
for TensorCodec. For NeuKron, we set $h$ to $14$ so that it uses slightly more
space than TensorCodec. However, NeuKron’s compression did not terminate
within 24 hours, so we used the result obtained at the time limit. The indices
of the first two modes of the dataset indicate regions of New York City (see
Section III of [27]).
As visualized in Figure 7. the reordering process of TensorCodec assigns
nearby locations to similar mode indices. This is particularly evident in
Manhattan and Staten Island, where the colors of nearby locations are similar
to each other. However, the mode indices obtained by NeuKron do not exhibit
any clear patterns. This result demonstrates the effectiveness of the
reordering technique used in TensorCodec.
(a) Reordering by TensorCodec
(b) Reordering by NeuKron
Figure 7: In the NYC dataset, where mode indices correspond to locations
within New York City, TensorCodec assigns nearby locations to similar mode
indices, as shown in (a), resulting in comparable colors for nearby areas,
particularly in Manhattan and Staten Island. However, the order obtained by
NeuKron in (b) does not exhibit any clear patterns.
Expressiveness: We evaluated the expressiveness of TensorCodec by analyzing
the number of parameters required for typical tensor decomposition methods to
accurately approximate tensors generated by TensorCodec. To conduct the
experiment, we created two tensors, one with size $256\times 256\times 256$
and another with size $128\times 128\times 128\times 128$, by unfolding
higher-order tensors generated by TensorCodec.777The TT rank $R$ and hidden
dimension $h$ of NTTD were both set to $5$, and the parameters of NTTD were
randomly initialized.
The results, depicted in Figure 8, demonstrate that TensorCodec is capable of
expressing high-rank tensors that require a large number of parameters for
typical tensor decomposition methods to approximate accurately.
(a) 3-order tensor (b) 4-order tensor
Figure 8: Expressiveness of TensorCodec. It is capable of generating high-
rank tensors, which require a large number of parameters to be precisely
approximated by traditional tensor decomposition methods.
### V-F Q5. Compression Speed
We compare the total compression time of all considered methods. We use the
experimental setups in Section V-B. For each dataset, we chose the setting of
$R$ and $h$ with the smallest number of parameters. For the competitors, we
used the settings with the fitness most similar to our method. As shown in
Figure 9, while TensorCodec takes less time than NeuKron, which sometimes
takes more than 24 hours, it is still slower than the non-deep-learning-based
methods.
In Sections VII of [27], we provide additional experimental results, including
hyperparameter sensitivity analysis.
## VI Conclusions
Figure 9: The total compression time of TensorCodec and the competitors.
NeuKron runs out of time on some datasets, taking more than 24 hours.
In this work, we study the problem of lossy compression of tensors and devise
TensorCodec that compactly compresses tensors without strong data assumptions.
Our main ideas are (a) Neural Tensor-Train Decomposition, (b) high-order
tensor folding, and (c) mode-index reordering. Using $8$ real-world tensors,
we show the following advantages of TensorCodec:
* •
Concise: It offers up to $7.38\times$ more compact compression than the best
competitor with similar reconstruction error.
* •
Accurate: When compression ratios are comparable, TensorCodec achieves up to
$3.33\times$ lower reconstruction error than the competitor with the smallest
error.
* •
Scalable: Empricially, its compression time is linear in the number of
entries. It reconstructs each entry in logarithmic time with respect to the
largest mode size.
We plan to improve the speed of TensorCodec and theoretically analyze its
approximation accuracy in future work.
## Acknowledgements
This work was funded by the Korea Meteorological Administration Research and
Development Program “Developing Intelligent Assistant Technology and Its
Application for Weather Forecasting Process” (KMA2021-00123). This work was
supported by Institute of Information & Communications Technology Planning &
Evaluation (IITP) grant funded by the Korea government (MSIT) (No.
2022-0-00157, Robust, Fair, Extensible Data-Centric Continual Learning) (No.
2019-0-00075, Artificial Intelligence Graduate School Program (KAIST))
(No.2021-0-02068, Artificial Intelligence Innovation Hub) This work was
supported by the National Research Foundation of Korea (NRF) grant funded by
the Korea government (MSIT) (No. NRF-2020R1C1C1008296) (No.
NRF-2021R1C1C1008526).
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|
Compressible Euler Equations with nonlocal Interactions]Global solutions of the one-dimensional compressible Euler equations with nonlocal interactions
via the inviscid limit
J.A. Carrillo]Jos$\acute{\mathrm{E}}$ A. Carrillo
J.A. Carrillo: Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
G.-Q. Chen]Gui-Qiang G. Chen
Gui-Qiang G.Chen: Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
D. F. Yuan]Difan Yuan
D. F. Yuan: School of Mathematical Sciences, Beijing Normal University and Laboratory of
Mathematics and Complex Systems; Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
<EMAIL_ADDRESS><EMAIL_ADDRESS>
E. Zatorska]Ewelina Zatorska
E. Zatorska: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
We are concerned with the global existence of finite-energy entropy solutions of the one-dimensional
compressible Euler equations with (possibly) damping, alignment forces, and nonlocal interactions: Newtonian repulsion and quadratic confinement.
Both the polytropic gas law and the general gas law are analyzed.
This is achieved by constructing a sequence of solutions of the one-dimensional compressible Navier-Stokes-type
equations with density-dependent viscosity under the stress-free boundary condition
and then taking the vanishing viscosity limit.
The main difficulties in this paper arise from the appearance of the nonlocal terms.
In particular, some uniform higher moment estimates for the compressible Navier-Stokes equations on expanding intervals with stress-free boundary conditions are obtained by careful design of the approximate initial data.
[2010] 35Q35, 35Q31, 35B25, 35B44, 35L65, 35L67, 76N10, 35R09, 35R35, 35D30, 76X05, 76N17
§ INTRODUCTION
Hydrodynamic models of collective behavior provide a comprehensive framework for characterizing
the behavior of vast assemblies of interacting individuals.
In most of the interesting cases, these models can only be formally derived from the particle-type
systems capturing precise interactions between the individuals.
In cases where full mathematical rigor is attainable,
it is usually established through
the mean-field limit techniques or the BBGKY hierarchies;
see [6, 27, 4]
and the classical references therein on the matter.
In this paper, we are interested in a specific
one-dimensional (1-D) example of such models, which captures
local repulsion, and nonlocal attraction and repulsion forces, as well as nonlocal alignment.
This model is described by the compressible Euler equations (CEEs) with the corresponding interaction forces:
\begin{align}\label{1.1}
\begin{cases}
\rho_t+m_x=0,\\[1mm]
m_t+(\frac{m^2}{\rho}+P(\rho))_x =\lambda m-\rho \partial_xW\ast\rho+\displaystyle\int_{\R}\varpi(x-y)\big(\rho(x) m(y)-\rho(y) m(x)\big)\,\dd y,
\end{cases}
\end{align}
for $t>0$ and $x\in \R$,
where $\rho\geq0$ is the density, $m$ is the momentum, $P$ is the pressure.
The nonlocal attraction-repulsion interaction forces are described by potential $W$ of the form:
\begin{equation*}%\label{defW}
\end{equation*}
The long-range attraction between the individuals is captured by the quadratic confinement part $\frac{1}{2}x^2,$ while the short-range repulsion is described by the Newtonian part of the potential $-|x|.$
The consensus in velocities (the alignment) among individuals is described by the weight $\varpi$ that satisfies:
$$\varpi\in L^{\infty}(\R),\quad \varpi(x)=\varpi(-x), \quad \varpi\geq0.$$
Finally, the linear term $\lambda m$ in our system stands for damping, if the coefficient $\lambda$ is non-positive.
§.§ The equation of state
We consider a general pressure law
$P(\rho)$ satisfying hypotheses ($\mathcal{H}$) formulated below:
($\mathcal{H}.1$) The pressure function $P(\rho)\in C^1([0,\infty))\cap C^4((0,\infty))$ satisfies the hyperbolic and genuinely nonlinear conditions:
\begin{equation}\label{pressure2}
P'(\rho)>0,\quad 2P'(\rho)+\rho P''(\rho)>0\qquad\,\, \text{for }\rho>0.
\end{equation}
($\mathcal{H}.2$) There exist constants $\rho_{*}>0$, $\gamma_1\in (1,3)$, and $\kappa_1>0$ such that
\begin{equation}\label{pressure3}
P(\rho)=\kappa_{1} \rho^{\gamma_1}\big(1+\mathcal{P}_1(\rho)\big) \qquad \text { for } \rho \in[0, \rho_{*}),
\end{equation}
for some function $\mathcal{P}_1(\rho)\in C^4((0,\infty))$
\[|\mathcal{P}_1^{(j)}(\rho)|\leq C_{*}\rho^{\gamma_1-1-j} \qquad \text { for }\rho\in (0,\rho_{*}),\ \text { and } j=0,\cdots,4,\]
where $C_{*}>0$ depends only on $\rho_{*}$.
($\mathcal{H}.3$) There exist constants $\rho^{*}> \rho_{*}>0$, $\gamma_2\in (1,\gamma_{1}]$, and $\kappa_{2}>0$, such that
\begin{equation}\label{pressure4}
P(\rho)=\kappa_2\rho^{\gamma_2}\big(1+\mathcal{P}_2(\rho)\big) \qquad \text { for } \rho \in[\rho^{*},\infty),
\end{equation}
for some function $\mathcal{P}_2(\rho)\in C^{4}((0,\infty))$ satisfying
\[|\mathcal{P}_{2}^{(j)}(\rho)|\leq C^{*}\rho^{-\epsilon-j}\qquad \text { for }\rho\in [\rho^{*},\infty),\ \epsilon>0,\ \text { and } j=0,\cdots,4,\]
where $C^{*}>0$ depends only on $\rho^{*}$.
Without loss of generality, by scaling, $\kappa_1$ may be chosen to be equal to $\kappa_1=\frac{(\gamma_1-1)^2}{4\gamma_1}$.
A special example of such an equation of state is the pressure-density relation for the polytropic gases:
\begin{equation}\label{pressure1}
P(\r)=\kappa\r^{\gamma}\qquad \text{ for $\gamma\in(1,\infty)$},
\end{equation}
where $\gamma$ is the adiabatic exponent. In this case, we can choose $\kappa=\frac{(\gamma-1)^2}{4\gamma}$.
§.§ Initial data
We consider the Cauchy problem for (<ref>) with the initial data:
\begin{equation}\label{1.6}
(\rho, m)|_{t=0}=(\rho_0, m_0)(x)
\end{equation}
such that $\rho_0\in L^1_+(\mathbb{R})$ and $\frac{m_0}{\sqrt{\rho_0}}\in L^2(\mathbb{R})$
with finite initial total mass and initial second moment:
\begin{equation}\label{1.7}
M:=\int_{\mathbb{R}} \rho_0(x)\,\dd x<\infty, \qquad M_2:=\int_{\R}x^2\rho_0(x)\,\dd x<\infty.
\end{equation}
We further assume that the initial data are of finite energy:
\begin{align}\label{2.1}
\mathcal{E}_0:=
\int_{\mathbb{R}}\Big(\frac{1}{2} \Big|\frac{m_0}{\sqrt{\rho_0}}\Big|^2
+ \rho_0e(\rho_0)+\frac{1}{2}\rho_0 W\ast\r_0\Big)(x) \,\dd x<\infty,
\end{align}
where $e(\rho)$ is the internal energy:
\begin{equation*}%\label{internal2}
\frac{P(\s)}{\s^2} \,\dd \s.
\end{equation*}
In particular, for the polytropic case (<ref>),
\begin{equation*}%\label{internal1}
\end{equation*}
§.§ Main objectives of the paper
The goal of the present paper is to establish the global-in-time existence of finite-energy entropy solutions
of system (<ref>) without restriction on the size of initial data. Our method of choice is
the vanishing viscosity limit for the strong solutions of the compressible
Navier-Stokes equations (CNSEs). More precisely, we construct the solution by means of a sequence of approximate problems:
\begin{align}\label{1.8}
\begin{cases}
\dis \r^{\varepsilon}_t+ (\rho^{\varepsilon}u^{\varepsilon})_x=0,\\[1mm] \dis (\r^{\varepsilon}u^{\varepsilon})_t+\big(\rho^{\varepsilon}(u^{\varepsilon})^2+P(\rho^{\varepsilon})\big)_x= \varepsilon(\mu(\r^{\varepsilon})u^{\varepsilon}_x)_x+\lambda\r^{\varepsilon}u^{\varepsilon}-\r^{\varepsilon}\partial_xW\ast\r^{\varepsilon}\\[1mm]
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\displaystyle+\rho^{\v}\int_{\R}\varpi(x-y)(u^{\v}(y)-u^{\v}(x))\rho^{\v}(y)\,\dd y,
\end{cases}
\end{align}
on bounded intervals:
\begin{equation}\label{3.1}
\Omega_T^\varepsilon=\big\{(t,x)\ :\, b^-_\varepsilon(t)\leq r\leq b^+_\varepsilon(t),\,0\leq t\leq T \big\},
\end{equation}
with the free boundaries $b^\pm_\varepsilon(t)$, expanding to infinity as $\varepsilon\rightarrow 0^+$.
In the above system, the viscosity coefficient is equal to
$\varepsilon\mu(\rho)=\varepsilon\rho^{\alpha},$ $\alpha\geq0$,
and the parameter $\v\in (0,1]$ denotes the inverse of the Reynolds number.
A derivation of CNSEs (<ref>) from the Boltzmann equations (without the nonlocal term)
may be found in
Liu-Xin-Yang [36], in which
the viscosity is not constant but depends on the temperature. This dependence
can be translated into the dependence of the viscosity on the density for the isentropic flow.
§.§ The state of the art
There is a huge literature devoted to the study of the existence of solutions to CEEs
either via the analysis of the vanishing artificial viscosity limit (see [1, 23, 29] for instance),
or by constructing the finite difference scheme.
The global existence of solutions with large initial data in $L^{\infty}$
was first established by DiPerna [26]
for $\gamma=1+\frac{2}{2n+1}$ with $n\geq2$ integer
by using the artificial viscosity method for the density equation.
For the general interval $1<\gamma\leq\frac{5}{3},$
the global existence problem was solved by Ding-Chen-Luo [24, 25] and Chen [11] by approximation via the Lax-Friedrichs scheme. The case of adiabatic exponent $\gamma>\frac{5}{3}$
was covered by Lions-Perthame-Tadmor [35] and
Lions-Perthame-Souganidis [34] via the vanishing viscosity method.
We also refer to [33] for blow-up results on CEEs and
to Chen-LeFloch [14, 15]
for relevant results on CEEs with general pressure law.
Concerning approximation by the CNSEs, the $L^{\infty}$ estimates for the approximate solutions
are not expected when the initial data only are of finite energy.
Therefore, we work with the finite-energy framework, which was first considered
by LeFloch-Westdickenberg [32] for $ 1<\gamma<\frac{5}{3}$, and was late
by Chen-Perepelitsa [16]
to the whole physical range of adiabatic exponents $\gamma>1.$
In particular, they established
a compensated compactness framework (based on the earlier work by Tartar [42] and Murat [38])
and proved the vanishing viscosity limit of the solutions of the 1-D CNSEs
to the corresponding finite-energy solutions
of CEEs for large initial data for all $\gamma>1$ in [16].
In this result, the initial density at the far-field is allowed to be positive, and the viscosity
is independent of the density, i.e., $\alpha=0.$
This framework has been subsequently extended by Chen-Schrecker [18] and
Chen-Wang [19] to study spherically symmetric Euler equations.
More recently, Chen-He-Wang-Yuan [12] established the global existence of finite-energy solutions
of the multi-dimensional Euler-Poisson equations for both compressible gaseous stars and plasmas
with large spherically symmetric initial data.
Adapting the approach developed in [12], He-Wang [28] proved the vanishing viscosity limit
for the 1-D CNSEs.
For CEEs with general pressure law, Shrecker-Schulz [40, 41]
proved the vanishing viscosity limit for the 1-D CNSEs under asymptotically isothermal assumptions.
More recently, Chen-Huang-Li-Wang-Wang [13] proved the vanishing physical viscosity
limit for CNSPEs with general pressure law with large spherically symmetric initial data,
in which an $L^p$ compensated compactness framework for general pressure law was also established.
Finally, the existence of solutions for a general class of density-degenarate viscous models has been
studied in [21, 22].
The main challenges tackled in this paper in comparison to the previous results
are due to the presence of the nonlocal terms. The existence of global-in-time solutions to the nonlocal Euler system (<ref>) on the whole line has never been proven before. In contrast, it is known that the classical solutions of (<ref>) without pressure and alignment, i.e., $P(\rho)=\varpi=0$, may blow up in finite time [5] and that they can be approximated by a degenerate vanishing viscosity method [8]. More recently, Carrillo-Galtung [9] showed that for such pressureless Euler systems in 1-D, the Lagrangian and entropy solutions are equivalent, which was then used to explore the long-time-asymptotics of the solutions. Moreover, Chaudhuri et al. [10] studied the two-velocity reformulation of system (<ref>) and derived an energy-type inequality, in the spirit of the Bresch-Desjardins estimate [2, 3]. It was then used to construct the weak solutions and to study their long-time behaviour leading to the same density profiles as in [5] or [9].
The weak solutions to CNSEs with nonlocal attraction-repulsion forces in three dimensions were recently constructed in [37], see also [7] for the constant viscosity case, where the long-time behavior of the solutions was considered.
Finally, global-in-time well-posedness theory for the Euler equations with Riesz interactions and linear damping was recently established in [20] for the initial data near the equilibrium state and on the torus. We construct global entropy solutions to (<ref>) via global weak solutions to the nonlocal Navier-Stokes system (<ref>) with stress-free boundary conditions.
The proof that these solutions are uniformly bounded, in terms of $\varepsilon$,
despite the presence of nonlocal terms is another novelty. A new procedure for approximating the initial data, with the bounded second moment of the density, is implemented to obtain the propagation in time of the second moment of the density so that the confinement is controlled; see Lemma <ref>.
The paper has the following structure. In <ref>, we first introduce the definitions of finite-energy entropy solutions for CEEs with nonlocal interactions
and present the main result – Theorem <ref>. We then describe the construction of the approximate solutions for CNSEs and state the inviscid limit theorem, Theorem <ref>. The rest of the paper is then devoted to the proof of this result for two types of equations of state: the polytropic equation of state (<ref>), and the general pressure satisfying hypotheses ($\mathcal{H}$).
In <ref>, we derive the basic estimates for both the density and velocity, and prove the $H^{-1}_{\rm loc}$–compactness of entropy dissipation measures for the approximate solutions for the polytropic case.
In <ref>, we prove the global existence of finite-energy entropy weak solutions for this case, $\it{i.e.},$ we prove Theorem <ref>.
In <ref>, we develop the arguments from <ref> and <ref> to prove the global existence of finite-energy entropy solutions for CEEs for the general pressure law case. In this case, we prove the inviscid limit by proving the $W^{-1,p}_{\rm loc}$ compactness of entropy dissipation measures for the approximate solutions for $1\leq p<2$. Finally, in Appendix <ref>, we explain how the approximate initial data sequences $(\rho^\v_0(x),\rho^\v_0u^{\v}_0(x))$ can be constructed with desired estimates, regularity, and boundary compatibility for the polytropic equation of state.
§ DEFINITIONS AND MAIN THEOREMS
In this section, we define the notion of entropy solutions of system (<ref>)
and then formulate our main results of this paper.
§.§ Entropy and entropy flux pairs for CEEs
Recall that a pair
is called an entropy-entropy flux pair (entropy pair, for short) of the Euler system (<ref>)
if they satisfy
\begin{equation}\label{entropypair}
\nabla q(\rho,m)=\nabla \eta(\rho,m)\nabla \Big(\begin{matrix}m\\
\frac{m^2}{\rho}+P(\rho)\end{matrix}\Big);
\end{equation}
see Lax [31].
Furthermore, $\eta(\rho,m)$ is called a weak entropy if
\begin{align}\nonumber
\eta|_{\rho=0}=0\qquad\,\, \text{for any fixed $u=\frac{m}{\rho}$}.
\end{align}
From [35],
it is well known (cf. [16, 17, 35])
that, for the polytropic case, any weak entropy $(\eta,q)$ can be represented by
\begin{align}\label{weakentropy}
\begin{cases}
\displaystyle \eta^\psi(\rho,m)=\eta(\rho,\rho u)=\int_{\R}\chi(\rho;s-u)\psi(s)\,\dd s,\\[3mm]
\displaystyle q^\psi(\rho,m)=q(\rho,\rho u)=\int_{\R}(\theta s+(1-\theta)u)\chi(\rho;s-u)\psi(s)\, \dd s,
\end{cases}
\end{align}
where $\chi(\rho;s-u)=[\rho^{2\theta}-(s-u)^2]_{+}^{\fb}$
with $\fb=\frac{3-\gamma}{2(\gamma-1)}>-\frac{1}{2}$
and $\theta=\frac{\gamma-1}{2}$ is the weak entropy kernel.
In particular, when $\psi(s)=\frac{1}{2}s^2,$
the entropy pair is the pair of the mechanical energy and the associated flux:
\begin{align*}\label{m-entropy}
\eta^{*}(\rho,m)=\frac{m^2}{2\rho}+\rho e(\rho),\quad q^{*}(\rho,m)=\frac{m^3}{2\rho^2}+m (\rho e(\rho))',
\end{align*}
From (<ref>), any entropy satisfies
\begin{equation}\label{2.7}
\eta_{\rho\rho}-\frac{P'(\rho)}{\rho^{2}}\eta_{uu}=0
\end{equation}
with $u=\frac{m}{\rho}$. It has been proved in [14, 15, 34, 35]
that any regular weak entropy can be generated by the convolution of a smooth function $\psi(x)$
with a fundamental solution $\chi(\rho,u,s)$ of the entropy equation (<ref>), i.e.,
\begin{equation}\label{2.8}
\eta^{\psi}(\rho,u)=\int_{\R}\chi(\rho,u,s)\psi(s)\,\mathrm{d}s.
\end{equation}
The corresponding entropy flux is generated from the flux kernel $\sigma(\rho,u,s)$ (see (<ref>)), i.e.,
\begin{align}\label{2.9}
\end{align}
The entropy kernel $\chi=\chi(\r,u,s)$ is a fundamental solution of the entropy equation (<ref>):
\begin{equation}\label{6.1}
\left\{\begin{aligned}
\dis&\chi_{\r\r}-\frac{P'(\r)}{\r^2}\chi_{uu}=0,\\
\dis&\chi\vert_{\r=0}=0,\quad \chi_{\r}\vert_{\r=0}=\delta_{u=s}.
\end{aligned}
\right.
\end{equation}
As pointed out in [14] that equation (<ref>)
is invariant under the Galilean transformation, which implies that
For simplicity, we write it as $\chi(\r,u,s)=\chi(\r,u-s)$ below
when no confusion arises.
The corresponding entropy flux kernel $\sigma(\r,u,s)$ satisfies the Cauchy problem
for $\sigma-u\chi$:
\begin{equation}\label{2.10}
\left\{\begin{aligned}
\dis&(\sigma-u\chi)_{\r\r}-\frac{P'(\r)}{\r^2}(\sigma-u\chi)_{uu}=\frac{P''(\r)}{\r}\chi_{u},\\
\dis&(\sigma-u\chi)\vert_{\r=0}=0,\quad (\sigma-u\chi)_{\r}\vert_{\r=0}=0.
\end{aligned}
\right.
\end{equation}
It can be checked that $P(\rho)$, described by (<ref>)–(<ref>),
satisfies all the conditions in [14, 15]. In particular, $\sigma-u\chi$
is Galilean invariant; see [14].
§.§ Existence of solutions to CEEs
We are now ready to define the notion of solutions to system (<ref>) and to formulate our main result.
A pair of functions $(\rho, m)(t,x)$ with $\rho \in L^\infty(\mathbb{R}_+; L^1_+(\mathbb{R}))$
and $\frac{m}{\sqrt{\rho}}
\in L^\infty(\mathbb{R}_+; L^2(\mathbb{R}))$ for $\mathbb{R}_+:=(0,\infty)$
is a finite-energy entropy solution of the Cauchy problem (<ref>) and (<ref>), with $P(\rho)$ satisfying hypotheses ($\mathcal{H}$), if the following conditions hold:
The total mass is conserved:
\int_{\mathbb{R}} \rho(t,x)\,\dd x=\int_{\mathbb{R}} \rho_0(x)\,\dd x=:M \qquad\mbox{ a.e. $t\geq 0$},
and $(\frac{m}{\sqrt{\rho}})(t,x)=0$ a.e. on
the vacuum states $\{(t,x)\,:\, \rho(t,x)=0\}$.
(ii) For a.e. $t>0$, the total energy is not increasing:
\begin{align*}
\int_{\mathbb{R}}\Big(\frac{1}{2} \Big|\frac{m}{\sqrt{\rho}}\Big|^2
+\rho e(\rho)+\frac{1}{2} \rho W\ast\r\Big)(t,x)\, \dd x
\leq \mathcal{E}_0.
\end{align*}
(iii) For any $\Psi(t,x)\in C^1_0([0,\infty)\times\mathbb{R})$,
\begin{align}
&\int_{\mathbb{R}^{2}_+} \big(\rho \Psi_t + m\Psi_x\big)\,\dd x\dd t
+\int_{\mathbb{R}} \rho_0(x) \Psi(0,x)\,\dd x=0,\label{2.5}\\[1mm]
&\int_{\mathbb{R}^{2}_+}\Big(m\Psi_t +\big(\frac{m^2}{\r}
+P(\rho)\big)\Psi_x\Big)\,\dd x\dd t
+\int_{\mathbb{R}} m_0(x)\Psi(0,x)\,\dd x\nonumber\\[-1mm]
&\quad=\int_{{\mathbb{R}^2_+}}\Big(-\lambda m+\r \partial_xW\ast\r-\int_{\R}\varpi(x-y)\big(\rho(x) m(y)-\rho(y) m(x)\big)\,\dd y\Big) \Psi \,\dd x\dd t.\label{2.6}
\end{align}
For any convex function $\psi(s)$ with subquadratic growth at infinity and any entropy pair $(\eta^{\psi},q^{\psi})$ defined in (<ref>)–(<ref>),
\begin{align*}
&-\eta^{\psi}_m\Big(\lambda m+\rho \int_{\R}\varpi(x-y)\big(\rho(x) m(y)-\rho(y) m(x)\big)\,\dd y-\rho \partial_x W\ast\rho \Big)\leq 0
\end{align*}
is satisfied in the sense of distributions.
We now state the main result of this paper.
Consider problem (<ref>) with initial data (<ref>) satisfying (<ref>)–(<ref>). Let $P(\rho)$ satisfy hypotheses ($\mathcal{H}$).
Then there exists a global-in-time finite-energy entropy solution $(\rho, m )(t,x)$ of problem (<ref>) and (<ref>),
in the sense of Definition <ref>.
In particular, there exists a global-in-time finite-energy entropy solution for
problem (<ref>) and (<ref>) with polytropic gases equation of state (<ref>).
The interaction potential can also be extended to the more general form:
$W(x)=-|x|+\frac{x^2}{2}+\tilde W (x),$ with
$|\tilde W (x)|\leq C x^2$ at infinity and smooth.
Even more, we can consider
$W(x)=-|x|+\frac{x^2}{2}+\frac{|x|^{\nu}}{\nu}$ for $1<\nu<2.$
We can allow the communication kernel $\varpi(x)\in C(\R\backslash\{0\})$, and
$$\varpi(x)\mathbf{I}_{B(0,R)}\in L^{\frac{\gamma}{\gamma-1}}(\R), \quad \varpi(x)\mathbf{I}_{\R\backslash B(0,R)}\in L^{\infty}(\R) \qquad \text{for any } R>0.$$
For example, we can choose the singular communication weight $\varpi(x)$ as $\varpi(x)=\frac{1}{|x|^{a}}$ for $a<\frac{\gamma-1}{\gamma}.$
Our results also hold for an asymptotically isothermal pressure law $(\gamma_2=1)$, that is, $P(\rho)/\rho= O(1)$ in the limit $\rho\rightarrow \infty$; see [40, 41].
§.§ Existence of solutions to CNSEs and their inviscid limit
In this paper, we do not prove Theorem <ref> directly. Instead, the solution from Definition <ref> is obtained as a limit of the regular solutions of CNSEs (<ref>)
on a truncated domain $\Omega_T^\varepsilon$ with the moving boundary. We define
\begin{equation}\label{3.1}
\Omega_T^\varepsilon=\big\{(t,x)\ :\, b^-_\varepsilon(t)\leq r\leq b^+_\varepsilon(t),\,0\leq t\leq T \big\},
\end{equation}
where $\{x=b^{\pm}_\varepsilon(t)\,:\,0<t\leq T\}$ are the free boundaries determined by
\begin{equation}\label{3.2}
\displaystyle \frac{\dd }{\dd t}b^\pm_\varepsilon(t)=u^\varepsilon(t,b^\pm_\varepsilon(t)), \quad b^\pm_\varepsilon(0)=\pm b_\varepsilon \qquad\quad \mbox{for $t>0$}.
\end{equation}
On the free boundaries $x=b^\pm_\varepsilon(t)$,
the stress-free boundary condition is imposed:
\begin{equation}\label{3.3}
\big(P(\rho^{\v})-\v\mu(\rho^{\varepsilon}) u^{\varepsilon}_x\big)(t,b^\pm_\varepsilon(t))
=0 \qquad \mbox{for $t>0$}.
\end{equation}
The initial intervals $[-b_\varepsilon, b_\varepsilon]$ approximate the whole line when $\v\to0^+$. More specifically, we assume explicitly that
b_\varepsilon=\varepsilon^{-p}\quad \text{with}\quad p>\left\{\begin{array}{ll}\frac{\gamma}{\gamma-\alpha} & \quad\text{for polytropic gases,}\\
\frac{\gamma_1}{\gamma_1-\alpha}& \quad\text{for general pressure law.}
\end{array}\right.
Let $(\rho_0, m_0)(x)$ satisfy (<ref>)–(<ref>).
The initial data for the approximate system are given by
\begin{equation}\label{3.4}
(\rho^{\varepsilon},\rho^{\varepsilon} u^{\varepsilon})(0,x)
=(\r_0^{\v},\rho_0^{\v}u_0^{\v})(x)\qquad\,\mbox{for $x\in [-b_\varepsilon,b_\varepsilon]$},
\end{equation}
where $(\rho_0^{\v}, u_0^{\v})(x)$ are obtained by the smooth approximation of
the original data $(\rho_0,m_0)$, constructed in Appendix A. In particular, they satisfy Lemmas <ref>–<ref>.
We further define
\begin{align}\label{3.5}
\mathcal{E}_{0}^{\v}:&=
\int_{-b_\varepsilon}^{b_\varepsilon}\Big(\frac{1}{2}\rho_0^{\v}|u_0^{\v}|^2+ \rho_0^{\v}e(\rho_0^{\v})+\frac{1}{2}\rho_0^{\v}W\ast\r^{\varepsilon}_0\Big)\,
\dd x,\,\,\, &\\[1.5mm]
\mathcal{E}_1^{\v}
:&= \v^2 \int_{-b_\varepsilon}^{b_\varepsilon} (\rho^{\varepsilon}_0)^{2\alpha-3}|\rho^{\varepsilon}_{0x}|^2\,\dd x,\label{3.6}
\end{align}
where the convolution in (<ref>) is understood as $\int^{b_\varepsilon}_{-b_\varepsilon}W(x-y)\rho_0^{\varepsilon}(y)\,\dd y$.
We are now ready to state the main theorem of our paper.
Let the hypotheses of be satisfied.
Let $\gamma_1\geq\gamma_2>1$ and $\frac{2}{3}<\alpha\leq1$, and let
$\{(\rho^{\v},\rho^{\v}u^{\v})\}_{\v\in (0,1]}$ be a sequence of the strong solutions to (<ref>) with initial and boundary data specified in (<ref>)–(<ref>) and satisfying:
\begin{align}
& 0< C^{-1}_{\varepsilon}\leq \r^{\varepsilon}_0(x)\leq C_{\varepsilon},\quad
(\r^{\varepsilon}_0(b))^{\gamma_1} b_\v= (\r^{\varepsilon}_0(-b_\varepsilon))^{\gamma_1} b_\varepsilon \leq C_0
\qquad\mbox{ for $\varepsilon \in (0,1]$},\nonumber\\[2mm]
&(\rho^\v_0,\rho_0^\v u^{\v}_0)(x) \rightarrow (\rho_0,m_0)(x)
\qquad \mbox{as $\v\to 0^+$ in $L^{q}_{\rm loc}(\R)\times L^1_{\rm loc}(\R)$} \mbox{ for $q\in\{1,\gamma_2\}$},\nonumber\\[2mm]
&(\mathcal{E}_0^\v,\mathcal{ E}_1^\v)\rightarrow (\mathcal{E}_0,0) \qquad \mbox{as $\v\to 0^+$}, \nonumber\\[2mm]
&\int_{-\infty}^{\infty}\rho^\v_0(x)\,\dd x= M,
\qquad \mathcal{E}^{\v}_0+\v^{-1}\mathcal{E}_1^\v\le C_0,
\quad \nonumber \\[2mm]
&\int^{\infty}_{-\infty}x^2\rho^{\v}_0(x)\,\dd x\,\rightarrow \,\int^{\infty}_{-\infty}x^2\rho_0 (x)\,\dd x=M_2\qquad\mbox{as $\v\to 0^+$},\nonumber
\end{align}
where $C_0>0$ is a constant independent of $\varepsilon\in (0,1]$ which may depend on $(\mathcal{E}_0, M, M_2, \gamma,\alpha)$, while
$C_\v>0$ is a constant depending on $\v>0$.
Then there exist both a subsequence (still denoted) $(\rho^{\v},m^{\v} )(t,x)$ with $m^{\v}=\rho^\v u^\v$, and a vector-valued function $(ρ, m)(t,x)$
such that, as $→̌0^+$,
\begin{align}\nonumber
\begin{split}
&(\rho^{\v},m^{\v})(t,x)\rightarrow (\rho,m)(t,x)\qquad
\mbox{a.e. $(t,x)\in \R_+\times\R$},\\
&(\rho^{\v},m^{\v})(t,x)\rightarrow (\rho,m)(t,x)\qquad
\mbox{in $L^{q_1}_{\rm loc}(\R^2_+)\times L^{q_2}_{\rm loc}(\R^2_+)$ with $q_1\in[1,\gamma_2+1)$ and $q_2\in [1,\frac{3(\gamma_2+1)}{\gamma_2+3})$}.
\end{split}
\end{align}
Moreover, $(ρ,m)(t,x)$
is a global finite-energy entropy
solution of the Cauchy problem \eqref{1.1} and \eqref{1.6}
in the sense of {\rm Definition \ref{definition-Euler}}.
In particular, the result is true for the polytropic case \eqref{pressure1}.
\end{theorem}
\begin{remark}
In {\rm Theorem \ref{thm3.3}}, we need $\frac{2}{3}<\alpha\leq 1$; see \eqref{A.4222}. However, note that the condition $\frac{2}{3}<\alpha<\gamma$ is needed only for obtaining the uniform estimates
for the approximate solutions{\rm ;} see {\rm Lemmas \ref{lem4.1}--\ref{lem4.2}} and
{\rm Lemmas \ref{lem5.2}--\ref{lem5.9}}.
In particular, our results cover the Saint-Venant model for shallow water ($\alpha=1$ and $\gamma=2$),
even with the nonlocal terms.
\end{remark}
\begin{remark} The regular solutions for the compressible Navier-Stokes problem \eqref{1.8} with the new nonlocal terms and with stress-free boundary conditions \eqref{3.2}--\eqref{3.3} can be obtained by following the arguments similar to {\rm [30, 39]}; see also {\rm[10]} for the Dirichlet boundary conditions.
\end{remark}
\section{Uniform Estimates of Approximate Solutions}\label{section3}
This section is dedicated to several uniform estimates with respect to $∈̌(0, 1]$
for the regular solutions of \eqref{1.8} on the bounded time-dependent domain \eqref{3.1} with corresponding boundary conditions. Although some of the estimates hold also for general $P(ρ)$ specified through hypotheses $(ℋ)$, for the purposes of this section, we restrict ourselves to the polytropic equation of state \eqref{pressure1}. The general pressure case is discussed in \S\ref{section5}.
We drop the $$̌-superscript of approximate solutions when no confusion arises.
In all of the estimates from now on,
$C>0$ is a universal constant independent of $\v$, which may depend on $ℰ_0, M, M_2, C_0,T,γ$, and $α$, and can be different at each occurrence. For abbreviation, we also denote
\begin{equation}\label{DefV}
V=V(t,x):=\int_{\R}\varpi(x-y)\Big(m(y)-\frac{m(x)}{\rho(x)}\rho(y)\Big)\,\dd y
=\int^{b^+(t)}_{b^-(t)}\varpi(x-y)\Big(m(y)-\frac{m(x)}{\rho(x)}\rho(y)\Big)\,\dd y
\end{equation}
for $x∈(b^-(t),b^+(t))$.
\subsection{Conservation of mass and Lagrangian coordinates}
We present the reformulation of IBVP \eqref{1.8} and \eqref{3.1}--\eqref{3.4}
in the so-called Lagrangian coordinates,
which is equivalent to the Eulerian formulation for sufficiently regular solutions.
It follows from $(<ref>)_1$ and \eqref{3.2} that
\begin{equation*}
\frac{\dd}{\dd t}\int_{b^-(t)}^{b^+(t)}\rho(t,x)\,\dd x= (\rho u)(t,b^+(t))-(\rho u)(t,b^-(t))-\int_{b^-(t)}^{b^+(t)}(\rho u )_x(t,x)\,\dd x=0,
\end{equation*}
which, due to \eqref{A.23-1}, yields that
\begin{equation}\label{3.7}
\int_{b^-(t)}^{b^+(t)}\rho(t,x)\,\dd x=\int_{b^-(t)}^{b^+(t)}\rho^{\v}_0(x)\,\dd x= M \qquad\,\, \mbox{for any $t\geq0$}.
\end{equation}
For $x∈[b^-(t),b^+(t)]$ and $t∈[0,T]$, we define the Lagrangian
coordinates $(τ,ξ)$ as
\begin{equation*}
\xi=\int_{b^-(t)}^x \rho(t,y)\,\dd y,\qquad \tau=t,
\end{equation*}
which transform the domain with the free boundary: $[0,T]×[b^-(t),b^+(t)]$
into the fixed domain: $[0,T]×[0,M]$.
A direct calculation shows that
\begin{align*}\label{2.9}
\begin{cases}
\displaystyle \frac{\partial \xi}{\partial t}=-\rho u,\quad \frac{\partial \xi}{\partial x}=\rho,\quad\frac{\partial\tau}{\partial t}=1, \quad\frac{\partial\tau}{\partial x}=0,\\[3mm]
\displaystyle \frac{\partial x}{\partial \tau}=u,\quad \frac{\partial x}{\partial\xi}=\frac{1}{\rho},\quad \frac{\partial t}{\partial\tau}=1,\quad \frac{\partial t}{\partial \xi}=0.
\end{cases}
\end{align*}
Applying the Euler-Lagrange transformation, IBVP \eqref{1.8} and \eqref{3.1}--\eqref{3.4}
\begin{equation}\label{3.8}
\begin{cases}
\displaystyle\rho_\tau+\rho^2u_{\xi}=0,\\[2mm]
\displaystyle u_\tau+ P(\rho)_\xi-\varepsilon (\mu(\rho)\rho u_{\xi})_{\xi}-\lambda u-V+\rho(W\ast\rho)_{\xi}=0,
\end{cases}
\end{equation}
for $(τ,ξ)∈[0,T]×[0,M]$, and
\begin{equation}\label{3.9}
\big(P(\rho)-\v \mu(\rho)\rho u_{\xi}\big)(\tau,0)=0, \quad\big(P(\rho)-\v \mu(\rho)\rho u_{\xi}\big)(\tau,M)=0
\qquad\,\, \mbox{ for $\tau\in[0,T]$}.
\end{equation}
With a slight abuse of notation, we denote by $V$ a function defined in \eqref{DefV}, taken as a function of $ξ$.
The fixed boundaries $ξ=0,M$ correspond to the free boundaries $x=b^±(t)$ in the Eulerian coordinates.
\subsection{Basic energy estimate}
To begin with, we obtain the basic energy estimate for problem \eqref{1.8}
and \eqref{3.1}--\eqref{3.4}.
\begin{lemma}\label{lem4.1}
For a smooth solution of problem \eqref{1.8} and \eqref{3.1}--\eqref{3.4}, the following estimate holds{\rm :}
\begin{align}
&\int^{b^+(t)}_{b^-(t)}\Big(\frac{1}{2}\rho u^2+\rho e(\rho)
+\frac{1}{2}\rho W\ast\rho\Big)(t,x)\,\dd x
\lr{\v\rho^{\alpha}|u_x|^2-\lambda \rho u^2}(s,x)\,\dd x\dd s\nonumber\\
&+\frac{1}{2}\int^t_0\int^{b^+(s)}_{b^-(s)}\int^{b^+(s)}_{b^-(s)}\varpi(x-y)\rho(x)\rho(y)|u(y)-u(x)|^2\,\dd y\dd x\dd s
= \mathcal{E}^{\varepsilon}_0\le C_0.\label{4.1}
\end{align}
Here and hereafter, $C_0>0$ is the constant from the statement of {\rm Theorem \ref{thm3.3}}, independent of $\varepsilon\in (0,1]$.
\end{lemma}
\noindent{\bf Proof.}
It follows from $(<ref>)_1$ that
\begin{align}\label{4.2}
\end{align}
Multiplying $(<ref>)_2$ by $u,$ we have
\begin{align}\label{4.3}
\Big(\frac{u^2}{2}+e(\rho)\Big)_{\tau}+\varepsilon\rho^{1+\alpha}(u_{\xi})^2
+ \big((P(\rho)-\varepsilon\mu(\rho)\rho u_{\xi})u\big)_{\xi}
-\lambda u^2-u V+\rho u (W\ast\rho)_{\xi}=0.
\end{align}
Notice that
\begin{align}
-\int^M_0u V\,\dd \xi
\big(u(y)-u(x)\big)\rho(y)\,\dd y\Big)\dd x\nonumber\\
\rho(y)\big(|u(x)|^2-u(x)u(y)\big)\,\dd y\dd x\nonumber\\
&=\frac{1}{2}\int^{b^+(t)}_{b^-(t)}\int^{b^+(t)}_{b^-(t)}\varpi(x-y)\rho(x)\rho(y)|u(y)-u(x)|^2\,\dd y\dd x,\nonumber
\end{align}
where we have used the symmetry of $ϖ(x)$.
For the other nonlocal term, our aim is to prove that
\begin{equation}\label{result}
\int^M_0\rho u(W \ast \rho)_{\xi}\,\dd \xi=\frac{\dd}{\dd t}\int^{b^+(t)}_{b^-(t)}\frac{1}{2}\rho W
\ast \rho \,\dd x.
\end{equation}
First we rewrite the right-hand-side (RHS) to obtain:
\begin{align}\label{4.8a}
&\frac{\dd}{\dd t}\int^{b^+(t)}_{b^-(t)}\frac{1}{2}\rho W\ast\rho \,\dd x\nonumber\\
&\,\,=\int^{b^+(t)}_{b^-(t)}\Big(\frac{1}{2}\rho W\ast\rho\Big)_t\,\dd x
+\frac{1}{2}(\rho W\ast\rho)(t,b^+(t))\frac{\dd b^+(t)}{\dd t}
-\frac{1}{2}(\rho W\ast\rho)(t,b^-(t))\frac{\dd b^-(t)}{\dd t}\nonumber\\
&\,\,=\frac{1}{2}\int^{b^+(t)}_{b^-(t)}\lr{\rho_t W\ast\rho+\rho W\ast\rho_t}\,\dd x
+\frac{1}{2}(u\rho W\ast\rho)(t,b^+(t))
-\frac{1}{2}(u\rho W\ast\rho)(t,b^-(t)).
\end{align}
Using the change of the order of integration
and the explicit form of $W(x)=-|x|+|x|^2/2$ (in particular $W(x-y)=W(y-x)$), we obtain
\begin{align*}
&\int^{b^+(t)}_{b^-(t)}\rho W\ast\rho_t\,\dd x\nonumber\\
&=\int^{b^+(t)}_{b^-(t)}\rho(t,x)\big(\int^{b^+(t)}_{b^-(t)}W(x-y)\rho(t,y)\,\dd y\big)_t\,\dd x\nonumber\\
&=\int^{b^+(t)}_{b^-(t)}\rho_t(t,y)\big(\int^{b^+(t)}_{b^-(t)}W(x-y)\rho(t,x)\,\dd x\big)\,\dd y
+(\rho u W\ast \rho)(t,b^+(t))-(\rho u W\ast \rho)(t,b^-(t))\nonumber\\
&=\int^{b^+(t)}_{b^-(t)}\rho_t(t,y)\big(\int^{b^+(t)}_{b^-(t)}W(y-x)\rho(t,x)\,\dd x\big)\,\dd y
+(\rho u W\ast \rho)(t,b^+(t))-(\rho u W\ast \rho)(t,b^-(t))\nonumber\\
&=\int^{b^+(t)}_{b^-(t)}\rho_t(t,x)\big(\int^{b^+(t)}_{b^-(t)}W(x-y)\rho(t,y)\,\dd y\big)\,\dd x
+(\rho u W\ast \rho)(t,b^+(t))-(\rho u W\ast \rho)(t,b^-(t))\nonumber\\
&=\int^{b^+(t)}_{b^-(t)}\rho_t W\ast\rho \,\dd x+(\rho u W\ast \rho)(t,b^+(t))-(\rho u W\ast \rho)(t,b^-(t)).
\end{align*}
Thus, it follows that the first term on the RHS of \eqref{4.8a} is equal to
\frac{1}{2}\int^{b^+(t)}_{b^-(t)}\big(\rho_t W\ast\rho+\rho W\ast\rho_t\big)\,\dd x
=\int^{b^+(t)}_{b^-(t)}\rho_t W\ast\rho \,\dd x+\frac{1}{2}(\rho u W\ast \rho)(t,b^+(t))-\frac{1}{2}(\rho u W\ast \rho)(t,b^-(t)).
Therefore, we have proven that
\begin{align}\label{new3.10}
&\frac{\dd}{\dd t}\int^{b^+(t)}_{b^-(t)}\frac{1}{2}\rho W\ast\rho \,\dd x=\int^{b^+(t)}_{b^-(t)}\rho_t W\ast\rho\,\dd x
+(u\rho W\ast\rho)(t,b^+(t))
-(u\rho W\ast\rho)(t,b^-(t)).
\end{align}
On the other hand, it follows directly from $(<ref>)_1$ that
\begin{align}
\int^M_0\rho u (W\ast\rho)_{\xi}\,\dd \xi
&=\int^{b^+(t)}_{b^-(t)}\rho u (W\ast\rho)_x\,\dd x\nonumber\\
&=(\rho u W\ast\rho)(t,b^+(t))-(\rho u W\ast\rho)(t,b^-(t))-\int^{b^+(t)}_{b^-(t)}(\rho u)_x W\ast\rho\,\dd x\nonumber\\
&=(\rho u W\ast\rho)(t,b^+(t))-(\rho u W\ast\rho)(t,b^-(t))+\int^{b^+(t)}_{b^-(t)}\rho_t W\ast\rho\,\dd x.\label{4.4}
\end{align}
Comparing \eqref{new3.10} with \eqref{4.4}, we deduce \eqref{result}.
Since $W=-|x|+|x|^2/2,$
we have that $W+1/2≥0.$ and so, using \eqref{3.7}, we obtain
\frac{\dd}{\dd t}\int^{b^+(t)}_{b^-(t)}\frac{1}{2}\rho W\ast\rho \,\dd x=\frac{\dd}{\dd t}\int^{b^+(t)}_{b^-(t)}\frac{1}{2}\rho\Big(W+\frac{1}{2}\Big)\ast\rho \,\dd x.
Integrating \eqref{4.3} over $[0,τ]×[0,M]$, using the Gr\"{o}nwall inequality
and the stress-free boundary conditions \eqref{3.9}, pulling the resultant equation
back to the Eulerian coordinates, we obtain \eqref{4.1}.
This completes the proof.
\smallskip
\subsection{Second moment estimate}
We now derive the second moment estimate for the density.
\begin{lemma}\label{lemma4.2}
There exists $C=C>0$, independent of $\v$, such that
\begin{align}\label{4.10}
\int^{b^+(t)}_{b^-(t)}x^2\rho(t,x) \,\dd x\leq C.
\end{align}
\end{lemma}
\noindent{\bf Proof.}
Using $\eqref{1.8}_1,$ we have
\begin{align*}%\label{4.15}
\frac{\dd}{\dd t}\int^{b^+(t)}_{b^-(t)}x^2\rho \,\dd x
&=(b^+(t))^2(\rho u)(t,b^+(t))
-(b^- (t))^2(\rho u)(t,b^-(t))+\int^{b^+(t)}_{b^-(t)} x^2\rho_t\,\dd x\\
&=2\int^{b^+(t)}_{b^-(t)} x\rho u\,\dd x,
\end{align*}
such that
\frac{\dd}{\dd t}\int^{b^+(t)}_{b^-(t)}x^2\rho \,\dd x
\leq\int^{b^+(t)}_{b^-(t)}x^2\rho \,\dd x+\int^{b^+(t)}_{b^-(t)}\rho u^2 \,\dd x.
Then we conclude \eqref{4.10} by using the Gr\"{o}nwall inequality.
\subsection{Higher-order estimates of the density and the pressure}
In this section, we derive several higher-order estimates
for the density and the pressure.
To start, we analyze the behavior of density $\rho$ on the free boundary.
It follows from $\eqref{3.8}_1$ and \eqref{3.9} that
\begin{align*}
\frac{\dd}{\dd\tau}(\rho^{\alpha-\gamma}(\tau,M))=-\frac{\kappa(\alpha-\gamma)}{\varepsilon}.
\end{align*}
Then we have
\begin{align}\nonumber
\rho(\tau,M)=\rho_0(M)\,\Big(1+\frac{\kappa(\gamma-\alpha)}{\varepsilon}(\rho_0(M))^{\gamma-\alpha}\tau\Big)^{-\frac{1}{\g-\alpha}},
\end{align}
so that
\begin{align}\nonumber
\rho(t,b^+(t))=\rho_0(b)\,\Big(1+\frac{\kappa(\gamma-\alpha)}{\varepsilon}(\rho_0(b))^{\gamma-\alpha}t\Big)^{-\frac{1}{\g-\alpha}}.
\end{align}
We obtain
\begin{align}\label{4.29}
\rho(t,b^+(t))=\rho(t,b^-(t))=\rho_0(b)\Big(1+\frac{\kappa(\g-\alpha)}{\v}(\r_0(b))^{\gamma-\alpha}t\Big)^{-\frac{1}{\g-\alpha}}\leq \rho_0(b),
\end{align}
where we have used that $\rho_0(b)=\rho_0(-b).$
From \eqref{4.29}, it follows in particular that the third and fourth term
on the left-hand side (LHS) of \eqref{4.30} below is nonnegative.
\begin{lemma} \label{lem2.2}
Let $\alpha\geq0$ and $\rho^{\gamma}_0(\pm b)b\leq C_0$.
Then, for any given $T>0$, there exists $C>0$ such that, for any $t\in[0,T]$,
\begin{align}\label{4.30}
&\varepsilon^2\int^{b^+(t)}_{b^-(t)}(\rho^{2\alpha-3}\rho^2_x)(t,x)\,\dd x
+\varepsilon\kappa\gamma\int^t_0\int^{b^+(t)}_{b^-(t)}(\rho^{\alpha+\gamma-3}\rho^2_x)(s,x)\,\dd x\dd s\nonumber\\
\Big(\rho^{2\gamma-\alpha}(s,b^+(s))b^+(s)-\rho^{2\gamma-\alpha}(s,b^-(s))b^-(s)\Big)\,\dd s\nonumber\\
&+\kappa\Big(\rho^{\gamma}(t,b^+(t))b^+(t)-\rho^{\gamma}(t,b^-(t))b^-(t)\Big)\leq C .
\end{align}
\end{lemma}
\noindent{\bf Proof.}
It is direct to calculate from $\eqref{3.8}_2$ that
\begin{align}\label{4.31}
\Big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\Big)_{\tau}+P(\rho)_{\xi}-\lambda u-V+\rho(W\ast\rho)_{\xi}=0,
\end{align}
where we have used the fact that
\begin{align}\nonumber
\mu(\rho)\rho u_{\xi}=\rho^{1+\alpha}u_{\xi}=-\rho^{\alpha-1}\rho_{\tau}=-\frac{1}{\alpha}(\rho^{\alpha})_{\tau}.
\end{align}
Multiplying \eqref{4.31} by $u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi},$ we obtain
\begin{align}\label{4.32}
+\frac{\kappa\rho^{\gamma-1}}{\gamma-1}+\frac{1}{2}\rho W\ast\rho\Big)_{\tau}
&+\big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\big)\big(-\lambda u-V+\rho(W\ast\rho)_{\xi}\big)=0.
\end{align}
Using $\eqref{3.8}_1$ and \eqref{3.9}, we have
\begin{align}\nonumber
\rho_{\tau}(\tau,M)=-\frac{\kappa}{\varepsilon}\rho^{\gamma+1-\alpha}(\tau,M),
\qquad \rho_{\tau}(\tau,0)=-\frac{\kappa}{\varepsilon}\rho^{\gamma+1-\alpha}(\tau,0).
\end{align}
Then we obtain
\begin{align}
\big(P(\rho)u\big)(\tau,M)-\big(P(\rho)u\big)(\tau,0)
&=\kappa\rho^{\gamma}(\tau,M)\frac{\dd b^+(\tau)}{\dd \tau}-\kappa\rho^{\gamma}(\tau,0)\frac{\dd b^-(\tau)}{\dd \tau}\nonumber\\
\end{align}
Integrating \eqref{4.32} over $[0,M]$ yields
%, it has
\begin{align}\label{4.34}
+\frac{\kappa\rho^{\gamma-1}}{\gamma-1}\Big)\,\dd \xi
+\varepsilon\kappa\gamma\int^M_0\rho^{\alpha+\gamma-2}(\rho_{\xi})^2\,\dd \xi \nonumber\\
&\,\,\, +\kappa\Big(\rho^{\gamma}(\tau,M)b^+(\tau)-\rho^{\gamma}(\tau,0)b^-(\tau)\Big)_{\tau}
&=\int^M_0\big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\big)\big(-\lambda u-V+\rho(W\ast\rho)_{\xi}\big)\,\dd \xi.
\end{align}
For the last term in \eqref{4.34}, we have
\begin{align}
&\int^M_0\big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\big)\big(-\lambda u-V+\rho(W\ast\rho)_{\xi}\big)\,\dd \xi\nonumber\\
&\leq\int^M_0\big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\big)^2\,\dd \xi
+\int^M_0\big(-\lambda u-V+\rho(W\ast\rho)_{\xi}\big)^2\,\dd \xi\nonumber\\
&\leq \int^M_0\big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\big)^2\,\dd \xi
+C\int^M_0u^2\,\dd \xi+C\int^M_0 V^2\,\dd \xi+ C\int^M_0\big(\rho(W\ast\rho)_{\xi}\big)^2\,\dd \xi\nonumber\\
&\leq \int^M_0\big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\big)^2\,\dd \xi
+ C\int^{b^+(t)}_{b^-(t)}\rho\big(\partial_xW\ast\rho\big)^2\,\dd x+C,\nonumber
\end{align}
where we have used the fact that
\begin{align}
&\int^M_0 V^2\,\dd \xi=\int^{b^+(t)}_{b^-(t)}\rho V^2\,\dd x\nonumber\\
&=\int^{b^+(t)}_{b^-(t)}\rho(t,x)\Big(\int^{b^+(t)}_{b^-(t)}\varpi(x-y)\big(u(t,y)-u(t,x)\big)\rho(t,y)\,\dd y\Big)^2\,\dd x\nonumber\\
&\leq C\int^{b^+(t)}_{b^-(t)}\bigg(\rho(t,x)\Big(\int^{b^+(t)}_{b^-(t)}\big(\rho(t,y)+(\rho u^2)(t,y)\big)\,\dd y\Big)^2
+u^2(t,x)\bigg)\,\dd x
\leq C.\nonumber
\end{align}
Notice that
\begin{align}\label{4.36}
=\int^{b^+(t)}_{b^-(t)}\big(1-2H(x-y)+x-y\big)\rho(t,y)\,\dd y,
\end{align}
where $H(\cdot)$ is the Heaviside function.
We now rewrite the RHS of \eqref{4.36} using
\begin{align*}
\int^{b^+(t)}_{b^-(t)}\rho(t,y)\,\dd y=M,\qquad \int^{b^+(t)}_{b^-(t)}H(x-y)\rho(t,y)\,\dd y=\int^x_{b^-(t)}\rho(t,y)\,\dd y.
\end{align*}
We obtain
\begin{align}\nonumber
(\partial_xW\ast\rho)(t,x)=M-2\int^x_{b^-(t)}\rho(t,y)\,\dd y+xM-\int^{b^+(t)}_{b^-(t)}y\rho (t,y)\,\dd y.
\end{align}
It follows from \eqref{3.7} and \eqref{4.10} that
$$\Big|\int^{b^+(t)}_{b^-(t)}|x|\rho(t,x) \,\dd x\Big|\leq\int^{b^+(t)}_{b^-(t)}(x^2+1)\rho(t,x) \,\dd x\leq C.$$
Then we have
\begin{align}\nonumber
\int^{b^+(t)}_{b^-(t)}\rho(t,x)(\partial_xW\ast\rho)^2(t,x)\,\dd x
\leq\int^{b^+(t)}_{b^-(t)}\rho\Big(3M+|x|M+\int^{b^+(t)}_{b^-(t)}|y|\rho(t,y)\,\dd y\Big)^2\,\dd x\leq C.
\end{align}
\smallskip
Integrating \eqref{4.32} over $[0,\tau]$ leads to
%we have
\begin{align}\nonumber
+\frac{\kappa}{\gamma-1}\rho^{\gamma-1}\Big)\,\dd \xi
+\varepsilon\kappa\gamma\int^{\tau}_0\int^M_0\rho^{\alpha+\gamma-2}(\rho_{\xi})^2\,\dd \xi \dd s\nonumber\\
&\,\,\,+\kappa\Big(\rho^{\gamma}(\tau,M)b^+(\tau)-\rho^{\gamma}(\tau,0)b^-(\tau)\Big)+\frac{\kappa\gamma}{\varepsilon}\int^{\tau}_0\Big(\rho^{2\gamma-\alpha}(s,M)b^+(s)-\rho^{2\gamma-\alpha}(s,0)b^-(s)\Big)\,\dd s\nonumber\\
&\leq C\int^M_0\Big(\frac{1}{2}\big(u_0+\frac{\varepsilon}{\alpha}(\rho^{\alpha}_0)_{\xi}\big)^2
+\frac{\kappa}{\gamma-1}\rho^{\gamma-1}_0\Big)\,\dd \xi
\end{align}
This completes the proof.
%of Lemma \ref{lem2.2}.
\medskip
Our next aim is to show that the domain $\Omega_T$ expands to the whole physical space $[0,T]\times\R$, that is,
$\displaystyle\inf_{t\in[0,T]}b(t)\rightarrow\infty$ as $b\rightarrow\infty$. We have the following result.
\begin{lemma}[[28], Lemma 2.3]\label{lem4.4}
Choose $0\leq\alpha<\gamma,$ $p>\frac{\gamma}{\gamma-\alpha},$ and $b:=\varepsilon^{-p}.$
Then, for any given $T>0$, there exists $\varepsilon_0>0$ such that, for $\v\in(0,\v_0]$,
%there holds
\begin{align*}
\pm b^\pm(t)\geq \frac12 b
%,\qquad\,\,b^-(t)\leq -\frac12 b,
\qquad \mbox{for $t\in[0,T]$}.
\end{align*}
\end{lemma}
In particular, the proof of Lemma \ref{lem4.4} is independent of the nonlocal terms. Similar result for the general pressure law is proven later; see Lemma \ref{lem5.6}.
\begin{lemma}[\bf Higher integrability of the density]\label{lem4.5}
Let $(\r,u)$ be the smooth solution of \eqref{1.8} and \eqref{3.1}--\eqref{3.4},
and let the assumption of {\rm Lemma \ref{lem4.4}} hold.
Then, for any $K\Subset[b^-(t),b^+(t)]$ for any $t \in[0,T]$, there exists
$C(K)>0$ independent of $\v\in (0,1]$ such that
\begin{align}\label{4.39}
\int_0^T\int_K\r^{\g+1}(t,x)\,\dd x \dd t\leq C(K).
\end{align}
\end{lemma}
\noindent{\bf Proof.} We divide the proof into two steps.
\smallskip
\noindent {\emph{Step 1.}} For given $K\Subset[b^-(t),b^+(t)]$ for any $t\in[0,T]$,
there exist $r_1$ and $r_2$ such that $K\Subset (r_1,r_2)\Subset[b^-(t),b^+(t)]$.
Let $w(x)$ be a smooth function with $\text{supp}\,w\subseteq(r_1,r_2)$ and $w(x)=1$ for $x\in K$.
Multiplying $\eqref{1.8}_2$ by $w(x)$, we have
\begin{align}\label{4.40}
&(\r u w)_t+\big((\r u^2+P(\rho))w\big)_x\nonumber\\
&=(\rho u^2+P(\rho))w_x+\varepsilon(\rho^{\alpha}w u_x)_x-\varepsilon\rho^{\alpha}u_xw_x+\lambda \rho uw+\rho Vw-\rho\partial_xW\ast\rho w.
\end{align}
Integrating \eqref{4.40} over $[r_1,x)$ to obtain
\begin{align}\label{4.41}
(\rho u^2+P(\rho))w
&=\varepsilon\rho^{\alpha}wu_x+\int^{x}_{r_1}\Big((\rho u^2+P(\rho))w_y-\varepsilon
\rho^{\alpha}u_yw_y\Big)\,\dd y-\frac{\dd}{\dd t}\int^x_{r_1}\rho uw\,\dd y\nonumber\\
&\quad+\int^x_{r_1}\lambda \rho uw\,\dd y-\int^x_{r_1}\rho w \partial_x W\ast\rho \,\dd y+\int^x_{r_1}\rho w V\,\dd y.
\end{align}
Multiplying \eqref{4.41} by $\rho w$ and performing a direct calculation, we obtain
\begin{align}\label{4.42}
\r P(\rho) w^2&=\v \rho^{\alpha+1}w^2u_x-\Big(\rho w\int^x_{r_1}\rho uw\,\dd y\Big)_t-\Big(\rho uw\int^x_{r_1}\rho uw\,\dd y\Big)_x\nonumber\\
&\quad+\rho uw_x\int^x_{r_1}\rho uw\,\dd y+\rho w\int^x_{r_1}\Big((\rho u^2+P(\rho))w_y-\varepsilon\rho^{\alpha}u_yw_y\Big)\,\dd y\nonumber\\
&\quad+\lambda \rho w\int^x_{r_1}\rho u w\,\dd y-\rho w\int^x_{r_1}\rho\partial_xW\ast\rho w\,\dd y+\rho w\int^x_{r_1}\rho Vw\,\dd y:=\sum^{8}_{i=1}K_{i}.
\end{align}
\noindent{\emph{Step 2.}} To estimate $K_i, i=1,\cdots, 8$, in \eqref{4.42}, we first notice that
\begin{align}
\int_{b^{-}(t)}^{b^+(t)}\rho|u|\, \dd x\leq \int_{b^-(t)}^{b^+(t)} (\rho+\rho u^2)\,\dd x\leq C.\nonumber
\end{align}
Then it follows from \eqref{3.7} and \eqref{4.1} that
\begin{align}
\left| \int_0^T\int_{r_1}^{r_2} K_2\, \dd x\dd t \right|&=\left| \int_0^T\int_{r_1}^{r_2} \Big(\rho w\int^x_{r_1}\rho uw\,\dd y\Big)_t\, \dd x\dd t \right|\nonumber\\
&\leq\left|\int^{r_2}_{r_1}\Big(\rho w\int^x_{r_1}\rho u w\,\dd y\Big)(T,x)\,\dd x\right|+\left|\int^{r_2}_{r_1}\Big(\rho w\int^x_{r_1}\rho u w\,\dd y\Big)(0,x)\,\dd x\right|\leq C.\nonumber
\end{align}
Similarly to \cite[Lemma 3.5]{Chen2021}, we obtain
%have that
\begin{align}
\left|\int_0^T\int_{r_1}^{r_2} K_1\,\dd x\dd t \right|&\leq C(r_1,r_2)
+\v\int^T_0\int^{r_2}_{r_1}\rho^{\gamma+1}w^2\,\dd x\dd t,\nonumber\\
\left| \int_0^T\int_{r_1}^{r_2}K_3\, \dd x\dd t \right|&=\left| \int_0^T\int_{r_1}^{r_2}
\Big(\rho uw\int^x_{r_1}\rho uw\,\dd y\Big)_x\, \dd x\dd t \right|=0,\nonumber\\
\left| \int_0^T\int_{r_1}^{r_2} K_4\, \dd x\dd t \right|&=\left| \int_0^T\int_{r_1}^{r_2} \Big(\rho uw_x\int^x_{r_1}\rho uw\,\dd y\Big)\, \dd x\dd t \right|\leq C.\nonumber
\end{align}
Using the fact that $\alpha\leq\gamma,$ we have
\begin{align}
\left| \int_0^T\int_{r_1}^{r_2} K_5\, \dd x\dd t \right|&=\left| \int_0^T\int_{r_1}^{r_2} (\rho uw\int^x_{r_1}\Big((\rho u^2+P(\rho))w_y-\v\rho^{\alpha}w_yu_y\,\dd y\Big)\, \dd x\dd t \right|\nonumber\\
&\leq C
+\left|\v\int^T_0\int^{r_2}_{r_1}\rho w\int^x_{r_1}\rho^{\alpha}w_yu_y\,\dd x\dd t\right|\nonumber\\
&\leq C(r_1,r_2),\nonumber
\end{align}
where we have used the following estimate
\begin{align}
\left|\v\int_0^T\int_{r_1}^{r_2} \rho w\int^x_{r_1}\rho^{\alpha}w_yu_y\, \dd y\dd x\dd t \right|&\leq \v\int^T_0\int^{r_2}_{r_1}|\rho^{\alpha}w_xu_x|\,\dd x\dd t\nonumber\\
&\leq \Big(\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}|u_x|^2\,\dd x \dd t\Big)^{\frac{1}{2}}\Big(\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}|w_x|^2\,\dd x\dd t\Big)^{\frac{1}{2}}\nonumber\\
&\leq C\Big(\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}\,\dd x\dd t\Big)^{\frac{1}{2}}
\leq C\Big(\v\int^T_0\int^{r_2}_{r_1}\rho^{\gamma}\,\dd x\dd t\Big)^{\frac{1}{2}}\nonumber\\
&\leq C(r_1,r_2).\nonumber
\end{align}
Moreover, we have
\begin{align}
\left|\int_0^T\int_{r_1}^{r_2} K_6\,\dd x\dd t \right|&\leq C(,\nonumber
\end{align}
\begin{align*}%\label{4.51}
&\int_0^T\int_{r_1}^{r_2} K_7\,\dd x\dd t \\
&\,\,\,=\int_0^T\int_{r_1}^{r_2} \rho(t,x) w(x)\Big(\int^x_{r_1}\rho(t,y) (\partial_xW\ast\rho)(t,y) w(y) \,\dd y\Big) \dd x\dd t \nonumber\\
&\,\,\,= \int_0^T\int_{r_1}^{r_2} \rho(t,x) w(x)\int^x_{r_1}\rho(t,y)
\Big( \int^{b^+(t)}_{b^-(t)}\big(1-2H(x-z)+x-z\big)\rho(t,z)\,\dd z\Big)w(y)\,\dd y\dd x\dd t \nonumber\\[2mm]
&\int_0^T\int_{r_1}^{r_2} K_8\,\dd x\dd t\nonumber\\
&\,\,\, =\int^T_0\int^{r_2}_{r_1}\rho(t,x)w(x)\int^x_{r_1}\rho(t,y)
\Big(\int_{b^{-}(t)}^{b^+(t)}\varpi(y-z)\big(u(z)-u(y)\big)\rho(t,z)\,\dd z\Big)w(y)\,\dd y\dd x\dd t,\nonumber\\
\end{align*}
which are clearly bounded by $C$.
Integrating \eqref{4.42} over $[0,T]\times[r_1, r_2],$ and collecting all the estimates in this step,
we can obtain \eqref{4.39}.
The proof
is completed.
\medskip
\section{Inviscid Limit for Polytropic Gases}\label{section4}
To prove Theorem \ref{thm3.3}, we intend to apply the compensated compactness framework
from Chen-Perepelitsa [16].
For clarity of presentation, we first focus on the polytropic case \eqref{pressure1}, while
the general pressure case will be discussed in \S 5 below.
We first explore some important properties of several special entropy entropy flux pairs.
\subsection{Choice of a special entropy and entropy flux pair}
Taking $\psi(s)=\frac12 {s}{|s|}$ in \eqref{weakentropy}, the corresponding entropy and entropy flux are represented as
\begin{align}\label{4.52}
\begin{cases}
\displaystyle\eta^{\#}(\r,\r u)=\f12 \r \int_{-1}^1 (u+\r^{\t} s) |u+\r^{\t}s| [1-s^2]_+^{\fb} \dd s,\\[3mm]
\displaystyle q^{\#}(\r, \r u)=\f12 \r \int_{-1}^1 (u+\theta\r^{\t}s)(u+\r^{\t} s) |u+\r^{\t}s| [1-s^2]_+^{\fb} \dd s.
\end{cases}
\end{align}
A direct calculation shows that
\begin{align}\label{4.53}
|\eta^{\#}(\r,\r u)|\leq C_{\g} \big(\r |u|^2+\r^{\g}\big), \qquad\,\, q^{\#}(\r,\r u)\geq C_{\g}^{-1} \big(\r |u|^3+\r^{\g+\t}\big),
\end{align}
where and whereafter $C_\g>0$ is a universal constant depending only on $\g>1$.
We regard $\eta^{\#}$ as a function of $(\rho,m)$ to obtain
\begin{align*}%\nonumber
\eta^{\#}_\r=\int_{-1}^1 \big(-\f12 u+(\t+\f12)\r^{\t} s\big)\,|u+\r^{\t}s| [1-s^2]_+^{\fb} \dd s,\qquad
\displaystyle \eta^{\#}_m=\int_{-1}^1|u+\r^{\t}s| [1-s^2]_+^{\fb} \dd s.
\end{align*}
It is direct to check that
\begin{align}\label{4.54}
|\eta^{\#}_m|\leq C_\g \big(|u|+\rho^\theta\big),\qquad\,\, |\eta^{\#}_\rho|\leq C_\g\big(|u|^2+\rho^{2\theta}\big).
\end{align}
\subsection{Higher integrability of the velocity}
The special entropy pair introduced in \S 4.1 allows us to derive a better estimate for
the integrability of the velocity vector field.
The following lemma is important to control the trace estimates for the higher integrability on the velocity; see \eqref{4.63}.
In fact, we have the boundary parts $(u \eta^{\#})(t,b^\pm(t))$ and $q^{\#}(t,b^\pm(t))$,
and it is impossible to have the uniform trace bound (independent of $\v$) for each of them.
\begin{lemma}[[12], Lemma 3.6]\label{lem2.5}
For the entropy pair defined in \eqref{4.52}, the following cancelation property holds:
\begin{align}\label{4.55}
|q^{\#}-u\eta^{\#}|\leq C_\g \big(\rho^{\gamma} |u|+ \rho^{\gamma+\theta} \big).
\end{align}
\end{lemma}
\medskip
\begin{lemma}\label{lem4.2}
Let $(\r,u)$ be the smooth solution of \eqref{1.8} and \eqref{3.1}--\eqref{3.4},
and let the assumption of {\rm Lemma \ref{lem4.4}} hold.
Then, for any $(r_1,r_2)\Subset [b^-(t), b^+(t)]$, there exists $C(r_1,r_2)>0$ indepedent of $\v\in (0,1]$
such that
\begin{align}\label{4.56}
\int_0^T\int_{r_1}^{r_2} \big(\rho|u|^3+\rho^{\gamma+\theta}\big)(t,x)\, \dd x \dd t\leq C(r_1,r_2).
\end{align}
\end{lemma}
\noindent{\bf Proof.} We divide the proof into four steps.
\smallskip
\noindent{\emph{Step 1.}} Multiplying $\eqref{1.8}_1$ by $\eta^{\#}_\r$ and $\eqref{1.8}_2$ by $ \eta^{\#}_m$,
we have
\begin{align}\label{4.57}
& \eta^{\#}_t+q^{\#}_x = \eta^{\#}_m\, \Big(\v (\rho^{\alpha} u_x)_x+\lambda \rho u+\rho V-\rho \partial_xW\ast\rho\Big).
\end{align}
Using \eqref{2.6}, a direct calculation shows that
\begin{align}
\frac{\dd}{\dd t}\int_x^{b^+(t)}\eta^{\#}\,\dd y
& =\eta^{\#}(t,b^+(t)) \frac{\dd }{\dd t}b^+(t)+\int_x^{b^+(t)}\eta^{\#}_t(t,y)\,\dd y= (u\eta^{\#})(t, b^+(t))+\int_x^{b^+(t)} \eta^{\#}_t(t,y)\,\dd y.\nonumber
\end{align}
Integrating \eqref{4.57} over $[x,b^+(t))$, we have
\begin{align}\label{4.59}
q^{\#}(t,x) &=\Big( \int_x^{b^+(t)}\eta^{\#}(t,y)\,\dd y\Big)_t+\big(q^{\#}-u\eta^{\#}\big)(t,b^+(t))\nonumber\\
&\quad-\v \int_x^{b^+(t)} \eta^{\#}_m(\rho^{\alpha} u_y)_y\,\dd y-
\lambda \int_x^{b^+(t)} \eta^{\#}_m\rho u\,\dd y+\int_x^{b^+(t)} \eta^{\#}_m \rho\partial W\ast\rho\,\dd y-\int^{b^+(t)}_x\eta^{\#}\rho V\,\dd y\nonumber\\
\end{align}
We now bound each term of the RHS of \eqref{4.59}.
\smallskip
\noindent{\emph{Step 2.}} First, for the term involving the trace estimates (second term) in \eqref{4.59}, it follows from \eqref{4.55} and
Lemmas \ref{lem4.1}, \ref{lem2.2}, and \ref{lem2.5}
\begin{align}\label{4.60}
\big|(q^{\#}-u\eta^{\#})(t,b^+(t))\big|\,
\dd x\dd t\leq C(r_1,r_2)\int_0^T\big(\rho^{\gamma+\theta}(t,b^+(t))+(\rho^{\gamma}|u|)(t, b^+(t))\big)\,\dd t.
\end{align}
It follows from \eqref{4.29} that
\begin{align}\label{4.61}
&\int_0^T\big(\rho(t,b^+(t)))^{\gamma+\theta}\dd t
+\frac{\kappa(\gamma-\alpha)}{\varepsilon}\rho^{\gamma-\alpha}_0(b)t\big)^{-\frac{1}{\gamma-\alpha}}\Big)^{\gamma+\theta}\,\dd t
\leq (\rho_0(b))^{\gamma+
\theta}T\leq C.
\end{align}
Similar with argument in [12, 28], we have
\begin{align}\label{4.62}
\big(\rho^{\gamma}|u|\big)(t,b^+(t))\,
\dd x\dd t\leq C\v^{\frac{p(\gamma-\alpha)-\gamma}{2\gamma}}
\leq C,
\end{align}
where $p>\frac{\gamma}{\gamma-\alpha}$.
Substituting \eqref{4.61}--\eqref{4.62} into \eqref{4.60}, we obtain
\begin{align}\label{4.63}
&\Big|\int^T_0\int^{r_2}_{r_1}I_2\,\dd x\dd t\Big|=\Big|\int^T_0\int^{r_2}_{r_1}
\dd x\dd t\Big|\leq C(r_1,r_2).
\end{align}
For the first term of the RHS of \eqref{4.59}, using \eqref{4.1} and \eqref{4.30}, we see that, for $I_1,$
\begin{align}\label{4.64}
&\Big|\int^T_0\int^{r_2}_{r_1}I_1\,\dd x\dd t\Big|=\Big|\int_0^T\int_{r_1}^{r_2}\Big(\int_x^{b^+(t)}\eta^{\#}(\rho,\rho u)\, \dd y\Big)_t\,\dd x\dd t\Big| \nonumber\\
&\leq \Big|\int_{r_1}^{r_2}\int_{b^-(t)}^{b^+(t)}\eta^{\#}(\rho,\rho u)(T,y)\, \dd y\dd x\Big|
+\Big|\int_{r_1}^{r_2}\int_{b^-(t)}^{b^+(t)}\eta^{\#}(\rho_0,\rho_0 u_0)\, \dd y\dd x\Big|\nonumber\\[2mm]
&\leq C(r_1,r_2).
\end{align}
\noindent{\emph{Step 3.}} For $I_3,$ we integrate by parts to obtain
\begin{align}
-\v\int_x^{b^+(t)}\eta^{\#}_m(\rho^{\alpha} u_y)_y\,\dd y = &\,
-\v\Big(\eta^{\#}_m(t,b^+(t))\, (\rho^{\alpha} u_x)(t,b^+(t))-\eta^{\#}_m(t,x)(\rho^{\alpha} u_x)(t,x) \Big) \nonumber\\
&\,\,+\v\int_x^{b^+(t)}\rho^{\alpha} u_y(\eta^{\#}_{mu}u_y+\eta^{\#}_{m\rho}\rho_y)\,\dd y:= J_1+J_2.\label{4.65}
\end{align}
Now, we discuss $J_2$ first:
\begin{align*} %\label{4.66}
|J_2|&=\Big|\v\int^{b^+(t)}_x\Big(\eta^{\#}_{m\rho}\rho^{\alpha}u_y\rho_y+\eta^{\#}_{mu}\rho^{\alpha}u^2_{y}\Big)\,\dd y\Big|\leq C\v\Big|\int^{b^+(t)}_x\rho^{\theta+\alpha-1}u_y\rho_y\,\dd y\Big|+\v\int^{b^+(t)}_x\rho^{\alpha}u^2_{y}\,\dd y\nonumber\\
&\leq \v\int^{b^+(t)}_x\rho^{\gamma+\alpha-3}\rho^2_y\,\dd y+\v\int^{b^+(t)}_x\rho^{\alpha}u^2_{y}\,\dd y,
\end{align*}
where we have used the fact that $|\eta^{\#}_{mu}|\leq C$ and $|\eta^{\#}_{m\r}|\leq C\r^{\t-1}$.
Then we obtain
\begin{align}\label{4.68}
\int^T_0\int^{r_2}_{r_1}|J_2|\,\dd x\dd t&\leq\varepsilon\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\rho^{\gamma+\alpha-3}\rho^2_y\,\dd y\dd x\dd t+\varepsilon\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\rho^{\alpha}u^2_y\,\dd y\dd x\dd t\nonumber\\
&\leq C(r_1,r_2).
\end{align}
For $J_1,$ we have
\begin{align}\label{4.69}
\varepsilon\Big|\int^T_0\int^{r_2}_{r_1} \eta^{\#}_m\rho^{\alpha}u_x\,\dd x\dd t\Big|
&\leq\varepsilon\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}u^2_x\,\dd x\dd t+\varepsilon\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}(|u|+\rho^{\theta})^2\,\dd x\dd t\nonumber\\
&\leq C
+\v\int^T_0\int^{r_2}_{r_1}\rho^{2\theta+\alpha}\,\dd x\dd t+\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}u^2\,\dd x\dd t.
\end{align}
Similar to the argument as in [12, 28], to control \eqref{4.69}, we need that
\begin{align}\label{4.70}
\v\int^T_0\int^{r_2}_{r_1}\rho^{2\theta+\alpha}\,\dd x\dd t
&=\v\int^T_0\int^{r_2}_{r_1}\rho^{\gamma-1+\alpha}\,\dd x\dd t
\leq C\int^T_0\int^{r_2}_{r_1}\rho^{\frac{\gamma-1}{2}}\,\dd x\dd t\nonumber\\
&\leq C\int^T_0\int^{r_2}_{r_1}\rho^{\gamma}\,\dd x\dd t+C(r_1,r_2)
\nonumber\\
&\leq C(r_1,r_2).
\end{align}
It follows from \eqref{4.1} and \eqref{4.30} that, for $\beta=\alpha+\frac{\gamma-1}{2},$
\begin{align}\label{4.71}
\v \rho^{\beta}(t,x)&=\v\Big(\rho^{\beta}(t,x)-\rho^{\beta}(t,b^-(t))+\rho^{\beta}(t,b^-(t))\Big)\nonumber\\
&\leq\v \beta\int^{b^+(t)}_{b^-(t)}\rho^{\beta-1}|\rho_x|\,\dd x+\v\rho^{\beta}_0(b)\nonumber\\
&\leq\beta\Big(\v^2\int^{b^+(t)}_{b^-(t)}\rho^{2\alpha-3}\rho^2_x \,\dd x\Big)^{\frac{1}{2}}\Big(\int^{b^+(t)}_{b^-(t)}\rho^{2(\beta-\alpha)+1}\,\dd x\Big)^{\frac{1}{2}}+\v\rho^{\beta}_0(b)\nonumber\\
&\leq C\Big(\int^{b^+(t)}_{b^-(t)}\rho^{2(\beta-\alpha)+1}\,\dd x\Big)^{\frac{1}{2}}
\leq C.
\end{align}
Using \eqref{4.71}, we have
\begin{align}\label{4.72}
\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}u^2\,\dd x\dd t&\leq \Big(\int^T_0\int^{r_2}_{r_1}\rho|u|^3\,\dd x\dd t\Big)^{\frac{2}{3}}\,\Big(\int^T_0\int^{r_2}_{r_1}\varepsilon^3\rho^{3\alpha-2}\,\dd x\dd t\Big)^{\frac{1}{3}}\nonumber\\
&\leq \Big(\int^T_0\int^{r_2}_{r_1}\rho|u|^3\,\dd x\dd t\Big)^{\frac{2}{3}}\,\Big(\int^T_0\int^{r_2}_{r_1}\varepsilon^3\rho^{3\beta}\,\dd x\dd t+C(r_1,r_2)\Big)^{\frac{1}{3}}\nonumber\\
&\leq C(r_1,r_2)
\Big(\int^T_0\int^{r_2}_{r_1}\rho|u|^3\,\dd x\dd t\Big)^{\frac{2}{3}},
\end{align}
where we have used the assumption: $\alpha\geq\frac{2}{3}.$
Inserting \eqref{4.70} and \eqref{4.72} into \eqref{4.69}, we find that, for $\delta>0,$
\begin{align}\label{4.73}
&\varepsilon\Big|\int^T_0\int^{r_2}_{r_1} \eta^{\#}_m\rho^{\alpha}u_x\,\dd x\dd t\Big|
\leq C(r_1,r_2)
\frac{1}{\delta}+\delta\int^T_0\int^{r_2}_{r_1}\rho|u|^3\,\dd x\dd t.
\end{align}
Using \eqref{4.54} and \eqref{3.3}, we have
%obtain that
\leq C_{\gamma}\big(\rho^{\gamma}|u|+\rho^{\gamma+\theta}\big)(t,b^+(t)).
Similar again to the argument as in [12, 28], we obtain
%it has
\begin{align}\label{4.74}
\int^T_0\int^{r_2}_{r_1}|\v(\eta^{\#}_m\rho^{\alpha}u_x)(t,b^+(t))|\,\dd x\dd t&\leq C(r_1,r_2)
\int^T_0(\rho^{\gamma}|u|+\rho^{\gamma+\theta})(t,b^+(t))\,\dd t\nonumber\\
&\leq C(r_1,r_2).
\end{align}
Combining \eqref{4.73} and \eqref{4.74} yields
%, we have%that
\begin{align}\label{4.75}
\Big|\int^T_0\int^{r_2}_{r_1}J_1\,\dd x\dd t\Big|&\leq \delta
\int^T_0\int^{r_2}_{r_1}\rho |u|^3\,\dd x\dd t
\nonumber\\
\end{align}
Inserting \eqref{4.75} and \eqref{4.68} into \eqref{4.65}, we obtain
\begin{align}\label{4.76}
\Big|\int^T_0\int^{r_2}_{r_1}I_3\,\dd x\dd t\Big|&\leq \delta
\int^T_0\int^{r_2}_{r_1}\rho |u|^3\,\dd x\dd t
\end{align}
\smallskip
\noindent{\emph{Step 4.}} We also obtain the estimates for $I_4$ and $I_5:$
\begin{align*}
\Big|\int^T_0\int^{r_2}_{r_1}I_4\,\dd x\dd t\Big|
&=\Big|\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\rho |u|(|u|+\rho^{\theta}) \,\dd y\dd x\dd t\Big|\nonumber\\
&=\Big|\int^T_0\int^{r_2}_{r_1}\rho u^2\,\dd x\dd t\Big|+\Big|\int^T_0\int^{r_2}_{r_1}\rho^{\gamma}\,\dd x\dd t\Big|
\leq C(r_1,r_2),
\end{align*}
\begin{align*}
&\Big|\int^T_0\int^{r_2}_{r_1}I_5\,\dd x\dd t\Big|\nonumber\\
&=\Big|\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\rho \eta^{\#}_m\partial_xW\ast \rho \,\dd y\dd x\dd t\Big|\nonumber\\
&=\Big|\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\rho\eta^{\#}_m\Big(M-2\int^{y}_{b^-(t)}\rho(t,z)\,\dd z+yM-\int^{b^+(t)}_{b^{-}(t)}z\rho(t,z) \,\dd z\Big)\,\dd y\dd x\dd t\Big|
\leq C(r_1,r_2).
\end{align*}
\begin{align}\label{4.782}
&\Big|\int^T_0\int^{r_2}_{r_1}I_6\,\dd x\dd t\Big|\nonumber\\
&=\Big|\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\eta^{\#}_m\rho V\,\dd y\,\dd x\dd t\Big|\nonumber\\
\Big(\int^{b^+(t)}_{b^-(t)}\varpi(y-z)(u(y)-u(z))\rho(t,z)\,\dd z\Big)\,\dd y\,\dd x\dd t\Big|
\leq C(r_1,r_2).
\end{align}
Combining estimates \eqref{4.63}--\eqref{4.64} with estimates \eqref{4.76}--\eqref{4.782},
we conclude \eqref{4.56} from \eqref{4.53} and \eqref{4.59}.
% \eqref{4.77}, \eqref{4.78} and
\medskip
\subsection{$H_{\rm loc}^{-1}(\mathbb{R}_+^2)$--Compactness}\label{section4.2}
In this section we use the uniform estimates obtained in \S 4.2 to prove the following key lemma, which states the
$H_{\rm loc}^{-1}(\mathbb{R}_+^2)-$compactness of the dissipation measures for the approximate solutions.
\begin{lemma}\label{lemma4.8}
Let $\alpha\in [\frac23, \gamma]$, and let
$(\eta^{\psi},q^{\psi})$ be the weak entropy pair generated by any $\psi\in C_0^2(\mathbb{R})$,
defined in \eqref{weakentropy}.
Then, for the solution sequence $(\rho^{\varepsilon},u^{\varepsilon})$ with
$m^{\varepsilon}=\rho^{\varepsilon}u^{\varepsilon}$ of CNSEs \eqref{1.8} and \eqref{3.1}--\eqref{3.4},
the following sequence
\begin{eqnarray}\nonumber
\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x
\qquad \mbox{is compact in $H_{\rm loc}^{-1}(\mathbb{R}_+\times \mathbb{R} )$.}
\end{eqnarray}
\end{lemma}
\noindent\textbf{Proof}. To prove this lemma, we first recall the following results
for the entropy pair
$(\eta^{\psi},q^{\psi})$ generated by $\psi\in C_0^2(\mathbb{R})$;
see also [17, 16] for details.
For a $C^2$--function $\psi:\mathbb{R}\rightarrow\mathbb{R}$,
compactly supported on the interval $[a,b]$, we have
\begin{eqnarray}\nonumber
{\rm supp}(\eta^{\psi}),\,{\rm supp}(q^{\psi})\subset \left\{{(\rho,m)=(\rho,\rho
u)\,:\, u+\rho^{\theta}\geq a,\quad u-\rho^{\theta}\leq b}\right\}.
\end{eqnarray}
Furthermore, from \cite[Lemma 2.1]{Perepelitsa}, there exists $C_{\psi}>0$ such that, for any
$\rho\geq0$ and $u\in\mathbb{R}$, we have the following facts:
%we have
\begin{enumerate}
\item [(\rmnum{1})] For $\gamma\in(1,3]$,
\begin{align}\label{4.79}
|\eta^{\psi}(\rho,m)|+|q^{\psi}(\rho,m)|\leq C_{\psi}\rho.
\end{align}
\item [(\rmnum{2})] For $\gamma\in(3,\infty)$,
\begin{align}\label{4.80}
|\eta^{\psi}(\rho,m)|\leq C_{\psi}\rho,\quad \ |q^{\psi}(\rho,m)|\leq
\end{align}
\item [(\rmnum{3})] If $\eta^{\psi}$ is considered as a function of $(\rho,m)$,
$m=\rho u$, then
\begin{align}\label{4.81}
|\eta^{\psi}_m(\rho,m)|\leq C_{\psi},\quad\, |\eta^{\psi}_{\rho}(\rho,m)|\leq C_{\psi}(1+\rho^{\theta}),
\end{align}
and, if $\eta^{\psi}_m$ is considered as a
function of $(\rho, u)$, then
\begin{align}\label{4.82}
u)|+|\rho^{1-\theta}\eta^{\psi}_{m\rho}(\rho,\rho u)|\leq C_{\psi}.
\end{align}
\end{enumerate}
Now we are going to prove Lemma \ref{lemma4.8}.
A direct computation on $\eqref{1.8}_1\times\eta^{\psi}_{\rho}(\rho^{\varepsilon},m^{\varepsilon})
\begin{align}\label{4.83}
\displaystyle\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x
\end{align}
For any compact set $K\Subset[b^-(t),b^+(t)]$, using \eqref{4.82} and the Cauchy-Schwartz
inequality, we have
\begin{equation}%\label{4.84}
\begin{aligned}
\displaystyle\int_{0}^{T}\!\!\!\int_{K} & \big|\eta^{\psi}_{mu}(\rho^{\varepsilon},m^{\varepsilon})(\rho^{\varepsilon})^{\alpha}
+\eta^{\psi}_{m\rho}(\rho^{\varepsilon},m^{\varepsilon})(\rho^{\varepsilon})^{\alpha}\rho_x^{\varepsilon}u_x^{\varepsilon}\big|\,\dd x\dd t\\
C_{\psi}\int_{0}^{T}\!\!\!\int_{K}(\rho^{\varepsilon})^{\alpha}(u_x^{\varepsilon})^2\, \dd x\dd t
+C_{\psi}\int_{0}^{T}\!\!\!\int_{K}(\rho^{\varepsilon})^{\alpha+\gamma-3}(\rho_x^{\varepsilon})^2\, \dd x\dd t
\leq C,
\nonumber\\[2mm]
\displaystyle\int_{0}^{T\!\!\!}\int_{K} & \big|\eta^{\psi}_m
\big(\lambda\rho^{\v}u^{\v}+\rho^{\v} V-\rho^{\v}\partial_xW\ast\rho^{\v}\big)\big|\,\dd x\dd t\\
C_{\psi}\int_{0}^{T}\!\!\!\int_{K}\rho^{\v}(u^{\v})^2+\rho^{\v} \,\dd x\dd t
+C_{\psi}\int_{0}^{T}\!\!\!\int_{K}|\rho^{\v}\partial_xW\ast\rho^{\v}| \,\dd x\dd t \leq C(K).
\nonumber
\end{aligned}
\end{equation}
This implies that
\begin{align}\label{4.86}
&\,\,\mbox{is uniformly bounded in $L^1([0,T]\times K)$},
\end{align}
so that it is compact in
$W_{\rm loc}^{-1,p_1}(\mathbb{R}_+^2)$ for $1<p_1<2$.
If $2\alpha\leq\gamma+1,$ then
%we obtain that
\begin{align}\label{4.87}
\v^{\frac{4}{3}}\int^T_0\int_{K}(\rho^{\v})^{2\alpha}\,\dd x\dd t\leq C(K)
\v^{\frac{4}{3}}.
\end{align}
If $2\alpha\geq\gamma+1,$ $\alpha<\gamma,$ we see from [28] that
\begin{align}\label{4.88}
\v^{\frac{4}{3}}\int^T_0\int_{K}(\rho^{\v})^{2\alpha}\,\dd x\dd t
&\leq C(K)
\v^{\frac{1}{3}}\int^T_0\int_K(\rho^{\v})^{\alpha-\frac{\gamma-1}{2}}\,\dd x\dd t\nonumber\\
&\quad+C(K )\v^{\frac{1}{3}}\int^T_0\int_K(\rho^{\v})^{\gamma+1}\,\dd x\dd t.
\end{align}
It follows from \eqref{4.81} and \eqref{4.87}--\eqref{4.88} that
%, we have
\begin{equation}\label{4.89}
\begin{aligned}
\displaystyle\int_{0}^{T}\int_{K}\Big(\varepsilon
\eta^{\psi}_m(\rho^{\varepsilon},m^{\varepsilon})
(\rho^{\varepsilon})^{\alpha}u_x^{\varepsilon}\Big)^{\frac{4}{3}}\, \dd x\dd t
(\rho^{\varepsilon})^{^{\frac{4\alpha}{3}}}|u_x^{\varepsilon}|^{\frac{4}{3}}\,\dd x\dd t\\
&\displaystyle\leq C\varepsilon^{\frac{4}{3}}\int_{0}^{T}\int_{K}
(\rho^{\varepsilon})^{\alpha}|u_x^{\varepsilon}|^2\,\dd x\dd t+C\varepsilon^{\frac{4}{3}}\int_{0}^{T}\int_{K}
(\rho^{\varepsilon})^{2\alpha}\,\dd x\dd t\\
&\displaystyle\leq C(K)\varepsilon^{\frac{1}{3}}+C\varepsilon^{\frac{4}{3}}\int_{0}^{T}\int_{K}
(\rho^{\varepsilon})^{\gamma+1}\,\dd x\dd t\\
&\displaystyle\leq C(K)\varepsilon^{\frac{1}{3}}\rightarrow 0\qquad \ \text{as $\varepsilon\rightarrow0^+$}.
\end{aligned}
\end{equation}
Then \eqref{4.86} and \eqref{4.89} yield
\begin{align}\label{4.90}
\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x
\qquad \mbox{is compact in $W_{\rm loc}^{-1,p_2}(\mathbb{R}_+^2)$} \ \
\mbox{for some $1<p_2<2$}.
\end{align}
Furthermore, for $\gamma\in (1,3],$ using \eqref{4.79} and Lemma \ref{lem4.5}, we have
\begin{align}
\int^T_0\int_K\big(|\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})|+|q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})|\big)^{\gamma+1}\,\dd x\dd t
\leq C\int^T_0\int_K(\rho^{\v})^{\gamma+1}\,\dd x\dd t\leq C(K).
\nonumber
\end{align}
For $\gamma\in (3, \infty),$ using \eqref{4.80} and Lemma \ref{lem4.2}, we have
%one has
\begin{align}
\int^T_0\!\!\!\int_K\big(|\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})|
+|q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})|\big)^{\frac{\gamma+\theta}{\theta+1}}\,\dd x\dd t
\leq C\int^T_0\!\!\!\int_K\big(|\rho^{\varepsilon}|^{\frac{\gamma+\theta}{\theta+1}}
+|\rho^{\v}|^{\gamma+\theta}\big)\,\dd x\dd t\leq C(K).
\nonumber
\end{align}
Thus, using the last two estimates, we obtain
\begin{align}
(\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon}),\, q^{\psi}(\rho^{\varepsilon},m^{\varepsilon}))
\qquad\, \mbox{is uniformly bounded in $L_{\rm loc}^{p_3}(\mathbb{R}_+^2)$} \ \
\mbox{for}\ p_3>2,\nonumber
\end{align}
where $p_3=\gamma+1>2$ when $\gamma\in(1,3]$, and
$p_3=\frac{\gamma+\theta}{1+\theta}>2$ when $\gamma\in(3,\infty)$. This implies that
\begin{align}\label{4.94}
\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x
\qquad\, \mbox{is uniformly bounded in $W_{\rm loc}^{-1, p_3}(\mathbb{R}_+^2)$ for $p_3>2$}.
\end{align}
Then, using \eqref{4.90}--\eqref{4.94} and the interpolation compactness
theorem $(${\it cf}. {\rm [24, 25]$)$}, we conclude Lemma $\ref{lemma4.8}$.
\subsection{Proof of Theorem \ref{thm3.3} for the polytropic equation of state}\label{section4.3}
Recall the following compactness theorem established by Chen-Perepelitsa [16]:
\begin{theorem}[Chen-Perepelitsa [16]]\label{th4.9}
Let $(\eta^{\psi},q^{\psi})$ be a weak
entropy pair generated by $\psi\in C_0^2(\mathbb{R})$.
Assume that the sequence
$(\rho^{\varepsilon},u^{\varepsilon})(t,x)$ defined on
$\mathbb{R}_+\times\mathbb{R}$ with
$m^{\varepsilon}=\rho^{\varepsilon}u^{\varepsilon}$ satisfies the
following conditions{\rm :}
\begin{enumerate}
\item [(\rmnum{1})] For any $-\infty<r_1<r_2<\infty$ and
\begin{align}%\label{4.95}
\int_{0}^{T}\int_{r_1}^{r_2}(\rho^\varepsilon)^{\gamma+1}\,\dd x\dd t\leq
\end{align}
where $C>0$ is independent of $\varepsilon$.
\item [(\rmnum{2})] For any set $K\Subset\mathbb{R}$,
\begin{align}%\label{4.96}
\int_{0}^{T}\int_{K}\big((\rho^\varepsilon)^{\gamma+\theta}+\rho^\varepsilon|u^\varepsilon|^3\big)\,\dd x\dd t\leq
\end{align}
where $C(K)>0$ is independent of $\varepsilon$.
\item [(\rmnum{3})] The sequence of entropy dissipation measures
\begin{align}%\label{4.97}
\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})
_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x \qquad\, \mbox{is
compact in $H_{\rm loc}^{-1}(\mathbb{R}_+^2)$}.\nonumber
\end{align}
\end{enumerate}
Then there exist both a subsequence $($still denoted$)$
$(\rho^{\varepsilon},m^{\varepsilon})(t,x)$ and a vector-valued function $(\rho, m)(t,x)$ such that
\begin{align*}%\label{4.98}
(\rho^{\varepsilon},m^{\varepsilon})\rightarrow (\rho, m)\qquad \ a.e. \text{ as $\varepsilon\rightarrow0^+$}.
\end{align*}
\end{theorem}
The uniform estimates and compactness properties obtained in \S\ref{section3}--\S\ref{section4} yields that,
for the sequence of solutions $(\rho^\varepsilon, m^\varepsilon)$ satisfying \eqref{1.8}, \eqref{3.1}--\eqref{3.4},
and the compensated compactness framework established in [16] (see Theorem \ref{th4.9}),
there exist both a subsequence (still denoted) $(\rho^{\varepsilon},m^{\varepsilon})(t,x)$
and a vector-valued function $(\rho, m)(t,x)$ such that
\begin{align}%\label{4.99}
(\rho^{\varepsilon},m^{\varepsilon})\rightarrow (\rho, m)\qquad \ a.e.\,\, (t,x)\in \R_+\times\R\quad
\text{as $\varepsilon\rightarrow0^+$}.\nonumber
\end{align}
Notice that
|m|^{\frac{3(\gamma+1)}{\gamma+3}}\leq C\big(\rho|u|^3+\rho^{\gamma+1}\big).
Then, using Lemma \ref{lem4.2} and \ref{lem4.5}, we obtain
\begin{align}%\label{4.100}
(\rho^{\v},m^{\v})\rightarrow(\rho,m) \qquad \text{ in } L^{q_1}_{\rm loc}(\R_+\times\R)\times L^{q_2}_{\rm loc}(\R_+\times\R),\nonumber
\end{align}
for $q_1\in[1,\gamma+1)$ and $q_2\in[1,\frac{3(\gamma+1)}{\gamma+3}).$
Using again Lemma \ref{lem4.2} and \ref{lem4.5}, we have
%it has
\begin{align}\label{4.101}
\eta^{\ast}(\rho^{\v},m^{\v})\rightarrow \eta^{\ast}(\rho,m)\qquad \text{ in $L^1_{\rm loc}(\R_+\times\R)$}\quad \text{ as $\varepsilon\rightarrow0^+$}.
\end{align}
Since $\eta^{\ast}$ is a positive convex function, we use
\eqref{3.7}, \eqref{4.1}, \eqref{4.101}, and Fatou's lemma to see that, for all $t_2\geq t_1\geq0,$
%we have
\int^{t_2}_{t_1}\int_{\R}\big(\eta^{\ast}(\rho,m)+\frac{1}{2}\rho (W+\frac{1}{2})\ast \rho \big)(t,x)\,\dd x \dd t
\leq(t_2-t_1)\int_{\R}\big(\eta^{\ast}(\rho_0,m_0)+\frac{1}{2}\rho_0 (W+\frac{1}{2})\ast \rho_0\big)\,\dd x.
Then, by the Lebesgue point theorem, we obtain
\begin{align}%\label{4.102}
\int_{\R}\big(\eta^{\ast}(\rho,m)+\frac{1}{2}\rho W\ast \rho\big)(t,x) \,\dd x
\leq \int_{\R}\big(\eta^{\ast}(\rho_0,m_0)+\frac{1}{2}\rho_0 W\ast \rho_0\big)(x)\,\dd x:=\mathcal{E}_0.\nonumber
\end{align}
We now prove that $(\rho,m)$ is an entropy solution of the Cauchy problem \eqref{1.1} and \eqref{pressure1}--\eqref{1.6}
for the polytropic case.
Let $\Psi(t,x)\in C^1(\R_+\times\R)$ be a function with compact support and
supp$\,\Psi(t,\cdot)\Subset(b^-(t),b^+(t))$ for any $t\in[0,T]$.
\int_{\R^2_+}\big(\rho^{\v}\Psi_t+\rho^{\v}u^{\v}\Psi_x\big)\,\dd x \dd t+\int_{\R}\rho^{\v}_0(x)\Psi(0,x)\,\dd x=0.
By the Lebesgue dominated convergence theorem, through taking limit $\varepsilon\rightarrow 0^+,$ up to a subsequence, we have
\int_{\R^2_+}\big(\rho\Psi_t+\rho u\Psi_x\big)\,\dd x\dd t+\int_{\R}\rho_0(x)\Psi(0,x)\,\dd x=0.
Next, we consider the momentum equation.
First, we have
\begin{align}\label{4.103}
+\rho^{\v}\big(\lambda u^{\v}+V +\partial_xW\ast\rho^{\v}\big)\Psi\Big)\,\dd x\dd t
+\int_{\R}(\rho^{\v}_0 u^{\v}_0)(x)\Psi(0,x)\,\dd x\nonumber\\
&=\v\int_{\R^2_+}(\rho^{\varepsilon})^{\alpha}u^{\varepsilon}_x\Psi_x\,\dd x\dd t.
\end{align}
%It is noted that
\begin{align*}%\label{4.104}
\v\Big|\int_{\R^2_+}(\rho^{\varepsilon})^{\alpha}u^{\varepsilon}_x\Psi_x\,\dd x\dd t\Big|& \leq C\sqrt{\varepsilon}\Big(\int^T_0\int_K\v(\rho^{\v})^{\alpha}(u^{\v}_x)^2\,\dd x\dd t\Big)^{\frac{1}{2}}
\Big(\int^T_0\int_K(\rho^{\v})^{\alpha}\,\dd x\dd t\Big)^{\frac{1}{2}}\nonumber\\
&\leq C(K)
\sqrt{\varepsilon}\rightarrow 0
\qquad \text{ as $\varepsilon\rightarrow0^+$}, %\nonumber
\end{align*}
then it follow from \eqref{4.103} that
\begin{align}%\label{4.105}
\int_{\R^2_+}\Big(m\Psi_t+\big(\frac{m^2}{\rho}+P(\rho)\big)\Psi_x\Big)\,\dd x\dd t
+\int_{\R}m_0(x)\Psi(0,x)\,\dd x=\int_{\R^2_+}(-\lambda m-\rho V +\rho \partial_xW\ast\rho)\Psi \,\dd x\dd t.\nonumber
\end{align}
The verification of entropy inequality is direct. Therefore, the proof of Theorem
\ref{thm3.3} and hence Theorem \ref{thm:merged} for the polytropic case is completed.
\section{Proof of Theorem \ref{thm3.3} for the General Pressure Law}\label{section5}
This section is dedicated to the essential improvements of the arguments in \S\ref{section3}--\S\ref{section4}
necessary to prove Theorem \ref{thm3.3} for the general pressure case.
\subsection{Properties of the general pressure law and the related internal energy}\label{section5.1}
In this section, we present some useful estimates
involving the general pressure $P(\r)$ with
\eqref{pressure2}--\eqref{pressure4} and the corresponding internal energy $e(\rho).$
\begin{equation}\label{5.1}
k(\rho):=\int_{0}^{\rho}\frac{\sqrt{P'(y)}}{y}\,\mathrm{d} y.
\end{equation}
By direct calculation, we recall the following asymptotic behaviors of $P(\rho)$, $e(\rho)$, and $k(\rho)$.
\begin{lemma}[[13], Lemma 3.1]\label{lemA.1}
The constant $\rho_{*}$ in \eqref{pressure3} can be chosen small enough, and the constant $\rho^{*}$ in \eqref{pressure4}
large enough, so that the following estimates hold{\rm :}
\begin{enumerate}
\item [(\rmnum{1})] When $\rho\in (0,\rho_{*}]$,
\begin{equation}\label{5.2}
\left\{\begin{aligned}
&\underline{\kappa}_{1}\rho^{\gamma_1}\leq P(\rho)\leq \bar{\kappa}_{1}\rho^{\g_1},\\
&\underline{\kappa}_{1}\gamma_1\rho^{\gamma_1-1}\leq P'(\rho)\leq \bar{\kappa}_{1}\g_1\rho^{\g_1-1},\\
&\underline{\kappa}_{1}\gamma_1(\gamma_1-1)\rho^{\gamma_1-2}\leq P''(\rho)\leq \bar{\kappa}_{1}\gamma_1(\gamma_1-1)\rho^{\g_1-2},
\end{aligned}
\right.
\end{equation}
and when $\rho\in [\rho^{*},\infty)$,
\begin{equation}\label{5.3}
\left\{\begin{aligned}
&\underline{\kappa}_{2}\rho^{\gamma_2}\leq P(\rho)\leq \bar{\kappa}_{2}\rho^{\g_2},\\
&\underline{\kappa}_{2}\gamma_2\rho^{\gamma_2-1}\leq P'(\rho)\leq \bar{\kappa}_{2}\g_2\rho^{\g_2-1},\\ &\underline{\kappa}_{2}\gamma_2(\gamma_2-1)\rho^{\gamma_2-2}\leq P''(\rho)\leq \bar{\kappa}_{2}\gamma_2(\gamma_2-1)\rho^{\g_2-2},
\end{aligned}
\right.
\end{equation}
where we have denoted $\underline{\kappa}_{i}:=(1-\mathfrak{a}_0)\kappa_{i}$
and $\bar{\kappa}_{i}:=(1+\mathfrak{a}_0) \kappa_{i}$ with $\mathfrak{a}_0=\frac{3-\g_1}{2(\g_1+1)}$ and $i=1,2$.
\item [(\rmnum{2})] For $e(\rho)$ and $k(\rho)$, there exists
$C>0$ depending on
$(\gamma_1, \gamma_2, \k_1,\k_2, \rho_{*}, \rho^{*})$ such that
\begin{align}
&C^{-1}\rho^{\g_1-1}\leq e(\rho)\leq C\rho^{\gamma_1-1},\quad
C^{-1}\rho^{\g_1-2}\leq e'(\rho)\leq C\rho^{\g_1-2}
\quad\,\, \text{ for }\rho\in (0,\rho_{*}],\nonumber\\%\label{A.8-1}\\
&C^{-1}\rho^{\g_2-1}\leq e(\rho)\leq C\rho^{\gamma_2-1},\quad
C^{-1}\rho^{\g_2-2}\leq e'(\rho)\leq C\rho^{\g_2-2}
\quad\,\, \text{ for }\rho\in [\rho^{*},\infty), \label{A.9-1}
\end{align}
and, for $i=0,1$,
\begin{align*}
&\qquad C^{-1}\rho^{\theta_{1}-i}\leq k^{(i)}(\rho)\leq C\rho^{\theta_{1}-i},\,\,
C^{-1}\rho^{\theta_{1}-2}\leq |k''(\rho)|\leq C\rho^{\theta_1-2}\,\,\,\, \text{for } \rho\in (0,\rho_{*}],\\
&\qquad C^{-1}\rho^{\theta_{2}-i}\leq k^{(i)}(\rho)\leq C\rho^{\theta_{2}-i},
\,\, C^{-1}\rho^{\theta_{2}-2}\leq |k''(\rho)|\leq C\rho^{\theta_2-2} \,\,\,\,\text{for } \rho\in [\rho^{*},\infty),
\end{align*}
where $\t_{1}=\frac{\g_1-1}{2}$ and $\t_2=\frac{\g_2-1}{2}$.
\end{enumerate}
\end{lemma}
\subsection{Uniform estimates of the approximate solutions}\label{section5.2}
We start with the basic energy estimate.
\begin{lemma}[\bf Basic energy estimate]\label{lem5.2}
For smooth solution $(\rho, u)(t,x)$ of problem \eqref{1.8} and \eqref{3.1}--\eqref{3.4},
the following estimate holds{\rm :}
\begin{align}\label{5.4}
&\int^{b^+(t)}_{b^-(t)}\Big(\frac{1}{2}\rho u^2+\rho e(\rho)+\frac{1}{2}\rho W\ast\rho\Big)(t,x)\,\dd x
+\int_0^t\int_{b^-(s)}^{b^+(s)} (\v\rho^{\alpha}|u_x|^2-\lambda \rho u^2)(s,x)\,\dd x\dd s\nonumber\\
&+\frac{1}{2}\int^t_0\int^{b^+(s)}_{b^-(s)}\int^{b^+(s)}_{b^-(s)}\varpi(x-y)\rho(x)\rho(y)|u(y)-u(x)|^2\,\dd y\dd x\dd s
= \mathcal{E}^{\varepsilon}_0\le C_0.
\end{align}
\end{lemma}
The proof is almost the same as Lemma \ref{lem4.1}, with \eqref{4.2} replaced by
\begin{align*}%\label{5.5}
\end{align*}
Using Lemma \ref{5.2}, we obtain
\begin{corollary}\label{cor5.3}
It follows from \eqref{A.9-1} and {\rm Lemma \ref{lem5.2}} that
\begin{equation}\nonumber
\int_{b^-(t)}^{b^+(t)}\rho^{\g_2}(t,x)\,\mathrm{d}x
\leq C
\int_{b^-(t)}^{b^+(t)}\big(\rho+\r e(\r)\big)(t,x)\,\mathrm{d}x\leq C
\qquad \text{ for $t\geq 0$}.
\end{equation}
\end{corollary}
\begin{lemma}[\bf Higher moment estimate]\label{lemma5.3}
The following estimate holds{\rm :}
\begin{align*}%\label{5.6}
\int^{b^+(t)}_{b^-(t)}x^2\rho(t,x) \,\dd x\leq C.
\end{align*}
\end{lemma}
\noindent{\bf Proof.} Following the same argument as in Lemma \ref{lemma4.2},
it suffices to show that
\begin{align*}
&\Big|\int^T_0\int^{b^+(s)}_{b^-(s)}P(\rho)(s,x)\,\dd x\dd s\Big|\\
&\leq \Big|\int^T_0\int^{b^+(s)}_{b^-(s)}\big(P(\rho)-\rho e(\rho)\big)(s,x)\,\dd x\dd s\Big|
+\Big|\int^T_0\int^{b^+(s)}_{b^-(s)}\big(\rho e(\rho)\big)(s,x)\,\dd x\dd s\Big|\\
&\leq\Big|\int^T_0\int^{b^+(s)}_{b^-(s)}\big(\rho^2 e'(\rho)-\rho e(\rho)\big)(s,x)\,\dd x\dd s\Big|+C
\leq C.
\end{align*}
For later use, we analyze the behavior of density $\rho$ on the free boundary. It follows from $\eqref{3.8}_1$ and \eqref{3.9} that
\begin{align*}%\label{5.7}
\rho_{\tau}(\tau,M)=-\frac{1}{\varepsilon}\lr{\frac{\rho P}{\mu(\rho)}}(\tau,M)\leq0.
\end{align*}
This yields that $\rho(\tau, M)\leq \rho_{0}(M)$.
In the Eulerian coordinates, it is equivalent to
\begin{equation}\label{5.8}
\rho(t,b^+(t))\leq \rho_0(b).
\end{equation}
Moreover, noting that $\rho^{\gamma_1}_0(\pm b)b\leq C_0$ and $b\geq (\rho_{*})^{-\g_1}$,
we see that $\rho(t,b^+(t))\leq \rho_0(b)\leq \rho_{*}$ for all $t\geq 0$.
From $\eqref{1.8}_1$ and \eqref{5.2}, there exists a positive constant $\tilde{C}$
depending only on $(\g_1, \k_1)$ such that
\begin{equation*}
\rho_{\tau}(\tau,M)=-\frac{1}{\v}\, \big(\frac{\rho P}{\mu(\rho)}\big)(\tau,M)
\geq -\frac{\tilde{C}}{\v}\big(\rho(\tau,M)\big)^{\g_1+1-
\alpha},
\end{equation*}
which implies
\begin{equation*}
\rho(\tau,M)\geq \rho_{0}(M)
\Big(1+\frac{\tilde{C}(\g_1-\alpha)}{\v}\big(\rho_{0}(M)\big)^{\g_1-\alpha}\tau\Big)^{-\frac{1}{\g_1-\alpha}}.
\end{equation*}
Therefore, in the Eulerian coordinates,
\begin{equation}\label{5.9}
\rho(t,b^+(t))\geq \rho_{0}(b)\Big(1+\frac{\tilde{C}(\g_1-\alpha)}{\v} (\rho_{0}(b))^{\g_1-\alpha}t\Big)^{-\frac{1}{\g_1-\alpha}}
\qquad \mbox{for $t\geq 0$}.
\end{equation}
Notice that, in the Lagrangian coordinates, $\rho_0(M)=\rho_0(0)$ since $\rho_0(b)=\rho_0(-b)$ in the Eulerian coordinates.
Therefore, by the uniqueness of solutions of the ordinary differential equation:
$\rho_{\tau}(\tau,\cdot)=-\frac{1}{\v}\lr{\frac{\rho P}{\mu(\rho)}}(\tau,\cdot)$ with the same initial data, we conclude that
\begin{equation}\label{5.10}
\rho(t,b^+(t))=\rho(t,b^-(t)).
\end{equation}
This implies that $P(\rho(t,b^+(t)))=P(\rho(t,b^-(t)))$.
This indicates, in particular, that the boundary term in \eqref{5.11} below is nonnegative.
\begin{lemma} \label{lem5.5}
$\rho^{\gamma_1}_0(\pm b)b\leq C_0$.
Then, for any given $T>0$ and for any $t\in[0,T]$, the following holds{\rm :}
\begin{align}\label{5.11}
&\varepsilon^2\int^{b^+(t)}_{b^-(t)}\big(\rho^{2\alpha-3}\rho^2_x\big)(t,x)\,\dd x
+\varepsilon\int^t_0\int^{b^+(t)}_{b^-(t)}\big(P'(\rho)\rho^{\alpha-2}\rho^2_x\big)(s,x)\,\dd x\dd s\nonumber\\
-\big(\frac{P'(\rho)P(\rho)\rho}{\mu(\rho)}\big)(s,b^-(s))b^-(s)\Big)\,\dd s\nonumber\\
&+\big(P(\rho(t,b^+(t)))b^+(t)-P(\rho(t,b^-(t)))b^-(t)\big)\leq C.
\end{align}
\end{lemma}
\noindent{\bf Proof.}
Similar with the argument in \eqref{4.32}, we obtain
\begin{align}\label{5.12}
&\Big(\frac{(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi})^2}{2}+e(\rho)+\frac{1}{2}\rho W\ast\rho\Big)_{\tau}
+\big(P(\rho)u\big)_{\xi}+\varepsilon P'(\rho)\rho^{\alpha-1}(\rho_{\xi})^2\nonumber\\
&+\big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\big)\big(-\lambda u-V+\rho(W\ast\rho)_{\xi}\big)=0.
\end{align}
Using $\eqref{3.8}_1$ and \eqref{3.9}, we have
\begin{align}\nonumber
\rho_{\tau}(\tau,M)=-\frac{1}{\v}\Big(\frac{\rho P}{\mu(\rho)}\Big)(\tau,M),
\qquad \rho_{\tau}(\tau,0)=-\frac{1}{\v}\Big(\frac{\rho P}{\mu(\rho)}\Big)(\tau,0),
\end{align}
so that
\begin{align}%\label{5.13}
&=P(\rho)(\tau,M)\frac{\dd}{\dd \tau}b^+(\tau)-P(\rho)(\tau,0)\frac{\dd}{\dd \tau}b^-(\tau)\nonumber\\
&\quad\, +\Big(P'(\rho)(\tau,0)\rho_{\tau}(\tau,0)b^{-}(\tau)-P'(\rho)(\tau,M)\rho_{\tau}(\tau,M)b^{+}(\tau)\Big)\nonumber\\
&\quad\, +\frac{1}{\v}\Big(\frac{P'(\rho)P(\rho)\rho}{\mu(\rho)}\Big)(\tau,M)b^+(\tau)
\end{align}
Integrating \eqref{5.12} over $[0,M]$, we arrive at
\begin{align}\label{5.14}
&\frac{\dd}{\dd \tau}\int^{M}_0\Big(\frac{1}{2}\big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\big)^2+e(\rho)\Big)\,\dd \xi
+\varepsilon\int^M_0P'(\rho)\rho^{\alpha-1}(\rho_{\xi})^2\,\dd \xi \nonumber\\
&=\int^M_0\big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\big)\big(-\lambda u-V+\rho(W\ast\rho)_{\xi}\big)\,\dd \xi.
\end{align}
Integrating \eqref{5.14} over $[0,\tau],$ we have
\begin{align}\label{5.142}
&\int^M_0\Big(\frac{1}{2}\big(u+\frac{\varepsilon}{\alpha}(\rho^{\alpha})_{\xi}\big)^2+e(\rho)\Big)\,\dd \xi
+\varepsilon\int^{\tau}_0\int^M_0P'(\rho)\rho^{\alpha-1}(\rho_{\xi})^2\,\dd \xi \dd s\nonumber\\
-\Big(\frac{P'(\rho)P(\rho)\rho}{\mu(\rho)}\Big)(\tau,0)b^-(\tau)\Big)\,\dd s\nonumber\\
&\leq C\int^M_0\Big(\frac{1}{2}\big(u_0+\frac{\varepsilon}{\alpha}(\rho^{\alpha}_0)_{\xi}\big)^2
+e(\rho_0)\Big)\,\dd \xi+C\big(P(\rho_0(M))+P(\rho_0(0))\big)\, b+C.
\end{align}
Using $b\geq\rho^{-\gamma_1}_{\ast},$ we see that $P(\rho_0(\pm b))\, b\leq C_0$ so that the second term
on the RHS of \eqref{5.142} is uniformly bounded.
This completes the proof.
%of Lemma \ref{lem2.2}.
\medskip
Motivated by [12], to take the limit: $b\rightarrow\infty$,
we need to make sure that domain $\Omega_T=[b^-(t),b^+(t)]$ can expand to the whole physical space $\R$.
\begin{lemma}\label{lem5.6}
Let $0\leq\alpha<\gamma_1,$ $p>\frac{\gamma_1}{\gamma_1-\alpha},$ and $b:=\varepsilon^{-p}$. Then, given any $T>0$,
there exists $\v_0=\v_0(\alpha,\gamma_1,\gamma_2,T)>0$
such that, for any $\v\in(0,\v_0]$,
%there holds
\pm b^\pm(t)\geq \frac12 b \qquad\,\, \mbox{for $t\in[0,T]$}.
\end{lemma}
\noindent{\bf Proof.}
We divide the proof into three steps:
\smallskip
\noindent {\emph{Step 1.}} Using \eqref{5.1} and \eqref{5.11}, we have
%obtain that
\begin{align}\label{5.16}
\v\rho^{\beta}(t,x)&=\v\big(\rho^{\beta}(t,x)-\rho^{\beta}(t,b^-(t))+\rho^{\beta}(t,b^-(t))\big)\nonumber\\
&\leq \v\beta\int^{b^+(t)}_{b^-(t)}\rho^{\beta-1}|\rho_x|\,\dd x+\v\rho^{\beta}_0(b)\nonumber\\
&\leq\beta\Big(\v^2\int^{b^+(t)}_{b^-(t)}\rho^{2\alpha-3}\rho^2_x\,\dd x\Big)^{\frac{1}{2}}\Big(\int^{b^+(t)}_{b^-(t)}\rho^{2(\beta-
\alpha)+1}\,\dd x\Big)^{\frac{1}{2}}+\v\rho^{\beta}_0(b)\nonumber\\
&\leq C(\mathcal{E}^{\v}_1)\Big(\int^{b^+(t)}_{b^-(t)}\rho^{2(\beta-
\alpha)+1}\,\dd x\Big)^{\frac{1}{2}}+C\le C,
\end{align}
where we have used \eqref{5.8} and $\beta=\alpha+\frac{\gamma_2-1}{2}.$
\smallskip
\noindent {\emph{Step 2.}} It follows from the boundary condition \eqref{3.2} that
\begin{equation}\label{5.17}
b^+(t)-b^-(t)=2b+\int^{t}_0\big(u(s,b^+(s))-u^-(s,b^-(s))\big)\,\dd s.
\end{equation}
Using \eqref{5.10}, we obtain
\begin{align}
=\frac{1}{\rho^l(t,b^+(t))}\Big|\int^{b^+(t)}_{b^-(t)}(\rho^l u)_x\,\dd x\Big|\nonumber\\
&=\frac{1}{\rho^l(t,b^+(t))}\Big|\int^{b^+(t)}_{b^-(t)}\big(l \rho^{l-1}\rho_xu+\rho^l u_x\big)\,\dd x\Big|\nonumber\\
&\leq\frac{1}{\rho^l(t,b^+(t))}\bigg\{l\Big(\int^{b^+(t)}_{b^-(t)}\v P'(\rho)\rho^{\alpha-2}\rho^2_x\,\dd x\Big)^{\frac{1}{2}}\Big(\int^{b^+(t)}_{b^-(t)}\varepsilon^{-1}\frac{\rho^{2l-\alpha}}{P'(\rho)}u^2\,\dd x\Big)^{\frac{1}{2}}\nonumber\\
&\qquad\qquad\qquad\quad +\Big(\int^{b^+(t)}_{b^-(t)}\v\rho^{\alpha}u^2_x\,\dd x\Big)^{\frac{1}{2}}
\Big(\int^{b^+(t)}_{b^-(t)}\v^{-1}\rho^{2l-\alpha}\,\dd x\Big)^{\frac{1}{2}}\bigg\}.\label{5.18}
\end{align}
Taking $2l-(\alpha+\gamma_2)+1=1,$ $\it{i.e.,}$ $l=\frac{\alpha+\gamma_2}{2},$ so that $2l-\alpha=\gamma_2.$
We also have
\begin{align}\label{5.19}
\Big|\int^{b^+(t)}_{b^-(t)}\v^{-1}\frac{\rho^{2l-\alpha}}{P'(\rho)}u^2\,\dd x\Big|
&\leq\Big|\int^{b^+(t)}_{b^-(t)}\v^{-1}\frac{\rho^{2l-\alpha}}{P'(\rho)}u^2\mathbf{I}_{\{\rho\leq\rho_{\ast}\}}\,\dd x\Big|+\Big|\int^{b^+(t)}_{b^-(t)}\v^{-1}\frac{\rho^{2l-\alpha}}{P'(\rho)}u^2\mathbf{I}_{\{\rho_{\ast}\leq\rho\leq\rho^{\ast}\}}\,\dd x\Big|\nonumber\\
&\quad\,\,+\Big|\int^{b^+(t)}_{b^-(t)}\v^{-1}\frac{\rho^{2l-\alpha}}{P'(\rho)}u^2\mathbf{I}_{\{\rho\geq\rho^{\ast}\}}\,\dd x\Big|\nonumber\\
&\leq C(\rho^{\ast},\rho_{\ast}).
\end{align}
It follows from \eqref{5.18}--\eqref{5.19}, \eqref{3.7}, and \eqref{5.4} that
%, we have
\begin{align}\nonumber
\leq\frac{C\v^{-\frac{1}{2}}}{\rho^l(t,b^+(t))}
\bigg\{\Big(\int^{b^+(t)}_{b^-(t)}\v P'(\rho)\rho^{\alpha-2}\rho^2_x\,\dd x\Big)^{\frac{1}{2}}+\Big(\int^{b^+(t)}_{b^-(t)}\v\rho^{\alpha}u^2_x\,\dd x\Big)^{\frac{1}{2}}\bigg\}.
\end{align}
Using \eqref{5.1} and \eqref{5.11}, we obtain
\begin{align}\label{5.20}
&\int^t_0\big|u(s,b^+(s)-u(s,b^-(s))\big|\,\dd s\nonumber\\
&\leq C\v^{-\frac{1}{2}}\Big(\int^t_0(\rho(s,b^+(s)))^{-\alpha-\gamma_2}\,\dd s\Big)^{\frac{1}{2}}\nonumber\\
&\quad\,\times \bigg\{\Big(\int^t_0\int^{b^+(s)}_{b^-(s)}\v P'(\rho)\rho^{\alpha-2}\rho^2_x\,\dd x\dd s\Big)^{\frac{1}{2}}
+\Big(\int^t_0\int^{b^+(s)}_{b^-(s)}\v\rho^{\alpha}u^2_x\,\dd x\dd s\Big)^{\frac{1}{2}}\bigg\}\nonumber\\
&\leq C(\v^{-\frac{1}{2}}\Big(\int^t_0(\rho(s,b^+(s)))^{-\alpha-\gamma_2}\,\dd s\Big)^{\frac{1}{2}}.
\end{align}
Since $\rho_0(b)\leq Cb^{-\frac{1}{\gamma_1}},$ we take $b\gtrsim(\frac{1}{\v})^{\frac{\gamma_1}{\gamma_1-\alpha}}$ to obtain
%we obtain that
\begin{align}\label{5.21}
\frac{\tilde{C}(\gamma_1-\alpha)}{\v}(\rho_0(b))^{\gamma_1-\alpha}\leq\frac{C\tilde{C}(\gamma_1-\alpha)}{\v}b^{-\frac{\gamma_1-\alpha}{\gamma_1}}\leq C.
\end{align}
It follows from \eqref{5.9} and \eqref{5.21} that, for $0\leq s\leq T,$
\begin{align*}%\label{5.22}
\rho(s,b^+(s))\geq \rho_{0}(b)\Big(1+\frac{\tilde{C}(\g_1-\alpha)}{\v} (\rho_{0}(b))^{\g_1-\alpha}s\Big)^{-\frac{1}{\g_1-\alpha}}\geq\frac{1}{C}(1+T)^{-\frac{1}{\gamma_1-\alpha}}\rho_0(b).
\end{align*}
Then we have
%obtain that
\begin{align}\label{5.23}
\int^t_0(\rho(s,b^+(s)))^{-\alpha-\gamma_2}\,\dd s\leq C(1+T)^{\frac{\alpha+\gamma_2}{\gamma_1-\alpha}}(\rho_0(b))^{-\alpha-\gamma_2}.
\end{align}
Take $b=\v^{-p}$ with $p>\frac{\gamma_1}{\gamma_1-\alpha}$. For any given $T\geq1$,
choose $\v_1:=\big(C_1(1+T)^{\frac{\alpha+\gamma_2}{2(\gamma_1-\alpha)}}\big)^{-\frac{2\gamma_1}{p(\gamma_1-\alpha)-\gamma_1}},$
where $C_1\geq1$ is a large constant depending only on $C_0$.
Then, for any $\v\in(0,\v_1],$ it follows from \eqref{5.20} and \eqref{5.23} that
\begin{align*}
\int^t_0\big|u(s,b^+(s))-u(s,b^-(s))\big|\,\dd s
&\leq C_1\v^{-\frac{1}{2}}(1+T)^{\frac{\alpha+\gamma_2}{2(\gamma_1-\alpha)}}(\rho_0(b))^{-\frac{\alpha+\gamma_2}{2}}\nonumber\\
&\leq b\, C_1(1+T)^{\frac{\alpha+\gamma_2}{2(\gamma_1-\alpha)}}\v^{p-\frac{1}{2}-\frac{p(\alpha+\gamma_2)}{2\gamma_1}}\nonumber\\
&\leq b\, (1+T)^{\frac{\alpha+\gamma_2}{2(\gamma_1-\alpha)}}\v^{\frac{p(\gamma_1-\alpha)-\gamma_1}{2\gamma_1}}_1\leq b.\nonumber\\
\end{align*}
Using \eqref{5.17}, we conclude
\begin{equation}\label{5.24}
b\leq b^+(t)-b^-(t)\leq3b.
\end{equation}
\smallskip
\noindent {\emph{Step 3.}} There exists $x_0(t)\in(b^-(t),b^+(t))$ such that
(\rho u^2)(t,x_0(t))\,\big(b^+(t)-b^-(t)\big)
=\int^{b^+(t)}_{b^-(t)}(\rho u^2)(t,x)\,\dd x\leq C.
Using \eqref{5.24}, we have
\begin{align}\label{5.25}
(\rho u^2)(t,x_0(t))\leq C(\mathcal{E}^{\v}_0)\,\big(b^+(t)-b^-(t)\big)^{-1}\leq C)b^{-1}.
\end{align}
It follows from \eqref{3.2} that
\begin{align}\label{5.26}
b^+(t)=b+\int^t_0u(s,b^+(s))\,\dd s=b+\int^t_0\frac{1}{\rho^l(s,b^+(s))}(\rho^lu)(s,b^+(s))\,\dd s,
\end{align}
where $l=\frac{\alpha+\gamma_2}{2}.$
Similar to those as in \eqref{5.18}, we use \eqref{5.16} and \eqref{5.25} to obtain
\begin{align}\label{5.27}
(\rho^lu)(s,b^+(s))&=(\rho^lu)(s,b^+(s))-(\rho^lu)(s,x_0(s))+(\rho^l u)(s,x_0(s))\nonumber\\
&=\int^{b^+(s)}_{b^-(s)}\big(l\rho^{l-1}\rho_xu+\rho^lu_x\big)\,\dd x+(\rho^lu)(s,x_0(s))\nonumber\\
&\leq\Big(\int^{b^+(s)}_{b^-(s)}P'(\rho)\rho^{\alpha-2}\rho^2_x\,\dd x\Big)^{\frac{1}{2}}\Big(\int^{b^+(s)}_{b^-(s)}\frac{\rho^{2l-\alpha}}{P'(\rho)}u^2\,\dd x\Big)^{\frac{1}{2}}\nonumber\\
&\quad\,+\Big(\int^{b^+(s)}_{b^-(s)}\rho^{\alpha}u^2_x\,\dd x\Big)^{\frac{1}{2}}\Big(\int^{b^+(s)}_{b^-(s)}\rho^{2l-\alpha}\,\dd x\Big)^{\frac{1}{2}}%\nonumber\\&\quad\,
&\leq C
\v^{-\frac{1}{2}}\bigg\{\Big(\v\int^{b^+(s)}_{b^-(s)}P'(\rho)\rho^{\alpha-2}\rho^2_x\,\dd x\Big)^{\frac{1}{2}}+\Big(\v\int^{b^+(s)}_{b^-(s)}\rho^{\alpha}u^2_x\,\dd x\Big)^{\frac{1}{2}}\bigg\}
\v^{-1}b^{-\frac{1}{2}}.
\end{align}
Using \eqref{5.4}, \eqref{5.11}, \eqref{5.23}, and \eqref{5.27}, we have
%one has that
\begin{align}\label{5.28}
&\Big|\int^t_0\frac{1}{\rho^l(s,b^+(s))}(\rho^lu)(s,b^+(s))\,\dd s\Big|\nonumber\\
&\leq C\v^{-\frac{1}{2}}\Big(\int^t_0(\rho(s,b^+(s)))^{-(\alpha+\gamma_2)}\,\dd s\Big)^{\frac{1}{2}}\nonumber\\
\bigg\{\Big(\int^t_0\int^{b^+(s)}_{b^-(s)}P'(\rho)\rho^{\alpha-2}\rho^2_x\,\dd x \dd s\Big)^{\frac{1}{2}}
+\Big(\int^t_0\int^{b^+(s)}_{b^-(s)}\v\rho^{\alpha}u^2_x\,\dd x\dd s\Big)^{\frac{1}{2}}\bigg\}\nonumber\\
&\quad +C\v^{-1}b^{-\frac{1}{2}}\int^t_0(\rho(s,b^+(s)))^{-\frac{\alpha+\gamma_2}{2}}\,\dd s\nonumber\\
&\leq C
\bigg\{\v^{-\frac{1}{2}}\Big(\int^t_0(\rho(s,b^+(s)))^{-(\alpha+\gamma_2)}\,\dd s\Big)^{\frac{1}{2}}
+\v^{-1}b^{-\frac{1}{2}}\int^t_0(\rho(s,b^+(s)))^{-\frac{\gamma_2+\alpha}{2}}\,\dd s\bigg\}\nonumber\\
&\leq b\,C
\Big(\v^{-\frac{1}{2}}b^{-1}(1+T)^{\frac{\alpha+\gamma_2}{2(\gamma_1-\alpha)}}
+\v^{-1}b^{-\frac{3}{2}}(1+T)^{\frac{3\gamma_2-\alpha}{2(\gamma_1-\alpha)}}\Big)\, b^{\frac{\alpha+\gamma_2}{2\gamma_1}}\nonumber\\
&\leq b\, C_2(1+T)^{\frac{3\gamma_2-\alpha}{2(\gamma_1-\alpha)}}\v^{\frac{3}{2}p-1-p\frac{\alpha+\gamma_2}{2\gamma_1}}\nonumber\\
&\leq b\, C_2(1+T)^{\frac{3\gamma_2-\alpha}{2(\gamma_1-\alpha)}}\v^{\frac{p(\gamma_1-\alpha)-\gamma_1}{2\gamma_1}}
\leq \frac{1}{2}b
\end{align}
for any $\v\in(0,\v_2]$ with
Take $\tilde{C_0}:=2\max\{C_1,C_2\}$ and
Then, for any $\v\in(0,\v_0]$, we obtain from \eqref{5.26} and \eqref{5.28} that
Similarly, it can be proved that $b^-(t)\leq-\frac{1}{2}b.$
This completes the proof.
\begin{lemma}[\bf Higher integrability of the density]\label{lemma5.6}
Let $(\r,u)$ be the smooth solution of \eqref{1.8} and \eqref{3.1}--\eqref{3.4}.
Then, under the assumption of {\rm Lemma \ref{lem5.6}},
\begin{align}\label{5.30}
\int_0^T\int_K (\r P(\rho))(t,x)\,\dd x \dd t\leq C(K)
\qquad\mbox{for any $K\Subset[b^-(t),b^+(t)]\,\,$ for any $t \in[0,T]$}.
\end{align}
\end{lemma}
\noindent{\bf Proof.} We divide the proof into two steps.
\smallskip
\noindent {\emph{Step 1.}} For given $K\Subset[b^-(t),b^+(t)]$ for any $t\in[0,T]$,
there exist $r_1$ and $r_2$ such that $K\Subset (r_1,r_2)\Subset[b^-(t),b^+(t)]$.
Let $w(x)$ be a smooth, compactly supported function with $\text{supp}\,w\subseteq(r_1,r_2)$ and $w(x)=1$ for $x\in K$.
Multiplying $\eqref{1.8}_2$ by $w(x)$, we have
\begin{align}\label{5.31}
&(\r u w)_t+\big((\r u^2+P(\rho))w\big)_x\nonumber\\
&=(\rho u^2+P(\rho))w_x+\varepsilon(\rho^{\alpha}w u_x)_x-\varepsilon\rho^{\alpha}u_xw_x+\lambda \rho uw+\rho Vw-\rho\partial_xW\ast\rho w.
\end{align}
Integrating \eqref{5.31} over $[r_1,x)$ to obtain
\begin{align}\label{5.32}
(\rho u^2+P(\rho))w
&=\varepsilon\rho^{\alpha}wu_x+\int^{x}_{r_1}\Big((\rho u^2+P(\rho))w_y-\varepsilon
\rho^{\alpha}u_yw_y\Big)\,\dd y-\frac{\dd}{\dd t}\int^x_{r_1}\rho uw\,\dd y\nonumber\\
&\quad+\int^x_{r_1}\lambda \rho uw\,\dd y-\int^x_{r_1}\rho \partial_x W\ast\rho w\,\dd y+\int^x_{r_1}\rho Vw\,\dd y .
\end{align}
Multiplying \eqref{5.32} by $\rho w$ and performing a direct calculation, we have
\begin{align}\label{5.33}
\r P(\rho) w^2&=\v \rho^{\alpha+1}w^2u_x-\Big(\rho w\int^x_{r_1}\rho uw\,\dd y\Big)_t-\Big(\rho uw\int^x_{r_1}\rho uw\,\dd y\Big)_x\nonumber\\
&\quad+\rho uw_x\int^x_{r_1}\rho uw\,\dd y+\rho w\int^x_{r_1}\Big((\rho u^2+P(\rho))w_y-\varepsilon\rho^{\alpha}u_yw_y\Big)\,\dd y\nonumber\\
&\quad+\lambda \rho w\int^x_{r_1}\rho u w\,\dd y-\rho w\int^x_{r_1}\rho\partial_xW\ast\rho w\,\dd y+\rho w\int^x_{r_1}\rho Vw\,\dd y\nonumber\\
\end{align}
\noindent {\emph{Step 2.}} To estimate $K_i, i=1,\cdots, 8$, in \eqref{5.33}, we first notice that
\begin{align}\label{5.34}
\int_{b^{-}(t)}^{b^+(t)}\rho|u|\, \dd x\leq \int_{b^-(t)}^{b^+(t)} (\rho+\rho u^2)\,\dd x\leq C.
\end{align}
Then it follows from \eqref{5.34} that
\begin{align}\label{5.35}
\left| \int_0^T\int_{r_1}^{r_2} K_2\, \dd x\dd t \right|&=\left| \int_0^T\int_{r_1}^{r_2} \Big(\rho w\int^x_{r_1}\rho uw\,\dd y\Big)_t\, \dd x\dd t \right|\nonumber\\
&\leq\left|\int^{r_2}_{r_1}\Big(\rho w\int^x_{r_1}\rho u w\dd y\Big)(T,x)\,\dd x\right|+\left|\int^{r_2}_{r_1}\Big(\rho w\int^x_{r_1}\rho u w\dd y\Big)(0,x)\,\dd x\right|\nonumber\\[1mm]
&\leq C.
\end{align}
For $K_1$, we have
\begin{align*}%\label{5.36}
\left|\int_0^T\int_{r_1}^{r_2} K_1\,\dd x\dd t \right|&=\left|\int^T_0\int^{r_2}_{r_1}\rho^{\alpha+1}\omega^2u_x\,\dd x\dd t\right|\nonumber\\
&\leq\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}|u_x|^2\omega^2\,\dd x\dd t+\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha+2}w^2\,\dd x\dd t\nonumber\\
&\leq C
+\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha+2}w^2\,\dd x\dd t.
\end{align*}
We now estimate
For any fixed $t\in [0,T]$,
A(t):=\{x\in [r_1,r_2]\,:\, \rho(t,x)\geq \rho^{*}\},
then it follows from \eqref{3.7} that $|A(t)|\leq C(r_1,r_2,\rho^{*})M$.
For any $x\in A(t)$, let $x_{0}$ be the closest point to $r$ so that
$\rho(t,x_{0})=\rho^{*}$ with $|x-x_{0}|\leq |A(t)|\leq C(r_1,r_2,\rho^{*})M$.
Then, for any smooth function $f(\rho)$,
\begin{align}
\sup_{x\in A(t)}f(\rho(t,x))\omega^{2}(x)&\leq f(\rho(t,x_0))\omega^2(x_0)
&\leq C(\|\omega\|_{C^1})|f(\rho^{*})|
\end{align}
Recalling \eqref{5.3} and \eqref{A.9-1},
we notice that $P(\rho)\cong \r^{\g_2}$ and $e(\rho)\cong \r^{\g_2-1}$ for any $x\in A(t)$.
\begin{align*}%\label{5.37}
&= \v\int_{0}^{T}\int_{r_1}^{r_2}\rho^{\alpha+2}{\bf I}_{\{\rho\leq \rho^{\ast}\}}\omega^2\,\mathrm{d}x\mathrm{d}t
+ \v\int_{0}^{T}\int_{r_1}^{r_2}\rho^{\alpha+2}
{\bf I}_{\{\rho\geq \rho^{\ast}\}}\omega^2\,\mathrm{d}x\mathrm{d}t\nonumber\\
&\leq C(\rho^{*})
+ C
\, \v\int_{0}^{T}\Big(\int_{r_1}^{r_2}\rho e(\rho)\dd x\Big)
\sup_{x\in A(t)} \Big(\frac{\rho^{\alpha+1}}{e(\rho)}\omega^2\Big)\mathrm{d}t\nonumber\\
&\leq C(\rho^{*})+ C\, \v\int_{0}^{T}\int_{A(t)}\Big\vert\Big(\frac{\rho^{\alpha+1}}{e(\rho)}\omega^2\Big)_{x}\Big\vert
\,\mathrm{d}x\mathrm{d}t\nonumber\\
&\leq C\,\v\int_{0}^{T}\int_{A(t)}\Big(\big(\frac{(\alpha+1)\rho^{\alpha}}{e(\rho)}-\frac{\rho^{\alpha-1}P(\rho)}{e(\rho)^2}\big)
\end{align*}
A direct calculation shows that
\begin{align*}%\label{5.38}
&\leq \int_{0}^{T}\int_{A(t)}\v\frac{P'(\rho)}{\rho^{2-\alpha}}|\rho_{x}|^2\omega^2\,\mathrm{d}x\mathrm{d}t
\nonumber\\
&\leq C+\int_{0}^{T}\int_{A(t)}\v\rho^{5+\alpha-3\gamma_2}\omega^2\,\mathrm{d}x\mathrm{d}t\nonumber\\
\begin{cases}
C+\v\int^T_0\int^{r_2}_{r_1}(\rho^{\beta}+1)\,\dd x\dd t \quad &\text{if } 5+\alpha-3\gamma_2\leq\beta,\\[1mm]
+\frac{\v}{2}\int^T_0\int^{r_2}_{r_1}\rho^{\alpha+2}\omega^2\,\dd x\dd t\,\,\,&\text{if }5+\alpha-3\gamma_2\geq\beta.
\end{cases}
\end{align*}
We also have
\begin{align*}%\label{5.39}
&\int^T_0\int_{A(t)}\v\frac{\rho^{\alpha+1}}{e(\rho)}\omega|\omega_x|\,\dd x\dd t\nonumber\\
&\leq\int^T_0\int^{r_2}_{r_1}\v\rho^{\alpha-\gamma_1+2}\omega\,\dd x\dd t
\leq\frac{\v}{2}\int^T_0\int^{r_2}_{r_1}\rho^{\alpha+2}\omega^2\,\dd x\dd t+C(r_1,r_2,\rho^{\ast}).
\end{align*}
Notice that
\begin{align*}
&\left| \int_0^T\int_{r_1}^{r_2}K_3\, \dd x\dd t \right|=\left| \int_0^T\int_{r_1}^{r_2} \Big(\rho uw\int^x_{r_1}\rho uw\dd y\Big)_x\, \dd x\dd t \right|=0,\\
&\left| \int_0^T\int_{r_1}^{r_2} K_4\, \dd x\dd t \right|=\left| \int_0^T\int_{r_1}^{r_2} \rho uw_x\Big(\int^x_{r_1}\rho uw\dd y\Big)\, \dd x\dd t \right|\leq C.
\end{align*}
Using the fact that $\alpha\leq\gamma_2,$ we have
\begin{align*}
\left| \int_0^T\int_{r_1}^{r_2} K_5\, \dd x\dd t \right|
&=\left|\int_0^T\int_{r_1}^{r_2} \Big(\rho uw\int^x_{r_1}\big((\rho u^2+P(\rho))w_y-\v\rho^{\alpha}w_yu_y\big)\dd y\Big)\,
\dd x\dd t \right|\nonumber\\
&\leq C
+\left|\v\int^T_0\int^{r_2}_{r_1}\rho w\Big(\int^x_{r_1}\rho^{\alpha}w_yu_y\,\dd y\Big)\dd x\dd t\right|\nonumber\\
&\leq C(r_1,r_2),
\end{align*}
where we have used the fact that
\begin{align*}
\left|\v\int_0^T\int_{r_1}^{r_2} \rho w\Big(\int^x_{r_1}\rho^{\alpha}w_yu_y\, \dd y\Big)\dd x\dd t \right|
&\leq \v\int^T_0\int^{r_2}_{r_1}|\rho^{\alpha}w_xu_x|\,\dd x\dd t\nonumber\\
&\leq \Big(\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}|u_x|^2\,\dd x \dd t\Big)^{\frac{1}{2}}\Big(\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}|w_x|^2\,\dd x\dd t\Big)^{\frac{1}{2}}\nonumber\\
&\leq C
\Big(\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}\,\dd x\dd t\Big)^{\frac{1}{2}}\nonumber\\
&\leq C
\Big(\v\int^T_0\int^{r_2}_{r_1}(\rho^{\gamma_2}+1)\,\dd x\dd t\Big)^{\frac{1}{2}}\nonumber\\
&\leq C(r_1,r_2).
\end{align*}
We also have
\begin{align}%\label{5.44}
\left|\int_0^T\int_{r_1}^{r_2} K_6\,\dd x\dd t \right|&\leq C,
\nonumber
\end{align}
\begin{align*}%\label{5.45}
&\left|\int_0^T\int_{r_1}^{r_2} K_7\,\dd x\dd t \right|\nonumber\\
&=\left|\int_0^T\int_{r_1}^{r_2} \rho(t,x) w(x)\Big(\int^x_{r_1}\rho(t,y) \partial_xW\ast\rho w(y) \,\dd y\Big)\dd x\dd t \right|\nonumber\\
&= \left|\int_0^T\int_{r_1}^{r_2} \rho(t,x) w(x)\Big(\int^x_{r_1}\rho(t,y) \big(\int^{b^+(t)}_{b^-(t)}(1-2H(x-z)+x-z)\rho(z)\,\dd z\big)w(y)\,\dd y\Big)\dd x\dd t \right|
\leq C,
\nonumber
\end{align*}
\begin{align}
&\left|\int_0^T\int_{r_1}^{r_2} K_8\,\dd x\dd t \right|\nonumber\\
&=\left|\int^T_0\int^{r_2}_{r_1}\rho(t,x)w(x)\int^x_{r_1}\rho(t,y)\Big(\int_{b^{-}(t)}^{b^+(t)}\varpi(y-z)(u(z)-u(y))\rho(t,z)\,\dd z\Big)w(y)\,\dd y\dd x\dd t\right|
\leq C.\label{5.452}
\end{align}
Integrating \eqref{5.33} over $[0,T]\times[r_1, r_2],$ and utilizing \eqref{5.35}--\eqref{5.452},
we can obtain \eqref{5.30}. This completes the proof.
\smallskip
\begin{corollary}\label{cordensity}
Under the assumptions of {\rm Lemma \ref{lem5.6}}, it follows from {\rm Lemma \ref{lemma5.6}} and \eqref{5.3} that
\begin{equation}\nonumber
\int_{b^-(t)}^{b^+(t)}\rho^{\g_2+1}(t,x)\,\mathrm{d}x
\leq C
\int_{b^-(t)}^{b^+(t)}(\rho+\r P(\r))(t,x)\,\mathrm{d}x\leq C(r_1,r_2)
\qquad \text{ for $t\geq 0$}.
\end{equation}
\end{corollary}
\medskip
\subsection{A special entropy pair}
Compared with the polytropic gas case in [12],
there is no explicit formula of the entropy kernel for the general pressure law
\eqref{pressure2}--\eqref{pressure4}
so that we have to analyze the entropy equation \eqref{2.7} carefully to obtain several desired estimates.
In order to obtain the higher integrability of the velocity, we use a special entropy pair constructed in [13]
such that $\rho|u|^3$ can be controlled by the entropy flux.
Indeed, such a special entropy $\hat{\eta}(\rho,u)$ is constructed as
\begin{equation*}%\label{5.46}
\hat{\eta}(\rho,u)=\begin{cases}
\frac{1}{2}\rho u^2+\rho e(\rho)\quad&\text{for }u\ge k(\rho),\\
-\frac{1}{2}\rho u^2-\rho e(\rho)\,\,\, &\text{for }u\le -k(\rho),
\end{cases}
\end{equation*}
for $k(\rho)=\int_{0}^{\rho}\frac{\sqrt{P'(y)}}{y}\,\mathrm{d}y$
and, in the intermediate region $-k(\rho)\le u\le k(\rho)$,
$\hat{\eta}(\rho,u)$ is the unique solution of the Goursat problem
of the entropy equation \eqref{2.7}:
\begin{equation}\label{5.47}
\left\{\begin{aligned}
\dis&\eta_{\rho\rho}-k'(\rho)^2\eta_{uu}=0 \qquad \mbox{for $-k(\rho)\le u\le k(\rho)$},\\
\dis&\eta(\rho,u)\vert_{u=\pm k(\rho)}=\pm \big(\frac{1}{2}\rho u^2+\rho e(\rho)\big){\rm .}
\end{aligned}
\right.
\end{equation}
Let us recall the following lemma, which will be used in the proof of Lemma \ref{lem5.9}.
\begin{lemma}[[13], Lemma 4.1]\label{lem5.8}
The Goursat problem \eqref{5.47} admits a unique solution $\hat{\eta}\in C^2(\R_{+} \times \R)$ such that
\begin{itemize}
\item [(\rmnum{1})]
$|\hat{\eta}(\rho,u)|\leq C(\rho|u|^2+\rho^{\gamma(\rho)})$ for $(\rho,u)\in \R_{+}\times \R$,
where $\gamma(\rho)=\gamma_1$ if $\rho\in [0,\rho_{*}]$ and $\gamma(\rho)=\gamma_2$ if $\rho\in (\rho_{*},\infty)$.
\smallskip
\item [(\rmnum{2})] If $\hat{\eta}$ is regarded as a function of $(\r, u)$,
\begin{align*}
\qquad |\hat{\eta}_{\rho}(\rho,u)|\leq C(|u|^2+\rho^{2\theta(\rho)}),\quad|\hat{\eta}_{u}(\rho,u)|\leq C(\rho|u|+\rho^{\theta(\rho)+1})
\qquad\text{for }(\rho,u)\in \R_{+}\times \R,
\end{align*}
and, if $\hat{\eta}$ is regarded as a function of $(\r, m)$,
\begin{align*}
\qquad |\hat{\eta}_{\rho}(\r,m)|\leq C(|u|^2+\rho^{2\t(\r)}),\quad|\hat{\eta}_{m}(\r,m)|\leq C(|u|+\rho^{\t(\r)})
\qquad\,\text{for } (\rho,m)\in \R_{+}\times \R,
\end{align*}
where $\theta(\rho):=\frac{\gamma(\rho)-1}{2}$.
\smallskip
\item [(\rmnum{3})] If $\hat{\eta}_{m}$ is regarded as a function of $(\r,u)$,
\begin{align*}
\qquad |\hat{\eta}_{m\rho}(\rho,u)|\leq C\rho^{\theta(\rho)-1},\quad |\hat{\eta}_{mu}(\rho,u)|\leq C,
\end{align*}
and, if $\hat{\eta}_{m}$ is regarded as a function of $(\r,m)$,
|\hat{\eta}_{m\rho}(\rho,m)|\leq C\rho^{\theta(\rho)-1},\quad |\hat{\eta}_{mm}(\rho,m)|\leq C\rho^{-1}.
\item[(\rmnum{4})] If $\hat{q}$ is the corresponding entropy flux determined by \eqref{entropypair},
then $\hat{q}\in C^2(\R_{+}\times \R)$ and
\begin{align*}
&\hat{q}(\rho,u)=\frac{1}{2}\rho |u|^3\pm \rho u(e(\rho)+\rho e'(\rho))\qquad\quad\text{for }\pm u\geq k(\rho),\\
&|\hat{q}(\rho,u)|\leq C\r^{\g(\rho)+\t(\r)}\qquad\qquad\qquad\qquad\,\,\,\,\,\,\text{for }|u|<k(\rho),\\
&\hat{q}(\rho,u)\geq \frac{1}{2}\rho |u|^3\qquad\qquad\qquad\qquad\qquad\quad\,\,\,\text{for } |u|\geq k(\rho),\\
&|\hat{q}-u\hat{\eta}|\leq C(\rho^{\g(\rho)}|u|+\rho^{\g(\rho)+\t(\rho)})\qquad\qquad\text{for } (\rho,u)\in \R_{+}\times \R.
\end{align*}
\end{itemize}
\end{lemma}
We are now ready to prove the better integrability of the velocity.
\begin{lemma}[\bf Higher integrability of the velocity]\label{lem5.9}
Let $(\r,u)$ be the smooth solution of \eqref{1.8} and \eqref{3.1}--\eqref{3.4}.
Then, under the assumption of {\rm Lemma \ref{lem5.6}},
\begin{align*}
\int_0^T\int_{r_1}^{r_2} \rho|u|^3(t,x) \,\dd x \dd t\leq C(r_1,r_2)
\end{align*}
for any $(r_1,r_2)\Subset [b^-(t), b^+(t)]$.
\end{lemma}
\noindent{\bf Proof.} Multiplying $\eqref{1.8}_1$ by $\hat{\eta}_\r$ and $\eqref{1.8}_2$ by $ \hat{\eta}_m$,
we have
\begin{align}\label{5.49}
& \hat{\eta}_t+\hat{q}_x = \hat{\eta}_m\, \Big(\v (\rho^{\alpha} u_x)_x+\lambda \rho u+\rho V-\rho \partial_xW\ast\rho\Big).
\end{align}
A direct calculation shows that
\begin{align}\label{5.50}
\frac{\dd}{\dd t}\int_x^{b^+(t)}\hat{\eta}\,\dd y
& =\hat{\eta}(t,b^+(t)) \frac{\dd}{\dd t}b^+(t)+\int_x^{b^+(t)}\hat{\eta}_t(t,y)\,\dd y= (u\hat{\eta})(t, b^+(t))+\int_x^{b^+(t)} \hat{\eta}_t(t,y)\,\dd y.
\end{align}
Integrating \eqref{5.49} over $[x,b^+(t))$, we have
\begin{align}\label{5.51}
\hat{q}(t,x) &=\Big( \int_x^{b^+(t)}\hat{\eta}(t,y)\,\dd y\Big)_t+\big(\hat{q}-u\hat{\eta}\big)(t,b^+(t))\nonumber\\
&\quad-\v \int_x^{b^+(t)} \hat{\eta}_m(\rho^{\alpha} u_y)_y\,\dd y-
\lambda \int_x^{b^+(t)} \hat{\eta}_m\rho u\,\dd y+\int_x^{b^+(t)}\hat{ \eta}_m \rho\partial W\ast\rho\,\dd y-\int^{b^+(t)}_x\hat{\eta}_m\rho V\,\dd y\nonumber\\
\end{align}
\smallskip
We now estimate each term $I_i, i=1,\cdots, 6$, in \eqref{5.51}.
for $I_2$ involving the trace estimates in \eqref{5.51}, it follows from \cite[Lemma 3.6]{Chen2021}, Lemma \ref{lem5.8}
\begin{align}\label{5.52}
\big|(\hat{q}-u\hat{\eta})(t,b^+(t))\big|\,
\dd x\dd t\leq C(r_1,r_2)\int_0^T\big(\rho^{\gamma_1+\theta_1}(t,b^+(t))+(\rho^{\gamma_1}|u|)(t, b^+(t)\big)\,\dd t.%\nonumber\\
\end{align}
It follows from \eqref{5.8} that
\begin{align}\label{5.53}
&\int_0^T\big(\rho(t,b^+(t))^{\gamma_1+\theta_1}\,\dd t=\int^T_0(\rho_0(b))^{\gamma_1+\theta_1}\,\dd t\leq\rho^{\gamma_1+\theta_1}_{\ast}T\leq C.
\end{align}
It is noted from \eqref{5.4}, \eqref{5.8}, and \eqref{5.11} that
\begin{align}\label{5.54}
\big(\rho^{\gamma_1}|u|\big)(t,b^+(t))\,
\dd x\dd t\nonumber\\
&=\int^T_0\rho^{\frac{\gamma_1-\alpha}{2}}(t,b^+(t))|(\rho^{\frac{\gamma_1+\alpha}{2}}u)(t,b^+(t))|\,\dd t\nonumber\\
&\leq(\rho_0(b))^{\frac{\gamma_1-\alpha}{2}}\int^T_0\bigg\{\Big(\int^{b^+(t)}_{b^-(t)}P'(\rho)\rho^{\alpha-2}\rho^2_x\,\dd x\Big)^{\frac{1}{2}}+\Big(\int^{b^+(t)}_{b^-(t)}\rho^{\alpha}u^2_x\,\dd x\Big)^{\frac{1}{2}}+\v^{-1}b^{-\frac{1}{2}}\bigg\}\,\dd t\nonumber\\
&\leq C
&\quad\,\,\times \bigg\{\v^{-\frac{1}{2}}b^{-\frac{1}{2}}T+\Big(\varepsilon\int^T_0\int^{b^+(t)}_{b^-(t)}P'(\rho)\rho^{\alpha-2}\rho^2_x\,\dd x\dd t\Big)^{\frac{1}{2}}(\frac{T}{\v})^{\frac{1}{2}}
+\Big(\v\int^T_0\int^{b^+(t)}_{b^-(t)}\rho^{\alpha}u^2_x\,\dd x\dd t\Big)^{\frac{1}{2}}(\frac{T}{\v})^{\frac{1}{2}}\bigg\}\nonumber\\
&\leq C
\big(\v^{-1}b^{-\frac{1}{2}}+\v^{-\frac{1}{2}}\big)
\leq C
\big(\v^{-1}b^{-\frac{1}{2}}+\v^{-\frac{1}{2}}\big)\nonumber\\
&\leq C
\v^{\frac{\gamma_1-\alpha}{2\gamma_1}p-\frac{1}{2}}
\leq C,
\end{align}
where $p>\frac{\gamma_1}{\gamma_1-\alpha}.$
Substituting \eqref{5.53}--\eqref{5.54} into \eqref{5.52}, we obtain
\begin{align}\label{5.55}
&\Big|\int^T_0\int^{r_2}_{r_1}I_2\,\dd x\dd t\Big|=\Big|\int^T_0\int^{r_2}_{r_1}
\dd x\dd t\Big|\leq C(r_1,r_2).
\end{align}
For $I_1$, using Lemma \ref{lem5.8}, we have
\begin{align}\label{5.56}
&\Big|\int^T_0\int^{r_2}_{r_1}I_1\,\dd x\dd t\Big|=\Big|\int_0^T\int_{r_1}^{r_2}\Big(\int_x^{b^+(t)}\hat{\eta}(\rho,\rho u)\, \dd y\Big)_t\,\dd x\dd t\Big| \nonumber\\
&\leq \Big|\int_{r_1}^{r_2}\int_{b^-(t)}^{b^+(t)}\hat{\eta}(\rho,\rho u)(T,y)\, \dd y\dd x\Big|
+\Big|\int_{r_1}^{r_2}\int_{b^-(t)}^{b^+(t)}\hat{\eta}(\rho_0,\rho_0 u_0) \,\dd y\dd x\Big|
\leq C(r_1,r_2).
\end{align}
For $I_3,$ we integrate by parts to obtain
\begin{align}\label{5.57}
-\v\int_x^{b^+(t)}\hat{\eta}_m(\rho^{\alpha} u_y)_y\,\dd y
&=-\v\Big(\hat{\eta}_m(t,b^+(t))\, (\rho^{\alpha} u_x)(t,b^+(t))-\hat{\eta}_m(t,x)(\rho^{\alpha} u_x)(t,x) \Big) \nonumber\\
&\quad\,\,+\v\int_x^{b^+(t)}\rho^{\alpha} u_y(\hat{\eta}_{mu}u_y+\hat{\eta}_{m\rho}\rho_y)\,\dd y:= J_1+J_2.
\end{align}
We first estimate $J_2$:
\begin{align*}
|J_2|=&\Big|\v\int^{b^+(t)}_x\Big(\hat{\eta}_{m\rho}\rho^{\alpha}u_y\rho_y+\hat{\eta}_{mu}\rho^{\alpha}u^2_{y}\Big)\,\dd y\Big|\leq C\v\Big|\int^{b^+(t)}_x\rho^{\theta(\rho)+\alpha-1}u_y\rho_y\,\dd y\Big|+\v\int^{b^+(t)}_x\rho^{\alpha}u^2_{y}\,\dd y\nonumber\\
&\leq \v\int^{b^+(t)}_xP'(\rho)\rho^{\alpha-2}\rho^2_y\,\dd y+\v\int^{b^+(t)}_x\frac{\rho^{\alpha+2\theta(\rho)}}{P'(\rho)}u^2_y\,\dd y+\v\int^{b^+(t)}_x\rho^{\alpha}u^2_{y}\,\dd y\nonumber\\
&\leq \v\int^{b^+(t)}_xP'(\rho)\rho^{\alpha-2}\rho^2_y\,\dd y+\v\int^{b^+(t)}_x\rho^{\alpha}u^2_{y}\,\dd y,
\end{align*}
where we have used the fact that
$|\hat{\eta}_{mu}|\leq C$, $|\hat{\eta}_{m\r}|\leq C\r^{\theta(\rho)-1}$,
\begin{align*}
&\Big|\v\int^{b^+(t)}_x\frac{\rho^{\alpha+2\theta(\rho)}}{P'(\rho)}u^2_y\,\dd y\Big|\nonumber\\
&\leq\v\int^{b^+(t)}_x\frac{\rho^{\alpha+2\theta(\rho)}}{P'(\rho)}u^2_y{\bf I}_{\{\rho\leq \rho_{\ast}\}}\,\dd y+\v\int^{b^+(t)}_x\frac{\rho^{\alpha+2\theta(\rho)}}{P'(\rho)}u^2_y{\bf I}_{\{\rho_{\ast}\leq\rho\leq \rho^{\ast}\}}\,\dd y+\v\int^{b^+(t)}_x\frac{\rho^{\alpha+2\theta(\rho)}}{P'(\rho)}u^2_y{\bf I}_{\{\rho\geq \rho^{\ast}\}}\,\dd y\nonumber\\
&\leq\v C\int^{b^+(t)}_x\rho^{\alpha+2\theta(\rho)-(\gamma_1-1)}u^2_y\,\dd y+\v C\int^{b^+(t)}_x\rho^{\alpha+2\theta(\rho)-(\gamma_2-1)}u^2_y\,\dd y
\leq \v C\int^{b^+(t)}_x\rho^{\alpha}u^2_y\,\dd y.
\end{align*}
Then we have
%obtain that
\begin{align}\label{5.61}
\int^T_0\int^{r_2}_{r_1}|J_2|\,\dd x\dd t&\leq\varepsilon\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_xP'(\rho)\rho^{\alpha-2}\rho^2_y\,\dd y\dd x\dd t+\varepsilon\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\rho^{\alpha}u^2_y\,\dd y\dd x\dd t\nonumber\\
&\leq C(r_1,r_2).
\end{align}
For $J_1,$ we have
\begin{align}
&\varepsilon\Big|\int^T_0\int^{r_2}_{r_1} \hat{\eta}_m\rho^{\alpha}u_x\,\dd x\dd t\Big|\nonumber\\
&\leq\varepsilon\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}u^2_x\,\dd x\dd t
+\varepsilon\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}\big(|u|+\rho^{\theta(\rho)}\big)^2\,\dd x\dd t\nonumber\\
&\leq C
+\v\int^T_0\int^{r_2}_{r_1}\rho^{2\theta(\rho)+\alpha}\,\dd x\dd t+\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}u^2\,\dd x\dd t.\label{5.62}
\end{align}
To control \eqref{5.62}, we need
\begin{align}\label{5.63}
\v\int^T_0\int^{r_2}_{r_1}\rho^{2\theta(\rho)+\alpha}\,\dd x\dd t
&=\v\int^T_0\int^{r_2}_{r_1}\rho^{\gamma_2-1+\alpha}\,\dd x\dd t\leq C(\mathcal{E}^{\v}_0,\mathcal{E}^{\v}_1)\int^T_0\int^{r_2}_{r_1}\rho^{\frac{\gamma_2-1}{2}}\,\dd x\dd t\nonumber\\
&\leq C
\int^T_0\int^{r_2}_{r_1}\rho^{\gamma_2}\,\dd x\dd t+C(r_1,r_2)
\leq C(r_1,r_2).
\end{align}
Then we have
\begin{align}\label{5.64}
\v\int^T_0\int^{r_2}_{r_1}\rho^{\alpha}u^2\,\dd x\dd t
&\leq \Big(\int^T_0\int^{r_2}_{r_1}\rho|u|^3\,\dd x\dd t\Big)^{\frac{2}{3}}
\,\Big(\int^T_0\int^{r_2}_{r_1}\varepsilon^3\rho^{3\alpha-2}\,\dd x\dd t\Big)^{\frac{1}{3}}\nonumber\\
&\leq \Big(\int^T_0\int^{r_2}_{r_1}\rho|u|^3\,\dd x\dd t\Big)^{\frac{2}{3}}
\,\Big(\int^T_0\int^{r_2}_{r_1}\varepsilon^3\rho^{3\beta}\,\dd x\dd t+C(r_1,r_2)\Big)^{\frac{1}{3}}\nonumber\\
&\leq C(r_1,r_2)
\Big(\int^T_0\int^{r_2}_{r_1}\rho|u|^3\,\dd x\dd t\Big)^{\frac{2}{3}},
\end{align}
where we have used $\alpha\geq\frac{2}{3}.$
Inserting \eqref{5.63} and \eqref{5.64} into \eqref{5.62}, we see that, for $\delta>0$,
\begin{align*}
&\varepsilon\Big|\int^T_0\int^{r_2}_{r_1} \hat{\eta}_m\rho^{\alpha}u_x\,\dd x\dd t\Big|
\leq C(r_1,r_2)
\frac{1}{\delta}+\delta\int^T_0\int^{r_2}_{r_1}\rho|u|^3\,\dd x\dd t.
\end{align*}
Using Lemma \ref{lem5.8} and \eqref{3.3}, we obtain
\leq C_{\gamma}\big(\rho^{\gamma_1}|u|+\rho^{\gamma_1+\theta_1}\big)(t,b^+(t)).
Similar again to the argument as in [12, 28] yields
\begin{align}\label{5.66}
\int^T_0\int^{r_2}_{r_1}|\v(\hat{\eta}_m\rho^{\alpha}u_x)(t,b^+(t))|\,\dd x\dd t&\leq C(r_1,r_2)
\int^T_0\big(\rho^{\gamma_1}|u|+\rho^{\gamma_1+\theta_1}\big)(t,b^+(t)))\,\dd t\nonumber\\
&\leq C(r_1,r_2).
\end{align}
Combining \eqref{5.72} and \eqref{5.66}, we have that
\begin{align}\label{5.67}
\Big|\int^T_0\int^{r_2}_{r_1}J_1\,\dd x\dd t\Big|&\leq \delta
\int^T_0\int^{r_2}_{r_1}\rho |u|^3\,\dd x\dd t+C(r_1,r_2).
\end{align}
Inserting \eqref{5.61} and \eqref{5.67} into \eqref{5.57}, we obtain
\begin{align}\label{5.68}
\Big|\int^T_0\int^{r_2}_{r_1}I_3\,\dd x\dd t\Big|&\leq \delta
\int^T_0\int^{r_2}_{r_1}\rho |u|^3\,\dd x\dd t+C(r_1,r_2).
\end{align}
We also obtain the estimates for $I_i, i=4,5,6$:
\begin{align*}
\Big|\int^T_0\int^{r_2}_{r_1}I_4\,\dd x\dd t\Big|
&=\Big|\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\rho |u|\big(|u|+\rho^{\theta(\rho)}\big) \,\dd y\dd x\dd t\Big|\nonumber\\
&=\Big|\int^T_0\int^{r_2}_{r_1}\rho u^2\,\dd x\dd t\Big|+\Big|\int^T_0\int^{r_2}_{r_1}\rho^{\gamma_2}\,\dd x\dd t\Big|
\leq C(r_1,r_2),
\end{align*}
\begin{align*}
\Big|\int^T_0\int^{r_2}_{r_1}I_5\,\dd x\dd t\Big|&=\Big|\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\rho \hat{\eta}_m\partial_xW\ast \rho \,\dd y\dd x\dd t\Big|\nonumber\\
&=\Big|\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\rho\hat{\eta}_m\Big(M-2\int^{y}_{b^-(t)}\rho(t,z)\,\dd z+yM-\int^{b^+(t)}_{b^{-}(t)}z\rho(z) \,\dd z\Big)\,\dd y\dd x\dd t\Big|\nonumber\\
&\leq C(r_1,r_2),
\end{align*}
\begin{align}\label{5.702}
\Big|\int^T_0\int^{r_2}_{r_1}I_6\,\dd x\dd t\Big|&=\Big|\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\hat{\eta}_m\rho V\,\dd y\,\dd x\dd t\Big|\nonumber\\
&=\Big|\int^T_0\int^{r_2}_{r_1}\int^{b^+(t)}_x\hat{\eta}_m\rho \Big(\int^{b^+(t)}_{b^-(t)}\varpi(y-z)(u(y)-u(z))\rho(t,z)\,\dd z\Big)\,\dd y\,\dd x\dd t\Big|\nonumber\\
&\leq C(r_1,r_2).
\end{align}
Combining estimates \eqref{5.55}--\eqref{5.56} and \eqref{5.68}--\eqref{5.702} leads to
\int_{0}^{T}\int_{r_1}^{r_2}\hat{q}\,\mathrm{d}x\mathrm{d}t\leq C(r_1, r_2),
which gives
\begin{align}\label{5.71}
\int_{0}^{T}\int_{[r_1, r_2]\cap \{x: |u|\geq k(\rho)\}}\rho |u|^3\,\mathrm{d}x\mathrm{d}t
&\leq 2\int_{0}^{T}\int_{[r_1, r_2]\cap \{x: |u|\geq k(\rho)\}}\hat{q}\,\mathrm{d}x\mathrm{d}t\nonumber\\
&=2\int_{0}^{T}\int_{r_1 }^{r_2}\hat{q}\,\mathrm{d}x\mathrm{d}t
-2\int_{0}^{T}\int_{[r_1, r_2]\cap \{x: |u|<k(\rho)\}}\hat{q}\,\mathrm{d}x\mathrm{d}t\nonumber\\
&\leq C(r_1, r_2)
&\leq C(r_1, r_2)
\end{align}
by using Lemma \ref{lem4.1}. On the other hand, we have
\begin{align}\label{5.72}
\int_{0}^{T}\int_{[r_1, r_2]\cap \{x:\,|u|\leq k(\rho)\}}\rho |u|^3\,\mathrm{d}x\mathrm{d}t
&\leq C\int_{0}^{T}\int_{r_1}^{r_2}\rho^{\g(\rho)+\t(\rho)}\,\mathrm{d}x\mathrm{d}t
\nonumber\\ &
\leq C\int_{0}^{T}\int_{r_1}^{r_2}\big(\rho+\rho P(\rho)\big)\,\mathrm{d}x\mathrm{d}t
\leq C.
\end{align}
Combining \eqref{5.71} with \eqref{5.72}, we obtain that
$\int_{0}^{T}\int_{r_1}^{r_2}\rho |u|^3\,\mathrm{d}x\mathrm{d}t\leq C(r_1, r_2)$.
This completes the proof.
\medskip
\subsection{$W^{-1,p}_{\rm loc}(\mathbb{R}_+^2)-$Compactness}\label{section5.4}
In this section, we use the uniform estimates obtained
in \S 5.3 to prove the following key lemma, which
states the
$W^{-1,p}_{\rm loc}(\mathbb{R}_+^2)-$compactness of entropy dissipation measures for the approximate
solution sequence.
\begin{lemma}\label{lemm5.10}
Let $\frac23\leq\alpha\leq\gamma_2$, and let $(\eta^{\psi},q^{\psi})$ be a weak entropy pair
generated by $\psi\in C_0^2(\mathbb{R})$ defined in \eqref{2.8}--\eqref{2.9}.
Then, for the solutions $(\rho^{\varepsilon},u^{\varepsilon})$ with
$m^{\varepsilon}=\rho^{\varepsilon}u^{\varepsilon}$ of CNSEs \eqref{1.8} and \eqref{3.1}--\eqref{3.4},
\begin{align}\label{5.73}
\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x
\qquad \mbox{is compact in $W^{-1,p}_{\rm loc}(\mathbb{R}_+\times \mathbb{R})\,$ for any $p\in[1,2)$}.
\end{align}
\end{lemma}
\noindent
\textbf{Proof}. To prove this lemma, we first
recall the entropy pair
$(\eta^{\psi},q^{\psi})$ generated by $\psi\in C_0^2(\mathbb{R})$
({\it cf.} [13]).
Given any $\psi\in C_{0}^2(\R)$, a regular weak entropy pair $(\eta^{\psi},\,q^{\psi})$ is given by
\begin{align}\label{5.74}
\eta^{\psi}(\r,u)=\int_{\R} \psi(s)\,\chi(\r,u,s)\, \mathrm{d}s,\qquad
q^{\psi}(\r,u)=\int_{\R}\psi(s)\,\sigma(\r,u,s) \,\mathrm{d}s,
\end{align}
and satisfies the following properties
([13]; also see [14, 15, 40, 41]):
\iffalse
It follows from \eqref{6.1} that
\begin{equation}\label{5.75aa}
\left\{\begin{aligned}
\dis&\eta_{\r\r}^{\psi}-k'(\r)^2\eta_{uu}^{\psi}=0,\\
\dis&\eta^{\psi}\vert_{\r=0}=0,\quad \eta_{\r}^{\psi}\vert_{\r=0}=\psi(u).
\end{aligned}
\right.
\end{equation}
We recall the following lemma for the weak entropy pair $(\eta^{\psi},q^{\psi})$. ({\it cf.} [14, 15, 40, 41]).
For any weak entropy $(\eta^{\psi},q^{\psi})$ defined in \eqref{5.74},
\fi
There exists a constant $C_{\psi}>0$ depending only on $\rho^{*}$ and $\psi$
such that
\begin{enumerate}
\item[(i)]
|q^{\psi}(\rho,u)|\leq C_{\psi}\rho\, $ for all $\rho\in [0,2\rho^{*}]$.
\item[(ii)] If $\eta^{\psi}$ is regarded as a function of $(\rho,m)$, then
\begin{align*}
|\eta_{m}^{\psi}(\rho,m)|+|\r\eta_{mm}^{\psi}(\rho,m)|\leq C_{\psi},\qquad
|\eta_{\r}^{\psi}(\r,m)|\leq C_{\psi}(1+\r^{\t_1}).
\end{align*}
\item[(iii)] If $\eta_m^{\psi}$ is regarded as a function of $(\r,u)$, then
\begin{equation}\nonumber
|\eta_{mu}^{\psi}(\r,u)|+|\rho^{1-\t_1}\eta_{m\rho}^{\psi}(\rho,u)|\leq C_{\psi}.
\end{equation}
\end{enumerate}
A direct computation on $\eqref{1.8}_1\times\eta^{\psi}_{\rho}(\rho^{\varepsilon},m^{\varepsilon})
\begin{align}
\displaystyle\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x
\end{align}
Let $K\Subset[b^-(t),b^+(t)]$ be compact. Using properties (ii)-(iii) of the weak entropy pair $(\eta^{\psi},q^{\psi})$
the Cauchy-Schwartz
inequality, we have
\begin{equation*}
\begin{aligned}
+\eta^{\psi}_{m\rho}(\rho^{\varepsilon},m^{\varepsilon})(\rho^{\varepsilon})^{\alpha}\rho_x^{\varepsilon}u_x^{\varepsilon}|\,\dd x\dd t\\
C_{\psi}\varepsilon\int_{0}^{T}\int_{K}(\rho^{\varepsilon})^{\alpha}(u_x^{\varepsilon})^2 \,\dd x\dd t
+C_{\psi}\varepsilon\int_{0}^{T}\int_{K}(\rho^{\varepsilon})^{\theta_2-1+\alpha}(\rho_x^{\varepsilon})^2 \,\dd x\dd t
\leq C(K),
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
&\displaystyle\int_{0}^{T}\int_{K}\Big|\eta^{\psi}_m\Big(\lambda\rho^{\v}u^{\v}+\rho^{\v}V-\rho^{\v}\partial_xW\ast\rho^{\v}\Big)\Big| \,\dd x\dd t\\
C_{\psi}\int_{0}^{T}\int_{K}\rho^{\v}(u^{\v})^2+\rho^{\v} \,\dd x\dd t
+C_{\psi}\int_{0}^{T}\int_{K}|\rho^{\v}\partial_xW\ast\rho^{\v}| \,\dd x\dd t
\leq C(K).
\end{aligned}
\end{equation*}
This implies that
\begin{eqnarray*}
\end{eqnarray*}
is uniformly bounded in $L^1([0,T]\times K)$, and thus it is compact in
$W_{\rm loc}^{-1,p_1}(\mathbb{R}_+^2)$, for $1<p_1<2$.
If $2\alpha\leq\gamma_2+1,$ then we obtain that
\begin{align}\label{5.75}
\v^{\frac{4}{3}}\int^T_0\int_{K}(\rho^{\v})^{2\alpha}\,\dd x\dd t\leq C(K)\v^{\frac{4}{3}}.%\nonumber
\end{align}
If $2\alpha\geq\gamma_2+1,$ $\alpha<\gamma_2,$ it yields that,
\begin{align}\label{5.75b}
\v^{\frac{4}{3}}\int^T_0\int_{K}(\rho^{\v})^{2\alpha}\,\dd x\dd t
&\leq C(K)\v^{\frac{1}{3}}\int^T_0\int_K(\rho^{\v})^{\alpha-\frac{\gamma_2-1}{2}}\,\dd x\dd t\nonumber\\
&\quad+C(K)\v^{\frac{1}{3}}\int^T_0\int_K(\rho^{\v})^{\gamma_2+1}\,\dd x\dd t.
\end{align}
Moreover, it follows from \eqref{5.75} and \eqref{5.75b} that
%, one has
\begin{equation*}%\label{5.76}
\begin{aligned}
\displaystyle\int_{0}^{t}\int_{K}\Big(\varepsilon
\eta^{\psi}_m(\rho^{\varepsilon},m^{\varepsilon})
(\rho^{\varepsilon})^{\alpha}u_x^{\varepsilon}\Big)^{\frac{4}{3}}\, \dd x\dd t
(\rho^{\varepsilon})^{^{\frac{4\alpha}{3}}}|u_x^{\varepsilon}|^{\frac{4}{3}}\,\dd x\dd t\\
&\displaystyle\leq C\varepsilon^{\frac{4}{3}}\int_{0}^{t}\int_{K}
(\rho^{\varepsilon})^{\alpha}|u_x^{\varepsilon}|^2\,\dd x\dd t+C\varepsilon^{\frac{4}{3}}\int_{0}^{t}\int_{K}
(\rho^{\varepsilon})^{2\alpha}\,\dd x\dd t\\
&\displaystyle\leq C(K)\varepsilon^{\frac{1}{3}}\rightarrow 0\qquad\text{ as $\varepsilon\rightarrow0^+$}.
\end{aligned}
\end{equation*}
Then, \eqref{4.86} and \eqref{4.89} yield that
\begin{eqnarray}\label{5.77}
\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x
\ \mbox{is compact in $W_{\rm loc}^{-1,p_2}(\mathbb{R}_+^2)$} \ \
\mbox{for some}\ p_2\in(1,2).
\end{eqnarray}
which also implies that
\begin{equation}\label{5.78}
\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x\qquad \text{is compact in
$W_{\mathrm{loc}}^{-1,p}(\R_{+}^2)$ with $1\leq p\leq p_{2}$}.
\end{equation}
On the other hand, for $\gamma_2\in (1,3),$ using property (i) of the weak entrpy pair $(\eta^{\psi}, q^{\psi})$,
we have
\begin{eqnarray*}
\int^T_0\int_K\big(|\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})|^{\gamma_2+1}+|q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})|^2\big)\,\dd x\dd t
\leq C_{\psi}\int^T_0\int_K(\rho^{\v})^{\gamma_2+1}\,\dd x\dd t\leq C(K),
\end{eqnarray*}
so that $\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})$ and $q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})$
are uniformly bounded in $L_{\rm loc}^{2}(\mathbb{R}_+^2)$. This yields
\begin{eqnarray}\label{5.81}
\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x
\qquad \mbox{is uniformly bounded in $W_{\rm loc}^{-1, 2}(\mathbb{R}_+^2)$}.
\end{eqnarray}
Then the interpolation compactness theorem ({\it cf.} [11, 24, 25]) indicates that,
for $p_{2}>1, p_{1} \in\left(p_{2}, \infty\right]$, and $p_{0} \in\left[p_{2}, p_{1}\right)$,
\begin{align}
\big(\text{compact set of }W_{\mathrm{loc}}^{-1, p_{2}}(\mathbb{R}_{+}^{2})\big)
\cap \big(\text{bounded set of }W_{\mathrm{loc}}^{-1, p_{1}}(\mathbb{R}_{+}^{2})\big)
\subset \big(\text{compact set of }W_{\mathrm{loc}}^{-1, p_{0}}(\mathbb{R}_{+}^{2})\big),\nonumber
\end{align}
which is a generalization of Murat's lemma in [38, 42].
Combining this theorem for $1<p_{2}<2$ and $p_{1}=2$ with the facts in \eqref{5.81} and \eqref{5.77}, we conclude that
\begin{equation}\label{6.73-2}
\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x\qquad
\text{is compact in $W_{\mathrm{loc}}^{-1,p}(\R_{+}^2)$ with $p_2\leq p<2$}.
\end{equation}
Combining \eqref{6.73-2} with \eqref{5.78}, we conclude \eqref{5.73}.
\subsection{Proof of Theorem \ref{thm3.3}}\label{section5.5}
Recall the following $L^p$ compensated compactness theorem established by Chen-Huang-Li-Wang-Wang [13]{\rm :}
\begin{theorem}[[13], Lemma 2.2]\label{thm5.11}
Let $(\rho^{\v},m^{\v})(t,x)=(\rho^\v, \rho^\v u^\v)(t,x)$ be a sequence of measurable functions
with $\rho^{\v}\geq 0$ a.e. on $\R_{+}^2$
satisfying the following two conditions{\rm :}
\begin{enumerate}
\item [(\rmnum{1})] For any $T>0$ and $K\Subset\mathbb{R}_+$, there exists $C(K)>0$ independent of $\v$ such that
\int_0^T\int_K\big((\rho^\v)^{\g_2+1}+\rho^{\v}|u^{\v}|^3\big)\,\mathrm{d}x\mathrm{d}t\le C(K).
\item [(\rmnum{2})]
For any entropy pair $(\eta^\psi,q^\psi)$ defined in \eqref{5.74} with
any smooth function $\psi(s)$ of compact support on $\mathbb{R}$,
\eta^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_t+q^{\psi}(\rho^{\varepsilon},m^{\varepsilon})_x
\qquad\mbox{is compact in $ W_{\mathrm{loc}}^{-1,1}(\mathbb{R}_{+}^{2})$}.
\end{enumerate}
Then there exists a subsequence $($still denoted$)$ $(\rho^{\v},m^{\v})(t,x)$ and a vector-valued
function $(\rho,m)(t,x)$ such that, as $\v\to 0^+$,
\begin{equation*}
\begin{aligned}
&\rho^{\v}(t,x)\to \rho(t,x)~~ \mbox{in}~L^{q_1}_{\rm loc}(\R_{+}^2)\qquad\,\,\,\,\text{for }q_1\in [1,\g_2+1),\\
& m^{\v}(t,x)\to m(t,x)~~ \mbox{in}~L^{q_2}_{\rm loc}(\R_{+}^2)\qquad \text{for }q_2\in [1,\frac{3(\g_2+1)}{\g_2+3}),
\end{aligned}
\end{equation*}
where $L_{\mathrm{loc}}^{p}(\mathbb{R}_{+}^{2})$ represents $L^{p}([0, T] \times K)$ for any $T>0$
and compact set $K \Subset \mathbb{R}_+$.
\end{theorem}
The uniform estimates and compactness properties obtained in \S\ref{section5.1}--\S\ref{section5.4} yields that
the sequence of solutions
$(\rho^\varepsilon, m^\varepsilon)$ of problem \eqref{1.8} and \eqref{3.1}--\eqref{3.4} satisfies
the compensated compactness framework established in [13] so that there exist both a subsequence (still denoted)
$(\rho^{\varepsilon},m^{\varepsilon})$ and a vector function $(\rho, m)$ such that
\begin{eqnarray*}
(\rho^{\varepsilon},m^{\varepsilon})\rightarrow (\rho, m)\qquad\,\, a.e.\,\, (t,x)\in \R_+\times\R\,\,\,
\text{ as $\varepsilon\rightarrow0^+$}.
\end{eqnarray*}
|m^{\v}|^{\frac{3(\gamma_2+1)}{\gamma_2+3}}\leq C\big(\rho^{\v}|u^{\v}|^3+(\rho^{\v})^{\gamma_2+1}\big),
it follows from Lemma \ref{lem5.9} and Corollary \ref{cordensity} that
\begin{eqnarray*}
(\rho^{\v},m^{\v})\rightarrow(\rho,m)\qquad \text{ in $L^{q_1}_{\rm loc}(\R_+\times\R)\times L^{q_2}_{\rm loc}(\R_+\times\R)$}
\end{eqnarray*}
for $q_1\in[1,\gamma_2+1)$ and $q_2\in[1,\frac{3(\gamma_2+1)}{\gamma_2+3}).$
Moreover, We have
\begin{align*}
\eta^{\ast}(\rho^{\v},m^{\v})\rightarrow \eta^{\ast}(\rho,m)\qquad
\text{ in $L^1_{\rm loc}(\R_+\times\R)$\quad as $\varepsilon\rightarrow0^+$}.
\end{align*}
Then it is direct to check that $(\rho,m)$ is a finite-energy entropy solution of the Cauchy problem \eqref{1.1}.
Therefore, the proof of Theorem
\ref{thm3.3} and hence Theorem \ref{thm:merged} is mow completed for the general pressure case.
\appendix
\section{Construction of the Approximate Initial Data Sequences}\setcounter{equation}{0}\label{SecA}
We now construct the approximate initial data sequences $(\rho^\v_0(x),\rho^\v_0u^{\v}_0(x))$ with desired estimates,
regularity, and boundary compatibility for the polytropic case.
For the general pressure law case, the construction arguments are similar.
\smallskip
\subsection{The initial density}
Let $J(x)$ be the standard mollification function and
$\dis J_{\delta}(x):=\frac{1}{\delta}J(\frac{x}{\delta})$ for $\delta\in(0,1)$.
For later use, we take $\delta=\v^{\frac12}$ and define $\tilde{\rho}_{0}^{\v}(x)$ as
\begin{align}\label{A.1}
\tilde{\rho}_{0}^{\v}(x)
:=\Big(\int_{\mathbb{R}} \big((\rho_0\mathbf{I}_{[-b+1,b-1]})(x-y)\big)^{\alpha-\frac{1}{2}}J_{\sqrt{\varepsilon}}(y)\,\dd y
+\v e^{-x^2}\Big)^\frac{2}{2\alpha-1},
\end{align}
where we have denoted $\mathbf{I}_{[-b+1,b-1]}$ to be the characteristic function of $\{x\in\mathbb{R}\,:\, -b+1\leq|x|\leq b-1\}$.
Since $\alpha>\frac{2}{3}$, then $\tilde{\rho}_{0}^\v(x)\geq \v^{\frac{2}{2\alpha-1}} e^{-\frac{2}{2\alpha-1}x^2}>0$.
\begin{lemma}\label{lemA.2}
Let $q\in \{1, \gamma, 2\alpha-1\}$ and $\frac{2}{3}<\alpha\leq 1$. Then $\tilde{\rho}_{0}^\v(x)$ satisfy the following properties{\rm :}
There exists $C_0>0$ independent of $\v$, but may depend on $\mathcal{E}_0, M, M_2, \gamma$, and $\alpha$, such that
%there holds
\begin{align}
&\tilde{\rho}^{\v}_0(b)=\tilde{\rho}^{\v}_0(-b)\leq C_0 b^{-\frac{1}{\gamma}}, \label{A.388}\\[1.5mm]
&\|\tilde{\rho}_0^\v\|_{L^q}\leq C_0\big(\|\rho_0\|_{L^q}+ \v^{\frac{2}{2\alpha-1}}\big) \qquad\,\mbox{for $\v\in(0,1]$},\label{A.2}\\[1.5mm]
&\lim_{\v\rightarrow0+} \big(\|\tilde{\rho}_0^\v-\rho_0\|_{L^q}
&\v^2\int_{\R}\Big|\big((\tilde{\rho}^{\v}_0(x))^{\alpha-\frac{1}{2}}\big)_x\Big|^2\dd x
\rightarrow0 \qquad\mbox{as $\v\rightarrow0^+$},\label{A.4}\\[1.5mm]
&\int_{\R} x^2|\tilde{\rho}^{\v}_0(x)|\,\dd x\leq C_0\Big(\int_{\R} x^2|\rho_0(x)|\,\dd x+\v\Big),\label{A.5}\\[1.5mm]
&\lim_{\v\rightarrow0+} \int_{\R} x^2|\tilde{\rho}_0^\v(x)-\rho_0(x)|\,\dd x= 0,\label{A.6}\\[1.5mm]
&\int_{\R}\big|\tilde{\rho}^{\v}_0(x)(W\ast\tilde{\rho}^{\v}_0)(x)-\rho_0(x)(W\ast\rho_0)(x)\big|\,\dd x\rightarrow0 \qquad\text{ as $\varepsilon\rightarrow0^+$}.\label{A.62}\\[1.5mm]
&\int_{\R}\tilde{\rho}^{\v}_0(x)(W\ast\tilde{\rho}^{\v}_0)(x) \,\dd x\leq C_0\big(1+\v^{\beta})\qquad
\text{ for $\beta:=\frac{2}{2\alpha-1}\in[2,6)$}.\label{A.63}
\end{align}
\end{lemma}
\noindent{\bf Proof.} We divide the proof into six steps. In the proof below,
for simplicity,
$C>0$ is a universal constant independent of $\v$, which may depend on $\mathcal{E}_0, M, M_2,\gamma$, and $\alpha$, and
may be different at each occurrence so that $C_0>0$ can be chosen eventually, depending only on this $C>0$ for the statement in this lemma.
\medskip
\noindent{\emph{Step 1.}} The proof of \eqref{A.2}--\eqref{A.4} are similar as in [12], so we omit them for brevity.
Here, we remark that $\frac{2}{3}<\alpha\leq1,$ then $\beta:=\frac{2}{2\alpha-1}\in [2,\,6)$. Since in the proof of \eqref{A.4}, using Young's inequality, one has
\begin{align}\label{A.4222}
\v^2\int_{\R}\Big|\big((\tilde{\rho}^{\v}_0(x))^{\alpha-\frac{1}{2}}\big)_x\Big|^2\,\dd x&\leq C\v\Big(\|\rho_0(x)\|^{2\alpha-1}_{L^{2\alpha-1}}+1\Big) \nonumber\\
&\leq C\v\Big(\int_{\R} (1+x^2)\rho_0(x)\,\dd x+\int_{\R}\frac{1}{(1+x^2)^{\frac{2\alpha-1}{2-2\alpha}}}\,\dd x+1\Big)\nonumber\\
&\leq C_0\v.
\end{align}
\smallskip
\noindent{\emph{Step 2.}} We first prove \eqref{A.388}, which is needed in the proof of Lemmas \ref{lem2.2} and \ref{lem4.5} by \eqref{A.1} since $\v<1$:
\begin{align*}
\tilde{\rho}^{\v}_0(b)=\tilde{\rho}^{\v}_0(-b)=(\v e^{-b^2})^{\frac{2}{2\alpha-1}}\leq C b^{-\frac{1}{\gamma}}.
\end{align*}
For $b=\v^{-p}$ and $0<\v<1,$ it is clear that there exists a positive constant C such that
(\v e^{-\v^{-2p}})^{\frac{2}{2\alpha-1}}\leq C\v^{\frac{p}{\gamma}},
where $C$ is independent of $\v\in (0,1]$.
\smallskip
\noindent{\emph{Step 3.}} We are now ready to prove \eqref{A.5}. By Young's inequality, we have
\begin{align}\label{A.7}
\tilde{\rho}_{0}^{\v}(x)
&=\Big(\int_{\mathbb{R}} \big((\rho_0\mathbf{I}_{[-b+1,b-1]})(x-y)\big)^{\alpha-\frac{1}{2}}J_{\sqrt{\varepsilon}}(y)\,\dd y
+\v e^{-x^2}\Big)^\frac{2}{2\alpha-1}\nonumber\\
&\leq 2^{\frac{2}{2\alpha-1}}\bigg(\Big(\int_{\mathbb{R}}
\big((\rho_0\mathbf{I}_{[-b+1,b-1]})(x-y)\big)^{\alpha-\frac{1}{2}}J_{\sqrt{\varepsilon}}(y)\,\dd y \Big)^\frac{2}{2\alpha-1}
+(\v e^{-x^2})^{\frac{2}{2\alpha-1}}\bigg)\nonumber\\
&\leq 2^{\frac{2}{2\alpha-1}}\Big(\int_{\mathbb{R}} J_{\sqrt{\varepsilon}}(y)\rho_0(x-y)\,\dd y +(\v e^{-x^2})^{\frac{2}{2\alpha-1}}\Big).
\end{align}
Therefore, using \eqref{A.7}, we obtain
x^2\tilde{\rho}_{0}^{\v}(x)\leq Cx^2\Big(\int_{\mathbb{R}} J_{\sqrt{\varepsilon}}(y)\rho_0(x-y)\,\dd y +(\v e^{-x^2})^{\frac{2}{2\alpha-1}}\Big),
so that
\int_{\R} x^2|\tilde{\rho}_{0}^{\v}(x)|\, \dd x
\leq C\Big(\int_{\mathbb{R}}\int_{\mathbb{R}} J_{\sqrt{\varepsilon}}(y)x^2\rho_0(x-y)\,\dd y\dd x +\int_{\mathbb{R}} x^2(\v e^{-x^2})^{\frac{2}{2\alpha-1}}\,\dd x\Big).
Notice that
\begin{align}\label{A.8}
\int_{\mathbb{R}^2} J_{\sqrt{\varepsilon}}(y)x^2\rho_0(x-y)\,\dd y\dd x |
11in8.5in 11in8.5in* 0in0in 1.3in1.3in* 1.3in1.3in* 13pt26pt *13pt*
subsubsection default
# Sample Complexity of Robust Learning against Evasion Attacks
###### Abstract
It is becoming increasingly important to understand the vulnerability of
machine learning models to adversarial attacks. One of the fundamental
problems in adversarial machine learning is to quantify how much training data
is needed in the presence of so-called evasion attacks, where data is
corrupted at test time. In this thesis, we work with the exact-in-the-ball
notion of robustness and study the feasibility of adversarially robust
learning from the perspective of learning theory, considering sample
complexity.
We start with two negative results. We show that no non-trivial concept class
can be robustly learned in the distribution-free setting against an adversary
who can perturb just a single input bit. We then exhibit a sample-complexity
lower bound: the class of monotone conjunctions and any superclass on the
boolean hypercube has sample complexity at least exponential in the
adversary’s budget (that is, the maximum number of bits it can perturb on each
input). This implies, in particular, that these classes cannot be robustly
learned under the uniform distribution against an adversary who can perturb
$\omega(\log n)$ bits of the input.
As a first route to obtaining robust learning guarantees, we consider
restricting the class of distributions over which training and testing data
are drawn. We focus on learning problems with probability distributions on the
input data that satisfy a Lipschitz condition: nearby points have similar
probability. We show that, if the adversary is restricted to perturbing
$O(\log n)$ bits, then one can robustly learn the class of monotone
conjunctions with respect to the class of log-Lipschitz distributions. We then
extend this result to show the learnability of $1$-decision lists, 2-decision
lists and monotone $k$-decision lists in the same distributional and
adversarial setting. We finish by showing that for every fixed $k$ the class
of $k$-decision lists has polynomial sample complexity against a
$\log(n)$-bounded adversary. The advantage of considering intermediate
subclasses of $k$-decision lists is that we are able to obtain improved sample
complexity bounds for these cases.
As a second route, we study learning models where the learner is given more
power through the use of _local_ queries. The first learning model we consider
uses local membership queries (LMQ), where the learner can query the label of
points near the training sample. We show that, under the uniform distribution,
the exponential dependence on the adversary’s budget to robustly learn
conjunctions and any superclass remains inevitable even when the learner is
given access to LMQs in addition to random examples. Faced with this negative
result, we introduce a local _equivalence_ query oracle, which returns whether
the hypothesis and target concept agree in a given region around a point in
the training sample, as well as a counterexample if it exists. We show a
separation result: on the one hand, if the query radius $\lambda$ is strictly
smaller than the adversary’s perturbation budget $\rho$, then distribution-
free robust learning is impossible for a wide variety of concept classes; on
the other hand, the setting $\lambda=\rho$ allows us to develop robust
empirical risk minimization algorithms in the distribution-free setting. We
then bound the query complexity of these algorithms based on online learning
guarantees and further improve these bounds for the special case of
conjunctions. We follow by giving a robust learning algorithm for halfspaces
on $\left\\{0,1\right\\}^{n}$. Finally, since the query complexity for
halfspaces on $\mathbb{R}^{n}$ is unbounded, we instead consider adversaries
with _bounded precision_ and give query complexity upper bounds in this
setting as well.
###### Acknowledgements
I would first like to express my most sincere gratitude to my supervisors
James (Ben) Worrell, Varun Kanade, and Marta Kwiatkowska.
Marta, I am extremely grateful for your guidance, support and generosity. Your
help has been invaluable in setting a research agenda and navigating my DPhil.
I very much value the time you make for your students, your involvement and
reliability.
Varun, thank you for taking a chance working with me in my second year, and
for introducing me to learning theory and interesting problems in the field.
Your expertise and knowledge have been beyond helpful. I am tremendously
grateful for our discussions, and greatly appreciate your insightful comments
and approach to research. I value your mentorship immensely.
Ben, your enthusiasm for research is inspiring. Working with you, I have
learned so much on how to approach and solve research problems. You have
helped me look at research as a ludic and collaborative endeavour – a
perspective that I hope will last throughout my career. I cannot thank you
enough for your generosity with your time, energy and ideas.
I would also like to thank my masters supervisors, Prakash Panangaden and
Doina Precup, for their help and support which has lasted to this day and has
greatly contributed to my academic path.
To my amazing friends, I am forever grateful for your support, care and
kindness. Friends from Montréal, Pearson UWC, Oxford, and beyond: you know who
you are, and I love and cherish every one of you.
I would like to thank Gabrielle, Rick and Joanie from the Institut des
Commotions Cérébrales, without whom I would most likely never have finished my
degree.
I would also like to acknowledge the financial support provided to me during
my DPhil: the Clarendon Fund (Oxford University Press) for the Clarendon
Scholarship, the Natural Sciences and Engineering Research Council of Canada
(NSERC) for the Postgraduate Scholarship, and the European Research Council
(ERC) for funding under the European Union’s Horizon 2020 research and
innovation programme (FUN2MODEL, grant agreement No. 834115).
Finally, I would like to thank my family, especially my parents, Caroline and
Richard, who have supported me in ways that words will never do justice to. A
very special thank you to my grandmother, Colette, whose love and wisdom are
always with me. Raymonde and Pierre, I so deeply wish I could share this
moment with you.
###### Contents
1. 1 Introduction
1. 1 Main Contributions
2. 2 Thesis Structure
3. 3 Statement of Contribution
2. 2 Literature Review
1. 4 The Learning Theory Landscape
1. 4.1 Classification
2. 4.2 Learning with Queries
2. 5 Adversarial Machine Learning
1. 5.1 Evasion Attacks
3. 3 Background
1. 6 Learning Theory: Classification
1. 6.1 The PAC Framework
2. 6.2 Complexity Measures
3. 6.3 Some Concept Classes and PAC Learning Algorithms
4. 6.4 Online Learning: The Mistake-Bound Model
5. 6.5 Learning with Membership and Equivalence Queries
2. 7 Probability Theory
1. 7.1 Log-Lipschitz Distributions
2. 7.2 Concentration Bounds and Martingales
3. 8 Fourier Analysis
4. 4 Robustness & Monotone Conjunctions
1. 9 Defining Robust Learnability
1. 9.1 Two Notions of Robustness
2. 9.2 A Separation between PAC and Robust Learning
2. 10 The Distribution-Free Assumption
3. 11 An Adversarial Sample Complexity Lower Bound
4. 12 Logarithmically-Bounded Adversary
1. 12.1 Log-Lipschitz Distributions
2. 12.2 A Robustness Guarantee
5. 13 Summary
5. 5 Robustness Thresholds: Random Examples
1. 14 Exact Learning
1. 14.1 Parity Functions
2. 14.2 Majority Functions
2. 15 Decision Lists
1. 15.1 1-Decision Lists
2. 15.2 Generalizing from 1-DL to $2$-DL and Monotone $k$-DL
3. 15.3 Non-Monotone Decision Lists
3. 16 Decision Trees
4. 17 Summary of Results and Open Problems
6. 6 Robust Learning with Local Queries
1. 18 Two Local Query Models
2. 19 Robust Learning with Local Membership Queries
3. 20 Robust Learning with Local Equivalence Queries
1. 20.1 Impossibility of Distribution-Free Robust Learning for $\lambda<\rho$
2. 20.2 Sample Complexity Upper Bounds
3. 20.3 General Query Complexity Upper Bounds
4. 20.4 Improved Query Complexity Bounds for Conjunctions
5. 20.5 Bounds for Linear Classifiers
4. 21 Precision-Bounded Adversaries
5. 22 Lower Bounds on Robust Learning with $\mathsf{LEQ}$
1. 22.1 General Query Complexity Lower Bounds
2. 22.2 Bounds on the Restricted VC and Littlestone Dimensions
6. 23 Further Comparing the Local Query Models
1. 23.1 Local Membership and Equivalence Queries
2. 23.2 A Two-Way Separation between $\mathsf{LEQ}$ and $\mathsf{EQ}$
7. 24 Summary and Open Problems
1. 24.1 Final Remarks on Local Query Oracles
2. 24.2 Future Work
7. 7 Conclusion
1. 25 Future Work
8. 8 Proofs from Chapter 5
1. 8.A Proof of Lemma 5.12
2. 8.B Proof of Corollary 5.24
9. 9 Proofs from Chapter 6
1. 9.A Proof of Lemma 6.6
2. 9.B Proof of Lemma 6.8
3. 9.C Bounds on the Restricted VC dimension
10. 10 Discussions from Chapter 6
1. 10.A A Closer Look at $\mathsf{VC}(\mathcal{L}_{\rho}(\mathcal{C},\mathcal{H}))$
2. 10.B A Lower Bound Based on $\mathsf{VC}(\mathcal{L}_{\rho}(\mathcal{C},\mathcal{H}))$
###### List of Figures
1. 1 A school bus is classified as an ostrich after a small perturbation is applied to the original image (Szegedy et al.,, 2013).
2. 2 A visual representation of sample-efficient PAC learning. $S_{c}$ means that the sample $S$ has been labelled with the ground truth $c$.
3. 3 A set $X$ of three points in $\mathbb{R}^{2}$ that is shattered by linear classifiers. Subfigures (a)-(d) represent different dichotomies on $X$; note that (b) and (c) are not the only labellings with one and two positively labelled points, respectively, but the other cases are symmetric.
4. (a)
5. (b)
6. (c)
7. (d)
8. 4 Any set of four points cannot be shattered by linear classifiers. Indeed, we distinguish two cases: either (a) one point is strictly in the convex hull of the three other points, and is the only point of its label (or all points are on the same line, which gives a similar argument) or (b) all points are on the boundary of the convex hull, in which case labelling opposite points with the same label gives an unachievable labelling. This argument is a special case for $\mathbb{R}^{2}$ which can be generalized to $\mathbb{R}^{n}$ using Radon’s theorem.
9. (a)
10. (b)
11. 5 The natural point $x$ has robust loss of 1 with respect to both notions of robustness: $z_{1}$ is a counterexample for exact-in-the-ball robustness (as $c(z_{1})\neq h(z_{1})$), and $z_{2}$ for constant-in-the-ball robustness (as $c(x)\neq h(z_{2})$).
12. 6 In all the examples above, the circles represent the support of the distribution, and the shaded region, its $\rho$-expansion (i.e., the points at a distance at most $\rho$ from points in the support of the distribution). (a) The support of the distribution is such that $\mathsf{R}^{C}_{\rho}(h,c)=0$ can only be achieved if $c$ is constant. (b) The $\rho$-expansion of the support of the distribution and target $c$ admit hypotheses $h$ such that $\mathsf{R}^{C}_{\rho}(h,c)=0$ (i.e., any $h$ that does not cross the shaded regions). (c) An example where $\mathsf{R}^{C}_{\rho}$ and $\mathsf{R}^{E}_{\rho}$ differ. The red concept, which crosses the shaded regions, is the target; the blue one is the hypothesis. The diamonds represent perturbed inputs which cause $\mathsf{R}^{E}_{\rho}(c,h)>0$, while $\mathsf{R}^{C}_{\rho}(h,c)=0$.
13. (a)
14. (b)
15. (c)
16. 7 Images from the CIFAR-10 (above) and MNIST (below) datasets, respectively from (Krizhevsky and Hinton,, 2009) and (LeCun,, 1998). While the margin assumption generally holds for CIFAR (e.g., the “boat” and “dog” classes are well-separated), this is not necessarily the case for MNIST (the three above could easily be transformed into an eight, and the left-hand side picture could be a one or a seven).
17. 8 A unifying result. When $\rho=\log(n)$, $\mathsf{SAT}_{\rho}(\varphi)$, the $\rho$-expansion of the error region, is not too large compared to the set $\mathsf{SAT}(\varphi)$.
18. 9 The dotted line is the hypothesis $h$, and the solid line, the target $c$. The adversary has precision $\tau$. The shaded regions represent the set $B_{\tau}(z_{i})$. The counterexample $z_{1}$ is valid as $c$ and $h$ disagree on all of $B_{\tau}(z_{1})$ and both functions are constant in this region, but $z_{2}$ is not as $c$ and $h$ agree on part of $B_{\tau}(z_{2})$.
19. 10 A visual representation of the proof of Theorem 6.28. The dotted lines on either side of the target $c$ represent a margin of $\tau/2$. Any hypothesis within the dotted lines in the (shaded) perturbation region ensures that an adversary of bounded precision $\tau$ cannot return any counterexamples. Finally, counterexamples must be labelled according to the target $c$, and both $h$ and $c$ are not constant on $B_{\rho}(x)$.
###### List of Tables
1. 1 The pros and cons of the two robust risk functions. The last line refers to the behaviour of hypotheses minimizing the robust risk as the perturbation region increases. At the extreme case, when the perturbation region is the whole space, the robust risk minimizer for the constant-in-the-ball risk is a constant function, while it is the target for the exact-in-the-ball risk (as we require exact learning).
2. 2 The robustness thresholds of concept classes from Chapters 4 and 5, and open problems.
3. 3 Comparing the VC dimension and the $\rho$-restricted VC dimension for given concept classes. The $\widetilde{\Theta}$ notation hides the logarithmic factors. Unless otherwise stated, we assume $\rho\geq 1$.
## Chapter 1 Introduction
In the standard theoretical analysis of machine learning, the learning process
uses and is evaluated on clean, unperturbed examples. Moreover, many machine
learning tasks are evaluated according to predictive accuracy alone, e.g.,
maximizing the accuracy of a classifier with respect to the ground truth which
labels the data. Though there remain existing knowledge gaps in the literature
(e.g., explaining the success of deep neural networks), machine learning
theory has generally been successful at designing algorithms and deriving
guarantees to explain generalization in this framework, even in the presence
of noise.
It is natural to ask whether similar results can be derived when the learning
objectives go beyond standard accuracy. This could be when the learning
process allows for the presence of a malicious adversary–which is more
powerful than simply adding random noise to the data– and thus requires
_robustness_. The study of robustness in machine learning falls under the more
general umbrella of _trustworthiness_ of machine learning models, where other
considerations such as privacy, interpretability or fairness come into play,
see, e.g., (Dwork,, 2008; Doshi-Velez and Kim,, 2017; Kleinberg et al.,,
2017). The trustworthiness of machine learning models is of utmost importance,
especially considering the speed at which new technology is currently
deployed. Crucially, learning theory can provide us with valuable tools to
explain, evaluate and guarantee the behaviour of safety-critical machine-
learning applications.
The focus of this thesis is on the robustness of machine learning algorithms
to _evasion attacks_ , which happen at test time after a model is trained
(without the presence of an adversary). This is in contrast to _poisoning
attacks_ , which happen at training time with the goal of reducing the test-
time accuracy of a machine learning algorithm. The distinction between these
two settings was proposed by Biggio et al., (2013), who independently observed
the phenomenon of adversarial examples presented by Szegedy et al., (2013),
who coined the latter term.
One of the main challenges in the theory of adversarial machine learning is to
analyse the intrinsic difficulty of learning in the presence of an adversary
that can modify the data. The present work studies various assumptions in a
learning problem, such as properties of the distribution underlying the data,
how the learner obtains data, limitations of the adversary, etc., and
determines whether robust learning is feasible with a reasonable amount of
data. Here, reasonable means that the _sample complexity_ of a robust learning
algorithm, i.e., the amount of data needed to enable guarantees, is
_polynomial_ in the input space dimension and the learning parameters (e.g.,
an algorithm’s _confidence_ and the desired _robust accuracy_ of a hypothesis
output by the learning algorithm).
### 1 Main Contributions
This thesis focuses on the existence of adversarial examples in classification
tasks. An adversarial example is obtained from a natural example at test time
by adding a perturbation, in the malicious goal of causing a
misclassification. We work under the _exact-in-the-ball_ notion of
robustness,111Also known as _error region_ risk in Diochnos et al., (2018).
which relies on the existence of a ground truth function (i.e., there exists a
concept that labels the data correctly). A misclassification occurs when the
hypothesis returned by the learning algorithm and the ground truth _disagree_
in the perturbation region. This is in contrast to the _constant-in-the-ball_
notion of robustness222Also known as _corrupted input_ robustness from the
work of Feige et al., (2015). which requires that the unperturbed point be
labelled correctly, and that the hypothesis remain _constant_ in the
perturbation region. Guarantees derived for the constant-in-the-ball notion of
robustness imply that the hypothesis returned has a certain stability (perhaps
at the cost of accuracy in certain cases, as demonstrated in Tsipras et al.,
(2019)), as an optimal algorithm would return a hypothesis that limits the
probability of a label change in the perturbation region. On the other hand,
guarantees derived for the exact-in-the-ball notion of robustness usually give
stronger accuracy, as we want to be _correct_ with respect to the ground truth
in the perturbation region. Deciding which notion of robustness to use depends
on the learning problem at hand, and what kind of guarantees one wishes to
ensure. We gave in (Gourdeau et al.,, 2019, 2021) a thorough comparison
between these two notions of robustness, and remarked that the exact-in-the-
ball notion of robustness is much less studied than the constant-in-the-ball
one.
Our motivation in this thesis is to study the intrinsic robustness of learning
algorithms from a learning theory perspective in the probably approximately
correct (PAC) learning model of Valiant, (1984). We investigate how different
learning settings enable robust learning guarantees, or, to the contrary, give
rise to hardness results. In this sense, our main aim is to delineate the
frontier of robust learnability in various learning models. We conceptually
divide our contributions based on the learning models we have studied.
##### Random examples.
In this model, as in the PAC framework, the learner has access to a random-
example oracle which samples a point from an underlying distribution, and
returns the point along with its label. We exhibit an impossibility result
(Gourdeau et al.,, 2019), stating that the distribution-free guarantees for
(standard) PAC learning cannot be achieved for robust learning under the
exact-in-the-ball definition of robustness, highlighting a key obstacle in
adversarial machine learning compared to its standard counterpart. Here,
distribution-free means that the learning guarantees hold for any distribution
that generates the data, provided that the training and testing data are both
drawn independently from the same distribution.
The above impossibility result is obtained by choosing a badly-behaved, and
quite unnatural distribution on the data. But we show that, even when looking
at natural distributions and simple concept classes, robust learning can have
high sample complexity. Indeed, we prove that there is no efficient robust
learning algorithm that learns monotone conjunctions under the uniform
distribution if the adversary can perturb $\rho=\omega(\log n)$ bits of a test
point in $\\{0,1\\}^{n}$; the maximum number $\rho$ of bits the adversary is
allowed to perturb at test time is called the _perturbation budget_. This is
particularly striking as the class of monotone conjunctions is one the
simplest non-trivial concept classes on the boolean hypercube. We extend this
result to establish a general sample complexity lower bound of
$\Omega(2^{\rho})$ (Gourdeau et al., 2022a, ), highlighting an _exponential_
dependence on the adversary’s budget $\rho$ in the sample complexity of robust
learning. Since linear classifiers and decision lists subsume this class of
functions, the lower bound holds for them as well. To complement these
results, we show that, under distributional assumptions and against a
_logarithmically-bounded_ adversary (i.e., with budget $\rho=O(\log n)$),
efficient robust learning is possible for various concept classes. We require
that the underlying distribution be _log-Lipschitz_ ; this notion encapsulates
the idea that nearby instances should have similar probability masses and
includes as particular instances product distributions with bounded means. We
show the above-mentioned result for conjunctions (Gourdeau et al.,, 2019),
monotone decision lists (Gourdeau et al.,, 2021), and non-monotone decision
lists (Gourdeau et al., 2022a, ). We define the term _robustness threshold_ to
mean a function $f(n)$ of the input dimension $n$ for which it is possible to
efficiently robustly learn against an adversary with budget $f(n)$, but
impossible if the adversary’s budget is $\omega(f(n))$ (with respect to a
given distribution family). The robustness threshold of these concept classes
is thus $\log(n)$ under log-Lipschitz distributions.
In general, the above-mentioned results rely on a proof of independent
interest: an upper bound on the $\log(n)$-expansion of subsets of the
hypercube defined by $k$-CNF formulas. This result relies on concentration
bounds for martingales, as well as properties of the resolution proof system.
In all the cases above, as well as for decision trees (Gourdeau et al.,,
2021), the error region between a hypothesis and a target333That is, for
target $c$ and hypothesis $h$ on input space $\mathcal{X}$, the set of points
$x\in\mathcal{X}$ such that $c(x)\neq h(x)$. can be expressed as a union of
$k$-CNF formulas. By controlling the standard risk, we can bound the robust
risk and, as a result, use PAC learning algorithms as black boxes for robust
learning.
##### Local membership queries.
In this model, introduced by Awasthi et al., (2013), the learner has access to
the random-example oracle and can query the label of points that are near the
randomly-drawn training sample. We show that at least $\Omega(2^{\rho})$ local
membership queries are needed for robustly learning conjunctions under the
uniform distribution against an adversary that can perturb $\rho$ bits of the
input (Gourdeau et al., 2022b, ). We thus have the same exponential dependence
in the adversary’s budget as with random examples only, implying that adding
local membership queries cannot, in general, improve the robustness threshold
of this concept class (and any superclass)., e.g. linear classifiers and
decision lists.
##### Local equivalence queries.
Faced with the lower bound for robust learning with a local membership query
oracle, we introduce a learning model where the learner is allowed to query
whether the hypothesis is _correct_ in a specific region of the space and get
a counterexample if not, which we call local equivalence queries in (Gourdeau
et al., 2022b, ), following the work of Angluin, (1987).
We first establish that, when the query budget is strictly smaller than the
perturbation budget (hence the adversary can access regions of the instance
space that the learner cannot), distribution-free robust learning with random
examples and local equivalence queries is in general impossible for monotone
conjunctions and any superclass thereof. However, when the query and
perturbation budgets coincide, a query to the local equivalence query oracle
is equivalent to querying the robust loss and getting a counterexample if it
exists. As a result, the local equivalence query oracle becomes the exact-in-
the-ball analogue of the Perfect Attack Oracle of Montasser et al., (2021). In
this case, efficient distribution-free robust learning becomes possible for a
wide variety of concept classes. Indeed, we show random-example and local-
equivalence-query upper bounds, which we refer to as sample and query
complexity, respectively. We demonstrate that the query complexity depends on
mistake bounds from online learning, and the sample complexity on the VC
dimension of the robust loss of a concept class, a notion of complexity that
we have adapted from Cullina et al., (2018) to the exact-in-the-ball notion of
robustness. We also show that the local equivalence query bound can be
improved in the special case of conjunctions. We moreover establish that the
VC dimension of the robust loss between linear classifiers on $\mathbb{R}^{n}$
is $O(n^{3})$.
Since the query complexity of linear classifiers is in general unbounded, we
study the setting in which we restrict the adversary’s _precision_ (e.g., the
number of bits needed to express an adversarial example). We use and adapt
tools and techniques from Ben-David et al., (2009), which pertain to the study
of margin-based classifiers in the context of online learning, for our
purposes and exhibit finite query complexity bounds. We then exhibit expected
local equivalence query lower bounds that are linear in the _restricted_
Littlestone dimension of a concept class (we require that a set of potential
counterexamples be in a specific region of the instance space), and show that,
for a wide variety of concept classes, they coincide asymptotically with the
local equivalence query upper bounds derived in Gourdeau et al., 2022b .
Finally, we offer a more nuanced discussion of the local membership and
equivalence query oracles. In particular, we show that the local equivalence
query and its global counterpart, the equivalence query, are in general
incomparable.
### 2 Thesis Structure
#### Chapter 2
This chapter consists of the literature review. We first review foundational
work on classification in the learning theory literature. We then turn our
attention to the more recent related work on adversarial robustness in machine
learning, particularly in the context of evasion attacks. We mainly focus on
work that is foundational in nature, as it is the lens with which we study
adversarial robustness.
#### Chapter 3
We review necessary technical background to the understanding of the technical
contributions of this thesis, which largely focuses on classification in the
following models: the PAC framework of Valiant, (1984), the exact learning
framework of Angluin, (1987), and the online learning setting. We also review
some probability theory and Fourier analysis.
#### Chapter 4
We motivate the study of adversarial robustness for classification tasks under
the exact-in-the-ball notion of robustness. We rigorously discuss the
different notions of robust risk and their significance, particularly the
impossibility of obtaining distribution-free guarantees in our setting. We
initiate our study of efficient robust learnability (from a sample-complexity
point of view) with monotone conjunctions. We show a sample complexity lower
bound that is exponential in the adversary’s budget under the uniform
distribution, ruling out the existence of efficient robust learning algorithms
against adversaries with a budget super-logarithmic in the input dimension in
this setting. We show, however, that it is possible to robustly learn monotone
conjunctions under log-Lipschitz distributions against a logarithmically-
bounded adversary.
The material in this chapter is based on the following papers:
* •
Pascale Gourdeau, Varun Kanade, Marta Kwiatkowska, and James Worrell, “On the
hardness of robust classification,” in _33rd Conference on Neural Information
Processing Systems (NeurIPS)_ , 2019.
* •
Pascale Gourdeau, Varun Kanade, Marta Kwiatkowska, and James Worrell, “Sample
complexity bounds for robustly learning decision lists against evasion
attacks,” in _International Joint Conference on Artificial Intelligence
(IJCAI)_ , 2022.
#### Chapter 5
In this chapter, we study the _robustness thresholds_ of various concept
classes under distributional assumptions. We show the exact learning of
parities under log-Lipschitz distributions and of majority functions under the
uniform distribution, giving a robustness threshold of $n$ for these classes.
We then show a robustness threshold of $\log(n)$ for the class of $k$-decision
lists, which is parametrized by the size $k$ of a conjunction at each node in
the list. Since our aim is to bound the sample complexity of robustly
learning, we study various restrictions of decision lists: 1-decision lists,
2-decision lists, monotone $k$-decision lists and finally (non-monotone)
$k$-decision lists. The proofs not only rely on different technical tools, but
they more importantly yield much better sample complexity bounds for the
simpler subclasses. We finish by relating the standard and robust errors of
decision trees under log-Lipschitz distributions.
This chapter is based on the following two papers, the first one being the
journal version of the NeurIPS 2019 paper presented in the previous chapter:
* •
Pascale Gourdeau, Varun Kanade, Marta Kwiatkowska, and James Worrell, “On the
hardness of robust classification,” in _Journal of Machine Learning Research
(JMLR)_ , 2021.
* •
Pascale Gourdeau, Varun Kanade, Marta Kwiatkowska, and James Worrell, “Sample
complexity bounds for robustly learning decision lists against evasion
attacks,” in _International Joint Conference on Artificial Intelligence
(IJCAI)_ , 2022.
#### Chapter 6
We consider learning models in which the learner has access to local queries
in addition to random examples. We first show that local membership queries do
not increase the robustness threshold of conjunctions under the uniform
distribution. We then study local equivalence queries, and show that
distribution-free robust learning is impossible for a wide variety of concept
classes if the query budget is strictly smaller than the adversarial budget.
We demonstrate, however, that when the two coincide, distribution-free robust
learning becomes possible. We exhibit general sample and query complexity
upper bounds as well as tighter bounds in the special case of conjunctions. We
also give explicit bounds for linear classifiers on the boolean hypercube. We
then study linear classifiers in the continuous case and establish a general
sample complexity upper bound, as well as a query complexity upper bound when
we limit the adversary’s precision. We complement the upper bounds by showing
general lower bounds on the expected number of queries to the local
equivalence query oracle and instantiate them for specific concept classes. We
finish by comparing the local membership and equivalence query oracles, as
well as how they compare with the membership and equivalence query oracles.
Sections 18, 19 and 20 are based on the following publication:
* •
Pascale Gourdeau, Varun Kanade, Marta Kwiatkowska, and James Worrell, “When
are local queries useful for robust learning?” in _36th Conference on Neural
Information Processing Systems (NeurIPS)_ , 2022.
Sections 21, 22 and 23 are based on work that we are currently preparing for
submission.
#### Chapter 7
We conclude by summarizing our contributions and drawing a picture of robust
learnability in the learning models we have studied. Finally, we outline
various avenues for future work.
### 3 Statement of Contribution
The publications mentioned in the previous section have largely been my own
work, with direction from my supervisors James Worrell, Varun Kanade and Marta
Kwiatkowska. While NeurIPS 2019/JMLR 2021 papers addressed research questions
posed by my supervisors, I lead the research – including the technical aspect
by deriving the proofs, and wrote most of the paper. For the IJCAI 2022 paper,
I continued to lead in the technical development and writing up of the
manuscript. In addition to this, I played a major role in formulating the
research questions and positioning the work in a wider context. For the
NeurIPS 2022 paper and subsequent ongoing work, I did most of the work on my
own – from finding and defining the research problem and learning model,
providing insights on the problem at hand, deriving the proofs and writing the
whole paper. I was of course supported by my supervisors: they referred me to
a paper and suggested a way to prove a particular bound, they strengthened the
paper by providing helpful feedback through nuanced discussions, and reviewed
many iterations of the draft.
## Chapter 2 Literature Review
This chapter gives an overview of the literature relevant to this thesis. We
start by reviewing classical learning theory results, focusing on
classification. We finish with a review of adversarial machine learning. While
we mention work pertaining to other views on robustness, our focus is the
study of robustness to evasion attacks, particularly from a foundational
viewpoint.
The results in this chapter are presented at a high level. However, readers
who are not familiar with learning theory may find it beneficial to refer to
Chapter 3, which gives a thorough technical introduction to various frameworks
and complexity measures discussed in this chapter.
### 4 The Learning Theory Landscape
We start with an overview of the established literature in classification in
the probably approximately correct and online learning frameworks, and then
move to learning with access to membership and equivalence queries.
#### 4.1 Classification
The probably approximately correct (PAC) learning model of Valiant, (1984) is
one of the most well-studied classification models in learning theory. In this
framework, the learner has access to the example oracle, which returns a point
$x\sim D$ sampled from an underlying distribution and its label $c(x)$, where
$c$ is the target concept (ground truth). The goal is to output a hypothesis
$h$ from a hypothesis class $\mathcal{H}$ such that $h$ has low error with
high probability.444In the realizable setting, where it is possible to achieve
zero risk, we want the risk to be as close as possible to zero. In the
agnostic setting, we compare the risk of the hypothesis output by an algorithm
to the risk of the optimal function from the hypothesis class. Remarkably,
there exists a complexity measure, namely the VC dimension of Vapnik and
Chervonenkis, (1971), that characterizes the learnability of a hypothesis
class. Indeed, it is possible to get both upper and lower bounds for the
number of samples needed for learning (i.e., the sample complexity) that are
_linear_ in the VC dimension. The upper bound is due to Vapnik, (1982); Blumer
et al., (1989), and the lower bounds to Blumer et al., (1989); Ehrenfeucht et
al., (1989). These bounds are tight up to a
$\log\left(\frac{1}{\epsilon}\right)$ factor, where $\epsilon$ is the
parameter controlling the accuracy of the hypothesis output by the learning
algorithm.
The one-inclusion graph of Haussler et al., (1994), which also enjoys an upper
bound that is linear in the VC dimension, was conjectured to be optimal (in
the sense that the upper and lower bounds on sample complexity are tight) by
Warmuth, (2004) until the recent work of Aden-Ali et al., (2023) showing that
this is not the case. However, the breakthrough work of Hanneke, (2016) showed
it is in general possible to get rid of the
$\log\left(\frac{1}{\epsilon}\right)$ factor with a majority-vote classifier,
following important advances made by Simon, (2015).
Another popular learning setting is that of online learning, introduced in the
seminal work of Littlestone, (1988) and in which a learning algorithm competes
against an adversary. At each iteration, the learner is presented with an
instance to predict, and afterwards the adversary reveals the true label of
the instance. The goal is to make as few mistakes as possible. Littlestone,
(1988) studied the _realizable setting_ , where there always is a function
that makes zero mistakes on the learning sequence, and showed that a notion of
complexity (the Littlestone dimension) characterizes online learnability in
this framework. The algorithm achieving this is called the standard optimal
algorithm (SOA), which was later adapted by Ben-David et al., (2009) to the
_agnostic setting_ , where there need not exist a function that makes zero
mistakes; the algorithm’s performance is instead compared with the best
hypothesis _a posteriori_. There is a vast literature on online learning, and
we refer the reader to the book of Cesa-Bianchi and Lugosi, (2006) for a
technical overview and references therein.
#### 4.2 Learning with Queries
The works mentioned in the previous section studied classification when the
learner has access to random examples. Active learning is another learning
framework in which the learner is given more power, often through the use of
membership and equivalence queries. Membership queries allow the learner to
query the label of any point in the input space $\mathcal{X}$, namely, if the
target concept is $c$, the membership query ($\mathsf{MQ}$) oracle returns
$c(x)$ when queried with $x\in\mathcal{X}$. On the other hand, the equivalence
query ($\mathsf{EQ}$) oracle takes as input a hypothesis $h$ and returns
whether $h=c$, and provides a counterexample $z$ such that $h(z)\neq c(z)$
otherwise. The goal in the $\mathsf{MQ}+\mathsf{EQ}$ model is usually to learn
the target $c$ exactly, which is in contrast to the PAC setting which requires
to learn with high confidence a hypothesis with low error.
The seminal work of Angluin, (1987) showed that deterministic finite automata
(DFA) are exactly learnable with a polynomial number of queries to
$\mathsf{MQ}$ and $\mathsf{EQ}$ in the size of the DFA. Follow-up work
generalized these results. E.g., Bshouty, (1993) showed that poly-size
decision trees are efficiently learnable in this setting as well; Angluin,
(1988) later investigated other types of queries and also showed that $k$-CNFs
and $k$-DNFs are exactly learnable with access to membership queries; Jackson,
(1997) showed that, in the $\mathsf{PAC}+\mathsf{MQ}$ setting, the class of
$\mathsf{DNF}$ formulas is learnable under the uniform distribution. But even
these powerful learning models have limitations: learning DFAs only with
$\mathsf{EQ}$ is hard (Angluin,, 1990) and, under cryptographic assumptions,
DFAs are also hard to learn solely with the $\mathsf{MQ}$ oracle (Angluin and
Kharitonov,, 1995).
On a more applied note, the $\mathsf{MQ}+\mathsf{EQ}$ model has recently been
used for recurrent and binarized neural networks (Weiss et al.,, 2018, 2019;
Okudono et al.,, 2020; Shih et al.,, 2019), and interpretability (Camacho and
McIlraith,, 2019). It is also worth noting that the $\mathsf{MQ}$ learning
model has been criticized by the applied machine learning community, as labels
can be queried in the whole input space, irrespective of the distribution that
generates the data. In particular, Baum and Lang, (1992) observed that query
points generated by a learning algorithm on the handwritten characters often
appeared meaningless to human labellers. Awasthi et al., (2013) thus offered
an alternative learning model to Valiant’s original model, the PAC and local
membership query ($\mathsf{EX}+\mathsf{LMQ}$) model, where the learning
algorithm is only allowed to query the label of points that are close to
examples from the training sample. Bary-Weisberg et al., (2020) later showed
that many concept classes, including DFAs, remain hard to learn in the
$\mathsf{EX}+\mathsf{LMQ}$ model.
### 5 Adversarial Machine Learning
There has been considerable interest in adversarial machine learning since the
seminal work of Szegedy et al., (2013), who coined the term _adversarial
example_ to denote the result of applying a carefully chosen perturbation that
causes a classification error to a previously correctly classified datum. This
work was largely experimental in nature and presented a striking instability
of deep neural networks, where for example a correctly-classified image of a
school bus was labelled as an ostrich after a perturbation (imperceptible to
the human eye) was applied, as in Figure 1. Biggio et al., (2013)
independently observed this phenomenon with experiments on the MNIST (LeCun,,
1998) dataset. However, as pointed out by Biggio and Roli, (2018), adversarial
machine learning has been considered much earlier in the context of spam
filtering (Dalvi et al., (2004); Lowd and Meek, 2005a ; Lowd and Meek, 2005b ;
Barreno et al., (2006)). Their survey also distinguished two settings:
_evasion attacks_ , where an adversary modifies data at test time, and
_poisoning attacks_ , where the adversary modifies the training data. For an
in-depth review and definitions of different types of attacks, the reader may
refer to (Biggio and Roli,, 2018; Dreossi et al.,, 2019). For an introduction
to adversarial defences in practice, see, e.g., (Goodfellow et al.,, 2015;
Zhang et al.,, 2019).
Figure 1: A school bus is classified as an ostrich after a small perturbation
is applied to the original image (Szegedy et al.,, 2013).
As our work pertains to the robustness of machine learning algorithms to
evasion attacks in classification tasks from a learning theory perspective,
our review of related work will mainly concern this topic (Section 5.1).
Before discussing this body of work, we will briefly mention other views on
robustness.
Many works have studied the robustness of learning algorithms to poisoning
attacks, in which an adversary can modify the training data in order to
increase the (standard) error at test time, one of the earliest being that of
Kearns and Li, (1988). Various types of poisoning attacks have been put
forward since then, especially as the study of robustness has garnered
interest in recent years. Clean-label attacks, proposed by Shafahi et al.,
(2018), are a distinct form of poisoning attacks where the poisoned examples
are labelled correctly, i.e., by the target function, and not adversarially.
For a learning-theoretic approach and results on this problem, see
(Mahloujifar and Mahmoody,, 2017, 2019; Mahloujifar et al.,, 2018, 2019;
Etesami et al.,, 2020; Blum et al.,, 2021) (non-exhaustive). In case there is
no restriction on the label of poisoned data, see, e.g., the works of (Barreno
et al.,, 2006; Biggio et al.,, 2012; Papernot et al.,, 2016; Steinhardt et
al.,, 2017) (non-exhaustive). Finally, for work on defences against poisoning
attacks, we refer the reader to (Goldblum et al.,, 2022).
Another view on robustness is out-of-distribution detection, where the goal is
to identify outliers at test time. We refer the reader to the textbook
(Quinonero-Candela et al.,, 2008) for an introduction on dataset shifts, and
to (Fang et al.,, 2022) for a study on out-of-distribution detection from a
PAC-learning perspective, as well as references therein for the empirical work
on the matter. A more general view on distributional discrepancies at test-
time is that of distribution shift. See (Wiles et al.,, 2022) for a taxonomy
on various distribution shifts and a review of important work in the area
(mostly from an empirical perspective).
#### 5.1 Evasion Attacks
We now turn our attention to the focus of this thesis: robustness to evasion
attacks. For ease of reading, we have thematically split the related work in
this section.
##### Defining Robustness.
The majority of the guarantees and impossibility results for evasion attacks
are based on the existence of adversarial examples. However, what is
considered to be an adversarial example has been defined in different, and in
some respects contradictory, ways in the literature. What we refer to as the
_exact-in-the-ball_ notion of robustness in this work (also known as _error
region_ risk in (Diochnos et al.,, 2018)) requires that the hypothesis and the
ground truth agree in the perturbation region around each test point; the
ground truth must thus be specified on all input points in the perturbation
region. On the other hand, what we refer to as the constant-in-the-ball notion
of robustness (which is also known as _corrupted input_ robustness from the
work of Feige et al., (2015)) requires that the unperturbed point be correctly
classified and that the points in the perturbation region share its label,
meaning that we only need access to the test point labels; the works Diochnos
et al., (2018); Dreossi et al., (2019); Pydi and Jog, (2021) offer thorough
discussions on the subject and also compare robustness definitions. Moreover,
Chowdhury and Urner, (2022) have studied settings where a model’s change of
label is justified by looking at robust-Bayes classifiers and their standard
counterparts.
We note that Suggala et al., (2019) proposed an alternative definition of
robustness, where a perturbation is deemed adversarial if it causes a label
change in the hypothesis _while the target classifier’s label remains
constant_. The existence of a ground truth is thus explicitly assumed (which
is not in general necessary for constant-in-the-ball robustness).
Rather than studying the existence of a misclassification in the perturbation
region, Pang et al., (2022) define robustness using the Kullback-Leibler (KL)
divergence. The robust loss at a given unperturbed point $x$ is the maximal KL
divergence over perturbations $z$ between the underlying labelling function
($\underset{{}}{\Pr}\left(y\;|\;z\right)$) of $z$ and the hypothesis’ label
for $z$ (which could also be non-deterministic). The authors proposed this
definition of robustness in an effort to avoid the trade-off between accuracy
and robustness observed in prior work, e.g., (Tsipras et al.,, 2019).
In the remainder of this section, whenever the robust risk is not explicitly
mentioned, the results will hold for the constant-in-the-ball notion of
robustness, as it is the most widely used in the literature.
##### Existence of Adversarial Examples.
There is a considerable body of work that studies the inevitability of
adversarial examples, e.g., (Fawzi et al.,, 2016; Fawzi et al., 2018a, ; Fawzi
et al., 2018b, ; Gilmer et al.,, 2018; Shafahi et al.,, 2019; Tsipras et al.,,
2019). These papers characterize robustness in the sense that a classifier’s
output on a point should not change if a perturbation of a certain magnitude
is applied to it. These works also study geometrical characteristics of
classifiers and statistical characteristics of classification data that lead
to adversarial vulnerability. It has been shown that, in many instances, the
vulnerability of learning models to adversarial examples is inevitable due to
the nature of the learning problem. Notably, Bhagoji et al., (2019) study
robustness to evasion attacks from an optimal transport perspective, obtaining
lower bounds on the robust error. Moreover, many works exhibit a trade-off
between standard accuracy and robustness in this setting, e.g., (Tsipras et
al.,, 2019; Dobriban et al.,, 2020).
As for the exact-in-the-ball definition of robustness, Diochnos et al., (2018)
consider the robustness of monotone conjunctions under the uniform
distribution. Their results concern the ability of an adversary to magnify the
missclassification error of _any_ hypothesis with respect to _any_ target
function by perturbing the input.555We will draw an explicit comparison with
the work of Diochnos et al., (2018) in Section 11. Mahloujifar et al., (2019)
generalized the above-mentioned result to Normal Lévy families and a class of
well-behaved classification problems (i.e., ones where the error regions are
measurable and average distances exist).
##### Computational Complexity of Robust Learning.
The computational complexity of robust learning is an active research area.
Bubeck et al., (2018) and Degwekar et al., (2019) have shown that there are
concept classes that are hard to robustly learn under cryptographic
assumptions, even when robust learning is information-theoretically feasible.
(Bubeck et al.,, 2019) established super-polynomial lower bounds for robust
learning in the statistical query framework. Diakonikolas et al., (2019) study
the more specific problem of (standard) proper learning of halfspaces with
noise and large $\ell_{2}$ margins in the agnostic PAC setting, focussing on
the computational complexity of this learning problem. They remark that these
guarantees can apply to robust learning. In follow-up work (Diakonikolas et
al.,, 2020), they explicitly study robustness to $\ell_{2}$ perturbations and
generalize their previous results. In particular, they obtain computationally-
efficient algorithms using an online learning reduction, and building on a
hardness result in (Diakonikolas et al.,, 2019), and provide tight running
time lower bounds. Finally, Awasthi et al., (2019) draw connections between
robustness to evasion attacks and polynomial optimization problems, obtaining
a computational hardness result. On the other hand, they exhibit
computationally efficient robust learning algorithms for linear and quadratic
threshold functions in the realizable case.
##### Sample Complexity of Robust Learning.
Despite being a relatively recent research area, there already exists a vast
literature on the sample complexity of robust learning to evasion attacks. One
of the earlier works is that of Cullina et al., (2018), who define the notion
of adversarial VC dimension to derive sample complexity upper bounds for
robust empirical risk minimization (ERM) algorithms, with respect to the
constant-in-the-ball robust risk. They also study the special case of
halfspaces under $\ell_{p}$ perturbations and show the adversarial VC
dimension is in general incomparable with its standard counterpart. Shortly
after, Attias et al., (2019) adopted a game-theoretic framework to study
robust learnability for classification and regression in a setting where the
adversary is limited to a fixed number $k$ of perturbations per input. They
obtain sample complexity bounds that are linear in both $k$ and the VC
dimension of a hypothesis class. The work of Montasser et al., (2019) later
provided a more complete picture of robust learnability. The authors show
sample complexity upper bounds for robust ERM algorithms that are polynomial
in the VC and dual VC dimensions of concept classes, giving general upper
bounds that are exponential in the VC dimension. They also exhibit sample
complexity lower bounds linear in the robust shattering dimension, a notion of
complexity introduced therein. The gap between the upper and lower bounds was
closed in their later work (Montasser et al.,, 2022), where they fully
characterize the sample complexity of robust learning with arbitrary
perturbation functions. The robust learning algorithm achieving the upper
bound is a generalization of the one-inclusion graph algorithm of Haussler et
al., (1994). Their robust variant of the one-inclusion graph is defined for
the constant-in-the-ball _realizable_ setting,666I.e., there exists a
hypothesis that has zero constant-in-the-ball robust loss. but the agnostic-
to-realizable reduction from previous work (Montasser et al.,, 2019) can be
applied. The (random-example) sample complexity characterizing robust
learnability is a notion of _dimension_ defined through the edges on the graph
structure.
The above bounds consider the supervised setting, where the learner has access
to labelled examples. Since the cost of obtaining data is at times largely due
to its labelling,777Think for example of obtaining images vs needing humans to
label them. studying semi-supervised learning, where the learner has access to
both unlabelled as well as labelled examples, is of general interest. Ashtiani
et al., (2020) build on the work of Montasser et al., (2019) (who showed that
proper robust learning, where the learner is required to output a hypothesis
from the same class as the potential target concept, is sometimes impossible)
and delineate when proper robust learning is possible. They moreover draw a
more nuanced picture of proper robust learnability with access to unlabelled
random examples. Attias et al., (2022) also study the sample complexity of
robust learning in the semi-supervised framework. Notably, in the realizable
setting, their labelled sample complexity bounds are linear in a variant of
the VC dimension where, for a shattered set, the perturbation region around a
given point must share the same label.888This complexity measure is always
upper bounded by the VC dimension, and the gap can be arbitrarily large. The
unlabelled sample complexity is linear in the sample complexity of
_supervised_ learning. The authors also extend their results to the agnostic
setting.
While it is worthwhile to study robust learnability for arbitrary perturbation
regions, focussing on specific perturbation functions that are more faithful
to real-world problems is of high interest, especially if this can provide
better guarantees or a clearer picture of robustness in this setting. In this
vein, Shao et al., (2022) study the robustness to evasion attacks under
_transformation invariances_. This terminology comes from group theory: the
transformations applied to instances form a group, and an invariant hypothesis
will give the same label to points in the orbit of every instance in the
support of the distribution generating the data.999E.g., rotating an image of
a cat will still result in an image of a cat, while rotating an image of a six
can result in an image of nine. Transformation invariances are thus problem
specific. As a characterization of robust learnability in these settings,
they propose two combinatorial measures that are variants of the VC dimension
that take into account the orbits of points in the shattered set, and prove
nearly-matching upper and lower bounds.
All the works mentioned above study sample complexity through the VC dimension
of a concept class, or variants adapted to robust learnability. On the other
hand, Khim et al., (2019); Yin et al., (2019); Awasthi et al., (2020) instead
use the _adversarial_ Rademacher complexity to study robust learning. These
works give results for ERM on linear classifiers and neural networks.
As for the exact-in-the-ball definition of robustness, Diochnos et al., (2020)
study sample complexity lower bounds. They show that, for a wide family of
concept classes, any learning algorithm that is robust against all attacks
with budget $\rho=o(n)$ must have a sample complexity that is at least
exponential in the input dimension $n$. They also show a superpolynomial lower
bound in case $\rho=\Theta(\sqrt{n})$. This, along with the previously-
mentioned works of Diochnos et al., (2018); Mahloujifar et al., (2019) are to
our knowledge the only other works apart from ours that consider the sample
complexity of exact-in-the-ball robust learning from a theoretical
perspective.
##### Relaxing Robustness Requirements.
Most adversarial learning guarantees and impossibility results in the
literature have focused on all-powerful adversaries. Recent works have studied
learning problems where the adversary’s power is curtailed. One way to do this
is to consider _computationally-bounded_ adversaries. E.g, Mahloujifar and
Mahmoody, (2019) and Garg et al., (2020) study the robustness of classifiers
to polynomial-time attacks. They show that, for product distributions, an
initial constant error implies the existence of a (black-box) polynomial-time
attack for adversarial examples that are $O(\sqrt{n})$ bits away from the test
instances. However, Garg et al., (2020) show a separation result for a
learning problem where a classifier can be successfully attacked by a
computationally-unbounded adversary, but not by a polynomial-time bounded
adversary subject to standard cryptographic hardness assumptions.
It is also possible to relax the optimality condition when evaluating a
hypothesis. Ashtiani et al., (2023) and Bhattacharjee et al., (2023) both
study _tolerant robust learning_ , where the learner is evaluated relative to
the hypothesis with the best robust risk under a slightly larger perturbation
region. Ashtiani et al., (2023) show that this setting enables better sample
complexity bounds that the standard robust setting for metric spaces
$(\mathcal{X},d)$ in case the perturbation region is a ball with respect to
the metric $d$. Bhattacharjee et al., (2023) build on their work and instead
consider problems with a geometric niceness property called _regularity_ to
get more general perturbation regions. They obtain matching sample complexity
bounds to (Ashtiani et al.,, 2023) as well as propose a variant of robust ERM
as a simpler robust learning algorithm for this problem.
Another relaxation of the robust learning objective is a probabilistic variant
of robust learning. Viallard et al., (2021) derive PAC-Bayesian generalization
bounds (where the output is a posterior distribution over hypotheses after
seeing the data) for the averaged risk on the perturbations, rather than
working in a worst-case scenario. (Robey et al.,, 2022) also consider
probabilistic robustness, where the aim is to output a hypothesis that is
robust to _most_ perturbations.
##### Increasing the Learner’s Power.
To improve robustness guarantees, it is also possible to give the learner
access to more powerful oracles than the random-example one. Montasser et al.,
(2020, 2021) study robust learning with access to a (constant-in-the-ball)
robust loss oracle, which they call the Perfect Attack Oracle (PAO). For a
perturbation type $\mathcal{U}:\mathcal{X}\rightarrow 2^{\mathcal{X}}$,
hypothesis $h$ and labelled point $(x,y)$, the PAO returns the constant-in-
the-ball robust loss of $h$ in the perturbation region $\mathcal{U}(x)$ and a
counterexample $z\in\mathcal{U}(x)$ where $h(z)\neq y$ if it exists. In the
constant-in-the-ball _realizable_ setting, the authors use online learning
results to show sample and query complexity bounds that are linear and
quadratic in the Littlestone dimension of concept classes, respectively
(Montasser et al.,, 2020). Montasser et al., (2021) moreover use the algorithm
from (Montasser et al.,, 2019) to get sample and query complexity upper bounds
that respectively have a linear and exponential dependence on the VC and dual
VC dimensions of the hypothesis class at hand. Finally, they extend their
results to the agnostic setting and derive lower bounds.
## Chapter 3 Background
In this chapter, we introduce the necessary background and notation for the
main contributions of this thesis. We start by reviewing standard learning
theory concepts in Section 6, before moving to probability theory in Section
7. We finish with an overview of Fourier analysis in Section 8.
##### Notation.
Throughout this text, we will use $[n]$ to denote the set
$\left\\{1,\dots,n\right\\}$. The symbol $\Delta$ will represent the symmetric
difference between two sets: $I\Delta J=\left\\{x\;|\;x\in I\setminus J\text{
or }x\in J\setminus I\right\\}$. We will use the asymptotic notation
($o,O,\omega,\Omega,\Theta$), with the convention that the symbol
$\;\widetilde{\;}$ (e.g., $\widetilde{O}$) omits the logarithmic factors.
Given a metric space $(\mathcal{X},d)$ and $\lambda\in\mathbb{R}$, we denote
by $B_{\lambda}(x)$ the ball
$\left\\{z\in\mathcal{X}\;|\;d(x,z)\leq\lambda\right\\}$ of radius $\lambda$
centred at $x$. We will use the symbol $\mathbf{1}[\cdot]$ for the indicator
function. Finally, for a given formula $\varphi$ and instance $x$, we denote
by $x\models\varphi$ the event that $x$ satisfies $\varphi$.
### 6 Learning Theory: Classification
Learning theory offers an elegant abstract framework to analyse the behaviour
of machine learning algorithms, as well as to provide performance and
correctness guarantees or show impossibility results. There exist various
learning settings, depending on assumptions on how the data is obtained and on
the learning objectives. This thesis is primarily concerned with _binary
classification_ , where, given an input space $\mathcal{X}$, the goal is to
output a function $h:\mathcal{X}\rightarrow\\{0,1\\}$ called a _hypothesis_ ,
which upon being given an instance $x\in\mathcal{X}$ outputs a label
$h(x)\in\\{0,1\\}$. The more general task of learning a function
$\mathcal{X}\rightarrow\mathcal{Y}$ is called _multiclass classification_ when
$\mathcal{Y}$ is a discrete finite set, and _regression_ when
$\mathcal{Y}=\mathbb{R}$.
In this section, we give an overview of three learning settings for binary
classification: learning with random examples in the Probably Approximately
Correct (PAC) framework, the mistake-bound model of online learning, and
learning with membership and equivalence queries. For each setting, we discuss
various notions of complexity that control the amount of data needed to learn,
i.e., the _sample complexity_. In all cases, we will be using the terms
learning algorithm, learner and learning process interchangeably to denote a
process of data acquisition and analysis resulting in outputting a hypothesis
$h$ as above. For a more in-depth introduction to the concepts presented in
this section, we refer the reader to Mohri et al., (2012) and Shalev-Shwartz
and Ben-David, (2014), both excellent introductory textbooks on learning
theory.
#### 6.1 The PAC Framework
The Probably Approximately Correct (PAC) framework of Valiant, (1984),
depicted in Figure 2, formalises the desired behaviour of a learning
algorithm. In this learning setting, a learning algorithm has access to
_random examples_ drawn in an i.i.d. fashion from an underlying distribution
$D$, and we wish to output a hypothesis that has small _error_ with high
_confidence_. The error $\text{err}_{D}(h,c)$ of a hypothesis with respect to
$D$ is measured against a _ground truth function_ or _target concept_
$c:\mathcal{X}\rightarrow\\{0,1\\}$ which labels the data, and is defined as
$\text{err}_{D}(h,c)=\underset{{x\sim D}}{\Pr}\left(c(x)\neq
h(x)\right)\enspace.$
The set of points $x\in\mathcal{X}$ such that $c(x)\neq h(x)$ is often
referred to as the _error region_. We sometimes model the sampling process by
having access to the random example oracle $\mathsf{EX}(c,D)$. The “probably”
part of the PAC learning framework speaks to the confidence of the learning
algorithm, and allows for the possibility that a sample $S\sim D^{m}$ of size
$m$ drawn from the underlying distribution $D$ is not representative of $D$.
The “approximately” part of PAC learning refers to the requirement that the
hypothesis have sufficiently high accuracy, a relaxation from learning
_exactly._ Both the confidence and accuracy parameters are inputs to the
learning algorithm, and are _learning parameters_.
Another important parameter that affects the sample complexity is how _large_
the instance size is, e.g., the larger the number of pixels for image
classification is, the larger the amount of data needed to learn could be.
This is usually controlled by the _dimension $n$ of the input space_, in
reference to $\left\\{0,1\right\\}^{n}$ and $\mathbb{R}^{n}$. To this end we
consider a collection of pairs of input space and concepts classes
$\mathcal{X}_{n}$ and $\mathcal{C}_{n}$ for each dimension $n$, where
$\mathcal{C}_{n}$ is a set of functions
$c:\mathcal{X}_{n}\rightarrow\\{0,1\\}$.
We are now ready to formally define the PAC learning setting.
###### Definition 3.1 (PAC Learning, Realizable Setting).
For all $n\in\mathbb{N}$, let $\mathcal{C}_{n}$ be a concept class over
$\mathcal{X}_{n}$ and let
$\mathcal{C}=\bigcup_{n\in\mathbb{N}}\mathcal{C}_{n}$. We say that
$\mathcal{C}$ is _PAC learnable using hypothesis class $\mathcal{H}$_ and
sample complexity function $m(\cdot,\cdot,\cdot,\cdot)$ if there exists an
algorithm $\mathcal{A}$ that satisfies the following: for all
$n\in\mathbb{N}$, for every $c\in\mathcal{C}_{n}$, for every $D$ over
$\mathcal{X}_{n}$, for every $0<\epsilon<1/2$ and $0<\delta<1/2$, if whenever
$\mathcal{A}$ is given access to $m\geq
m(n,1/\epsilon,1/\delta,\text{size}(c))$ examples drawn i.i.d. from $D$ and
labeled with $c$, $\mathcal{A}$ outputs an $h\in\mathcal{H}$ such that with
probability at least $1-\delta$,
$\text{err}_{D}(h,c)=\underset{{x\sim D}}{\Pr}\left(c(x)\neq
h(x)\right)\leq\epsilon\enspace.$
We say that $\mathcal{C}$ is statistically efficiently PAC learnable if $m$ is
polynomial in $n,1/\epsilon$, $1/\delta$ and size$(c)$, and computationally
efficiently PAC learnable if $\mathcal{A}$ runs in polynomial time in
$n,1/\epsilon$, $1/\delta$ and size$(c)$ and $h$ is polynomially evaluatable.
Probably Approximately Correct Learning
$\mathcal{A}$$S_{c}\sim D^{m}$
$m=\text{poly}(n,\frac{1}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\varepsilon}},\frac{1}{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta}})$$h\in\mathcal{H}$
s.t.
err${}_{D}(h,c)\leq{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\varepsilon}$w.p.
$>1-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta}$
Figure 2: A visual representation of sample-efficient PAC learning. $S_{c}$
means that the sample $S$ has been labelled with the ground truth $c$.
##### Size$(c)$ and polynomial evaluatability.
Two additional requirements from accuracy and confidence are introduced in the
above definition: these are a sample complexity function dependent on the size
$size(c)$ of the target concept $c$, and, if one requires computational
efficiency, the fact that $h$ is _polynomially evaluatable_. The size of a
concept is defined through a _representation scheme_. Essentially, there could
exist several representations of a function, e.g., a function can be computed
by many different boolean circuits. Assuming that there exists a function
measuring the size of a representation, the size of a concept $c$ is the
minimal size of a representation of $c$. The second requirement is natural: if
the hypothesis is not required to be polynomially evaluatable, then the
learner could simply “offload” the learning process at test time (there is
nothing to do at training, so it would be considered “efficient”), and overall
require arbitrarily high computational complexity.
##### Proper vs improper learning.
The setting where $\mathcal{C}=\mathcal{H}$ is called _proper learning_ , and
_improper learning_ if $\mathcal{C}\subseteq\mathcal{H}$. While requiring
proper learning does not affect the sample complexity of learning very
much,101010It is possible to get rid of the $\log 1/\epsilon$ factor of
Theorem 3.8 as shown by the recent breakthrough of Hanneke, (2016) with an
improper learner, but, aside from this, the sample complexity bounds in
Theorems 3.8 and 3.9 are tight for any consistent learner. it can affect its
computational efficiency. Indeed, unless $\mathsf{RP}=\mathsf{NP}$, which is
widely believed not to be the case, it is impossible to computationally
efficiently _properly_ learn the class of 3-term formulas in disjunctive
normal form (DNF), i.e., formulas of the form $T_{1}\vee T_{2}\vee T_{3}$
where the $T_{i}$’s are conjunctions of arbitrary lengths. However, it is
possible to computationally efficiently PAC learn 3-CNF formulas properly
(formulas in conjunctive normal form where each term is a disjunction of at
most 3 literals), and this class subsumes 3-term DNFs. Hence, one can use the
PAC-learning algorithm for 3-CNF to (improperly) PAC learn 3-term DNFs in a
computationally efficient manner.
##### The distribution-free assumption.
PAC learning is _distribution-free_ , in the sense that no assumptions are
made about the distribution from which the data is generated. As long as the
training data is sampled i.i.d. from a given distribution $D$, and that the
algorithm is tested on independent examples drawn from $D$, the learning
guarantees hold. Of course, this is sometimes not a sensible assumption to
make in practice. Many lines of work consider learning settings that allow for
this and provide a more realistic learning framework, e.g., when noise is
added to the data, or when the training and testing distributions differ
(i.e., distribution shift), as outlined in Chapter 2.
##### Realizable vs agnostic learning.
The _realizability assumption_ of Definition 3.1, where there always exists a
concept with zero error, does not always hold. In the presence of noise, or
more generally in the absence of a _deterministic_ labelling function $c$
representing the ground truth (e.g., there is a joint distribution on
$\mathcal{X}\times\mathcal{Y}$), we instead work in the _agnostic setting_. In
this setting, the goal is rather to learn a hypothesis that does well compared
to the best concept in the concept class:
###### Definition 3.2 (PAC Learning, Agnostic Setting).
Let $\mathcal{C}_{n}$ be a concept class over $\mathcal{X}_{n}$ and let
$\mathcal{C}=\bigcup_{n\in\mathbb{N}}\mathcal{C}_{n}$. We say that
$\mathcal{C}$ is _agnostically PAC learnable using $\mathcal{H}$_ with sample
complexity function $m(\cdot,\cdot,\cdot,\cdot)$ if there exists an algorithm
$\mathcal{A}$ that satisfies the following: for all $n\in\mathbb{N}$, for
every $D$ over $\mathcal{X}_{n}\times\\{0,1\\}$, for every $0<\epsilon<1/2$
and $0<\delta<1/2$, if whenever $\mathcal{A}$ is given access to $m\geq
m(n,1/\epsilon,1/\delta,s)$ labelled examples drawn i.i.d. from $D$, where
$s=\underset{c\in\mathcal{C}_{n}}{\sup}\;\text{size(c)}$, $\mathcal{A}$
outputs an $h\in\mathcal{H}$ such that with probability at least $1-\delta$,
$\text{err}_{D}(h)\leq\underset{c\in\mathcal{C}_{n}}{\inf}\text{err}_{D}(c)+\epsilon\enspace,$
where $\text{err}_{D}(h)=\underset{{(x,y)\sim D}}{\Pr}\left(h(x)\neq
y\right)$. We say that $\mathcal{H}$ is statistically efficiently agnostically
learnable if $m$ is polynomial in $n,1/\epsilon$, $1/\delta$ and $s$, and
computationally efficiently agnostically learnable if $\mathcal{A}$ runs in
polynomial time in $n,1/\epsilon$, $1/\delta$ and $s$, and $h$ is polynomially
evaluatable.
The definition above allows for improper learning ($\mathcal{C}$ is usually
called the “touchstone” class), but we can recover proper learning by setting
$\mathcal{C}=\mathcal{H}$. In this work, unless otherwise stated, we will
assume the realizability of a learning problem, and the sample complexity
bounds will be derived for this setting. Note that there exist PAC guarantees
for classes of finite VC dimension in the agnostic setting as well, at the
cost of a multiplicative factor of $1/\epsilon$ in the sample complexity. See
(Kearns et al.,, 1994; Haussler,, 1992) for original work on the matter and
the textbook (Mohri et al.,, 2012) for an introduction on the topic.
#### 6.2 Complexity Measures
While it is possible to derive sample complexity bounds for specific
hypothesis classes, one can take a more general approach with the use of
_complexity measures_. Indeed, a complexity measure assigns to each hypothesis
class $\mathcal{H}$ a function (w.r.t. the size $n$ of the instance space)
quantifying its richness. Intuitively, as the complexity measure increases,
more data should be needed to identify a candidate hypothesis that would
generalize well on unseen data. We briefly note that the standard theory
outlined in this chapter has failed to explain the recent success of
overparametrised deep neural networks in practice which in many ways remains
an open problem in the learning theory literature.
The first complexity measure we will study is perhaps the simplest one: the
size of $\mathcal{H}$. Similarly to $\mathcal{C}$, the class $\mathcal{H}$ is
defined as the union $\bigcup_{n\in\mathbb{N}}\mathcal{H}_{n}$, and the size
of $\mathcal{H}$ is a function of $n$. The theorem below, known as Occam’s
razor, gives an upper bound on the sample complexity of learning with finite
hypothesis classes, given access to a _consistent_ learner. A consistent
learner is a learning algorithm that outputs a hypothesis that has zero
empirical loss on the training sample, i.e., a hypothesis that correctly
classifies all the points in the training sample.
###### Theorem 3.3 (Occam’s Razor (Blumer et al.,, 1987)).
Let $\mathcal{C}$ and $\mathcal{H}$ be a concept and hypothesis classes,
respectively. Let $\mathcal{A}$ be a consistent learner for $\mathcal{C}$
using $\mathcal{H}$. Then, for all $n\in\mathbb{N}$, for every
$c\in\mathcal{C}_{n}$, for every $D$ over $\mathcal{X}_{n}$, for every
$0<\epsilon<1/2$ and $0<\delta<1/2$, if whenever $\mathcal{A}$ is given access
to
$m\geq\frac{1}{\epsilon}\left(\log(|\mathcal{H}_{n}|)+\log(1/\delta)\right)$
examples drawn i.i.d. from $D$ and labeled with $c$, then $\mathcal{A}$ is
guaranteed to output an $h\in\mathcal{H}_{n}$ such that
$\text{err}_{D}(h,c)<\epsilon$ with probability at least $1-\delta$.
Furthermore, if $\log(|\mathcal{H}_{n}|)$ is polynomial in $n$ and size$(c)$,
and $h$ is polynomially evaluatable, then $\mathcal{C}$ is statistically
efficiently PAC-learnable using $\mathcal{H}$.
While the theorem above can be useful if $\mathcal{H}_{n}$ is finite for all
$n$, it does not tell us much when $\mathcal{H}$ is infinite. To this end, one
would want to consider complexity measures that are meaningful for infinite
concept classes as well. In the PAC setting, a useful complexity measure is
the Vapnik Chervonenkis (VC) dimension of a hypothesis class, from the work of
Vapnik and Chervonenkis, (1971). It turns out that this measure fully
_characterizes_ the learnability of a concept class, in the sense that one can
obtain upper _and_ lower bounds on the sample complexity that are both
_linear_ in the VC dimension of $\mathcal{H}$.
In order to define the VC dimension of a concept class, we must first define
the notion of _shattering_ of a set. In Figure 3, we give an example of a set
being shattered by linear classifiers in $\mathbb{R}^{2}$.
###### Definition 3.4 (Shattering).
Given a class of functions $\mathcal{F}$ from input space $\mathcal{X}$ to
$\left\\{0,1\right\\}$, we say that a set $S\subseteq\mathcal{X}$ is
_shattered by $\mathcal{F}$_ if all the possible dichotomies of $S$ (i.e., all
the possible ways of labelling the points in $S$) can be realized by some
$f\in\mathcal{F}$.
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$
(a)
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+}$
(b)
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$
(c)
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+}$
(d)
Figure 3: A set $X$ of three points in $\mathbb{R}^{2}$ that is shattered by
linear classifiers. Subfigures (a)-(d) represent different dichotomies on $X$;
note that (b) and (c) are not the only labellings with one and two positively
labelled points, respectively, but the other cases are symmetric.
We are now ready to define the VC dimension of a class.
###### Definition 3.5 (VC Dimension).
The VC dimension of a hypothesis class $\mathcal{H}$, denoted
$\mathsf{VC}(\mathcal{H})$, is the size $d$ of the largest set that can be
shattered by $\mathcal{H}$. If no such $d$ exists then
$\mathsf{VC}(\mathcal{H})=\infty$.
Figure 4 illustrates the argument that no set in $\mathbb{R}^{2}$ of size 4
can be shattered by linear classifiers.
An important property of the VC dimension is that it is upper bounded by
$\log\left|\mathcal{H}\right|$. Indeed, a shattered set $S$ of size $m$ needs
$2^{m}$ distinct functions to achieve all its possible labellings.
It also is possible to define the VC dimension through the _growth function_
of a concept class. For some finite set of instances $S$, we denote by
$\Pi_{\mathcal{C}}(S)=\\{c|_{S}\;|\;c\in\mathcal{C}\\}$ the set of distinct
restrictions of concepts in $\mathcal{C}$ on the set $S$, which is referred to
as the set of all possible dichotomies on $S$ induced by $\mathcal{C}$. Then a
shattered set $S$ satisfies
$\left|\Pi_{\mathcal{C}}(S)\right|=2^{\left|S\right|}$, and the VC dimension
is thus the largest set satisfying this relationship.
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$
(a)
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+}$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-}$
(b)
Figure 4: Any set of four points cannot be shattered by linear classifiers.
Indeed, we distinguish two cases: either (a) one point is strictly in the
convex hull of the three other points, and is the only point of its label (or
all points are on the same line, which gives a similar argument) or (b) all
points are on the boundary of the convex hull, in which case labelling
opposite points with the same label gives an unachievable labelling. This
argument is a special case for $\mathbb{R}^{2}$ which can be generalized to
$\mathbb{R}^{n}$ using Radon’s theorem.
###### Definition 3.6 (Growth Function).
For any natural number $m\in\mathbb{N}$, the growth function is defined as
$\Pi_{\mathcal{C}}(m)=\max\left\\{\left|\Pi_{\mathcal{C}}(S)\right|\;|\;\left|S\right|=m\right\\}$.
Denote by $\Phi_{d}(m)$ the summation $\sum_{i=0}^{d}{m\choose i}$. The growth
function of a concept class $\mathcal{C}$ can be bounded as follows, as a
function of $m$ and the VC dimension $d$.
###### Lemma 3.7 (Sauer-Shelah).
Let $\mathcal{C}$ be a concept class of VC dimension $d$. Then
$\Pi_{\mathcal{C}}(m)\leq\Phi_{d}(m)\leq\left(\frac{em}{d}\right)^{d}\enspace.$
As previously mentioned, the VC dimension characterizes PAC learnability. We
start with a sample complexity upper bound that is linear in the VC dimension,
due to Vapnik, (1982) and Blumer et al., (1989).
###### Theorem 3.8 (VC Dimension Sample Complexity Upper Bound).
Let $\mathcal{C}$ be a concept class. Let $\mathcal{A}$ be a consistent
learner for $\mathcal{C}$ using a hypothesis class $\mathcal{H}$ of VC
dimension $\mathsf{VC}(\mathcal{H})=d$. Then $\mathcal{A}$ is a PAC-learning
algorithm for $\mathcal{C}$ using $\mathcal{H}$ provided it is given an i.i.d.
sample $S\sim D^{m}$ drawn from some $D$ and labelled with some
$c\in\mathcal{C}$, where
$m\geq\kappa_{0}\cdot\frac{1}{\epsilon}\left(d\log\frac{1}{\epsilon}+\log\frac{1}{\delta}\right)\enspace,$
for some universal constant $\kappa_{0}$.
We now have a sample complexity lower bound that is also linear in the VC
dimension, due to Blumer et al., (1989) and Ehrenfeucht et al., (1989). The
proofs of both Theorems 3.8 and 3.9 appear in reference textbooks such as
(Mohri et al.,, 2012) and (Shalev-Shwartz and Ben-David,, 2014).
###### Theorem 3.9 (VC Dimension Sample Complexity Lower Bound).
Let $\mathcal{C}$ be a concept class with VC dimension $d$. Then any PAC-
learning algorithm for $\mathcal{C}$ requires
$\Omega\left(\frac{d}{\epsilon}+\frac{1}{\epsilon}\log\frac{1}{\delta}\right)$
examples.
While the bounds of Theorems 3.8 and 3.9 are tight up to a
$\log\frac{1}{\epsilon}$, the breakthrough work of Hanneke, (2016) recently
showed the existence of a specific learning algorithm that is optimal in the
sense that its sample complexity matches that of Theorem 3.9 up to constant
factors, and thus avoids the $\log\frac{1}{\epsilon}$ dependence.
#### 6.3 Some Concept Classes and PAC Learning Algorithms
In this section, we introduce various concept classes that have been studied
in the learning theory literature, along with PAC learning algorithms. All the
algorithms outlined below are consistent on a given training sample, given we
are working in the realizable setting. A bound on the VC dimension of these
concept classes directly gives sample complexity upper bounds as per Theorem
3.8. We start with concept classes defined on the boolean hypercube
$\mathcal{X}=\left\\{0,1\right\\}^{n}$.
##### Singletons.
For an input space $\mathcal{X}$, the class of singletons is the class of
functions
$\left\\{x\mapsto\mathbf{1}[x=x^{*}]\;|\;x^{*}\in\mathcal{X}\right\\}$.
##### Dictators.
The class of dictators on $\left\\{0,1\right\\}^{n}$ is the class of functions
determined by a single bit, i.e., functions of the form $h(x)=x_{i}$ or
$h(x)=\bar{x_{i}}$ for $i\in[n]$. Dictators are subsumed by conjunctions.
Monotone dictators are dictators where negations are not allowed, i.e.,
functions of the form $h(x)=x_{i}$.
##### Conjunctions.
Conjunctions, which we denote CONJUNCTIONS, are perhaps one of the simplest
non-trivial concept classes one can study on the boolean hypercube. A
conjunction $c$ over $\\{0,1\\}^{n}$ is a logical formula over a set of
literals $l_{1},\dots,l_{k}$ from
$\left\\{x_{1},\bar{x_{1}},\dots,x_{n},\bar{x_{n}}\right\\}$, where, for
$x\in\mathcal{X}_{n}$, $c(x)=\bigwedge_{i=1}^{k}l_{i}$. The _length_ of a
conjunction $c$ is the number of literals in $c$.111111We use the term
_length_ for conjunctions that are not equivalent to the constant function 0.
For example, $c(x)=x_{1}\wedge\bar{x_{2}}\wedge{x_{5}}$ is a conjunction of
length 3. Monotone conjunctions are the subclass of conjunctions where
negations are not allowed, i.e., all literals are of the form $l_{i}=x_{j}$
for some $j\in[n]$. Note that this implies that monotone conjunctions do not
include the constant function 0.
Algorithm 1 PAC-learning algorithm for conjunctions
$S_{c}\sim D^{m}$
$L\leftarrow\left\\{x_{1},\bar{x_{1}},\dots,x_{n},\bar{x_{n}}\right\\}$
$h(x)=\bigwedge_{l\in L}l$ $\triangleright$ $h=0$
for $(x,c(x))\in S$ do
if $c(x)\neq h(x)$ then $\triangleright$ Only happens if $c(x)=1$
$L\leftarrow L\setminus\left\\{l\in L\;|\;l(x)=0\right\\}$
end if
end for
The standard PAC learning algorithm to learn conjunctions is as outlined in
Algorithm 1. We start with the constant hypothesis $h(x)=\bigwedge_{i\in
I_{h}}(x_{i}\wedge\bar{x_{i}})\equiv 0$, where $I_{h}=[n]$. To ensure
consistency, for each example $x$ in the training sample, we remove a literal
$l$ from $h$ if $c(x)=1$ and $l(x)=0$, as if $l$ is in the conjunction, $h$
must evaluate to $0$ on $x$. After seeing all the examples in the training set
$S$, the resulting hypothesis will thus be consistent on $S$. Note that
$\mathsf{VC}(\textsf{CONJUNCTIONS}_{n})=n$ (Natschläger and Schmitt,, 1996).
Finally, Algorithm 1 can also be used for monotone conjunctions, but where the
initial hypothesis is $h(x)=\bigwedge_{i\in[n]}x_{i}$.
##### CNF and DNF formulas.
A formula $\varphi$ in the conjunctive normal form (CNF) is a conjunction of
clauses, where each clause is itself a disjunction of literals. A $k$-CNF
formula is a CNF formula where each clause contains at most $k$ literals. For
example, $\varphi=(x_{1}\vee x_{2})\wedge(\bar{x_{3}}\vee
x_{4})\wedge\bar{x_{5}}$ is a 2-CNF. On the other hand, a DNF formula is a
disjunction of clauses, where each clause is itself a conjunction of literals.
A $k$-DNF is defined analogously to a $k$-CNF.
##### Decision lists.
Given a positive integer $k$, a $k$-decision list $f\in k$-$\mathsf{DL}$ is a
list $(K_{1},v_{1}),\dots,(K_{r},v_{r})$ of pairs where $K_{j}$ is a term in
the set of all conjunctions of size at most $k$ with literals drawn from
$\left\\{x_{1},\bar{x_{1}},\dots,x_{n},\bar{x_{n}}\right\\}$, $v_{j}$ is a
value in $\left\\{0,1\right\\}$, and $K_{r}$ is $\mathtt{true}$. The output
$f(x)$ of $f$ on $x\in\left\\{0,1\right\\}^{n}$ is $v_{j}$, where $j$ is the
least index such that the conjunction $K_{j}$ evaluates to $\mathtt{true}$ on
$x$. Decision lists subsume conjunctions. Indeed, a conjunction
$c(x)=\bigwedge_{i=1}^{k}l_{i}$ can be expressed as the following 1-decision
list: $(\neg l_{1},0),\dots,(\neg l_{k},0),(\mathtt{true},1)$.
The PAC-learning algorithm for decision lists, introduced by Rivest, (1987),
is outlined in Algorithm 2. The sample size $m$ is given by Theorem 3.8 and an
observation that the size of the class is
$O\left(3^{\left|C_{n,k}\right|}\left|C_{n,k}\right|!\right)$, where $C_{n,k}$
is the set of conjunctions of length at most $k$ on $n$ variables, giving a VC
dimension bound of $O(n^{k}\log n)$. Note that, as we consider $k$ to be a
fixed constant, the sample complexity bound is polynomial in $n$ and the
learning parameters.
Algorithm 2 PAC-learning algorithm for 1-decision lists from Rivest, (1987)
$S\sim D^{m}$
$L:=\left\\{x_{i},\bar{x_{i}}\right\\}_{i=1}^{n}$ $\triangleright$ Set of all
literals
$h=\emptyset$ $\triangleright$ Empty decision list
while $S\neq\emptyset$ do
if $\exists b\in\left\\{0,1\right\\}$ s.t. $\forall(x,y)\in S,\;y=b$ then
$S\leftarrow\emptyset$
append $(\mathtt{true},b)$ to $h$
else
for $l\in L$ s.t. $\exists(x,y)\in S$ s.t. $l(x)=1$ do $\triangleright$ $l$ is
true for some $x$
if $\exists b\in\left\\{0,1\right\\}$ s.t. $\forall(x,y)\in
S\;\left(l(x)=1\Rightarrow y=b\right)$ then
append $(l,b)$ to $h$
$S\leftarrow S\setminus\left\\{(x,y)\in S\;|\;l(x)=1\right\\}$
end if
end for
end if
end while
Note that, while the algorithm above is for 1-decision lists, it is sufficient
to only consider this case. Indeed, if we are dealing with $k$-decision lists,
we can draw our attention to the set $C_{n,k}$ of conjunctions of length at
most $k$ on $n$ variables by defining the following injective map:
$\Phi:\left\\{0,1\right\\}^{n}\rightarrow\left\\{0,1\right\\}^{C_{n,k}}\enspace,$
(1)
where $\Phi(x)_{c_{i}}=\mathbf{1}[x\models c_{i}]$ for $c_{i}\in C_{n,k}$,
i.e. whether $x$ satisfies clause $c_{i}$. Now, any distribution $D$ on
$\left\\{0,1\right\\}^{n}$ induces a well-defined distribution $D^{\prime}$ on
$\left\\{0,1\right\\}^{C_{n,k}}$. Moreover, since
$\left|{C_{n,k}}\right|=O(n^{k})$, an input $x\in\left\\{0,1\right\\}^{n}$ and
a 1-decision $h$ on $\left\\{0,1\right\\}^{n}$ can respectively be transformed
into $\Phi(x)\in\left\\{0,1\right\\}^{C_{n,k}}$ and a $k$-decision list
$h^{\prime}$ on $\left\\{0,1\right\\}^{C_{n,k}}$ in polynomial time, for a
fixed $k$, and vice-versa in the case of going from $h^{\prime}$ to $h$. It
also follows that
$\text{err}_{D}(h,c)=\text{err}_{D^{\prime}}(h^{\prime},c^{\prime})$, where
$c^{\prime}$ is the $k$-decision list on $\left\\{0,1\right\\}^{C_{n,k}}$
induced by $c$. Hence, an efficient learning algorithm for 1-decision lists
can be used as a black box to efficiently learn $k$-decision lists.
Finally, the class of $k$-decision lists subsume $k$-CNF and $k$-DNF (Rivest,,
1987).
##### Decision trees.
A decision tree $T$ is a binary tree whose nodes are positive literals in
$\left\\{x_{1},\dots,x_{n}\right\\}$. For a given node with variable $x_{i}$,
the edge to its left child node is labelled with 0 and the edge to its right
child node is labelled as 1, representing the value of the $x_{i}$ for a given
instance $x\in\left\\{0,1\right\\}^{n}$. The leaves take label in
$\left\\{0,1\right\\}$; a given $x\in\left\\{0,1\right\\}^{n}$ induces a path
from the root to a leaf in $T$, which will give the label $T(x)$. Decision
trees generalize 1-decision lists: a 1-decision list is a decision tree with
each node having at most one child. Note that it is currently unknown whether
polynomial-sized decision trees are PAC learnable.
##### Parities.
Parities are defined with respect to a subset $I\subseteq[n]$ of indices as
$f_{I}(x)=\left(\sum_{i\in I}x_{i}\right)\bmod 2$, i.e. the output is whether
adding the bits at indices in $S$ results in an odd or even sum. Learning
parities amounts to learning the set $S$. Given a set of examples
$(X,Y)\subseteq\left\\{0,1\right\\}^{n}\times\left\\{0,1\right\\}$, where each
$(x,y)\in(X,Y)$ is a labelled example, finding this set is equivalent to
finding a solution $a\in\left\\{0,1\right\\}^{n}$ to the system of linear
equations $Xa=Y$ in the finite field $\mathbb{F}_{2}$. The set
$J:=\left\\{j\in[n]\;|\;a_{j}=1\right\\}$ gives a hypothesis
$f_{J}(x)=\left(\sum_{j\in J}x_{j}\right)\bmod 2$ consistent with the data.
This can be done using Gaussian elimination, provided a solution exists (this
is guaranteed by the realizability assumption). See (Helmbold et al.,, 1992;
Goldberg,, 2006) for details.
Note that, when working in $\left\\{-1,1\right\\}^{n}$ instead of
$\left\\{0,1\right\\}^{n}$, we can define the parity function as
$f_{I}(x)=\prod_{i\in I}x_{i}$ instead. This representation will be especially
relevant in Section 8 when we introduce Fourier analysis concepts.
##### Majorities.
Similarly to parities, majorities are defined with respect to a set $I$ of
indices, as follows: $\text{maj}_{I}(x)=\mathbf{1}\left[\sum_{i\in
I}x_{i}\geq\left|I\right|/2\right]$. Again, when working in
$\left\\{-1,1\right\\}^{n}$ instead of $\left\\{0,1\right\\}^{n}$, majority
functions are defined as $\text{maj}_{I}(x)=\text{sgn}\left(\sum_{i\in
I}x_{i}\right)$. Clearly, from the representations above, majorities are
subsumed by linear classifiers, which are defined further below.
##### Linear classifiers.
The class of linear classifiers (also known as halfspaces and linear threshold
functions) on input spaces $\mathcal{X}=\left\\{0,1\right\\}^{n}$ or
$\mathcal{X}=\mathbb{R}^{n}$ are defined as
$\left\\{x\mapsto\text{sgn}\left(w\cdot
x+b\right)\;|\;w\in\mathbb{R}^{n},b\in\mathbb{R}\right\\}$, where the
$w_{i}\in w$ are the _weights_ and $b$ is the _bias_. When the instance space
is the reals, we will denote the class as $\mathsf{LTF}_{\mathbb{R}^{n}}$.
Moreover, we will denote by $\mathsf{LTF}_{\left\\{0,1\right\\}^{n}}^{W}$ the
class of linear threshold functions on $\left\\{0,1\right\\}^{n}$ with integer
weights such that the sum of the absolute values of the weights and the bias
is bounded above by $W$, and $W^{+}$ when the weights are positive. Finally,
when the weights and the bias are binary, i.e., $w_{i},b\in\\{0,1\\}$ for all
$i$, the class is called _boolean threshold functions_.
The VC dimension of halfspaces is $n+1$. The upper bound of $n+1$ can be shown
by using Radon’s theorem (any set of size $n+2$ in $\mathbb{R}^{n}$ can be
partitioned into two subsets whose convex hulls intersect), and the lower
bound can be obtained by showing that the set
$\left\\{\mathbf{e}_{i}\right\\}_{i=1}^{n}\cup\mathbf{0}$ can be shattered.
The support vector machine (SVM) algorithm, or solving a system of linear
inequalities with linear programming, can be used as a consistent learner for
this concept class. Finally, the class of conjunctions is subsumed by linear
classifiers: a conjunction $f(x)=\bigwedge_{i=1}^{k}l_{i}$ can be represented
as the linear classifier $g(x)=\text{sgn}\left(\sum_{i\in
I^{+}}x_{i}-\sum_{i\in I^{-}}x_{i}-\left|I\right|+1\right)$, where
$I^{+}=\left\\{j\in[n]\;|\;\exists i\;.\;l_{i}=x_{j}\right\\}$ and
$I^{-}=\left\\{j\in[n]\;|\;\exists i\;.\;l_{i}=\bar{x_{i}}\right\\}$.
#### 6.4 Online Learning: The Mistake-Bound Model
In online learning, the learner is given access to examples _sequentially_. At
each time step $t$, the learner receives an example $x_{t}$, predicts its
label $\hat{y}_{t}$ using a given hypothesis class $\mathcal{H}$, receives the
true label $y_{t}$ and can update its hypothesis, typically when
$\hat{y}_{t}\neq y_{t}$. A fundamental distinction between the PAC- and
online-learning models is that, in the latter, there are usually no
distributional assumptions on the data.121212Some lines of work in online
learning look at mild distributional assumptions in the learning problem in
order to get better guarantees, but the basic mistake-bound online learning
set-up assumes that examples (or more generally losses in the regret
framework) can be given in an adversarial and adaptive manner. Thus, we need
to evaluate the learner’s performance with different benchmark than the error
$\text{err}_{D}(h)$ from the (offline) PAC setting.
In the mistake-bound model, examples and their labels can be given in an
adversarial fashion. The performance of the learner is evaluated with respect
to the number of mistakes it makes compared to the ground truth; we again
assume the _realizability_ of the learning problem, meaning that there is a
target concept $c\in\mathcal{C}$ such that $c(x_{t})=y_{t}$ for all $t$.
Crucially, the target concept need not be chosen a priori: the only
requirement is that, at every time $t$, there exists a concept
$c\in\mathcal{C}$ that is consistent on the past sequence of points
$(x_{1},y_{1}),\dots,(x_{t},y_{t})$. The goal of the learner is to learn the
target exactly.
We now formally define the mistake-bound model of online learning.
###### Definition 3.10 (Mistake Bound).
For a given hypothesis class $\mathcal{C}$ and instance space
$\mathcal{X}=\bigcup_{n}\mathcal{X}_{n}$, we say that an algorithm
$\mathcal{A}$ learns $\mathcal{C}$ with mistake bound $M$ if $\mathcal{A}$
makes at most $M$ mistakes on any sequence of samples consistent with a
concept $c\in\mathcal{C}$.
In the mistake bound model, we usually require that $M$ be polynomial in $n$
and size$(c)$. A good example where this holds is the online learning
algorithm for conjunctions, outlined in Algorithm 3, which is immediately
adapted from its PAC-learning counterpart. Indeed, Algorithm 1 only changes
its hypothesis whenever it sees a positive example $(x,1)$ such that $h(x)=0$,
and works through the sample sequentially.
Algorithm 3 PAC-learning algorithm for conjunctions, online version
$L\leftarrow\left\\{x_{1},\bar{x_{1}},\dots,x_{n},\bar{x_{n}}\right\\}$
$h(x)=\bigwedge_{l\in L}l$ $\triangleright$ $h=0$
for $t=1,2,\dots$ do
Receive $x_{t}$
Predict $h(x_{t})$
Receive true label $y_{t}$
if $h(x_{t})\neq y_{t}$ then $\triangleright$ Only happens if $y=1$
$L\leftarrow L\setminus\left\\{l\in L\;|\;l(x)=0\right\\}$
end if
end for
Unlike with conjunctions, the vast majority of PAC-learning algorithms cannot
be so straightforwardly tailored to online learning, resulting in a rich
literature on algorithms, benchmarks and guarantees specific to this setting.
One of the simplest general-purpose algorithms for online learning in the
realizable mistake-bound model is the halving algorithm, outlined in Algorithm
4.
Algorithm 4 Halving algorithm
A hypothesis class $\mathcal{H}$
for $t=1,2,\dots$ do
Receive example $x_{t}$
$V^{(b)}_{t}\leftarrow\left\\{h\in V_{t}\;|\;h(x_{t})=b\right\\}$
$\hat{y_{t}}=\arg\max_{b}\left|V_{t}^{(b)}\right|$ $\triangleright$ Predict
label acc. to a majority vote
Receive true label $y_{t}$
$V_{t+1}\leftarrow V^{(y_{t})}_{t}$
end for
At each time step, the learner predicts the label of a new point according to
the majority vote of the hypotheses consistent with the sequence of data seen
so far, which is denoted as $V_{t}$. It is easy to see that the halving
algorithm will make at most $\log\left|\mathcal{H}\right|$ mistakes: every
time the learner makes a mistake on $(x_{t},y_{t})$, at least half of the
hypotheses are not consistent with $(x_{t},y_{t})$, and are thus eliminated.
There are two significant disadvantages to this learning algorithm: (i) its
computational complexity, with a runtime $\Omega(\left|\mathcal{H}\right|)$,
as it requires iterating through the whole hypothesis class to get a majority
vote and (ii) it can only be used on _finite_ concept classes. Note that these
drawbacks can be addressed by instead drawing a hypothesis at random from the
version space, as argued in (Maass,, 1991). We will now address the second
drawback and turn our attention to potentially _infinite_ concept classes.
We have seen that, in PAC learning, the VC dimension of a concept class
characterizes its learnability, enabling learning guarantees for infinite
concept classes that have finite VC dimension. One may wonder whether there
exists an analogous complexity measure to the VC dimension when working in the
mistake-bound model. It turns out that such a measure exists in this setting:
the Littlestone dimension, defined and proved to characterize online
learnability in (Littlestone,, 1988). In order to define the Littlestone
dimension, we must first define Littlestone trees.
###### Definition 3.11 (Littlestone Tree).
A Littlestone tree for a hypothesis class $\mathcal{H}$ on $\mathcal{X}$ is a
complete binary tree $T$ of depth $d$ whose internal nodes are instances
$x\in\mathcal{X}$. Each edge is labeled with $0$ or $1$ and corresponds to the
potential labels of the parent node. Each path from the root to a leaf must be
consistent with some $h\in\mathcal{H}$, i.e. if $x_{1},\dots,x_{d}$ with
labelings $y_{1},\dots,y_{d}$ is a path in $T$, there must exist
$h\in\mathcal{H}$ such that $h(x_{i})=y_{i}$ for all $i$.
We are now ready to define the Littlestone dimension.
###### Definition 3.12 (Littlestone Dimension).
The Littlestone dimension of a hypothesis class $\mathcal{H}$, denoted
$\mathsf{Lit}(\mathcal{H})$, is the largest depth $d$ of a Littlestone tree
for $\mathcal{H}$. If no such $d$ exists then
$\mathsf{Lit}(\mathcal{H})=\infty$.
##### Relationship to other complexity measures.
Before showing that the Littlestone dimension characterizes online
learnability in this setting, we will study some of its properties. First, the
Littlestone dimension is an upper bound on the VC dimension. Indeed, it is
possible to convert any shattered set $X=\left\\{x_{1},\dots,x_{d}\right\\}$
of size $d$ into a Littlestone tree of depth $d$, where the nodes at depth $i$
are all $x_{i}$ and every path from the root to a leaf corresponds to a
dichotomy on $X$.
Moreover, from the definition of Littlestone trees, since each path from the
root to a leaf of a tree is achievable by a distinct function
$h\in\mathcal{H}$, the Littlestone dimension is bounded above by the logarithm
of the size of $\mathcal{H}$. We then have the following inequality for all
$\mathcal{H}$
$\mathsf{VC}(\mathcal{H})\leq\mathsf{Lit}(\mathcal{H})\leq\log(\left|\mathcal{H}\right|)\enspace.$
(2)
It can be shown that the gaps between the terms in Equation 2 can be
arbitrarily large. To show the gap between $\mathsf{VC}(\mathcal{H})$ and
$\mathsf{Lit}(\mathcal{H})$, consider the set
$\mathsf{THRESHOLDS}=\bigcup_{a\in\mathbb{R}}\mathbf{1}[x\geq a]$ of threshold
functions on $\mathbb{R}$. The VC dimension of $\mathsf{THRESHOLDS}$ is 1, as
a set of one point can be shattered, but a set of two points
$x_{1}<x_{2}\in\mathbb{R}$ cannot achieve the labelling $(1,0)$. However, its
Littlestone dimension is infinite: consider the interval $[0,1]$. At each
depth $i$ of the Littlestone tree, the set of nodes from left to right is
$\left\\{\frac{j+1}{2^{i}}\right\\}_{j=0}^{2^{i-1}}$, and the labelling of all
the left edges is $1$ and $0$ for right edges. For a given depth $i$, a path
$p$ from the root to node $x_{i,j}:=\frac{j+1}{2^{i}}$ for some
$j\in\left\\{0,1,\dots,2^{i-1}\right\\}$ (including $x_{i,j}$’s label) is thus
consistent with the threshold function $\mathbf{1}[x\geq x^{*}]$ where $x^{*}$
is the deepest node in $p$ (inclusive of $x_{i,j}$) that is positively
labelled. This infinite gap between the VC and Littlestone dimensions clearly
illustrates that online and offline (PAC) learnability are fundamentally
different from each other, as some concept classes are PAC learnable but not
online learnable. To show the other arbitrary large gap between
$\mathsf{Lit}(\mathcal{H})$ and $\log(\left|\mathcal{H}\right|)$, consider the
singletons on $\mathbb{R}$, i.e. the class of functions
$\bigcup_{a\in\mathbb{R}}\mathbf{1}[x=a]$. While the class is infinite, any
Littlestone tree, which must be complete, has depth 1, as each hypothesis in
the class labels a unique point (the target $a$) positively. Thus
$\mathsf{Lit}(\mathcal{H})=1$.
We now show that the Littlestone dimension lower bounds the number of mistakes
any online learning makes.
###### Theorem 3.13.
(Littlestone,, 1988) Any online learning algorithm for $\mathcal{C}$ has
mistake bound $M\geq\mathsf{Lit}(\mathcal{C})$.
###### Proof.
Let $\mathcal{A}$ be any online learning algorithm for $\mathcal{C}$. Let $T$
be a Littlestone tree of depth $\mathsf{Lit}(\mathcal{C})$ for $\mathcal{C}$.
Clearly, an adversary can force $\mathcal{A}$ to make
$\mathsf{Lit}(\mathcal{C})$ mistakes by sequentially and adaptively choosing a
path in $T$ in function of $\mathcal{A}$’s predictions. ∎
As previously suggested, the Littlestone dimension can also upper bound the
number of mistakes made by an online learning algorithm. This bound is
achieved for arbitrary concept classes with finite Littlestone dimension by
the Standard Optimal Algorithm from Littlestone, (1988), outlined in Algorithm
5.
Algorithm 5 Standard Optimal Algorithm from Littlestone, (1988)
A hypothesis class $\mathcal{\mathcal{C}}$
for $t=1,2,\dots$ do
Receive example $x_{t}$
$V^{(b)}_{t}\leftarrow\left\\{h\in V_{t}\;|\;h(x_{t})=b\right\\}$
$\hat{y_{t}}=\arg\max_{b}\mathsf{Lit}(V^{(b)}_{t})$
Receive true label $y_{t}$
$V_{t+1}\leftarrow V^{(y_{t})}_{t}$
end for
The SOA works in a similar fashion as the halving algorithm, only considering
at time $t$ the version space $V_{t}$ of hypotheses that are consistent with
the sequence of examples so far. However, instead of taking the majority vote,
the algorithm predicts the label $\hat{y_{t}}$ of a new point according to the
subclass (w.r.t. a label prediction $b\in\left\\{0,1\right\\}$) with larger
Littlestone dimension. The theorem below completes the proof that the
Littlestone dimension characterizes online learnability.
###### Theorem 3.14.
The Standard Optimal Algorithm from Littlestone, (1988) makes at most
$\mathsf{Lit}(\mathcal{C})$ mistakes in the mistake-bound model.
###### Proof.
We will show that, at every mistake, the Littlestone dimension of the subclass
$V_{t}$ decreases by at least 1 after receiving the true label $y_{t}$.
Suppose that, at time $t$, $y_{t}=\arg\min_{b}\mathsf{Lit}(V^{(b)}_{t})$. Note
that $V_{t+1}=V^{(y_{t})}_{t}$. Now, consider any two Littlestone trees
$T_{y_{t}}$ and $T_{\hat{y}_{t}}$ of maximal depths for $V^{(y_{t})}_{t}$ and
$V^{(\hat{y})}_{t}$, respectively. By definition, neither tree can contain
$x_{t}$, so it is possible to construct a Littlestone tree $T$ for $V_{t}$ of
depth $\min_{b}\mathsf{Lit}(V^{(b)}_{t})+1$ (recall that $T$ must be
complete). Then
$\mathsf{Lit}(V_{t})\geq\mathsf{Lit}(V^{({y_{t}})}_{t})+1=\mathsf{Lit}(V_{t+1})+1$,
as required.131313Note that the Littlestone dimension does not necessarily
decrease when $y_{t}=\hat{y}_{t}$, as we could have $V_{t}=V_{t}^{y_{y}}$. ∎
While the SOA has an optimal mistake bound and is defined for arbitrary
concept classes, it remains highly inefficient in general, as computing the
Littlestone dimension of the concept subclasses could be very costly. For the
remainder of this section, we will consider online learning algorithms for
specific concept classes in order to circumvent some of these issues.141414We
will discuss the algorithms and their mistake bounds here, but we refer the
reader to the references for the algorithms themselves and their analysis.
The first algorithm we will look at is Winnow, which is for linear threshold
functions with bounded weights in the boolean hypercube. This algorithm and
its analysis are due to Littlestone, (1988).
We now recall the mistake upper bound for Winnow in the special case of
$\mathsf{LTF}_{\left\\{0,1\right\\}^{n}}^{W+}$, where the weights are positive
integers.151515See https://www.cs.utexas.edu/~klivans/05f7.pdf for a full
derivation.
###### Theorem 3.15 (Winnow Mistake Bound).
The Winnow algorithm for learning the class
$\mathsf{LTF}_{\left\\{0,1\right\\}^{n}}^{W+}$ makes at most $O(W^{2}\log(n))$
mistakes.
We now look at the perceptron algorithm, which first appeared in Rosenblatt,
(1958), and whose first proofs of convergence were shown in Block, (1962) and
Novikoff, (1963).
While it is not possible to have a mistake bound for linear classifiers in
$\mathbb{R}^{n}$, as the Littlestone dimension is infinite, requiring a
_margin_ on the data ensures a finite mistake bound with the perceptron
algorithm, as stated below.
###### Theorem 3.16 (Mistake Bound for Perceptron, Margin Condition; Theorem
7.8 in Mohri et al., (2012)).
Let $\mathbf{x}_{1},\dots,\mathbf{x}_{T}\in\mathbb{R}^{n}$ be a sequence of
$T$ points with $\left\|\mathbf{x}_{t}\right\|\leq r$ for all $1\leq t\leq T$
for some $r>0$. Assume that there exists $\gamma>0$ and
$\mathbf{v}\in\mathbb{R}^{n}$ such that for all $1\leq t\leq T$,
$\gamma\leq\frac{y_{t}(\mathbf{v}\cdot\mathbf{x}_{t})}{\left\|\mathbf{v}\right\|}$.
Then, the number of updates made by the Perceptron algorithm when processing
$\mathbf{x}_{1},\dots,\mathbf{x}_{T}$ is bounded by $r^{2}/\gamma^{2}$.
#### 6.5 Learning with Membership and Equivalence Queries
So far, we have studied models where the learner does not have any control
over the data it gets: in the PAC setting, labelled instances are received
i.i.d. from the random example oracle, and in the online setting, the new
points can be given adversarially. In this sense the learner is quite passive
during the learning process. We will now turn our attention towards learning
models where the learner is more active, and, in addition to receiving random
examples, can make queries to an oracle, also sometimes referred to as
teacher.
For simplicity, we will for now assume that there is no distribution
underlying the data. Hence, similarly to the mistake-bound model of online
learning, the goal is to learn the target concept _exactly_ on the instance
space. We will start by defining two different types of queries: membership
and equivalence queries.
###### Definition 3.17.
A _membership oracle_ $\mathsf{MQ}(c)$ defined for a concept $c\in\mathcal{C}$
returns the value $c(x)$ when queried with an instance $x\in\mathcal{X}$.
The terminology refers to the fact that $\mathcal{C}$ is a class of boolean
functions, which can be interpreted as a subsets of $\mathcal{X}$. Then, a
membership query returns whether an instance $x$ is in the target subset of
$\mathcal{X}$. In the case of real-valued functions, a _value oracle_ might be
a more appropriate term.
###### Definition 3.18.
An _equivalence query oracle_ $\mathsf{EQ}(c)$ defined for a target concept
$c\in\mathcal{C}$ takes as input a representation of a hypothesis $h$ and
returns whether or not $h$ agrees with $c$ on the input space $\mathcal{X}$.
If $h\neq c$ on $\mathcal{X}$, $\mathsf{EQ}(c)$ also returns an instance
$x\in\mathcal{X}$, called a _counterexample_ , such that $h(x)\neq c(x)$.
With these two types of queries, we will now present the exact learning model
for concept classes in this setting, where the goal is to learn a hypothesis
$h$ such that for all $x\in\mathcal{X}$, $h(x)=c(x)$. We formally define this
model below, where we will assume that the learning algorithm is
deterministic.
###### Definition 3.19.
A concept class $\mathcal{C}$ is _efficiently exactly learnable_ using
membership and equivalence queries if there exists a polynomially-evaluatable
hypothesis class $\mathcal{H}$, a learning algorithm $\mathcal{A}$ and a
polynomial $p(\cdot,\cdot)$ such that for all $n\geq 1$, $c\in\mathcal{C}$,
whenever $\mathcal{A}$ is given access to the $\mathsf{MQ}(c)$ and
$\mathsf{EQ}(c)$ oracles, it halts in time $p(n,\text{size}(c))$ and outputs
some $h\in\mathcal{H}_{n}$ such that $h(x)=c(x)$ for all instances
$x\in\mathcal{X}$. Furthermore, every query made to $\mathsf{EQ}(c)$ by
$\mathcal{A}$ must made with some $h\in\mathcal{H}_{n}$.
The exact learning model with access to $\mathsf{MQ}$ and $\mathsf{EQ}$ has a
long history, particularly in automata theory, where the seminal work of
Angluin, (1987) presented an exact learning algorithm, called $L^{*}$, to
exactly learn deterministic finite automata.
Before going further, a few remarks are in order. First, the efficiency in
this definition is with respect to the computational complexity of the
problem. This entails requiring statistical efficiency as well, in the sense
that the number of queries to the $\mathsf{MQ}$ and $\mathsf{EQ}$ oracles be
also polynomial in $n$ and size$(c)$.
Second, it may seem that having access to an equivalence oracle is an
impractical requirement. After all, while it makes sense to consider
membership oracles, as they can often be simulated by human “experts” (e.g.,
captioning done by internet users), it could perhaps be unrealistic to expect
humans or automated systems to simulate the equivalence oracle in practice.
However, the following result shows that, if the exact learning requirement
can be relaxed to PAC learning, i.e., allowing for accuracy and confidence
parameters, then one can simply work in the $\mathsf{EX}+\mathsf{MQ}$ learning
model, and forgo equivalence queries.
###### Theorem 3.20.
Let $\mathcal{C}$ be exactly efficiently learnable using membership and
equivalence queries. Then $\mathcal{C}$ is efficiently PAC-learnable using
random examples and membership queries.
The proof, omitted for brevity, relies on the fact that it is possible to
simulate (with sufficient accuracy) the $\mathsf{EQ}$ oracle with access to
random examples.
Third, we have assumed that the learning algorithm is deterministic. It would
be possible to accommodate randomized learning algorithms with the addition of
a confidence parameter $\delta$ as in PAC learning. In this case, the
probability of failure would not come from the randomness in sampling the
data, but rather from the fact that we are working with an algorithm with
internal randomization, which could result in computational gains.
Now, note that it is possible to efficiently exactly learn conjunctions in the
$\mathsf{MQ}+\mathsf{EQ}$ model (just by using the $\mathsf{EQ}$ oracle). We
simply need to use the online learning version of the algorithm (Algorithm 3)
and, instead of receiving an instance and predicting its label, the learner
gives the hypothesis $h$ to $\mathsf{EQ}(c)$ and receives a counterexample if
$h\neq c$. The number of calls to $\mathsf{EQ}$ is upper bounded by the
mistake bound (the reasoning is the same as in the online setting).
A more interesting class of functions to study is the class
$\mathsf{MONOTONE\text{-}DNF}$, i.e., functions of the form
$T_{1}\vee\dots\vee T_{r}$ where each $T_{i}$ is a monotone conjunction
$\bigwedge_{j\in S_{i}}x_{j}$. It is not known whether
$\mathsf{MONOTONE\text{-}DNF}$ is PAC learnable. However, it can be shown that
this class can be exactly learned in the $\mathsf{MQ}+\mathsf{EQ}$ model (and
thus is PAC learnable when the learner has additional access to $\mathsf{MQ}$
by Theorem 3.20).
We finish this section by formally introducing _local_ membership queries
(LMQ), which were mentioned in Chapter 2. They were introduced by Awasthi et
al., (2013) and shown to circumvent some impossibility results in the standard
PAC setting (or impossibility conjectures). Here, given a sample $S$ drawn
from the example oracle $\mathsf{EX}(c,D)$, a membership query for a point $x$
is $\lambda$-_local_ if there exists $x^{\prime}\in S$ such that $x\in
B_{\lambda}(x^{\prime})$, i.e., an algorithm can only query the label of
points within distance $\lambda$ of the training sample.
###### Definition 3.21 (PAC Learning with $\lambda$-$\mathsf{LMQ}$ ).
Let $\mathcal{X}$ be the instance space equipped with a metric $d$,
$\mathcal{C}$ a concept class over $\mathcal{X}$, and $\mathcal{D}$ a class of
distributions over $\mathcal{X}$. We say that $\mathcal{C}$ is $\rho$-robustly
learnable using $\lambda$-local membership queries with respect to
$\mathcal{D}$ if there exists a learning algorithm $\mathcal{A}$ such that for
every $\epsilon>0$, $\delta>0$, for every distribution $D\in\mathcal{D}$ and
every target concept $c\in\mathcal{C}$, the following hold:
1. 1.
$\mathcal{A}$ draws a sample $S$ of size
$m=\text{poly}(n,1/\delta,1/\epsilon,\text{size}(c))$ using the example oracle
$\mathsf{EX}(c,D)$
2. 2.
Each query $x^{\prime}$ made by $\mathcal{A}$ to the $\mathsf{LMQ}$ oracle is
$\lambda$-local with respect to some example $x\in S$
3. 3.
$\mathcal{A}$ outputs a hypothesis $h$ that satisfies
$\text{err}_{D}(h,c)\leq\epsilon$ with probability at least $1-\delta$
4. 4.
The running time of $\mathcal{A}$ (hence also the number of oracle accesses)
is polynomial in $n$, $1/\epsilon$, $1/\delta$, $\text{size}(c)$ and the
output hypothesis $h$ is polynomially evaluable.
We conclude this section by remarking that learnability in the above setting
is with respect to a family $\mathcal{D}$ of distributions, rather than the
distribution-free setting of PAC learning. This is because LMQs have mostly
been used in the literature for learning problems which require distributional
assumptions.
### 7 Probability Theory
In this section, we first present log-Lipschitz distributions, a family of
distributions that will be studied throughout the text. We then introduce
martingales, which are sequences of random variables satisfying certain
properties. They can be used to give concentration bounds for random variables
which are not necessarily independent, such as bits in instances from
$\left\\{0,1\right\\}^{n}$ sampled from log-Lipschitz distributions.
#### 7.1 Log-Lipschitz Distributions
While it is natural to consider product distributions on the input space
$\left\\{0,1\right\\}^{n}$, such as the uniform distribution, independence
among the values of the bits of an input is seldom a reasonable assumption to
make in practice (e.g., two features may be correlated). By working with log-
Lipschitz distributions, we can still operate in a regime where some
distributional assumptions hold, but where the requirements are less stringent
than for product distributions. A distribution is log-Lipschitz if the
logarithm of the density function is $\log(\alpha)$-Lipschitz with respect to
the Hamming distance:
###### Definition 3.22.
A distribution $D$ on $\left\\{0,1\right\\}^{n}$ is said to be
$\alpha$-$\log$-Lipschitz if for all input points
$x,x^{\prime}\in\left\\{0,1\right\\}^{n}$, if $d_{H}(x,x^{\prime})=1$, then
$|\log(D(x))-\log(D(x^{\prime}))|\leq\log(\alpha)$.
The intuition behind $\log$-Lipschitz distributions is that points that are
close to each other must not have frequencies that greatly differ from each
other. From the definition, it is straightforward to see that if two points
$x,x^{\prime}$ differ only by one bit, then $D(x)/D(x^{\prime})\leq\alpha$.
Thus, neighbouring points in $\left\\{0,1\right\\}^{n}$ have probability
masses that differ by at most a multiplicative factor of $\alpha$. This
implies that the decay of probability mass along a chain of neighbouring
points is at most exponential. Not having sharp changes to the underlying
distribution is a very natural assumption, and weaker than many other
distributional assumptions in the literature. Again note that features are
allowed a small dependency between each other and, by construction, log-
Lipschitz distributions are supported on the whole input space. Log-Lipschitz
distributions have been studied in Awasthi et al., (2013), and their variants
in Feldman and Schulman, (2012); Koltun and Papadimitriou, (2007).
##### Examples of log-Lipschitz distributions.
The uniform distribution is $\log$-Lipschitz with parameter $\alpha=1$.
Another example of $\log$-Lipschitz distributions is the class of product
distributions where the probability of drawing a $0$ (or equivalently a $1$)
at index $i$ is in the interval
$\left[\frac{1}{1+\alpha},\frac{\alpha}{1+\alpha}\right]$. For an example
where some of the bits are not independent, let $\eta\in(1/2,1)$ and let the
input space be $\left\\{0,1\right\\}^{n}$ again. We first draw $x_{1}$
uniformly at random (u.a.r.), and then let $x_{2}$ be $x_{1}$ with probability
$\eta$ and $\bar{x_{1}}$ with probability $1-\eta$. The remaining bits are
drawn u.a.r. Then, this distributions is $\frac{\eta}{1-\eta}$-log-Lipschitz.
##### Properties.
Log-Lipschitz distributions have the following useful properties, which we
will often refer to in our proofs.
###### Lemma 3.23.
Let $D$ be an $\alpha$-$\log$-Lipschitz distribution over
$\left\\{0,1\right\\}^{n}$. Then the following hold:
1. 1.
For $b\in\\{0,1\\}$, $\frac{1}{1+\alpha}\leq\underset{{x\sim
D}}{\Pr}\left(x_{i}=b\right)\leq\frac{\alpha}{1+\alpha}$.
2. 2.
For any $S\subseteq[n]$, the marginal distribution $D_{\bar{S}}$ is
$\alpha$-$\log$-Lipschitz, where
$D_{\bar{S}}(y)=\sum_{y^{\prime}\in\\{0,1\\}^{S}}D(yy^{\prime})$.
3. 3.
For any $S\subseteq[n]$ and for any property $\pi_{S}$ that only depends on
variables $x_{S}$, the marginal with respect to $\bar{S}$ of the conditional
distribution $(D|\pi_{S})_{\bar{S}}$ is $\alpha$-$\log$-Lipschitz.
4. 4.
For any $S\subseteq[n]$ and $b_{S}\in\\{0,1\\}^{S}$, we have that
$\left(\frac{1}{1+\alpha}\right)^{|S|}\leq\underset{{x\sim
D}}{\Pr}\left(x_{i}=b\right)\leq\left(\frac{\alpha}{1+\alpha}\right)^{|S|}$.
###### Proof.
To prove (1), fix $i\in[n]$ and $b\in\\{0,1\\}$ and denote by $x^{\oplus i}$
the result of flipping the $i$-th bit of $x$. Note that
$\displaystyle\underset{{x\sim D}}{\Pr}\left(x_{i}=b\right)$
$\displaystyle=\sum_{\begin{subarray}{c}z\in\left\\{0,1\right\\}^{n}:\\\
z_{i}=b\end{subarray}}D(z)$
$\displaystyle=\sum_{\begin{subarray}{c}z\in\left\\{0,1\right\\}^{n}:\\\
z_{i}=b\end{subarray}}\frac{D(z)}{D(z^{\oplus i})}D(z^{\oplus i})$
$\displaystyle\leq\alpha\sum_{\begin{subarray}{c}z\in\left\\{0,1\right\\}^{n}:\\\
z_{i}=b\end{subarray}}D(z^{\oplus i})$ $\displaystyle=\alpha\underset{{x\sim
D}}{\Pr}\left(x_{i}\neq b\right)\enspace.$
The result follows from solving for $\underset{{x\sim
D}}{\Pr}\left(x_{i}=b\right)$.
Without loss of generality, let $\bar{S}=\\{1,\dots,k\\}$ for some $k\leq n$.
Let $x,x^{\prime}\in\\{0,1\\}^{\bar{S}}$ with $d_{H}(x,x^{\prime})=~{}1$.
To prove (2), let $D_{\bar{S}}$ be the marginal distribution. Then,
$D_{\bar{S}}(x)=\sum_{y\in\\{0,1\\}^{S}}D(xy)=\sum_{y\in\\{0,1\\}^{S}}\frac{D(xy)}{D(x^{\prime}y)}D(x^{\prime}y)\leq\alpha\sum_{y\in\\{0,1\\}^{S}}D(x^{\prime}y)=\alpha
D_{\bar{S}}(x^{\prime})\enspace.$
To prove (3), denote by $X_{\pi_{S}}$ the set of points in $\\{0,1\\}^{S}$
satisfying property $\pi_{S}$, and by $xX_{\pi_{S}}$ the set of inputs of the
form $xy$, where $y\in X_{\pi_{S}}$. By a slight abuse of notation, let
$D(X_{\pi_{S}})$ be the probability of drawing a point in
$\left\\{0,1\right\\}^{n}$ that satisfies $\pi_{S}$. Then,
$D(xX_{\pi_{S}})=\sum_{y\in X_{\pi_{S}}}D(xy)=\sum_{y\in
X_{\pi_{S}}}\frac{D(xy)}{D(x^{\prime}y)}D(x^{\prime}y)\leq\alpha\sum_{y\in
X_{\pi_{S}}}D(x^{\prime}y)=\alpha D(x^{\prime}X_{\pi_{S}})\enspace.$
We can use the above to show that
$(D|\pi_{S})_{\bar{S}}(x)=\frac{D(xX_{\pi_{S}})}{D(x^{\prime}X_{\pi_{S}})}\frac{D(x^{\prime}X_{\pi_{S}})}{D(X_{\pi_{S}})}\leq\alpha(D|\pi_{S})_{\bar{S}}(x^{\prime})\enspace.$
Finally, (4) is a corollary of (1)–(3).
∎
#### 7.2 Concentration Bounds and Martingales
Let us start with some notation and probability theory basics. A random
variable $X$ on a sample space $\Omega$, which represents the set of all
possible outcomes, is a real-valued measurable function
$X:\Omega\rightarrow\mathbb{R}$. Turning our attention to discrete random
variables, the conditional probability of $X$ given a random variable $Y$ is
defined as
$\underset{{}}{\Pr}\left(X=x\;|\;Y=y\right)=\frac{\underset{{}}{\Pr}\left(X=x\wedge
Y=y\right)}{\underset{{}}{\Pr}\left(Y=y\right)}\enspace.$
We can now use this to define the conditional expectation as
$\underset{{}}{\mathbb{E}}\left[X\;|\;Y=y\right]=\sum_{x}\underset{{}}{\Pr}\left(X=x\;|\;Y=y\right)\enspace$,
where $\underset{{}}{\Pr}\left(Y=y\right)$ is assumed to be non-zero. While
these are defined for discrete random variables, they can be extended to
continuous random variables. Moreover, note that the conditional expectation
$\underset{{}}{\mathbb{E}}\left[X\;|\;Y\right]$ is itself a random variable.
##### Useful facts.
The law of total expectation, which in full generality states that
$\underset{{}}{\mathbb{E}}\left[X\right]=\underset{{}}{\mathbb{E}}\left[\underset{{}}{\mathbb{E}}\left[X\;|\;Y\right]\right]$,
can also be formulated as
$\underset{{}}{\mathbb{E}}\left[X\right]=\sum_{y}\underset{{}}{\Pr}\left(Y=y\right)\underset{{}}{\mathbb{E}}\left[X\;|\;Y=y\right]\enspace.$
Moreover, the linearity of expectation also holds under conditioning, i.e.,
$\underset{{}}{\mathbb{E}}\left[X+Z\;|\;Y\right]=\underset{{}}{\mathbb{E}}\left[X\;|\;Y\right]+\underset{{}}{\mathbb{E}}\left[Z\;|\;Y\right]\enspace.$
Concentration inequalities and tail bounds are key tools to provide guarantees
in machine learning. Among the most commonly used and well-known bounds are
the Hoeffding inequality and the Chernoff bound, stated below.
###### Theorem 3.24 (Hoeffding, (1963)).
Let $X_{1},\dots,X_{n}$ be $n$ independent random variables such that
$X_{i}:\Omega\rightarrow[0,1]$. Denote by
$\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}$ their arithmetic mean and let
$\mu=\underset{{}}{\mathbb{E}}\left[\bar{X}\right]$. Then, for every $t\geq
0$,
$\underset{{}}{\Pr}\left(\left|\bar{X}-\mu\right|\geq t\right)\leq
2\exp\left(-2mt^{2}\right)\enspace.$ (3)
The Chernoff bound is the multiplicative form of Hoeffding’s inequality.
###### Theorem 3.25 (Chernoff, (1952)).
Let $X_{1},\dots,X_{n}$ be $n$ independent random variables such that
$X_{i}:\Omega\rightarrow\left\\{0,1\right\\}$. Denote their sum by
$\bar{X}=\sum_{i=1}^{n}X_{i}$ and let
$\mu=\underset{{}}{\mathbb{E}}\left[\bar{X}\right]$. Then, for every
$0\leq\delta\leq 1$,
$\displaystyle\underset{{}}{\Pr}\left(\bar{X}\leq(1-\delta)\mu\right)$
$\displaystyle\leq\exp\left(-\delta^{2}\mu/2\right)\enspace,$
$\displaystyle\underset{{}}{\Pr}\left(\bar{X}\geq(1+\delta)\mu\right)$
$\displaystyle\leq\exp\left(-\delta^{2}\mu/3\right)\enspace.$
Both results rely on the _independence_ of the random variables, which is not
always a reasonable assumption to make. Which tools are available to us when
independence cannot be guaranteed?
Martingales offer us the opportunity to weaken assumptions on the random
variables, which are allowed to depend on each other. To this end, we consider
a _sequence_ of random variables, where the value of a given random variable
is a function of the preceding ones. Additional requirements on their
expectation and conditional expectation are given in order to get meaningful
mathematical objects to study.
###### Definition 3.26.
A _martingale_ is a sequence of random variables $X_{0},X_{1},\dots$ of
bounded expectation, i.e.,
$\underset{{}}{\mathbb{E}}\left[\left|X_{i}\right|\right]<\infty$, for all
$i$, such that, for every $i\geq 0$,
$\underset{{}}{\mathbb{E}}\left[X_{i+1}\;|\;X_{0},\dots,X_{i}\right]=X_{i}$.
More generally, a sequence of random variables $Z_{0},Z_{1},\dots$ is a
martingale with respect to the sequence $X_{0},X_{1},\dots$ if for all $n\geq
0$
1. (i)
$Z_{n}$ is a function of $X_{0},\dots,X_{n}$,
2. (ii)
$\underset{{}}{\mathbb{E}}\left[\left|Z_{n}\right|\right]<\infty$,
3. (iii)
$\underset{{}}{\mathbb{E}}\left[Z_{n+1}\;|\;X_{0},\dots,X_{n}\right]=Z_{n}$.
When $\underset{{}}{\mathbb{E}}\left[Z_{n+1}\;|\;X_{0},\dots,X_{n}\right]\leq
Z_{n}$ the sequence is a supermartingale, and when
$\underset{{}}{\mathbb{E}}\left[Z_{n+1}\;|\;X_{0},\dots,X_{n}\right]\geq
Z_{n}$, the sequence is a submartingale.
###### Example 3.27 (Gambler’s fortune.).
Suppose a gambler plays a sequence of fair games, meaning that
$\underset{{}}{\mathbb{E}}\left[X_{i}\;|\;X_{0},\dots,X_{i-1}\right]=0$, where
$X_{i}$ is the gains (or losses) incurred at every game $i$. We are interested
in the cumulative gains $Z_{n}=\sum_{i=0}^{n}X_{i}$, the gambler’s total gains
at the end of the $n$-th game. If
$\underset{{}}{\mathbb{E}}\left[\left|X_{i}\right|\right]<\infty$ for all
games $i$, then
$\underset{{}}{\mathbb{E}}\left[\left|Z_{n}\right|\right]<\infty$ as well.
Moreover,
$\underset{{}}{\mathbb{E}}\left[Z_{n+1}\;|\;X_{0},\dots,X_{n}\right]=\underset{{}}{\mathbb{E}}\left[X_{n+1}\;|\;X_{0},\dots,X_{n}\right]+\underset{{}}{\mathbb{E}}\left[Z_{n}\;|\;X_{0},\dots,X_{n}\right]=Z_{n}\enspace,$
together implying that the sequence $Z_{0},Z_{1},\dots$. is a martingale. Note
that the assumptions are quite permissive: the gambler’s strategy can fully
depend on the history of the previous games.
Now, when bounding the difference between two consecutive random variables,
one can obtain a powerful concentration bound, known as the Azuma-Hoeffding
inequality.
###### Theorem 3.28 (Azuma-Hoeffding Inequality).
Let $X_{0},\dots,X_{n}$ be (super)martingales such that
$\left|X_{i}-X_{i+1}\right|\leq c_{i}$. Then for any $\lambda>0$:
$\displaystyle\underset{{}}{\Pr}\left(X_{n}-X_{0}\geq\lambda\right)\leq\exp\left(-\frac{\lambda^{2}}{2\sum_{i=1}^{n}c_{i}^{2}}\right)\enspace,$
$\displaystyle\underset{{}}{\Pr}\left(X_{n}-X_{0}\leq-\lambda\right)\leq\exp\left(-\frac{\lambda^{2}}{2\sum_{i=1}^{n}c_{i}^{2}}\right)\enspace.$
Note that this inequality is similar in form to the Chernoff bounds, though
the gain in generality results in a weaker bound.
As previously mentioned, martingales and the Azuma-Hoeffding inequality will
be valuable when considering log-Lipschitz distributions, where the values of
the bits in an instance are not assumed to be independent.
### 8 Fourier Analysis
In this section, we introduce basic Fourier analysis concepts for boolean
functions, i.e., functions of the form
$f:\left\\{0,1\right\\}^{n}\rightarrow\left\\{0,1\right\\}$, which comprise a
large part of the functions studied in this thesis. As previously mentioned,
it is also possible to look at functions of the form
$f:\left\\{-1,1\right\\}^{n}\rightarrow\left\\{-1,1\right\\}$. In fact, this
is what we will do in this section as it eases analyses and notation. For
various reasons, the encoding
$\varphi:\left\\{0,1\right\\}\rightarrow\left\\{-1,1\right\\}$ satisfying
$\varphi(0)=1$ and $\varphi(1)=-1$ for both the input and output spaces is
usually preferred. In general, one can also consider real-valued functions
$f:\left\\{-1,1\right\\}^{n}\rightarrow\mathbb{R}$. The type of functions for
a given theorem will be featured in the theorem statements, unless it is clear
from the context. A thorough introduction to the Fourier analysis of boolean
functions, as well as the proofs omitted in this section, can be found in the
textbook by O’Donnell, (2014).
Fourier analysis relies on considering functions’ _Fourier expansion_ : their
representation as real multilinear polynomials. We start with some notation. |
# User Attitudes to Content Moderation in Web Search
Aleksandra Urman University of ZurichSwitzerland<EMAIL_ADDRESS>, Aniko
Hannak University of ZurichSwitzerland<EMAIL_ADDRESS>and Mykola
Makhortykh University of BernSwitzerland<EMAIL_ADDRESS>
###### Abstract.
Internet users highly rely on and trust web search engines, such as Google, to
find relevant information online. However, scholars have documented numerous
biases and inaccuracies in search outputs. To improve the quality of search
results, search engines employ various content moderation practices such as
interface elements informing users about potentially dangerous websites and
algorithmic mechanisms for downgrading or removing low-quality search results.
While the reliance of the public on web search engines and their use of
moderation practices is well-established, user attitudes towards these
practices have not yet been explored in detail. To address this gap, we first
conducted an overview of content moderation practices used by search engines,
and then surveyed a representative sample of the US adult population (N=398)
to examine the levels of support for different moderation practices applied to
potentially misleading and/or potentially offensive content in web search. We
also analyzed the relationship between user characteristics and their support
for specific moderation practices. We find that the most supported practice is
informing users about potentially misleading or offensive content, and the
least supported one is the complete removal of search results. More
conservative users and users with lower levels of trust in web search results
are more likely to be against content moderation in web search.
web search, content moderation, user study, survey
††copyright: none††ccs: Human-centered computing Empirical studies in
HCI††ccs: Applied computing Law, social and behavioral sciences††ccs:
Information systems Users and interactive retrieval
## 1\. Introduction
The amount of information available online nowadays necessitates the use of
web search engines (SEs) that filter and rank information in response to user
queries. Internet users turn to SEs on a daily basis and put high trust in the
information they find through web search (Schultheiß et al., 2018; Urman and
Makhortykh, 2021). At the same time, while SEs are often perceived as
impartial mechanisms for information retrieval (Tripodi, 2022b), scholars have
documented numerous biases and inaccuracies in web search outputs over the
years (e.g., (Kay et al., 2015; Noble, 2018; Zhang et al., 2015)). Others have
highlighted the differences across SEs and their localized outputs in the
prevalence of low-quality content such as materials promoting conspiracy
theories (Urman et al., 2022) or the availability of crucial information such
as suicide helpline numbers (Scherr et al., 2022). The observed discrepancies
partially stem from the differences in the search algorithms employed by
different SEs and the availability of certain content in different languages
and can potentially in part be attributed to the ways content moderation is
implemented for individual SEs.
In scholarly research, content moderation (CM) is discussed primarily in the
context of social media but other online platforms, including SEs, also employ
it - though in a highly intransparent manner (Gillespie, 2018; Gorwa et al.,
2020; Urman and Makhortykh, 2023). The official documents of the most popular
search engines confirm this as they outline how SEs utilize the practices of
either informing users about potentially dangerous websites, downgrading low-
quality outputs or removing them altogether (Google, 2022a; Microsoft, 2022;
Yandex, 2022b, a; DuckDuckGo, 2022; Google, 2020). Importantly, generally
within this paper - for instance, when describing search companies’ moderation
practices - we understand content moderation broadly, similarly to (Gorwa et
al., 2020). That is, we discuss CM including the moderation of content that is
illegal, not just content that the search companies themselves regard as
necessary to moderate. However, our examination of user attitudes to CM does
not concern illegal content since the types of content deemed illegal and
forms of its moderation - i.e., its removal - are not determined by search
companies and are outside their control. Thus, when it comes to illegal
content, user attitudes to this specific form of moderation are less
consequential, and arguably need to be explored in relation to the users’
perceptions of relevant laws in their countries, not search companies’
policies and practices.
Content moderation on online platforms becomes an increasingly salient
political issue, at least in the Western democracies (Alizadeh et al., 2022)
since its implementation can directly affect users’ access to information and
thus socio-political processes. At the same time, user support for content
moderation is imperative for its successful implementation. For these reasons,
numerous studies have examined the determinants of support for content
moderation online. However, to date, this, to the best of our knowledge, has
been examined only in the context of social media, and despite the high
reliance of the public on SEs and the active use of moderation by search
engines, moderation practices in web search have not been systematized and
user attitudes to them have not been examined. Since SEs and social media are
distinctly different types of platforms used by different groups of users and
for different purposes, we believe that the findings on content moderation
from social media domain do not necessarily translate directly into the SE
domain. Thus, the lack of research on user perceptions of content moderation
in web search specifically constitutes a clear research gap that we aim to
address with the present study.
We use the data from a survey of a demographically representative sample of
the US adult population (N=398) to examine the levels of user support for
different content moderation practices in web search in relation to
potentially misleading and potentially offensive content. We analyze which
user characteristics and opinions such as demographics, ideology, or trust in
SEs are associated with higher/lower support for specific moderation
practices. In order to construct our survey questions in a way that covers
actual moderation practices that are currently in use by search engines, we
first systematize these practices based on the search companies’ official
documents and media statements. As such systematization has not been done
before, to the best of our knowledge, we suggest that the resulting overview
is a contribution on its own. We hope it will be helpful for other scholars
examining user interactions and information quality in web search as well as
content moderation across different types of online platforms. We present this
overview preceding the study design. We also discuss our findings juxtaposing
them against the actual CM practices of SEs and the findings on the
relationships between user characteristics and support for content moderation
previously documented by scholars in the context of social media.
In the next sections, we first outline relevant observations from the previous
work on the usage of SEs and the quality of search outputs. Then, we present
an overview of content moderation practices in web search and shortly
systematize them. This is followed by an overview of related work on user
attitudes to CM in the context of other types of online platforms such as
social media. After that, we detail specific Research Questions and Hypotheses
building on the related work and the systematization of content moderation
practices presented in the previous sections. Finally, we outline the
methodology, describe and discuss our results.
## 2\. Related work on web search usage and quality of search outputs
Individuals regularly use search engines to gather information on a variety of
topics and facilitate navigation through contemporary high-choice media
environments (Urman and Makhortykh, 2021). The fact that Google - the biggest
SE by market share - is one of the most frequented websites worldwide further
highlights how much people rely on web search in their daily lives. Further,
not only do people regularly use SEs, they also trust their outputs as much as
the information from journalistic media (Edelman, 2021). This is not a recent
phenomenon - high trust in search outputs has been consistently observed by
scholars for over a decade (Hargittai et al., 2010; Pan et al., 2007;
Schultheiß et al., 2018). Together with the increasing abundance of online
information that is almost impossible to navigate without SEs, this high trust
turns SEs into major information gate-keepers.
While trust in search outputs is high and has remained stable over time,
numerous studies showed that search results are prone to inaccuracies and
biases. For example, research has demonstrated that SE outputs exhibit
different forms of gender and/or racial bias in search results about specific
social groups (Kay et al., 2015; Noble, 2018; Ulloa et al., 2022; Urman and
Makhortykh, 2022; Metaxa et al., 2021). Recent scholarship also shows that
exposure to such stereotyped or biased representations of people via SEs can
increase people’s prejudices against the groups portrayed in a biased manner
(Vlasceanu and Amodio, 2022).
One domain where the prevalence of misleading information in web search
results is particularly concerning, and thus has attracted a lot of scholarly
attention, is public health. While the share of inaccurate or low-quality
outputs varies by specific health domain, scholars highlight that overall, the
quality of health-related outputs remains problematic (see (Zhang et al.,
2015) for a systematic literature review prior to 2015 or (Ghenai, 2017; Cuan-
Baltazar et al., 2020) for more recent evidence). In domains other than
health, recent comparative studies show that the prevalence of low-quality
information such as results promoting conspiracy theories or distorting
historical facts differs drastically by SE and the language in which the
search is performed (Makhortykh et al., 2021, 2022a; Urman et al., 2022). The
language-based differences in the quality of search results specifically on
Google are further documented by a number of other recent studies (Toepfl et
al., 2022; Arendt et al., 2020; Scherr et al., 2022, 2019). Such cross-engine
and cross-language differences can, in turn, contribute to digital divides
between users (Scherr et al., 2022). The documented differences are likely
attributed to the differences in the availability of specific sources across
languages and discrepancies in web search algorithms. However, it is possible
that some of the differences in the share of low-quality (e.g., misleading,
conspiratorial, or offensive) content have to do with the differences in the
content moderation practices of SE companies across languages and contexts.
Content moderation in web search is especially crucial given users’ high trust
in and reliance on search outputs as well as a common belief that search
engines present ”unbiased” information (Tripodi, 2022a). Relevant research
provides evidence that search results can affect individual opinions or
(perceived) knowledge (Epstein and Robertson, 2015; Vlasceanu and Amodio,
2022; Fisher et al., 2015; Xu et al., 2021; Knobloch-Westerwick et al., 2015).
Hence, low-quality content can effectively contribute to the spread of
misinformation and the propagation of harmful stereotypes, and thus arguably
needs to be moderated. On the other hand, there exists a risk of
overmoderation or the abuse of content moderation practices resulting in de-
facto censorship of certain search results as is the case in some
authoritarian regimes that have tight control over local search engines
(Makhortykh et al., 2022b). In the next section, we provide an overview of the
state of content moderation across web search engines.
## 3\. Overview and systematization of content moderation practices in web
search
SEs formally fit the criteria commonly used to define online platforms
(Gillespie, 2018): they host and organize users’ content without having
produced or commissioned that content and their infrastructure enables
organization and distribution of information, including for-profit uses of
user data (e.g., for advertising). Another common criterion used to define
platforms is: ”platforms do, and must, moderate the content and activity of
users using some logics of detection, review, and enforcement” (Gillespie,
2018).
In the case of SEs, content moderation (CM) practices can take different
forms. One of them relates to the prioritization of specific types of
information sources. Today’s search engine outputs are typically structured in
the form of vertically organized lists. This contributes to the users’
likelihood to perceive top results as more important or reliable (Pan et al.,
2007; Tripodi, 2022a), and to click on top results more often (Pan et al.,
2007; Urman and Makhortykh, 2021). There is evidence that presenting search
results in a different form - e.g., as a tabular ”grid” rather than a list, -
mitigates these tendencies and leads to users searching in a more focused
manner (Kammerer and Gerjets, 2013). Thus, the decision to organize outputs as
lists itself affects user behavior. The list-based organization of information
increases the importance of the way search results are ranked. It is not only
important which results are displayed in response to a search query, but how
\- i.e., in what order, - they are displayed. Thus, in web search results not
only removal but downgrading of certain outputs - and thus the reduction of
their visibility (Gillespie, 2022) \- is a highly viable moderation practice
that many SEs actively employ.
In contrast to the substantive volume of scholarship on social media content
moderation (Gillespie, 2020, 2022; Gerrard, 2018; Ganesh and Bright, 2020;
Riedl et al., 2022; Morrow et al., 2022; Myers West, 2018), web search content
moderation remains a rather under-studied subject. We have not been able to
find empirical studies examining the ways different moderation practices work
across SEs or the ways they are implemented. Hence, we provide some background
on web search moderation based on the information from the documentation and
statements by SEs representatives. To infer whether and how SEs moderate their
outputs, we have checked the statements made by the companies in the official
documentation and the claims coming from their official representatives -
e.g., through social media and news media comments. Our analysis here is
limited, and we provide only more general information since the detailed
examination of related documents and statements arguably merits a standalone
paper and is out of the scope of the present study.
Importantly, we focus on the general moderation practices and do not cover
anything specific to the so-called SafeSearch mode that is implemented by some
engines. Further, our overview originally corresponded to the practices
employed by SEs in the second half of 2022 - to align with the time when the
survey for our study was conducted. As such practices and policies change
overtime, we revisited this section in September 2023 when preparing the final
version of the paper, and have documented the observed changes (or lack
thereof) in the companies’ policies and practices.
We focus on the major SEs by market share in the US (Statcounter, 2022) since
our study is US-focused. Notably, the same engines are the most popular ones
in most Western countries. This includes Google, Bing, Yahoo!, DuckDuckGo,
Yandex, and Ecosia according to (Statcounter, 2022).
### 3.1. Content moderation on Google
Google has published a White Paper on the way it moderates content across its
services (Google, 2020). This includes not only web search but also other
services such as Google Maps (with a bulk of the report devoted to YouTube).
Among the actions Google takes to limit the spread of harmful or misleading
content are removals and reduction of exposure to it (e.g., through not
recommending such content). It is unclear how these are applied in web search.
It is known that ”quality” is one of the characteristics taken into account by
Google when ranking content. The operationalization of quality, however, is
ambiguous. Google employs 14000 (as of 2022 (Google, 2022d)) ”Quality Raters”
across the world that rate different aspects of web pages resurfacing in
search results, including whether these pages are potentially harmful - e.g.,
offensive or containing misinformation (see detailed guidelines and
definitions from Google as of 2022 (Google, 2022a); also see (Meisner et al.,
2022) for more details on the work of Quality Raters). At the same time, it is
ambiguous how these ratings impact the ranking of pages deemed harmful in
search results. Google simply states111The statement was originally accessed
in 2022, and was still available in the same form in September 2023. ”We work
with external Search Quality Raters to measure the quality of Search results
on an ongoing basis. Raters assess how well content fulfills a search request,
and evaluate the quality of results based on the expertise, authoritativeness,
and trustworthiness of the content. These ratings do not directly impact
ranking, but they do help us benchmark the quality of our results and make
sure these meet a high bar all around the world.” (Google, 2022c). The hidden
labor of Quality Raters is entangled with the different ideological and
economic layers of the algorithmic development (Bilić, 2016), and it remains
unclear how this affects the actual composition of search results.
Additionally, Google states that it attaches warning notes to website links
that can be potentially dangerous for the users and their computers - i.e.,
those suspected of phishing or spreading malware (Google, 2022b).
### 3.2. Content moderation on Yahoo!
On Yahoo! the implementation of content moderationis even more opaque than on
Google based on the company’s official documents. For instance, in a FAQ page
on search result removal Yahoo! states that it has no control over what is
published outside of its network (Yahoo!, 2022a). However, there is a note
that if users’ personal information is published, they can seek assistance
from Yahoo! to remove the website publishing such information from search
results (Yahoo!, 2022a). In addition, the company’s description of its
international Search Services privacy practices includes a statement that
”Users who are European residents can request that certain URLs be blocked
from search results in certain circumstances.” (Yahoo!, 2022b). The specific
circumstances however are not specified.
Based on this information, it can be implied that Yahoo! sometimes removes
search results (e.g., when it comes to illegal content or personal
information), but it is unclear how such decisions take place and whether the
search engine additionally removes or downgrades any links containing
misinformation or offensive content. We did not find any updates on this in
Yahoo!’s documentation as of September 2023.
### 3.3. Content moderation on Bing
Microsoft, the owner of Bing, as of 2022 clearly stated that it removes search
results under certain circumstances which include, for example, government
requests or requests from companies/individuals when it comes to content that
is illegal - e.g., content dealing with child abuse or copyright infringing -
or in the cases of spam (Microsoft, 2022). The company also noted that when it
removes content, it mentions this at the bottom of the search results page
(Microsoft, 2022). In 2022, we did not find information on the downranking of
search results, we did find a statement from Microsoft that in some cases
instead of removing a result, the company accompanies it with a warning to the
users - e.g., for the websites that potentially contain malware or sell
illegal pharmaceuticals (Microsoft, 2022). How exactly the decisions on the
addition of warnings or content removal are made is unclear. In September
2023, the information provided by Microsoft regarding content moderation was
slightly different than that we originally read in 2022. Specifically, the
company has now added mentions of downranking as a form of content moderation
”where the content violates local law, or Microsoft’s policies or core values”
(Microsoft, 2022). The company as of September 2023 mentions it strives for
such actions to be ”narrowly tailored” (Microsoft, 2022), however, how exactly
such decisions are made is still not clarified.
### 3.4. Content moderation on DuckDuckGo
It is unclear whether and how DuckDuckGo moderated search results up to 2022.
Several analyses in 2021 found that DuckDuckGo outputs often promote
conspiratorial content (Thompson, 2022; Urman et al., 2022). However, shortly
after Russia invaded Ukraine in February 2022 DuckDuckGo’s CEO and founder,
Gabriel Weinberg, announced that the search engine has been ”rolling out
search updates that down-rank sites associated with Russian disinformation”
(Gabriel Weinberg [@yegg], 2022). DuckDuckGo’s official webpage at the time of
writing also states that low-quality news media are downgraded in the search
results - albeit users should still be able to find the links to them as only
illegal content is completely removed (DuckDuckGo, 2022). The company also
clarifies that in their assessment of the quality of news media they ”rely on
multiple non-governmental and non-political organizations that specialize in
objectively assessing journalistic standards. To take any ranking action using
this factor, we must see at least three of these organizations independently
assess a site as having extremely low journalistic standards and also see that
none of these organizations has assessed the same site as having even somewhat
robust journalistic standards” (DuckDuckGo, 2022). It is unclear, however, in
which countries these organizations function and whether this applies only to
the US-based and/or English-speaking media or those in other languages and/or
other parts of the world. As of September 2023, the company has added an
additional explanation about its moderation processes in the section about
”common misconceptions” regarding DuckDuckGo. Specifically, DuckDuckGo, in
connection to the potential censorship of search results, states ”Our search
ranking is strictly non-political, meaning we don’t evaluate or otherwise take
into account any potential political bias or leanings of websites in our
search result rankings.” (DuckDuckGo, 2023b). Additionally, on a page devoted
to a misconception about Russian search results, the company states ”We also
do not evaluate the “truth” of any particular news story or narrative.”
(DuckDuckGo, 2023a). The latter is a notable distinction between DuckDuckGo’s
policies and that of other engines such as Google that state they provide
warnings with regard to misleading content - and thus implicitly evaluate the
”truth” of different sites and narratives.
### 3.5. Content moderation on Ecosia
We could not find information on content moderation on Ecosia in the SE’s
official documents and statements or news reports neither in 2022 nor in 2023.
### 3.6. Content moderation on Yandex
Yandex states that for certain violations of its policies, it might remove a
link from search results completely, demote it in results and/or also
accompany it with a warning to the users - e.g., that a website might be
potentially dangerous (Yandex, 2022b, a). The decision depends on the type of
policy violation with the correspondence between demotion/deletion/warning and
violation types clearly outlined (Yandex, 2022b, a). We found the same was
true as of September 2023. In a way, Yandex is more transparent than other SEs
about the content moderation practices it employs. At the same time, it is a
Russian search engine, and according to reports, it sometimes removes or
alters content in ways that favor the Russian government (Makhortykh et al.,
2022b; Lomas, 2022).
### 3.7. Summary
Overall, the content moderation policies of the most popular SEs are rather
opaque. At the same time, we can systematize the information about existing
practices and derive 3 main types of CM practices that are currently used by
the SEs:
* •
Informing users \- for instance, through adding ”warning labels” to certain
types of content such as misleading content. We found confirmations that this
is done by Google, Bing and Yandex, according to their official statements
(Yandex, 2022b, a; Microsoft, 2022; Google, 2022a).
* •
Reducing the reach of certain content \- mostly through downgrading it in
search results. This practice is explicitly mentioned by DuckDuckGo, Google
and Yandex (Yandex, 2022b, a; DuckDuckGo, 2022; Google, 2020).
* •
Removing certain content \- in certain cases, SEs remove content from search
results altogether. This practice is confirmed to be used by Google, Bing,
DuckDuckGo, Yahoo and Yandex (Google, 2020; Microsoft, 2022; Yahoo!, 2022a;
Yandex, 2022b, a; DuckDuckGo, 2022). Most often, based on what we inferred
from the cited companies’ documents and statements, removals take place in the
cases when the indexed content violates local laws.
In addition, we observe that at least according to the companies’ official
statements, SEs currently focus on moderating two main types of content:
illegal content and misleading content. This is in contrast to other platforms
(e.g., social media) which typically also moderate offensive content such as
hate speech (Gillespie, 2018). It is unclear what drives the difference
between SEs and social media in this regard - the difference in the nature of
the platforms, company cultures, or perceived user expectations.
The observations on the SEs’ content moderation practices outlined in this
section inform our research questions (RQs) as detailed below.
## 4\. Related work on user attitudes towards content moderation
While all online platforms including SEs moderate content in one way or
another (Gillespie, 2018), thus directly influencing information exposure and
experiences of their users, the user attitudes towards content moderation
practices so far have been explored only to a limited extent and, to the best
of our knowledge, exclusively for social media platforms. There is thus a
clear research gap with regard to the user attitudes towards content
moderation practices in web search that we aim to address. Before outlining
concrete RQs and hypotheses in the next section, we first present a summary of
the findings on attitudes towards content moderation on social media as they
inform our research.
A 2019 survey by YouGov showed that around 45% of respondents in the US
support the idea of CM by social media in general (Ballard, 2019). The same
survey however also demonstrated the drastic differences in the attitudes to
content moderation between liberals and conservatives with the former being
more likely to support content moderation than the latter (Ballard, 2019). A
similar observation was made in a different study from 2022 (Kozyreva et al.,
2022). Another study conducted in the US did not find a relationship between
political partisanship and support for CM but found that age and level of
education are significantly related to CM support with older and higher-
educated users more likely to be in favor of it (Riedl et al., 2022). Yet
another analysis conducted in the US has shown that there is bipartisan
support for labeling certain content (i.e., informing users) as a form of
content moderation (Wihbey et al., 2021). In one study, sex and race of the
respondents were not associated with attitudes towards CM (Riedl et al.,
2022). At the same time, a survey among the US youth found that young women
were more likely than young men to support CM (Schoenebeck et al., 2021).
Other analyses on the topic - some of which relied on in-depth interviews
rather than surveys - have concluded that opposition to content moderation
often is connected to the users’ low trust in the companies’ ability to make
fair and transparent moderation decisions and/or beliefs that CM processes and
outcomes are biased in a certain way (e.g., affected by political or business
interests) (Jhaver et al., 2019; Duffy and Meisner, 2022; Myers West, 2018;
Saltz et al., 2021). Another factor that previous research has found to be
associated with lower/higher support for content moderation and specific
moderation decisions in relation to offensive content specifically is the
exact wording used in a social media post that is to be moderated (Pradel et
al., 2022).
Additionally, researchers have found that users’ attitudes to content
moderation differ, depending on who - or what, in the case of algorithms -
makes a decision to moderate certain content. For instance, an experimental
study of Facebook users found that the participants perceived moderation
decisions taken by expert panels as more legitimate than those taken by the
algorithms or juries (Pan et al., 2022). Further, (Ozanne et al., 2022)
established that social media users have less trust in moderation decisions
that are coming from AI, as compared to when the moderation decision is taken
by a human or when the moderation source is ambiguous. A similar observation
was described by (Calleberg, 2021). In addition, experimental research has
shown that users’ perceptions of fairness and accountability in the context of
CM decisions taken by the algorithms are not influenced by the presence of the
right to appeal, regardless of the appeal formats tested by the researchers
(Vaccaro et al., 2020). At the same time, users’ levels of trust in moderation
decisions taken by the algorithms vs humans are related to their ideological
orientation - e.g., researchers established that conservatives in the US are
more likely to trust moderation decisions when they are taken by AI rather
than humans (Molina and Sundar, 2022), once again highlighting the relation of
ideology to the users’ attitudes towards content moderation.
As this overview demonstrates, there is a lot of conflicting evidence
regarding user attitudes toward content moderation in the context of social
media platforms. Despite the apparent contradictions, however, several
patterns emerge: user demographics, political opinions, and trust in the
platforms tend to be associated with the users’ support for CM (on social
media). We rely on these findings in formulating our research questions and
hypotheses.
## 5\. Research questions and hypotheses
In the previous sections, we have shown that 1) SEs are highly trusted and
relied on by the users for retrieving correct and ”unbiased” information, yet
there is consistent evidence of biases and inaccuracies being present in web
search results and varying across SEs; 2) SEs engage in diverse forms of
content moderation - informing users, reducing the reach of content or
removing content - in relation to illegal or false content, but not to
offensive content despite it being moderated by other types of online
platforms; 3) there is no evidence regarding user preferences on CM in web
search, but research about user attitudes towards CM on other platforms shows
that these attitudes are influenced by demographic characteristics, political
opinions and trust in platforms. Based on this, we formulate specific RQs and
hypotheses to address the existing research gap with regard to user attitudes
to content moderation in web search.
For the first RQ, we aim to examine general user attitudes towards different
forms of content moderation in web search. Here and in other RQs we focus on
two specific types of content that might be subject to moderation:
misleading/false content and potentially offensive content. This is informed
by the fact that these two types of content are currently moderated by online
platforms such as social media (in addition to content that is explicitly
illegal) but only one of them (i.e., false content) seems to be moderated by
SEs. Answering our RQs will allow us to establish whether this divergence in
moderation practices corresponds to the user expectations.
While we formulate the RQs below in general terms, we in fact examine user
preferences for CM and their relation to user demographics and opinions with a
breakdown of user preferences for three distinct practices employed by SEs as
identified in the previous sections: informing users; reducing the reach of
content; removing content. Hence, we examine user preferences separately for
each of these practices of moderating misleading or offensive content.
Importantly, we note that we interpret the reduction of the reach of specific
content here only as a moderation practice. The reach of certain content would
always be reduced (or, conversely, amplified) by search engines as they rank
search outputs. However, we do not interpret the reduction of reach of some
content in this case as moderation. We treat the reduction of reach as a form
of moderation (Gillespie, 2022) when a company specifically configures its
algorithm to downrank certain sites in search output due to the nature of the
content there, as compared to other websites that do not contain the content
of that type (e.g., offensive or misleading).
The RQs are formulated as follows:
* •
RQ1: What are users’ preferences on content moderation in web search?
This RQ is divided into two sub-RQs corresponding to two different types of
content: false/misleading content and content which some users might find
offensive.
* –
RQ1a: What are users’ preferences on content moderation in web search in
relation to misleading or false content?
* –
RQ1b: What are users’ preferences on content moderation in web search in
relation to potentially offensive content?
In RQs 2 and 3 we go beyond the descriptive analysis of CM preferences and
evaluate how these preferences relate to different user characteristics.
Within RQ2 we focus on misleading/false content; within RQ3 we focus on
potentially offensive content.
* •
RQ2: How do user preferences for the moderation of misleading or false content
in web search relate to user characteristics?
* •
RQ3: How do user preferences for the moderation of potentially offensive
content in web search relate to user characteristics?
The two RQs are divided into sub-questions focused on specific user
characteristics. Specific characteristics we choose to examine as being
potentially relevant for CM preferences are informed by the prior research on
user support for CM on other types of platforms and include user demographics
(age, sex, race, level of education), political leaning (on the left-right
spectrum), trust in the platforms and the perceived independence of the
platforms. Additionally, motivated by the findings that the prevalence of
biased and/or false information differs drastically across SEs, we also
examine how the use of specific SEs is related to CM support. Since the
examined characteristics and opinions are the same for both RQ2 and RQ3, we
list dedicated sub-RQs only once (e.g., as RQ2/3a, RQ2/3b, etc).
* •
RQ2/3a: How do user preferences for content moderation in web search relate to
users’ demographic characteristics (age, sex, race, level of education)?
* •
RQ2/3b: How do user preferences for content moderation in web search relate to
users’ political (left-right) leaning?
* •
RQ2/3c: How do user preferences for content moderation in web search relate to
users’ trust in web search?
* •
RQ2/3d: How do user preferences for content moderation in web search relate to
users’ frequency of use of specific search engines?
* •
RQ2/3e: How do user preferences for content moderation in web search relate to
users’ assessments of web search platforms’ independence from undue political
and business interests?
As findings on the relationship between users’ demographic characteristics or
political opinions and support for CM on other platforms are contradictory, we
do not formulate hypotheses in relation to this relationship and rather aim to
explore the potential relationships in the context of web search. However,
since previous research consistently shows that trust in platforms is related
to the users’ likelihood to support platforms’ CM practices, while a belief
that CM practices are biased due to political or business interests is related
to lower support for CM, we hypothesize that the same effects will be present
in the context of web search and formulate the following hypotheses connected
to RQ2/3c and RQ2/3e:
* •
H1: Users with higher levels of trust in SEs will be more likely to be in
favor of CM in web search.
* •
H2: Users with higher levels of confidence in SE’s independence will be more
likely to be in favor of CM in web search.
## 6\. Methodology
To address the research questions outlined above, we conducted a survey of a
representative (in terms of age, sex, ethnicity) sample (N=398) of the US
adult population, administered through Qualtrics and recruited through
Prolific using the platform’s representative sampling functionality (see
(Prolific, 2022)). We chose to focus on the US as it is a democratic country
with a high internet penetration rate; further, most of the research on CM-
related attitudes on other types of platforms (social media) so far focused on
the US (e.g., (Kozyreva et al., 2022; Riedl et al., 2022; Ballard, 2019)),
thus conducting analysis in the US enables us to connect our findings to those
from other platforms. All responses were collected on August 22, 2022. In our
sample, 50.2% of respondents were female; mean age = 45.87, median = 46;
13.57% of respondents were 18-25 years old (y.o.), 18.84% - 26-35 y.o.; 32.91%
- 36-55 y.o.; 21.61% - 56-65 y.o.; 13.07% - 65+ y.o.; 76.13% self-reported to
be White, 12.81% Black, 5.79% Asian, 2.51% Mixed, 2.76% Other.
### 6.1. RQ1
To measure user attitudes towards different web search content moderation
practices and thus address RQ1, we have adapted survey items used in (Atreja
et al., 2022) in the context of social media. For the content moderation
practices in relation to misleading information, we used the following
question:
”Some websites on the internet contain misleading content. When it comes to
displaying links to such sites, search engines can take one of the following
actions:
* •
Inform users. For example, by showing a “misleading” icon next to the link to
a misleading site in web search results.
* •
Reduce the audience that can see links to misleading websites without removing
them. For example, by showing the link only on the second or third page of
search results but not on the first page.
* •
Remove links to misleading websites from search results.
How much do you personally support or oppose taking any of these actions when
it comes to websites with misleading content?”
Then, the respondents were presented with a response matrix where they could
mark their level of support for each of the measures on a 7-point Likert scale
(see example in Fig. 1).
Figure 1. Survey response matrix for survey item on content moderation of
misleading content.
To measure the participants’ attitudes to content moderation of potentially
offensive content, we used a similarly formulated question followed by a
response matrix similar to that in Fig.1. The difference here was that instead
of the term ”misleading content” in this case we used ”content that some users
can find offensive or disturbing”.
The responses to the questions on CM practices were used to calculate
descriptive statistics necessary to answer RQ1a, RQ1b. In addition, to
establish whether the discrepancies in the levels of support towards different
types of content moderation observed through descriptive analysis are
statistically significant, we performed a Kruskall-Wallis rank sum test
followed by pairwise comparisons using Wilcoxon signed rank test with
Bonferroni adjustment to control for multiple comparisons (core team, 2023).
We opted for these tests instead of, e.g., MANCOVA, as our variables are
ordinal in nature and not normally distributed. Thus, MANCOVA assumptions
would have been violated (French et al., 2008), and the chosen tests are
suitable for our data.
### 6.2. RQs 2, 3, Hypotheses 1, 2
To answer RQ2 and RQ3 with all the corresponding sub-RQs as well as test
hypotheses H1 and H2, we used regression analysis. Specifically, we ran
ordinal logistic regression models using the 6 variables on the attitudes to
CM practices (3 for each practice in relation to misleading content and 3 for
each practice for potentially offensive content as described in relation to
RQ1) as dependent variables. The independent variables included in the models
correspond to specific sub-RQs 2/3 and H1, H2.
We chose ordinal logistic regression as the most appropriate model for the
discrete ordinal dependent variables such as the Likert-scale survey responses
as in the case of the present study. It has to be noted, however, that recent
research suggests models such as GLM (generalized linear model) can be used
with Likert-scale data as well (Harpe, 2015). The benefit of using GLM
compared to ordinal logistic regression would be in the fact that it is easier
to interpret. However, we opted for ordinal logistic regression as model
diagnostics showed in our case several assumptions for GLM (specifically,
linearity, homoskedasticity, and normality) were not met. Hence, the use of
the GLM would have been inappropriate in this case. Ordinal logistic
regression is not constrained by these assumptions. Instead, the assumptions
for it include the absence of multicollinearity and proportional odds. We
tested if the no multicollinearity assumption is met using VIF scores
(Thompson et al., 2017). The goodness of fit of the models was assessed using
an ordinal version of the Hosmer-Lemeshow test and the Lipsitz test (Fagerland
and Hosmer, 2017). The proportional odds assumption for each model was first
tested using Brant’s test (Brant, 1990). However, this test is highly
anticonservative - meaning that often the statistical significance of the test
does not correspond to practical significance, especially when the number of
predictors is high, sample size is large, or at least one continuous variable
is used as a predictor (Allison, 1999; Peterson and Harrell, 1990; Kim, 2003;
Das and Rahman, 2011). Hence, in line with other research employing the
methodology (Kim, 2003; Das and Rahman, 2011), we have also used the graphical
method to assess the practical significance of the assumption violation when
Brant’s test indicated statistical violation of the assumption. We discuss the
models when this was the case and the implications for the interpretations of
our findings at the end of this subsection.
#### 6.2.1. RQ2/3a: demographic characteristics
In correspondence with RQ2/3a, we included the following independent variables
on the demographic characteristics of the respondents: age (measured in
numbers, data from the metadata on respondents collected and provided to us by
Prolific), sex (binary category female/male222In addition to including a
binary sex independent variable, we also included a non-binary gender variable
(woman/man/non-binary). In the main text of the paper, we discuss only the
models with sex as an independent variable - those allow us to contextualize
our findings against those about CM attitudes on other platforms as those
studies included sex, not gender, as an independent variable. However, we
reran all our models using a non-binary gender variable instead of the binary
sex variable. The models with gender are included in the Appendix, and the
analysis shows that all our observations hold in those models as well., data
collected and provided by Prolific), race (data collected and provided by
Prolific; for the regression analysis we recoded the variable to a binary
(White/non-White) variable), level of education (data collected and provided
by Prolific).
#### 6.2.2. RQ2/3b: political leaning
To measure the respondents’ political leaning and include it as an independent
variable in the model, we have used the following survey item adapted from
(Kroh, 2007): ”Political views are often seen as a spectrum between extremely
liberal (left) to extremely conservative (right). Where would you place
yourself on this scale where 0 means extremely liberal and 10 means extremely
conservative?”
#### 6.2.3. RQ2/3c, H1: trust in web search
To measure the respondents’ trust in web search and include it as an
independent variable in the model to answer RQ2/3c and test H1, we used a
composite measure of trust in search outputs adapted from (Strömbäck et al.,
2020). The measure was constructed based on the survey items formulated as
follows:
”Generally speaking, to what extent do you agree or disagree with the
following statements about the information you find in web search engine
results?
* •
The selection of information I find in web search results tends to be fair and
neutral
* •
The information I find in web search results tends to be accurate
* •
The information I find in web search results tends to be relevant for me”
The respondents could select their level of agreement with each of the
statements on a 7-point Likert scale. Then, to construct the measure of the
overall level of trust in web search, we calculated the mean of the responses
to the three items (Cronbach’s alpha = 0.789 indicating good item
reliability).
#### 6.2.4. RQ2/3d: search engine use frequency
To measure the frequency of use of specific search engines, we asked the
respondents how often they use each of the search engines that are the most
popular in the US (Statcounter, 2022): Google, Bing, Yahoo, Yandex,
DuckDuckGo, Ecosia. The exact question was formulated as follows: ”How often
do you use each of the following search engines?” The responses were measured
on a 7-point Likert scale.
#### 6.2.5. RQ2/3e, H2: search engines’ independence
To measure the degree to which the participants believe that search engines
are independent of political or government influence, we have constructed a
composite variable based on the mean of the participants’ level of agreement
(on a 7-point Likert scale) with each of the following two statements:
* •
”Search engines are independent from undue political or government influence
most of the time
* •
Search engines are independent from undue business or commercial influence
most of the time”
This item was adapted from (Institute, 2016). Cronbach’s alpha = 0.84
indicates good item reliability.
#### 6.2.6. A note on the violation of the proportional odds assumption
As noted at the beginning of the subsection, we have relied on a combination
of Brant’s test (Brant, 1990) and the graphical method to evaluate the
practical and statistical significance of the violation of the proportional
odds assumption across our models. There was a practically significant
violation of the assumption for the Google use variable in the models where
the dependent variable related to misleading content. Specifically, the
graphical analysis indicated that more frequent use of Google is associated
with slightly lower likelihood of the users indicating that they somewhat
support/support content moderation compared to strongly supporting it, and
much lower likelihood of them indicating that they oppose content moderation
to any degree. This has to be taken into account when interpreting the
findings. In addition, in an attempt to address this limitation, we have run
partial proportional odds models (Peterson and Harrell, 1990) that allow
relaxing the proportional odds assumption for certain variables; however,
these models indicated a very poor fit; hence, we opted not to use them.
Instead, in addition to the models reported in the main text of the article,
we have also run the models where content moderation preferences for
misleading content are a dependent variable, omitting the Google use variable.
These additional models are reported in the Appendix in Table 4. Omitting
Google use variable slightly changes the results - specifically, the Trust in
SE variable has somewhat higher coefficients, indicating a stronger
relationship to the DV, especially for the Reduction of reach of misleading
information preferences, and in the case of this DV the use of DuckDuckGo
emerges as a significant predictor when Google use is omitted. Since the
results change only in minor ways, and the models with Google use omitted
indicate a goodness-of-fit similar to those with Google use included, we opted
for the inclusion of the full models with Google use included in the main text
of the article to allow for consistent interpretation of the results. But here
and below, in the results section, we emphasize the implications of the
partial violation of proportional odds assumption for our findings.
### 6.3. Ethics statement
We obtained informed consent from the survey respondents for participation in
the study and informed the respondents about the goals of the study and the
ways in which their data will be used. The full statement to which the
respondents consented is available in the Appendix. The respondents were
remunerated for participation in accordance with Prolific’s terms (as the
survey took around 15-20 minutes, we compensated the respondents with a 1/3 of
the average hourly wage as determined by Prolific). We used only anonymized
data and did not collect any personal information that would allow us or
others to infer the identities of the respondents.
## 7\. Results
### 7.1. RQ1: User preferences for different content moderation options -
descriptive analysis
In Fig. 2 we provide information on the shares of respondents supporting
specific CM practices in web search. We observe that the option to inform
users about potentially misleading or offensive content retrieved via web
search is overwhelmingly supported with 84% of respondents supporting333In
this section we combine all support options - somewhat
support/support/strongly support - to calculate overall support, same applies
for the opposing options. A more fine-grained breakdown of the responses and
corresponding share of survey participants selecting them is demonstrated in
Fig. 2. it for misleading content and 85% for offensive content. Only 10% of
respondents oppose this option for misleading content and 8% for offensive
content.
Two other CM practices - to reduce the reach of certain content or to remove
it from search results altogether - attracted less support from the
respondents. For these practices, the shares of undecided users and those who
only somewhat support/oppose the practice are higher than for informing users.
Still, 64% of respondents support reducing the reach of misleading content,
and 54% support reducing the reach of offensive content; the shares of
respondents opposing this option is 22% and 32%, respectively. 58% of survey
participants also support removing misleading results from the outputs
altogether, while 30% oppose this option. In the case of offensive content,
the removal of results seems to be a highly divisive issue - 43% support this
option, while 41% oppose it.
When it comes to the statistical significance of the observed discrepancies,
the result of the Kruskall-Wallis test provided a p¡0.00, indicating a
statistically significant difference between user preferences for different
types of CM. As shown in Table 1, the observed differences in the levels of
support for different types of CM in web search when comparing different
options pairwise are statistically significant for all pairs of options except
informing users about misleading vs potentially offensive content and removing
misleading results vs reducing the reach of potentially offensive content.
| Mis: Inform | Mis: Reduce | Mis: Remove | Off: Inform | Off: Reduce
---|---|---|---|---|---
Mis: Reduce | $<0.00$ | - | - | - | -
Mis: Remove | $<0.00$ | 0.00 | - | - | -
Off: Inform | 1.00 | $<0.00$ | $<0.00$ | - | -
Off: Reduce | $<0.00$ | 0.00 | 1.00 | $<0.00$ | -
Off: Remove | $<0.00$ | 0.00 | 0.00 | $<0.00$ | 0.00
Table 1. P-values corresponding to pairwise comparisons of user preferences
regarding different types of CM in web search (Wilcoxon signed rank test)
We discuss the implications of our findings and how our observations
correspond to the actual content moderation practices in web search in a
dedicated Discussion section.
Figure 2. Share of survey respondents supporting/opposing each CM option.
### 7.2. RQs2,3, H1,2: Predictors of support for different CM practices
In Table 2 we present the results of the regression analysis examining the
relationship between the participants’ characteristics and their support for
different CM practices. The coefficients are exponentiated, and statistically
significant coefficients are highlighted in red. The very bottom coefficients
in Table 2 refer to the intercepts for each category of the dependent variable
in the ordered logistic regression models. In ordinal regression, since the
dependent variable has multiple ordered categories, separate intercepts are
estimated for each category transition. These intercept coefficients provide
information about the relative likelihood of being in each category compared
to the reference category. They capture the inherent differences in the
baseline odds of the different response categories before considering the
effects of the predictor variables. For instance, the coefficients
corresponding to 6—7 indicate the odds of the respondents selecting option 6
on the Likert scale (”Support”) compared to option 7 (”Strongly support”).
Since these coefficients are not relevant for our RQs and analysis, we do not
interpret them below, and just keep them in the table for reference.
| Mis: Inform | Mis: Reduce | Mis: Remove | Off: Inform | Off: Reduce | Off: Remove
---|---|---|---|---|---|---
Age | $0.00$ | $-0.01$ | $0.01$ | $0.00$ | $-0.01$ | $0.00$
| $(0.01)$ | $(0.01)$ | $(0.01)$ | $(0.01)$ | $(0.01)$ | $(0.01)$
Sex (Male) | $0.08$ | $-0.22$ | $-0.15$ | $-0.02$ | ${\color[rgb]{1,0,0}-0.51^{**}}$ | $-0.24$
| $(0.21)$ | $(0.20)$ | $(0.19)$ | $(0.21)$ | $(0.19)$ | $(0.19)$
Education | $-0.06$ | $-0.00$ | $-0.04$ | $0.01$ | $0.01$ | $0.11$
| $(0.08)$ | $(0.07)$ | $(0.07)$ | $(0.08)$ | $(0.07)$ | $(0.07)$
Ethnicity (White) | $0.34$ | $-0.09$ | $-0.19$ | $0.28$ | $-0.20$ | ${\color[rgb]{1,0,0}-0.48^{*}}$
| $(0.24)$ | $(0.22)$ | $(0.22)$ | $(0.24)$ | $(0.22)$ | $(0.22)$
Trust in SE | ${\color[rgb]{1,0,0}0.54^{***}}$ | ${\color[rgb]{1,0,0}0.28^{*}}$ | $0.15$ | ${\color[rgb]{1,0,0}0.24^{*}}$ | $0.04$ | $0.10$
| $(0.13)$ | $(0.12)$ | $(0.12)$ | $(0.12)$ | $(0.11)$ | $(0.11)$
SE independence | $0.06$ | ${\color[rgb]{1,0,0}0.22^{**}}$ | ${\color[rgb]{1,0,0}0.26^{***}}$ | $0.14$ | ${\color[rgb]{1,0,0}0.31^{***}}$ | ${\color[rgb]{1,0,0}0.31^{***}}$
| $(0.08)$ | $(0.07)$ | $(0.07)$ | $(0.08)$ | $(0.07)$ | $(0.07)$
Political ideology | | | | | |
(left-right) | ${\color[rgb]{1,0,0}-0.25^{***}}$ | ${\color[rgb]{1,0,0}-0.21^{***}}$ | ${\color[rgb]{1,0,0}-0.21^{***}}$ | ${\color[rgb]{1,0,0}-0.15^{***}}$ | ${\color[rgb]{1,0,0}-0.10^{**}}$ | $-0.04$
| $(0.04)$ | $(0.04)$ | $(0.04)$ | $(0.04)$ | $(0.04)$ | $(0.04)$
Google use | ${\color[rgb]{1,0,0}0.17^{*}}$ | ${\color[rgb]{1,0,0}0.28^{**}}$ | ${\color[rgb]{1,0,0}0.17^{*}}$ | ${\color[rgb]{1,0,0}0.20^{*}}$ | $0.13$ | $0.09$
| $(0.08)$ | $(0.08)$ | $(0.08)$ | $(0.08)$ | $(0.08)$ | $(0.08)$
DDG use | $-0.06$ | $-0.09$ | ${\color[rgb]{1,0,0}-0.14^{*}}$ | $-0.08$ | ${\color[rgb]{1,0,0}-0.13^{*}}$ | ${\color[rgb]{1,0,0}-0.18^{***}}$
| $(0.06)$ | $(0.05)$ | $(0.05)$ | $(0.06)$ | $(0.05)$ | $(0.05)$
Yandex use | $-0.13$ | $0.01$ | $-0.02$ | $-0.22$ | $-0.01$ | $0.02$
| $(0.14)$ | $(0.14)$ | $(0.15)$ | $(0.14)$ | $(0.14)$ | $(0.14)$
Yahoo use | $0.01$ | $0.06$ | $0.07$ | $0.03$ | $0.12$ | ${\color[rgb]{1,0,0}0.13^{*}}$
| $(0.07)$ | $(0.07)$ | $(0.06)$ | $(0.07)$ | $(0.06)$ | $(0.06)$
Bing use | $0.10$ | $0.04$ | $0.05$ | ${\color[rgb]{1,0,0}0.12^{*}}$ | $0.04$ | $-0.03$
| $(0.06)$ | $(0.06)$ | $(0.05)$ | $(0.06)$ | $(0.05)$ | $(0.05)$
Ecosia use | $-0.16$ | $-0.22$ | $-0.07$ | $-0.18$ | $-0.05$ | $0.12$
| $(0.12)$ | $(0.13)$ | $(0.14)$ | $(0.12)$ | $(0.12)$ | $(0.13)$
1—2 | $-0.86$ | $-0.43$ | $-0.51$ | $-1.57$ | $-1.46$ | $0.20$
| $(0.91)$ | $(0.85)$ | $(0.84)$ | $(0.91)$ | $(0.83)$ | $(0.82)$
2—3 | $-0.14$ | $0.33$ | $0.29$ | $-0.78$ | $-0.56$ | $1.00$
| $(0.90)$ | $(0.85)$ | $(0.84)$ | $(0.89)$ | $(0.82)$ | $(0.82)$
3—4 | $0.49$ | $0.93$ | $0.79$ | $-0.46$ | $0.04$ | $1.55$
| $(0.90)$ | $(0.85)$ | $(0.84)$ | $(0.88)$ | $(0.82)$ | $(0.83)$
4—5 | $1.24$ | ${\color[rgb]{1,0,0}1.78^{*}}$ | $1.37$ | $0.46$ | $0.78$ | ${\color[rgb]{1,0,0}2.30^{**}}$
| $(0.90)$ | $(0.85)$ | $(0.84)$ | $(0.88)$ | $(0.82)$ | $(0.83)$
5—6 | ${\color[rgb]{1,0,0}1.99^{*}}$ | ${\color[rgb]{1,0,0}2.52^{**}}$ | ${\color[rgb]{1,0,0}2.00^{*}}$ | $1.08$ | $1.42$ | ${\color[rgb]{1,0,0}2.78^{***}}$
| $(0.90)$ | $(0.85)$ | $(0.84)$ | $(0.88)$ | $(0.82)$ | $(0.83)$
6—7 | ${\color[rgb]{1,0,0}3.22^{***}}$ | ${\color[rgb]{1,0,0}3.44^{***}}$ | ${\color[rgb]{1,0,0}2.74^{**}}$ | ${\color[rgb]{1,0,0}2.37^{**}}$ | ${\color[rgb]{1,0,0}2.51^{**}}$ | ${\color[rgb]{1,0,0}3.48^{***}}$
| $(0.90)$ | $(0.85)$ | $(0.85)$ | $(0.89)$ | $(0.83)$ | $(0.84)$
AIC | $1042.80$ | $1341.55$ | $1385.60$ | $1047.01$ | $1440.70$ | $1443.81$
BIC | $1118.11$ | $1416.86$ | $1460.91$ | $1122.32$ | $1516.00$ | $1519.12$
Log Likelihood | $-502.40$ | $-651.78$ | $-673.80$ | $-504.50$ | $-701.35$ | $-702.90$
Deviance | $1004.80$ | $1303.55$ | $1347.60$ | $1009.01$ | $1402.70$ | $1405.81$
Num. obs. | $389$ | $389$ | $389$ | $389$ | $389$ | $389$
${}^{***}p<0.001$; ${}^{**}p<0.01$; ${}^{*}p<0.05$
Table 2. Outputs of regression models on the association between respondents’
characteristics and their level of support for different CM practices for
misleading (Mis) and Offensive (Off) content. Statistically significant
coefficients are highlighted in red.
#### 7.2.1. RQs2/3a: support for CM and users’ demographic characteristics
We find no significant relationships between the age and level of education of
the respondents and their levels of support for any CM practice in the context
of web search. However, in a few cases, we observe a significant relationship
between the respondents’ sex and race and their support for CM for offensive
content. Specifically, we find that male users are significantly more likely
to oppose reducing the reach of potentially offensive content as compared to
female users. Besides, White respondents are significantly more likely to
oppose the removal of potentially offensive web search results than non-White
ones.
#### 7.2.2. RQs2/3b: support for CM and users’ political orientation
We observe that respondents’ political leaning is associated with their
support for CM practices in all the examined cases with the exception of the
removal of potentially offensive content. Similarly to the earlier
observations in the context of social media (Kozyreva et al., 2022; Ballard,
2019), we find that more conservative users are less likely to support CM
measures. This effect was stronger for misleading information than for
offensive content.
#### 7.2.3. RQs2/3c, H1: support for CM and trust in web search results
Based on earlier research about the relationship between trust in platforms
and support for CM practices on these platforms, we hypothesized that the same
relationship would be observed in the context of web search (H1). This
hypothesis was only partially confirmed. Specifically, we find that trust is
associated with increased support for informing users about both misleading
and potentially offensive content with the effect being stronger for
misleading content. Additionally, trust in web search results is related to
the increased support for reducing the reach of misleading content; notably,
this relationship is somewhat stronger in the models in which Google use is
omitted (see Methodology and Table 4). However, there was no association
between trust in web search and support for removing results or reducing the
reach of potentially offensive content.
#### 7.2.4. RQs2/3d: support for CM and usage of specific search engines
Figure 3. Distribution of the shares of respondents who report different
frequencies of use of specific search engines.
We observe multiple statistically significant associations between the use of
specific web search engines and support for CM practices. It is worth noting,
however, that the frequency of use of different SEs is drastically as one
might expect based on the information about their respective market shares
(Statcounter, 2022). We present the distribution of the frequencies of SEs’
use in Fig.3. Unsurprisingly, Google is the most used SE with almost all
participants reporting using it at least a couple of times a year, and more
than two-thirds stating they use it on a daily basis. Google is followed by
Yahoo and Bing which are used at least once a year by around 50% of the users,
then comes DuckDuckGo with ca. 40% respondents using it at least once a year.
Ecosia and Yandex are used only by a small share of respondents.
We find that Google use frequency is positively associated with support for
most CM practices with the exception of the reduction of reach and removal of
potentially offensive content. However, as noted in the methodology, it is
necessary to interpret the coefficients with caution in this case due to the
violation of the proportional odds assumption in the case of misleading
content-related dependent variables. Specifically, our analysis during the
model diagnostics stage revealed that more frequent use of Google is
associated with slightly lower likelihood of the users somewhat
supporting/supporting content moderation compared to strongly supporting it,
and much lower likelihood of them opposing content moderation to any degree.
Bing use frequency is positively related to the support for informing users
about offensive content while Yahoo use frequency is associated with increased
support for the removal of potentially offensive content. On the contrary,
more frequent DuckDuckGo users are significantly less likely to support
certain content moderation policies, specifically the removal of misleading or
offensive content and the reduction of reach of the offensive content; when
the use of Google variable is removed from the model, this relationship is
also significant for the reduction of the reach of misleading content, see
Table 4. We find it important to highlight that all these observations emerge
even when controlling for user demographics and political views, and discuss
their implications in a dedicated section below.
#### 7.2.5. RQs2/3e, H2: support for CM and belief in the independence of SEs
Our hypothesis (H2) that users’ beliefs in the independence of search engines
from political or business interests are associated with increased support for
CM is confirmed in the case of the removal or reduction of reach of both
misleading and offensive content. At the same time, there is no significant
relationship between support for informing users about such content and belief
in SE independence.
## 8\. Discussion
Our observations show that there is a lot of divergence in the levels of the
US adult respondents’ support for different CM practices in web search for
misleading or offensive content, with some of the differences explained by
user characteristics, in particular political attitudes, frequency of SE use
and trust in SE.
### 8.1. User attitudes to CM and actual SE moderation practices
One CM practice that seems to be largely uncontroversial - as it is supported
by an overwhelming majority of respondents - is informing users. Further,
there is no statistically significant difference between the levels of user
support for informing about misleading vs potentially offensive content.
Currently, of the six most popular SEs, only Bing, Yandex and Google,
according to their official statements and documents, inform users of some
potentially problematic content (e.g., through dedicated warning labels),
including misleading content. However, to the best of our knowledge, no such
labels are attached to offensive content by any of the most popular SEs. This
seems to be in clear contradiction with the US respondents’ attitudes to CM:
our analysis shows that informing users in the context of offensive content is
supported even by slightly more respondents than informing users about
misleading content (77% vs 74% of respondents) - albeit the difference is not
statistically significant. Notably, respondents who use Bing and Google more
frequently are more likely to support informing web search users about
offensive content, suggesting that the implementation of such measures might
be especially desired by the users of these two SEs.
We observe another apparent contradiction between the level of support for CM
practices in web search among our respondents and SE’s actual practices. All
aforementioned SEs except Ecosia remove search results altogether in certain
cases - albeit mostly when it comes to explicitly illegal content - however,
this practice is the one least supported by the respondents. While it still
receives the support of a considerable share of respondents, the practice of
removal is the only one to be opposed by over 10% of users. Our analysis also
reveals that the support for the complete removal of search results is
significantly lower among more frequent users of DuckDuckGo for both
misleading and offensive content. DuckDuckGo - at least based on the official
statements and media reports we found - completely removes the results only
when it comes to explicitly illegal content (DuckDuckGo, 2022). Thus, the
engine’s policies seem to be largely in alignment with the preferences of its
more frequent users. The same applies in the case of downranking certain
content: DuckDuckGo acknowledges the downgrading of ”low quality” news
websites (DuckDuckGo, 2022) but does not mention downgrading offensive
content, and explicitly states it does not downrank content based on its
”truthfulness” (DuckDuckGo, 2023a). Its more frequent users are at the same
time significantly more likely to oppose downranking offensive but not
misleading content, which thus is to a degree in contradiction with the SE’s
practices.
### 8.2. Predictors of user support for CM: web search vs social media
Previous research has examined support for CM practices in the context of
social media. We suggest it is worthwhile to compare our observations on CM
attitudes in the context of web search to those regarding social media as such
a comparison will reveal whether CM attitudes are similar across these
different types of platforms.
We find that more conservative users are significantly less likely to support
all forms of CM with the exception of the removal of offensive content than
more liberal users. In the context of social media, scholars have observed a
similar division in CM attitudes along ideological lines (Ballard, 2019;
Kozyreva et al., 2022) (though see (Riedl et al., 2022) that finds no
association between partisanship and support for CM in the US).
(Riedl et al., 2022) found no association between the US respondents’ race or
sex and preferences for content moderation on social media, while (Schoenebeck
et al., 2021) showed that young women are more likely to support CM than young
men. Our findings are broadly in line with both these observations. For most
CM practices in web search, there is no association with the survey
participants’ sex and race. However, we find that men are significantly less
likely than women to support the downgrading of offensive content in search
outputs while White participants are significantly less likely than non-White
respondents to support the complete removal of offensive results. We suggest
this contextual difference might stem from the fact that women and non-White
internet users encounter hate speech and other types of offensive content
directed against them more often online (Chetty and Alathur, 2018). Thus, they
might be more in favor of reducing the reach or completely removing such
content. Nonetheless, this explanation needs to be further examined and
confirmed in future work.
Our findings are in contrast to the observations of (Riedl et al., 2022) about
the association between the users’ age and education level and support for CM
on social media. We find no such association in web search. It remains to be
confirmed in future work that this is not due to the different
operationalizations of CM (as the findings of (Riedl et al., 2022) contradict
those of other scholars with regard to the relationship between social media
CM attitudes and political ideology or sex).
Similarly to the findings of other scholars regarding support for CM on social
media (Jhaver et al., 2019; Duffy and Meisner, 2022; Myers West, 2018; Saltz
et al., 2021), we observe that higher trust in search engines and belief that
they are independent from business or political influence are both
significantly associated with stronger levels of support for CM practices.
However, we find that only the reduction of reach of misleading content in
search outputs is significantly related to both trust and belief in
independence. Support for informing users about both misleading and offensive
content is significantly related only to the general trust in search outputs
while removing misleading and offensive content or reducing the reach of
offensive content is significantly related only to the belief in the
independence of SEs. One potential explanation is that even users who trust
SEs support more drastic - in terms of the impact on information availability
to search users - CM practices only if they also believe that SEs are
independent and thus can make unbiased and fair decisions. This explanation
would be in line with the observations regarding social media CM attitudes but
is yet to be tested specifically in the context of web search.
To sum up, our findings are broadly in line with those regarding the support
for CM on social media, suggesting that the mechanisms driving users’ support
and opposition to CM on online platforms are similar across different platform
types.
### 8.3. Limitations and future work
Our study is not without limitations. First, we looked only at the respondents
from the US thus our findings hold only for this context. While this allowed
us to contextualize our findings against the studies about social media CM
attitudes that were mainly conducted in the US, we believe it is necessary to
examine attitudes to CM in other contexts as well, preferably through
comparative analysis in order to draw more general and meaningful conclusions.
We suggest that a comparative analysis of CM attitudes on both social media
and SEs - and possibly other types of platforms - across national contexts
would be a particularly fruitful direction for future work. Second, we did not
present the survey respondents with specific definitions or examples of either
misleading or offensive content. We did it on purpose to gauge the
respondents’ general attitudes to CM, relying on their own perceptions of what
constitutes misleading or offensive information. However, research
demonstrates that users can have different views on whether certain content is
misleading or offensive (Ruokolainen and Widén, 2020). Our study does not
account for such differences but we suggest that it would be important to
examine how they are related to support for CM practices in the future. The
latter limitation is especially relevant in the context of the actual
implementation of CM by search engines. Even if SEs, for instance, start
informing users about misleading or offensive content - as there is broad
support for this measure as we show - defining what constitutes such content
and harmonizing this definition taking into account potentially diverging
opinions of different groups of users will be a major challenge. We suggest it
would also be important in future work not only to examine what different
users perceive as offensive or misleading but also to examine different
mechanisms that would allow for the implementation of CM in a way that is both
supported by diverse groups of users and is conducive to fostering well-
informed society. In addition, since users’ declared preferences might not
always match their actual behavior, we suggest it would be worthwhile in
future work to evaluate how users in fact perceive more or less moderated
search results. This can be done, for example, by relying on experimental
methods. Finally, in this paper, we have only explored and described users’
preferences towards content moderation in web search and examined their
predictors. Future work could additionally explore what are the most effective
moderation measures in web search and whether or not users’ preferences are in
alignment with the most effective techniques. We highlight that while user
preferences should be taken into account when designing moderation policies,
they do not necessarily always correspond to what the actual most effective
practices for moderation would be. Thus, for content moderation design it
would be inappropriate to simply reflect user preferences - rather, it would
be worthwhile to further explore them and their consequences and engage with
them critically.
## 9\. Conclusion
In this paper, we aimed to address the gap in the understanding of public
attitudes to CM practices in web search in application to potentially
misleading and potentially offensive content based on a survey of a
representative sample of the US adult population. In addition to examining the
user attitudes to different content moderation practices, we have first
conducted an overview of the actual practices employed by different search
engines and systematized them, identifying three main practices: informing the
users about certain types of content; reducing the reach of certain content;
removing certain content altogether. In terms of user attitudes towards these
practices, we find that there is broad support for informing users about
misleading/offensive content among the respondents. The attitudes towards
reducing the reach of such information through downgrading it in search
results and completely removing such information are more divided. While high
shares of respondents are in support of these practices, the support is not as
broad as for informing users. Further, over 10% of respondents strongly oppose
removing search results altogether. We also find that levels of support for
content moderation are significantly associated with the respondents’
political ideology - more conservative users are less likely to support CM
practices - and trust in web search as well as belief in the independence of
SEs - users who trust SEs more and have a stronger belief in their
independence are more likely to support CM in search. In addition, we find
that male users are less likely to support the downgrading of potentially
offensive information in search results, while White users are less likely to
support its complete removal. Our findings on the associations between user
characteristics and attitudes to content moderation in web search are broadly
in line with those previously made by scholars in the context of social media.
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* Yandex (2022b) Yandex. 2022b. Why are pages excluded from the search? https://yandex.com/support/webmaster/site-indexing/excluded-pages.html
* Zhang et al. (2015) Yan Zhang, Yalin Sun, and Bo Xie. 2015. Quality of health information for consumers on the web: A systematic review of indicators, criteria, tools, and evaluation results. _Journal of the Association for Information Science and Technology_ 66, 10 (2015), 2071–2084. https://doi.org/10.1002/asi.23311 _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/asi.23311.
## Appendix A Appendix
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### A.2. Appendix 2: Additional regression models
| Mis: Inform | Mis: Reduce | Mis: Remove | Off: Inform | Off: Reduce | Off: Remove
---|---|---|---|---|---|---
Age | $0.01$ | $-0.01$ | $0.01$ | $0.00$ | $-0.01$ | $0.00$
| $(0.01)$ | $(0.01)$ | $(0.01)$ | $(0.01)$ | $(0.01)$ | $(0.01)$
G - Non-binary | $0.58$ | $0.60$ | $0.76$ | $1.02$ | $0.40$ | $0.26$
| $(0.77)$ | $(0.65)$ | $(0.64)$ | $(0.76)$ | $(0.60)$ | $(0.63)$
G - Woman | $-0.08$ | $0.26$ | $0.18$ | $0.01$ | $0.42^{*}$ | $0.17$
| $(0.22)$ | $(0.20)$ | $(0.20)$ | $(0.21)$ | $(0.20)$ | $(0.19)$
Education | $-0.06$ | $0.00$ | $-0.04$ | $0.01$ | $0.02$ | $0.11$
| $(0.08)$ | $(0.07)$ | $(0.07)$ | $(0.08)$ | $(0.07)$ | $(0.07)$
Race (White) | $0.32$ | $-0.11$ | $-0.22$ | $0.25$ | $-0.21$ | $-0.50^{*}$
| $(0.24)$ | $(0.23)$ | $(0.22)$ | $(0.24)$ | $(0.22)$ | $(0.22)$
Trust in SE | $0.55^{***}$ | $0.29^{*}$ | $0.15$ | $0.25^{*}$ | $0.03$ | $0.10$
| $(0.13)$ | $(0.12)$ | $(0.12)$ | $(0.12)$ | $(0.11)$ | $(0.11)$
SE independence | $0.06$ | $0.22^{**}$ | $0.26^{***}$ | $0.14$ | $0.31^{***}$ | $0.31^{***}$
| $(0.08)$ | $(0.07)$ | $(0.07)$ | $(0.08)$ | $(0.07)$ | $(0.07)$
Political ideology | $-0.25^{***}$ | $-0.20^{***}$ | $-0.20^{***}$ | $-0.14^{***}$ | $-0.10^{**}$ | $-0.04$
| $(0.04)$ | $(0.04)$ | $(0.04)$ | $(0.04)$ | $(0.04)$ | $(0.04)$
Google use | $0.18^{*}$ | $0.28^{***}$ | $0.17^{*}$ | $0.20^{*}$ | $0.13$ | $0.10$
| $(0.09)$ | $(0.08)$ | $(0.08)$ | $(0.08)$ | $(0.08)$ | $(0.08)$
DDG use | $-0.06$ | $-0.09$ | $-0.14^{*}$ | $-0.08$ | $-0.14^{**}$ | $-0.19^{***}$
| $(0.06)$ | $(0.05)$ | $(0.05)$ | $(0.06)$ | $(0.05)$ | $(0.05)$
Yandex use | $-0.12$ | $0.02$ | $0.00$ | $-0.20$ | $-0.02$ | $0.02$
| $(0.14)$ | $(0.14)$ | $(0.15)$ | $(0.14)$ | $(0.14)$ | $(0.14)$
Yahoo use | $0.01$ | $0.06$ | $0.07$ | $0.03$ | $0.12$ | $0.13^{*}$
| $(0.07)$ | $(0.07)$ | $(0.06)$ | $(0.07)$ | $(0.06)$ | $(0.06)$
Bing use | $0.11$ | $0.04$ | $0.05$ | $0.13^{*}$ | $0.04$ | $-0.03$
| $(0.06)$ | $(0.06)$ | $(0.05)$ | $(0.06)$ | $(0.05)$ | $(0.05)$
Ecosia use | $-0.18$ | $-0.25$ | $-0.10$ | $-0.21$ | $-0.05$ | $0.12$
| $(0.12)$ | $(0.13)$ | $(0.14)$ | $(0.13)$ | $(0.13)$ | $(0.13)$
1—2 | $-0.83$ | $-0.06$ | $-0.19$ | $-1.36$ | $-0.96$ | $0.43$
| $(0.94)$ | $(0.87)$ | $(0.87)$ | $(0.93)$ | $(0.85)$ | $(0.85)$
2—3 | $-0.11$ | $0.70$ | $0.61$ | $-0.58$ | $-0.06$ | $1.23$
| $(0.93)$ | $(0.87)$ | $(0.87)$ | $(0.91)$ | $(0.85)$ | $(0.85)$
3—4 | $0.52$ | $1.30$ | $1.11$ | $-0.25$ | $0.53$ | $1.78^{*}$
| $(0.92)$ | $(0.87)$ | $(0.87)$ | $(0.91)$ | $(0.85)$ | $(0.85)$
4—5 | $1.27$ | $2.15^{*}$ | $1.69$ | $0.66$ | $1.27$ | $2.53^{**}$
| $(0.92)$ | $(0.88)$ | $(0.87)$ | $(0.91)$ | $(0.85)$ | $(0.86)$
5—6 | $2.01^{*}$ | $2.89^{**}$ | $2.32^{**}$ | $1.28$ | $1.91^{*}$ | $3.01^{***}$
| $(0.92)$ | $(0.88)$ | $(0.87)$ | $(0.91)$ | $(0.85)$ | $(0.86)$
6—7 | $3.24^{***}$ | $3.82^{***}$ | $3.07^{***}$ | $2.58^{**}$ | $2.99^{***}$ | $3.70^{***}$
| $(0.93)$ | $(0.88)$ | $(0.88)$ | $(0.91)$ | $(0.86)$ | $(0.87)$
AIC | $1044.04$ | $1342.68$ | $1386.32$ | $1046.94$ | $1444.95$ | $1446.63$
BIC | $1123.31$ | $1421.95$ | $1465.59$ | $1126.22$ | $1524.22$ | $1525.91$
Log Likelihood | $-502.02$ | $-651.34$ | $-673.16$ | $-503.47$ | $-702.47$ | $-703.32$
Deviance | $1004.04$ | $1302.68$ | $1346.32$ | $1006.94$ | $1404.95$ | $1406.63$
Num. obs. | $389$ | $389$ | $389$ | $389$ | $389$ | $389$
${}^{***}p<0.001$; ${}^{**}p<0.01$; ${}^{*}p<0.05$
Table 3. Additional regression models (using gender (G) rather than sex as one of the independent variables). | Mis: Inform | Mis: Reduce | Mis: Remove
---|---|---|---
Age | $0.00$ | $-0.01$ | $0.01$
| $(0.01)$ | $(0.01)$ | $(0.01)$
Sex (Male) | $0.04$ | $-0.23$ | $-0.16$
| $(0.21)$ | $(0.19)$ | $(0.19)$
Education | $-0.06$ | $0.01$ | $-0.04$
| $(0.08)$ | $(0.07)$ | $(0.07)$
Ethnicity (White) | $0.40$ | $0.01$ | $-0.14$
| $(0.24)$ | $(0.22)$ | $(0.22)$
Trust in SE | $0.57^{***}$ | $0.36^{**}$ | $0.19$
| $(0.12)$ | $(0.12)$ | $(0.12)$
SE Independence | $0.08$ | $0.21^{**}$ | $0.26^{***}$
| $(0.08)$ | $(0.07)$ | $(0.07)$
Political ideology | $-0.25^{***}$ | $-0.21^{***}$ | $-0.21^{***}$
| $(0.04)$ | $(0.04)$ | $(0.04)$
DDG Use | $-0.07$ | $-0.12^{*}$ | $-0.15^{**}$
| $(0.06)$ | $(0.05)$ | $(0.05)$
Yandex Use | $-0.13$ | $0.01$ | $-0.02$
| $(0.14)$ | $(0.14)$ | $(0.15)$
Yahoo Use | $0.02$ | $0.07$ | $0.08$
| $(0.07)$ | $(0.06)$ | $(0.06)$
Bing Use | $0.10$ | $0.03$ | $0.04$
| $(0.06)$ | $(0.06)$ | $(0.05)$
Ecosia Use | $-0.15$ | $-0.20$ | $-0.06$
| $(0.12)$ | $(0.13)$ | $(0.14)$
1—2 | $-1.74^{*}$ | $-1.76^{*}$ | $-1.35$
| $(0.80)$ | $(0.74)$ | $(0.73)$
2—3 | $-1.03$ | $-1.03$ | $-0.56$
| $(0.79)$ | $(0.74)$ | $(0.73)$
3—4 | $-0.41$ | $-0.44$ | $-0.06$
| $(0.78)$ | $(0.73)$ | $(0.73)$
4—5 | $0.33$ | $0.40$ | $0.52$
| $(0.77)$ | $(0.73)$ | $(0.73)$
5—6 | $1.07$ | $1.12$ | $1.14$
| $(0.77)$ | $(0.73)$ | $(0.73)$
6—7 | $2.30^{**}$ | $2.04^{**}$ | $1.88^{**}$
| $(0.78)$ | $(0.73)$ | $(0.73)$
AIC | $1044.99$ | $1350.49$ | $1387.76$
BIC | $1116.33$ | $1421.83$ | $1459.10$
Log Likelihood | $-504.49$ | $-657.25$ | $-675.88$
Deviance | $1008.99$ | $1314.49$ | $1351.76$
Num. obs. | $389$ | $389$ | $389$
${}^{***}p<0.001$; ${}^{**}p<0.01$; ${}^{*}p<0.05$
Table 4. Statistical models with Google Use omitted
|
# On relation between generalized diffusion equations and subordination
schemes
A. Chechkin<EMAIL_ADDRESS>Institute of Physics and Astronomy,
Potsdam University, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
Akhiezer Institute for Theoretical Physics, Akademicheskaya Str. 1, 61108
Kharkow, Ukraine I.M. Sokolov<EMAIL_ADDRESS>Institut für
Physik and IRIS Adlershof, Humboldt Universität zu Berlin, Newtonstraße 15,
12489 Berlin, Germany
###### Abstract
Generalized (non-Markovian) diffusion equations with different memory kernels
and subordination schemes based on random time change in the Brownian
diffusion process are popular mathematical tools for description of a variety
of non-Fickian diffusion processes in physics, biology and earth sciences.
Some of such processes (notably, the fluid limits of continuous time random
walks) allow for either kind of description, but other ones do not. In the
present work we discuss the conditions under which a generalized diffusion
equation does correspond to a subordination scheme, and the conditions under
which a subordination scheme does possess the corresponding generalized
diffusion equation. Moreover, we discuss examples of random processes for
which only one, or both kinds of description are applicable.
## I Introduction
In his seminal paper of 1961 Robert Zwanzig has introduced a generalized non-
Markovian Fokker-Planck equation Zwanzig with a memory kernel, a GFPE in what
follows. The work was much cited, due to the fact that the approach to the
derivation of this equation based on the projection operator formalism has
found its application in the variety of problems in non-equilibrium
statistical physics Grabert . The equation itself was however hardly used,
except for obtaining Markovian approximations. The situation changed when the
generalized Fokker-Planck equations with power-law memory kernels gained
popularity in different fields. This kind of GFPEs, called fractional Fokker-
Planck equations (FFPEs) describe the continuous (long time-space) limit of
continuous time random walks (CTRW) with power-law waiting times Hilfer ;
Compte ; MeerScheff2019 , which gives a physical foundation and explains broad
applicability of FFPE. Such equations proved useful for description of
anomalous transport processes in different media. Applications range from
charge transport in amorphous semiconductors to underground water pollution
and motion of subcellular units in biology, see, e.g., the reviews
MetzlerKlafter ; MeKla-2 ; PhysToday and references therein, as well as the
Chapters in collective monographs AnoTrans ; FracDyn . Further important
generalizations involve kernels consisting of mixtures of power laws, which
correspond to distributed-order fractional derivatives CheGorSok2002 ;
ChechkinFCAA ; CheKlaSok2003 ; APPB ; Naber2004 ; SoKla2005 ; SokChe2005 ;
UmaGor2005 ; MeerScheff2005 ; MeerScheff2006 ; Langlands ; Hanyga ; MaiPag2007
; MaiPaGo2007 ; CheGorSok2008 ; Kochubei2008 ; Meer2011 ; CheSoKla2012 , or
truncated (tempered) power laws SokCheKlaTruncated ; Stanislavsky1 ; MeerGRL ;
Baeumer . Other kernels in use include combinations of power laws with Mittag-
Leffler functions and with generalized Mittag-Leffler functions (Prabhakar
derivatives) TriChe2015 ; StanWer2016 ; Trifce1 ; Trifce2 ; StanWer2018 ;
Trifce3 ; StanWer2019 .
It is well-known that the Markovian, ”normal” Fokker-Planck equation, can be
obtained from the Langevin equation, the stochastic differential equation for
a Brownian motion under the action of an external force Chandra . Similarly,
the FFPE follows from two Langevin equations, giving a parametric
representation of the time dependence of the coordinate. These equations
describe the evolution of the coordinate and of the physical time in an
internal time-like variable (operational time) Fogedby ; Baule1 ; Baule2 ;
Kleinhans ; Hofmann ; Trifce4 . Such an approach is closely related to the
concept of subordination, i.e. random time change in a random process: A
process $X(\tau(t))$ is said to be subordinated to the process $X(\tau)$ under
operational time $\tau(t)$ being a random process with non-negative increments
Feller . The FFPEs discussed above thus describe the Brownian motion, possibly
in a force field, under a random time change, i.e. a process subordinated to a
Brownian motion with or without drift Meerschaert1 ; Stanislavsky2 ; Gorenflo1
; Gorenflo2 ; Gorenflo . Subordination schemes not only deliver a method of
analytical solution of FFPE SaiZasl ; Eli ; Meerschaert4 ; Meerschaert2 or
GFPE SokSub by its integral transformation to the usual, Markovian,
counterpart, but also give the possibility of stochastic simulations of the
processes governed by GFPEs Kleinhans ; SaiUt1 ; MaiPa2003 ; Piryatinska ;
MaiPa2006 ; GoMai2007 ; Marcin1 ; Marcin2 ; Gajda ; Meerschaert3 ;
Stanislavsky3 ; Annunziato ; Marcin3 ; Marcin4 ; Stanislavsky4 ; Stanislavsky5
.
The subordination approach was also used in BNG ; Vittoria1 ; Vittoria2 in
describing, within the diffusive diffusivity model, a recently discovered but
widely spread phenomenon of Brownian yet non-Gaussian diffusion, a kind of
diffusion process in which the mean squared displacement (MSD) grows linearly
in time, like in normal, Fickian diffusion, but the probability density of the
particles’ displacements shows a (double-sided) exponential rather than
Gaussian distribution, at least at short or intermediate times Wang1 ; Wang2 ;
Chub ; Sebastian ; Cherail ; Grebenkov1 ; Grebenkov2 ; Korean ; Sandalo ;
RalfEPJB .
This broad use of generalized (not necessarily fractional) Fokker-Planck
equations on one hand, and of random processes subordinated to the Brownian
motion, on the other hand, urges us to put a question on the relation of these
two kinds of description. In other words, the following questions arise: (i)
given a GFPE (with a certain memory kernel), can one find the corresponding
subordinator or show that none exists, and (ii) given a subordination scheme,
can one find the corresponding GFPE (if any), or show that none exists. These
two questions are addressed in our paper.
Our main statements are summarized as follows. Not all valid GFPEs correspond
to subordination schemes. Not all subordination schemes possess a
corresponding GFPE. In our paper we give criteria to check, whether a
subordination scheme possesses a GFPE, and whether a particular GFPE
corresponds to a subordination scheme or not. We moreover discuss examples of
particular stochastic processes of interest by themselves, having one or
another description, or both.
## II Generalized Fokker-Planck equations and subordination schemes
Let us first present the objects of our investigation: The GFPEs of a specific
form, and the two kinds of subordination schemes as they are discussed in the
literature cited above.
### II.1 Generalized Fokker-Planck equations
In present paper we discuss equations of the form
$\frac{\partial}{\partial
t}P(\mathbf{x},t)=\hat{\Phi}\mathcal{L}P(\mathbf{x},t),$ (1)
with the linear integrodifferential operator $\hat{\Phi}$ acting on time
variable, and ${\cal L}$ is a time-independent linear operator acting on a
function of spatial variable(s) $\mathbf{x}$. We note that Eq.(1) is the most
popular, but not the most general form of such equations; in Ref. Zwanzig a
more general form was derived. This includes the possible additional
coordinate or time dependence of $\hat{\Phi}$ and ${\cal L}$, respectively. In
the case of time dependence of ${\cal L}$ or of position-dependence of
$\hat{\Phi}$ the operators may not commute. Such situations were discussed
e.g. in Refs. SokKla and Inhomogeneous , and are not a topic of present
investigation.
In the time domain the corresponding equations are always representable as a
GFPE
$\frac{\partial}{\partial
t}P(\mathbf{x},t)=\int_{0}^{t}\Phi(t-t^{\prime})\mathcal{L}P(\mathbf{x},t^{\prime})dt^{\prime}$
(2)
where the memory kernel $\Phi(t-t^{\prime})$ can be a generalized function
(contain delta-functions or derivatives thereof). Sometimes the corresponding
equations come in a form
$\frac{\partial}{\partial t}P(\mathbf{x},t)=\frac{\partial}{\partial
t}\int_{0}^{t}M_{R}(t-t^{\prime}){\cal L}P(\mathbf{x},t^{\prime})dt^{\prime},$
(3)
or
$\int_{0}^{t}M_{L}(t-t^{\prime})\frac{\partial}{\partial
t^{\prime}}P(\mathbf{x},t^{\prime})dt^{\prime}={\cal L}P(\mathbf{x},t),$ (4)
however, Eqs. (3) and (4) can be reduced to Eq.(2). Indeed, in the Laplace
domain Eq.(2) reads
$u\tilde{P}(\mathbf{x},u)-P(\mathbf{x},0)=\tilde{\Phi}(u)\mathcal{L}\tilde{P}(\mathbf{x},t)$
with
$\tilde{\Phi}(u)=\left\\{\begin{array}[]{ll}u\tilde{M}_{R}(u)&\mbox{for the
case Eq.(\ref{NMFPE1})}\\\ 1/\tilde{M}_{L}(u)&\mbox{for the case
Eq.(\ref{NMFPE2}),}\end{array}\right.$ (5)
where $\tilde{M}_{...}(u)=\int_{0}^{\infty}M_{...}(t)e^{-ut}dt$. We note that
for special cases of integral kernels corresponding to fractional or
distributed-order derivatives Eqs.(3) and (4) were called “modified” and
“normal” forms of generalized Fokker-Planck equation, respectively APPB .
Essentially, in the most cases the equation can be expressed in the either
form, with the left or the right memory kernel, but sometimes one of the forms
is preferable APPB ; SoKla2005 ; CheSoKla2012 ; Trifce4 . Finally, one can
also consider schemes with integral kernels on both sides, for which case
$\tilde{\Phi}(u)=u\frac{\tilde{M}_{R}(u)}{\tilde{M}_{L}(u)}.$
From now on we will use one-dimensional notation for $x$. Generalization to
higher spatial dimensions is straightforward. In all our examples we will
concentrate mostly on the case of free diffusion
$\mathcal{L}=D\frac{\partial^{2}}{\partial x^{2}}.$ (6)
The coefficient $D$ has a dimension of the normal diffusion coefficient,
$[D]=[\mathrm{L}^{2}/\mathrm{T}]$, the operator $\hat{\Phi}$ is therefore
dimensionless, and its integral kernel has a dimension of the inverse time. We
note however that concentration on free diffusion is not a restriction for the
generality of our approach since it is only about temporal operators and
temporal parts of the subordination procedures.
### II.2 Kernel of GFPE uniquely defines MSD in free diffusion
Here we show that the form of the memory kernel uniquely defines the mean
squared displacement (MSD) in free diffusion, and vice versa (under mild
restrictions). Let us consider the MSD in free diffusion (i.e. without
external force and in absence of external boundaries). We multiply both parts
of Eq.(1) by $x^{2}$ and integrate over $x$ to get
$\frac{d}{dt}\int_{-\infty}^{\infty}x^{2}P(x,t)dx=D\hat{\Phi}\int_{-\infty}^{\infty}x^{2}\frac{\partial^{2}}{\partial
x^{2}}P(x,t)dx.$ (7)
Integrating the right hand side of Eq.(7) by parts twice and assuming the PDF
$P(x,t)$ to vanish at infinity together with its first derivative we get for
the r.h.s.
$\int_{-\infty}^{\infty}x^{2}\frac{\partial^{2}}{\partial x^{2}}P(x,t)dx=2,$
so that the evolution of the MSD is governed by
$\frac{d}{dt}\langle x^{2}(t)\rangle=2D\hat{\Phi}1,$ (8)
with operator $\hat{\Phi}$ acting on a numeric constant. Passing to the
Laplace representation we obtain
$u\langle x^{2}(u)\rangle=2D\tilde{\Phi}(u)\frac{1}{u}$
where we assumed to start from an initial condition concentrated at the
origin, $\langle x^{2}(t=0)\rangle=0$. This uniquely defines $\tilde{\Phi}(u)$
via the MSD:
$\tilde{\Phi}(u)=\frac{1}{2D}u^{2}\langle x^{2}(u)\rangle.$ (9)
Let us consider our first example. If the MSD in free motion grows linearly in
time, $\langle x^{2}(t)\rangle=2Dt$, we have $\langle
x^{2}(u)\rangle=2D/u^{2}$ and therefore $\tilde{\Phi}(u)\equiv 1$ (a unit
operator, an integral operator with a $\delta$-functional kernel). This means
that the only GFPE leading to the linear growth of the MSD is a usual,
Fickian, diffusion equation, for which the PDF is Gaussian at all times.
Therefore, the GFPE is not a valid instrument to describe the BnG diffusion,
contrary to what is claimed in Korean . On the other hand, the subordination
schemes of Chub ; BNG ; Grebenkov1 do describe the phenomenon.
Now we can turn to the main topic of our work: which processes can and which
can not be described by the GFPEs. To this end we first discuss a specific
integral representation of the solution to a GFPE and its relation to
subordination schemes.
### II.3 Subordination schemes
Let $X(\tau)$ be a random process parametrized by a “time-like” variable
$\tau$. Let $\tau$ by itself be a random process, parametrized by the physical
time, or clock time, $t$. The random process $X(\tau)$ is called the parent
process, the random variable $\tau$ is called the operational time, and the
random process $\tau(t)$ the directing process, or subordinator. The process
$t(\tau)$ is called the leading process of the subordination scheme, see
Gorenflo for the consistent explanation of the terminology used. The process
$X(\tau(t))$ is said to be subordinated to $X(\tau)$ under the operational
time $\tau$. The properties of $X(\tau)$ and $\tau(t)$ (or alternatively,
$t(\tau)$) fully define the properties of the composite process $X(\tau(t))$.
The fact that the variable $\tau$ is time-like means that the directing
process $\tau(t)$ preserves the causality: from $t_{2}>t_{1}$ it must follow
that $\tau(t_{2})\geq\tau(t_{1})$: the directing process of a subordination
scheme is increasing at least in a weak sense, that is possesses non-negative
increments. It is moreover assumed that $\tau(0)=0$: the count of operational
time starts together with switching the physical clock.
The composite random function $X(\tau(t))$ can be defined in two ways. The
first way corresponds to an explicit definition by defining the (stochastic)
equations governing $X(\tau)$ and $\tau(t)$. The second way defines the
function parametrically, so that the equations for $X(\tau)$ and $t(\tau)$ are
given.
A process subordinated to a Brownian motion with drift is a process whose
parent process is defined by a stochastic differential equation
$\frac{d}{d\tau}x(\tau)=F(x(\tau))+\sqrt{2D}\xi(\tau)$ (10)
with white Gaussian noise $\xi(\tau)$, with $\langle\xi(\tau)\rangle=0$,
$\langle\xi(\tau_{1})\xi(\tau_{1})\rangle=\delta(\tau_{1}-\tau_{2})$, whose
strength is given by a diffusion coefficient $D$, and with deterministic drift
$F(x(\tau))$ which will be considered absent in our examples concentrating on
free diffusion.
As example of the explicit, or direct, subordination scheme we name the
minimal diffusing diffusivity model,
$\displaystyle\frac{dx(\tau)}{d\tau}$ $\displaystyle=$
$\displaystyle\sqrt{2}\xi(\tau),$ $\displaystyle\frac{d\tau}{dt}$
$\displaystyle=$ $\displaystyle D(t),$
where the random diffusion coefficient $D(t)$ is a squared Ornstein-Uhlenbeck
process BNG .
In the parametric subordination scheme the dependence of $t(\tau)$ is given,
again in a form of an SDE, or via an additional transform of its solution. The
classical Fogedby scheme Fogedby corresponds to a stochastic differential
equation
$\frac{dt}{d\tau}=\lambda(\tau)$ (11)
with $\lambda(t)$ being a one-sided Lévy noise. The clock time given by the
solution of this equation is
$t(\tau)=\int_{0}^{\tau}\lambda(\tau^{\prime})d\tau^{\prime}$, , and the PDF
$q(t,\tau)$ of the process $t(\tau)$ is given by a one-sided $\alpha$-stable
Lévy law, such that its Laplace transform in $t$ variable reads
$\tilde{q}(u,\tau)=\exp(-u^{\alpha}\tau),0<\alpha<1$. This scheme corresponds
to a diffusive limit of CTRW with a power law waiting time distribution.
The correlated CTRW model of Ref. Hofmann is another example of the
parametric scheme where $t(\tau)$ is obtained by an additional integration of
$\lambda(\tau)$ above:
$\frac{dt}{d\tau}=\int_{0}^{\tau}\Psi(\tau-\tau^{\prime})\lambda(\tau^{\prime})d\tau^{\prime}.$
(12)
The previous case is restored if the kernel $\Psi(\tau)$ is a
$\delta$-function.
The attractiveness of subordination schemes lays in the fact that if the
solution for the PDFs of the parent process at a given operational time, and
of the directing process at a given physical time are known, the PDF $P(x,t)$
of the subordinated process can be obtained simply by applying the Bayes
formula. Let $f(x,\tau)$ be the PDF of $x=X(\tau)$ for a given value of the
operational time $\tau$, and $p(\tau,t)$ the PDF of the operational time for
the given physical time $t$. Than the PDF of $x=X(t)=X(\tau(t))$ is given by
$P(x,t)=\int_{0}^{\infty}f(x,\tau)p(\tau,t)d\tau,$ (13)
which in this context is called the integral formula of subordination Feller .
The PDF $p(\tau,t)$ of the operational time at a given clock time is delivered
immediately by explicit schemes, and can be obtained for parametric
subordination schemes by using an additional transformation Baule1 ; Gorenflo
, see below. Note that the PDF $f(x,\tau)$ for a process subordinated to a
Brownian motion always satisfies a usual, Markovian Fokker-Planck equation
$\frac{\partial}{\partial\tau}f(x,\tau)={\cal L}f(x,\tau),$ (14)
with $\mathcal{L}f(x,\tau)=-\frac{\partial}{\partial
x}F(x)f(x,\tau)+D\frac{\partial^{2}}{\partial x^{2}}f(x,\tau)$ by virtue of
Eq.(10) of the subordination schemes.
## III The sufficient condition for GFPE to have a subordination scheme
In Ref. SokSub it was shown that the formal solution of the GFPE (2) can be
obtained in a form of an integral decomposition
$P(x,t)=\int_{0}^{\infty}f(x,\tau)T(\tau,t)d\tau,$ (15)
where $f(x,\tau)$ is a solution of a Markovian FPE with the same Fokker-Planck
operator ${\cal L}$, Eq.(14), and for the same initial and boundary
conditions. Here the function $T(\tau,t)$ is normalized in its first variable
and connected with the memory kernel of the GFPE, as it is discussed in this
Section below. The corresponding form of solution was obtained in SaiZasl ;
Eli for the fractional diffusion and Fokker-Planck equations, and was applied
to the fractional Kramers equation in BarSil ; its more general discussion
followed in Sok1 . Equation (15) is akin to the integral formula of
subordination, Eq.(13). However, the PDF $P(x,t)$ in Eq. (15) may or may not
correspond to a PDF of a random process subordinated to the Brownian motion
with drift (as described by the ordinary FPE), since $T(\tau,t)$ may or may
not be a conditional probability density of $\tau$ at time $t$, e.g.
$T(\tau,t)$ may get negative. In Ref. SokSub the kernels corresponding to
subordination schemes with non-negative $T(\tau,t)$ were called ”safe”, while
the kernels not corresponding to any subordination scheme, for which
$T(\tau,t)$ oscillate, were called “dangerous”. For safe kernels the non-
negativity of solutions of GFPE corresponding to non-negative initial
conditions is guaranteed by virtue of Eq.(13). The sufficient condition for
Eq. (15) to correspond to a subordination scheme will be considered later.
If one assumes that the solution of GFPE (2) can be obtained in the form of
integral decomposition (15) and then insert such form in Eq. (2), one gets the
Laplace transform of the function $\tilde{T}(\tau,t)$ in its second variable,
$\tilde{T}(\tau,u)=\int_{0}^{\infty}T(\tau,t)e^{-ut}dt$, as SokSub
$\tilde{T}(\tau,u)=\frac{1}{\tilde{\Phi}(u)}\exp\left[-\tau\frac{u}{\tilde{\Phi}(u)}\right].$
(16)
This however, does not answer the questions what are the conditions under
which such a solution holds and whether it is unique. Below we discuss these
issues in some detail by presenting an alternative derivation, i.e. explicitly
constructing the solution.
Let us start from our Eq.(2) and integrate its both parts over time getting
$\displaystyle\int_{0}^{t}\frac{\partial}{\partial
t^{\prime\prime}}P(x,t^{\prime\prime})dt^{\prime\prime}=P(x,t)-P(x,0)$
$\displaystyle\qquad=\int_{0}^{t}dt^{\prime\prime}\int_{0}^{t^{\prime\prime}}\Phi(t^{\prime\prime}-t^{\prime}){\cal
L}P(x,t^{\prime})dt^{\prime}.$
Now we exchange the sequence of integrations in $t^{\prime}$ and
$t^{\prime\prime}$ on the r.h.s.,
$\displaystyle\int_{0}^{t}dt^{\prime\prime}\int_{0}^{t^{\prime\prime}}K(t^{\prime\prime}-t^{\prime}){\cal
L}P(x,t^{\prime})dt^{\prime}$
$\displaystyle\qquad=\int_{0}^{t}dt^{\prime}{\cal
L}P(x,t^{\prime})\int_{t^{\prime}}^{t}\Phi(t^{\prime\prime}-t^{\prime})dt^{\prime\prime},$
getting the integral form
$P(x,t)-P(x,0)=\int_{0}^{t}K(t-t^{\prime}){\cal L}P(x,t^{\prime})dt^{\prime},$
(17)
with the integral kernel
$K(t)=\int_{0}^{t}\Phi(t^{\prime\prime})dt^{\prime\prime}$ whose Laplace
transform is equal to $\tilde{\Phi}(u)/u$. Using the condition $\tau(t=0)=0$
we substitute the assumed solution form, Eq.(15), into Eq. (17):
$\displaystyle\int_{0}^{\infty}f(x,\tau)T(\tau,t)d\tau-P(x,0)=$
$\displaystyle\qquad\int_{0}^{t}dt^{\prime}K(t-t^{\prime})\int_{0}^{\infty}d\tau\mathcal{L}f(x,\tau)T(\tau,t^{\prime}).$
Now we use the assumption that $f(x,\tau)$ is the solution of a Markovian
Fokker-Planck equation, and make the substitution
$\mathcal{L}f(x,\tau)=\frac{\partial}{\partial\tau}f(x,\tau)$. We get:
$\displaystyle\int_{0}^{\infty}f(x,\tau)T(\tau,t)d\tau-P(x,0)=$
$\displaystyle\qquad\int_{0}^{t}dt^{\prime}K(t-t^{\prime})\int_{0}^{\infty}d\tau\left(\frac{\partial}{\partial\tau}f(x,\tau)\right)T(\tau,t^{\prime}).$
Performing partial integration in the inner integral on the r.h.s., and
interchanging the sequence of integrations in $t^{\prime}$ and in $\tau$ in
the integral which appears in the r.h.s. we arrive at the final expression
$\displaystyle\int_{0}^{\infty}f(x,\tau)T(\tau,t)d\tau-P(x,0)=$
$\displaystyle\qquad\int_{0}^{t}K(t-t^{\prime})\left[f(x,\infty)T(\infty,t^{\prime})-f(x,0)T(0,t^{\prime})\right]dt^{\prime}$
$\displaystyle\qquad-\int_{0}^{\infty}d\tau
f(x,\tau)\frac{\partial}{\partial\tau}\int_{0}^{t}K(t-t^{\prime})T(\tau,t^{\prime})dt^{\prime}.$
Now we request that the l.h.s. and the r.h.s. are equal at any time $t$ for
all admissible functions $f(x,\tau)$ satisfying the Fokker-Planck equation
$\frac{\partial}{\partial\tau}f(x,\tau)=\mathcal{L}f(x,\tau)$ irrespective of
the particular form of the linear operator $\mathcal{L}$ and of the boundary
and initial conditions. This gives us three conditions:
$\displaystyle\int_{0}^{\infty}f(x|\tau)T(\tau,t)d\tau=$
$\displaystyle\qquad-\int_{0}^{\infty}d\tau
f(x|\tau)\frac{\partial}{\partial\tau}\int_{0}^{t}K(t-t^{\prime})T(\tau,t^{\prime}),$
$\displaystyle-P(x,0)=-f(x,0)\int_{0}^{t}K(t-t^{\prime})T(0,t^{\prime}),$
$\displaystyle 0=\int_{0}^{t}K(t-t^{\prime})f(x,\infty)T(\infty,t^{\prime}),$
which can be rewritten as conditions on $T(\tau,t)$ only:
$\displaystyle
T(\tau,t)=-\frac{\partial}{\partial\tau}\int_{0}^{t}K(t-t^{\prime})T(\tau,t^{\prime})dt^{\prime},$
(18) $\displaystyle\int_{0}^{t}K(t-t^{\prime})T(0,t^{\prime})dt^{\prime}=1,$
(19)
$\displaystyle\int_{0}^{t}K(t-t^{\prime})T(\infty,t^{\prime})dt^{\prime}=0.$
(20)
In the Laplace domain Eq.(18) turns to a simple linear ODE
$-\frac{\tilde{\Phi}(u)}{u}\frac{\partial}{\partial\tau}\tilde{T}(\tau,u)=\tilde{T}(\tau,u)$
whose general solution is
$\tilde{T}(\tau,u)=C\cdot\exp\left(-\tau\frac{u}{\tilde{\Phi}(u)}\right)$
with the integration constant $C$. This integration constant is set by the
second equation, Eq.(19), which in the Laplace domain reads
$\frac{\tilde{\Phi}(u)}{u}\tilde{T}(0,u)=\frac{1}{u},$
so that $C=1/\Phi(u)$, and therefore
$\tilde{T}(\tau,u)=\frac{1}{\tilde{\Phi}(u)}\exp\left(-\tau\frac{u}{\tilde{\Phi}(u)}\right),$
which is our Eq.(16). The function $T(\tau,t)$ is normalized in its first
variable, which follows by the direct integration of Eq.(16):
$\int_{0}^{\infty}\tilde{T}(\tau,u)d\tau=\frac{1}{u},$ (21)
so that its inverse Laplace transform to the time domain is unity:
$\int_{0}^{\infty}T(\tau,t)d\tau=1$ (22)
for any $t>0$.
The third condition, Eq.(20), is fulfilled automatically provided
$T(\infty,t^{\prime})=0$ which implies $\tilde{T}(\infty,u)=0$. This is e.g.
always the case for non-negative kernels $\Phi(t)$ (as encountered in all our
examples) whose Laplace transform $\tilde{\Phi}(u)$ is positive for all $u$.
For non-positive kernels the property has to be checked explicitly.
Let us stress again that the solution in form of Eq.(16), which, as we have
seen, is applicable for a wide range of memory kernels $\Phi$, may or may not
correspond to some subordination scheme. We note, however, the fact that the
kernel does not correspond to a subordination scheme does not devaluate the
corresponding GFPE by itself, and does not mean that this leads to negative
probability densities. As an example of a “dangerous” kernel let us consider a
simple exponential kernel, $\Phi(t)=re^{-rt}$ (where the prefactor $r$ of the
exponential is added to keep the correct dimension of $\Phi$). For example, a
generalized diffusion equation with an exponential kernel,
$\frac{\partial}{\partial
t}p(x,t)=\int_{0}^{t}re^{-r(t-t^{\prime})}D\frac{\partial^{2}}{\partial
x^{2}}p(x,t^{\prime})dt^{\prime},$ (23)
is essentially the Cattaneo equation for the diffusion with finite propagation
speed, i.e. a kind of a telegrapher’s equation, as can be seen by taking a
derivative of its both sides w.r.t. $t$:
$\frac{\partial^{2}}{\partial t^{2}}p(x,t)=r\frac{\partial}{\partial
t}p(x,t)+rD\frac{\partial^{2}}{\partial x^{2}}p(x,t).$ (24)
The solutions to Eq.(24) for non-negative initial conditions are known to be
non-negative on the whole real line, but changing the operator ${\cal L}$ from
a diffusion to a more general one (e.g. to diffusion in presence of the
constant force) may lead to oscillating solutions SokSub .
The reason for this, within our line of argumentation, is that the function
$\tilde{T}(\tau,u)$ for equation (23),
$\tilde{T}(\tau,u)=(1+ur^{-1})\exp[-\tau u(1+r^{-1}u)]$ (25)
is not a Laplace transform of a non-negative function. We remind that the
function $\tilde{\phi}(u),0\leq u\leq\infty$, is a Laplace transform of a non-
negative function $\phi(t)$ defined on the non-negative half-axis, if and only
if $\tilde{\phi}(u)$ is completely monotone, i.e. its derivatives satisfy
$(-1)^{n}\phi^{(n)}(u)\geq 0$
for $n=0,1,2,...$ Feller . On the other hand, it is easy to see that the
second derivative of $\tilde{T}(\tau,u)$ changes its sign. Moreover, using the
mean value theorem, it is not hard to show that the Laplace transform of any
non-negative function integrable to unity cannot decay for $u\to\infty$ faster
than exponentially (see Appendix A), which is not true for Eq.(25).
Summarizing the result of this Section, we see that not all GFPEs can
correspond to subordination schemes. The sufficient condition to have such
scheme is the following: The kernel $\Phi(t)$ in the GFPE (2) is such that the
function $\tilde{T}(\tau,u)$ given by Eq.(16) is completely monotone as a
function of $u$. This always corresponds to subordination scheme, for which
$T(\tau,t)$ can be interpreted as a probability density function of the
operational time $\tau$ for given physical time $t$, $T(\tau,t)\equiv
p(\tau,t)$.
## IV What subordination schemes do have a GFPE?
Let us first perform some simple manipulations while assuming that the
function $T(\tau,t)$ in the integral decomposition formula (15) does
correspond to a subordination scheme, and thus has a meaning of the PDF of
operational time $\tau$ for a given physical time $t$, $T(\tau,t)\equiv
p(\tau,t)$. Then, in the Laplace domain the function $\tilde{p}(\tau,u)$,
Eq.(16), can be represented as
$\tilde{p}(\tau,u)=-\frac{d}{d\tau}u^{-1}\exp\left[-\tau\frac{u}{\tilde{\Phi}(u)}\right],$
(26)
so that in the time domain we have
$p(\tau,t)=-\frac{d}{d\tau}\int_{0}^{t}q(t^{\prime},\tau)dt^{\prime},$
where the function $q(t,\tau)$ is given by the inverse Laplace transform of
$\tilde{q}(u,\tau)=\exp\left[-\tau\frac{u}{\tilde{\Phi}(u)}\right].$ (27)
Thus
$\int_{\tau_{0}}^{\infty}p(\tau,t_{0})d\tau=\int_{0}^{t_{0}}q(t,\tau_{0})dt.$
(28)
Now we proceed to show that, as discussed already in Baule1 ; Gorenflo , the
function $q(t,\tau)$ has a clear physical meaning: this is namely the PDF of
clock times corresponding to the given operational time $\tau$. We note here
that in spite of the fact that Eq.(28) was obtained for a specific form of the
PDF $p(\tau,t)$ given by Eq.(26), it is more general and gives the relation
between the PDFs of a (weakly) increasing process and its inverse.
Indeed, let us consider a set of monotonically non-decaying functions, either
continuous (diffusive limit) or of càdlàg type (genuine continuous time random
walks) on a $(t,\tau)$ plane, see Fig. 1. The integral on the l.h.s. of
Eq.(28) counts all functions (with their probability weights) which cross the
horizontal segment $t\in[0,t_{0}),\tau=\tau_{0}$, the integral on the r.h.s.
counts all functions crossing the semi-infinite vertical segment
$t=t_{0},\tau\in[\tau_{0},\infty)$. The set of these functions is the same:
any monotonically non-decaying function crossing the horizontal segment or
passing from its one side to another side on a jump has to cross the vertical
one. No non-decaying function which never crossed the horizontal segment can
cross the vertical one. Therefore such a monotonicity implies that
$\textrm{Prob}(\tau>\tau_{0}|t_{0})=\textrm{Prob}(t<t_{0}|\tau_{0}),$
where the probabilities are defined on the set of the corresponding
trajectories. The physical meaning of the functions
$\textrm{Prob}(\tau>\tau(t))$ and $\textrm{Prob}(t<t(\tau))$ is that they
represent the survival probability and the cumulative distribution function
for the operational time and for the clock time, respectively. In the
continuous case the PDF of a clock time given operational time is then given
by
$\displaystyle q(t,\tau)$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\textrm{Prob}(t<t(\tau))=\frac{d}{dt}\textrm{Prob}(\tau>\tau(t))$
(29) $\displaystyle=$
$\displaystyle\frac{d}{dt}\int_{\tau}^{\infty}p(\tau^{\prime},t)d\tau^{\prime}.$
This statement allows for immediate transition between direct (random variable
change $\tau(t)$) and parametric (inverse) subordination schemes. This also
gives a necessary condition for a process obeying GFPE to be a subordinated
one.
Figure 1: A schematic picture explaining the nature of Eq.(28). Black lines:
the case of continuous time traces. Here all monotonically non-decaying traces
crossing the horizontal segment $[0,\tau_{0})$ (shown in blue online) also
cross the vertical one $[\tau_{0},\infty)$ (shown in red online), and
therefore contribute equally to the r.h.s. and to the l.h.s. of Eq.(28). A
gray dashed line shows exemplary a càdlàg piecewise constant function with
jumps, the genuine operational time of a CTRW. Any càdlàg trace passing at a
jump from below to above the horizontal segment has to cross the vertical
segment during the waiting time.
To obtain such a necessary condition, let us fix two subsequent non-
intersecting operational time intervals $\tau_{1}$ and $\tau_{2}$
corresponding to the physical time intervals $t_{1}$ and $t_{2}$. Than
$t(\tau_{1}+\tau_{2})=t(\tau_{1})+t(\tau_{2})=t_{1}+t_{2}$. Than it follows
from Eq.(27) that
$\tilde{q}(u,\tau_{1}+\tau_{2})=\exp\left[-\tau_{1}\frac{u}{\tilde{\Phi}(u)}\right]\cdot\exp\left[-\tau_{2}\frac{u}{\tilde{\Phi}(u)}\right],$
or, denoting the Laplace characteristic functions of $t_{1}$ and $t_{2}$ by
$\tilde{\theta}_{t_{1}}(u)$ and $\tilde{\theta}_{t_{2}}(u)$, and the
characteristic function of their sum by $\tilde{\theta}_{t_{1}+t_{2}}(u)$ we
get
$\tilde{\theta}_{t_{1}+t_{2}}(u)=\tilde{\theta}_{t_{1}}(u)\cdot\tilde{\theta}_{t_{2}}(u),$
(30)
that is the random variables $t_{1}$ and $t_{2}$ are sub-independent Hamedani
. This property puts a necessary condition for the possibility to describe the
subordination scheme by a generalized FPE. The condition is always fulfilled
when $t_{1}$ and $t_{2}$ are independent (e.g. for a parametric Fogedby scheme
Fogedby ).
Therefore, the following statement can be made: Only subordination schemes in
which the increments of physical time $t(\tau)$ are sub-independent can be
described by GFPEs.
Combining Eqs.(27) and (30) we find the final criteria for the existence of
GFPE for a given subordination scheme.
* •
A direct subordination scheme does posses a GFPE only if the Laplace transform
of the PDF $p(\tau,t)$ in its $t$ variable has the form
$\tilde{p}(\tau,u)=\frac{f(u)}{u}\exp(-\tau f(u)).$ (31)
The kernel of the ensuing GFPE is then $\tilde{\Phi}(u)=u/f(u)$. Here it might
be easier to check that the double Laplace transform has a form
$\tilde{\tilde{p}}(s,u)=\int_{0}^{\infty}d\tau\int_{0}^{\infty}dte^{-s\tau}e^{-ut}p(\tau,t)=\frac{f(u)}{u[s+f(u)]},$
(32)
i.e. the function $F(s,u)=1/\tilde{\tilde{p}}(s,u)$ is a linear function in
$s$: $F(s,u)=a(u)s+1$. Using this criterion one can show that the BnG model of
Ref. BNG does not posses a corresponding GFPE, see Appendix 2.
* •
A parametric subordination scheme does posses a GFPE only if the
characteristic function (Laplace transform) of the PDF $q(t,\tau)$ in its $t$
variable has a form
$\tilde{q}(u,\tau)=\exp(-\tau f(u)),$ (33)
i.e. the $\tau$-dependence of this function must be simple exponential. The
$t$-variables corresponding to non-intersecting $\tau$ intervals are thus sub-
independent. The kernel of the ensuing GFPE is then $\tilde{\Phi}(u)=u/f(u)$.
Using the last criterion one can easily show that there is no GFPE
corresponding to correlated CTRW of Ref. Hofmann . The distribution of $t$ as
a function of $\tau$ in this model has a Laplace characteristic function
$\tilde{q}(u,\tau)=\exp\left[-u^{\alpha}\phi(\tau)\right]$ (34)
with
$\phi(\tau)=\int_{0}^{\tau}d\tau^{\prime}\left[\int_{\tau^{\prime}}^{\tau}d\tau^{\prime\prime}\Psi(\tau^{\prime\prime}-\tau^{\prime})\right]^{\alpha}$
where $\Psi(\tau)$ denotes the memory function for waiting times along the
trajectory expressed as a function of the number of steps, cf. Eq.(26) of Ref.
Hofmann . To correspond to any GFPE the argument of the exponential in Eq.(34)
has to be linear in $\tau$, i.e. $\phi(\tau)=a\tau$, where $a$=const. This
means that the square bracket in the expression for $\phi$ must be a constant
(equal to $a$) and therefore $\Psi(\tau)=a^{1/\alpha}\delta(\tau)$, which
corresponds to a standard, non-correlated CTRW.
## V Conclusions
Growing awareness of the complexity of non-Markovian diffusion processes in
physics, earth sciences and biology gave rise to a spark of interest to
mathematical tools capable to describe such non-standard diffusion processes
beyond the Fick’s law. Generalized diffusion equations on one hand, and
subordination schemes, on the other hand, are the two classes of such
instruments, which were successfully used for investigation of a broad variety
of anomalous diffusion processes. For several situations, notably, for
decoupled continuous time random walks, both are applicable, and stand in the
same relation to each other as the Fokker-Planck equation and the Langevin
equation do for the case of normal, Fickian diffusion. In the present work we
address the question, whether this is always the case. The answer to this
question is negative: some processes described by the generalized diffusion
equations do not possess an underlying subordination scheme, i.e. cannot be
described by a random time change in the normal diffusion process. On the
other hand, many subordination schemes do not possess the corresponding
generalized diffusion equation. The example for the first situation is the
Cattaneo equation, which can be represented as a generalized diffusion with
exponential memory kernel, for which no subordination scheme exists. The
example of the second situation is the minimal model of Brownian yet non-
Gaussian diffusion and correlated CTRW, the subordination schemes for which we
show that no corresponding generalized diffusion equation can be put down. We
discuss the conditions under which one or the other description is applicable,
i.e. what are the properties of the memory kernel of the diffusion equation
sufficient for its relation with subordination, and what are the properties of
the random time change in the subordination scheme necessary for existence of
the corresponding generalized diffusion equation.
## VI Acknowledgements
AC acknowledges support from the NAWA Project PPN/ULM/2020/1/00236.
## Appendix A Laplace transform of a non-negative function
Let us show that the Laplace transform $\widetilde{\phi}(u)$ of any non-
negative function $\phi(t)$ integrable to a constant, i.e. with $\phi(t)\geq
0$ and with $0<\int_{0}^{\infty}\phi(t)dt<\infty$, cannot decay faster than
exponentially with the Laplace variable $u$:
$\widetilde{\phi}(u)=\int_{0}^{\infty}\phi(t)e^{-ut}dt\geq Ae^{-Bu},$
with a positive constant $A$ and a non-negative constant $B$.
To see this we use the following chain of relations:
$\displaystyle\int_{0}^{\infty}\phi(t)e^{-ut}dt$ $\displaystyle\geq$
$\displaystyle\int_{0}^{C}\phi(t)e^{-ut}dt$ $\displaystyle=$ $\displaystyle
e^{-Bu}\int_{0}^{C}\phi(t)dt=Ae^{-Bu}.$
Here $C$ is some cut-off value which is chosen such that
$\int_{0}^{C}\phi(t)dt>0$, which is always possible due to our assumptions
about the integrability of $\phi(t)$ to a positive constant. The inequality
follows from the mean value theorem for integrals: by assumptions $\phi(t)$ is
non-negative and integralble, and $e^{-ut}$ is, evidently, continuous. The
value of $B$ then follows the inequality $0\leq B\leq C$, i.e. is non-
negative.
For our function $T(\tau,t)$, Eq.(25), the Laplace transform $T(\tau,u)$ is
defined for $u=0$ so that $\int_{0}^{\infty}T(\tau,t)dt=1>0$ for any $\tau$.
Eq.(25) essentially corresponds to a Laplace transform of a function strongly
oscillating at small $t$.
## Appendix B The minimal model of BNG diffusion
In the BnG model of Ref. BNG the PDF $p(\tau,t)$ (denoted there as
$T(\tau,t))$ is defined via its Laplace transform in $\tau$ variable,
$\displaystyle\tilde{p}(s,t)=$
$\displaystyle\frac{e^{t/2}}{\sqrt{\frac{1}{2}\left(\sqrt{1+2s}+\frac{1}{\sqrt{1+2s}}\right)\mathrm{sinh}(t\sqrt{1+2s})+\mathrm{cosh}(t\sqrt{1+2s})}}.$
Let us take the double Laplace transform
$\tilde{\tilde{p}}(s,u)=\int_{0}^{\infty}\tilde{p}(s,t)e^{-ut}dt$ (35)
and check whether it has the form of Eq.(32).
We first denote $\alpha=\sqrt{1+2s}>1$ and rewrite the function
$\tilde{p}(s,t)$ as
$\displaystyle\tilde{p}(s,t)$ $\displaystyle=$
$\displaystyle\frac{e^{t/2}}{\sqrt{\frac{1}{2}\left(\alpha+\frac{1}{\alpha}\right)\mathrm{sinh}(\alpha
t)+\mathrm{cosh}(\alpha t)}}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{\alpha}e^{-\frac{\alpha-1}{2}t}}{(\alpha+1)^{2}}\left[1+\left(\frac{\alpha-1}{\alpha+1}\right)^{2}e^{-2\alpha
t}\right]^{-\frac{1}{2}}.$
Denoting $A=2\sqrt{\alpha}/(\alpha+1)^{2}$ and
$\zeta=\left[(\alpha-1)/(\alpha+1)\right]^{2}$ we get
$\tilde{p}(s,t)=Ae^{-\frac{\alpha-1}{2}t}\left(1+\zeta e^{-2\alpha
t}\right)^{-\frac{1}{2}}.$
Substituting this expression into Eq. (35) and denoting
$\beta=(\alpha-1+2u)/2$, we obtain
$\tilde{\tilde{p}}(s,u)=A\int_{0}^{\infty}\left(1+\zeta e^{-2\alpha
t}\right)^{-\frac{1}{2}}e^{-\beta t}dt.$
Now we change the variable of integration to $x=e^{-2\alpha t}$ to arrive to
the expression
$\tilde{\tilde{p}}(s,u)=\frac{A}{2\alpha}\int_{0}^{1}\left(1+\zeta
x\right)^{-\frac{1}{2}}x^{\frac{\beta}{2\alpha}-1}dx.$
This can be compared with the integral representation of the hypergeometric
function:
$\;{}_{2}F_{1}(a,b;c;z)=$
$\displaystyle\qquad\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_{0}^{1}x^{b-1}(1-x)^{c-b-1}(1-xz)^{-a}dx,$
Eq.(15.3.1) of Ref. AbraSteg , from which we get $a=\frac{1}{2}$,
$b=\frac{\beta}{2\alpha}$, $c=b+1$, and $z=-\zeta$ so that,
$\tilde{\tilde{p}}(s,u)=\frac{A}{\beta}\;_{2}F_{1}\left(\frac{1}{2},\frac{\beta}{2a},1+\frac{\beta}{2a},-\zeta\right).$
Substituting the values of parameters we obtain:
$\displaystyle\tilde{\tilde{p}}(s,u)=\frac{4(1+2s)^{\frac{1}{4}}}{(1+\sqrt{1+2s})^{2}(\sqrt{1+2s}+2u-1)}\times$
$\qquad{}_{2}F_{1}\left[\frac{1}{2},\frac{\sqrt{1+2s}-1+2u}{4\sqrt{1+2s}},\frac{5\sqrt{1+2s}-1+2u}{4\sqrt{1+2s}},\right.$
$\displaystyle\qquad\qquad\left.-\left(\frac{\sqrt{1+2s}-1}{\sqrt{1+2s}+1}\right)^{2}\right].$
The function $F(s,u)=1/\tilde{\tilde{p}}(s,u)$ is not a linear function of $s$
for fixed $u$. This can be clearly seen when plotting the $s$-derivative of
this function for fixed $u$ with the help of Mathematica, see Fig. 2.
Figure 2: The $s$-derivative of the function $F(s,u)$ for $u=0.5,1$ and $2$,
shown by solid, dashed and dotted lines, respectively.
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|
# Multiple Clues for Dayside Aerosols and Temperature Gradients in WASP-69 b
from a Panchromatic JWST Emission Spectrum
Everett Schlawin Steward Observatory, 933 North Cherry Avenue, Tucson, AZ
85721, USA Sagnick Mukherjee Department of Astronomy and Astrophysics,
University of California, Santa Cruz, CA, USA Kazumasa Ohno Division of
Science, National Astronomical Observatory of Japan, Tokyo, Japan Department
of Astronomy and Astrophysics, University of California, Santa Cruz, CA, USA
Taylor Bell Bay Area Environmental Research Institute, NASA’s Ames Research
Center, Moffett Field, CA 94035, USA Space Science and Astrobiology Division,
NASA’s Ames Research Center, Moffett Field, CA 94035, USA Thomas G. Beatty
Department of Astronomy, University of Wisconsin–Madison, Madison, WI 53703,
USA Thomas P. Greene Space Science and Astrobiology Division, NASA’s Ames
Research Center, Moffett Field, CA, USA Michael Line School of Earth and
Space Exploration, Arizona State University, Tempe, AZ, USA Ryan C. Challener
Department of Astronomy, Cornell University, 122 Sciences Drive, Ithaca, NY
14853, USA Vivien Parmentier Université de la Côte d’Azur, Observatoire de la
Côte d’Azur, CNRS, Laboratoire Lagrange, France Jonathan J. Fortney
Department of Astronomy and Astrophysics, University of California, Santa
Cruz, CA, USA Emily Rauscher Department of Astronomy, University of Michigan,
1085 S. University Ave., Ann Arbor, MI 48109, USA Lindsey Wiser School of
Earth and Space Exploration, Arizona State University, Tempe, AZ, USA Luis
Welbanks School of Earth and Space Exploration, Arizona State University,
Tempe, AZ, USA Matthew Murphy Steward Observatory, 933 North Cherry Avenue,
Tucson, AZ 85721, USA Isaac Edelman Bay Area Environmental Research
Institute, NASA’s Ames Research Center, Moffett Field, CA 94035, USA Natasha
Batalha Space Science and Astrobiology Division, NASA’s Ames Research Center,
Moffett Field, CA, USA Sarah E. Moran Lunar and Planetary Laboratory,
University of Arizona, Tucson, AZ 85721, USA Nishil Mehta Université de la
Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange,
France Marcia Rieke Steward Observatory, 933 North Cherry Avenue, Tucson, AZ
85721, USA
(Accepted May 31, 2024)
###### Abstract
WASP-69 b is a hot, inflated, Saturn-mass planet (0.26 MJup) with a zero-
albedo equilibrium temperature of 963 K. Here, we report the JWST 2 to 12 µm
emission spectrum of the planet consisting of two eclipses observed with
NIRCam grism time series and one eclipse observed with MIRI LRS. The emission
spectrum shows absorption features of water vapor, carbon dioxide and carbon
monoxide, but no strong evidence for methane. WASP-69 b’s emission spectrum is
poorly fit by cloud-free homogeneous models. We find three possible model
scenarios for the planet: 1) a Scattering Model that raises the brightness at
short wavelengths with a free Geometric Albedo parameter 2) a Cloud Layer
model that includes high altitude silicate aerosols to moderate long
wavelength emission and 3) a Two-Region model that includes significant
dayside inhomogeneity and cloud opacity with two different temperature-
pressure profiles. In all cases, aerosols are needed to fit the spectrum of
the planet. The Scattering model requires an unexpectedly high Geometric
Albedo of 0.64. Our atmospheric retrievals indicate inefficient redistribution
of heat and an inhomogeneous dayside distribution, which is tentatively
supported by MIRI LRS broadband eclipse maps that show a central concentration
of brightness. Our more plausible models (2 and 3) retrieve chemical
abundances enriched in heavy elements relative to solar composition by
6$\times$ to 14$\times$ solar and a C/O ratio of 0.65 to 0.94, whereas the
less plausible highly reflective scenario (1) retrieves a slightly lower
metallicity and lower C/O ratio.
stars: atmospheres — stars: individual (WASP-69 b (catalog ))
††facilities: JWST(NIRCam), JWST(MIRI)††software: astropy (Astropy
Collaboration et al., 2013), photutils v0.3 (Bradley et al., 2016), matplotlib
(Hunter, 2007), numpy (van der Walt et al., 2011), scipy (Virtanen et al.,
2020), starry (Luger et al., 2019), batman (Kreidberg, 2015), ThERESA
(Challener & Rauscher, 2022), MC3 (Cubillos et al., 2017), pymc3 (Salvatier et
al., 2016), celerite2 (Foreman-Mackey, 2018), emcee (Foreman-Mackey et al.,
2013), webbpsf (Perrin et al., 2014),
## 1 Introduction
### 1.1 The Value of Spectra of Moderately Irradiated Hot Jupiters
JWST is opening up a new window into giant irradiated planet atmospheres to
reveal their chemical, physical and dynamical processes (e.g. Ahrer et al.,
2023; Alderson et al., 2022; Rustamkulov et al., 2022; Feinstein et al., 2023;
Bean et al., 2023; Grant et al., 2023; Xue et al., 2023). Emission spectra and
phase curves are particularly valuable for understanding the temperature
structure, composition and 3D effects in the atmospheres of hot planets (Bell
et al., 2023a; Coulombe et al., 2023a). Most JWST studies thus far have probed
either cooler, smaller planets (GJ 1214 b Teq=600 K, Kempton et al. (2023);
Gao et al. (2023); LHS 475 b Teq= 590 K, Lustig-Yaeger & Fu et al. (2023); GJ
486 b Teq=700 K, Moran & Stevenson et al. (2023)) or more highly irradiated
Jupiters (WASP-39 b Teq=1170 K (JWST Transiting Exoplanet Community Early
Release Science Team et al., 2023), WASP-43 b Teq =1400 K (Bell et al.,
2023a), HD 149026 b Teq=1700 K (Bean et al., 2023), HD 209458 b Teq=1450 K
(Xue et al., 2023)), WASP-77 A b Teq=1700 K (August et al., 2023). However,
the study of giant planets in the 800-1000 K temperature range is potentially
a valuable window in the chemical equilibrium of the methane (CH4) and ammonia
(NH3) molecules in exoplanet atmospheres, due to both photochemistry and
mixing from deeper layers (Fortney et al., 2020).
Methane and ammonia are expected to be the dominant Carbon-bearing and
Nitrogen-bearing molecules below 900 K to 1000 K for hot Jupiter atmospheres
observed at 0.1 bar (Moses et al., 2013) so their abundances are critical for
evaluating the level of Carbon and Nitrogen in exoplanet atmospheres. Recently
methane has been definitively detected for the first time via low resolution
space-based spectroscopy with JWST, which opens up this molecule as a tracer
of dynamical and chemical processes in exoplanets. Methane was found in the
moderately-irradiated Jupiter, WASP-80 b ($T_{eq}=825K$), (Bell et al., 2023c)
as well as the lower temperature sub-Neptunes K2-18 b T${}_{eq}=270K$
(Madhusudhan et al., 2023) and TOI-270 d T${}_{eq}=354$K (Benneke et al.,
2024; Holmberg & Madhusudhan, 2024). The moderately irradiated Jupiter
HAT-P-18 b (Teq=850 K) may also have methane, as found by retrieval modeling,
but there are no strong, high signal-to-noise absorption features in the SOSS
bandpass from 0.6 µm to 2.8 µm (Fu et al., 2022) and a subsequent re-analysis
did not find evidence for methane in HAT-P-18 b with the same data (Fournier-
Tondreau et al., 2023). Additional spectra of irradiated giant planets in the
800-1000 K temperature regime will be valuable probes of the atmospheric
physics. This temperature regime efficiently covers the transition between
CH4-dominated versus CO-dominated reservoirs of carbon in planet atmospheres.
The presence of CH4 in planets below $\sim$950 K is also predicted to supply
the building blocks for photochemical hazes in planets that can explain some
of the muted atmospheric features below 950 K as observed by HST transmission
spectra (Gao et al., 2020).
### 1.2 WASP-69 b
WASP-69 b is an inflated Saturn mass transiting planet (0.26 $M_{Jup}$, 1.06
$R_{Jup}$) orbiting a K-type star (4715 K, 0.83 M⊙, 0.81 R⊙, Age $\sim$ 2
Gyr), with a zero-albedo full redistribution equilibrium temperature of 963 K
(Anderson et al., 2014). The host star shows evidence for metal enrichment
with metallicities reported from [Fe/H]= 0.15 $\pm$ 0.08 (Anderson et al.,
2014) to 0.29 $\pm$ 0.04 (Sousa et al., 2021). The equilibrium temperature
regime of WASP-69 b is where CH4 can potentially become a dominant carbon-
bearing molecule in chemical equilibrium (Moses et al., 2013), but the cooling
history of a planet and vertical mixing can potentially alter the composition
of the planet from equilibrium expectations (Fortney et al., 2020).
Atoms and molecules have been previously detected in WASP-69 b’s atmosphere
and give initial insights into its composition and aerosols. Searches for Na
in the atmosphere of WASP-69 b with high resolution spectroscopy have yielded
a range of results, but most of the studies find that Na is significantly
detected in the Na D2 line but less significantly detected in the Na D1 line,
potentially from obscuration by atmospheric aerosols (Casasayas-Barris et al.,
2017; Deibert et al., 2019; Langeveld et al., 2022; Khalafinejad et al.,
2021).111 Specifically, Casasayas-Barris et al. (2017) report that the Na D2
line was detected at a 5$\sigma$, while the Na D1 was not detected. Casasayas-
Barris et al. (2017) find that the Na D2 has an excess transit depth of 0.53
$\pm$0.14% for a a 1.5 Å bandpass and a line contrast of 5.8 $\pm$ 0.3% from a
Gaussian fit the transit profile. The slightly different oscillator strengths
of the two lines makes it possible for Na D1 to be more obscured (Huitson et
al., 2012). Deibert et al. (2019), by contrast, do not find significant
absorption by WASP-69 b with the GRACES instrument for either the Na D2 or Na
D1 lines. Langeveld et al. (2022) find that WASP-69 b has a significant Na
line contrast of 3.28 $\pm$ 0.70% for the Na D2 line and 1.26 $\pm$ 0.61 % for
the Na D1 line. Muted but significant detections of both the Na D2 and Na D1
lines were detected by Khalafinejad et al. (2021). The ratio of the two lines
Na D2/ Na D1 of 2.5 $\pm$ 0.7 suggests that there is aerosol opacity that
muted the Na D1 line (Khalafinejad et al., 2021). Thus, the majority of Na
line absorption studies at high spectral resolution mostly suggest an
atmosphere with significant sodium but the Na D1 line is at least partially
obscured by aerosols as compared to Na D2. WASP-69 b’s Na detections have also
been theorized to be potentially influenced by a volcanically active moon (Oza
et al., 2019); so far no moons have been detected (Narang et al., 2023).
Other atmospheric features of WASP-69 b found with transmission spectroscopy
include its scattering slope, tentative evidence for TiO, Carbon-bearing and
Oxygen-bearing molecules and an escaping outflow, all described below. Murgas
et al. (2020) find a spectral slope in the GTC OSIRIS optical transmission
spectrum that is consistent with Rayleigh scattering from hydrogen. However,
the spectral slope is also consistent with stellar activity and the transit
light source effect (e.g. Rackham et al., 2018). Estrela et al. (2021) find an
even stronger spectral slope than Murgas et al. (2020) with HST’s Space
Telescope Imaging Spectrograph (STIS). The large transit depths down to
$\sim$0.4 µm are best explained by aerosols at high altitudes ($\mu$bar
pressure levels) (Estrela et al., 2021). There is further evidence of Raleigh
scattering in ground-based spectroscopy from 0.5 to 0.9 µm with tentative
evidence of TiO in the spectrum (Ouyang et al., 2023). Extended Helium has
also been detected in WASP-69 b with a signal-to-noise ratio of 18 that is
blue-shifted by several km/s (Nortmann et al., 2018). This was confirmed with
narrowband photometry, constraining the mass loss rate to be 10-4 to 10-3
MJup/Gyr (Vissapragada et al., 2020). The envelope escaping WASP-69 b extends
several radii and is confined to a tail like structure that extends up to 7.5
planetary radii behind the planet’s direction of motion (Tyler et al., 2023).
However, the inferred mass loss rate from the observed escaping tail ($\sim
10^{-3}$ MJup/Gyr) is not a significant fraction of the overall mass (0.26
MJup) to measurably alter the deeper atmosphere.
In transit transmission, 5 molecules were detected with high resolution cross
correlation (Guilluy et al., 2022): CO, CH4, NH3, H2O and C2H2. In the same
analysis, the detection of H2O and CH4 and the absence of CO2 disfavored
models larger than 10$\times$ solar enrichment of heavy elements. If the
atmosphere is assumed to be solar composition, the presence of the 5 detected
molecules favors models that have high C/O ratios, perhaps larger than 1.0.
The presence of C2H2 in the terminator also implies some disequilibrium
processes.
Baxter et al. (2020) and Wallack et al. (2019) used secondary eclipse
measurements at the Spitzer 3.6 µm and 4.5 µm bandpasses to constrain the
composition and temperature profile of WASP-69 b. Baxter et al. (2020) found
that WASP-69 b and other planets below $\lesssim 1660$K lack temperature
inversions, ie. their gas temperatures decrease as a function of altitude. The
spectral slope of the Spitzer 3.6 µm and 4.5 µm eclipse depths was also fit
with chemically equilibrated atmospheric models. The low spectral slope
observed in the planet ($\lesssim 20ppm/\micron$) favors a high atmospheric
metallicity of the planet above $\sim 30\times$ solar (Wallack et al., 2019).
However, the two photometric eclipse depths cannot uniquely constrain the
composition of WASP-69 b due to the correlation with the planet’s temperature-
pressure profile, aerosols, abundance ratios and possible disequilibrium
effects (Wallack et al., 2019). JWST observations covering a wide range of
wavelengths that encompass many molecular features and probe different
pressure levels can better constrain the abundances, dynamics, chemical
processes and aerosols in WASP-69 b and similar planets’ atmospheres.
#### 1.2.1 Paper Outline
In Section 2, we describe the observations of 3 eclipses spanning 2 µm to 12
µm. In Section 3, we describe three independent analyses to extract the
spectrum of WASP-69 b from the raw data and also describe the absolute flux of
the star as compared to a stellar model. In Section 4, we present the planet’s
dayside spectrum and its overall properties. We introduce several types of
models, the results of atmospheric retrievals and why they must incorporate
aerosols and 3D effects in Section 5. We find some evidence for an
inhomogeneous dayside temperature distribution, which we investigate
independently with broadband eclipse mapping in Section 6. We discuss the
energy budget of the planet’s dayside and the properties of atmospheric
aerosols and synthesize WASP-69 b’s inferred composition across all of our
models in Section 7. Finally, we conclude in Section 8 that WASP-69 b is
chemically enriched relative to solar composition, has dayside aerosols and
temperature gradients away from the substellar point.
## 2 Observations
Figure 1: Dynamic Spectra for WASP-69 b’s show the eclipse as a horizontal
dark band. The eclipse depth (as seen by the darkness of the band) increases
from short wavelengths toward long wavelengths and also shows a dip where
there is strong CO2 atmospheric absorption near 4.3 µm. The dynamic spectra
are the extracted spectra for each detector column (NIRCam) and row (MIRI) at
full spectral and full time resolution without binning. The top panels of each
dynamic spectrum show the average spectrum in electrons for one integration.
WASP-69 b was observed as part of the MIRI And NIRCam Assay for the
Transmission and Emission of Exoplanets (MANATEE), which combines observations
from GTO program 1177 (Observation 3) and GTO program 1185 (Observation 6 and
7) to build up near-infrared to mid-infrared spectra on a wide variety of
planets. Table 1 contains a summary of observations.
For NIRCam, we used the readout pattern with the most number of reads per
integration to mitigate the effects of 1/f noise (Schlawin et al., 2020) while
still keeping data volume manageable222With BRIGHT2, our NIRCam observations
had data volume excess above the sustainable rate (0.87 MB/s) with an excess
of 7 (9) GB for the F322W (F444W) filters, respectively, but below the medium
volume excess threshold of 15 GB. With the usually-favored RAPID readout
pattern (1 frame per group), the observations would exceed the medium
threshold with an excess of 24 GB., which led us to the BRIGHT2 readout
pattern, which has no skipped frames and 2 co-added frames per group. We
selected the F210M filter and WLP8 pupil with the short wavelength arm of
NIRCam to measure the the eclipse at 2.1 $\mu$m for both NIRCam observations.
For MIRI, we used the FASTR1 read pattern with the slit-less Low Resolution
Spectrometer (LRS) mode. The 3 separate observations spanned from 372 to 373
minutes in duration, or 2.8 times the eclipse duration for the planet. This
long baseline ensured significant detector settling time for persistence
effects like charge trapping (Zhou et al., 2017) and MIRI detector upward and
downward ramps (Bell et al., 2023b; Bouwman et al., 2023) to settle. The
observation details were specified before Cycle 1 measurements of these
settling constants, but the on-orbit measurements of exoplanet lightcurves
show that the time constants for the charge trapping ramp-up are short for the
near-infrared instruments that employ HgCdTe H2RG detectors ($\lesssim$ 15
min) (e.g. Ahrer et al., 2023; Feinstein et al., 2023; Espinoza et al., 2022;
Schlawin et al., 2023b), so future near-infrared observations may succeed with
shorter pre-transit or pre-eclipse baselines.
The pixel-level lightcurves for the three observations are shown in Figure 1
along with the detected counts in electrons per integration. These dynamic
spectra are normalized by the median spectrum per integration to show the
change as a function of time and wavelength. The eclipse as the planet goes
behind the star is clearly detected in all three observations at this pixel-
level resolution, as seen by the dark horizontal band across the dynamic
spectra. As expected, the eclipse depth rises from the shortest to longest
wavelengths because the planet has a cooler temperature than the star. Also
visible in the dynamic spectra is the shallower eclipse depth near 4.3 µm due
to carbon dioxide in the planet’s atmosphere.
Some systematic correlated noise is also visible in the dynamic spectra in
both the NIRCam and MIRI data. For NIRCam, horizontal stripes and bands are
visible due to the 1/f noise present in the detector, which arises from the
electronics that read the detector in the horizontal (wavelength) direction
for NIRCam (Schlawin et al., 2020; Ahrer et al., 2023). 1/f noise appears as
horizontal striping in the individual detector groups that manifests as
horizontal striping in the dynamic spectra for the two NIRCam observations.
The 1/f noise was already mitigated by row-by-row subtraction using background
pixels (e.g. Schlawin et al., 2023b), as described in Section 3. However, the
row-by-row subtraction is only possible with background pixels of the right-
most amplifier for the F322W2 NIRCam observation and the left two amplifiers
of the F444W NIRCam observation, when they are oriented in the Data Management
System orientation that is used in MAST. 1/f noise that is present in the
middle two amplifiers can be mitigated by subtracting the median of rows of
the other amplifiers, but this can only subtract 1/f noise that is common to
the shared SIDECAR ASIC device and not noise that is unique to each
amplifier’s p-type field-effect transistor (PFET) (Schlawin et al., 2020).
Therefore, residuals 1/f noise is visible in Figure 1, mostly at wavelengths
shorter than 3.89 µm for the the F322W2 observation 6 and longer than 4.09 µm
for observation 7. Also noticeable in the dynamic spectra is a downward flux
trend, most prominently in the NIRCam F322W2 spectrum near 3.1 µm. As will be
described in Section 3, we further mitigate and account for 1/f noise with co-
variance weighted extraction, fitting excess noise and model the time-
dependent trend with either a polynomial or the detector housing temperature.
For MIRI, there are also two systematic effects visible in Figure 1. There is
a ramp visible in the first several hundred integrations which is typical of
MIRI/LRS observations, especially at the shortest wavelengths (e.g., Bell et
al., 2023b, 2024; Bouwman et al., 2023). There is also an odd-even effect in
alternating spectral-axis pixels that appears as vertical striping in the
dynamic spectrum; this is caused by the simultaneous resetting of pairs of
rows (in MIRI’s 90-degree rotated reference frame), and such odd-even striping
is also typical of MIRI/LRS observations (e.g., Bell et al., 2024). This odd-
even effect, the 390 Hz noise seen in the data from some MIRI subarrays
(Bouwman et al. 2023; Bell et al. 2024), and other still-unknown noise sources
result in excessively noisy MIRI/LRS spectra at high-resolutions which is
significantly improved by spectral binning (Bell et al., 2024; Welbanks & et
al., 2023). As a result, we used wider bin sizes (0.25 $\mu$m full width) to
average over these effects and give more reliable uncertainty estimates.
Table 1: Summary of WASP-69 b observations UT Exp Date | Prog | Obs # | Instrument | SW Filter | Spectroscopic | Spec Wave Range | Nint | Ngroup | Readout
---|---|---|---|---|---|---|---|---|---
YYYY-mm-dd | | | | | Description | µm | | | Pattern
2022-10-25 | 1185 | 6 | NIRCam | F210M | F322W2 | 2.4 - 4.01 | 2367 | 3 | BRIGHT2
2023-06-06 | 1185 | 7 | NIRCam | F210M | F444W | 3.88 - 4.98 | 1507 | 5 | BRIGHT2
2023-05-02 | 1177 | 3 | MIRI | $\cdots$ | LRS | 5.0 - 14 | 8276 | 16 | FASTR1
Table 2: Summary of WASP-69 b Properties and Priors
Parameter | Prior/Existing Value | Prior/Existing Source | TESS Posterior (this work)
---|---|---|---
t0 | 2455748.83344 $\pm$ 1.8e-4 | Casasayas-Barris et al. (2017) | 2455748.83345 $\pm$ 1.8e-4
P | 3.8681382 $\pm$ 1.7e-6 d | Casasayas-Barris et al. (2017) | 3.86813942 $\pm$ 1.8e-7 d
$i$ | 86.71∘ $\pm$ 0.20 | Casasayas-Barris et al. (2017) | 86.631∘ $\pm$ 0.029
a/R∗ | 12.0 $\pm$ 0.46 | Casasayas-Barris et al. (2017) | 11.919 $\pm$ 0.05
Rp/R∗ | 0.1336 $\pm$ 0.005 | Casasayas-Barris et al. (2017) | 0.1285 $\pm$ 0.00025
e | 0.0 | fixed | fixed
R∗ | 0.813 $\pm$ 0.028 R⊙ | Anderson et al. (2014) | $\cdots$
log g∗ (cgs) | 4.535 $\pm$ 0.023 | Anderson et al. (2014) | $\cdots$
Distance | 50.29 $\pm$ 0.04 pc | Gaia Collaboration et al. (2023) | $\cdots$
Note. — For the spectroscopic analysis, we use the prior values from
Casasayas-Barris et al. (2017), which come from Anderson et al. (2014). We
also fit the publicly available TESS data and report the posteriors (right) to
improve the orbital parameter precisions for eclipse mapping.
## 3 Data Analysis and Extraction
### 3.1 Reduction and Lightcurve Extraction
#### 3.1.1 tshirt
We extract spectra in a similar manner as in previous JWST data analyzed with
tshirt and published (Ahrer et al., 2023; Bell et al., 2023c; Welbanks & et
al., 2023), but with the exception that we allow for the small curvature and
tilt of the trace (about 4 pixels across the spectrum with NIRCam, but
negligibly for MIRI LRS). For NIRCam data, we begin with the _uncal.fits stage
0 data products from MAST and process them with a modified version of the jwst
pipeline (Bushouse et al., 2023) version 1.10.2, except for the F210M
photometry on Observation 6, which was extracted soon after observation with
an older jwst version 1.6.0. We ran the jwst pipeline with CRDS context
jwst_1077.pmap for Program 1185 observation 6’s spectroscopy and CRDS context
jwst_1093.pmap for observation 7’s spectroscopy, but checked that the used
reference files were the same for both observations’ contexts. For the
photometry, we used jwst_1009.pmap for both observations. We also used tshirt
to extract the spectrum of the MIRI LRS data in Program 1177 observation 3.
For MIRI LRS data, we used SDP version 2023_4a, jwst version 1.13.4, CRDS
version 11.17.15 and CRDS context jwst_1225.pmap. For reference on different
version numbers, we have found negligible differences in extracted tshirt
photometry and spectroscopy $\lesssim$ 10 ppm depths between JWST version
numbers and reference file contexts from JWST versions 1.8 to 1.13 and pmap
contexts 1039 to 1188.
Our main modification to the jwst pipeline for NIRCam is to replace the
reference pixel correction with a row-by-row, odd-even by amplifier (ROEBA)
correction to mitigate 1/f noise in the photometry and spectroscopy (e.g.
Schlawin et al., 2023b). For MIRI LRS data, we do not use the ROEBA
correction. We also skip the dark current step for NIRCam because it adds
noise for subarray modes. We also set the jump sigma rejection threshold for
cosmic rays to 6 $\sigma$ for NIRCam and 7 $\sigma$ for MIRI and stop
processing the jwst pipeline after the _rateints.fits products. We manually
divide by the NIRCam spectral images by the flat field reference file for
imaging in CRDS (jwst_nircam_flat_0313.fits for the F444W filter and
jwst_nircam_flat_0266.fits for the F322W2 filter). The jwst pipeline was run
on NIRCam data with the JWST Time Series Observation Wrapper (jtow) version
0.1.4 (https://github.com/eas342/jtow.git) and the spectra and photometry were
extracted with the Time Series Helper and Integration Reduction Tool tshirt
version 0.3 (https://github.com/eas342/tshirt). For MIRI LRS data, we used an
updated tshirt version 0.4 that includes wavelength calibration of MIRI LRS.
We do not apply a flat field correction to the MIRI LRS data.
For the photometric F210M data, we use a circular aperture of 92 pixels and a
background annulus from 93 to 129 pixels, as shown in Appendix C. For
spectroscopy, we fit the traces of the spectrum for one image and use this to
define the aperture and background for all images. For the trace centroids, we
fit the profile with a Gaussian in each column and then fit the centroids with
a 3rd order polynomial with iterative 3 $\sigma$ clipping of outlier points.
For the long wavelength extraction, we next subtract the background using a
linear fit along the Y (spatial) direction for all the pixels that are more
than 7 pixels away from the source (rounded to the nearest whole pixel) and
also located in a rectangular region from pixels Y=5 to 65. We use a
covariance-weighted extraction (Schlawin et al., 2020) for the non-NaN pixels
that are within 5 pixels from the source trace, assuming a read noise of 14
$e^{-}$ and a correlation of 0.08 between pixels. We use the interpolated PSF
to estimate the missing flux in the spatial direction for pixels that are
marked as NaN or outliers more than 30$\sigma$ from a spline fit and multiply
the flux to account for the missing pixels’ fractional flux. The extracted
average spectrum in ($e^{-}$) and spectroscopic lightcurves at the pixel
resolution are shown in Figure 1. We next bin the spectra in wavelength for
lightcurve analysis to approximately 0.010 µm wide wavelength bins, but
rounded to the nearest whole pixel.
#### 3.1.2 Eureka!
Our Eureka! NIRCam and MIRI reductions used version 0.10 of the Eureka!
pipeline (Bell et al., 2022); jwst package version 1.10.2 (Bushouse et al.,
2023); CRDS version 11.17.0; and CRDS context 1094 for NIRCam/F322W2 and
NIRCam/F444W, and 1097 for MIRI/LRS. Our NIRCam reduction methods closely
follow those used in previous Eureka! NIRCam spectroscopy analyses (Ahrer et
al., 2023; Bell et al., 2023c; Welbanks & et al., 2023), and our MIRI/LRS
reduction method generally followed the Eureka! v1 method described by Bell et
al. (2023a) and the Eureka! MIRI/LRS analysis of Welbanks & et al. (2023).
Eureka! Control Files and Eureka! Parameter Files that can be used to
reproduce this work are available for download
(https://doi.org/10.5281/zenodo.11168833; Schlawin et al. 2024), but the
important parameters are summarized below.
Eureka!’s Stages 1 and 2 make use of the jwst pipeline’s (Bushouse et al.,
2023) Stages 1 and 2 but allow for some modifications. For both NIRCam
spectroscopy datasets, the only change to Stage 1 was increasing the jump
rejection threshold to 6 to avoid excessive false positives when ramp fitting.
For MIRI/LRS, we similarly increased the jump rejection threshold to 7 and
also turned on the firstframe and lasstframe steps (which skip the first and
last frames of each integration) as these frames are subject to excessive
noise and generally increase the final noise level in the data. Finally,
during the Stage 1 processing of the MIRI/LRS data we also applied Eureka!’s
newly developed 390 Hz noise removal and group-level background subtraction
technique described in detail by Welbanks & et al. (2023); in general, this
step removes the wavelength correlated noise in raw MIRI/LRS SLITLESSPRISM
subarray data first pointed out by Bouwman et al. (2023). Our Stage 2
processing followed the jwst standard processing with the exception of turning
off the photom and extract1d steps which are not required for time-series
observations. Later, we re-analyzed the data for checking the stellar absolute
flux against a model.
Our Stage 3 processing of the NIRCam data closely followed the Eureka! NIRCam
reductions of Welbanks & et al. (2023). In summary, we cropped the frames to
only include the relevant pixels, masked outlier pixels, corrected for the
curvature of the spectral trace, performed a linear column-by-column
background subtraction using pixels $>$13 px away from the source center,
computed the position and width of the source for later use when fitting the
lightcurves, and performed optimal spectral extraction (Horne, 1986)
considering only the pixels within 5 pixels of the source center and using a
cleaned median integration to compute our variance-weighted spatial profile.
Our Stage 3 processing of the MIRI data also closely followed the Eureka! MIRI
reduction of Welbanks & et al. (2023) which differs only slightly from the
NIRCam reduction steps. In particular, the differences from the NIRCam
reduction steps include rotating the MIRI/LRS data to have wavelength
increasing toward the right, manually applying an estimated gain of 3.1 e/DN
(Bell et al., 2023b, a), using pixel indices 11–61 (excluding pixels within 9
px of the spectral trace) to perform a column-by-column background
subtraction, and only using pixels within 3 px of the source center when
performing optimal spectral extraction.
In Stage 4, we spectrally binned the data and sigma-clipped temporal outliers.
For the NIRCam data we used 0.01 $\mu$m spectral bins across 2.45–3.95 $\mu$m
for F322W2 and 3.89–4.97 $\mu$m for F444W, while for the MIRI data we used
coarser 0.25 $\mu$m spectral bins from 5–12 $\mu$m following the
recommendation of Bell et al. (2023a). We did not see evidence of the
“shadowed region effect” described by Bell et al. (2023a), and therefore
include the 10.6–11.8 $\mu$m data in our reduction. For all three
observations, we performed 4-sigma clipping comparing each point to a smoothed
version of the data computed with a 20-integration wide boxcar filter which
helped to removed otherwise missed temporal outliers while ensuring not to
mask the eclipse ingress or egress.
#### 3.1.3 Pegasus
We also reduced the NIRCam observations using the Pegasus pipeline
(https://github.com/TGBeatty/PegasusProject). Our Pegasus reduction began with
background subtraction on the rateint files provided by version 1.10.2 of the
jwst pipeline using CRDS version 11.17.0. We began by applying a basic
background subtraction to each rateint file by fitting a two-dimensional,
second-order spline to each integration, covering the full 256$\times$2048
rateint images. For the spline fitting, we masked out image rows 5 through 75
to prevent the self-subtraction of light from the WASP-69 system. We then
performed a single round of $3\,\sigma$ clipping on the unmasked portions of
the image. Next, we fit individual splines to each amplifier region on the
image using the unmasked, unclipped pixel values with a median box size of 20
pixels. This per-amplifier spline fitting was necessary to eliminate residual
bias differences across the amplifier areas. We extrapolated the combined
background spline for the whole image over the masked portions near WASP-69
and subtracted it from the original image values. Visual inspection of the
rateint images showed that in roughly 2% of the integrations the reference
pixel correction failed for at least one of the amplifier regions, so after
the spline fitting and subtraction we re-ran the reference pixel correction
using hxrg-ref-pixel (https://github.com/JarronL/hxrg_ref_pixels). Finally, we
attempted to remove some of the red-noise caused by the NIRCam readout
electronics which is present along detector rows, by calculating the robust
mean of each row using pixels from column 1800 onwards for the F322W2 images
and up to column 600 for F444W (both chosen to avoid light from WASP-69), and
then subtracting this mean from each row. Visually, this removed most of the
horizontal banding typical for NIRCam grismr images.
We then extracted spectroscopic lightcurves from our background-subtracted
images. To do so, we fit the spectral trace using a fourth-order polynomial
and then used optimal extraction to measure the 1D spectrum in each image. We
performed three rounds of iterative profile estimation for the optimal
extraction routine, after which we judged the profile fit to have converged.
Using the resultant 1D spectra, we extracted a broadband lightcurve from 2.45
$\mu$m to 3.95 $\mu$m at F322W2 and from 3.89 $\mu$m to 4.97 $\mu$m at F444W.
For the spectroscopic lightcurves we subdivided each of these wavelength
regions into individual 0.01µm-wide spectral channels. During this extraction
process, we linearly interpolated over each spectral column to account for
partial-pixel effects in the 0.01µm wavelength bins.
### 3.2 Lightcurve Fitting
#### 3.2.1 tshirt
We fit the lightcurves with a starry (Luger et al., 2019) lightcurve model,
assuming a uniform intensity map characterized by an amplitude term only. We
use the probabilistic programming suite pymc3 (Salvatier et al., 2016) and the
pymc3-ext tools from Foreman-Mackey et al. (2021) to calculate the posterior
distributions of the model with No U-Turns sampling. For the orbital parameter
priors, we use the values from Casasayas-Barris et al. (2017) listed in Table
2. For NIRCam, we bin the time series to a resolution of 300 equally spaced
time bins at a cadence of 1.24 minutes to speed up model evaluation and also
provide an empirical estimate of noise. We estimate the errors by taking the
standard error in the man of all integrations within each 1.24 minute bin and
then taking the median of all bins’ standard deviations and adopting this
across all time bins. edit1For MIRI data, we trim the first 1000 integrations
(ie. 45 minutes) to discard much of the initial ramp-up behavior (Bell et al.,
2023b) and bin the data to 100 equally spaced time bins.
We have found that the measured flux of many different targets anti-correlates
with NIRCam’s detector housing temperature. Figure 18 in Section C shows the
Focal Plane Housing Temperature and the out-of-transit and out-of-eclipse
fluxes of many different exoplanet observations and how they anti-correlate.
While the mechanism is not exactly understood, we can model the flux as a
linear function of the focal plane housing temperature deviation. For both the
short wavelength photometry and the long wavelength grism spectroscopy, we
collect the detector housing temperature from the MAST engineering database.
The short wavelength and long wavelength temperatures are accessible by
querying the IGDP_NRC_A_T_SWFPAH1 and IGDP_NRC_A_T_LWFPAH1 telemetry
mnemonics, respectively. We find the temperature deviation by subtracting all
temperatures by the median. The temperature sensors have significant noise, so
we smooth both the short wavelength and long wavelength housing temperatures
first by fitting them with a 5th order polynomial and using this as an
interpolation function for the temperature at each integration. Finally, we
model the NIRCam flux as a linear function of the smoothed temperature
deviation.
In addition to the drift with focal plane housing temperature for NIRCam, we
include a linear slope in time to account for other drifts such as stellar
rotation. We assume a prior slope of 0$\pm$1.6%/hr for the linear drift. For
the MIRI data, we also include an exponential baseline function to fit for the
ramp behaviors of MIRI LRS lightcurves (Bell et al., 2023b). For the
exponential ramp, we assume the behavior decays exponentially from the first
integration with a timescale prior with a lognormal prior that has a timescale
of 1 minute with a wide geometric mean of 2 and amplitude of 0.1%.
Additionally, a strong level of correlated noise is visible in both NIRCam
observations using the same F210M photomeric filter, as seen in Figure 2.
Therefore, for the short wavelength F210M lightcurves only, we also model the
time series with a Gaussian Process (e.g. Gibson et al., 2012). We use
celerite2 code (Foreman-Mackey, 2018) with a stochastically-driven damped
harmonic oscillator kernel that has a fixed quality factor of Q=0.25 and a
log-normal prior on the period of the oscillator $\rho$ that has a geometric
mean equal to the duration of the observation.
We iteratively sigma clipped any fluxes that deviated by the model by more
than 5 $\sigma$ and iterated 2 times to maximize the a priori probability
using the starry, pymc3 and pymc3-ext software suites. The resulting maximum a
priori lightcurve solutions are shown in Figure 2. This maximum a prior
solution was used as an initial set of parameters for the Hamiltonian No
UTurns Sampler (NUTS). The sampling was tuned with 3000 steps and samples for
another 3000 steps with two different chains.
Figure 2: Top Half-Panels: Broadband eclipse Lightcurves with a Best-Fit
Model. Bottom Half-Panels: Residuals of the model.
We first found the posterior distribution of the broadband eclipse lightcurve.
We then fix the semi-major axis, inclination, period, transit time (linked to
the eclipse time) at the broadband fit value for all spectroscopic lightcurve
fits. We allow the linear baseline and FPAH temperature deviation terms to be
free parameters with the same priors as applied to the broadband.
We note that the short wavelength F210M lightcurves exhibited significant
correlated noise, as seen in Figure 2, upper left, especially for Observation
7. Ignoring this noise and fitting with polynomial de-trending vectors results
in significantly discrepant eclipse depths by 3.6$\sigma$. However, when we
use a GP fit to both observations, we find agreement to within 0.7$\sigma$.
The correlated noise seen in these observations is larger than found for
HAT-P-14 b with the same defocused photometry mode (Schlawin et al., 2023b).
The increased noise for WASP-69 b may be related to activity in the host star,
which has been found to have spots and a rotation period of $\sim$24 days
(Khalafinejad et al., 2021; Anderson et al., 2014). Another possibility is
JWST primary mirror flexures because the defocused photometry is highly
sensitive to JWST’s wavefront (e.g. Schlawin et al., 2023b; McElwain et al.,
2023). The longer wavelength spectroscopic observations (grism time series and
MIRI LRS) show less of this correlated noise from either of these effects
because stellar spot contrast decreases as function of wavelength and these
modes are not as sensitive to JWST’s wavefront.
The resulting NIRCam emission spectrum of WASP-69 b is shown in Figure 3.
There is a rise from 2.5 to 5.0 µm expected for a $\sim$963 K zero-albedo
equilibrium temperature planet orbiting a 4715 K star. Additionally, there is
a deep CO2 absorption feature visible at 4.3 µm, but no obvious CH4 feature at
3.3 µm. A comparison with the Eureka! and Pegasus reductions (described below)
is also shown in Figure 3, which show consistency within 1 $\sigma$ errors,
but some subtle 30-50 ppm offsets between reduction methods.
Figure 3: Left: Eclipse spectrum with three independent extraction and
lightcurve fitting methods on a common $\sim$0.010 µm-spaced grid. All spectra
agree within the $\sim$50-80 ppm errorbars. Right: Eclipse Spectrum with MIRI
with two independent extraction and lightcurve fitting methods .
#### 3.2.2 Eureka!
As was done with tshirt, our Eureka! lightcurve fitting also used a starry
(Luger et al., 2019) astrophysical model assuming a uniform map. For the
orbital parameter priors, we fixed all parameters except $t_{0}$ to the values
from Casasayas-Barris et al. (2017). For $t_{0}$, we used a Normal prior based
on the posterior of Casasayas-Barris et al. (2017) when fitting the MIRI/LRS
broadband (5–12 $\mu$m) lightcurve which resulted in an inferred
$t_{0}=2455748.83473\pm 0.00012$; we then adopted this new value and fixed
$t_{0}$ for the spectroscopic fits to all lightcurves. We also assumed a
stellar radius of 0.813 $R_{\odot}$ (Casasayas-Barris et al., 2017) combined
with $a/R_{*}$ to account for the light travel delay when computing the
expected time of eclipse. For our systematic noise model, we used a polynomial
in time (quadratic for NIRCam and linear for MIRI), a linear decorrelation
against the spatial position and PSF-width of the spectral trace, and for MIRI
an exponential ramp following the recommendations of Bell et al. (2023a). We
also removed the first 800 integrations from the MIRI observations to reduce
the impact of the strong initial exponential ramp. We also fitted an error
inflation parameter which multiples our estimated uncertainties by a constant
factor; this ensures a reduced chi-squared of 1.0 and avoids issues caused by
incorrectly estimated gain values in Stage 3. For all our systematic noise
models, we used minimally informative priors.
We used pymc3’s No U-Turns Sampler (Salvatier et al., 2016) to sample our
posteriors using two independent chains with a target acceptance rate of 0.85.
For the MIRI spectra we used 3000 tuning steps and 1500 posterior samples,
while for the NIRCam spectra we used 4000 tuning steps and 3000 posterior
samples. For all fits, the Gelman-Rubin statistic (Gelman & Rubin, 1992) was
at or below 1.01, ensuring the chains had converged. We then used the 16th,
50th, and 84th percentiles of the posterior samples to estimate the best-fit
parameter values and their corresponding uncertainties. The fits to our NIRCam
data showed minimal residual red noise, and the white noise in our residuals
was $\sim$20–30% above the estimated photon limit for F322W2 and $\sim$10–20%
above the estimated limit for F444W. Meanwhile, the MIRI data did exhibit
significant residual red noise in some channels (especially between 6–8
$\mu$m), and for MIRI the white noise level in our residuals ranged from
$\sim$20–30% above the estimated photon limit from 6–10 $\mu$m with a gradual
increase to 80% above the limit longward of 10 $\mu$m and a steep increase to
140% above the limit shortward of 6 $\mu$m (the cause of which could not be
identified). To account for the impact of unmodelled red noise on our final
spectra, we used the $\beta$ error inflation method developed by Winn et al.
(2008); in particular, we computed $\beta$ at timescales spanning 22–32
minutes (within 5 minutes of WASP-69b’s $\sim$27 minute ingress/egress
duration) and multiplied our emission spectra uncertainties by the computed
$\beta$ value. For most spectroscopic channels this inflated the uncertainties
by $<$20% but significantly increased the uncertainties of three channels
(6.125, 6.875, and 7.75 $\mu$m).
#### 3.2.3 Pegasus
Our Pegasus analysis used a BATMAN eclipse model (Kreidberg, 2015) to fit the
spectroscopic lightcurves extracted via the Pegasus pipeline. We fixed the
orbital parameters of WASP-69b to those measured in Casasayas-Barris et al.
(2017), which left the free parameters in our spectroscopic lightcurve fitting
to be the secondary eclipse depth and the slope and normalization of a
background linear trend. We did not impose a prior on any of these parameters.
We fit each spectral channel individually.
To fit the spectroscopic lightcurves we performed an initial likelihood
maximization using a Nelder-Mead sampler followed by MCMC likelihood sampling.
We used the maximum likelihood point identified by the Nelder-Mead
maximization as the starting locus about which we initialized the MCMC chains.
To perform the MCMC runs, we used the emcee Python package (Foreman-Mackey et
al., 2013) using 12 walkers with a 2,000-step burn-in and then a 4,000-step
production run for each spectral channel. We checked that the MCMC had
converged by verifying that the Gelman-Rubin statistic was below 1.1 for each
parameter in each spectral channel.
We additionally checked the goodness-of-fit and statistical properties of our
eclipse modeling in each spectral channel. We did so by first verifying that
the average of the per-point flux uncertainties in each channel’s lightcurve
matched the standard deviation of the residuals to the best fit eclipse model.
We also computed the Anderson-Darling statistic for each channel’s lightcurve
residuals to check that the residuals themselves appeared Gaussian. We did not
find statistically significant non-Gaussianity in the residuals to our
spectroscopic lightcurve fits.
### 3.3 Comparison of Extracted Spectra
A comparison of our three NIRCam reductions on a common $\sim$0.010µm
wavelength grid is visible in Figure 3. The tshirt, Eureka! and Pegasus
reductions all agree well within the error bars of 50-80 ppm. Even though each
spectral extraction treated 1/f noise and there were differences in treatment
of the systematic trends, the three reductions give very similar absorption
features, slopes and overall depths. A moving average of the observations
showed agreement to better than the 50 ppm level. Given the close agreement
between reductions, we proceed with the tshirt reduction, which had the most
1/f noise mitigations and smallest broadband out-of-eclipse lightcurve
scatter. We also did two independent reductions of the MIRI LRS spectrum shown
in Figure 3. Again, the agreement for both reductions is broadly consistent
with only 2 points differing by 2.6 and 2.8 times the 1$\sigma$ uncertainties
of the Eureka! reduction. These two wavelengths do not correspond to any
strong opacity sources in our models. We use the Eureka! data for our modeling
because it has a longer heritage of analysis with MIRI LRS data (e.g. Bell et
al., 2024; Kempton et al., 2023).
### 3.4 Absolutely Calibrated Stellar Spectrum
Figure 4: Top: Calibrated absolute flux for 3 observations of the host star
WASP-69 A and a 4750 K BOSZ model (Bohlin et al., 2017). Additionally,
photometry from 2MASS KS and WISE bands 1-3 is shown as data points with error
bars. The black circles are synthetic photometry of the BOSZ model for those
same photometric bands. Bottom: The ratio of the data over the model shows
closest agreement in the F322W2 filter (where a direct ratio with a calibrator
was possible) and some 1-3% discrepancies where stellar CO absorption is
present.. No circumstellar debris disks are present above $\gtrsim 3\%$ flux
levels that could dilute the planet’s flux significantly.
We also create an absolutely calibrated stellar spectrum across the NIRCam and
MIRI wavelengths to find the intrinsic dayside planet flux as well as verify
if a stellar model provides an accurate estimate for the star. We first
calculated calibrated stellar spectra and compared them to stellar models that
most closely matched the stellar parameters from Anderson et al. (2014):
T∗,eff = 4700 $\pm$ 50 K, log(g∗)=4.5$\pm$0.15, [Fe/H] = 0.15 $\pm$ 0.08,
R=0.813 R⊙ and a Gaia DR3 distance of 50.29 $\pm$ 0.04 pc (Gaia Collaboration
et al., 2023). For the BOSZ model (Bohlin et al., 2017), the parameters are
T=4750 K, log(g)=4.5 and [Fe/H]=0.0.
For the NIRCam observations, we extracted stellar spectra of the solar-analog
calibrator GSPC P-330E with the same extraction parameters as WASP-69 b to
minimize errors due to aperture corrections and/or field-dependence
differences with the NIRCam Wide Field Slitless calibration observations. We
used calibration program 1076 observations 1 and 2 for the F322W2 and F444W
filters, respectively. We used the CALSPEC model for GSPC P-330E
(p330e_mod_006.fits) to convert from DN/s to physical flux units by dividing
the stellar model by the observed count rates.
Finally, we multiplied this calibration factor by the observed count rate to
calculate the calibrated NIRCam stellar spectrum of WASP-69 b. Program 1076
observation 2 was take at a different field point that was separated by 94
pixels in the detector X direction from the science observations, so it
necessitated interpolating the F444W calibrator to the same wavelengths, which
was a shift of about 0.092 µm. An improved model for P-330E is being prepared
(Rieke & et al., 2023) that modifies the CALSPEC model by up to 1.5% at 2 µm
but this is below the level of systematic errors, such as non-linearities in
our measured absolute fluxes. We also compared the F322W2 calibration to the
value when extracting the spectrum with the jwst pipeline with the photometry
step applied, an imaging flat field jwst_nircam_flat_0610.fits, aperture
correction jwst_nircam_apcorr_0003.fits, and pixel area
jwst_nircam_area_0054.fits from CRDS and find agreement to within 3% with the
jwst pipeline correction being 3% higher than the method with a ratio to GSPC
P-330E.
For the MIRI observations, we re-ran Eureka!’s Stage 2 with the photometric
calibration step turned on and then used Eureka!’s Stage 3 optimal extraction
on the _cal.fits calibrated data product using the exact same procedure we
used to produce the MIRI lightcurve. The Eureka! optimal extraction allowed
for better interpolation over bad pixels than a simple box extraction.
Additionally, we analyzed the MIRI/LRS SLITLESS calibration observations of HD
167060, HD 106252, and HD 37962 using the exact same methods as for our
WASP-69b science data to improve the flux-calibration of our observations. We
describe the use of these standard star observations below.
We compared the measured absolute fluxes of the star from before eclipse to
stellar models, as shown in Figure 4. While the measured fluxes from before
eclipse include the planet emission, the planet contributes less than 0.3% to
the total flux so we ignore it at the level of calibration and systematic
errors (several %). We use the stellar modeled flux at the photosphere of the
star (i.e. $\pi$ times the intensity) and multiply it by the square of the
ratio of the stellar radius (0.813 R⊙, Anderson et al., 2014) to the Gaia DR3
distance (50.2871 pc, Gaia Collaboration et al., 2023). We were able to
confirm that a BOSZ model accurately predicted the spectrum to a 5% level, but
required an additional multiplicative factor to best match the data, which is
1.0266. In other words, the combined flux multiplicative factor was
1.36399$\times 10^{-19}$ times the modeled flux at the stellar photospheric
“surface”, which is $\pi$ times the modeled intensity. The multiplicative
offset of 1.027 can either be due to calibration flux uncertainties or
physical system parameter differences such as the distance to the system and
radius. Given 2% uncertainties in the flux calibration of the NIRCam slitless
grism mode and MIRI LRS mode (STScI, 2016, accessed on 2024-02-01), this
multiplication factor is within the calibration errors of both modes. The
strength of the CO absorption feature near 4.5 µm is also slightly deeper by
1-3% in the model as compared to the measured flux, either due to model
systematics or differences in stellar abundances. We also used our extracted
and calibrated spectra for HD 167060, HD 106252 and HD 37962 to check for
systematic calibration errors. We used the average calibration factor (from Jy
to DN/s) for these three stars using CALSPEC models and used this as an
alternative calibration factor for WASP-69 b’s calibration. In this re-
normalized spectrum, the drop from 5-7 µm is still present, the residuals
between 10-12 µm are smaller and the overall spectrum is shifted upward by 5%.
Given that the stellar model does not show any large ($>3\%$) discrepancies
with the calibrated spectrum, we can rule out the possibility that debris
disks can significantly dilute the spectrum of the planet at the measured
precisions in $\sigma_{F_{\rm p}/F_{*}}$ (3-16%, where $F_{\rm p}$ is the flux
of the planet and $F_{*}$ is the flux of the star and $\sigma$ represents the
uncertainty on the ratio of the two fluxes at the binned resolution).
We also checked our absolutely calibrated spectrum against 2MASS (Skrutskie et
al., 2006) and WISE (Wright et al., 2010) photometry. The 2MASS and WISE
photometry show slightly elevated fluxes as compared with the NIRCam grism
spectra. It is possible that there are residual non-linearity effects or the
brighter-fatter effect (Plazas et al., 2018) that degrades the linearity
correction of individual pixels. A general analysis of the absolute
calibration well below JWST requirements of 10% accuracy for spectroscopy
(Gordon et al., 2022) is beyond the scope of this work. However, given that
there are no large deviations from the stellar model, no significant evidence
for dust excesses, a small discontinuity between NIRCam F444W and MIRI LRS and
future improvements to the JWST flux calibration ahead (Gordon et al., 2022),
we proceed in using the stellar model and multiply it by $F_{\rm p}/F_{*}$ to
derive the planet’s dayside flux.
## 4 General Properties of WASP-69 b’s Emission Spectrum
Figure 5 shows the combined Eureka! MIRI and tshirt NIRCam emission spectrum,
with NIRCam data binned to lower resolution for visualization purposes. The
bins for visualization are about 0.05 µm for NIRCam, as compared to the
original lightcurve bins of about 0.01 µm with the MIRI LRS binning of 0.25 µm
already sufficiently low enough for visualization purposes. Figure 5 also
shows some representative blackbody planet spectra. For these blackbody
spectra, we multiply the Planck function by the planet-to-star radius ratio
squared from Table 2 and divide this by the stellar intensity from our BOSZ
stellar model (Bohlin et al., 2017) with a stellar effective temperature of
4750 K, log(g)=4.5 log(cm s-2), described in Section 3.4. The measured
spectrum clearly is not well fit by a blackbody and shows absorption features
and crosses from high to low blackbody temperature from short to long
wavelengths.
We also show the Spitzer secondary eclipse values for the 3.6 µm and 4.5 µm
bandpasses in Figure 5 (421 $\pm$ 29, 463 $\pm$ 39; Wallack et al., 2019) and
that they are shallower than the NIRCam spectroscopy. We calculate the
synthetic photometric eclipse depths using our JWST emission spectra and a
photometric response curve interpolated to a common grid to compare the
results between JWST and Spitzer. We find synthesized eclipse depths of 542
$\pm$ 7 ppm and 644 $\pm$ 7 ppm for the 3.6 µm and 4.5 µm bandpasses
respectively, assuming that each pixel is statistically independent. In both
bands, the JWST NIRCam eclipses are significantly deeper than the Spitzer
values (Wallack et al., 2019) of 421 $\pm$ 29 ppm and 463 $\pm$ 39 ppm for the
3.6 µm and 4.5 µm bands, respectively. For context, a previous comparison with
the Spitzer IRAC 3.6 µm eclipse depth and the synthetic NIRCam IRAC 3.6 µm
eclipse depth for WASP-39 b indicated consistency to within the 1 $\sigma$
uncertainty (Ahrer et al., 2023); however the Spitzer uncertainty on this
transit depth is 176 ppm in Sing et al. (2016), so the 120 and 180 ppm offsets
we find in WASP-69 b would not be significant for WASP-39 b. Bean et al.
(2023) find a 4.4$\sigma$ discrepancy with the 3.6 µm eclipse depth from Zhang
et al. (2018), ie 84 ppm $\pm$ 19 ppm for the hot Jupiter HD 149026 b. Some of
the difference we find between NIRCam and Spitzer IRAC for WASP-69 b could be
due to red noise in the Spitzer lightcurves, which exhibited deviations from
$1/\sqrt{N}$ statistics (where the noise for $N$ averaged data points is
$1/\sqrt{N}$ times the non-averaged data). This was observed for the second
visits in both the 3.6 µm and 4.5 µm filters (Wallack et al., 2019). Another
possibility is stellar or planet variability and/or a different set of
systematics with JWST NIRCam versus Spitzer IRAC.
Given the large dynamic range of the eclipse depth (Fp/F∗) across our full
spectrum from 2 µm to 12 µm, it hard to see how individual gases affect the
spectrum and see where atmospheric models perform well. Instead, it is
preferable for visualization purposes to compare the brightness temperature of
the planet with the brightness temperature of atmospheric models because its
dynamic range is much smaller. We calculate the brightness temperature
spectrum for WASP-69 b in the following manner: First, we average the
intensity of the star from the BOSZ stellar model described in Section 4
across each wavelength bin where the intensity is evaluated in wavelength
space. Second, we multiply this stellar intensity by the planet-to-star flux
ratio and by the square of the star-to-planet radius ratio to evaluate the
planet intensity. Third, we invert the Planck function in wavelength space to
calculate the associated blackbody temperature that corresponds to the planet
intensity. Fourth, we propagate intensity error intervals to brightness
temperature error intervals by linearizing the inverse Planck function (with a
first order Taylor expansion) and evaluating the width of the temperature
uncertainty at its median intensity. The brightness temperature (as well as
the binning) are used for illustrative purposes to better view the measured
spectrum and model over a wide dynamic range, but the original lightcurve
wavelength bins (0.01 µm bins, rounded to the nearest pixel for NIRCam
spectroscopy, a filter bandpass from F210M and 0.25 µm for MIRI LRS) were used
to fit the models and perform retrievals using the eclipse depth (Fp/F∗).
The brightness temperature spectrum of WASP-69 b is shown in Figure 5 (second
panel from the top and bottom panel) for the binned version of the spectrum
for visual clarity. Additionally, we show the relative opacities for abundant
and observable gases at the temperature of WASP-69 b in the second from the
bottom panel. The brightness temperature spectrum more clearly shows the
spectral features of H2O near 2.8 µm and 6.7 to 8.0 µm, and CO2 near 4.3 µm.
The red edge of the CO2 feature also shows excess absorption due to CO near
4.7 µm. CH4 is expected to be highly abundant in chemical equilibrium at the
zero albedo full redistribution equilibrium temperature of WASP-69 b (963 K)
for near-solar composition at 0.1 bar (Moses et al., 2013). However, our data
do not show any strong CH4 features near 3.3 µm nor 7.7 µm.
Figure 5: The emission spectrum of WASP-69 b shows significant absorption
features from molecules in its atmosphere and a trend from high to low
brightness temperature from 2.1 µm to 12µm. Top: Binned emission spectrum of
WASP-69 b in terms of planet-to-star flux for the NIRCam and MIRI data,
colored by their respective filters. The full higher resolution NIRCam data
(which are used for atmospheric retrieval) are shown as points and the binned
spectra are shown for illustrative purposes as lines with error bars. The MIRI
full resolution is the same as the binned one due to its intrinsically lower
resolution. Synthetic photometry is calculated for the IRAC 3.6 and 4.5
channels (gray circles) indicates larger eclipse depths than Spitzer published
measurements (purple points with error bars Wallack et al., 2019). Blackbody
planetary spectra for example temperatures are shown as black lines. Middle:
The spectrum is poorly fit by a Clear 1D Homogeneous model (dashed blue line)
where it over-predicts the flux at MIRI wavelengths (5-12 µm), over-predicts
the 3.9 µm flux and under-predicts the shortest wavelengths (2.5-3.2 µm). It
is necessary to include the effects of clouds (thin orange line) as well as
dayside inhomogeneities (thick green line) to fit the spectrum of the planet.
Bottom: The brightness temperature spectrum shows absorption signatures of
H2O, CO2 and CO. The relative pressure level where each molecule has an
optical depth of 1 is shown for each model in the style of Mukherjee et al.
(2024). We plot 2 models (described in Section 5, which can alternatively fit
the spectrum of WASP-69 b with either significant reflection (Scattering
Model) or a high altitude cloud (Cloud Layer Model).
## 5 Atmospheric Retrievals
### 5.1 Initial Fits with 1D Atmospheric Models for a Homogeneous Dayside
We interpret the measured emission spectrum with theoretical atmospheric
models beginning with the simplest assumptions. We first attempted to fit the
spectrum under the assumption that the disk-averaged emission from a planetary
dayside can be represented by a single 1D atmospheric column, ie. a
homogeneous dayside:
1. 1.
We initially do not include any aerosols in this model. We term this model the
“Clear One-Region Model.”
As will be described below, the “Clear One-Region Model” provides a poor fit
to the data, so that is why we also include a cloud for the dayside following
results of the transmission spectra that show evidence for aerosols (e.g.
Khalafinejad et al., 2021; Estrela et al., 2021).
1. 2.
In a second fit we term the “Cloudy One-Region Model”, we assume a vertically
uniform gray cloud opacity.
We computed a grid of self-consistent radiative-convective equilibrium
temperature-pressure (TP) profiles using the Extrasolar Giant Planet (EGP)
code (Marley & McKay, 1999; Fortney et al., 2005; Marley & Robinson, 2015;
Thorngren et al., 2019) for possible combinations of atmospheric metallicity
of [M/H]=0.0, 0.5, 1.0, 1.5 and 2.0, C/O ratio of 0.25$\times$, 0.5$\times$,
1.0$\times$ and 2.0$\times$ solar C / O value, and the heat redistribution
factor of $f_{\rm redist}=0.5,0.6,0.7,0.8,0.9,1.0$. The redistribution factor
is $f_{\rm redist}=0.5$ in the case of full redistribution of absorbed stellar
radiation over a solid angle of $4\pi$, whereas $f_{\rm redist}=1.0$ is
dayside-only redistribution of absorbed stellar radiation to a solid angle of
$2\pi$ (ie. a hemisphere).333The Dayside Effective Temperature is
$T_{day}=T_{*}\sqrt{R_{*}/a}(1-A_{\rm B})^{1/4}(0.5f_{\rm redis})^{1/4}$,
where T∗ is the stellar effective temperature, $R_{*}$ is the stellar radius,
$a$ is the semi-major axis and $A_{\rm B}$ is the Bond Albedo. We then use the
open-source radiative transfer code CHIMERA (Line et al., 2013a) to compute
the emission spectrum from the atmospheric RCE grid, where the molecular
abundances at each pressure level are computed by a precomputed equilibrium
chemistry table implemented by CHIMERA in default. We used PyMultinest
(Buchner et al., 2014), a python implementation of the Multinest tool (Feroz
et al., 2009), to obtain posterior distributions for [M/H], C/O, and $f_{\rm
redist}$. The temperature at each pressure level for sub-grid point values of
${\rm[M/H]}$, C/O, and $f_{\rm redist}$ are interpolated from the RCE grid
with the RegularGridInterpolator in Python scipy library, as conducted in Bell
et al. (2023c). We assumed a uniform prior for each parameter and set the
Nested Sampling live points to $500$.
As shown in the second from the top panel of Figure 5, the Clear One-Region
model tested here is unable to explain the emission spectrum of WASP-69b. The
best-fit spectrum has a $\tilde{\chi}^{2}$ of 14.5. The difficulty originates
from the difference in brightness temperature between the NIRCam and MIRI
bandpass. The data reveal the average brightness temperature of $\sim
1050\leavevmode\nobreak\ {\rm K}$ at $2.1$–$4\leavevmode\nobreak\
{\rm{\mu}m}$, which is much hotter than the temperature of $\sim
950\leavevmode\nobreak\ {\rm K}$ at $>5\leavevmode\nobreak\ {\rm{\mu}m}$. The
Clear One-Region model underestimates the brightness temperature at
$2.1$–$3.3\leavevmode\nobreak\ {\rm{\mu}m}$ and overestimates it at
$>5\leavevmode\nobreak\ {\rm{\mu}m}$. This result causes a dilemma for the
one-region model in fitting the data: the atmosphere needs to be hotter to fit
the planet’s emission at $2.1$–$4$ µm but it further worsens the discrepancy
at $>5\leavevmode\nobreak\ $µm. The addition of a gray cloud opacity in the
“Cloudy One-Region Model” shown in Figure 5 helps lower the flux at long
wavelengths and at 3.9 µm but still cannot achieve a high enough brightness
temperature from 2.1 to 3.3 µm and simultaneously low enough brightness
temperature from 6 to 9 µm. In the following sections, we overcome the
difficulty found here by investigating more detailed atmospheric properties.
### 5.2 Models With Additional Parameters and Complexity
Given that our initial fit does not match the observed dayside spectrum of
WASP-69 b well, we added additional complexity to the models. We considered a
variety assumptions about aerosols, reflected light, the temperature-pressure
profiles and dayside inhomogeneities. We used two different radiative transfer
codes to simulate the emission spectra of WASP-69 b that each take as inputs
temperature-pressure profiles and chemical abundances: PICASO (Batalha et al.,
2019; Mukherjee et al., 2023) and CHIMERA (Line et al., 2013b). The
independent sets of radiative transfer models provide robustness to the
results to a particular radiative transfer implementation. We include the
following types of reflection assumptions, aerosols and 3D effects, which are
described in more detail later in this section:
1. 3.
A “Cloudy Two-region model” that includes two different temperature-pressure
profiles and sets of cloud properties between two geographically distinct
regions. Scattering is not significant. (7 free parameters)
2. 4.
A “Scattering” model that includes a wavelength-independent geometric albedo
parameter as a free “knob” to the model, which can approximate the effects of
aerosols without a specific assumption about their composition or sizes. This
adds relatively more flux at short wavelengths than long wavelengths because
the thermal contribution decreases at short wavelengths but the scattering
term remains constant. (10 free parameters)
3. 5.
A “Cloud Layer” model that includes a distribution of silicate condensates
that can emit at a different brightness temperature than the underlying
molecular emission. There is no free parameter for the albedo like Model 4,
but there is a modest level of scattering due to molecules and silicate
condensates. (14 free parameters)
For model 3 we use the Extrasolar Giant Planet (EGP) code (Marley & McKay,
1999; Fortney et al., 2005; Marley & Robinson, 2015; Thorngren et al., 2019)
with radiative transfer calculated by the CHIMERA code (e.g. Line et al.,
2013b). For models 4 and 5 we use the PICASO code (e.g. Batalha et al., 2019;
Mukherjee et al., 2023). We introduce more detailed model descriptions in what
follows.
#### 5.2.1 Cloudy Two-Region Model
It has been known that modeling exoplanet daysides with a single TP profile,
as adopted in the One-Region models, has an inability to fit the emission
spectrum if there is a strong temperature contrast in the dayside (Feng et
al., 2016; Taylor et al., 2020). In fact, the mock retrieval of Feng et al.
(2016) demonstrated that a single TP profile cannot produce bright emission at
$\sim 2$–$3\leavevmode\nobreak\ {\rm{\mu}m}$ produced by a dayside with strong
temperature contrast (see their Figure 4), which is reminiscent of the
difficulty found in Section 5.1. Fitting a homogeneous dayside model to planet
with an inhomogeneous dayside can also result in biased abundances and
spurious molecular detections (Feng et al., 2016; Taylor et al., 2020).
To account for the dayside inhomogeneity, we introduce the Two-Region model
that computes the emission spectrum using 2 TP profiles. The setup of the
model is largely the same as the one-region model presented in Section 5.1. We
utilize the grid of radiative-convective equilibrium TP profiles used in the
one-region models (see Section 5.1), although we have extended the heat
redistribution factor to $f_{\rm redist}=2$ because the dayside now has a
local region that is hotter than the dayside average. We split the dayside
into hot and cool regions and compute the the emission spectrum from each
region using the CHIMERA. The observable spectrum is then computed as
$F_{\rm obs}=x_{\rm hot}F_{\rm p}(f_{\rm redist,hot},\kappa_{\rm
cld,hot})+(1-x_{\rm hot})F_{\rm p}(f_{\rm redist,cold},\kappa_{\rm
cld,cold}),$ (1)
where $x_{\rm hot}$ is the areal fraction of the hotter region of the dayside,
and we have assigned different values of the heat redistribution factors
$f_{\rm redist,hot}$ and $f_{\rm redist,cold}$ for each hot and cool regions.
$\kappa_{\rm{cld,hot}}$ and $\kappa_{\rm{cld,cold}}$ are the gray cloud
opacities of the hot region and cold region, respectively. Since the cloud
properties can also be very different at each region due to the distinct TP
profiles, we also independently assign the gray cloud opacity for hot and cold
regions. We note that the gray cloud is treated as an purely isotropic
scattering opacity source in CHIMERA. The atmospheric metallicity and C/O
ratio are expected to be horizontally uniform and thus are assumed to be the
same in the hot and cool regions. As in the One-Region models, we use
PyMultinest with 400 live points to obtain the posterior distribution of 5
climate parameters: [M/H], C/O, $f_{\rm redist,hot}$, $f_{\rm redist,cold}$,
and $x_{\rm hot}$ and 2 cloud parameters: $\kappa_{\rm redist,hot}$,
$\kappa_{\rm redist,cold}$ for a total of 7 parameters. We note that the model
includes a gray cloud for computing the spectrum, but ignores the radiative
effects of clouds on TP profiles. For the priors on the temperature of the
cold region, we restrict $f_{\rm redist,cold}$ to be between 0.5 and 1.0 (ie
between full and and no heat re-distribution). For the hot region, we set a
wider prior on $f_{\rm redist,hot}$, from 0.5 and 2.0. The upper bound
corresponds to the substellar point temperature in pure radiative equilibrium.
This is higher than the no redistribution case discussed above, because the
latter corresponds to the averaged dayside temperature, whereas the $f_{\rm
redist,hot}=2$ bound corresponds to the maximum possible local temperature.
#### 5.2.2 Scattering Model
We use the open-source PICASO model (Batalha et al., 2019; Mukherjee et al.,
2021) to model the planet’s atmosphere within a 1D Bayesian retrieval
framework. The temperature-pressure profile is assumed to be in an analytic
form of Line et al. (2013b), with five free parameters: the equilibrium
temperature Teq (simply a parameter and not necessarily consistent with the
planet’s equilibrium temperature), log(g), Planck mean infrared opacity
$\kappa_{\rm IR}$, internal temperature $T_{\rm int}$ and the relative
fraction of the second visible stream in the two-stream approximation
$\alpha$. We assume thermochemical equilibrium throughout the atmosphere in
our retrieval framework. For a given $T(P)$ profile, atmospheric metallicity,
and C/O ratio, the FASTCHEM chemical equilibrium model (Stock et al., 2022) is
used to generate the abundance profiles of gases like CO2, CO, CH4, NH3, H2O,
etc. The solar composition elemental abundances from Lodders et al. (2009) are
scaled for different metallicities and C/O ratios within the retrieval. The
C/O ratio for a given metallicity is varied by scaling the C and O elemental
abundances such that the total C+O remains unaltered. In addition to
thermochemical equilibrium gases, we also retrieve on a constant (with
altitude) SO2 abundance to capture possibility of photochemically produced SO2
(Tsai et al., 2022). This SO2 abundance is not self-consistent with the
chemical equilibrium chemical profiles, but it gives a clue whether
photochemical calculations such as with VULCAN (Tsai et al., 2021) may be
necessary. These chemical abundances along with the $T(P)$ profile are then
used to generate the 1D emission spectrum of the planet with PICASO. To
account for the horizontal temperature contrast, the model also adds a
dilution parameter ($s_{\mathrm{dilute}}$), which is the fractional area of
the planet’s dayside that is emitted by this 1D model. This smaller emitting
area can compensate for the 2D structure of the planet and mitigate the biases
in retrievals (Taylor et al., 2020).
In the scattering model retrieval setup, we include a constant reflected light
term within our retrieval model in addition to the thermal component to fit
the eclipse spectrum. The planet-to-star flux ratio spectrum in this case is
defined as,
$\dfrac{F_{\rm planet}(\lambda)}{F_{\rm star}(\lambda)}=A_{\rm g}\dfrac{R_{\rm
p}^{2}}{a^{2}}+\dfrac{F_{\rm planet,thermal}(\lambda)}{F_{\rm star}(\lambda)}$
(2)
where $A_{\rm g}$ is the geometric albedo of the planet, Rp is the planet
radius, and $a$ is the semi-major axis. We choose the $A_{\rm g}$ term to be
wavelength-independent to simply asses the amount of starlight reflection that
would be needed to explain the near-infrared data, even though such a
wavelength-independent reflected light term is not very realistic and we do
not include the effect that scattering can have on radiative transfer. For the
prior on the geometric albedo, we put in a strict upper limit of 0.67 on
$A_{\rm g}$. We use this upper limit because if the $A_{\rm g}$ is greater
than 0.67, then the Bond Albedo of the planet would be higher than 1 if the
planet is assumed to be Lambertian. We also note that this retrieval setup can
violate energy conservation if the thermal component plus the reflected light
component are larger than the total incident energy on the planet. We sample
the posterior distribution with the Dynasty code (Speagle, 2020).
#### 5.2.3 Cloud Layer Model
We use a second model setup with the same PICASO framework, equilibrium
chemistry, parameters for the metallicity, C/O ratio, and analytic Line et al.
(2013b) Temperature-Pressure profile but with a cloud deck. We also use the
Dynasty sampler (Speagle, 2020). Our cloud deck setup attempts to fit the data
without an arbitrary free Geometric Albedo that allows high values of
reflected light. Instead, we simulate the (small) contribution to scattering
light from molecular Raleigh scattering and condensate clouds and the (larger)
effects on radiative transfer of planet thermal emission that the clouds can
have. We include a parametric form of condensate cloud deck in this retrieval
setup, which can be used with any cloud species’ optical properties. The cloud
is parameterized assuming a log-normal cloud particle size distribution, where
$r_{\rm mean}$ is the geometric mean (and also the median) of the particle
size distribution and $\sigma$ is the geometric standard deviation the
particle size distribution as in Ackerman & Marley (2001) Equation 9. We
assume that $r_{\rm mean}$ and $\sigma$ remain the same for all cloudy
atmospheric layers and calculate the layer-by-layer cloud optical depth
$\tau_{\rm{cld}}$, asymmetry parameter $g_{0}$, and single scattering albedo
$\rm{w}_{0}$ using Mie scattering calculations. These layer-by-layer Mie
properties are then scaled using the normalization factor–$ndz$. This lets us
calculate the cloud Mie properties at the base of the cloud deck. We fit for a
base pressure – $P_{\rm base}$ deeper than which the cloud optical depth is
zero. At pressures smaller than $P_{\rm base}$, the layer-by-layer optical
depth of the cloud deck is calculated using $r_{\rm mean}$ and $ndz$ is scaled
using the scaling factor,
$f=e^{-{f_{\rm sed}}z/H}$ (3)
where $z$ is altitude relative to the cloud deck base. We set the reference
altitude $z$ arbitrarily and the scale height $H$ to be a constant throughout
the atmosphere and use $f_{\rm sed}$ as a free fitting parameter. We fit for
the five free cloud parameters $log(r_{\rm mean})$, $\sigma$, $P_{\rm base}$,
${f_{\rm sed}}$, and $ndz$ in this setup. We also include a dilution parameter
($s_{\mathrm{dilute}}$) to approximate the 3D effects of day-to-terminator
temperature contrasts, as for the Scattering model described in Section 5.2.2.
We apply our cloud layer retrieval setup using optical properties of Enstatite
(MgSiO3) (Scott & Duley, 1996). We initially tried other sulfide compositions
such as Na2S but found unrealistically high temperatures that deviated
significantly from radiative-convective equilibrium grids.
### 5.3 Retrieval Results
#### 5.3.1 Two-Region Retrieval
Figure 6: Retrieved Posterior Distribution for the Two-Region Model. The
physical constraints of the model give tight bounds on the metallicity and C/O
ratio of the planet. The hotter region fills about 68% of the dayside in this
retrieval. Figure 7: Left: The Two-Region Model can fit the spectrum with a
hot temperature-pressure profile that covers 68% of the dayside area and a
cooler cloudier region that contributes negligibly to the planet flux. The
binned data are shown as gray diamonds with error bars. The individual spectra
from the hot component (gray dotted line) and cold component (gray dashed
line) are combined via Equation 1 to give a best fit spectrum (thick cyan
line). A model with no reflected light and only thermal flux (thin red line)
and the hot component’s emission only (dotted orange line) show that reflected
light and the cold component’s emission are both negligible compared to the
thermal emission from the planet’s hot region. This also means that the
dilution parameter used in the Cloudy Layer and Scattering models should give
a very similar results as two separate regions. Right: Our measured spectrum
shows signatures of H2O, CO2 and CO molecules as well as clouds to both
suppress the emission from the cold region and the 4 µm and 9–11 µm emission
from the hot region. We remove one molecule at at time and plot the change in
spectrum as a shaded region spanning from the best fit model (cyan line) to
the molecule-removed and cloud-removed spectra (color-shaded regions). The
binned data data (gray diamonds) are the same as on the left plot. We note
that removing H2O can decrease the flux as compared to the best-fit model
because it increases the single scattering albedo and thus decreases the
emissivity of the photosphere.
The introduction of a dayside inhomogeneity greatly improves the model ability
to explain the data. Figure 5 shows the median emission spectrum of the two-
region model. While One-Region models over-predicts the long wavelengths and
under-predict the short wavelengths, the Two-Region model better fits the
overall shape of the spectrum (see Figure 5 and 7). This result demonstrates
that it is critical to accounting for the temperature contrast on the day
side, as suggested by Feng et al. (2016) and Taylor et al. (2020). Figure 6
shows the posterior distributions of the two-region model. We retrieved a
super-solar metallicity of [M/H] = 1.01${}^{+0.08}_{-0.06}$ and moderately
super-solar C/O ratio of $0.79^{+0.03}_{-0.03}$. It should be noted that the
uncertainties of the parameters retrieved here are likely underestimated, as
our two-region model relies on the TP profiles interpolated from the 1D RCE
grid, which has less flexibility to change the TP profile compared to
parameterized TP profiles. It is also interesting to mention that the cloud
opacity retrieved for the cold region is much higher than that for the hot
region, which indicates a strong cloudiness contrast on the dayside.
To further elaborate on the results of the two-region model, the left panel of
Figure 7 shows the median emission spectrum along with individual emission
spectra from the hot and cold regions. The data can be well fitted by the hot
region with an area fraction of $68\%$. A comparison between the median
spectrum with the diluted hot region spectrum further reveals that the
emission from the cold region has negligible impacts on the total thermal
emission from the dayside, which is attributed to thick clouds in cold
regions. Thus, our result indicates that the overall shape of the observed
spectrum is controlled by the emission from the hot regions where it is hotter
and less cloudy as compared to the cold regions. We also note that the median
spectrum is invariant if we omit the reflected light component, indicating
that the reflected light is negligible in the Two-Region model. Given that the
cold region’s flux is negligible due to thermal scattering in the atmosphere,
the temperature on the cold region largely unconstrained by the data. The
$f_{\rm redist,cold}$ parameter pushes up against our prior of 0.5 (full
redistribution of heat) as shown in Figure 6. Thus, we only find an upper
limit on the redistribution factor and the data do not probe the deeper layers
as will be discussed in the contribution functions described in Section 7.3.
That being said, the temperatures implied by $f_{\rm redist}$ (1220$\pm$12 K
in the hot region and $\sim$1030 K in the cold region) are roughly consistent
but slightly below the temperature contrast in a General Circulation Model for
WASP-69 b in a cloudless simulation (Mehta & Parmentier, 2024).
The observed spectrum shows the absorption features of several molecules. As
shown in the left panel of 7, the spectrum clearly shows the absorption
feature of CO2 at ${\sim}4.3\leavevmode\nobreak\ {\rm{\mu}m}$ that is a strong
indicator of a high metallicity atmosphere (e.g., Moses et al., 2011; JWST
Transiting Exoplanet Community Early Release Science Team et al., 2023). The
spectrum also indicates the presence of CO that is needed to explain the
relatively low flux at ${\sim}4.6\leavevmode\nobreak\ {\rm{\mu}m}$. The H2O
absorption and cloud opacity have comparable impacts and control the overall
shape of the spectrum. On the other hand, the spectrum does not show any
noticeable CH4 features. As will be discussed in Section 7.2 , the atmospheric
TP profile inferred from the observed spectrum leads to an equilibrium CH4
abundance of $\lesssim{10}^{-6}$ at $P<{10}^{-1}\leavevmode\nobreak\ {\rm
bar}$, which is orders of magnitude lower than the abundances of CO and H2O.
This makes the CH4 feature unnoticeable even without the disequilibrium
quenching from the deep hot atmosphere. However, this would not be obvious
under the assumption of a homogeneous model that is closer to the zero-albedo
full-redistribution equilibrium temperature of 963 K for WASP-69 b, which
would contain more significant CH4 absorption.
The observed spectrum indicates the presence of clouds in both the hot and the
cold regions. The hot region needs to be veiled by a moderate amount of
clouds; otherwise, the spectrum around $4\leavevmode\nobreak\ {\rm{\mu}m}$
becomes too bright to explain the observation. The cold regions should be
veiled by clouds much thicker than those in the hot regions to suppress the
emission from the cold region. The model fails to explain the faint emission
at $5\leavevmode\nobreak\ {\rm{\mu}m}$ if the cold regions have clear
atmospheres. We will discuss the potential cloud compositions in the hot and
cold regions in Section 7.3.
Our two-region model still struggles to explain the sudden increase of the
observed emission at ${\gtrsim}9\leavevmode\nobreak\ {\rm{\mu}m}$. Since
removing clouds from the hot region leads to thermal emission comparable to
the observed value at ${\gtrsim}9\leavevmode\nobreak\ {\rm{\mu}m}$, the data
potentially indicate that clouds in the hot region begin to be transparent at
${\sim}9\leavevmode\nobreak\ {\rm{\mu}m}$ through Mie scattering, which cannot
be modeled by our gray scattering clouds in the two-region model. If this is
true, the clouds in the hot region would mainly consist of ${\sim}(9/2\pi)$=
1.4 µm cloud particles. Another intriguing possibility is the sudden decrease
in the cloud’s single scattering albedo at ${>}9\leavevmode\nobreak\
{\rm{\mu}m}$, which we will further discuss in Section 7.3.
Figure 8: Posterior densities of retrieved atmospheric parameters from PICASO
for the scattering model (red) and the cloud layer model (blue). The modeled
parameters (from left to right and down are the atmospheric log metallicity
relative to solar composition ([M/H]), absolute carbon-to-oxygen ratio (C/O),
geometric albedo ($A_{g}$), scattering model only), dayside area dilution
($s_{\mathrm{dilute}}$), Five Temperature-Pressure Profile parameters from
from Line et al. (2013b): the $T_{eq}$ parameter (not consistent with the
planet’s equilibrium temperature, described in the text), $log(g)$,
$\kappa_{IR}$, the internal heat flux temperature $T_{int}$,$\alpha$, and
finally a free-floating SOs abundance (SO2). Stair-step 2D representations of
these posteriors can be found in Figures 16 and 17.
#### 5.3.2 Scattering Model Retrieval
We show the posterior distribution of the Scattering Model spectrum in the
bottom panel of Figure 5. The posterior distributions on different parameters
like atmospheric metallicity, C/O ratio, dilution factor, and geometric albedo
obtained from this retrieval setup is shown in the panels of Figure 8 in blue.
This retrieval setup constrains the atmospheric metallicity of the planet to
be supersolar at with [M/H] = +0.75$\pm$0.15 dex and the absolute C/O ratio at
0.42$\pm$0.16. The dilution parameter is significantly below 1.0, indicating
that the emission is dominated by a central region of higher brightness
approximated by this 1D model. There is also a peak in the posterior abundance
of SO2 near a mixing ratio of $10^{-6}$, which likely improves the fit near
7.5 µm, but it has a long tail extending to lower mixing ratios. Given that
there is no significant absorption feature in the brightness temperature
spectrum at 7.7 µm we do not consider this significant evidence for SO2.
In our scattering model setup, we find that the retrieved geometric albedo
pushes right up against our prior of 0.67 in order to fit the spectra at short
wavelengths, as shown in Figure 8. The retrieved very high reflected component
is much higher than the albedo inferred in other hot Jupiter atmospheres which
are, for example, 0.096 $\pm$ 0.02 for HD 209458 b (Brandeker et al., 2022)
and 0.076 $\pm$ 0.02 for HD 189733 b (Krenn et al., 2023), as measured by
CHEOPS secondary eclipse, with a range of 0.02 to 0.27 for 1500 K to 1700 K a
sample of hot Jupiters (Adams et al., 2022). Recently, an ultra-hot (1980 K)
Neptune LTT 9779 b was found to have a large geometric albedo of
0.80${}^{+0.10}_{-0.17}$ from its CHEOPS optical secondary eclipse depth
(Hoyer et al., 2023), and even at 1980 K, the thermal contribution contributed
negligibly ($<$10) ppm for a 115 ppm eclipse. However, a Neptune-mass planet
like LTT 9779 b may have a different composition than a warm Jupiter like
WASP-69 b so it is unknown if Jupiter-mass planets can attain such high
geometric albedos. The $T(P)$ profile constrained from this retrieval is shown
along with the abundances of key molecular absorbers of CH4, CO, CO2 and, H2O
using blue in Figure 9.
#### 5.3.3 Cloud Layer Retrieval
The fit to the brightness temperature spectra obtained with this setup is
shown in red in Figure 5. The posterior distributions of the parameters are
shown in Figure 8. This setup estimates the planet’s atmospheric metallicity
to be +0.96$\pm$0.2 above Solar and the absolute C/O ratio to be 0.75$\pm$0.1,
as seen in Figure 8. Thus, the compositional constraints depend on the
assumptions about aerosols and the level of reflection from WASP-69 b. As with
the Scattering retrieval described in Section 5.3.2, there is a peak in the
SO2 posterior, but it is at very low abundances where the 7.7 µm feature does
not show up significantly in the spectrum. Our constraints on the layer-by-
layer cloud optical depth, asymmetry parameter, and single scattering albedo
at a wavelength of 10 µm along with the constrained $T(P)$ profile is shown in
Figure 9. The retrieval prefers a silicate cloud deck near $\sim$ $10^{-4.5}$
bar pressure level extending up to $10^{-6}$ bar with a peak cloud optical
depth close to 0.1. The retrieved particle size is very small with a median
radius of 10-6 cm.
Figure 9 shows the retrieved temperature-pressure profiles for the Cloud Layer
model. The temperature-pressure profile crosses the saturation pressure curves
for Na2S and MgSiO3 (enstatite). This indicates that Na2S can form a cloud at
a similar altitude as the inferred cloud deck base $10^{-4.5}$ bar. However,
Na2S does not have the right optical properties to work in our Cloud Layer
model, which requires a high absorption coefficient at MIRI wavelengths ($5$
to $12$µm) as compared to short wavelengths. MgSiO3, on the other hand, has
the right optical properties (e.g. Taylor et al., 2021) to explain the low
($\sim$930 K) brightness temperatures at the MIRI wavelengths without
significantly absorbing the NIRCam wavelengths $2$ to $5$ µm. However, our
retrieved TP profiles for even the warmest Cloud Layer model does not cross
the MgSiO3 condensation curve shown in Figure 9, except at very low altitudes
(10 bars or deeper), so MgSiO3 particles would likely rain down from
observable pressures. The enstatite particles would need to be lofted to the
retrieved 10-4.5 to $10^{-6}$ bar pressure level from below 10 bars. This
would require an extreme level of vertical mixing (such as inferred for
WASP-107 b to loft high altitude silicate cloud particles (Dyrek et al.,
2024)).
The normalized contribution per wavelength is plotted in Figure 10, which
shows that the cloud dominates the emission at longer wavelengths (thus
explaining the lower brightness temperature) while gas near 10-1.5 bar
dominates the emission at short wavelengths (thus explaining the higher
brightness temperature). MgSiO3 clouds produce this wavelength dependence
because the absorption coefficient of these silicates is much larger at
wavelengths longer than 4 µm than at short wavelengths (e.g. Taylor et al.,
2021). The cloud emission at long wavelengths has a lower brightness
temperature because it is emitted by a cooler upper layer. The gas emission at
short wavelengths has a higher brightness temperatures because it comes from a
warmer layer below the cloud deck. We also include reflected light from
Rayleigh scattering by molecular gas and silicate clouds, but their Albedo is
negligible ($\sim$10 ppm or A${}_{g}\approx 0.08$) compared to the thermal
component or the high geometric Albedo inferred from the Scattering Model
(A${}_{g}\approx$0.64) described in Section 5.2.2.
Figure 9: Temperature-Pressure Profile (upper left) and cloud properties for
the silicate cloud model . The TP profile for the Cloud Layer model crosses
the condensation curve for Na2S (dashed gray line) at very high altitudes and
MgSiO3 (dashed pink line) at very deep altitudes, while the Scattering model
crosses neither of these cloud candidate’s condensation curves. While the
Cloud Layer crosses these two condensation curves, Na2S has inefficient
nucleation rates from theoretical models (Gao & Benneke, 2018) and MgSiO3
would require extreme verticle mixing to be lofted to the high altitudes
needed to match the observed brightness spectrum. The upper right 3 panels
show the optical properties of the retrieved silicate clouds: optical depth
$\tau_{cld}$, asymmetry $g_{0}$ and single scattering albedo $w_{0}$ at a
reference wavelength of 10 µm. Figure 10: Contribution functions for the Two
Region model (top row) and the Cloud Layer model (bottom plot). The individual
regions’ contributions are shown for the Two Region model (top left two plots)
as well as the combined full dayside (right plot). A high altitude ($10^{-4}$
to $10^{-5}$ bar) cloud dominate the longer wavelength emission (with smaller
brightness temperatures) while the warmer lower layers dominate at short
wavelengths (with larger brightness temperatures) in both models. The
posterior retrieved Temperature-Pressure Profile for the CHIMERA model (bottom
middle) is shown as a red curve with the temperature at the top axis.
#### 5.3.4 Summary of Model Retrieval Results
Table 3: Summary of retrieved atmospheric metallicity and C/O ratio from each model setup. The most plausible scenarios are the Two-Region (Cloudy) and Cloud Layer retrievals Model | Retrieval Code | Dayside Geometry | TP Profile | Reflected Light | Metallicity [M/H] | C/O ratio | $\tilde{\chi}^{2}$ | log$Z$
---|---|---|---|---|---|---|---|---
One-Region (Clear) | CHIMERA | Homogeneous | RCE Grid | Molecular | $1.30^{+0.02}_{-0.02}$ | $0.11^{+0.02}_{-0.01}$ | 14.5 | $-588$
One-Region (Cloudy) | CHIMERA | Homogeneous | RCE Grid | Molecular | $0.96^{+0.03}_{-0.04}$ | $0.65^{+0.03}_{-0.03}$ | 3.07 | $-439$
Two-Region (Cloudy) | CHIMERA | 2 TP | RCE Grid | Molecular | $1.01^{+0.08}_{-0.06}$ | $0.79^{+0.04}_{-0.04}$ | 1.79 | $-255$
Scattering | PICASO | Diluted | Line et al. (2013b) | Constant AG | 0.75${}^{+0.15}_{-0.15}$ | 0.42${}^{+0.16}_{-0.16}$ | 2.00 | $-253$
Cloud Layer | PICASO | Diluted | Line et al. (2013b) | Silicate | 0.96${}^{+0.20}_{-0.17}$ | 0.75${}^{+0.19}_{-0.10}$ | 1.65 | $-256$
The 5 models considered in this work are summarized in Table 3, with the major
assumptions, derived abundances, reduced chi-squared ($\tilde{\chi}^{2}$) and
Bayesian Evidence (log(Z)). The 1D homogeneous models that were originally
explored in Section 5.1 under-predict the short wavelength planet flux and
over-predicts the long wavelength planet flux thus giving large
$\tilde{\chi}^{2}$ values. The difference in log Bayesian evidence between the
1D Cloudy Homogeneous model and the Two-Region model is 184, corresponding to
a 19 $\sigma$ difference. The three models that best fit the data (with a
$\tilde{\chi}^{2}$ of 2.0 or less) are the the Scattering Model, the Two-
Region Model and the Cloud Layer Model with $\tilde{\chi}^{2}$ of 2.0, 1.79
and 1.65 respectively. The Bayesian evidence favors the Scattering model, but
the extremely high Geometric Albedo of 0.64 is likely unrealistic. The Two-
Region model and Cloud Layer models, by contrast have negligible stellar
reflected light, as shown in Figure 7. While we regard the Scattering Model as
less plausible, it is still consistent with existing JWST data and can be used
to assess the robustness of other conclusions like the composition of WASP-69
b’s atmosphere inferred from the Cloud Layer and Two-Region Models. We further
discuss the synthesized inferences about WASP-69 b’s cloud properties and
atmospheric composition across all models in Section 7.
## 6 Eclipse Mapping
Figure 11: MIRI LRS broadband lightcurves and Residuals for Lightcurve Fits to
the Uniform Map as well as the model difference between non-uniform map and
the uniform map. The non-uniform map allows for a gradient from the sub-
stellar point to the terminator through the Y2,0 and Y1,0 spherical harmonics.
The Bayesian Information Criterion favors the non-uniform map (BIC=184.1
versus BIC=191.1), but higher signal to noise is needed to robustly map
WASP-69 b to high confidence . Figure 12: Posterior maps for the non-uniform
map fits to WASP-69 b’s dayside. The maps only include the Y2,0 and Y1,0
spherical harmonics to sense temperature gradients, but do not reveal location
information. The mean map (Left) and Error (Right) favor a dayside brightness
distribution that is centrally concentrated and inefficient at redistributing
heat. The plus sign shows the peak brightness, which is forced to be at the
substellar point by the limited number of spherical harmonics. .
Motivated by the inhomogeneous dayside inferred by both the Two-Region
retrieval and the Scattering Model’s dilution parameter, we investigated
whether the lightcurves show further and independent evidence of an
inhomogeneous dayside. Eclipse mapping (e.g. Williams et al., 2006; Rauscher
et al., 2007; de Wit et al., 2012; Majeau et al., 2012; Coulombe et al.,
2023b), can use the star as a spatial scanner to map some non-uniform features
of the planet. We used the broadband MIRI lightcurve, which had the highest
signal-to-noise eclipse signal to map the dayside of the planet.
First, it is necessary to use the highest available precision orbital
parameters to decrease the correlations of these parameters with maps. This is
achieved by fitting the publicly available TESS lightcurve from sector 55
taken between 2022-08-05 and 2022-09-01. The posterior parameters we find are
listed in Table 2.
We model the MIRI LRS lightcuve a with starry (Luger et al., 2019) and include
a systematic trend for the lightcurves. The lightcurve is modeled as
$F(t)=F_{a}(t)\left(1+Ce^{-(t-t_{\rm{min}})/\tau}\right)(A+Bx),$ (4)
where $F_{a}(t)$ is the astrophysical variation modeled by starry, $t$ is the
time, $t_{\rm{min}}$ is the minimum (ie. start) time, $\tau$ is an exponential
time constant, $C$ is the exponential amplitude constant and $A$ and $B$ are
polynomial baseline terms. $x$ is the scaled time
$x=2(t-t_{mid})/(t_{max}-t_{min}),$ (5)
where tmid is the mid-time.
We fit the broadband MIRI LRS lightcurve with a uniform map, a spherical
harmonic degree 1 map and a spherical harmonic degree 2 map and a fourth fit
with the spherical harmonics corresponding to dayside brightness gradients
only. We sample the posterior with No-U-Turns sampling with pymc3 (Salvatier
et al., 2016) and the pymc3-ext tools from Foreman-Mackey et al. (2021). All
of our model fits results show that there is excess noise as compared to the
photon and read noise estimate. The best-fit (maximum a-priori) spherical
harmonic degree 2 lightcurve model has a $\tilde{\chi}^{2}$ = 1.05, which is
smaller than the Uniform model, which has $\tilde{\chi}^{2}$ = 1.11, where
$\tilde{\chi}^{2}$ is the reduced-chi-squared metric. We included an error
inflation factor that was fit as a hyper-parameter for the uniform fit and
then that same uncertainty is used in cross-model comparisons. We find that
the theoretical photon and read noise error should be 33 ppm, whereas the
inflated error is 56 ppm. The ratio of the inflated to theoretical error is on
the higher end as compared with other MIRI LRS lightcurves, but less than the
commissioning target L169-9 b (Bouwman et al., 2023).= We also test whether
the extra degrees of freedom in the spherical harmonic degree 2 lightcurve are
justified with the Bayesian Information Criterion (BIC) and find that it is
214.5 for the spherical harmonic degree 2 versus 191.1 for a uniform map,
which favors uniform map due to its 8 fewer free parameters. We repeated an
eclipse mapping fit but with only two free parameters in the spherical
harmonic map besides the overall amplitude: Y2,0, Y1,0, which create a
gradient from the sub-stellar point to the terminator - see Schlawin et al.
(2023a) Figure 5 or Luger et al. (2019) Figure 1, for example. This time the
non-uniform map has $\tilde{\chi}^{2}$ = 0.99 and a BIC=184.0, so both
statistics favor a non-uniform map with dayside temperature gradients. The
spherical harmonic degree 1 map, on the other hand, is not statistically
preferred over the Uniform map.
To better visualize the residuals, we also show the difference between the two
best-fit models and the residuals for a uniform eclipse map fit in Figure 11,
bottom. The lightcurve for the non-Uniform explains some of the 50-100 ppm
deviations in the residuals at ingress and egress. While the evidence for a
non-Uniform map is still at a similar level as the excess noise in the data,
the MIRI LRS broadband lightcurve supports the possibility that a non-Uniform
dayside model may explain the spectrum.
Figure 12 shows the posterior map distributions for the spherical harmonic
degree 2 fits to the lightcurves. The maps suggest significant day-to-
terminator contrasts on the planet. We also analyzed the lightcurve
independently with the eigenmapping method (Rauscher et al., 2018) using the
same systematic model as described in Equation 4 using the ThERESA mapping
code in its single-wavelength implementation (Challener & Rauscher, 2022). The
BIC preferred 2 eigenmap components up to 2nd order spherical harmonics. The
ThERESA maps also significantly favored a non-Uniform dayside map over a
uniform one by $\Delta$BIC of 53.6. The excess noise in the lightcurves beyond
photon and read noise means that caution is warranted when interpreting the
maps. Transit observations (planned for JWST Cycle 2 GO program 3712 and Cycle
3 GO program 5924) and a full phase curve will better measure if this high
contrast is indeed needed for the planet because the transmission spectrum
will constrain the properties around the terminator (ie longitudes of $\pm$ 90
degrees). The black points in Figure 12 are the locations of peak brightness
for random sample posterior draws from the pymc3 No U-Turns sampling of the
lightcurve. The location of peak brightness is consistent with the substellar
point (5$\pm 14^{\circ}$,-5$\pm 24^{\circ}$). The large variance in the peak
brightness longitude is due in part to the baseline we fit to the data in
Equation 4. Non-flat baselines can decrease the precision of the m=1 spherical
harmonic terms, which constrain the peak brightness longitude (Schlawin et
al., 2023a). The ThERESA analysis of the same lightcurve indicates an eastward
hotspot shift of $\sim 20^{\circ}$ with less uncertainty in the ThERESA maps
due to the fewer number of mapping terms and free parameters, but an improved
orbital ephemeris is needed to better constrain the hotspot shift. This
ephemeris will be improved with the upcoming JWST transit observations of
WASP-69 b. As with confirming the central concentration of brightness from the
eclipse maps, a full phase curve would better constrain the longitude of
WASP-69 b’s hotspot.
## 7 Discussion
### 7.1 Energy Balance
Figure 13: The dayside temperature from our JWST spectrum indicates that heat
is not fully re-circulated around the planet. The allowed ranges of dayside
temperature from full redistribution of heat around the planet (solid line) to
zero redistribution of heat (dotted line) for a given Bond Albedo are bounded
within the gray region. The integrated thermal flux from the planet
extrapolated to 0 µm and $\infty\leavevmode\nobreak\ \micron$ is calculated
for our better-fitting models (thick red line, an orange line and and thin
blue line). The high geometric albedo of the Scattering model (blue) which
pushes against the prior of 0.67 strongly indicates a high Bond albedo so we
show values below 0.4 with a dashed line. The Scattering Model barely
conserves energy at the highest Geometric Albedo and would result in a very
extreme dayside-to-nightside temperature contrast..
The total planet emission as measured over the NIRCam and MIRI wavelength
ranges can be used to estimate the dayside temperature, which constrains the
Bond albedo and the heat recirculation efficiency of the planet. For the
energy budget, we assume that internal heat flux is negligible for this
circularized hot Jupiter planet, which is also supported by our internal
temperature $T_{\rm int}$ of less than 200 K shown in Figure 8. We calculate
the dayside effective temperature of the planet for the Scattering model,
Cloud Layer Model and Two-Region models, which were the best fits in terms of
$\tilde{\chi}^{2}$. We integrate the thermal component of a best-fit model
spectrum from 0 to $\infty$, which requires extrapolation beyond our measured
wavelengths from 2.0 µm to 12 µm. We then find the temperature of a spherical
homogeneous blackbody radiator the size of the planet that has the same flux.
This dayside effective temperature can be compared to the incoming radiation
from the host star to derive a Bond albedo. However, the day/night heat
recirculation efficiency cannot be measured with only an eclipse lightcurve of
WASP-69 b. A full phase curve would be needed to measure the global effective
temperature, as has been done for GJ 1214 b (Kempton et al., 2023). We
therefore show the combined constraints on both the Bond albedo ($A_{B}$) and
heat redistribution efficiency in Figure 13. We mark the the allowed dayside
temperatures in the gray region in Figure 13, which are bounded by full heat
redistribution ($f=0.5$ or $\varepsilon=1.0$ in the parameterization of Cowan
& Agol (2011)) and no heat redistribution ($f=\frac{4}{3}$ or
$\varepsilon=0$).
We examine the constraints on the redistribution efficiency, model-by-model
from their dayside temperature. The Scattering Model spans a wide range of
possible redistribituion efficiencies that depend on the Bond Albedo. However,
the high constant-wavelength Geometric Albedo inferred from the Scattering
model 0.64 or higher indicates that the Bond Albedo is also likely large. For
a Lambertian sphere that scatters equally at all wavelengths, the Geometric
Albedo is $\frac{2}{3}$ times the Bond Albedo (Heng et al., 2021). In all four
of the Solar System’s giant planets, the infrared geometric albedo is less
than the Bond Albedo (de Pater & Lissauer, 2001). If the Bond albedo of
WASP-69 b is indeed larger than 0.64 (ie the right hand side of the blue line
in Figure 13, the redistribution efficiency would have to be near $f=1.33$
(ie. no redistribution of heat). However, it is possible (with a preference
for back-scattering starlight), that the geometric Albedo at some wavelengths
exceeds the Bond albedo. In the Solar System giant planets, the visual
Geometric Albedo is on average $\sim\frac{2}{3}$ times the Bond Albedo.
Therefore, we mark all Bond Albedos below 0.4 with a dashed line for the
Scattering model. Given the likely high Bond Albedo for the planet, we
conclude that the Scattering Model’s dayside temperature implies an
inefficient heat redistribution for the planet (with $f>0.87$ or
$\varepsilon<0.56$ in the parameterization of Cowan & Agol (2011)). The Bond
Albedo can be much smaller (near 0.0) for the Two Region and Cloud Layer
models shown in Figure 13 (ie the left hand side of the Figure). However, the
dayside temperatures inferred from these models are significantly warmer, so
even for a Bond Albedo of 0.05, the heat redistribution must be inefficient
($f>0.7$ or equivalently $\varepsilon<0.76$ ) for the Two Region Model and
($f>0.8$ or equivalently $\varepsilon<0.64$) for the Cloud Layer Model. Thus,
in all of the models we fit to the dayside spectrum of WASP-69 b, full
redistribution of heat is ruled out and some significant day-to-night
temperatures are expected for the planet.
Dynamical models can help inform whether inefficient heat redistribution and
high Bond Albedo (thinner blue line in Figure 13) are plausible for WASP-69 b.
In models of zero albedo equilibrium temperatures of 1061 K with no clouds or
hazes, circulation is efficient with temperature contrasts of less than 100 K
between the day and night sides at 1 mbar pressures, but clouds can
significantly enhance the day-to-night temperature contrast at the same
pressure to 500 K as well as alter the photosphere altitude (Roman et al.,
2021). Therefore, depending on the clouds in an atmosphere, there can be a
large variety of day-to-night contrasts and recirculation efficiencies
possible for a planet. The phase curve amplitudes of similar temperature gas
giant planets in short orbits measured by the Spitzer space telescope have
also shown a wide variety of longitudinal brightness distributions are
possible at 1100-1500 K equilibrium temperatures, potentially influenced by
clouds, with some cooler planets like WASP-34 b showing significant day/night
contrasts (May et al., 2022). WASP-43 b also shows evidence for nightside
clouds and a day-to-night brightness temperature that varies between 1520 and
860 K for the two hemispheres (Bell et al., 2024). This is similar for our
emerging picture of WASP-69 b, that there are significant aerosols in the
atmosphere that can enhance the day-to-night contrast of the planet.
Furthermore, the inferred dayside inhomogeneity from the Two Region model
(Section 5.3.1), dilution factor in the Scattering model (Section 5.3.2) and
tentative results of eclipse mapping (Section 6), all point to a central
concentration of high temperature gas as compared to the limbs of the planet,
which would be expected for inefficient heat redistribution.
### 7.2 WASP-69 b’s Atmospheric Composition
All of the models explored in this work give posterior distributions that are
enriched in heavy elements as compared to solar composition. This largely
comes from the constraints on the dominant molecular absorbers in the observed
2 µm to 12 µm region: H2O, CO and CO2. Even the lowest metallicity retrieval
from the Scattering model has [M/H]= 0.75$\pm$ 0.15. The two models which give
the lowest $\tilde{\chi}^{2}$ and (the Cloud Layer and Two-Region models) give
consistent compositional constraints within 1$\sigma$, in addition to showing
similar cloud effects described in Section 7.3. Despite different assumptions
about the 3D geometry of the planet and the temperature-pressure profile(s),
the Cloud Layer and Two-Region retrievals overlap well in inferred chemical
enrichment over solar of [M/H]=0.96${}^{+0.20}_{-0.17}$ and
[M/H]=1.01${}^{+0.08}_{-0.06}$, respectively. Similarly, the posterior C/O
ratios overlap with values of 0.75${}^{+0.19}_{-0.10}$ and
0.79${}^{+0.04}_{-0.04}$ respectively. The Cloud Layer model has broader
compositional constraints because of the more flexible temperature-pressure
profile whereas the Two-Region model uses fixed profiles based on radiative-
convective equilibrium calculations. We did not include radiative feedback
from the clouds in the Two-Region model so the very small standard deviations
in the posterior metallicity and C/O ratio likely under-estimate model
uncertainty. Therefore, we recommend adopting a metallicity of 6$\times$ to
14$\times$ and a C/O ratio of 0.65 to 0.94 (from the Cloud Layer model) for
1$\sigma$ (68%) posterior intervals for modeling and comparative planetology.
We also note that the Scattering Model is not completely ruled out (and has
the largest log(Z) or Bayesian evidence in Table 3) so WASP-69 b’s composition
may be less metal-enriched than the Two-Region and Cloud Layer retrievals
indicate. While we consider this a less likely scenario, it is worth
considering the possibility that WASP-69 b’s atmosphere has a metallicity of
4$\times$ to 8$\times$ solar and a C/O ratio of 0.26 to 0.58 for 1$\sigma$
(68%) posterior intervals. The Scattering Model can be ruled out or confirmed
by measuring the optical reflection from WASP-69 b, which diverges strongly
from the Cloud Layer and Two-Region models below wavelengths of 2 µm, as shown
in Figure 5. CHEOPS (archived), WFC3 UVIS (planned) or JWST NIRISS SOSS
lightcurves of WASP-69 b can help answer this question at short wavelengths.
Until those data are analyzed and published, we carry forward these two
possible compositions for WASP-69 b.
WASP-69 b’s host star also shows enrichment relative to solar composition with
[Fe/H]=0.144$\pm$0.077 (Anderson et al., 2014) and another determination of
[Fe/H]= 0.29 $\pm$ 0.04 (Sousa et al., 2021). Machine-learning
characterization of the spectrum indicates [Fe/H]=0.41 $\pm$ 0.03, [C/H] =
0.27 $\pm$ 0.05 and [O/H]=0.18 $\pm$ 0.07 (Polanski et al., 2022). If indeed
WASP-69 A has a super-solar C/O ratio, this could explain the slightly super-
solar C/O ratio for WASP-69 b using the Cloud Layer and Two Region models.
More characterization of the star should be performed to better constrain its
metallicity and heavy element enrichment. If the star is super-solar in heavy
element enrichment of [Fe/H]=0.29 $\pm$ 0.04 from Sousa et al. (2021), WASP-69
b may only be $\sim$0.7 dex (ie. 5$\times$) above stellar composition. This
would place WASP-69 b more similar in chemical enrichment to Jupiter
($\sim$5$\times$) than Saturn ($\sim$10$\times$) relative to the Sun (Atreya
et al., 2022).
The brightness temperature spectrum plotted in Figure 5 does not show any
obvious signs of a 7.7 µm SO2 feature, that can be very prominent due to
photochemical production of SO2 (Dyrek et al., 2024). The Two Region model
fits the data well in terms of both Bayesian Evidence and $\tilde{\chi}^{2}$
and this model assumes chemical equilibrium and thus essentially no SO2. The
Cloud Layer retrieval has a long tail in the posterior mixing ratio down 10-10
and the Cloud Layer model retrieves a mixing ratio of 10-8, where the feature
size becomes small. Thus, we do not see the significant abundances of SO2
inferred in the hot Jupiter WASP-39 b of 10-5.2 to 10-5.7 (Rustamkulov et al.,
2022; Alderson et al., 2022). One reason could be the moderately high C/O
ratio of 0.65 to 0.94 inferred for WASP-69 b in the Cloud Layer and Two Region
models, which decreases the expected SO2 abundances from photochemical models
(Tsai et al., 2022).
Our observations are potentially sensitive to absorption features from CH4
near 3.3 µm and 7.8 µm, but there are no such features in the emission
spectrum of WASP-69 b, as shown in Figure 5. This is well modeled in our
Scattering, Cloud Layer and Two-Region models, which assume chemical
equilibrium. At the high inferred temperatures of these models ($\gtrsim$1000
K at 10-3 bar), the CH4 abundance is low enough that H2O absorption dominates
and obscures any CH4 features. Note in Figure 5 (second panel from the bottom)
that CH4 doesn’t become optically thick until pressures deeper than 10-1 bar,
which is deeper than where H2O becomes optically thick. In cooler planets,
such as WASP-80 b (850 K), methane presents a strong feature that was observed
in absorption (Bell et al., 2023c). If the terminator and nightside of WASP-69
b is significantly cooler than the dayside, as inferred by our retrievals, CH4
will become a more significant absorber in WASP-69 b’s transmission spectrum.
Our inferred moderately large C/O ratio of 0.65 to 0.94 from the Cloud Layer
and Two Region models would significantly enhance the CH4 abundance in
chemical equilibrium. Continued analysis of existing spectra (Guilluy et al.,
2022; Khalafinejad et al., 2021) and future JWST measurements (e.g. GO
programs 3712 and 5924) will provide better insight into the chemical
processes shaping WASP-69 b’s atmosphere and CH4, now that the dayside
abundances have been constrained in this work.
One other consideration in our abundance inferences from atmospheric
retrievals is that there may be systematic offsets in the JWST emission
spectra. We do not expect significant non-linearity effects in our data
because the difference between the in-eclipse and the out-of-eclipse (0.1% to
0.25% in this case for MIRI, 0.02 to 0.08% for NIRCam) do not appreciably
change the detector well filling factor. However, we did experiment with a
retrieval that allows for the MIRI LRS data spectrum to have an offset
relative to the NIRCam data. We used the Cloud Layer model with an MIRI offset
with an allowed range of -500 ppm to +500 ppm. This retrieval results in a
bimodal posterior distribution with offsets of -80 ppm and +40 ppm that have
significantly different abundances. The -80 ppm offset resulted in a similar
metallicity as the Cloud Layer, Scattering and Two Region models ([M/H]
$\approx$ 1.0, with a moderately lower C/O ratio (C/O$\approx 0.6$). The
higher MIRI offset resulted in a very extended cloud and an order magnitude
higher metallicity. Therefore, if our MIRI data have a systematic offset at
all wavelengths, the inferred composition of the planet can change.
Fortunately, we do not find significant evidence for offsets in our
independent data reductions shown in Figure 3, with a mean MIRI LRS offset
between Eureka! and tshirt of 2 ppm, despite larger differences at individual
data points.
### 7.3 Aerosol Properties
Both the Cloud Layer and Two-Region Models show evidence for clouds that most
significantly affect the long wavelength radiation, as seen in Figure 10. The
Cloud Layer model has a combination of this infrared-absorbing cloud deck
located at 10-4.5 to 10-6 bar and gaseous deeper emission from 10-1 to 10-3
bar with a homogeneous (but slightly diluted) 1D model. The Two-Region model
instead fits the data with separate temperature-pressure profiles and gray
scattering clouds so that the hot region is dominated by the molecular
features near 10-1 to 10-3 bars, and cloud scattering suppresses the flux at
4.0 µm and 7 to 9 µm. The cold region of the Two-Region model has negligible
emission because of the combination of cold temperature and thermal scattering
from clouds (e.g. Taylor et al., 2021) that strongly suppresses the planet
emission. In either case, the averaged dayside properties are best explained
with the inclusion of clouds, and the clouds help reduce the flux at long
wavelengths. The next best model in terms of $\tilde{\chi}^{2}$ the Scattering
model, also would require aerosols to produce the significant reflection
inferred by the geometric albedo parameter A${}_{G}\approx$0.64. The Raleigh
scattering of molecules only provides a geometric albedo of A${}_{G}\lesssim
0.1$, so this would require reflective aerosols in the atmosphere of WASP-69 b
to achieve the inferred geometric albedo. Thus, in all three of the best
models with a $\tilde{\chi}^{2}$ $\lesssim 2.0$, we infer the presence of
aerosols in the dayside of WASP-69 b.
#### 7.3.1 Potential Mineral Cloud Compositions
We now discuss the possible compositions of aerosols that could exist in
WASP-69 b’s dayside, first considering cloud formation from commonly assumed
condensed minerals (e.g. Zhang, 2020; Gao et al., 2021). As shown in Figure 9,
the retrieved temperature-pressure profiles from the Cloud Layer model crosses
the the saturation vapor pressure curves for Na2S and MgSiO3 and the
Scattering model crosses the curve for Na2S within 1 $\sigma$. While the Cloud
Layer model uses MgSiO3 (enstatite) optical properties (Scott & Duley, 1996)
to fit the observed dayside spectrum of WASP-69 b, the inferred cloud that
extends from 10-4.5 to 10-6 bar is far higher than the $\sim$10 bar pressure
where MgSiO3 cloud is expected to form. The situation is the same for other
silicates like quartz (SiO2), which condenses at a similar temperature (Grant
et al., 2023) and forsterite (Mg2SiO4), which condenses at even higher
temperatures (e.g., Morley et al., 2012; Wakeford et al., 2017). This
situation is reminiscent of WASP-107b, for which a recent JWST-MIRI
transmission spectrum indicates the presence of silicate clouds at high
altitudes based on a 9-11 µm feature despite its cool equilibrium temperature
of 740 K. At these low temperatures, silicate clouds would be expected to rain
down to the deep atmosphere (Dyrek et al., 2024). An extreme lofting
mechanism, such as vigorous vertical mixing (Gao & Benneke, 2018; Ormel & Min,
2019) and/or nonspherical cloud particles with high porosity (Ohno et al.,
2020; Samra et al., 2020), would be needed to elevate these particles from the
$\sim 10\leavevmode\nobreak\ {\rm bar}$ to the
10-4.5–10${}^{-6}\leavevmode\nobreak\ {\rm bar}$ pressure levels inferred by
our model retrievals for aerosols in WASP-69 b.
Besides silicate clouds, salt clouds such as KCl have been predicted to form
efficiently thanks to their low surface energy which drastically enhance the
nucleation rate (Gao & Benneke, 2018; Lee et al., 2018; Gao et al., 2020).
However, as shown in Figure 9, the atmospheric temperature profile inferred
from the dayside emission spectrum is too hot to form KCl clouds. Instead, KCl
would have to be formed in the cooler terminator and nightside of the planet
and horizontally transported to the dayside.
As discussed in Section 5.3.3, the Cloud Layer model crosses the saturation
vapor pressure curve of Na2S at a similar pressure level as the inferred
cloud. Nucleation theory suggests that Na2S, ZnS, and MnS nucleate very
inefficiently due to their high surface energy, leading to the predction that
those sulfide clouds are absent in exoplanet atmospheres (Gao & Benneke, 2018;
Gao et al., 2020). However, one should keep in mind that the nucleation theory
involves many uncertainties, such as the contact angles of the nucleating
embryo on the condensation nuclei, and Na2S clouds may form if there are
condensation nuclei suitable for Na2S nucleation (for discussion, see Arfaux &
Lavvas, 2023).
To investigate the possibility of Na2S clouds, we attempted to fit the
spectrum of WASP-69 b with an alternative model with the same configuration as
the Cloud Layer model described in Section 5.2.3 but with Na2S’s optical
properties (Montaner et al., 1979). This Na2S Cloud Layer model required a
temperature-pressure profile with about twice the temperature from a
radiative-convective equilibrium calculation and thus was not a physically
plausible solution. One other challenge for Na2S is that WASP-69 b’s
transmission spectrum shows Na atomic features (Casasayas-Barris et al., 2017;
Langeveld et al., 2022; Khalafinejad et al., 2021), which should be suppressed
for significant levels of Na2S cloud formation. Thus overall, there are a few
candidate cloud particle compositions that could form directly on WASP-69 b’s
dayside at high altitudes but do not nucleate efficiently (Na2S and MnS) and
there are more readily-formed cloud particle candidates (KCl and silicates)
that could form in colder regions or deep in the dayside, respectively, and
transported to the observed dayside pressures. Photochemical haze particles
can also form in conditions like atmosphere of WASP-69 b but are expected to
create a temperature inversion not observed in WASP-69 b’s dayside. We discuss
photochemical hazes in Section 7.3.2.
We next consider atmospheric features of candidate cloud particles that may
explain some of the features in WASP-69 b’s dayside spectrum that are not
fully explained by the Two Region, Scattering and Cloud Layer models.
Inspection of Figure 5 reveals that the MIRI LRS spectrum shows a jump in
brightness temperature at a wavelength of 9.4 µm. This is not the same
wavelength as the known instrument artifact, the shadow-region effect, from
10.6 to 11.8 µm (Bell et al., 2023a), nor is there any evidence for a shadow-
region effect in the lightcurves. Two possibilities for this model-data
disagreement are either an absorption feature from WASP-69 b near 8.6 µm
against a rising continuum or a change in the optical properties of aerosols
at 9.4 µm. Evidence for SiO2 (Quartz) has been found in the hot exoplanet
WASP-17 b from its 8.6 µm absorption feature (Grant et al., 2023). WASP-69 b
also has an apparent 8.6 µm absorption feature compared to the Scattering and
Cloud Layer models as visible in Figure 5. Unfortunately, SiO2 is condensed at
temperatures above our inferred temperature-pressure profile for WASP-69 b,
except at deep pressures more than 10 bar, so it would require extreme
vertical mixing to reach the high altitudes near 10-4.5, as with the enstatite
and forsterite discussed above. Another possibility is that the scattering
properties of a cloud (like in the Two-Region model) change abruptly with
wavelength near 9.4 µm. The single scattering albedo of MgSiO3 also drops off
steeply from near 1.0 at wavelengths of 8 µm. If another type of particle
dropped in its single scattering albedo near 9.4 µm, we expect that it would
fit the spectrum of WASP-69 b well. Thus, we encourage future studies of
particles that can exist in solid form on the dayside temperatures and
pressures of WASP-69 b and also have an 8.6 µm absorption feature or changes
in single scattering albedo near 9.4 µm.
#### 7.3.2 A Lack of Photochemical Haze?
It has been suggested that hydrocarbon photochemical hazes tend to produce
temperature inversions in the atmospheres of exoplanets (Morley et al., 2015;
Lavvas & Arfaux, 2021; Arfaux & Lavvas, 2022; Steinrueck et al., 2023). Our
observed emission spectrum disfavors the presence of such a temperature
inversion, at least in the hot region that controls the spectral shape,
because the CO2 feature unambiguously appears in absorption rather than in
emission. We also independently computed an emission spectrum by taking into
account the haze’s radiative feedback using the TP profiles computed by the
EGP code coupled with a two-moment microphysical haze model (Ohno & Kawashima,
2020) and found that the addition of hazes tends to worsen the model fit for
haze optical constants tested. Namely, we used Titan-like haze compositions
(Khare et al., 1984) and soot (Lavvas & Koskinen, 2017) . The lack of
hydrocarbon hazes could be consistent with the apparent lack of CH4 in the
dayside spectrum and several modeling studies that predicted the decline in
the haze abundance at $T_{\rm eq}\gtrsim 950\leavevmode\nobreak\ {\rm K}$ due
to the conversion of CH4 into CO (Morley et al., 2015; Kawashima & Ikoma,
2019; Gao et al., 2020). On the other hand, WASP-69 b is known to show a steep
spectral slope in optical wavelengths (Murgas et al., 2020; Estrela et al.,
2021), and our results may challenge previous studies that attribute the
optical slope to photochemical hazes (e.g., Lavvas & Koskinen, 2017; Ohno &
Kawashima, 2020). However, we note that the impacts of photochemical hazes on
the TP profile should be sensitive to the actual optical properties of
exoplanetary hazes, which have been highly uncertain to date. Further
understanding of the haze optical properties, such as by experimental studies
(Corrales et al., 2023; He et al., 2023), would be needed to quantitatively
investigate to what extent photochemical hazes might exist in the atmosphere
of WASP-69b.
## 8 Conclusions
We analyzed a 2 µm to 12 µm JWST emission spectrum of WASP-69 b and find the
following:
* •
We observe molecular features in absorption, so the temperature profile
decreases with altitude
* •
We observe features of of CO2, CO and H2O in the atmosphere, but no features
of CH4.
* •
The shortest wavelengths (less than 3 µm) observed in NIRCam have unexpectedly
high brightness temperatures above 1000 K as compared to the longest
wavelengths (more than 5 µm)
* •
There is some marginal evidence of an inhomogeneous dayside for WASP-69 b from
the MIRI broadband eclipse maps.
We model the emission spectrum with a variety of assumptions about the
atmosphere and find the following:
* •
Cloudless 1D homogeneous chemical equilibrium models cannot fit our observed
spectrum well and imply that additional complexity is needed to explain
WASP-69 b’s dayside spectrum.
* •
Aerosols are present in WASP-69 b’s dayside atmosphere because they are needed
in our models to fit the shape of the planet’s 2 µm to 12 µm spectrum,
especially the high brightness temperature at short wavelengths as compared to
long wavelengths and the lack of a strong peak in brightness near 4.0 µm.
* •
We find three models that can explain WASP-69 b’s spectrum 1) a Scattering
Model with a free parameter for the geometric albedo, possibly due to aerosols
2) A Cloud Layer Model that has a high altitude silicate cloud layer and 3) A
Two-Region model that has an inhomogeneous dayside that is a combination warm
and cool components that both contain gray aerosols. The Scattering model
requires unexpectedly high albedos of 0.64, which is less plausible.
* •
The abundances of CO2, CO and H2O point to super-solar concentrations of heavy
elements in the atmosphere. Considering the close agrement between the Cloud
Layer and Two-Region Models, the metallicity of the atmosphere is enriched by
6$\times$ to 14$\times$ compared to solar composition and the C/O ratio is
0.65 to 0.94, but we cannot completely rule out the Scattering model, which
has an atmospheric metallicity of 4$\times$ to 8$\times$ and a C/O ratio of |
# Hydrodynamic limit of multiscale viscoelastic models for rigid particle
suspensions
Mitia Duerinckx, Lucas Ertzbischoff,
Alexandre Girodroux-Lavigne, Richard M. Höfer Université Libre de Bruxelles,
Département de Mathématique, 1050 Brussels, Belgium<EMAIL_ADDRESS>Department of Mathematics, Imperial College London, London, SW7 2AZ, United-
Kingdom<EMAIL_ADDRESS>University of Regensburg, Faculty of
Mathematics, Regensburg, Germany<EMAIL_ADDRESS>regensburg.de University of Regensburg, Faculty of Mathematics, Regensburg,
Germany<EMAIL_ADDRESS>
###### Abstract.
We study the multiscale viscoelastic Doi model for suspensions of Brownian
rigid rod-like particles, as well as its generalization by Saintillan and
Shelley for self-propelled particles. We consider the regime of a small
Weissenberg number, which corresponds to a fast rotational diffusion compared
to the fluid velocity gradient, and we analyze the resulting hydrodynamic
approximation. More precisely, we show the asymptotic validity of macroscopic
nonlinear viscoelastic models, in form of so-called ordered fluid models, as
an expansion in the Weissenberg number. The result holds for zero Reynolds
number in 3D and for arbitrary Reynolds number in 2D. Along the way, we
establish several new well-posedness and regularity results for nonlinear
fluid models, which may be of independent interest.
###### Contents
1. 1 Introduction
2. 2 Doi–Saintillan–Shelley theory
3. 3 Ordered fluid models
4. 4 Statement of main results
5. 5 Small-$\operatorname{Wi}$ expansion of Doi–Saintillan–Shelley theory
6. A Well-posedness of the Doi–Saintillan–Shelley system
7. B Perturbative well-posedness of ordered fluid equations
8. C Derivation of third-order fluid equations
## 1\. Introduction
### 1.1. General overview
Suspensions of rigid particles in Stokesian fluids are ubiquitous both in
nature and in applications and are known to typically exhibit non-Newtonian
behaviors. These systems can be described on different scales: either by
macroscopic non-Newtonian fluid models, or by so-called multiscale kinetic
models, or else on the microscopic scale as suspended particles moving in a
fluid flow. Let us briefly describe these different levels of physical
description:
1. $\bullet$
Macroscopic non-Newtonian fluid models:
An incompressible fluid flow is generally described by the Navier–Stokes
equations,
$\left\\{\begin{array}[]{l}\rho_{\operatorname{fl}}(\partial_{t}+u\cdot\nabla)u-{\operatorname{div}}(\sigma)+\nabla
p\,=\,h,\\\\[2.84526pt] {\operatorname{div}}(u)=0,\end{array}\right.$ (1.1)
where $u:\mathbb{R}^{+}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ is the fluid
velocity field, where the source term $h$ accounts for internal forces, where
$\rho_{\operatorname{fl}}$ stands for the fluid density (constant, say), and
where $\sigma$ is the deviatoric fluid stress (a trace-free symmetric matrix
field). For Newtonian fluids, the stress is linear in the strain rate, that
is,
$\sigma=2\mu\operatorname{D}(u),$ (1.2)
where the strain rate $\operatorname{D}(u)=\frac{1}{2}(\nabla u+(\nabla
u)^{T})$ is defined as the symmetric gradient of the fluid velocity, and where
the viscosity $\mu\geq 0$ is a constant. In contrast, non-Newtonian fluids are
characterized by more complex constitutive laws relating the stress $\sigma$
to the strain rate $\operatorname{D}(u)$, describing various possible types of
nonlinear and memory effects. In _macroscopic_ non-Newtonian models, these
laws are assumed to take some explicit form and are usually fitted
phenomenologically to experimental rheological measurements. In Reiner–Rivlin
and in generalized Newtonian fluid models, the stress $\sigma$ is simply taken
to be a local function of $\operatorname{D}(u)$, meaning that only nonlinear
effects are retained. To further describe memory effects of non-Newtonian
fluids, such as viscoelastic properties, more realistic models rather relate
$\sigma$ and $\operatorname{D}(u)$ via integral or differential equations,
which aim to take into account the dependence of the stress on the fluid
deformation history. Such models that are frame invariant are generically
called _simple fluids_ , of which the celebrated Oldroyd–B and FENE–P models
are particular cases. In the fast relaxation limit, simple fluid models reduce
to the hierarchy of so-called ordered fluid models. We refer to Section 3 for
details.
2. $\bullet$
Multiscale kinetic models for suspensions:
So-called multiscale or micro-macro models go one step away from the pure
phenomenology, towards a microscopically more accurate description of particle
suspensions. More precisely, the macroscopic fluid equation (1.1) gets coupled
via the stress $\sigma$ to a kinetic equation describing the evolution of the
suspended solid phase. The latter is modeled by a particle density function
$f(t,x,n)\in\mathbb{R}^{+}$ at time $t$, where $x$ is the position of
particles and $n$ is their ‘state’: for instance, $n\in\mathbb{R}^{d}$ may
describe the relative position of endpoints of elongated particles (hence
$n\in\mathbb{S}^{d-1}$ in case of rigid suspended particles as will be
considered in this work). The evolution of the particle density $f$ is then
modeled by a Fokker–Planck equation describing transport with the fluid and
diffusive effects. Finally, the coupling to the macroscopic fluid equation is
expressed through an explicit constitutive law
$\sigma=\sigma(f,\operatorname{D}(u))$, which is typically derived from formal
microscopic considerations in dilute regime. Such models describe how the
microscopic state of the particles adapts collectively to local fluid
deformations and how the macroscopic fluid flow gets itself effectively
impacted. Popular models include the kinetic FENE and Hookean dumbbell models
for dilute suspensions of flexible polymers, the so-called Doi model for
suspensions of Brownian rigid rod-like particles, and the
Doi–Saintillan–Shelley for corresponding active particles. We refer to Section
2 for details.
3. $\bullet$
Microscopic models:
At the particle scale, we can formulate a fully detailed hydrodynamic model
describing the motion of suspended particles in the Stokesian background fluid
flow. It takes form of equation (1.1) restricted to the fluid domain, with
Newtonian constitutive law (1.2), coupled with Newton’s equations of motion
for the particles. We refer for instance to [HS23] for a discussion of this
complex dynamics.
From the modeling perspective, while microscopic models are certainly
impractical due to the huge number of particles in real-life systems, one can
argue that multiscale kinetic models are more satisfactory than macroscopic
fluid models as they retain some information from the fluid-particle coupling
and therefore better reveal mechanisms leading to non-Newtonian behavior. Yet,
macroscopic models appeal through their simpler description: in particular,
they are much more accessible for numerical simulations and proner to
comparison with experimental rheological measurements.
Most previous works on particle suspensions have aimed either to study
properties of macroscopic non-Newtonian fluid models or multiscale kinetic
models, or else to derive multiscale kinetic models rigorously from
microscopic particle dynamics, which has indeed attracted considerable
interest in recent years; see Section 1.3 below for references. In the present
contribution, we propose to fill the gap in the micro-macro understanding of
non-Newtonian effects of particle suspensions by further studying the
derivation of explicit macroscopic fluid models from multiscale kinetic models
in suitable regimes.
In some exceptional cases, the kinetic equation describing the state of the
particles in micro-macro models can be integrated out and leads to a closed
equation for the stress $\sigma$ in terms of the strain rate
$\operatorname{D}(u)$. This is for instance the case for the kinetic Hookean
dumbbell model, which is well-known to be formally equivalent to the Oldroyd–B
macroscopic fluid model (see e.g. the recent rigorous analysis in [DS23]). In
general, however, no exact macroscopic closure is available and we can only
hope for perturbative closures to be valid in suitable asymptotic regimes.
In fact, in the regime of weak non-Newtonian effects, a fairly large class of
non-Newtonian macroscopic fluid models is believed to be well approximated by
a special family of models, called _ordered fluid_ models. More precisely, the
latter are expected to be good approximations for all viscoelastic fluids in
the fading memory regime, that is, in the regime when the elastic time-
dependent effects due to suspended particles in the fluid have an inherent
relaxation timescale that is much shorter than the overall timescale of the
fluid flow. This ratio of timescales is the so-called Weissenberg number
$\operatorname{Wi}$. In other words, ordered fluid models arise formally as
expansions of any viscoelastic fluid model at small $\operatorname{Wi}$, which
is sometimes referred to as the retarded motion expansion. A first-order fluid
is a Newtonian fluid, a second-order fluid is a non-Newtonian fluid where
effects of order $O(\operatorname{Wi}^{2})$ are neglected, a third-order fluid
amounts to neglecting effects of order $O(\operatorname{Wi}^{3})$, etc.
In the present work, we focus on the so-called Doi model, which is a
multiscale kinetic model describing suspensions of Brownian rigid rod-like
particles, and we further consider its generalization by Saintillan and
Shelley for active (self-propelled) particles. We rigorously analyze the
hydrodynamic limit of these models in the small-$\operatorname{Wi}$ expansion,
which was extensively studied on a formal level in the physics literature in
the 1970s, see [HL72, Bre74], and we confirm the asymptotic validity of
ordered fluid models in this setting. As a prerequisite, the justification of
the asymptotic expansion requires a careful study both of the
Doi–Saintillan–Shelley model and of ordered fluid models: in particular, we
establish some new well-posedness and regularity results for these nonlinear
viscoelastic fluid models, which we believe are nontrivial and of independent
interest. Moreover, as the Doi–Saintillan–Shelley model can itself be derived
from a microscopic hydrodynamic model (at least formally), see Section 1.3,
our derivation of macroscopic ordered fluid models comes together with
explicit expressions for rheological parameters in terms of microscopic
characteristics of the underlying particle suspension.
### 1.2. Informal statement of main results
We start from the following dimensionless Doi–Saintillan–Shelley model
describing suspensions of (active or passive) Brownian rigid rod-like
particles in a Stokesian fluid,
$\left\\{\begin{array}[]{l}\operatorname{Re}\big{(}\partial_{t}u_{\varepsilon}+(u_{\varepsilon}\cdot\nabla)u_{\varepsilon}\big{)}-\Delta
u_{\varepsilon}+\nabla
p_{\varepsilon}=h+\frac{1}{\varepsilon}{\operatorname{div}}(\sigma_{1}[f_{\varepsilon}])+{\operatorname{div}}(\sigma_{2}[f_{\varepsilon},\nabla
u_{\varepsilon}]),\\\\[2.84526pt]
\partial_{t}f_{\varepsilon}+{\operatorname{div}}_{x}\big{(}(u_{\varepsilon}+U_{0}n)f_{\varepsilon}\big{)}+{\operatorname{div}}_{n}\big{(}\pi_{n}^{\bot}(\nabla
u_{\varepsilon})nf_{\varepsilon}\big{)}=\tfrac{1}{\operatorname{Pe}}\Delta_{x}f_{\varepsilon}+\tfrac{1}{\varepsilon}\Delta_{n}f_{\varepsilon},\\\\[2.84526pt]
{\operatorname{div}}(u_{\varepsilon})=0,\end{array}\right.$ (1.3)
in terms of the elastic and viscous stresses
$\displaystyle\sigma_{1}[f]$ $\displaystyle:=$
$\displaystyle\lambda\theta\int_{\mathbb{S}^{d-1}}\big{(}n\otimes
n-\tfrac{1}{d}\operatorname{Id}\big{)}f(\cdot,n)\,dn,$ (1.4)
$\displaystyle\sigma_{2}[f,\nabla u]$ $\displaystyle:=$
$\displaystyle\lambda\int_{\mathbb{S}^{d-1}}(n\otimes n)(\nabla u)(n\otimes
n)\,f\,dn,$ (1.5)
where $\varepsilon:=\operatorname{Wi}>0$ stands for the Weissenberg number,
$\operatorname{Re}\geq 0$ is the Reynolds number, $\operatorname{Pe}>0$ is the
so-called Péclet number, and where the source term $h$ is taken to be smooth
and accounts for internal forces. The state variable $n$ in the kinetic
Fokker–Planck equation describes the orientation of rigid particles on the
unit sphere, hence $n\in\mathbb{S}^{d-1}$: we write $\Delta_{n}$ and
${\operatorname{div}}_{n}$ for the Laplace–Beltrami operator and the
divergence on the sphere $\mathbb{S}^{d-1}$, and we also use the short-hand
notation $\pi_{n}^{\bot}:=\operatorname{Id}-n\otimes n$ for the orthogonal
projection onto $n^{\bot}$. The above system is introduced in detail in
Section 2, where in particular the constants
$U_{0},\lambda,\theta\in\mathbb{R}$ are further described. The Doi model for
passive suspensions is recovered for the special choice $U_{0}=0$ and
$\theta=6$. We set this model for simplicity in a finite box
$\mathbb{T}^{d}=[0,1)^{d}$ with periodic boundary conditions, and consider
space dimension $d=2$ or $3$.
Note that we take into account a non-vanishing spatial diffusion
$O(\frac{1}{\operatorname{Pe}})$ in (1.3), which differs from the setting
usually considered in applications where $\operatorname{Pe}=\infty$: a
nontrivial spatial diffusion $\operatorname{Pe}<\infty$ is actually needed in
the present work for technical well-posedness reasons. Due to this spatial
diffusion, the structure of ordered fluid equations needs to be slightly
adapted, leading to the nonstandard definition of Rivlin–Ericksen tensors in
(1.13) below; see Section 3 for details. This is reminiscent of the version of
the Oldroyd–B model with stress diffusion that is often considered both for
analytical and numerical studies [RT21].
Our main result is the following asymptotic validity of second-order fluid
models. We refer to Theorem 4.2 in Section 4 for a more detailed statement,
including the precise well-preparedness requirement for kinetic initial data,
as well as the explicit expression for the effective coefficients
$\mu,\nu,\gamma_{1},\gamma_{2}$ and their rheological interpretation. New
results on the well-posedness of the kinetic model (1.3) and of the second-
order fluid model (1.13) are postponed to Sections 2 and 3, respectively.
###### Theorem 1.1 (Informal statement of the main result).
Consider either the Stokes case $\operatorname{Re}=0$ with $d\leq 3$, or the
Navier–Stokes case $\operatorname{Re}=1$ with $d=2$. Given an initial particle
density $f_{\varepsilon}^{\circ}\in
C^{\infty}\cap\mathcal{P}(\mathbb{T}^{d}\times\mathbb{S}^{d-1})$ that is well-
prepared in a sense that will be clarified later (see Assumption 4.1), and
given also an initial fluid velocity $u^{\circ}\in
C^{\infty}(\mathbb{T}^{d})^{d}$ with ${\operatorname{div}}(u^{\circ})=0$ in
the Navier–Stokes case, consider a weak global solution
$(u_{\varepsilon},f_{\varepsilon})$ of the Cauchy problem for the
Doi–Saintillan–Shelley model (1.3). For all $T>0$, provided that
$\varepsilon\ll 1\qquad\text{and}\qquad\lambda\theta(1+\operatorname{Pe})\ll
1$ (1.6)
are small enough, the fluid velocity $u_{\varepsilon}$ satisfies
$\displaystyle\begin{array}[]{rlll}\|\nabla(u_{\varepsilon}-\bar{u}_{\varepsilon})\|_{\operatorname{L}^{2}(0,T;\operatorname{L}^{2}(\mathbb{T}^{d})^{d^{2}})}&\lesssim&\varepsilon^{2},&\quad\text{if\leavevmode\nobreak\
\leavevmode\nobreak\ $\operatorname{Re}=0$},\\\\[2.84526pt]
\|u_{\varepsilon}-\bar{u}_{\varepsilon}\|_{\operatorname{L}^{\infty}(0,T;\operatorname{L}^{2}(\mathbb{T}^{d})^{d})}+\|\nabla(u_{\varepsilon}-\bar{u}_{\varepsilon})\|_{\operatorname{L}^{2}(0,T;\operatorname{L}^{2}(\mathbb{T}^{d})^{\small{d^{2}}})}&\lesssim&\varepsilon^{2},&\quad\text{if\leavevmode\nobreak\
\leavevmode\nobreak\ $\operatorname{Re}=1$},\end{array}$
and the particles’ spatial density
$\rho_{\varepsilon}:=\fint_{\mathbb{S}^{d-1}}f_{\varepsilon}(\cdot,n)\,\mathrm{d}n$
satisfies
$\|\rho_{\varepsilon}-\bar{\rho}_{\varepsilon}\|_{\operatorname{L}^{\infty}(0,T;\operatorname{L}^{2}(\mathbb{T}^{d}))}+\|\nabla(\rho_{\varepsilon}-\bar{\rho}_{\varepsilon})\|_{\operatorname{L}^{2}(0,T;\operatorname{L}^{2}(\mathbb{T}^{d})^{d})}\,\lesssim\,\varepsilon^{2},$
where $(\bar{u}_{\varepsilon},\bar{\rho}_{\varepsilon})$ solves the following
(non-standard) second-order fluid equation
$\displaystyle\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+\bar{u}_{\varepsilon}\cdot\nabla)\bar{u}_{\varepsilon}-{\operatorname{div}}(\bar{\sigma}_{\varepsilon})+\nabla\bar{p}_{\varepsilon}\,=\,h+O(\varepsilon^{2}),\\\\[2.84526pt]
(\partial_{t}+\bar{u}_{\varepsilon}\cdot\nabla)\bar{\rho}_{\varepsilon}-(\tfrac{1}{\operatorname{Pe}}+\varepsilon\nu)\Delta\bar{\rho}_{\varepsilon}=O(\varepsilon^{2}),\\\\[2.84526pt]
\bar{\sigma}_{\varepsilon}=(1+\mu\bar{\rho}_{\varepsilon})A_{1}(\bar{u}_{\varepsilon})+\varepsilon\gamma_{1}A_{2}^{\prime}(\bar{u}_{\varepsilon},\bar{\rho}_{\varepsilon})+\varepsilon\gamma_{2}\bar{\rho}_{\varepsilon}A_{1}(\bar{u}_{\varepsilon})^{2},\\\\[2.84526pt]
{\operatorname{div}}(\bar{u}_{\varepsilon})=0,\end{array}\right.$ (1.12)
for some explicit coefficients $\mu,\nu,\gamma_{1},\gamma_{2}\in\mathbb{R}$,
where $A_{1}(u):=2\operatorname{D}(u)=(\nabla u)^{T}+\nabla u$ is the strain
rate and where $A_{2}^{\prime}$ is a (non-standard) inhomogeneous, diffusive
version of the second Rivlin-Ericksen tensor, defined by
$A_{2}^{\prime}(\rho,u)\,:=\,\big{(}\partial_{t}-{\tfrac{1}{\operatorname{Pe}}}\Delta+u\cdot\nabla\big{)}\big{(}\rho
A_{1}(u)\big{)}+\rho\big{(}(\nabla u)^{T}A_{1}(u)+A_{1}(u)(\nabla
u)\big{)}.\qed$ (1.13)
###### Remark 1.2.
A few comments are in order:
1. (a)
The well-preparedness assumption for the initial kinetic data will be
clarified later on in Assumption 4.1. Informally, it amounts to assuming that
initial data are locally at equilibrium with respect to the dynamics of
orientations and are perturbatively compatible with the formal
$\varepsilon$-expansion. It allows to avoid initial boundary layers.
2. (b)
The existence of global weak solutions for the Doi–Saintillan–Shelley model
(1.3), which is assumed to hold in the above statement, is indeed proven in
Section 2.2 below. We further establish a new weak-strong uniqueness principle
for this system, which implies some stability that is at the very heart of the
above result.
3. (c)
Although the second-order fluid model (1.12) is well-known to be ill-posed
whenever $\gamma_{1}<0$ (which is indeed the case for relevant effective
parameters), we can define several well-posed notions of approximate solutions
that only satisfy (1.12) up to higher-order $O(\varepsilon^{2})$ errors. This
discussion is postponed to Section 3, where we present two approaches to fix
this issue: First, we introduce the notion of approximate hierarchical
solutions, which naturally appear as low-$\operatorname{Wi}$ expansions; see
Propositions 3.2 and 3.4. Second, by means of a Boussinesq-type perturbative
rearrangement, we additionally provide a reformulation of second-order fluid
equations in terms of a closed well-posed system; see Proposition 3.3.
4. (d)
The explicit expression for the effective second-order fluid coefficients
$\mu,\nu,\gamma_{1},\gamma_{2}$ is postponed to Section 4, where we further
describe the rheological properties of the obtained macroscopic fluid model.
The expressions for the coefficients agree with those computed in [HL72,
Bre74] in the case of passive suspensions. Moreover, they qualitatively match
experimental data and formal predictions on active suspensions.
5. (e)
A similar result could be obtained with the same approach when starting from
the co-rotational kinetic FENE model for elastic polymers (see e.g. [LM07] for
a review of this system). For conciseness, we do not repeat our analysis in
that setting and leave the adaptation to the reader. ∎
### 1.3. Previous results
We briefly review previous rigorous results related to the multiscale
description of particle suspensions and related systems.
#### Derivation of the Doi–Saintillan–Shelley model
The systematic theoretical study of the effective rheology of suspensions has
been initiated by Einstein [Ein06], who found that passive non-Brownian
spherical rigid particles effectively increase the fluid viscosity by
$\frac{5}{2}\phi\mu$, where $\phi$ is the volume fraction of the particles and
$\mu$ is the viscosity of the solvent. Jeffery [Jef22] studied the analogous
problem for ellipsoidal particles and found an increase of the viscosity
depending locally on orientations of the particles. By slender-body theory, in
the limit of very elongated particles, Jeffery’s viscous stress exactly takes
the form of $\sigma_{2}$ in (1.3); see e.g. [Bre74, KK13]. Starting in the
1930s, there is a vast literature in physics on the rheology of suspensions of
non-spherical rigid particles, see e.g. [Kuh32, Eis33, Pet38, Bur38], but this
early work was restricted to specific fluid flows like simple shear, and
Brownian effects were neglected. Brownian particles were first considered in
[Sim40, KK45, RK50, Sai51], by means of different models and justifications,
finally leading to the additional elastic stress $\sigma_{1}$ in (1.3). These
models were largely reviewed in [LH71, HL72, Bre74]: in particular, the
multiscale kinetic model (1.3) in the passive case ($U_{0}=0$, $\theta=6$)
then entered textbooks such as [DE88, Gra18] and became known in the
mathematical literature as the Doi model. The extension to active suspensions
has been proposed by Saintillan and Shelley [SS08]: by coarse-graining force
dipoles exerted by the particles on the fluid, they derived a further
contribution of the elastic stress $\sigma_{1}$ due to particles’ activity;
see also [HABK08, Hai+09, Sai10, PRB16, DVY19].
On the mathematical side, the derivation of the viscous stress $\sigma_{2}$
from microscopic models has received considerable attention in the last years.
When the fluid is modeled by the Stokes equations and the particle
distribution is given, the effective increase of the fluid viscosity is by now
well understood [HM12, NS20, GVH20, HW20, DG23, GV21, GVH21, DG21, GVM22,
Due22, DG23a]; see [DG22] for a review. In a similar setting, for active
suspensions, the elastic stress $\sigma_{1}$ has been derived in [GL22,
BDG22]. Yet, non-Newtonian effects originate from the retroaction of the fluid
on the particles, that is, from coupling the particle density to the fluid
flow: beyond the derivation of $\sigma_{1}$ and $\sigma_{2}$ in the static
setting, the derivation of kinetic equations for the particle density in the
time-dependent setting is of key interest. First steps in that direction have
been undertaken in [HS21, HMS24, Due23]. Finally, regarding the derivation of
the passive part of the elastic stress $\sigma_{1}$ for Brownian particles, we
refer to [HLM23], where the authors start from a simplified microscopic model
where the particle dynamics is given by Brownian motion and not coupled to the
fluid.
In a different direction, we also mention recent work [HS10, AO22, CZDGV22],
where the (linear) stability and mixing properties of the
Doi–Saintillan–Shelley model have been investigated (neglecting however the
viscous stress $\sigma_{2}$).
#### Macroscopic rheology and formal closures
Most of the above-mentioned works in the physics literature on the derivation
of the Doi model do not stop at multiscale models but also aim at macroscopic
non-Newtonian models, as well as at explicit calculations of stresses in
specific flows such as simple shear. In particular, they typically give
formulas for the shear viscosity in simple shear flow for very large or very
small Weissenberg number $\operatorname{Wi}$. Normal-stress differences (see
(3.6) below) have also been computed at small $\operatorname{Wi}$ by Giesekus
[Gie62], showing that the elastic stress $\sigma_{1}$ does not contribute to
the second normal-stress difference but that the viscous stress does. This was
in contradiction with Weissenberg’s original conjecture that all real-world
fluids must have vanishing second normal-stress difference, a conjecture that
was later falsified also through experiments. Hinch and Leal [LH71, HL72]
systematically computed expansions for the stress in simple shear flow both at
small and at large $\operatorname{Wi}$. They also noticed that their findings
at small $\operatorname{Wi}$ agree with second-order fluid models, but they
did not investigate whether this holds in more general fluid flows.
Although no exact macroscopic closure is available for the Doi model in the
general non-perturbative setting, formal approximate closures have been
studied for example in [DE88, Chapter 8.7]. A particular instance is the
Oldoyd–B model, which is well-known to be remarkably an exact macroscopic
closure for the kinetic Hookean dumbbell model; this formal connection was
made rigorous in [DS23] (in the stress-diffusive case). For the Doi and
Doi–Saintillan–Shelley models, an exact closure is not available, thus raising
the question whether macroscopic closures are at least asymptotically valid in
some scaling regimes: for instance, we will see that the validity of the
Oldoyd–B closure already fails at order $O(\operatorname{Wi})$ for the Doi
model in the small-$\operatorname{Wi}$ regime (see Remark 3.1). The situation
is very similar for kinetic FENE models for elastic polymers: no exact
macroscopic closure holds, but the so-called FENE–P model is still a very
popular formal approximate closure (see e.g. [BDJ80], where the name FENE–P is
attributed in reference to an earlier paper of Peterlin [Pet66]).
#### Small-$\operatorname{Wi}$ regime and hydrodynamic limits
At small $\operatorname{Wi}$, the Doi–Saintillan–Shelley system (1.3)
undergoes a strong diffusion in orientation (cf. factor
$\frac{1}{\varepsilon}=\frac{1}{\operatorname{Wi}}$ in front of $\Delta_{n}$):
to leading order, the orientations of the particles relax instantaneously to
the steady state, which corresponds to isotropic orientations. While this
leading order amounts to a trivial Newtonian behavior, next-order
$O(\operatorname{Wi})$ corrections encode nontrivial non-Newtonian effects
where the stress starts to depend on the local fluid deformation and its
history. The formal perturbative expansion in powers of $\operatorname{Wi}$ is
comparable to the Hilbert expansion method in the Boltzmann theory [Hil12,
Caf80, Gol05, SR09], where one looks for a solution of the Boltzmann equation
as a formal power series in terms of the Knudsen number $\mathrm{Kn}\ll 1$ and
where the leading-order approximation simply leads to compressible Euler
equations. From the closely related Chapman–Enskog asymptotics expansion, one
can (formally) obtain compressible Navier–Stokes equations as a
$O(\mathrm{Kn})$ correction to the compressible Euler system, up to
$O(\mathrm{Kn}^{2})$ errors (see e.g. [Gol05, Section 5.2] or [SR09, Section
2.2.2]). In a similar way, for the Doi model, second-order fluid equations are
obtained in this work as a $O(\operatorname{Wi})$ correction to the Stokes
equations, up to $O(\operatorname{Wi}^{2})$ errors.
Our results can be compared with corresponding results for the Doi–Onsager
model for liquid crystals, which indeed shares some similarities with the Doi
model that we consider in this work. In [EZ06, WZZ15], the macroscopic
Ericksen–Leslie system is derived from the Doi–Onsager model by means of a
Hilbert expansion. Note however that in that case the leading term in the
expansion already yields a non-trivial system, so that higher-order
corrections are not investigated in [EZ06, WZZ15]. We also mention recent
related work on hydrodynamic limits for alignment models [DM08, DFMA17,
DFMAT19, DFL22] and for flocking models [KMT15, KV15, FK18], as well as a
preliminary result for kinetic FENE models for elastic polymers [LPD02].
Note that hydrodynamic limits have been investigated also for various other
kinetic models for particle suspensions in different settings: for instance,
in the context of sedimentation for small inertial particles, let us mention
the inertialess limits studied in [Jab00, Höf18, HKM23, Ert23], as well as the
high-diffusion limit in velocity investigated in [GJV04, GJV04a, MV08, SY20].
### 1.4. Structure of the article
The article is split into five main sections, in addition to three appendices
containing proofs of secondary results:
1. $\bullet$
In Section 2, we give a detailed account of the Doi–Saintillan–Shelley model:
we describe the non-dimensionalization leading to its form (1.3), and we state
a new well-posedness result for this system, see Proposition 2.1, which we
prove in Appendix A.
2. $\bullet$
Section 3 provides an introduction to ordered fluid models, starting with
their basic non-Newtonian rheological properties. We then discuss the ill-
posedness of second-order fluid equations for the relevant range of
coefficients, and we present two approaches to fix this issue: we introduce
hierarchical solutions in Proposition 3.2 and we define a Boussinesq-type
perturbative rearrangement in Proposition 3.3. As we allow the suspended
particle density to be spatially inhomogeneous in general, and as we include a
non-vanishing spatial diffusion $O(\frac{1}{\operatorname{Pe}})$ in the model
for technical reasons, we actually derive ordered fluid models of the form
(1.12), which slightly differ from their standard version: we motivate and
introduce these nonstandard models in Section 3.4 and give the corresponding
definition of hierarchical solutions in Proposition 3.4. The proofs of
Propositions 3.2, 3.3 and 3.4 are postponed to Appendix B.
3. $\bullet$
In Section 4, we provide a more detailed formulation of our main result on the
derivation of second-ordered fluid equations from the Doi–Saintillan–Shelley
theory at small Weissenberg number, see Theorem 4.2. We further comment on the
derivation of higher-order fluid models, for which details are postponed to
Appendix C.
4. $\bullet$
In Section 5 we prove our main result, that is, the rigorous
$\varepsilon$-expansion of the solution $(u_{\varepsilon},f_{\varepsilon})$ of
the Doi–Saintillan–Shelley system (1.3).
### Notations
We summarize the main notations that we use in this work:
1. $\bullet$
We denote by $C\geq 1$ any constant that only depends on the dimension $d$ and
possibly on other controlled quantities to be specified. We use the notation
$\lesssim$ for $\leq C\times$ up to such a multiplicative constant $C$. We
write $\ll$ (resp. $\gg$) for $\leq C\times$ (resp. $\geq C\times$) up to a
sufficiently large multiplicative constant $C$. When needed, we add subscripts
to indicate dependence on other parameters.
2. $\bullet$
For a vector field $u$ and a matrix field $S$, we set $(\nabla
u)_{ij}:=\nabla_{j}u_{i}$, $S^{T}_{ij}:=S_{ji}$,
$\operatorname{D}(u):=\frac{1}{2}(\nabla u+(\nabla u)^{T})$, and
${\operatorname{div}}(S)_{i}:=\nabla_{j}S_{ij}$ (we systematically use
Einstein’s summation convention on repeated indices).
3. $\bullet$
We denote by $\mathrm{d}n$ the (not normalized) Lebesgue measure on the
$(d-1)$-dimensional unit sphere $\mathbb{S}^{d-1}$, and we denote its area by
$\omega_{d}:=|\mathbb{S}_{d-1}|$. Differential operators with a subscript $n$
(such as $\mathrm{div}_{n}$ and $\Delta_{n}$) refer to differential operators
on $\mathbb{S}^{d-1}$, endowed with the natural Riemannian metric.
4. $\bullet$
For $n\in\mathbb{S}^{d-1}$, we denote by
$\pi_{n}^{\bot}:=\operatorname{Id}-n\otimes n$ the orthogonal projection on
$n^{\bot}$.
5. $\bullet$
We let $\langle
g\rangle(x):=\fint_{\mathbb{S}^{d-1}}g(x,n)\,\mathrm{d}n:=\frac{1}{\omega_{d}}\int_{\mathbb{S}^{d-1}}g(x,n)\,\mathrm{d}n$
be the angular averaging of a function $g$ on
$\mathbb{T}^{d}\times\mathbb{S}^{d-1}$. We also use the short-hand notation
$P_{1}^{\bot}g:=g-\langle g\rangle$.
6. $\bullet$
We denote by $H^{k}(\mathbb{T}^{d})$ (resp.
$H^{k}(\mathbb{T}^{d}\times\mathbb{S}^{d-1})$) the standard
$\operatorname{L}^{2}$ Sobolev spaces for functions depending on
$x\in\mathbb{T}^{d}$ (resp. $(x,n)\in\mathbb{T}^{d}\times\mathbb{S}^{d-1}$),
and we use the notation $\|\cdot\|_{H^{k}_{x}}$ (resp.
$\|\cdot\|_{H^{k}_{x,n}}$) for the corresponding norms. For time-dependent
functions, given a Banach space $X$ and $t>0$, the norm of
$\operatorname{L}^{p}(0,t;X)$ is denoted by
$\|\cdot\|_{\operatorname{L}_{t}^{p}X}$.
7. $\bullet$
The space of probability measures on $\mathbb{T}^{d}$ (resp. on
$\mathbb{T}^{d}\times\mathbb{S}^{d-1}$) is denoted by
$\mathcal{P}(\mathbb{T}^{d})$ (resp. by
$\mathcal{P}(\mathbb{T}^{d}\times\mathbb{S}^{d-1})$).
## 2\. Doi–Saintillan–Shelley theory
For simplicity, we focus in this work on systems in a finite box
$\mathbb{T}^{d}=[0,1)^{d}$ with periodic boundary conditions. In terms of the
fluid velocity field
$u\colon\mathbb{R}^{+}\times\mathbb{T}^{d}\to\mathbb{R}^{d}$ and of the
particles’ position and orientation density function
$f\colon\mathbb{R}^{+}\times\mathbb{T}^{d}\times\mathbb{S}^{d-1}\to\mathbb{R}^{+}$,
we consider the so-called Doi–Saintillan–Shelley kinetic model for a
suspension of very elongated, rigid, active particles in an incompressible
viscous fluid flow (see [DE88, Chapter 8] and [Sai18]),
$\left\\{\begin{array}[]{l}\rho_{\operatorname{fl}}(\partial_{t}+u\cdot\nabla)u-\mu_{\operatorname{fl}}\Delta
u+\nabla
p\,=\,h+{\operatorname{div}}(\sigma_{1}[f])+{\operatorname{div}}(\sigma_{2}[f,\nabla
u]),\\\\[5.69054pt]
\partial_{t}f+{\operatorname{div}}_{x}\big{(}(u+V_{0}n)f\big{)}+{\operatorname{div}}_{n}\big{(}\pi_{n}^{\bot}(\nabla
u)nf\big{)}\\\ \hskip
142.26378pt=D_{\operatorname{tr}}{\operatorname{div}}_{x}\big{(}(\operatorname{Id}+n\otimes
n)\nabla_{x}f\big{)}+D_{\operatorname{ro}}\Delta_{n}f,\\\\[5.69054pt]
{\operatorname{div}}(u)=0,\end{array}\right.$ (2.1)
where elastic and viscous stresses are respectively given by
$\displaystyle\sigma_{1}[f]$ $\displaystyle:=$
$\displaystyle\left(3k_{B}\Theta+\alpha\mu_{\operatorname{fl}}|V_{0}|\ell^{2}\right)\int_{\mathbb{S}^{d-1}}\big{(}n\otimes
n-\tfrac{1}{d}\operatorname{Id}\big{)}f(\cdot,n)\,\mathrm{d}n,$
$\displaystyle\sigma_{2}[f,\nabla u]$ $\displaystyle:=$
$\displaystyle\tfrac{1}{2}\zeta_{\operatorname{ro}}\int_{\mathbb{S}^{d-1}}(n\otimes
n):\operatorname{D}(u)(n\otimes n)\,f(\cdot,n)\,\mathrm{d}n,$ (2.2)
where $h$ is an internal force, and where the different parameters
$\rho_{\operatorname{fl}},\mu_{\operatorname{fl}},D_{\operatorname{tr}},D_{\operatorname{ro}},\zeta_{\operatorname{ro}},V_{0},\ell$
are all assumed to be constant. In this kinetic description,
$\rho_{\operatorname{fl}}$ stands for the fluid density,
$\mu_{\operatorname{fl}}$ for the solvent viscosity, $D_{\operatorname{tr}}$
and $D_{\operatorname{ro}}$ for the translational and rotational diffusion
coefficients, $\zeta_{\operatorname{ro}}$ for the rotational resistance
coefficient, $V_{0}$ for the self-propulsion speed of the particles, and
$\ell$ for the length of the particles. The fluid flow is coupled to the
kinetic equation for the particle density via the additional stresses
$\sigma_{1}$ (elastic stress) and $\sigma_{2}$ (viscous stress), which make
the fluid equations non-Newtonian. We briefly describe the structure and
physical origin of these contributions (see Section 1.3 for references):
1. —
The viscous stress $\sigma_{2}$ arises from the rigidity of suspended
particles in the fluid flow: it is formally understood by homogenization of
the solid phase, viewed as inclusions with infinite shear viscosity in the
fluid. The above expression (2.2) is an approximation for very elongated
particles: for general axisymmetric particles, the viscous stress involves
additional terms depending on the precise shape of the particles (see e.g.
[HL72] for spheroids), but slender body theory indeed shows that it reduces to
the above form in the limit of very elongated particles (see e.g. [KK13,
Section 3.4]).
2. —
The elastic stress $\sigma_{1}$ contains a passive and an active contribution.
The passive part, proportional to the the Boltzmann constant $k_{B}$ and to
the absolute temperature $\Theta$, is created by the random torques that are
responsible for the rotational Brownian motion of the particles. These torques
indeed create stresses due to the rigidity and anisotropy of the particles. On
the other hand, the active part arises directly from the swimming mechanism,
which is encoded in the parameter $\alpha\in\mathbb{R}$: a so-called puller
particle corresponds to $\alpha>0$, and a pusher particle corresponds to
$\alpha<0$.
For $V_{0}=0$, the model (2.1) reduces to the classical Doi model for passive
Brownian particles [DE88]. Finally, for very elongated particles in 3D, we
also note that the translational and rotational diffusion and resistance
coefficients are asymptotically given as follows (see [Dho96, Section 5.15]),
$\begin{array}[]{rllrrll}D_{\operatorname{ro}}&=&\tfrac{k_{B}\Theta}{\zeta_{\operatorname{ro}}}+D_{\operatorname{act}},&&\zeta_{\operatorname{ro}}&=&\tfrac{\pi\mu_{\operatorname{fl}}\ell^{3}}{3\log(\ell/a)},\\\\[5.69054pt]
D_{\operatorname{tr}}&=&\tfrac{k_{B}\Theta}{\zeta_{\operatorname{tr}}},&&\zeta_{\operatorname{tr}}&=&\tfrac{2\pi\mu_{\operatorname{fl}}\ell}{\log(\ell/a)},\end{array}$
(2.3)
where $a$ is the width of the particles and where $D_{\operatorname{act}}$ is
some possible active contribution to the rotational diffusion (tumbling).
Compared to the above model (2.1), we henceforth make two minor
simplifications:
1. —
While the translational diffusion in (2.1) is proportional to
$\operatorname{Id}+n\otimes n$, hence is twice as strong in the direction of
particle orientation as in the orthogonal directions, we choose to neglect
this $O(1)$ difference and assume that the diffusion is isotropic. This choice
is for simplicity and does not change anything in the analysis of the model.
2. —
It has been observed in the seminal work [BB72] that, for the example of E.
coli bacteria and related microswimmers, the contribution of thermal rotation
and active tumbling is of the same order, and we therefore set
$D_{\operatorname{act}}=0$ for simplicity.
### 2.1. Non-dimensionalization and relevant regimes
We non-dimensionalize the above model (2.1) (after the two above-described
minor simplifications), in terms of the typical speed $u_{0}$ of the fluid,
the typical macroscopic length scale $L$, and the typical number $N$ of
particles in a cube of side length $L$. We further rescale time according to
the time scale of the fluid flow, that is, $T=L/u_{0}$. More precisely, we
define
$\displaystyle\hat{u}(t,x):=\tfrac{1}{u_{0}}u(Tt,Lx),$
$\displaystyle\hat{f}(t,x):=\tfrac{L^{d}}{N}f(Tt,Lx),$
$\displaystyle\hat{h}(t,x):=\tfrac{L^{2}}{\mu u_{0}}h(Tt,Lx).$
This leads to the following dimensionless model, dropping the hats for
simplicity,
$\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+u\cdot\nabla)u-\Delta
u+\nabla
p\,=\,h+{\operatorname{div}}(\sigma_{1}[f])+{\operatorname{div}}(\sigma_{2}[f,\nabla
u]),\\\\[5.69054pt]
\partial_{t}f+{\operatorname{div}}_{x}\big{(}(u+U_{0}n)f\big{)}+{\operatorname{div}}_{n}\big{(}\pi_{n}^{\bot}(\nabla
u)nf\big{)}=\frac{1}{\operatorname{Pe}}\Delta_{x}f+\frac{1}{\operatorname{Wi}}\Delta_{n}f,\\\\[5.69054pt]
{\operatorname{div}}(u)=0,\end{array}\right.$
where the dimensionless counterparts of the additional stresses
$\sigma_{1},\sigma_{2}$ now take the form
$\displaystyle\sigma_{1}[f]$ $\displaystyle=$
$\displaystyle(6\tfrac{\lambda}{\operatorname{Wi}}+\gamma)\int_{\mathbb{S}^{d-1}}\big{(}n\otimes
n-\tfrac{1}{d}\operatorname{Id}\big{)}f(\cdot,n)\,\mathrm{d}n,$
$\displaystyle\sigma_{2}[f,\nabla u]$ $\displaystyle=$
$\displaystyle\lambda\int_{\mathbb{S}^{d-1}}(n\otimes n)(\nabla u)(n\otimes
n)\,f(\cdot,n)\,\mathrm{d}n.$
Here, $\operatorname{Re},\operatorname{Pe},\operatorname{Wi}>0$ stand for the
so-called Reynolds, Péclet, and Weissenberg numbers, $\lambda>0$ depends only
on the shape and number density of the particles, and $\gamma\in\mathbb{R}$
accounts for the active contribution to the stress. More precisely, these
parameters are given by
$\begin{array}[]{rllrrll}\operatorname{Re}&=&\tfrac{\rho_{\operatorname{fl}}u_{0}L}{\mu_{\operatorname{fl}}},&&\lambda&=&\tfrac{\zeta_{\operatorname{ro}}N}{2\mu_{\operatorname{fl}}L^{d}},\\\\[5.69054pt]
\operatorname{Pe}&=&\tfrac{u_{0}L\zeta_{\operatorname{tr}}}{k_{B}\Theta},&&\operatorname{Wi}&=&\tfrac{u_{0}\zeta_{\operatorname{ro}}}{k_{B}\Theta
L},\\\\[5.69054pt] \gamma&=&\alpha
N\tfrac{|U_{0}|\ell^{2}}{L^{d-1}},&&U_{0}&=&\tfrac{V_{0}}{u_{0}},\end{array}$
and we briefly comment on their range and interpretation:
1. —
The Weissenberg number $\operatorname{Wi}$ is the ratio between convection and
relaxation time scales. For the kinetic viscoelastic models under
consideration, it coincides with the rotational Péclet number. For elongated
particles in 3D, as $\zeta_{\operatorname{ro}}$ is proportional to the cube of
the particle length, cf. (2.3), the regime when $\operatorname{Wi}$ is of
order $O(1)$ is very narrow, and lies under standard flow conditions at
particle lengths of around $10$ microns. In this work, we are interested in
the derivation of hydrodynamic approximations in case of very small
$\operatorname{Wi}\ll 1$: this means that we have applications in mind where
the particles have a length of a few microns and below, which is in particular
the case for many types of bacteria. For notational convenience, we rename the
Weissenberg number as
$\varepsilon\,:=\,\operatorname{Wi}\ll 1.$
2. —
The (translational) Péclet number $\operatorname{Pe}$ is typically much larger
than its rotational counterpart $\operatorname{Wi}$ since111The same holds
with an additional logarithmic correction in 2D.
$\tfrac{\operatorname{Pe}}{\operatorname{Wi}}\,=\,\tfrac{\zeta_{\operatorname{tr}}L^{2}}{\zeta_{\operatorname{ro}}}\,=\,\tfrac{6L^{2}}{\ell^{2}}\,\gg\,1.$
In fact, from the application-oriented perspective, it makes sense to consider
$\operatorname{Pe}\gg 1$. However, due to well-posedness and stability issues,
our analysis crucially relies on keeping $\operatorname{Pe}$ not too large,
and we shall generally assume $\operatorname{Pe}\simeq 1$. We can actually
allow for $\operatorname{Pe}$ to slightly diverge, but more slowly than the
inverse of the volume fraction of the particles, cf. (1.6).
3. —
The shape parameter $\lambda$ is typically quite small as it is proportional
to the volume fraction $NL^{-d}$ of the particles.
4. —
The prefactor in the viscous stress $\sigma_{1}$ reads
$6\tfrac{\lambda}{\operatorname{Wi}}+\gamma\,=\,\tfrac{\lambda}{\operatorname{Wi}}\Big{(}6+2\alpha\tfrac{|V_{0}|\ell^{2}\mu_{\operatorname{fl}}}{k_{B}\Theta}\Big{)},$
where the term $\frac{|V_{0}|\ell^{2}\mu_{\operatorname{fl}}}{k_{B}\Theta}$ is
of order $1-10$ for typical microswimmers like E. coli bacteria [BB72]. We
shall set for abbreviation
$\displaystyle\theta\,:=\,6+2\alpha\tfrac{|V_{0}|\ell^{2}\mu_{\operatorname{fl}}}{k_{B}\Theta}.$
(2.4)
Note in particular that for passive particles $V_{0}=0$ we find $\theta=6$.
5. —
The self-propulsion speed $V_{0}$ of active particles is typically around $10$
micron per second. This is so slow with respect to typical shear flows
considered in experimental settings that the ratio $U_{0}=\frac{V_{0}}{u_{0}}$
is typically tiny. However, extremely low shear rates leading to $u_{0}\sim
V_{0}$ can possibly also be achieved, as for instance in the experiments
reported by [Lóp+15]. For that reason, we choose not to neglect $U_{0}$ in the
equations and to keep track of its effects.
In conclusion, we are led to consider the following system,
$\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+u_{\varepsilon}\cdot\nabla)u_{\varepsilon}-\Delta
u_{\varepsilon}+\nabla
p_{\varepsilon}=h+\frac{1}{\varepsilon}{\operatorname{div}}(\sigma_{1}[f_{\varepsilon}])+{\operatorname{div}}(\sigma_{2}[f_{\varepsilon},\nabla
u_{\varepsilon}]),\\\\[5.69054pt]
\partial_{t}f_{\varepsilon}+{\operatorname{div}}_{x}\big{(}(u_{\varepsilon}+U_{0}n)f_{\varepsilon}\big{)}+{\operatorname{div}}_{n}\big{(}\pi_{n}^{\bot}(\nabla
u_{\varepsilon})nf_{\varepsilon}\big{)}=\tfrac{1}{\operatorname{Pe}}\Delta_{x}f_{\varepsilon}+\tfrac{1}{\varepsilon}\Delta_{n}f_{\varepsilon},\\\\[5.69054pt]
{\operatorname{div}}(u_{\varepsilon})=0,\\\\[5.69054pt]
\int_{\mathbb{T}^{d}}u_{\varepsilon}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{if $\operatorname{Re}=0$},\\\\[5.69054pt]
\sigma_{1}[f]=\lambda\theta\int_{\mathbb{S}^{d-1}}\big{(}n\otimes
n-\tfrac{1}{d}\operatorname{Id}\big{)}f(\cdot,n)\,\mathrm{d}n,\\\\[5.69054pt]
\sigma_{2}[f,\nabla u]=\lambda\int_{\mathbb{S}^{d-1}}(n\otimes n)(\nabla
u)(n\otimes n)\,f(\cdot,n)\,\mathrm{d}n,\end{array}\right.$ (2.5)
which is complemented by initial conditions
$\left\\{\begin{array}[]{l}f_{\varepsilon}|_{t=0}=f^{\circ}_{\varepsilon},\\\\[2.84526pt]
u_{\varepsilon}|_{t=0}=u^{\circ}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{if $\operatorname{Re}\neq 0$},\end{array}\right.$ (2.6)
and we consider the asymptotic limit $\varepsilon\downarrow 0$ in the regime
with $\lambda\leq 1$ and with $\lambda\theta\ll 1$ small enough. Regarding the
Reynolds number, we focus on $\operatorname{Re}\in\\{0,1\\}$ for simplicity.
Due to classical regularity issues for the Navier–Stokes equations, we
actually limit ourselves to the Stokes case $\operatorname{Re}=0$ for the 3D
model, but we also study the Navier–Stokes case $\operatorname{Re}=1$ in 2D.
Note that the 3D Navier–Stokes case is much more complicated and is not
discussed in this work as the well-posedness of the system is still open in
that case.
### 2.2. Well-posedness of the Doi–Saintillan–Shelley model
The system (2.5) is an instance of the more general class of
Fokker–Planck–Navier–Stokes systems, but we emphasize two main peculiarities:
1. —
We include in (2.5) the contribution of the viscous stress $\sigma_{2}$, which
arises from the rigidity of underlying suspended particles on the microscale
and effectively modifies the solvent viscosity.
2. —
We also include the effect of particle swimming via $U_{0}$. This creates
local changes in the spatial density
$\rho_{\varepsilon}=\fint_{\mathbb{S}^{d-1}}f_{\varepsilon}(\cdot,n)\,\mathrm{d}n$,
which no longer remains constant in general.
In contrast, most previous works have focused on the corresponding model
without viscous stress $\sigma_{2}$ and without particle swimming $U_{0}=0$.
In that simplified setting, for the 3D Stokes case, the existence of global
entropy solutions was proven in [OT08], and the global well-posedness of
smooth solutions was proven in [CS10] (without translational diffusion,
$\operatorname{Pe}=\infty$). In the Navier–Stokes case, corresponding well-
posedness results were obtained for instance in [CM08]. The model including
the viscous stress $\sigma_{2}[f_{\varepsilon},\nabla u_{\varepsilon}]$ but
without particle swimming $U_{0}=0$ was first studied in [LM07], where the
existence of global weak solutions was proven for the Navier–Stokes case in 2D
and 3D (without translational diffusion, $\operatorname{Pe}=\infty$). We also
refer to [La19] for the global well-posedness of smooth solutions in the 2D
Navier–Stokes case. Particle swimming $U_{0}\neq 0$ was first considered in
[CL13], where the authors studied the corresponding model without viscous
stress $\sigma_{2}$ and proved the global existence of weak entropy solutions
both for the Stokes and Navier–Stokes cases in 2D and 3D, as well as the
existence of energy solutions for the Stokes case and their uniqueness in 2D.
Building on similar ideas, we can actually establish the existence of global
energy solutions for the full model (2.5) in the 2D and 3D Stokes cases, as
well as in the 2D Navier–Stokes case, and we further obtain a weak-strong
uniqueness principle. To our knowledge, this is surprisingly the first result
where both the viscous stress and the swimming forces are included at the same
time. The proof is postponed to Appendix A.
###### Proposition 2.1.
Consider either the Stokes case $\operatorname{Re}=0$ with $d\leq 3$, or the
Navier–Stokes case $\operatorname{Re}=1$ with $d=2$. Given $\varepsilon>0$,
given
$h\in\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};\operatorname{L}^{2}(\mathbb{T}^{d}))$,
given an initial condition
$f^{\circ}_{\varepsilon}\in\operatorname{L}^{2}\cap\mathcal{P}(\mathbb{T}^{d}\times\mathbb{S}^{d-1})$,
and given also $u^{\circ}\in\operatorname{L}^{2}(\mathbb{T}^{d})^{d}$ with
${\operatorname{div}}(u^{\circ})=0$ in the Navier–Stokes case, the Cauchy
problem (2.5)–(2.6) admits a global weak solution
$(u_{\varepsilon},f_{\varepsilon})$ with:
1. (i)
in the Stokes case $\operatorname{Re}=0$, $d\leq 3$,
$\displaystyle u_{\varepsilon}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{1}(\mathbb{T}^{d})^{d}),$
$\displaystyle f_{\varepsilon}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};\operatorname{L}^{2}\cap\mathcal{P}(\mathbb{T}^{d}\times\mathbb{S}^{d-1}))\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{1}(\mathbb{T}^{d}\times\mathbb{S}^{d-1}));$
2. (ii)
in the Navier–Stokes case $\operatorname{Re}=1$, $d=2$,
$\displaystyle u_{\varepsilon}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};\operatorname{L}^{2}(\mathbb{T}^{2})^{2})\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{1}(\mathbb{T}^{2})^{2}),$
$\displaystyle f_{\varepsilon}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};\operatorname{L}^{2}\cap\mathcal{P}(\mathbb{T}^{2}\times\mathbb{S}^{1}))\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{1}(\mathbb{T}^{2}\times\mathbb{S}^{1})).$
In both cases, a weak-strong uniqueness principle further holds: if
$(u_{\varepsilon},f_{\varepsilon})$ and
$(u^{\prime}_{\varepsilon},f^{\prime}_{\varepsilon})$ are two such global weak
solutions with identical initial conditions, and if
$(u^{\prime}_{\varepsilon},f^{\prime}_{\varepsilon})$ has the following
additional regularity,
$\displaystyle u^{\prime}_{\varepsilon}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{2}_{\mathrm{loc}}(\mathbb{R}^{+};W^{1,\infty}(\mathbb{T}^{d})^{d}),$
$\displaystyle f^{\prime}_{\varepsilon}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\mathrm{loc}}(\mathbb{R}^{+};\operatorname{L}^{\infty}(\mathbb{T}^{d}\times\mathbb{S}^{d-1})),$
then we have
$(u_{\varepsilon},f_{\varepsilon})=(u^{\prime}_{\varepsilon},f^{\prime}_{\varepsilon})$.
∎
## 3\. Ordered fluid models
Various non-Newtonian fluid models have been considered in the literature,
taking into account nonlinear and memory effects in different ways on the
macroscopic scale. We focus here on so-called ordered fluid models, which will
indeed be shown to naturally appear as hydrodynamic approximations of the
Doi–Saintillan–Shelley theory. We start by defining such models and reviewing
their non-Newtonian properties; for more details on modelling aspects and
applications, we refer to [DR95, Böh87, BAH87, Jos13, PTMD13]. Next, we
develop a perturbative well-posedness theory and we introduce (non-standard)
inhomogeneous, diffusive versions of these models.
### 3.1. Standard ordered fluid models
Ordered fluid equations are of the form
$\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+u\cdot\nabla)u-{\operatorname{div}}(\sigma)+\nabla
p=h,\\\\[2.84526pt] {\operatorname{div}}(u)=0,\end{array}\right.$ (3.1)
where $\sigma$ is a function of the Rivlin–Ericksen tensors
$\\{A_{k}(u)\\}_{k}$ defined iteratively through
$\displaystyle A_{1}(u)\,:=\,2\\!\operatorname{D}(u)\,=\,\nabla u+(\nabla
u)^{T},$ $\displaystyle
A_{n+1}(u)\,:=\,(\partial_{t}+u\cdot\nabla)A_{n}(u)+(\nabla
u)^{T}A_{n}(u)+A_{n}(u)\nabla u,\qquad n\geq 1.$ (3.2)
In other words, $A_{n+1}(u)$ is the so-called lower-convected derivative of
$A_{n}(u)$, which ensures frame indifference of the equations.222In the
literature, _upper_ -convected instead of lower-convected derivatives are
sometimes used to define ordered fluids (see e.g. [BCAH87, Section 6]). This
is merely a choice of convention, as both lead to equivalent fluid equations
(although the value and interpretation of parameters of course depends on the
chosen convention). Indeed, the upper-convected derivative of $A_{n}(u)$ can
be rewritten in terms of the lower-convected derivative as
$(\partial_{t}+u\cdot\nabla)A_{n}(u)-A_{n}(u)(\nabla u)^{T}-(\nabla
u)A_{n}(u)\,=\,A_{n+1}(u)-\big{(}A_{1}(u)A_{n}(u)+A_{n}(u)A_{1}(u)\big{)}.$
While first-order fluids are of the form (3.1) with constitutive law
$\sigma=\eta_{0}A_{1}(u)$, thus coinciding with standard Newtonian fluids,
second-order fluids are of the form (3.1) with
$\displaystyle\sigma\,:=\,\eta_{0}A_{1}(u)+\alpha_{1}A_{2}(u)+\alpha_{2}A_{1}(u)^{2},$
(3.3)
see e.g. [Böh87, Eq. (8.48)], and third-order fluids amount to
$\sigma\,:=\,\eta_{0}A_{1}(u)+\alpha_{1}A_{2}(u)+\alpha_{2}A_{1}(u)^{2}\\\
+\beta_{1}A_{3}(u)+\beta_{2}\big{(}A_{1}(u)A_{2}(u)+A_{2}(u)A_{1}(u)\big{)}+\beta_{3}A_{1}(u){\operatorname{tr}}\big{(}A_{1}(u)^{2}\big{)},$
(3.4)
for some coefficients
$\eta_{0},\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},\beta_{3}\in\mathbb{R}$.333In
arbitrary dimension $d$, a further term $\beta_{4}A_{1}(u)^{3}$ should be
included in general in the stress $\sigma$ of third-order fluids, but it is
redundant in dimensions $d\leq 3$ as we then have
$B^{3}=\frac{1}{2}B\,{\operatorname{tr}}(B^{2})+\frac{1}{3}{\operatorname{tr}}(B^{3})$
by the Cayley–Hamilton theorem for any symmetric trace-free matrix $B$. Since
third- and higher-order fluid models are fairly complicated and involve a lot
of terms, the most widely used ordered fluid model for viscoelastic fluids is
the second-order model (3.3).
We briefly recall the basic rheological properties of these ordered fluid
models depending on the different parameter values; we focus here on the 3D
setting. The coefficient $\eta_{0}>0$ in (3.3) and (3.4) is the zero-shear
viscosity, as in the first-order (Newtonian) model, while other parameters
account for various non-Newtonian behaviors.
1. —
Shear-dependent viscosity: In a simple shear flow $u_{0}(t,x)=\kappa
x_{2}e_{1}$ with shear rate $\kappa>0$, the shear-dependent viscosity is
defined as
$\eta(\kappa)\,:=\,\frac{\sigma_{12}}{\kappa}.$ (3.5)
Noting that
$\displaystyle\qquad A_{1}(u_{0})=\kappa\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},$ $\displaystyle
A_{1}(u_{0})^{2}=\kappa^{2}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix},$ $\displaystyle
A_{2}(u_{0})=\kappa^{2}\begin{pmatrix}0&0&0\\\ 0&2&0\\\ 0&0&0\end{pmatrix},$
we compute for the second-order fluid that the shear-dependent viscosity
simply coincides with the zero-shear viscosity, $\eta(\kappa)=\eta_{0}$. For
the third-order model, in contrast, we find a nontrivial shear-dependent
relation,
$\eta(\kappa)=\eta_{0}+2\kappa^{2}(\beta_{2}+\beta_{3}).$
Most real-life viscoelastic fluids, and in particular passive dilute
suspensions, happen to be shear-thinning, meaning that the map
$\kappa\mapsto\eta(\kappa)$ is decreasing: this holds for the third-order
fluid model provided that the coefficients satisfy $\beta_{2}+\beta_{3}<0$.
2. —
Normal-stress differences: In a simple shear flow $u_{0}(t,x)=\kappa
x_{2}e_{1}$ with shear rate $\kappa\in\mathbb{R}$, non-Newtonian fluids
typically display non-zero normal stresses. This is responsible for a number
of phenomena, of which the rod-climbing effect is the best known; see e.g.
[Jos13, Chapter 17] and [BAH87, Chapter 6]. Normal stress coefficients are
defined as444Beware of different sign conventions for the normal stress
coefficients. We follow here the definition of [Böh87, Chapter 2.2].
$\displaystyle\nu_{10}\,:=\,\frac{\sigma_{11}-\sigma_{22}}{\kappa^{2}},$
$\displaystyle\nu_{20}\,:=\,\frac{\sigma_{22}-\sigma_{33}}{\kappa^{2}},$ (3.6)
and thus, for second- and third-order models,
$\displaystyle\nu_{10}\,=\,-2\alpha_{1},$
$\displaystyle\nu_{20}\,=\,2\alpha_{1}+\alpha_{2}.$ (3.7)
In other words, $\alpha_{1},\alpha_{2}$ are related to normal stress
coefficients via
$\displaystyle\alpha_{1}\,=\,-\tfrac{1}{2}\nu_{10},$
$\displaystyle\alpha_{2}\,=\,\nu_{10}+\nu_{20}.$ (3.8)
For most real-life viscoelastic fluids, and in particular for polymer
solutions, it is found experimentally that $\nu_{10}>0$, $\nu_{20}<0$, and
that $|\nu_{20}|$ is considerably smaller than $|\nu_{10}|$ (by a factor of
around $10$, see e.g. [Böh87, Section 2.2] and [PTMD13, Section2.2]), which
means in particular, in terms of second-order fluid coefficients,
$\displaystyle\alpha_{1}<0,$ $\displaystyle\alpha_{2}>0.$ (3.9)
3. —
Elongational viscosity: The apparent viscosity of a non-Newtonian fluid may be
completely different in an elongational flow. Given a uniaxial elongational
flow $u_{0}(t,x)=\kappa(x_{1}e_{1}-\frac{1}{2}(x_{2}e_{2}+x_{3}e_{3}))$ in the
direction $e_{1}$, the elongational viscosity is defined as
$\displaystyle\eta_{E}\,:=\,\frac{\sigma_{11}-\frac{1}{2}(\sigma_{22}+\sigma_{33})}{\kappa}.$
(3.10)
Noting that
$\displaystyle A_{1}(u_{0})=\kappa\begin{pmatrix}2&0&0\\\ 0&-1&0\\\
0&0&-1\end{pmatrix},$ $\displaystyle
A_{2}(u_{0})=A_{1}(u_{0})^{2}=\kappa^{2}\begin{pmatrix}4&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},$ (3.11)
we compute for the second-order model,
$\displaystyle\eta_{E}\,=\,3\eta_{0}+3\kappa(\alpha_{1}+\alpha_{2}).$
For real-life viscoelastic fluids, it is typically observed that the
elongational viscosity increases with the strain rate (so-called strain-
thickening behavior), which holds provided that coefficients satisfy
$\alpha_{1}+\alpha_{2}>0$.
4. —
Retardation phase shift in oscillatory flow: In a simple shear flow
$u_{0}(t,x)=\kappa(t)x_{2}e_{1}$ with oscillatory shear rate $\kappa(t)=\sin
t$, we compute
$A_{2}(u_{0})=\dot{\kappa}\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix}+\kappa^{2}\begin{pmatrix}0&0&0\\\ 0&2&0\\\
0&0&0\end{pmatrix},$
which gives rise to a phase shift for $\sigma_{12}$ in the second-order model,
in form of
$\sigma_{12}\,=\,\eta_{0}\kappa+\alpha_{1}\dot{\kappa}\,=\,(\eta_{0}^{2}+\alpha_{1}^{2})^{\frac{1}{2}}\sin\big{(}t+\arctan(\tfrac{\alpha_{1}}{\eta_{0}})\big{)}.$
This constitutes another typical (time-dependent) non-Newtonian feature, in
link with the dependence of the stress on the flow history.
###### Remark 3.1 (Connection to Oldroyd–B model).
The Oldroyd–B model is a special case of a simple fluid model, which is
particularly popular as a formal exact closure of the kinetic Hookean dumbbell
model. It is characterized by the following constitutive equation for the
stress tensor,
$\displaystyle\sigma=2\eta_{s}\operatorname{D}(u)+\eta_{p}\tau,$
$\displaystyle\tau+\operatorname{Wi}\left((\partial_{t}+u\cdot\nabla)\tau-\tau(\nabla
u)^{T}-(\nabla u)\tau\right)\,=\,2\operatorname{D}(u).$ (3.12)
By a formal expansion with respect to $\operatorname{Wi}\ll 1$, this model is
found to agree to order $\operatorname{Wi}^{k}$ with a $k$th-order fluid model
with some specific choice of parameters. In particular, to order
$\operatorname{Wi}^{2}$, we recover the second-order fluid model (3.3) with
$\alpha_{2}=-2\alpha_{1}$, which means in particular that the second normal
stress coefficient vanishes, $\nu_{20}=0$. We emphasize that this is not the
case for the second-order fluid model that we shall derive from the
Doi–Saintillan–Shelley theory. Hence, at small $\operatorname{Wi}$, the
viscoelastic effects of suspensions of rigid Brownian particles are not well
described by an Oldroyd–B model. ∎
### 3.2. Non-standard ordered fluid models at $\operatorname{Pe}<\infty$
In the case of a particle suspension with finite Péclet number
$\operatorname{Pe}<\infty$, as considered in this work, cf. (2.5), the
Rivlin–Ericksen tensors (3.2) naturally need to be modified as follows,
$\displaystyle A_{1}^{\prime}(u)\,:=\,A_{1}(u)\,=\,2\operatorname{D}(u),$
(3.13) $\displaystyle
A_{n+1}^{\prime}(u)\,:=\,(\partial_{t}-{\tfrac{1}{\operatorname{Pe}}}\Delta+u\cdot\nabla)A_{n}^{\prime}(u)+(\nabla
u)^{T}A_{n}^{\prime}(u)+A_{n}^{\prime}(u)(\nabla u),\qquad n\geq 1,$
hence in particular
$A_{2}^{\prime}(u):=A_{2}(u)-\frac{1}{\operatorname{Pe}}\Delta A_{1}(u)$. The
second-order fluid equations (3.1)–(3.3) are then replaced by
$\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+u\cdot\nabla)u-{\operatorname{div}}(\sigma)+\nabla
p=h,\\\\[2.84526pt]
\sigma=\eta_{0}A_{1}(u)+\alpha_{1}A_{2}^{\prime}(u)+\alpha_{2}A_{1}(u)^{2},\\\\[2.84526pt]
{\operatorname{div}}(u)=0.\end{array}\right.$ (3.14)
For higher-order fluid models, in this diffusive setting
$\operatorname{Pe}<\infty$, several additional tensors actually need to be
included in the stress: for the third-order model, instead of (3.4), the
stress rather needs to be chosen in general as
$\sigma\,=\,\eta_{0}A_{1}(u)+\alpha_{1}A_{2}^{\prime}(u)+\alpha_{2}A_{1}(u)^{2}+\beta_{1}A_{3}^{\prime}(u)+\beta_{1}^{\prime}B_{3}^{\prime}(u)\\\
+\beta_{2}\big{(}A_{1}(u)A_{2}^{\prime}(u)+A_{2}^{\prime}(u)A_{1}(u)\big{)}+\beta_{3}A_{1}(u){\operatorname{tr}}\big{(}A_{1}(u)^{2}\big{)},$
(3.15)
in terms of the following additional tensor,
$B_{3}^{\prime}(u)\,:=\,(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta+u\cdot\nabla)A_{1}(u)^{2}+\big{(}(\nabla
u)^{T}A_{1}(u)^{2}+A_{1}(u)^{2}(\nabla u)\big{)}.$
At infinite Péclet number $\operatorname{Pe}=\infty$, this additional tensor
is indeed redundant as it reduces to
$B_{3}^{\prime}(u)=A_{1}(u)A_{2}(u)+A_{2}(u)A_{1}(u)-A_{1}(u)^{3}$, so we
recover (3.4).555Recall that, as in (3.4), a further term
$\beta_{4}A_{1}(u)^{3}$ should always be included in the stress in arbitrary
dimension $d$, but it reduces to
$\frac{1}{2}\beta_{4}A_{1}(u){\operatorname{tr}}(A_{1}(u)^{2})$ in dimensions
$d\leq 3$.
### 3.3. Well-posedness of ordered fluid models
We focus for shortness on the second-order fluid model. There has actually
been a fair amount of confusion on the relevant range of parameters
$\alpha_{1},\alpha_{2}$: the sign condition $\alpha_{1}<0$ in (3.9) is
motivated by experimental normal stress measurements, but it actually appears
inconsistent with thermodynamics; see e.g. [DR95] for a detailed discussion
from the physics perspective. From the mathematical point of view, this
inconsistency leads to ill-posedness issues. The matter was investigated by
Galdi [Gal95], who showed the following for the second-order fluid equations
(3.1)–(3.3) at infinite Péclet number $\operatorname{Pe}=\infty$:
1. —
the local-in-time well-posedness holds whenever $1/\alpha_{1}>-\lambda_{1}$,
where $\lambda_{1}$ stands for the Poincaré constant in $\mathbb{T}^{d}$,
which is quite consistent with the choice (3.9) (although the case of a
negative $\alpha_{1}$ with a small absolute value is prohibited);
2. —
the long-time well-posedness, as well as the stability of steady solutions,
can only hold provided that $\alpha_{1}>0$.
For the corresponding system (3.14) with finite Péclet number
$\operatorname{Pe}<\infty$, the situation is even worse: even local-in-time
well-posedness actually fails whenever $\alpha_{1}<0$ because the equation
then behaves like a backwards heat equation. Yet, even though the kinetic Doi
model itself is known to be thermodynamically consistent (see [DE88, Chapter
8]), our analysis will precisely lead us to a second-order fluid with
$\alpha_{1}<0$, and it is thus crucial to determine what meaning should be
given to the model in that case. In fact, we shall derive (3.14) in the
small-$\operatorname{Wi}$ limit, $\varepsilon:=\operatorname{Wi}\ll 1$, with
$(\alpha_{1},\alpha_{2})=(\varepsilon\gamma_{1},\varepsilon\gamma_{2}),\qquad\text{for
some $\gamma_{1}<0,\leavevmode\nobreak\ \gamma_{2}\in\mathbb{R}$.}$ (3.16)
In this perturbative setting $\varepsilon\ll 1$, although the equation is ill-
posed for fixed $\varepsilon$, there are several ways to rearrange the
nonlinearity and define well-posed notions of approximate solutions that only
satisfy the system (3.14) up to a higher-order $O(\varepsilon^{2})$ remainder:
$\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+\bar{u}_{\varepsilon}\cdot\nabla)\bar{u}_{\varepsilon}-{\operatorname{div}}(\bar{\sigma}_{\varepsilon})+\nabla\bar{p}_{\varepsilon}=h+O(\varepsilon^{2}),\\\\[2.84526pt]
\bar{\sigma}_{\varepsilon}\,:=\,\eta_{0}A_{1}(\bar{u}_{\varepsilon})+\varepsilon\gamma_{1}A_{2}^{\prime}(\bar{u}_{\varepsilon})+\varepsilon\gamma_{2}A_{1}(\bar{u}_{\varepsilon})^{2},\\\\[2.84526pt]
{\operatorname{div}}(\bar{u}_{\varepsilon})=0.\end{array}\right.$ (3.17)
These notions of approximate solutions will be viewed as proper perturbative
ways to interpret second-order fluid equations and settle ill-posedness and
instability issues.
The simplest way is to define a notion of approximate _hierarchical
solutions_. It is particularly convenient for us in this work due to its close
relation to the Hilbert expansion method that we use for the hydrodynamic
approximation. The proof is straightforward and is postponed to Appendix B.
###### Proposition 3.2 (Hierarchical solutions).
Consider the system (3.17) in the regime $\varepsilon\ll 1$ with parameters
$\eta_{0},\operatorname{Pe}>0$ and $\gamma_{1},\gamma_{2}\in\mathbb{R}$. If
$v_{0},v_{1}$ are smooth solutions of the following two auxiliary systems,
$\left\\{\begin{array}[]{l}\operatorname{Re}\big{(}\partial_{t}v_{0}+(v_{0}\cdot\nabla)v_{0}\big{)}-\eta_{0}\Delta
v_{0}+\nabla p_{0}=h,\\\\[1.42262pt] {\operatorname{div}}(v_{0})=0,\\\
v_{0}|_{t=0}=u^{\circ}\quad\text{if $\operatorname{Re}\neq
0$},\quad\int_{\mathbb{T}^{d}}v_{0}=0\quad\text{if
$\operatorname{Re}=0$},\end{array}\right.$ (3.18)
$\left\\{\begin{array}[]{l}\operatorname{Re}\big{(}\partial_{t}v_{1}+(v_{0}\cdot\nabla)v_{1}+(v_{1}\cdot\nabla)v_{0}\big{)}-\eta_{0}\Delta
v_{1}+\nabla p_{1}\\\\[2.84526pt] \hskip
113.81102pt={\operatorname{div}}\big{(}\gamma_{1}A_{2}^{\prime}(v_{0})+\gamma_{2}A_{1}(v_{0})^{2}\big{)},\\\\[1.42262pt]
{\operatorname{div}}(v_{1})=0,\\\\[1.42262pt] v_{1}|_{t=0}=0\quad\text{if
$\operatorname{Re}\neq 0$},\quad\int_{\mathbb{T}^{d}}v_{1}=0\quad\text{if
$\operatorname{Re}=0$},\end{array}\right.$ (3.19)
then the superposition $\bar{u}_{\varepsilon}:=v_{0}+\varepsilon v_{1}$ indeed
satisfies the system (3.17) with some controlled error term
$O(\varepsilon^{2})$ (and with initial condition
$\bar{u}_{\varepsilon}|_{t=0}=u^{\circ}$ if $\operatorname{Re}\neq 0$). For
the well-posedness of (3.18) and (3.19), we separately consider the Stokes and
Navier–Stokes cases:
1. (i)
_Stokes case $\operatorname{Re}=0$, $d\leq 3$:_
Given $s>\frac{d}{2}-1$ and
$h\in\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{d})^{d})\cap
W^{1,\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s-2}(\mathbb{T}^{d})^{d})$,
there is a unique global solution
$v_{0}\in\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+2}(\mathbb{T}^{d})^{d})$
of (3.18), and a unique global solution
$v_{1}\in\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})^{d})$
of (3.19), leading to $\bar{u}_{\varepsilon}=v_{0}+\varepsilon
v_{1}\in\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})^{d})$.
2. (ii)
_Navier–Stokes case $\operatorname{Re}=1$, $d=2$:_
Given $s>0$,
$h\in\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{2})^{2})$,
and $u^{\circ}\in H^{s+1}(\mathbb{T}^{2})^{2}$ with
${\operatorname{div}}(u^{\circ})=0$, there is a unique global solution
$v_{0}\in\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{2})^{2})\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+2}(\mathbb{T}^{2})^{2})$
of (3.18), and a unique global solution
$v_{1}\in\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{2})^{2})\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{2})^{2})$
of (3.19), leading to $\bar{u}_{\varepsilon}=v_{0}+\varepsilon
v_{1}\in\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{2})^{2})\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{2})^{2})$.
∎
There are also non-hierarchical ways to perturbatively make sense of the ill-
posed second-order fluid model (3.17), which may be more desirable in
particular for stability issues. Comparing to corresponding ill-posedness
issues in the Boussinesq theory for water waves, we recall that there is a
standard way to rearrange the ill-posed Boussinesq equation perturbatively and
make it well-posed, see [CMV96]: in a nutshell, the idea is to replace
indefinite operators like $1+\varepsilon\Delta$ by corresponding positive
operators like $(1-\varepsilon\Delta)^{-1}$ up to $O(\varepsilon^{2})$ errors.
We show that a similar so-called Boussinesq trick can be used in the present
situation as well: for any value of $\gamma_{1},\gamma_{2}$, both at finite
and infinite Péclet number, it leads us to a perturbative rearrangement of the
second-order fluid equation that is well-posed and is indeed equivalent to
(3.14) up to $O(\varepsilon^{2})$ terms. The so-defined solution is easily
checked to differ from the corresponding hierarchical solution only by
$O(\varepsilon^{2})$. We focus here on the relevant ill-posed case
$\gamma_{1}\leq 0$. Note that the procedure is easily extended to higher-order
fluid equations (see also [ABV16, DGR23] in a different context). The proof is
postponed to Appendix B.
###### Proposition 3.3 (Boussinesq-like solutions).
Consider the system (3.17) in the regime $\varepsilon\ll 1$ with parameters
$\eta_{0},\operatorname{Pe}>0$, $\gamma_{1}\leq 0$, and
$\gamma_{2}\in\mathbb{R}$.
1. (i)
_Stokes case $\operatorname{Re}=0$, $d\leq 3$:_
Given $s>\frac{d}{2}$, $T_{0}>0$, and
$h\in\operatorname{L}^{\infty}(0,T_{0};H^{s+1}(\mathbb{T}^{d})^{d})\cap
W^{1,\infty}(0,T_{0};H^{s-1}(\mathbb{T}^{d})^{d})$, provided that
$\varepsilon\ll 1$ is small enough (depending on $s,T_{0},h$ and on all
parameters), the following nonlinear problem admits a unique solution
$\bar{u}_{\varepsilon}\in\operatorname{L}^{\infty}(0,T_{0};H^{s+1}(\mathbb{T}^{d})^{d})$,
$\qquad\left\\{\begin{array}[]{l}-\eta_{0}\Delta\bar{u}_{\varepsilon}+\nabla\bar{p}_{\varepsilon}=\big{(}1-\varepsilon\tfrac{\gamma_{1}}{\eta_{0}}(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta)\big{)}h+\varepsilon{\operatorname{div}}(F_{0}(\bar{u}_{\varepsilon})),\\\\[2.84526pt]
{\operatorname{div}}(\bar{u}_{\varepsilon})=0,\quad\int_{\mathbb{T}^{d}}\bar{u}_{\varepsilon}=0,\end{array}\right.$
(3.20)
where we have set for abbreviation
$\qquad
F_{0}(u)\,:=\,\gamma_{1}(u\cdot\nabla)2\\!\operatorname{D}(u)+\gamma_{1}\big{(}(\nabla
u)^{T}2\\!\operatorname{D}(u)+2\\!\operatorname{D}(u)(\nabla
u)\big{)}+\gamma_{2}(2\\!\operatorname{D}(u))^{2}.$ (3.21)
Moreover, the so-defined solution $\bar{u}_{\varepsilon}$ satisfies (3.17)
with $\operatorname{Re}=0$ for some controlled error term
$O(\varepsilon^{2})$.
2. (ii)
_Navier–Stokes case $\operatorname{Re}=1$, $d=2$:_
Further assume $\eta_{0}\geq\frac{1}{\operatorname{Pe}}$.666For
$\eta_{0}<\frac{1}{\operatorname{Pe}}$, the equations would need to be
rearranged differently; we skip it here for shortness. Given $s>1$, $T_{0}>0$,
$u^{\circ}\in H^{s}(\mathbb{T}^{2})^{2}$ with
${\operatorname{div}}(u^{\circ})=0$, and
$h\in\operatorname{L}^{2}(0,T_{0};H^{s+1}(\mathbb{T}^{2})^{2})$, provided that
$\varepsilon\ll 1$ is small enough (depending on $s,T_{0},u^{\circ},h$ and on
all parameters), the following nonlinear problem admits a unique solution
$\bar{u}_{\varepsilon}\in\operatorname{L}^{\infty}(0,T_{0};H^{s}(\mathbb{T}^{2})^{2})\cap\operatorname{L}^{2}(0,T_{0};H^{s+1}(\mathbb{T}^{2})^{2})$,
$\qquad\left\\{\begin{array}[]{l}(\partial_{t}+\bar{u}_{\varepsilon}\cdot\nabla)\bar{u}_{\varepsilon}-\eta_{0}\Delta\bar{u}_{\varepsilon}-\varepsilon\gamma_{1}(\eta_{0}-\frac{1}{\operatorname{Pe}})\Delta^{2}\bar{u}_{\varepsilon}+\nabla\bar{p}_{\varepsilon}\\\
\hskip
142.26378pt\,=\,(1+\varepsilon\gamma_{1}\Delta)h+\varepsilon{\operatorname{div}}(F_{1}(\bar{u}_{\varepsilon})),\\\
{\operatorname{div}}(\bar{u}_{\varepsilon})=0,\\\
\bar{u}_{\varepsilon}|_{t=0}=u^{\circ},\end{array}\right.$ (3.22)
where we have set for abbreviation
$\qquad F_{1}(u)\,:=\,2\gamma_{1}(\nabla u)^{T}(\nabla
u)+\gamma_{2}(2\\!\operatorname{D}(u))^{2}.$
Moreover, the so-defined solution $\bar{u}_{\varepsilon}$ satisfies (3.17)
with $\operatorname{Re}=1$ with some controlled error term
$O(\varepsilon^{2})$ and with initial condition
$\bar{u}_{\varepsilon}|_{t=0}=u^{\circ}$.∎
### 3.4. Models for inhomogeneous suspensions
The above formulation of ordered fluid models describes the behavior of
spatially homogeneous suspensions, that is, the behavior of fluids with a
constant density of suspended particles. This can be naturally generalized to
an inhomogeneous setting to describe non-uniform particle suspensions.
Surprisingly, we were not able to find any account of this generalization in
the literature. We emphasize however that such an inhomogeneous setting should
also arise naturally from the small-$\operatorname{Wi}$ expansion of the
inhomogeneous Oldroyd–B model, which is the formal exact closure of the
kinetic Hookean dumbbell model. Note that we are still considering an
homogeneous solvent fluid. At infinite Péclet number
$\operatorname{Pe}=\infty$, the adaptation is straightforward: the fluid
equations (3.1) are simply coupled to a conservation equation for the particle
density $\rho:=\fint_{\mathbb{S}^{d-1}}f(\cdot,n)\,\mathrm{d}n$,777The
physical particle density is rather given by
$x\mapsto\int_{\mathbb{S}^{d-1}}f(x,n)\,\mathrm{d}n$, but for notational
convenience we choose to normalize it by the area of $\mathbb{S}^{d-1}$. In
particular, we have $\rho\in\frac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d})$.
$(\partial_{t}+u\cdot\nabla)\rho=0,$
while the stress $\sigma$ is now a function both of $\rho$ and of the
Rivlin–Ericksen tensors $\\{A_{k}(u)\\}_{k}$ where non-Newtonian corrections
to the pure solvent viscosity $\eta_{0}$ are taken proportional to the
suspended particle density $\rho$. More precisely, the inhomogeneous second-
order fluid model takes the form
$\displaystyle\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+u\cdot\nabla)u-{\operatorname{div}}(\sigma)+\nabla
p=h,\\\\[2.84526pt] (\partial_{t}+u\cdot\nabla)\rho=0,\\\\[2.84526pt]
\sigma\,=\,(\eta_{0}+\eta_{1}\rho)A_{1}(u)+\alpha_{1}\rho
A_{2}(u)+\alpha_{2}\rho A_{1}(u)^{2},\\\\[2.84526pt]
{\operatorname{div}}(u)=0,\end{array}\right.$ (3.27)
for some coefficients $\eta_{1},\alpha_{1},\alpha_{2}\in\mathbb{R}$.
Inhomogeneous versions of higher-order fluid models are formulated similarly.
At finite Péclet number $\operatorname{Pe}<\infty$, on the other hand, the
particle density $\rho$ is no longer simply transported by the fluid, but
rather solves a transport-diffusion equation,
$(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta+u\cdot\nabla)\rho=0.$
In this diffusive setting, the structure of ordered fluid models becomes
slightly more complicated: due to diffusion, the transport-diffusion operator
that appears in the Rivlin–Ericksen tensors (3.13) at finite Péclet number
does not commute with multiplication with the particle density $\rho$. The
second-order fluid model then rather takes the form
$\displaystyle\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+u\cdot\nabla)u-{\operatorname{div}}(\sigma)+\nabla
p=h,\\\\[2.84526pt]
(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta+u\cdot\nabla)\rho=0,\\\\[2.84526pt]
\sigma\,=\,(\eta_{0}+\eta_{1}\rho)A_{1}(u)+\alpha_{1}A_{2}^{\prime}(\rho,u)+\alpha_{2}\rho
A_{1}(u)^{2},\\\\[2.84526pt] {\operatorname{div}}(u)=0,\end{array}\right.$
(3.32)
for some coefficients $\eta_{1},\alpha_{1},\alpha_{2}\in\mathbb{R}$, in terms
of the modified inhomogeneous second-order Rivlin–Ericksen tensor
$A_{2}^{\prime}(\rho,u)\,:=\,(\partial_{t}-{\tfrac{1}{\operatorname{Pe}}}\Delta+u\cdot\nabla)(\rho
A_{1}(u))+\rho\big{(}(\nabla u)^{T}A_{1}(u)+A_{1}(u)(\nabla u)\big{)}.$ (3.33)
Indeed, due to the diffusion, the latter quantity does not reduce to the
Rivlin–Ericksen tensor defined in (3.13): we have
$A_{2}^{\prime}(\rho,u)\neq\rho A_{2}^{\prime}(u)$ in general along solutions
— in contrast with the case of infinite Péclet number.888At infinite Péclet
number $\operatorname{Pe}=\infty$, as the particle density $\rho$ satisfies
$(\partial_{t}+u\cdot\nabla)\rho=0$, we indeed obtain
$(\partial_{t}+u\cdot\nabla)(\rho
A_{1}(u))=\rho(\partial_{t}+u\cdot\nabla)A_{1}(u)$, so that the system (3.32)
reduces to (3.27). Similarly, the inhomogeneous third-order fluid model
amounts to (3.32) with stress
$\sigma\,=\,(\eta_{0}+\eta_{1}\rho)A_{1}(u)+\alpha_{1}A_{2}^{\prime}(\rho,u)+\alpha_{2}A_{1}(u)^{2}+\beta_{1}A_{3}^{\prime}(\rho,u)+\beta_{1}^{\prime}B_{3}^{\prime}(\rho,u)\\\
+\beta_{2}\big{(}A_{1}(u)A_{2}^{\prime}(\rho,u)+A_{2}^{\prime}(\rho,u)A_{1}(u)\big{)}+\beta_{3}\rho
A_{1}(u){\operatorname{tr}}(A_{1}(u)^{2}),$
in terms of the modified inhomogeneous third-order Rivlin–Ericksen tensor
$A_{3}^{\prime}(\rho,u)\,:=\,(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta+u\cdot\nabla)A_{2}^{\prime}(\rho,u)+(\nabla
u)^{T}A_{2}^{\prime}(\rho,u)+A_{2}^{\prime}(\rho,u)(\nabla u),$
and in terms of the following additional quantity, which needs to be included
similarly as in (3.15) at finite Péclet number,
$B_{3}^{\prime}(\rho,u)\,:=\,(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta+u\cdot\nabla)(\rho
A_{1}(u)^{2})+\rho\big{(}(\nabla u)^{T}A_{1}(u)^{2}+A_{1}(u)^{2}(\nabla
u)\big{)}.$
We turn to the corresponding well-posedness question for the above
inhomogeneous models. As in the homogeneous setting, we focus for shortness on
the second-order model (3.32), which is again ill-posed whenever
$\alpha_{1}<0$. We consider the perturbative case of a weak nonlinearity,
$\alpha_{1}=\varepsilon\gamma_{1},\qquad\alpha_{2}=\varepsilon\gamma_{2},\qquad\varepsilon\ll
1,$
and we shall define a well-posed notion of approximate solutions that only
satisfy equation (3.32) up to higher-order $O(\varepsilon^{2})$ remainder:
$\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+\bar{u}_{\varepsilon}\cdot\nabla)\bar{u}_{\varepsilon}-{\operatorname{div}}(\bar{\sigma}_{\varepsilon})+\nabla\bar{p}_{\varepsilon}=h+O(\varepsilon^{2}),\\\\[2.84526pt]
(\partial_{t}+\bar{u}_{\varepsilon}\cdot\nabla)\bar{\rho}_{\varepsilon}-(\tfrac{1}{\operatorname{Pe}}+\varepsilon\mu_{0})\Delta\bar{\rho}_{\varepsilon}=O(\varepsilon^{2}),\\\\[2.84526pt]
\bar{\sigma}_{\varepsilon}\,=\,(\eta_{0}+\eta_{1}\bar{\rho}_{\varepsilon})A_{1}(\bar{u}_{\varepsilon})+\varepsilon\gamma_{1}A_{2}^{\prime}(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})+\varepsilon\gamma_{2}A_{1}(\bar{u}_{\varepsilon})^{2},\\\\[2.84526pt]
{\operatorname{div}}(\bar{u}_{\varepsilon})=0.\end{array}\right.$ (3.34)
For later purposes, note that we henceforth increase the diffusion of the
particle density by an additional constant $\mu_{0}\geq 0$, which will appear
in our setting as a possible effect of particle swimming velocities. Similarly
as in Proposition 3.2 for the homogeneous case, the simplest notion of well-
posed solutions takes the form of hierarchical solutions as described in the
following statement. The proof is postponed to Appendix B. Note that the
regularity theory for (3.35) below is quite delicate in the 3D Stokes case, as
a particularly careful stepwise argument is needed to first cover low-
regularity situations. The notion of Boussinesq-type solutions of Proposition
3.3 could also be easily extended to the present inhomogeneous setting, but we
skip the detail for conciseness.
###### Proposition 3.4 (Hierarchical solutions).
Consider equation (3.34) with parameters $\eta_{0},\operatorname{Pe}>0$,
$\eta_{1},\mu_{0}\geq 0$, and $\gamma_{1},\gamma_{2}\in\mathbb{R}$. If
$(u_{0},\rho_{0}),(u_{1},\rho_{1})$ are smooth solutions of the following two
auxiliary systems,
$\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+u_{0}\cdot\nabla)u_{0}-{\operatorname{div}}\big{(}2(\eta_{0}+\eta_{1}\rho_{0})\operatorname{D}(u_{0})\big{)}+\nabla
p_{0}=h,\\\\[2.84526pt]
(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta+u_{0}\cdot\nabla)\rho_{0}=0,\\\\[2.84526pt]
{\operatorname{div}}(u_{0})=0,\\\\[2.84526pt]
u_{0}|_{t=0}=u^{\circ}\quad\text{if $\operatorname{Re}\neq
0$},\quad\int_{\mathbb{T}^{d}}u_{0}=0\quad\text{if
$\operatorname{Re}=0$},\\\\[2.84526pt]
\rho_{0}|_{t=0}=\rho^{\circ},\end{array}\right.$ (3.35)
$\left\\{\begin{array}[]{l}\operatorname{Re}\big{(}(\partial_{t}+u_{0}\cdot\nabla)u_{1}+(u_{1}\cdot\nabla)u_{0}\big{)}-{\operatorname{div}}\big{(}2(\eta_{0}+\eta_{1}\rho_{0})\operatorname{D}(u_{1})\big{)}+\nabla
p_{1}\\\\[1.42262pt] \hskip
99.58464pt={\operatorname{div}}\big{(}2\eta_{1}\rho_{1}\operatorname{D}(u_{0})+\gamma_{1}A_{2}^{\prime}(\rho_{0},u_{0})+\gamma_{2}(2\operatorname{D}(u_{0}))^{2}\big{)},\\\\[2.84526pt]
(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta+u_{0}\cdot\nabla)\rho_{1}=\mu_{0}\Delta\rho_{0}-u_{1}\cdot\nabla\rho_{0},\\\\[2.84526pt]
{\operatorname{div}}(u_{1})=0,\\\\[2.84526pt] u_{1}|_{t=0}=0\quad\text{if
$\operatorname{Re}\neq 0$},\quad\int_{\mathbb{T}^{d}}u_{1}=0\quad\text{if
$\operatorname{Re}=0$},\\\\[2.84526pt] \rho_{1}|_{t=0}=0,\end{array}\right.$
(3.36)
then the superposition
$(\bar{u}_{\varepsilon},\bar{\rho}_{\varepsilon})=(u_{0}+\varepsilon
u_{1},\rho_{0}+\varepsilon\rho_{1})$ satisfies equation (3.34) with some
controlled remainder $O(\varepsilon^{2})$ and with initial condition
$\bar{\rho}_{\varepsilon}|_{t=0}=\rho^{\circ}$ (and
$\bar{u}_{\varepsilon}|_{t=0}=u^{\circ}$ if $\operatorname{Re}\neq 0$). For
the well-posedness of (3.35) and (3.36), we separately consider the Stokes and
Navier–Stokes cases:
1. (i)
_Stokes case $\operatorname{Re}=0$, $d\leq 3$:_
Given
$\rho^{\circ}\in\operatorname{L}^{2}(\mathbb{T}^{d})\cap\frac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d})$
and
$h\in\operatorname{L}^{\infty}_{{\operatorname{loc}}}(\mathbb{R}^{+};H^{-1}(\mathbb{T}^{d})^{d})$,
there exists a unique global solution $(u_{0},\rho_{0})$ of (3.35) with
$\displaystyle u_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{1}(\mathbb{T}^{d})^{d}),$
$\displaystyle\rho_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}\big{(}\mathbb{R}^{+};\operatorname{L}^{2}(\mathbb{T}^{d})\cap\tfrac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d})\big{)}\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{1}(\mathbb{T}^{d})).$
Moreover, for all integers $s\geq\frac{d}{2}+1$, provided that
$\rho^{\circ}\in H^{s}(\mathbb{T}^{d})$ and that $h$ belongs to
$\operatorname{L}^{\infty}_{{\operatorname{loc}}}(\mathbb{R}^{+};H^{s-1}(\mathbb{R}^{+}\times\mathbb{T}^{d})^{d})$,
this solution further satisfies
$\displaystyle u_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})^{d}),$
$\displaystyle\rho_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{d}))\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})).$
In that case, if furthermore $h\in
W^{1,\infty}_{{\operatorname{loc}}}(\mathbb{R}^{+};H^{s-3}(\mathbb{T}^{d})^{d})$,
there exists a unique global solution $(u_{1},\rho_{1})$ of (3.36) with
$\displaystyle u_{1}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s-1}(\mathbb{T}^{d})^{d}),$
$\displaystyle\rho_{1}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s-2}(\mathbb{T}^{d}))\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s-1}(\mathbb{T}^{d})).$
2. (ii)
_Navier–Stokes case $\operatorname{Re}=1$, $d=2$:_
Given
$\rho^{\circ}\in\operatorname{L}^{2}(\mathbb{T}^{2})\cap\frac{1}{2\pi}\mathcal{P}(\mathbb{T}^{2})$,
$u^{\circ}\in\operatorname{L}^{2}(\mathbb{T}^{2})^{2}$ which satisfies
${\operatorname{div}}(u^{\circ})=0$, and
$h\in\operatorname{L}^{2}_{{\operatorname{loc}}}(\mathbb{R}^{+};H^{-1}(\mathbb{T}^{2})^{2})$,
there exists a unique global solution $(u_{0},\rho_{0})$ of (3.35) with
$\displaystyle u_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};\operatorname{L}^{2}(\mathbb{T}^{2})^{2})\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{1}(\mathbb{T}^{2})^{2}),$
$\displaystyle\rho_{0}$ $\displaystyle\in$
$\displaystyle{\operatorname{L}^{\infty}_{\operatorname{loc}}\big{(}\mathbb{R}^{+};\operatorname{L}^{2}(\mathbb{T}^{2})\cap\tfrac{1}{2\pi}\mathcal{P}(\mathbb{T}^{2})\big{)}\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{1}(\mathbb{T}^{2})).}$
Moreover, for all integers $s\geq 2$, provided that $\rho^{\circ}\in
H^{s}(\mathbb{T}^{2})$, $u^{\circ}\in H^{s}(\mathbb{T}^{2})^{2}$, and
$h\in\operatorname{L}^{2}_{{\operatorname{loc}}}(\mathbb{R}^{+};H^{s-1}(\mathbb{T}^{2})^{2})$,
this solution further satisfies
$\displaystyle u_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{2})^{2})\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{2})^{2}),$
$\displaystyle\rho_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{2}))\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{2})).$
In that case, there exists a unique global solution $(u_{1},\rho_{1})$ of
(3.36) with
$\displaystyle u_{1}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\in\leavevmode\nobreak\ \leavevmode\nobreak\
\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s-2}(\mathbb{T}^{2})^{2})\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s-1}(\mathbb{T}^{2})^{2}),$
$\displaystyle\rho_{1}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \in\leavevmode\nobreak\ \leavevmode\nobreak\
\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s-2}(\mathbb{T}^{2}))\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s-1}(\mathbb{T}^{2})).\qed$
## 4\. Statement of main results
We turn to the precise statement of our main result, that is, the rigorous
hydrodynamic approximation of the Doi–Saintillan–Shelley theory in the
small-$\operatorname{Wi}$ regime. For simplicity, we focus on the first-order
approximation and the emergence of the second-order fluid model, but the same
analysis can be pursued to arbitrary order (see Section 4.2). More precisely,
we derive the second-order fluid model (3.34) with explicit coefficients
$\displaystyle\begin{split}\eta_{0}=1,\qquad\eta_{1}=\lambda\tfrac{(\theta+2)\omega_{d}}{2d(d+2)},\qquad\mu_{0}=\tfrac{1}{d(d-1)}U_{0}^{2},\\\\[2.84526pt]
\gamma_{1}=-\lambda\theta\tfrac{\omega_{d}}{4d^{2}(d+2)},\qquad\gamma_{2}=\lambda\tfrac{\omega_{d}}{2d^{2}(d+4)}(\theta+\tfrac{2d}{d+2}),\end{split}$
(4.1)
where we recall that $\lambda,\theta,U_{0}$ are parameters from the
Doi–Saintillan–Shelley system (2.5). Before formulating a precise result, we
introduce a suitable well-preparedness assumption for initial data. More
precisely, in order to avoid initial boundary layers due to the
$O(\frac{1}{\varepsilon})$ rotational diffusion, we first need to assume that
to leading order the initial density is invariant under this rotational
diffusion, which means that it is isotropic to leading order,
$f_{\varepsilon}(x,n)|_{t=0}\,=\,f_{\varepsilon}^{\circ}(x,n)\,=\,\rho^{\circ}(x)+O(\varepsilon),\qquad{\rho^{\circ}\in\tfrac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d})}.$
Yet, this is not sufficient: in order to avoid initial boundary layers in the
higher-order $\varepsilon$-expansion, we further need to assume that initial
data are compatible with the formal limiting hierarchy, which is precisely the
content of the assumption below. Similar issues are well known for higher-
order hydrodynamic expansions in the Boltzmann theory, see e.g. [Caf80, SK83,
Lac87].
###### Assumption 4.1 (Well-preparedness).
Let $h\in C^{\infty}(\mathbb{R}^{+}\times\mathbb{T}^{d})^{d}$, let
$\rho^{\circ}\in
H^{s}(\mathbb{T}^{d})\cap\frac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d})$ for
some $s\gg 1$, and let also $u^{\circ}\in H^{s}(\mathbb{T}^{d})^{d}$ with
${\operatorname{div}}(u^{\circ})=0$ in the Navier–Stokes case. We assume that
the initial condition $f_{\varepsilon}|_{t=0}=f_{\varepsilon}^{\circ}$ for the
Doi–Saintillan–Shelley system (2.5) is well-prepared in the following sense:
decomposing
$f_{\varepsilon}^{\circ}(x,n)\,=\,\rho_{\varepsilon}^{\circ}(x)+g_{\varepsilon}^{\circ}(x,n),\qquad\rho_{\varepsilon}^{\circ}\,:=\,\langle
f_{\varepsilon}^{\circ}\rangle\,=\,\fint_{\mathbb{S}^{d-1}}f_{\varepsilon}^{\circ}(\cdot,n)\,\mathrm{d}n\,\in\,\tfrac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d}),$
we have
$\rho_{\varepsilon}^{\circ}=\rho^{\circ}\qquad\text{and}\qquad\varepsilon^{\frac{1}{2}}\|g_{\varepsilon}^{\circ}-(\varepsilon
g_{1}+\varepsilon^{2}g_{2})|_{t=0}\|_{\operatorname{L}^{2}_{x,n}}\leq\,C_{0}\varepsilon^{3},$
for some constant $C_{0}<\infty$, where $g_{1}$ and $g_{2}$ are the solutions
of the hierarchical equations (5.3) and (5.4) below with initial data
$\rho^{\circ}$ in the Stokes case and $(u^{\circ},\rho^{\circ})$ in the
Navier–Stokes case. ∎
Note that this well-preparedness assumption is compatible with the positivity
$f_{\varepsilon}|_{t=0}\geq 0$ for $\varepsilon$ small enough, which is
necessary to ensure well-posedness of the Doi–Saintillan–Shelley system (2.5),
cf. Proposition 2.1. In these terms, we are now in position to state our main
result, thus finally providing a more detailed statement of Theorem 1.1. The
proof is given in Section 5.
###### Theorem 4.2 (Small-$\operatorname{Wi}$ expansion).
Let $h\in C^{\infty}(\mathbb{R}^{+}\times\mathbb{T}^{d})^{d}$, let
$\rho^{\circ}\in
H^{s}(\mathbb{T}^{d})\cap\frac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d})$ for
some $s\gg 1$, and let also $u^{\circ}\in H^{s}(\mathbb{T}^{d})^{d}$ with
${\operatorname{div}}(u^{\circ})=0$ in the Navier–Stokes case. Denote by
$(u_{\varepsilon},f_{\varepsilon})$ the global solution of the
Doi–Saintillan–Shelley model (2.5) as given by Proposition 2.1 with initial
condition
$f_{\varepsilon}|_{t=0}=f_{\varepsilon}^{\circ}\in\operatorname{L}^{2}\cap\mathcal{P}(\mathbb{T}^{d}\times\mathbb{S}^{d-1})$
satisfying the well-preparedness of Assumption 4.1. Further assume that
$\displaystyle\lambda\theta(1+\operatorname{Pe})\|\rho^{\circ}\|_{\operatorname{L}^{\infty}_{x}}\,\ll\,1$
is smaller than some universal constant.
1. (i)
_Stokes case $\operatorname{Re}=0$, $d\leq 3$:_
Let $(\bar{u}_{\varepsilon},\bar{\rho}_{\varepsilon})$ be the unique global
hierarchical solution of (3.34) as given by Proposition 3.4(i) with explicit
coefficients (4.1). Then we have for all $t\geq 0$,
$\displaystyle\|\nabla(u_{\varepsilon}-\bar{u}_{\varepsilon})\|_{\operatorname{L}^{2}_{t}\operatorname{L}^{2}_{x}}$
$\displaystyle\lesssim$ $\displaystyle\varepsilon^{2},$
$\displaystyle\|\rho_{\varepsilon}-\bar{\rho}_{\varepsilon}\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x}}+\|\nabla(\rho_{\varepsilon}-\bar{\rho}_{\varepsilon})\|_{\operatorname{L}^{2}_{t}\operatorname{L}^{2}_{x}}$
$\displaystyle\lesssim$ $\displaystyle\varepsilon^{2},$
where multiplicative constants depend on $\operatorname{Pe}$ and on an upper
bound on $t,\lambda,U_{0},$ and on controlled norms of $h$ and $\rho^{\circ}$.
2. (ii)
_Navier–Stokes case $\operatorname{Re}=0$, $d=2$:_
Let $(\bar{u}_{\varepsilon},\bar{\rho}_{\varepsilon})$ be the unique global
hierarchical solution of (3.34) as given by Proposition 3.4(ii) with explicit
coefficients (4.1). Then we have for all $t\geq 0$,
$\displaystyle\|u_{\varepsilon}-\bar{u}_{\varepsilon}\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x}}+\|\nabla(u_{\varepsilon}-\bar{u}_{\varepsilon})\|_{\operatorname{L}^{2}_{t}\operatorname{L}^{2}_{x}}$
$\displaystyle\leavevmode\nobreak\ \lesssim\leavevmode\nobreak\
\varepsilon^{2},$
$\displaystyle\|\rho_{\varepsilon}-\bar{\rho}_{\varepsilon}\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x}}+\|\nabla(\rho_{\varepsilon}-\bar{\rho}_{\varepsilon})\|_{\operatorname{L}^{2}_{t}\operatorname{L}^{2}_{x}}$
$\displaystyle\leavevmode\nobreak\ \lesssim\leavevmode\nobreak\
\varepsilon^{2},$
where multiplicative constants depend on $\operatorname{Pe}$ and on an upper
bound on $t,\lambda,U_{0},$ and on controlled norms of $h,\rho^{\circ},$ and
$u^{\circ}$. ∎
### 4.1. Non-Newtonian properties of hydrodynamic approximation
We comment on the rheological features of the obtained second-order fluid
system (3.34) with the explicit parameters
$\eta_{0},\eta_{1},\mu_{0},\gamma_{1},\gamma_{2}$ as defined in (4.1), briefly
describing the resulting non-Newtonian properties and how they depend on
microscopic features. We focus here on the physically relevant 3D setting, and
we point out that in the passive case the parameter values agree with [HL72,
Eqn (41) and Table 2] and [Bre74, Eqn (7.4)].
1. $\bullet$
Effective spatial diffusion:
The spatial diffusion $\frac{1}{\operatorname{Pe}}$ of the suspended particle
density is enhanced by particles’ activity even at infinite Péclet number: it
is replaced by $\frac{1}{\operatorname{Pe}}+\varepsilon\mu_{0}$ with
$\mu_{0}=\frac{1}{6}U_{0}^{2}$. This naturally follows from the coupling of
particles’ swimming velocity with their rotational diffusion. This phenomenon
of increased mixing has been observed in studies such as [SS07, Sai18].
2. $\bullet$
Modified zero-shear viscosity:
The presence of suspended particles leads to a non-trivial contribution to the
zero-shear viscosity: in the homogeneous setting
$\bar{\rho}_{\varepsilon}\equiv\frac{1}{\omega_{d}}$, we obtain a zero-shear
viscosity
$\displaystyle\tilde{\eta}_{0}\,:=\,\eta_{0}+\tfrac{1}{\omega_{d}}\eta_{1}=1+\tfrac{1}{30}\lambda(2+\theta).$
In particular, in case of passive particles, this zero-shear viscosity is
always larger than the plain fluid viscosity, $\tilde{\eta}_{0}>1$, while
particles’ activity can reverse this effect. For a precise description, first
recall that
$\theta=6+2\alpha\tfrac{|V_{0}|\ell\mu_{\operatorname{fl}}}{k_{B}\Theta}$, cf.
(2.4), where $\alpha$ characterizes the swimming mechanism:
1. —
for passive particles $\alpha=0$, the zero-shear viscosity is
$\tilde{\eta}_{0}=1+\frac{4}{15}\lambda>1$;
2. —
for so-called puller particles $\alpha>0$, the zero-shear viscosity is even
larger than for passive particles;
3. —
for so-called pusher particles $\alpha<0$, the zero-shear viscosity is smaller
than for passive particles, and it can even be smaller than the plain fluid
viscosity provided that the activity of the particles is strong enough: we
find $\tilde{\eta}_{0}<1$ if
$\alpha<-\frac{4k_{B}\Theta}{|V_{0}|\ell\mu_{\operatorname{fl}}}$.
This prediction is consistent with well-known experimental results, see e.g.
[SA09, RJP10, Lóp+15], and it has been largely confirmed in the literature
[HABK08, Hai+09, Sai10a, Sai10, GLAB11, AMES16]. In particular, for E. coli
bacteria (a typical pusher particle), we can assess the value of the parameter
$\alpha$ using the experimental measurements performed in [Dre+11]: this
yields $\alpha<-\frac{4k_{B}\Theta}{|V_{0}|\ell\mu_{\operatorname{fl}}}$ and
the experimental findings of [Gac+13] then confirm our prediction that the
effective zero-shear viscosity is smaller than the plain fluid viscosity.
3. $\bullet$
Normal-stress differences:
In the homogeneous setting
$\bar{\rho}_{\varepsilon}\equiv\frac{1}{\omega_{d}}$, we obtain the following
values for first and second-normal stress coefficients, as defined in (3.6),
$\nu_{10}=\tfrac{\theta}{90}\varepsilon\lambda\qquad\text{and}\qquad\nu_{20}=\tfrac{3-\theta}{315}\varepsilon\lambda.$
For passive particles ($\theta=6$), we thus obtain $\nu_{10}>0$, $\nu_{20}<0$,
and $\nu_{10}/|\nu_{20}|=7$, which agrees with experiments as discussed in
Section 3. The amplitude of these normal stress coefficients is even increased
in case of puller particles. In contrast, for pusher particles, normal stress
coefficients are reduced, and a very large activity could even result into
opposite effects: we find $\nu_{10}<0$ if $\theta<0$, and $\nu_{20}>0$ if
$\theta<3$. This behavior was also predicted in [Sai10, PRB16], but has yet to
be experimentally verified.
4. $\bullet$
Elongational viscosity:
In the homogeneous setting
$\bar{\rho}_{\varepsilon}\equiv\frac{1}{\omega_{d}}$, in a uniaxial
elongational flow in the direction $e_{1}$, that is,
$\bar{u}_{\varepsilon}=\kappa(x_{1}e_{1}-\tfrac{1}{2}(x_{2}e_{2}+x_{3}e_{3}))$,
we obtain the following value for the elongational viscosity, as defined in
(3.10),
$\eta_{E}=\big{(}3+\tfrac{1}{10}\lambda(2+\theta)\big{)}+\kappa\tfrac{1}{140}\varepsilon\lambda(4+\theta).$
This shows that passive suspensions ($\theta=6$) lead to a strain-thickening
behavior, which is even increased in case of puller particles. In contrast,
for pusher particles, the strain-thickening behavior is reduced, and a very
large activity could even result in the opposite effect: the system becomes
strain-thinning if $\theta<-4$ (that is,
$\alpha<-\tfrac{5k_{B}\Theta}{|V_{0}|\ell\mu_{\operatorname{fl}}}$).
### 4.2. Next-order description
The above result is easily pursued to higher orders in the
small-$\operatorname{Wi}$ expansion. For shortness, we stick here to a formal
discussion. First, we show in Appendix C that the next-order description of
the suspended particle density involves additional nontrivial transport and
anisotropic diffusion terms depending on the surrounding fluid flow: we find
$(\partial_{t}+\bar{u}_{\varepsilon}\cdot\nabla)\bar{\rho}_{\varepsilon}-{\operatorname{div}}\Big{(}\big{(}{\tfrac{1}{\operatorname{Pe}}}+\varepsilon\mu_{0}+\varepsilon^{2}\mu_{1}\operatorname{D}(\bar{u}_{\varepsilon})\big{)}\nabla\bar{\rho}_{\varepsilon}\Big{)}=\varepsilon^{2}\mu_{2}\,{\operatorname{div}}(\bar{\rho}_{\varepsilon}\Delta\bar{u}_{\varepsilon})+O(\varepsilon^{3}),$
(4.2)
with explicit coefficients
$\displaystyle\mu_{0}$ $\displaystyle:=$
$\displaystyle\tfrac{U_{0}^{2}}{d(d-1)},$ $\displaystyle\mu_{1}$
$\displaystyle:=$ $\displaystyle\tfrac{(3d+1)U_{0}^{2}}{d(d-1)^{2}(d+2)},$
$\displaystyle\mu_{2}$ $\displaystyle:=$
$\displaystyle\tfrac{U_{0}^{2}}{2d(d-1)(d+2)}.$
In particular, this shows that homogeneous spatial densities are still stable
to order $O(\varepsilon^{3})$, and we shall henceforth restrict for simplicity
to the homogeneous setting,
$\bar{\rho}_{\varepsilon}=\tfrac{1}{\omega_{d}}+O(\varepsilon^{3}).$
In addition, we shall focus on the case of infinite Péclet number and of
vanishing particle swimming velocity,
$\operatorname{Pe}=\infty,\qquad U_{0}=0,$
as this choice strongly simplifies the macroscopic equations and as it seems
anyhow to be the most relevant setting physically, cf. Section 2 (recall
however that our rigorous results do not hold for $\operatorname{Pe}=\infty$).
In this setting, we formally derive in Appendix C the following third-order
fluid equations,
$\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+\bar{u}_{\varepsilon}\cdot\nabla)\bar{u}_{\varepsilon}-{\operatorname{div}}(\bar{\sigma}_{\varepsilon})+\nabla\bar{p}_{\varepsilon}=h+O(\varepsilon^{3}),\\\\[2.84526pt]
{\operatorname{div}}(\bar{u}_{\varepsilon})=0,\end{array}\right.$ (4.3)
where the stress is given by
$\bar{\sigma}_{\varepsilon}=(1+\eta_{1})A_{1}(\bar{u}_{\varepsilon})+\varepsilon\gamma_{1}A_{2}(\bar{u}_{\varepsilon})+\varepsilon\gamma_{2}A_{1}(\bar{u}_{\varepsilon})^{2}\\\
\qquad\,+\varepsilon^{2}\kappa_{1}A_{3}(\bar{u}_{\varepsilon})+\varepsilon^{2}\kappa_{2}\big{(}A_{1}(\bar{u}_{\varepsilon})A_{2}(\bar{u}_{\varepsilon})+A_{2}(\bar{u}_{\varepsilon})A_{1}(\bar{u}_{\varepsilon})\big{)}+\varepsilon^{2}\kappa_{3}A_{1}(\bar{u}_{\varepsilon}){\operatorname{tr}}(A_{1}(\bar{u}_{\varepsilon})^{2})$
with explicit coefficients
$\displaystyle\eta_{1}$ $\displaystyle:=$
$\displaystyle\lambda\tfrac{1}{2d(d+2)}(\theta+2),$ $\displaystyle\gamma_{1}$
$\displaystyle:=$ $\displaystyle-\lambda\theta\tfrac{1}{4d^{2}(d+2)},$
$\displaystyle\gamma_{2}$ $\displaystyle:=$
$\displaystyle\lambda\tfrac{1}{2d^{2}(d+4)}(\theta+\tfrac{2d}{d+2}),$
$\displaystyle\kappa_{1}$ $\displaystyle:=$
$\displaystyle\lambda\theta\tfrac{1}{8d^{3}(d+2)},$ $\displaystyle\kappa_{2}$
$\displaystyle:=$
$\displaystyle-\lambda\tfrac{1}{8d^{3}(d+4)}(3\theta+\tfrac{2d}{d+2}),$
$\displaystyle\kappa_{3}$ $\displaystyle:=$
$\displaystyle\lambda\tfrac{1}{8d^{3}(d+2)^{2}(d+4)(d+6)}\Big{(}2d(3d^{2}+10d+6)+\theta(d+4)(3d^{2}+11d+12)\Big{)}.$
These third-order fluid coefficients coincide in the passive case ($\theta=6$)
with those computed by Brenner [Bre74, Eq. (7.4)] (once the notation is
properly identified). Regarding the non-Newtonian phenomena discussed in
Section 3, the main observation is that this third-order fluid model describes
the expected shear-thinning behavior of the suspension. Indeed, the shear-
dependent viscosity is given in 3D as follows, cf. Section 3.1,
$\displaystyle\kappa\leavevmode\nobreak\ \leavevmode\nobreak\
\mapsto\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle
1+\eta_{1}+2\varepsilon^{2}(\kappa_{2}+\kappa_{3})\kappa^{2}$
$\displaystyle\,\qquad\qquad=\leavevmode\nobreak\ $ $\displaystyle
1+\lambda\tfrac{\theta+2}{30}-\varepsilon^{2}\lambda\tfrac{19\theta-12}{18900}\kappa^{2},$
which is decreasing in $\kappa$ if and only if $\theta>\frac{12}{19}$. As
expected, this shows that passive suspensions ($\theta=6$) lead to a shear-
thinning behavior, which is even increased in case of puller particles. In
contrast, for pusher particles, the shear-thinning behavior is reduced, and a
very large activity could even result in the opposite effect: the system
becomes shear-thickening if $\theta<\frac{12}{19}$ (that is,
$\alpha<-\frac{51k_{B}\Theta}{19|V_{0}|\ell\mu_{\operatorname{fl}}}$). This
possible shear-thickening effect was indeed measured experimentally in
[Gac+13, Lóp+15] for suspensions of E. coli bacteria (pusher-type
microswimmers) with strong enough activity. We also refer to [Hai+09, GLAB11,
PRB16] for analytical and numerical results showing the same effect.
We note that for $U_{0}\neq 0$ the corresponding fluid equation for
$\bar{u}_{\varepsilon}$ would differ from the 3rd-order fluid model even at
infinite Péclet number and at uniform particle density. In particular, an
additional dispersive correction
$-\varepsilon^{2}\kappa_{4}\Delta^{2}\bar{u}_{\varepsilon}$ needs to be
included in the fluid equation. We skip the detail as the case $|U_{0}|\ll 1$
seems to be the most relevant physically.
## 5\. Small-$\operatorname{Wi}$ expansion of Doi–Saintillan–Shelley theory
This section is devoted to the small-$\varepsilon$ expansion of the solution
$(u_{\varepsilon},f_{\varepsilon})$ of the Doi–Saintillan–Shelley model (2.5),
as well as to the identification of second-order fluid equations satisfied by
the truncated expansion. We naturally split the particle density as
$f_{\varepsilon}(x,n)\,=\,\rho_{\varepsilon}(x)+g_{\varepsilon}(x,n),\qquad\rho_{\varepsilon}\,:=\,\langle
f_{\varepsilon}\rangle\,:=\,\fint_{\mathbb{S}^{d-1}}f_{\varepsilon}(\cdot,n)\,\mathrm{d}n\,\in\,\tfrac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d}),$
where $\rho_{\varepsilon}$ stands for the spatial density and where $\langle
g_{\varepsilon}\rangle=0$. Recall the well-preparedness condition of
Assumption 4.1: we assume in particular
$\rho_{\varepsilon}|_{t=0}=\rho^{\circ},\qquad
g_{\varepsilon}|_{t=0}=O(\varepsilon),\qquad\rho^{\circ}\in\tfrac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d}).$
(5.1)
meaning that we start from an initial density that is to leading order at
equilibrium with respect to the strong rotational diffusion in (2.5). In this
setting, we shall analyze the asymptotic behavior of the solution
$(u_{\varepsilon},f_{\varepsilon})$ and derive a hydrodynamic approximation in
the spirit of Hilbert’s expansion method in the Boltzmann theory [Hil12,
Caf80, Gol05, SR09]. We start from the ansatz
$\displaystyle\begin{split}u_{\varepsilon}&=u_{0}+\varepsilon
u_{1}+\varepsilon^{2}u_{2}+\ldots,\\\
\rho_{\varepsilon}&=\rho_{0}+\varepsilon\rho_{1}+\varepsilon^{2}\rho_{2}+\ldots,\\\
g_{\varepsilon}&=g_{0}+\varepsilon
g_{1}+\varepsilon^{2}g_{2}+\varepsilon^{3}g_{3}+\ldots,\end{split}$ (5.2)
with $\rho_{0}\in\frac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d})$, with
$\int_{\mathbb{T}^{d}}\rho_{n}=0$ for $n\geq 1$, and with $\langle
g_{n}\rangle=0$ for all $n\geq 0$. Inserting it into the system (2.5), and
identifying powers of $\varepsilon$, we are led formally to the following
hierarchy of coupled equations:
1. $\bullet$
_Order $O(\varepsilon^{-1})$:_ The equation for the particle density yields
$\Delta_{n}g_{0}=0$, and therefore, as by definition $\langle g_{0}\rangle=0$,
we must have
$g_{0}=0.$
On the other hand, the fluid equation yields
${\operatorname{div}}_{x}(\sigma_{1}[\rho_{0}+g_{0}])=0$, which is then
automatically satisfied as $g_{0}=0$ and as $\rho_{0}$ does not depend on $n$.
2. $\bullet$
_Order $O(\varepsilon^{0})$:_ The triplet $(u_{0},\rho_{0},g_{1})$ satisfies
$\left\\{\begin{array}[]{l}\operatorname{Re}(\partial_{t}+u_{0}\cdot\nabla)u_{0}-\Delta
u_{0}+\nabla
p_{0}=h+{\operatorname{div}}(\sigma_{1}[g_{1}])+{\operatorname{div}}(\sigma_{2}[\rho_{0},\nabla
u_{0}]),\\\\[2.84526pt]
\Delta_{n}g_{1}=U_{0}n\cdot\nabla_{x}\rho_{0}+{\operatorname{div}}_{n}\big{(}\pi_{n}^{\perp}(\nabla
u_{0})n\rho_{0}\big{)},\\\\[2.84526pt]
(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta+u_{0}\cdot\nabla)\rho_{0}=0,\\\\[2.84526pt]
{\operatorname{div}}(u_{0})=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\langle g_{1}\rangle=0,\\\\[2.84526pt]
u_{0}|_{t=0}=u^{\circ}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{if
$\operatorname{Re}\neq 0$},\ \
\int_{\mathbb{T}^{d}}u_{0}=0\leavevmode\nobreak\ \leavevmode\nobreak\ \text{if
$\operatorname{Re}=0$},\\\\[2.84526pt]
\rho_{0}|_{t=0}=\rho^{\circ}.\end{array}\right.$ (5.3)
3. $\bullet$
_Order $O(\varepsilon^{1})$:_ The triplet $(u_{1},\rho_{1},g_{2})$ satisfies
$\left\\{\begin{array}[]{l}\operatorname{Re}\big{(}(\partial_{t}+u_{0}\cdot\nabla)u_{1}+(u_{1}\cdot\nabla)u_{0}\big{)}-\Delta
u_{1}+\nabla p_{1}\\\\[2.84526pt] \hskip
56.9055pt\,=\,{\operatorname{div}}(\sigma_{1}[g_{2}])+{\operatorname{div}}\big{(}\sigma_{2}[\rho_{0},\nabla
u_{1}]+\sigma_{2}[\rho_{1}+g_{1},\nabla u_{0}]\big{)},\\\\[2.84526pt]
\Delta_{n}g_{2}=(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta_{x}+u_{0}\cdot\nabla_{x})g_{1}+P_{1}^{\bot}\big{(}U_{0}n\cdot\nabla_{x}(\rho_{1}+g_{1})\big{)}\\\\[2.84526pt]
\hskip 113.81102pt+{\operatorname{div}}_{n}\big{(}\pi_{n}^{\bot}(\nabla
u_{1})n\rho_{0}+\pi_{n}^{\bot}(\nabla
u_{0})n(\rho_{1}+g_{1})\big{)},\\\\[2.84526pt]
(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta_{x}+u_{0}\cdot\nabla_{x})\rho_{1}+u_{1}\cdot\nabla_{x}\rho_{0}+\langle
U_{0}n\cdot\nabla_{x}g_{1}\rangle=0,\\\\[2.84526pt]
{\operatorname{div}}(u_{1})=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\langle g_{2}\rangle=0,\\\\[2.84526pt] u_{1}|_{t=0}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{if $\operatorname{Re}\neq 0$},\ \
\int_{\mathbb{T}^{d}}u_{1}=0\leavevmode\nobreak\ \leavevmode\nobreak\ \text{if
$\operatorname{Re}=0$},\\\\[2.84526pt] \rho_{1}|_{t=0}=0.\end{array}\right.$
(5.4)
4. $\bullet$
_Order $O(\varepsilon^{2})$:_ The triplet $(u_{2},\rho_{2},f_{3})$ satisfies
$\left\\{\begin{array}[]{l}\operatorname{Re}\big{(}(\partial_{t}+u_{0}\cdot\nabla)u_{2}+(u_{1}\cdot\nabla)u_{1}+(u_{2}\cdot\nabla)u_{0}\big{)}-\Delta
u_{2}+\nabla p_{2}\\\\[2.84526pt] \hskip
28.45274pt=\,{\operatorname{div}}(\sigma_{1}[g_{3}])+{\operatorname{div}}\big{(}\sigma_{2}[\rho_{0},\nabla
u_{2}]+\sigma_{2}[\rho_{1}+g_{1},\nabla
u_{1}]+\sigma_{2}[\rho_{2}+g_{2},\nabla u_{0}]\big{)},\\\\[2.84526pt]
\Delta_{n}g_{3}=(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta_{x}+u_{0}\cdot\nabla_{x})g_{2}+u_{1}\cdot\nabla_{x}g_{1}+P_{1}^{\bot}\big{(}U_{0}n\cdot\nabla_{x}(\rho_{2}+g_{2})\big{)}\\\\[2.84526pt]
\hskip 42.67912pt\,+\,{\operatorname{div}}_{n}\big{(}\pi_{n}^{\bot}(\nabla
u_{2})n\rho_{0}+\pi_{n}^{\bot}(\nabla
u_{1})n(\rho_{1}+g_{1})+\pi_{n}^{\bot}(\nabla
u_{0})n(\rho_{2}+g_{2})\big{)},\\\\[2.84526pt]
(\partial_{t}-\tfrac{1}{\operatorname{Pe}}\Delta_{x}+u_{0}\cdot\nabla_{x})\rho_{2}+u_{1}\cdot\nabla_{x}\rho_{1}+u_{2}\cdot\nabla_{x}\rho_{0}+\langle
U_{0}n\cdot\nabla_{x}g_{2}\rangle=0,\\\\[2.84526pt]
{\operatorname{div}}(u_{2})=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\langle g_{3}\rangle=0,\\\\[2.84526pt] u_{2}|_{t=0}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \text{if $\operatorname{Re}\neq 0$},\ \
\int_{\mathbb{T}^{d}}u_{2}=0\leavevmode\nobreak\ \leavevmode\nobreak\ \text{if
$\operatorname{Re}=0$},\\\\[2.84526pt] \rho_{2}|_{t=0}=0.\end{array}\right.$
(5.5)
In view of this hierarchy, we understand that the condition (5.1) for initial
data needs to be further strengthened to avoid initial boundary layers: more
precisely, we need to assume that the initial condition
$f_{\varepsilon}|_{t=0}=f_{\varepsilon}^{\circ}$ is compatible with the above
hierarchy, meaning that $g_{\varepsilon}$ coincides initially with
$\varepsilon g_{1}+\varepsilon^{2}g_{2}+\varepsilon^{3}g_{3}$ up to higher
order errors. To accuracy $O(\varepsilon^{2})$, this well-preparedness is
precisely the content of Assumption 4.1.
The well-posedness and the propagation of regularity for the above hierarchy
are stated in the following result. We emphasize that we were not able to find
references for the types of systems that naturally appear in this
$\varepsilon$-expansion, and we believe that some of these new well-posedness
results may be of independent interest (see for instance the system (5.18)
below). Note that the hierarchy is triangular as we can eliminate the
densities $g_{1},g_{2},g_{3}$ in terms of the velocity fields
$u_{0},u_{1},u_{2}$ and of the spatial densities $\rho_{0},\rho_{1},\rho_{2}$.
We focus on the Stokes case $\operatorname{Re}=0$, while the 2D Navier–Stokes
case follows up to straightforward adaptations and is omitted for shortness.
The proof is displayed in Section 5.2.
###### Proposition 5.1 (Well-posedness of hierarchy).
Consider the Stokes case $\operatorname{Re}=0$, $d\leq 3$.
1. (i)
_Well-posedness for $(u_{0},\rho_{0},g_{1})$:_
Given integer $s\geq\tfrac{d}{2}+1$, $\rho^{\circ}\in
H^{s}(\mathbb{T}^{d})\cap\frac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d})$, and
$h\in\operatorname{L}^{\infty}_{{\operatorname{loc}}}(\mathbb{R}^{+};H^{s-1}(\mathbb{T}^{d})^{d})$,
there exists a unique solution $(u_{0},\rho_{0},g_{1})$ of the Cauchy problem
(5.3) with
$\displaystyle u_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})^{d}),$
$\displaystyle\rho_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}\big{(}\mathbb{R}^{+};H^{s}(\mathbb{T}^{d})\cap\tfrac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d})\big{)}\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})),$
$\displaystyle g_{1}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s-1}(\mathbb{T}^{d}\times\mathbb{S}^{d-1})),$
and $g_{1}$ is given by the explicit formula
$\displaystyle
g_{1}(\cdot,n)=-\tfrac{1}{d-1}U_{0}n\cdot\nabla\rho_{0}+\tfrac{1}{2}(n\otimes
n):\rho_{0}\operatorname{D}(u_{0}).$ (5.6)
Moreover, for all $r\geq 0$, provided that $h\in
W^{2,\infty}(\mathbb{R}^{+};H^{s-1}(\mathbb{T}^{d})^{d})$ and that $s$ is
chosen large enough, we also have
$\displaystyle\partial_{t}u_{0},\,\partial_{t}^{2}u_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{{\operatorname{loc}}}(\mathbb{R}^{+};H^{r}(\mathbb{T}^{d})^{d}),$
$\displaystyle\partial_{t}\rho_{0},\,\partial_{t}^{2}\rho_{0}$
$\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{{\operatorname{loc}}}(\mathbb{R}^{+};H^{r}(\mathbb{T}^{d})).$
2. (ii)
_Well-posedness for $(u_{1},\rho_{1},g_{2})$:_
Given $s\geq 0$, provided that the solution $(u_{0},\rho_{0})$ of item (i) is
such that
$\displaystyle u_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+3}(\mathbb{T}^{d})^{d})\cap
W^{1,\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})^{d}),$
$\displaystyle\rho_{0}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+2}(\mathbb{T}^{d}))\cap
W^{1,\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{d})),$
there exists a unique solution $(u_{1},\rho_{1},g_{2})$ of the Cauchy problem
(5.4) with
$\displaystyle u_{1}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})^{d}),$
$\displaystyle\rho_{1}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{d}))\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})),$
$\displaystyle g_{2}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{d}\times\mathbb{S}^{d-1})),$
and $g_{2}$ is given by the explicit formula
$\displaystyle\qquad g_{2}(\cdot,n)$ $\displaystyle=$
$\displaystyle-\tfrac{1}{d-1}U_{0}n\cdot\nabla\rho_{1}-\tfrac{1}{2d}(n\otimes
n-\tfrac{1}{d}\operatorname{Id}):\Big{(}\tfrac{1}{4}A_{2}^{\prime}(u_{0},\rho_{0})-\rho_{0}\operatorname{D}(u_{0})^{2}$
(5.7) $\displaystyle\hskip
113.81102pt-d\rho_{0}\operatorname{D}(u_{1})-d\rho_{1}\operatorname{D}(u_{0})-\tfrac{1}{d-1}U_{0}^{2}\nabla^{2}\rho_{0}\Big{)}$
$\displaystyle-\tfrac{1}{3(d-1)}U_{0}(\nabla\rho_{0})\cdot\Big{(}n\big{(}(n\otimes
n):\operatorname{D}(u_{0})\big{)}+\tfrac{4}{d-1}\operatorname{D}(u_{0})n\Big{)}$
$\displaystyle-\tfrac{1}{6(d+1)}U_{0}\rho_{0}\,{\operatorname{div}}_{x}\Big{(}n\big{(}(n\otimes
n):\operatorname{D}(u_{0})\big{)}+\tfrac{4}{d-1}\operatorname{D}(u_{0})n\Big{)}$
$\displaystyle+\tfrac{1}{8}\rho_{0}\Big{(}\big{(}(n\otimes
n):\operatorname{D}(u_{0})\big{)}^{2}-\tfrac{2}{d(d+2)}\operatorname{tr}(\operatorname{D}(u_{0})^{2})\Big{)},$
in terms of the (non-standard) Rivlin–Ericksen tensor $A_{2}^{\prime}$ defined
in (3.33). Moreover, for all $r\geq 0$, provided that $s$ is chosen large
enough, we also have
$\displaystyle\partial_{t}u_{1}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{{\operatorname{loc}}}(\mathbb{R}^{+};H^{r}(\mathbb{T}^{d})^{d}),$
$\displaystyle\partial_{t}\rho_{1}$ $\displaystyle\in$
$\displaystyle\operatorname{L}^{\infty}_{{\operatorname{loc}}}(\mathbb{R}^{+};H^{r}(\mathbb{T}^{d})).$
3. (iii)
_Well-posedness for $(u_{2},\rho_{2},g_{3})$:_
Given $s\geq 0$, provided that the solutions $(u_{0},\rho_{0})$ and
$(u_{1},\rho_{1})$ of items (i) and (ii) are such that
$\displaystyle(u_{0},\rho_{0})$ $\displaystyle\in$ $\displaystyle
W^{2,\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{r}(\mathbb{T}^{d})^{d+1}),$
$\displaystyle(u_{1},\rho_{1})$ $\displaystyle\in$ $\displaystyle
W^{1,\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{r}(\mathbb{T}^{d})^{d+1}),$
for some $r$ large enough, then there exists a unique solution
$(u_{2},\rho_{2},g_{3})$ of the Cauchy problem (5.5) with
$\displaystyle u_{2}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\in\leavevmode\nobreak\ \leavevmode\nobreak\
\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})^{d}),$
$\displaystyle\rho_{2}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \in\leavevmode\nobreak\ \leavevmode\nobreak\
\operatorname{L}^{\infty}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s}(\mathbb{T}^{d}))\cap\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s+1}(\mathbb{T}^{d})),$
$\displaystyle g_{3}$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\in\leavevmode\nobreak\ \leavevmode\nobreak\
\operatorname{L}^{2}_{\operatorname{loc}}(\mathbb{R}^{+};H^{s-1}(\mathbb{T}^{d}\times\mathbb{S}^{d-1})).$
Associated to all the above well-posedness results are estimates of the
corresponding norms of the solutions in terms of all the parameters and of the
controlled norms of the data. ∎
With the above construction of the hierarchy
$\\{u_{n},\rho_{n},g_{n+1}\\}_{n\geq 0}$, we can now turn to the justification
of the formal expansion (5.2). We stick to order $O(\varepsilon^{3})$ for
conciseness, but the proof could be pursued to arbitrary order. The proof is
displayed in Section 5.3. Note that the well-preparedness assumption (5.8)
below for initial data is one order stronger than in Assumption 4.1: indeed,
while our main result in Section 4 focusses on $O(\varepsilon)$ effects, only
deriving second-order fluid models, the present result further describes
$O(\varepsilon^{2})$ effects and therefore requires this strengthened well-
preparedness condition. Although not needed for the purposes of our main
result, the present next-order analysis is included to illustrate how the
$\varepsilon$-expansion can be pursued to arbitrary order without additional
mathematical difficulties, then leading to higher-order fluid models; we refer
to Appendix C for the corresponding derivation of third-order fluid models.
###### Proposition 5.2 (Error estimates for $\varepsilon$-expansion).
Let $h\in C^{\infty}(\mathbb{R}^{+}\times\mathbb{T}^{d})^{d}$, let
$\rho^{\circ}\in
H^{s}(\mathbb{T}^{d})\cap\frac{1}{\omega_{d}}\mathcal{P}(\mathbb{T}^{d})$ for
some $s\gg 1$, and let also $u^{\circ}\in H^{s}(\mathbb{T}^{d})^{d}$ with
${\operatorname{div}}(u^{\circ})=0$ in the Navier–Stokes case. Denote by
$(u_{\varepsilon},f_{\varepsilon})$ the solution of the Doi–Saintillan–Shelley
model (2.5) as given by Proposition 2.1, and assume that the initial condition
$f_{\varepsilon}|_{t=0}=f_{\varepsilon}^{\circ}$ is well-prepared in the
following sense: decomposing
$f_{\varepsilon}^{\circ}=\rho_{\varepsilon}^{\circ}+g_{\varepsilon}^{\circ}$
with $\rho_{\varepsilon}^{\circ}:=\langle f_{\varepsilon}^{\circ}\rangle$, we
have in terms of the functions $u_{0},u_{1},u_{2},g_{1},g_{2},g_{3}$ defined
in Proposition 5.1 with data $(h,u^{\circ},\rho^{\circ})$,
${\rho_{\varepsilon}^{\circ}\,=\,\rho^{\circ}}\qquad\text{and}\qquad\varepsilon^{\frac{1}{2}}\|g_{\varepsilon}^{\circ}-(\varepsilon
g_{1}+\varepsilon^{2}g_{2}+\varepsilon^{3}g_{3})|_{t=0}\|_{\operatorname{L}^{2}_{x,n}}\,\leq\,C_{0}\varepsilon^{4},$
(5.8)
for some constant $C_{0}<\infty$. Further assume that
$\displaystyle\lambda\theta(1+\operatorname{Pe})\|\rho^{\circ}\|_{\operatorname{L}^{\infty}_{x}}\,\ll\,1$
is smaller than some universal constant.
1. (i)
_Stokes case $\operatorname{Re}=0$, $d\leq 3$:_
For all $t\geq 0$, we have
$\displaystyle\hskip
28.45274pt\varepsilon^{\frac{1}{2}}\|\nabla(u_{\varepsilon}-u_{0}-\varepsilon
u_{1}-\varepsilon^{2}u_{2})\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x}}+\|\nabla(u_{\varepsilon}-u_{0}-\varepsilon
u_{1}-\varepsilon^{2}u_{2})\|_{\operatorname{L}^{2}_{t}\operatorname{L}^{2}_{x}}$
$\displaystyle\\!\\!\leq\\!\\!$ $\displaystyle\mathcal{C}(t)\varepsilon^{3},$
$\displaystyle\hskip
28.45274pt\varepsilon^{\frac{1}{2}}\|g_{\varepsilon}-\varepsilon
g_{1}-\varepsilon^{2}g_{2}-\varepsilon^{3}g_{3}\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x,n}}\\!+\|\nabla_{n}(g_{\varepsilon}-\varepsilon
g_{1}-\varepsilon^{2}g_{2}-\varepsilon^{3}g_{3})\|_{\operatorname{L}^{2}_{t}\operatorname{L}^{2}_{x,n}}$
$\displaystyle\\!\\!\leq\\!\\!$ $\displaystyle\mathcal{C}(t)\varepsilon^{4},$
$\displaystyle\hskip
28.45274pt\|\rho_{\varepsilon}-\rho_{0}-\varepsilon\rho_{1}-\varepsilon^{2}\rho_{2}\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x}}+\|\nabla(\rho_{\varepsilon}-\rho_{0}-\varepsilon\rho_{1}-\varepsilon^{2}\rho_{2})\|_{\operatorname{L}^{2}_{t}\operatorname{L}^{2}_{x}}$
$\displaystyle\\!\\!\leq\\!\\!$ $\displaystyle\mathcal{C}(t)\varepsilon^{3},$
provided that $\varepsilon\mathcal{C}(t)\ll 1$ is small enough, where the
multiplicative constant $\mathcal{C}(t)$ depends on $\operatorname{Pe}$ and on
an upper bound on
$\displaystyle\qquad t,\leavevmode\nobreak\ {C_{0},\leavevmode\nobreak\
\lambda,\leavevmode\nobreak\ U_{0},}\quad\|(\nabla u_{0},\nabla u_{1},\nabla
u_{2})\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{\infty}_{x}},\quad\|(\rho_{0},\rho_{1},\rho_{2})\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{\infty}_{x}},$
$\displaystyle\qquad\|(g_{1},g_{2},g_{3})\|_{\operatorname{L}^{\infty}_{t}W^{1,\infty}_{x,n}},\quad\|(\partial_{t}-\Delta_{x})g_{3}\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x,n}}.$
2. (ii)
_Navier–Stokes case $\operatorname{Re}=0$, $d=2$:_
For all $t\geq 0$, we have
$\displaystyle\hskip 28.45274pt\|u_{\varepsilon}-u_{0}-\varepsilon
u_{1}-\varepsilon^{2}u_{2}\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x}}+\|\nabla(u_{\varepsilon}-u_{0}-\varepsilon
u_{1}-\varepsilon^{2}u_{2})\|_{\operatorname{L}^{2}_{t}\operatorname{L}^{2}_{x}}$
$\displaystyle\\!\\!\leq\\!\\!$ $\displaystyle\mathcal{C}(t)\varepsilon^{3},$
$\displaystyle\hskip
28.45274pt\varepsilon^{\frac{1}{2}}\|g_{\varepsilon}-\varepsilon
g_{1}-\varepsilon^{2}g_{2}-\varepsilon^{3}g_{3}\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x,n}}\\!+\|\nabla_{n}(g_{\varepsilon}-\varepsilon
g_{1}-\varepsilon^{2}g_{2}-\varepsilon^{3}g_{3})\|_{\operatorname{L}^{2}_{t}\operatorname{L}^{2}_{x,n}}$
$\displaystyle\\!\\!\leq\\!\\!$ $\displaystyle\mathcal{C}(t)\varepsilon^{4},$
$\displaystyle\hskip
28.45274pt\|\rho_{\varepsilon}-\rho_{0}-\varepsilon\rho_{1}-\varepsilon^{2}\rho_{2}\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x}}+\|\nabla(\rho_{\varepsilon}-\rho_{0}-\varepsilon\rho_{1}-\varepsilon^{2}\rho_{2})\|_{\operatorname{L}^{2}_{t}\operatorname{L}^{2}_{x}}$
$\displaystyle\\!\\!\leq\\!\\!$ $\displaystyle\mathcal{C}(t)\varepsilon^{3},$
provided that $\varepsilon\mathcal{C}(t)\ll 1$ is small enough, where
$\mathcal{C}(t)$ now depends $\operatorname{Pe}$ and on an upper bound on
$\displaystyle\qquad t,\leavevmode\nobreak\ {C_{0},\leavevmode\nobreak\
\lambda,\leavevmode\nobreak\
U_{0},}\quad\|(u_{0},u_{1},u_{2})\|_{\operatorname{L}^{\infty}_{t}W^{1,\infty}_{x}},\quad\|(\rho_{0},\rho_{1},\rho_{2})\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{\infty}_{x}},$
$\displaystyle\qquad\|(g_{1},g_{2},g_{3})\|_{\operatorname{L}^{\infty}_{t}W^{1,\infty}_{x,n}},\quad\|(\partial_{t}-\Delta_{x})g_{3}\|_{\operatorname{L}^{\infty}_{t}\operatorname{L}^{2}_{x,n}}.\qed$
Finally, it remains to identify the equation satisfied by the truncated
$\varepsilon$-expansion
$(\bar{u}_{\varepsilon},\bar{\rho}_{\varepsilon}):=(u_{0}+\varepsilon
u_{1},\rho_{0}+\varepsilon\rho_{1})$: we show that this expansion precisely
coincides with the hierarchical solution of the second-order fluid model
(3.34) with some explicit choice of parameters. The proof is displayed in
Section 5.4.
###### Proposition 5.3 (From hierarchy to second-order fluids).
Given the solutions $(u_{0},\rho_{0},g_{1})$ and $(u_{1},\rho_{1},g_{2})$ of
the hierarchy (5.3)–(5.4), as constructed in Proposition 5.1 in the Stokes
case, the superposition
$(\bar{u}_{\varepsilon},\bar{\rho}_{\varepsilon}):=(u_{0}+\varepsilon
u_{1},\rho_{0}+\varepsilon\rho_{1})$ coincides with the unique hierarchical
solution of the second-order fluid model (3.34) in the sense of Proposition
3.2, with coefficients $\eta_{0},\eta_{1},\mu_{0},\gamma_{1},\gamma_{2}$
explicitly given by (4.1). ∎
The combination of Propositions 5.2 and 5.3 completes the proof of Theorem
4.2. Note however that we only appeal to Assumption 4.1 in the statement of
Theorem 4.2, which is one order weaker than the well-preparedness assumption
(5.8) required in Proposition 5.2. Indeed, as explained, we focus in Theorem
4.2 on $O(\varepsilon)$ effects, only deriving the second-order fluid model,
while in Proposition 5.2 we took care to further describe $O(\varepsilon^{2})$
effects. For the purposes of Theorem 4.2, the well-preparedness assumption
(5.8) can therefore simply be replaced by Assumption 4.1.
### 5.1. Computational tools for spherical calculus
In this section, we briefly recall several computational tools that will be
used throughout this work to compute derivatives and integrals on the sphere.
First, we recall that the Laplace–Beltrami operator $\Delta_{n}$ on the sphere
$\mathbb{S}^{d-1}$ ($d\geq 2$) can be computed as follows: given a smooth
function $g:\mathbb{S}^{d-1}\to\mathbb{R}$, we can extend it to
$\mathbb{R}^{d}\setminus\\{0\\}$ by setting
$G(x):=g\big{(}\tfrac{x}{|x|}\big{)}$, and we then have
$\Delta_{n}g\,=\,(\Delta_{x}G)|_{\mathbb{S}^{d-1}}.$ (5.9)
In particular, we can compute in this way
$\displaystyle\Delta_{n}(n_{i})$ $\displaystyle=$ $\displaystyle(1-d)n_{i},$
(5.10) $\displaystyle\Delta_{n}(n_{i}n_{j})$ $\displaystyle=$ $\displaystyle
2\delta_{ij}-2dn_{i}n_{j},$ $\displaystyle\Delta_{n}(n_{i}n_{j}n_{k})$
$\displaystyle=$ $\displaystyle
2(\delta_{ij}n_{k}+\delta_{ik}n_{j}+\delta_{jk}n_{i})-3(d+1)n_{i}n_{j}n_{k},$
$\displaystyle\Delta_{n}(n_{i}n_{j}n_{k}n_{l})$ $\displaystyle=$
$\displaystyle
2(\delta_{ij}n_{k}n_{l}+\delta_{kj}n_{i}n_{l}+\delta_{ki}n_{j}n_{l}+\delta_{il}n_{j}n_{k}+\delta_{lk}n_{i}n_{j}+\delta_{jl}n_{i}n_{k})$
$\displaystyle-4(d+2)n_{i}n_{j}n_{k}n_{l},$
and so on for higher-order polynomials. These formulas can be used to
explicitly invert $\Delta_{n}$ on mean-zero polynomial expressions: for any
trace-free symmetric matrix $A\in\mathbb{R}^{d\times d}$, we find for
instance,
$\displaystyle\Delta_{n}^{-1}(n)$ $\displaystyle=$
$\displaystyle-\tfrac{1}{d-1}n,$ (5.11)
$\displaystyle\Delta_{n}^{-1}\big{(}n\otimes n:A\big{)}$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2d}\,n\otimes n:A,$ (5.12)
$\displaystyle\Delta_{n}^{-1}\Big{(}n(n\otimes n:A)\Big{)}$ $\displaystyle=$
$\displaystyle-\tfrac{1}{3(d+1)}\Big{(}n(n\otimes
n:A)+\tfrac{4}{d-1}An\Big{)},$ (5.13)
$\displaystyle\Delta_{n}^{-1}\left((n\otimes
n:A)^{2}-\tfrac{2}{d(d+2)}\operatorname{tr}(A^{2})\right)$ $\displaystyle=$
$\displaystyle-\tfrac{1}{4(d+2)}\Big{(}(n\otimes n:A)^{2}+\tfrac{4}{d}n\otimes
n:A^{2}\hskip 28.45274pt$ (5.14) $\displaystyle\hskip
71.13188pt-\tfrac{2}{d}(\tfrac{1}{d+2}+\tfrac{2}{d})\,\operatorname{tr}(A^{2})\Big{)}.$
Henceforth, the pseudo-inverse $\Delta_{n}^{-1}$ is chosen to be defined as an
operator from mean-zero fields to mean-zero fields.
We also recall that the divergence of functions on $\mathbb{S}^{d-1}$ can be
computed similarly as the Laplace–Beltrami operator (5.9) by an extension
procedure: for any trace-free matrix $A$ and any smooth function
$g:\mathbb{S}^{d-1}\to\mathbb{R}$, we find for instance
$\displaystyle{\operatorname{div}}_{n}(\pi_{n}^{\bot}Ang)\,=\,\nabla_{n}g\cdot
An-d\,(n\otimes n):Ag.$ (5.15)
Finally, we further note that the above differential formulas (5.10) imply by
direct integration the following elementary integral identities for polynomial
expressions on the sphere,
$\displaystyle\int_{\mathbb{S}^{d-1}}n_{i}n_{j}\,\mathrm{d}n$ $\displaystyle=$
$\displaystyle\tfrac{\omega_{d}}{d}\delta_{ij},$ (5.16)
$\displaystyle\int_{\mathbb{S}^{d-1}}n_{i}n_{j}n_{k}n_{l}\,\mathrm{d}n$
$\displaystyle=$
$\displaystyle\tfrac{\omega_{d}}{d(d+2)}\big{(}\delta_{ij}\delta_{kl}+\delta_{kj}\delta_{il}+\delta_{ki}\delta_{jl}\big{)},$
$\displaystyle\int_{\mathbb{S}^{d-1}}n_{i}n_{j}n_{k}n_{l}n_{m}n_{p}\,\mathrm{d}n$
$\displaystyle=$
$\displaystyle\tfrac{\omega_{d}}{d(d+2)(d+4)}\big{(}\delta_{ij}\delta_{kl}\delta_{mp}+\ldots\big{)},$
and so on, where we recall the notation $\omega_{d}=|\mathbb{S}^{d-1}|$. These
identities imply in particular, for any trace-free symmetric matrices
$A,B\in\mathbb{R}^{d\times d}$,
$\displaystyle\int_{\mathbb{S}^{d-1}}\big{(}n\otimes
n-\tfrac{1}{d}{\operatorname{Id}}\big{)}\,\mathrm{d}n$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\int_{\mathbb{S}^{d-1}}\big{(}n\otimes
n-\tfrac{1}{d}\operatorname{Id}\big{)}\,(n\otimes n:A)\,\mathrm{d}n$ |
comment
# A Complete Criterion for Value of Information in Soluble Influence Diagrams
Chris van Merwijk*,1 Ryan Carey*,1 Tom Everitt2
###### Abstract
Influence diagrams have recently been used to analyse the safety and fairness
properties of AI systems. A key building block for this analysis is a
graphical criterion for value of information (VoI). This paper establishes the
first complete graphical criterion for VoI in influence diagrams with multiple
decisions. Along the way, we establish two important techniques for proving
properties of multi-decision influence diagrams: ID homomorphisms are
structure-preserving transformations of influence diagrams, while a Tree of
Systems is a collection of paths that captures how information and control can
flow in an influence diagram.
## 1 Introduction
One approach to analysing the safety and fairness of AI systems is to
represent them using variants of Bayesian networks (Everitt et al. 2019;
Kusner et al. 2017). Influence diagrams (IDs) can be viewed an extension of
Bayesian networks for representing agents (Howard et al. 2005; Everitt et al.
2021a). This graphical perspective offers a concise view of key relationships,
that abstracts away from much of the internal complexity of modern-day AI
systems.
Once a decision problem is represented graphically, key aspects can be
summarised. One well-studied concept is the _value of information_ (VoI)
(Howard 1966), which describes how much more utility an agent is able to
obtain if it can observe a variable in its environment, compared with if it
cannot. Other summary concepts includes “materiality”, “value of control”,
“response incentives”.
These concepts have been used to analyse the redirectability (Everitt et al.
2021b; Holtman 2020) of AI systems, fairness (Everitt et al. 2021a; Ashurst et
al. 2022), ambitiousness (Cohen, Vellambi, and Hutter 2020), and the safety of
reward learning systems (Armstrong et al. 2020; Everitt et al. 2019; Langlois
and Everitt 2021; Evans and Kasirzadeh 2021; Farquhar, Carey, and Everitt
2022). Typically, this analysis involves applying _graphical criteria_ , that
indicate which properties can or cannot occur in a given diagram, based on the
graph structure alone. Graphical criteria are useful because they enable
qualitative judgements even when the precise functional relationships between
variables are unknown or unspecified.
For the single-decision case, complete criteria have been established for all
four of the aforementioned concepts (Everitt et al. 2021a). However, many AI
applications such as reinforcement learning involve an agent making multiple
decisions. For the multi-decision case, multiple criteria for VoI have been
proposed (Nielsen and Jensen 1999; Shachter 1998; Nilsson and Lauritzen 2000),
but none proven complete.
$X$$D$$V$$X^{\prime}$$D^{\prime}$$Q^{\prime}$$U$chance nodedecision
nodeutility node Figure 1: Does $X$ has positive value of information for
$D$?
This means that for some graphs, it is not known whether a node can have
positive VoI. For example, in Fig. 1, it is not known whether it can be
valuable for $D$ to observe $X$. Specifically, the edge $X\to D$ does not meet
the criterion of _nonrequisiteness_ used by Nilsson and Lauritzen (2000), so
we cannot rule out that it contains valuable information. However, the
procedure that is used to prove completeness in the single-decision setting
(Everitt et al. 2021a) does not establish positive VoI.
We prove that the graphical criterion of Nilsson and Lauritzen (2000) is
complete, in that any environmental variable not guaranteed to have zero VoI
by their criterion must have positive VoI in some compatible ID. In the course
of the proof, we develop several tools for reasoning about soluble IDs. In
summary, our main contributions are:
* •
ID homomorphisms. These allow us to transform an ID into another with similar
properties, that may be more easily analysed (Section 4).
* •
Trees of systems. A system is a set of paths that make information valuable to
a decision. A tree of systems describes how those paths traverse other
decisions (Section 5.3).
* •
A complete VoI criterion. We prove the criterion in Section 5. In Section 6 we
explain why this criterion may be useful, how it may be used in an AI safety
application, and share an open source implementation.
## 2 Setup
color=blue!30]Ryan: Or “ all of the results transfer to a regular influence
diagram setting. footnote: the only difference is that if edges do not match
the direction of causation, then a node may be deemed valuable to control,
when controlling it is not in-fact useful (everitt2021, Appendix A) ”
Limited memory influence diagrams (also called LIMIDs) are graphical models
containing decision and utility nodes, used to model decision-making problems
(Howard 1966; Nilsson and Lauritzen 2000).
###### Definition 1 (Limited memory influence diagram graph; Nilsson and
Lauritzen 2000).
A _(limited memory) ID graph_ is a directed acyclic graph
${\mathcal{G}}\\!=\\!({\bm{V}},E)$ where the vertex set ${\bm{V}}$ is
partitioned into _chance-_ (${\bm{X}}$), _decision-_ (${\bm{D}}$), and
_utility nodes_ (${\bm{U}}$). Utility nodes lack children.
Since all of the influence diagram graphs in this paper have limited memory,
we will consistently refer to them simply as _influence diagram_ (ID) graphs.
We denote the parents, descendants,color=green!30]Chris: And ancestors? and
family of a node $V\in{\bm{V}}$ as $\mathrm{\mathbf{Pa}}(V),\textbf{Desc}(V)$,
and $\mathrm{\mathbf{{Fa}}}(V)=\mathrm{\mathbf{Pa}}(V)\cup\\{V\\}$. For
$Y\in{\bm{V}}$, color=green!30]Chris: Remove ”For $Y\in{\bm{V}}$,”? we denote
an edge by $V\to Y$, and a directed path by $V\dashrightarrow Y$.
To specify the precise statistical relationships, rather than just their
structure, we will use a model that attaches probability distributions to the
variables in an ID graph.
###### Definition 2.
An _influence diagram_ (ID) is a tuple
${\mathcal{M}}=({\mathcal{G}},\mathrm{dom},P)$ where ${\mathcal{G}}$ is an ID
graph, $\mathrm{dom}(X)$ is a finite domain for each node $X$ in
${\mathcal{G}}$ that is real-valued for utility nodes, and
$P(X|\mathrm{\mathbf{Pa}}(X))$ is a conditional probability distribution (CPD)
for each chance and utility node $X$ in ${\mathcal{G}}$. We will say that
${\mathcal{M}}$ is _compatible with_ ${\mathcal{G}}$, or simply that
${\mathcal{M}}$ is an ID _on_ ${\mathcal{G}}$.
The decision-making task is to maximize the sum of expected utilities by
selecting a CPD $\pi^{D}(D|\mathrm{\mathbf{Pa}}(D))$, called a _decision rule_
, for each decision $D\in{\bm{D}}$. A _policy_
${\pi}=\\{\pi^{D}\\}_{D\in{\bm{D}}}$ consists of one decision rule for each
decision. Once the policy is specified, this induces joint probability
distribution $P^{\mathcal{M}}_{\pi}$ over all the variables. We denote
expectations by ${\mathbb{E}}^{\mathcal{M}}_{\pi}$ and omit the superscript
when clear from context. A policy ${\pi}$ is called _optimal_ if it maximises
${\mathbb{E}}_{{\pi}}[\mathcal{U}]$, where
$\mathcal{U}\coloneqq\sum_{U\in{\bm{U}}}{U}$. Throughout this paper, we use
subscripts for policies, and superscripts for indexing. A lowercase
$v\in\mathrm{dom}(V)$ denotes an outcome of $V$.
Some past work has assumed “no-forgetting”, meaning that every decision $d$ is
allowed to depend on the value $v$ of any past decision $D^{\prime}$ or its
observations $\mathrm{\mathbf{Pa}}(D^{\prime})$, even when that variable
$V\in\mathrm{\mathbf{{Fa}}}(D^{\prime})$ is not a parent of the current
decision ($V\not\in\mathrm{\mathbf{Pa}}(D)$) (Shachter 1986). In contrast, we
follow the more flexible convention of limited memory IDs (Nilsson and
Lauritzen 2000), by explicitly indicating whether a decision $d$ can depend on
the value of an observation or decision $v$ by the presence (or absence) of an
edge $V\to D$, just as we would do with any variable that is not associated
with a past decision.
Within the space of limited memory IDs, this paper focuses on _soluble_ IDs
(Nilsson and Lauritzen 2000), also known as IDs with “sufficient recall”
(Milch and Koller 2008). The solubility assumption requires that it is always
possible to choose an optimal decision rule without knowing what decision
rules were followed by past decisions. The formal definition uses
$d$-separation.
###### Definition 3 (d-separation; Verma and Pearl (1988)).
A path $p$ is _blocked_ by a set of nodes ${\bm{Z}}$ if $p$ contains a
collider $X\to W\leftarrow Y$, such that neither $W$ nor any of its
descendants are in ${\bm{Z}}$, or $p$ contains a chain $X\to W\to Y$ or fork
$X\leftarrow W\to Y$ where $W$ is in ${\bm{Z}}$. If $p$ is not blocked, then
it is _active_. For disjoint sets ${\bm{X}}$, ${\bm{Y}}$, ${\bm{Z}}$, the set
${\bm{Z}}$ is said to _d-separate_ ${\bm{X}}$ from ${\bm{Y}}$,
${({\bm{X}}\perp{\bm{Y}}\mid{\bm{Z}})}$ if ${\bm{Z}}$ blocks every path from a
node in ${\bm{X}}$ to a node in ${\bm{Y}}$. Sets that are not d-separated are
called _d-connected_.
###### Definition 4 (Solubility; Nilsson and Lauritzen (2000)).
For an ID graph ${\mathcal{G}}$ let the _mapping extension_
${\mathcal{G}}^{\prime}$ be a modified version of ${\mathcal{G}}$ where a
chance node parent $\Pi^{i}$ is added to each decision $D^{i}$. Then
${\mathcal{G}}$ is _soluble_ if there exists an ordering $D^{1},\dots,D^{n}$
over the decisions, such that in the mapping extension
${\mathcal{G}}^{\prime}$, for all $i$:
${\Pi}^{<i}\perp{\bm{U}}(D^{i})\mid\mathrm{\mathbf{{Fa}}}(D^{i})$
where ${\Pi}^{<i}:=\\{\Pi^{j}\mid j<i\\}$ and
${\bm{U}}(D^{i}):={\bm{U}}\cap\textbf{Desc}(D^{i})$.
We will subsequently only consider ID graphs that are soluble. Solubility is
entailed by the popular more restrictive “no forgetting” assumption, where the
decision-maker remembers previous decisions and observations (Shachter 1986,
2016): in no forgetting, the family $\mathrm{\mathbf{{Fa}}}(D^{i})$ includes
$\mathrm{\mathbf{{Fa}}}(D^{j})$ for $j<i$, so every policy node $\Pi^{j}$ is
$d$-separated from
${\bm{V}}\setminus\mathrm{\mathbf{{Fa}}}(D^{j})\supseteq{\bm{U}}\cap\textbf{Desc}^{D^{j}}$.
However, solubility is more general, for example Fig. 1 is soluble, even
though past decisions are forgotten.
## 3 Value of Information
The VoI of a variable color=blue!30]Ryan: Add “is a widely studied property
that [cite cite]”? indicates how much the attainable expected utility
increases when a variable is observed compared to when it is not:
###### Definition 5 (Value of Information; Howard (1966)).
For an ID ${\mathcal{M}}$ and $X\\!\not\in\textbf{Desc}_{D}$,
color=blue!30]Ryan: I added a bit about it being a nondescendant. let
${\mathcal{M}}_{X\\!\not\\!\to\\!D}$ and ${\mathcal{M}}_{X\\!\to\\!D}$ be
${\mathcal{M}}$ modified by respectively removing and adding the edge
$X\\!\to\\!D$. Then, the _value of information_ of $X$ for $D$ is:
$\max_{\pi}{\mathbb{E}}^{{\mathcal{M}}_{X\to
D}}_{\pi}[\mathcal{U}]-\max_{\pi}{\mathbb{E}}^{{\mathcal{M}}_{X\not\to
D}}_{\pi}[\mathcal{U}].$
This is closely related to the concept of _materiality_ ; an observation
$X\in\mathrm{\mathbf{Pa}}(D)$ is called material if its VoI is positive.
The graphical criterion for VoI that we will use iteratively removes
information links that cannot contain useful information, based on a condition
called _nonrequisiteness_. If
$X\perp{\bm{U}}(D^{i})\mid\mathrm{\mathbf{{Fa}}}(D^{i})\setminus\\{X\\}$, then
both $X$ and the information link $X\to D^{i}$ are called _nonrequisite_ ,
otherwise, they are _requisite_. Intuitively, nonrequisite links contain no
information about influencable utility nodes, so the attainable expected
utility is not decreased by their removal. Removing one nonrequisite
observation link can make a previously requisite information link
nonrequisite, so the criterion involves iterative removal of nonrequisite
links. The criterion was first proposed by Nilsson and Lauritzen (2000), who
also proved that it is sound. Formally, it is captured by what we calll a
$d$-reduction:
###### Definition 6 ($d$-reduction).
The ID graph ${\mathcal{G}}^{\prime}$ is a _$d$ -reduction_ of ${\mathcal{G}}$
if ${\mathcal{G}}^{\prime}$ can be obtained from ${\mathcal{G}}$ via a
sequence
${\mathcal{G}}={\mathcal{G}}^{1},...,{\mathcal{G}}^{k}={\mathcal{G}}^{\prime}$
where each ${\mathcal{G}}^{i},i>1$ differs from its predecessor
${\mathcal{G}}^{i-1}$ by the removal of one nonrequisite information link. A
$d$-reduction is called _minimal_ if it lacks any nonrequisite information
links.
For any ID graph ${\mathcal{G}}$, there is only one minimal $d$-reduction
(Nilsson and Lauritzen 2000), i.e. the minimal $d$ reduction is independent of
the order in which edges are removed. We can therefore denote _the minimal
$d$-reduction_ of ${\mathcal{G}}$ as ${\mathcal{G}}^{*}$. Thus, Nilsson and
Lauritzen (2000, Theorem 3) states that _if_ an ID graph ${\mathcal{G}}$
contains $X\to D$ but ${\mathcal{G}}^{*}$ does not, then $X$ has zero VoI in
every ID compatible with ${\mathcal{G}}$. Our completeness result replaces
this with an _if and only if_ statement.
###### Theorem 7 (VoI Criterion).
Let ${\mathcal{G}}$ be a soluble ID graph containing an edge $X\to D$ from
chance node $X\in{\bm{X}}$ to decision $D\in{\bm{D}}$. There exists an ID
${\mathcal{M}}$ compatible with ${\mathcal{G}}$ such that $X$ has strictly
positive VoI for $D$ if and only if the minimal $d$-reduction contains $X\to
D$.
The VoI criterion is posed in terms of a graph ${\mathcal{G}}$ that contains
$X\to D$. To analyse a graph that does not, one can simply add the edge $X\to
D$ then apply the same criterion as long as the new ID graph is soluble
(Shachter 2016). color=blue!30]Ryan: Is this the correct shachter cite?
The proof will be given in Section 5, with details in Appendices C and D. We
note that this excludes the case of remembering a past decision
$X\in{\bm{D}}$, because Nilsson’s criterion is incomplete for this case. For
example, the simple ID graph with the edges $D\to D^{\prime}\to U$ and $D\to
U$, $D$ satisfies the graphical criterion of being requisite for $D^{\prime}$,
but $D^{\prime}$ has zero VoI because it is possible for the decision $D$ to
be deterministically assigned some optimal value. This means that there is no
need for $D^{\prime}$ to observe $D$.
## 4 ID Homomorphisms
To make the analysis easier, we will often want to transform an original ID
graph into a more structured one. Before describing the structure we will be
aiming for, we consider the general question of when a modified ID graph
retains important properties of the original. To this end, we will define the
concept of an _ID homomorphism_ , which we then use to define a class of
property-preserving ID transformations. (Proofs are supplied in Appendix B.)
color=blue!30]Ryan: Should be able to condense the bullets from 8 lines to 5
###### Definition 8 (ID homomorphism).
For ID graphs ${\mathcal{G}}\\!\\!=\\!({\bm{V}}\\!,E)$ and
${\mathcal{G}}^{\prime}\\!\\!=\\!({\bm{V}}^{\prime}\\!,E^{\prime})$, a map
$h\colon\\!{\bm{V}}^{\prime}\\!\\!\to\\!{\bm{V}}\\!$ is an _ID homomorphism_
from ${\mathcal{G}}^{\prime}$ to ${\mathcal{G}}$ iff:
1. (a)
(Preserves node types) $h$ maps each chance-, decision-, or utility-node to a
node of the same type;
2. (b)
(Preserves links) For every $A\to B$ in ${\mathcal{G}}^{\prime}$ either
$h(A)\to h(B)$ is in ${\mathcal{G}}$, or $h(A)=h(B)$;
3. (c)
(Covers all information links) If $h(N)\to h(D)$ is in ${\mathcal{G}}$ for
$D\in{\bm{D}}$, then $N\to D$ is in ${\mathcal{G}}^{\prime}$; and
4. (d)
(Combines only linked decisions) If $h(D_{1})\\!=\\!h(D_{2})$ for decisions
$D^{1}\neq D^{2}$ in ${\mathcal{G}}^{\prime}$ then ${\mathcal{G}}^{\prime}$
contains $D^{1}\\!\\!\to\\!D^{2}$ or $D^{2}\\!\to\\!D^{1}$.
$Y$$D$$U$${\mathcal{G}}$original$D$$U$${\mathcal{G}}^{\prime}$remove
$Y$$D$$D^{\prime}$$U$${\mathcal{G}}^{\prime\prime}$duplicate
$D$$D$$D^{\prime}$$U$${\mathcal{G}}^{\prime\prime\prime}$remove an edge Figure
2: A sequence of homorphic transformations showing how ${\mathcal{G}}$ can be
homorphically transformed into ${\mathcal{G}}^{\prime\prime\prime}$ by
composition of Lemmas 13 and 14. In the first step from ${\mathcal{G}}$ to
${\mathcal{G}}^{\prime}$, $Y$ is removed; in the step from
${\mathcal{G}}^{\prime}$ to ${\mathcal{G}}^{\prime\prime}$ a decision is
duplicated; and in the final step from ${\mathcal{G}}^{\prime\prime}$ to
${\mathcal{G}}^{\prime\prime\prime}$, a link is removed. Since the mapping at
each step (blue, green, and orange respectively) meets the definition of an ID
homomorphism, ${\mathcal{G}}^{\prime\prime\prime}$ must be an ID homorphism of
${\mathcal{G}}$ (Lemma 15).
An ID homomorphism is analogous to the notion of graph homomorphism from graph
theory, which essentially requires that edges are preserved along the map. An
ID homomorphism additionally requires that decisions in the two graphs have
equivalent parents (c), and that split decisions are connected (d). This
requirement maintains a direct correspondence between policies on the two
graphs, so that, as we will see, ID homomorphisms preserve VoI. Examples of ID
homorphisms are given in Fig. 2.
color=blue!30]Ryan: Cite graph homomorphisms? color=green!30]Chris: we could
add a citation to graph theory, Diestel, 2017. Though it’s also just a “well
known concept” and has a wikipedia page. color=blue!30]Ryan: This is where we
should have remarks about the intuition of this definition. But I don’t
understand what is being said about adding edges from each node to itself.
The following three lemmas establish properties that are preserved under ID
homorphisms.
###### Lemma 9 (Preserves Solubility).
Let ${\mathcal{G}}=({\bm{V}},E)$ and
${\mathcal{G}}^{\prime}=({\bm{V}}^{\prime},E^{\prime})$ be ID graphs. If
${\mathcal{G}}$ is soluble, and there exists a homomorphism
$h\colon{\bm{V}}^{\prime}\to{\bm{V}}$, then ${\mathcal{G}}^{\prime}$ is also
soluble.
color=blue!30]Ryan: Probably we should uniformise to either
${\mathcal{G}}^{\prime}/{\mathcal{M}}^{\prime}$ or
$\bar{\mathcal{G}}\bar{\mathcal{M}}$ throughout this section
Given a homomorphism $h$ from ${\mathcal{G}}^{\prime}$ to ${\mathcal{G}}$, we
can define a notion of equivalence between IDs (and policies) on each graph.
Roughly, two IDs are equivalent if the domain of every node is a cartesian
product of the domains of the nodes in its pre-image (or the sum, in the case
of a utility node). Formally:
###### Definition 10 (Equivalence).
${\mathcal{M}}_{\pi}$ on ${\mathcal{G}}$ and
${\mathcal{M}}^{\prime}_{\pi^{\prime}}$ on ${\mathcal{G}}_{\pi^{\prime}}$ are
_equivalent_ if each non-utility node $N$ in ${\mathcal{G}}$ has
$\mathrm{dom}(N):=\bigtimes_{N^{i}\in h^{-1}(N)}\mathrm{dom}(N^{i})$, and
$P^{\mathcal{M}}_{\pi}(N\\!=\\!(n^{1},...,n^{k}))=P^{{\mathcal{M}}^{\prime}}_{\pi^{\prime}}(N^{1}\\!=\\!n^{1},...,N^{k}\\!=\\!n^{k})$,
and each utility node has
$P^{\mathcal{M}}_{\pi}(U\\!=\\!u)=P^{{\mathcal{M}}^{\prime}}_{\pi^{\prime}}(\sum_{U^{i}\in
h^{-1}(U)}U^{i}\\!=\\!u)$.
###### Lemma 11 (Equivalence).
If there is an ID homomorphism $h$ from ${\mathcal{G}}^{\prime}$ to
${\mathcal{G}}$, then for any policy $\pi^{\prime}$ in any ID
${\mathcal{M}}^{\prime}$ on ${\mathcal{G}}^{\prime}$ there is a policy $\pi$
in a ID ${\mathcal{M}}$ on ${\mathcal{G}}$ such that ${\mathcal{M}}_{\pi}$ and
${\mathcal{M}}^{\prime}_{\pi^{\prime}}$ are equivalent. color=red!30]Tom: what
does it mean for two probability distributions to be equivalent?
color=blue!30]Ryan: Changed from $\tilde{M}$ to $M^{\prime}$ here. May need to
make corresponding change to proof in appendix
In this case, we will call ${\mathcal{M}}$ and $\pi$ the _ID and policy
transported along the homomorphism $h$_. In the appendix, we show that this
correspondence between policies on ${\mathcal{M}}^{\prime}$ and
${\mathcal{M}}$ is a bijection. color=blue!30]Ryan: The one-sentence
explanation above isn’t very explanatory. color=green!30]Chris: I just removed
it Intuitively, if there is an ID homomorphism
${\mathcal{G}}^{\prime}\to{\mathcal{G}}$, this means we have a particular way
to fit an ID on ${\mathcal{G}}^{\prime}$ into ${\mathcal{G}}$, while
preserving the information that the decisions can access. The basis of this
proof is that properties (c,d) of ID homomorphisms (Definition 8) require
decisions to have precisely the same information in $\cal M$ as in $\cal
M^{\prime}$.
color=blue!30]Ryan: “interpreted as” feels a bit too informal to me.
For our proof of Theorem 7, we will require that VoI is preserved under
homomorphism.
###### Lemma 12 (Preserves VoI).
Let $h\colon\\!{\mathcal{G}}^{\prime}\\!\\!\to\\!{\mathcal{G}}$ be an ID
homomorphism. If $X^{\prime}$ has positive VoI for $D^{\prime}$ in an ID
${\mathcal{M}}^{\prime}$ on ${\mathcal{G}}^{\prime}$, then $X\\!=\\!h(X)$ has
positive VoI for $D\\!=\\!h(D^{\prime})$ in the transported ID
${\mathcal{M}}\\!=\\!h({\mathcal{M}}^{\prime})$.
color=blue!30]Ryan: I thought materiality is defined with respect to just a
model, so I’ve changed this statement. I think the proof should remain
similar, and use transported model + policy?
The proof builds heavily on there being a precise correspondence between
policies on $\cal M$ and on $\cal M^{\prime}$. Since these two IDs are
equivalent (Lemma 11), if obtaining certain information in $\cal M^{\prime}$
has value, so does obtaining that information in $\cal M$. The formal details
are left to Appendix B.
We next present two transformation rulescolor=green!30]Chris: rudimentary
calculus is kind of a weird phrasing with which to modify any ID graph, which
are illustrated in Fig. 2. The first transformation obtains a new graph
${\mathcal{G}}^{\prime}$ by deleting or duplicating nodes, while preserving
all links. Under this transformation, the function that maps a node in
${\mathcal{G}}^{\prime}$ to its ‘originating node’ in ${\mathcal{G}}$ is an ID
homomorphism:
###### Lemma 13 (Deletion & Link-Preserving Copying).
Let ${\mathcal{G}}\\!\\!=\\!\\!({\bm{V}},E)$ be an ID graph and
${\mathcal{G}}^{\prime}\\!=\\!(\bigcup_{N\in{\bm{V}}}\mathrm{Copies}(N),E^{\prime})$
an ID graph where $\mathrm{Copies}$ maps nodes in ${\mathcal{G}}$ to disjoint
sets in ${\mathcal{G}}^{\prime}$, and where $E^{\prime}$ is a minimal set of
edges such that for any edge $A\to B$ in $E$ and $A^{i}\in\mathrm{Copies}(A)$
and $B^{i}\in\mathrm{Copies}(B)$ there is an edge $A^{i}\to B^{i}$, and if
$A^{i},A^{j}\in\mathrm{Copies}(A)$ are non-utility nodes then either $A^{i}\to
A^{j}$ or $A^{i}\leftarrow A^{j}$. Then the function $h$ that maps each
$V\in\mathrm{Copies}(N)$ to $N$ is an ID homomorphism.
color=blue!30]Ryan: I’ve simplified/shortened this a bit further. Feel free to
revert any changes is preferred.
Edges that are not information links can also be removed, while having a
homomorphism back to the original:
###### Lemma 14 (Link Pruning).
Let ${\mathcal{G}}=({\bm{V}},E)$ and
${\mathcal{G}}^{\prime}=({\bm{V}},E^{\prime})$ be ID graphs, where
$E^{\prime}\subseteq E$ and where for each decision node $D$ in ${\bm{V}}$,
every incoming edge $N\to D$ in $E$ is in $E^{\prime}$. Then the identity
function $h(N)=N$ on ${\bm{V}}$ is a homomorphism from
${\mathcal{G}}^{\prime}$ to ${\mathcal{G}}$.
Finally, we can chain together a sequence of such graph transformation steps,
and still maintain a homomorphism to the original. The justification for this
is that a composition of ID homomorphisms is again an ID homomorphism:
###### Lemma 15 (Composition).
If $h\colon{\mathcal{G}}^{\prime}\to{\mathcal{G}}$ and
$h^{\prime}\colon{\mathcal{G}}^{\prime\prime}\to{\mathcal{G}}^{\prime}$ are ID
homomorphisms then the composition $h\circ
h^{\prime}\colon{\mathcal{G}}^{\prime\prime}\to{\mathcal{G}}$ is an ID
homomorphism.
## 5 Completeness of the VoI Criterion
We will now prove that the _value of information_ (VoI) criterion of Nilsson
and Lauritzen (2000) is complete for chance nodes (details are deferred to
Appendices C and D).
### 5.1 Parameterising one system
To prove that the criterion from Theorem 7 is complete we must show that for
any graph where $X\to D$ is in the minimal d-reduction, $X$ has positive VoI
for $D$. For example, consider the graph in Fig. 3, which is its own
d-reduction, and contains $X\to D$. In this graph, we can choose for $X$ to be
Bernoulli distributed, for $D$ to have the boolean domain $\\{0,1\\}$, and for
$U$ to be equal to $1$ if and only if $X$ and $D$ match. Clearly, the policy
$d=x$ will obtain $\mathbb{E}[U]=1$. In contrast, if $X$ were not observed (no
link $X\to D)$, then no policy could achieve expected utility more than $0.5$;
so the VoI of $X$ in this ID is $0.5$.
$X$$x\sim\text{Bern}(0.5)$$D$$d\in\\{0,1\\}$$U$$u=\delta_{d=x}$ Figure 3: The
observation $X$ has positive VoI for $D$.
A general procedure for parameterising any single-decision ID graph meeting
the Theorem 7 criterion to exhibit positive VoI has been established by
Everitt et al. (2021a) and Lee and Bareinboim (2020). This procedure consists
of two steps: first, establish the existence of some paths, then choose CPDs
for the nodes on those paths. We call the paths found in the first step a
system, which will be a building block for our analysis of IDs with multiple
decisions. A fully-general illustration of a system is shown in Fig. 4.
color=blue!30]Ryan: Maybe move some of these definitions down to wherever
they’re used.
###### Definition 16 (System).
A _system_ $s$ in an ID graph ${\mathcal{G}}$ is a tuple
$(\mathrm{control}^{s},\mathrm{info}^{s},\mathrm{obs}^{s})$ where:
* •
The _control path_ , $\mathrm{control}^{s}$, is a directed path
$D^{s}\dashrightarrow U^{s}$ where $D^{s}\in\bm{D}$ and $U^{s}\in\bm{U}$,
* •
The _info path_ , $\mathrm{info}^{s}$, is a path
$\mathrm{\mathbf{Pa}}(D^{s})\ni X^{s}\;\hbox{- - -}\;U^{s}$, active given
$\mathrm{\mathbf{{Fa}}}(D^{s})\setminus\\{X^{s}\\}$,
* •
$\mathrm{obs}^{s}$ maps each collider $C^{i}$ in $\mathrm{info}^{s}$ to an
_obs path_ , a _minimal-length_ directed path
$C^{i}\\!\\!\dashrightarrow\\!D^{s}$.
color=blue!30]Ryan: Tweaked the formatting of this definition, old version
below
We denote the _information link of_ $s$, $X^{s}\\!\to\\!D^{s}$, by
$\mathrm{infolink}^{s}$ and the union of nodes in _all_ paths of $s$ by
${\bm{V}}^{s}$.
color=blue!30]Ryan: TODO: set nodes to be circular, except for labels, which
are rectangular
$F^{1}$$f^{1}\sim\text{Bern}(0.5)$$C^{1}$$c^{i}\\!\\!=\\!f^{i}\\!\oplus\\!f^{i+1}$$\ldots$$C^{n}$$Q^{s}$$q^{s}\sim\text{Bern}(0.5)$$O^{1}$$o^{i}=c^{i}$$O^{n}$$X^{s}$$x^{s}\\!\\!=\\!\\!f^{1}$$D^{s}$$d^{s}\in\\{0,1\\}$$U^{s}$$u^{s}=\delta_{q^{s}=d^{s}}$pathedge$\mathrm{info}^{s}$$\mathrm{obs}^{s}$$\mathrm{infolink}^{s}$$\mathrm{control}^{s}$
Figure 4: A system, annotated with a parameterization that has positive VoI
in the single-decision case. Dashed arrows can zero or more nodes.
color=blue!30]Ryan: Probably should try to make the text larger in this figure
if spare space
The existence of these paths follow from the graphical criterion of Theorem 7.
In particular, since $X\\!\to\\!D$ is in the minimal d-reduction of
${\mathcal{G}}$, there must exist a path from $X$ to some utility node
$U\in{\bm{U}}\cap\textbf{Desc}^{D^{s}}\\!$, active given
$\mathrm{\mathbf{{Fa}}}(D^{s})\\!\setminus\\!\\{X^{s}\\}$ (the “info path” in
Definition 16).
The second step is to choose CPDs for the nodes ${\bm{V}}^{s}$ in the system
$s$, as also illustrated in Fig. 4. The idea is to require the decision
$D^{s}$ to match the value of $Q^{s}$, by letting the utility $U^{s}$ equal
$1$ if and only if its parents along the control and information paths are
equal. If $X^{s}$ is observed, the decision $D^{s}=X^{s}\oplus O^{1}...\oplus
O^{n}=Q^{s}$ yields $\mathbb{E}[U^{s}]=1$, where $\oplus$ denotes _exclusive
or_ (XOR). Otherwise, the observations $O^{1},...,O^{n}$ are insufficient to
decrypt $Q^{s}$, giving $\mathbb{E}[U^{s}]<1$. So $X^{s}$ has positive VoI.
The intuitive idea is that $U^{s}$ tests whether $D^{s}$ knows $Q^{s}$, based
on the value $d^{s}$ transmitted along $\mathrm{control}^{s}$.
color=green!30]Chris: Above is a bit hard to follow I think.
### 5.2 Parameterising two systems
$X$$x\sim\text{Bern}(0.5)$$D$$d=0$$V$$v=d$$Q^{\prime}$$X^{\prime}$$x^{\prime}=v$$D^{\prime}$$d^{\prime}=x$$U$$U=\delta_{d^{\prime}=x}$
(a) The variable $X$ has zero VoI for $D$.
$X$$x\sim\text{Bern}(0.5)$$D$$d=x$$V$$v=d$$Q^{\prime}$$q^{\prime}\sim\text{Bern}(0.5)^{2}$$X^{\prime}$$x^{\prime}=(v,q^{\prime}[v])$$D^{\prime}$$d^{\prime}=x^{\prime}$$U$$U=\delta_{d^{\prime}[2]=q^{\prime}[d^{\prime}[1]]}$$+\delta_{d^{\prime}[1]=x}$
(b) The variable $X$ has positive VoI for $D$.
Figure 5: In (a), a parameterisation of nodes in a single (red) system fails
to exhibit that $X$ has positive VoI for $D$, whereas in (b), positive VoI is
exhibited by parameterising two (red and blue) systems.
When we have two decisions, however, it becomes insufficient to parameterise
just one system. For example, suppose that we try to apply the same scheme as
in the previous subsection to the graph of Fig. 5(a). Then, we would generate
a random bit at $X$ and stipulate that the utility is $U=1$ if the parents $X$
and $D^{\prime}$ on the red paths are equal. One might hope that this would
give $D$ an incentive to observe $X$, so that $d=x$ is copied through
$D^{\prime}$ to obtain $\mathbb{E}[U]=1$. And that is indeed one way to obtain
optimal expected utility. However, the presence of a second decision
$D^{\prime}$ means that maximal utility of $U=1$ may also be obtained using
the policy $d=0,d^{\prime}=x$, which does not require $X$ to be observed by
$D$.
To achieve positive VoI, it is necessary to parameterise two systems as shown
in Fig. 5(b). We first parameterise the second (blue) system to ensure that
$x^{\prime}$ is transmitted to $U$, and then parameterise the initial (red)
system.
To check that $X$ has positive VoI for $D$, we now solve the combined model.
Due to the solubility assumption, we know that the optimal decision rule at
$D^{\prime}$ does not depend on the decision rule taken at $D$. So let us
consider $D^{\prime}$ first. $D^{\prime}$ chooses a pair $(i,j)$ where $i$ is
interpreted as an index of the bits generated at $Q^{\prime}$, and $j$ is
interpreted as a claim about the $i\textsuperscript{th}$ bit of $Q^{\prime}$.
The first term of the utility $U$ is equal to $1$ if and only if the “claim”
made by $D^{\prime}$ is correct, i.e. if the $i\textsuperscript{th}$ bit
generated by $Q^{\prime}$ really is $j$. $X^{\prime}$ contains (only) the
$v\textsuperscript{th}$ digit of $Q^{\prime}$. Hence $D^{\prime}$ can only
ensure its “claim” is correct if it chooses
$d^{\prime}=x^{\prime}=(v,q^{\prime}[v])$, where $q^{\prime}[v]$ denotes the
$v\textsuperscript{th}$ bit of $q^{\prime}$. Having figured out the optimal
policy for $D^{\prime}$, we next turn our attention to $D$. Intuitively, the
task of $D$ is to match $X$, as in Fig. 3. The parameterization encodes this
task, by letting $D$ determine $V$, which in turn influences which bit of
$Q^{\prime}$ is revealed to $D^{\prime}$. This allows $U$ to check the output
of $D$ via the index outputted by $D^{\prime}$, and thereby check whether $D$
matched $X$. This means the second term of $U$ is 1 if and only if $D=X$ so
$d=x$ the optimal policy for $D$, with expected utility $\mathbb{E}[U]=2$.
In contrast, if $X$ were unobserved by $D$, then it would no-longer be
possible to achieve a perfect score on both terms of $U$, so
$\mathbb{E}[U]<2$. This shows that $X$ has positive VoI for $D$.
### 5.3 A tree of systems
In order to generalise this approach to arbitrary number of decisions, we need
a structure that specifies a system for each decision, and indicates what
downstream decisions that system may depend on. These relationships may be
represented by a tree.
###### Definition 17 (Tree of systems).
A _tree of systems_ on an ID graph ${\mathcal{G}}$ is a tuple
$T=({\mathcal{S}},\mathrm{pred})$ where:
* •
${\mathcal{S}}=(s^{0},...,s^{k})$ is a list of systems (which may include
duplicates).
* •
$\mathrm{pred}$ maps each $s^{i}$ to a pair $(s^{j},p)$, where
$s^{j}\in({\mathcal{S}}\setminus\\{s^{i}\\})$ is a system, $p$ is one of the
paths of $s^{j}$ (info, control, or obs), and $\mathrm{infolink}^{s^{i}}$ is
in the path $p$, except there is a unique “root system” $s^{\mathrm{root}}$
that is mapped to $(s^{\mathrm{root}},``\mathrm{None}")$.
Moreover, a _full tree of systems_ is one where for each information link
$X^{\prime}\to D^{\prime}$ in each path $p$ in each system $s$, there is
precisely one system $s^{\prime}$ whose information link equals $X^{\prime}\to
D^{\prime}$ and with $\mathrm{pred}(s^{\prime})=(s,p)$.
The idea of a tree of systems is that if a decision $D^{s^{\prime}}$ lies on a
path in the system $s$ of some decision $D^{s}$, then $s$ is a predecessor of
$s^{\prime}$. We will use this tree to parameterise the ID graph, and then we
will also use it to supply an ordering over the decisions (from leaf to root)
in which the model can be solved by backward induction. color=blue!30]Ryan:
Added some explanation here.
$X^{s}$$D^{s}$$Y$$X^{\prime}$$D^{\prime}$$U$$Q^{\prime}$ (a) $Y$ and $U$ occur
in both $s$ (red) and in $s^{\prime}$ (blue)
$X^{s}$$D^{s}$$Y$$X^{\prime}$$D^{\prime}$$U$$Y^{\prime}$$U^{\prime}$$Q^{\prime}$
(b) Copying $Y$ and $U$ ensures position-in-tree-uniqueness
$X^{s}$$D^{s}$$Y$$X^{\prime}$$D^{\prime}$$U$$O$$Y^{\prime}$$U^{\prime}$$Q^{\prime}$
(c) Making a copy $O$ of $X^{\prime}$, ensures no-backdoor-infopaths.
$X^{s}$$D^{s}$$Y$$X^{\prime}$$D^{\prime}$$U$$O$$Y^{\prime}$$U^{\prime}$$Q^{\prime}$
(d) Finally, links are removed, ensuring no-redundant-links
Figure 6: color=green!30]Chris: Obtaining a normal form tree via homomorphic
graph transformations: An ID graph (a) is homomorphically transformed via
graphs (b) and (c) into a graph (d) whose tree is in normal form.
In order to generalise the approach taken to parameterising two systems, we
need to reason about the systems independently, in reverse order. If the
systems overlap, however, this makes it harder to reason about them
independently. Thus it is useful to define a notion of systems called _normal
form_ that are well-behaved.
###### Definition 18 (Normal form tree).
A tree $T$ on ${\mathcal{G}}$ is in _normal form_ if all of the following
hold:
1. (a)
(position-in-tree-uniqueness) A node $N$ in $T$ can only be in multiple paths
$p^{1},...,p^{k}$ of systems in the tree, if splitting $N$ into
$\\{N,N^{\prime}\\}$ via Lemma 13 and obtaining $T^{\prime}$ from $T$ by
replacing $N$ with $N^{\prime}$ in one of those paths would make $T^{\prime}$
no longer a tree of systems.
2. (b)
(no-backdoor-infopaths) Every system $s$ in $T$ has an info path that starts
with an outgoing link from $X^{s}$.
3. (c)
(no-redundant-links) If $N\to N^{\prime}$ is an edge to a non-decision
$N^{\prime}$, where one of $N$ and $N^{\prime}$ is in a path in a system of
$T$, not including the nodes of the root information link, then $N\to
N^{\prime}$ is in a path of a system of $T$.
inline, color=blue!30]Ryan: We should define front-door or just say “starting
with a tail”
An arbitrarily chosen tree will not generally be in normal form. For example,
Fig. 6(a) contains two systems (a red root system for $X^{s}\to D^{s}$ and a
blue child system for $X^{\prime}\to D^{\prime}$) that constitute a tree, but
this tree fails all three requirements for being in normal form. However, by a
series of homomorphic transformations, it is possible to obtain a new graph
with a tree of systems that is in normal form (as in Fig. 6(d)).
###### Lemma 19 (Normal Form Existence).
Let ${\mathcal{G}}$ be a soluble ID graph whose minimal $d$-reduction
${\mathcal{G}}^{*}$ contains $X\to D$. Then there is a normal form tree
$T^{\prime}$ on a soluble ID graph ${\mathcal{G}}^{\prime}$, with a
homomorphism $h$ from ${\mathcal{G}}^{\prime}$ to ${\mathcal{G}}$ where the
information link $X^{\prime}\to D^{\prime}$ of the root system of
$T^{\prime}$, has $h(X^{\prime})=X$, $h(D^{\prime})=D$, and every node in
${\mathcal{G}}$ is also in ${\mathcal{G}}^{\prime}$ but the only nodes in
${\mathcal{G}}$ that are in $T^{\prime}$ are $X$ and $D$.
color=blue!30]Ryan: Is it actually meaningful for nodes in ${\mathcal{G}}$ to
be in $T^{\prime}$, or do we need to talk about applying $h^{-1}$ to them
first? color=blue!30]Ryan: Need to state that
${\mathcal{G}}^{3}={\mathcal{G}}^{\prime},T^{3}=T^{\prime},h^{\prime}=h^{0\leftarrow
3}$ in the proof. color=blue!30]Ryan: Maybe remark at this point that it’s
interesting that we can homomorphically modify the tree to obtain one that
behaves differently, describing how this relates to the unidirectionality of
the homomorphism property? / Address the apparent contradiction that
homomorphisms preserved the properties of a tree, but that we can analyse them
differently.
Essentially, the procedure for obtaining a normal form tree proceeds in four
steps:
1. 1.
Construct a tree of systems on $X\to D$: First, pick any system for $X\to D$.
Then, pick any system for every other information link $X^{\prime}\to
D^{\prime}$ in the existing system. Iterate until every link in the tree has a
system.
2. 2.
Make a copy (lemma 12) of each node for each position (basically, each path)
that node has in the tree. This ensures position-in-tree-uniqueness.
3. 3.
For systems whose infopath starts with an incoming link $X\leftarrow Y$, copy
$X$ (lemma 12), to obtain $X\to O\leftarrow Y$. This ensures no-backdoor-
infopaths.
4. 4.
Prune the graph (using lemma 13), by removing any (non-information) links
outside the tree of systems. This ensures no-redundant-links.
For example, in Figs. 6(a), 6(b), 6(c) and 6(d), three transformations are
performed, each of which makes the tree meet one additional requirement,
ultimately yielding a normal form tree (Fig. 6(d)) with a homomorphism to the
original.
### 5.4 Proving positive VoI given a normal form tree
The reason for using normal form trees is that they enable each system to be
parameterized and solved independently. In particular, we know that the
optimal policy for one system involves reproducing information from ancestor
nodes such as $Q^{s}$. As optimal policies can be found with backwards
induction in soluble graphs, our approach involves finding optimal policies in
reverse order. It will therefore suffice to prove that non-descendant systems
cannot provide information about ancestor nodes within the system. For
example, in Fig. 6(a), when solving for $\pi^{D^{\prime}}$, we would like to
know that $D^{s}$ cannot provide information about $Q^{\prime}$.
###### Lemma 20 (Subtree Independence).
Let $s$ be a system in a normal form tree $\mathcal{T}$ on a soluble ID graph
${\mathcal{G}}$. Let
$\mathrm{\mathbf{Pa}}^{-s}=\mathrm{\mathbf{Pa}}(D^{s})\setminus{\bm{V}}^{s}$
be $D^{s}$’s out-of-system parents,
$\mathrm{\mathbf{Pa}}^{s}={\mathrm{\mathbf{Pa}}(D^{s})\cap{\bm{V}}^{s}}$ be
the within-system parents of $D^{s}$, $\bf{{ObsDesc}}^{s}$ be the observation
nodes in descendant systems of $s$, and let
$\bf{Back}^{s}={\bm{V}}^{s}\cup(\textbf{Anc}(D^{s})\setminus\mathrm{\mathbf{{Fa}}}(D^{s}))$.
Then
$\bf{Back}^{s}\perp\mathrm{\mathbf{Pa}}^{-s}\setminus\bf{{ObsDesc}}^{s}\mid\mathrm{\mathbf{Pa}}^{s}\cup\bf{{ObsDesc}}^{s}$.
For example, Fig. 6(d), has a normal form tree, which implies the assurance
that $X^{\prime}$ cannot use information from the red system to tell it about
$Q^{\prime}$; formally, $Q^{\prime}\perp(Y\cup X^{s})\mid X^{\prime}$. Given
that each decision $D^{s}$ in the tree cannot use information from ancestor
systems, we can then prove that $D^{s}$ cannot know enough about $X^{s}$ and
$Q^{s}$ to perform optimally, without observing $X^{s}$. More formally:
###### Lemma 21 (VoI Given Normal Form Tree).
Let ${\mathcal{G}}$ be a soluble ID graph with a normal form tree with root
info link $X\to D$. Then there exists an ID compatible with ${\mathcal{G}}$
for which $X$ has positive VoI for $D$.
The formal proof is given in Section D.3. Informally, in order to show that
the decision of each system is forced to behave as intended despite there now
being a tree of systems full of other decisions, we use Lemma 20 to show that
the utility that a decision obtains in system $s$ only depends on the
information it obtains from within system $s$. This rules out that ancestor
decisions can observe and pass along relevant information via a path outside
the system. Moreover, we know by the solubility assumption that the optimal
decision rule at a later decision cannot depend on the decision rule followed
by earlier decisions. The argument then proceeds by backward induction. The
final decision $D^{s^{n}}$ must copy the value of $X^{s_{n}}$. Given that it
does so, the penultimate decision $D^{s^{n-1}}$ must do the same. And so on,
until we find that $D$ must copy $X$, and cannot do so in any way other than
by observing it, meaning that $X$ has positive VoI for $D$.
color=green!30]Chris: I think these explanatory paragraphs can be better.
color=blue!30]Ryan: Is this accurate & better?
Finally, we can prove our main result, that there exists an ID on
${\mathcal{G}}$ where $X$ has positive VoI.
###### Proof of Theorem 7 (completeness direction).
We know that the d-reduction ${\mathcal{G}}^{*}$ of ${\mathcal{G}}$ contains
$X\to D$. By Lemma 19, there exists an ID graph ${\mathcal{G}}^{\prime}$ with
normal form tree rooted at a link $X^{\prime}\to D^{\prime}$, with an ID
homomorphism from ${\mathcal{G}}^{\prime}$ to ${\mathcal{G}}$ that has
$h(X^{\prime})=X$ and $h(D^{\prime})=D$. By Lemma 21, since
${\mathcal{G}}^{\prime}$ has a normal form tree rooted at $X^{\prime}\to
D^{\prime}$, there exists an ID on ${\mathcal{G}}^{\prime}$ in which
$X^{\prime}$ has positive VoI for $D^{\prime}$. By Lemma 12, the presence of
the ID homomorphism $h$ from ${\mathcal{G}}^{\prime}$ to ${\mathcal{G}}$ means
that there also exists an ID ${\mathcal{M}}$ on ${\mathcal{G}}$ such that
$h(X^{\prime})=X$ has positive VoI for $h(D^{\prime})=D$, showing the result.
∎
## 6 Applications & Implementation
color=blue!30]Ryan: Change the figure to all-superscripts? color=blue!30]Ryan:
Cite frameworks paper. color=blue!30]Ryan: Illustrate nodes with +VoC?
Graphical criteria can help with modeling agents’ incentives in a wide range
of settings including (factored) Partially Observed Markov Decision Processes
(POMDPs) and Modified-action Markov Decision Processes (Langlois and Everitt
2021). For concreteness, we show how our contributions can aid in analysing a
supervision POMDP (Milli et al. 2017). In a supervision POMDP, an AI interacts
with its environment, given suggested actions from a human player. We will
assume that the human’s policy has already been selected, in order to focus on
the incentives of the AI system.
$R_{1}$$S_{1}$$A_{1}$$R_{2}$$S_{2}$$A_{2}$$R_{3}$$S_{3}$$A^{\mathrm{H}}_{1}$$A^{\mathrm{H}}_{2}$$\Theta^{H}$
Figure 7: A supervision POMDP with the human considered part of the
environment; we show 3 timesteps and 2 actions.
Given the graph in Fig. 7, we can apply the VoI criterion to each $A^{H}_{i}$,
the sole parent of $A^{i}$. The minimal d-reduction is identical to the
original graph, so since $A^{H}_{i}\\!\not\perp\\!R^{i+1}\\!\mid\\!\emptyset$,
the observation $A^{H}_{i}$ can have positive VoI. This formalises the claim
of Milli et al. (2017) that in a supervision POMDP, the agent “can learn about
reward through [the human’s] orders”. We can say the same about Cooperative
Inverse Reinforcement Learning (CIRL). CIRL differs from supervision POMDPs
only in that each human action $A^{H}_{i}$ directly affects the state
$S_{i+1}$. If ${\mathcal{G}}$ is modified by adding edges $A^{H}_{i}\to
S_{i+1}$, and the VoI criterion is applied at $A^{H}_{i}$ once again, we find
that $A^{H}_{i}$ may have positive VoI for $A^{i}$, thereby formalising the
claim that the robot is “incentivised to learn” (Hadfield-Menell et al. 2016,
Remark 1).
To facilitate convenient use of the graphical criterion, we have implemented
it in the open source ID library _pycid_ (Fox et al. 2021), whereas the
previous implementation was limited to single-decision IDs.111Code is
available at www.github.com/causalincentives/pycid.
## 7 Related Work
##### Value of information
The concept of value of information dates back to the earliest papers on
influence diagrams (Howard 1966; Matheson 1968). For a review of recent
advances, see Borgonovo and Plischke (2016).
Previous results have shown how to identify observations with zero VoI or
equivalent properties in various settings. In the no forgetting setting,
Fagiuoli and Zaffalon (1998) and Nielsen and Jensen (1999) identified
“structurally redundant” and “required nodes” respectively. In soluble IDs,
Nilsson and Lauritzen (2000) proved that optimal decisions need not rely on
nonrequisite nodes. Completeness proofs in a setting of one decision have been
discovered for VoI and its analogues by Zhang, Kumor, and Bareinboim (2020);
Lee and Bareinboim (2020); Everitt et al. (2021a). Finally, in insoluble IDs,
Lee and Bareinboim (2020) proved that certain nodes are “redundant under
optimality”. Of these works, only Nielsen and Jensen (1999) attempts a
completeness result for the multi-decision setting. However, as pointed out by
Everitt et al. (2021a), it falls short in two respects: Firstly, the criterion
$X\not\perp{\bm{U}}^{D}\mid\mathrm{\mathbf{Pa}}(D)$ is proposed, which differs
from nonrequisiteness in the conditioning set. Secondly, and more importantly,
the proof is incomplete because it assumes that positive VoI follows from
d-connectedness.
##### Submodel-trees
Trees of systems are loosely related to the “submodel-trees” of Lee,
Marinescu, and Dechter (2021). In both cases, the tree encodes an ordering in
which the ID can be solved, so the edges in a tree of systems are analogous to
those in a submodel-tree. The nodes, however, (i.e. systems and submodels)
differ. Whereas a submodel-tree aids with solving IDs, a tree of systems helps
with parameterising an ID graph. As a result, a submodel contains all nodes
relevant for $D$, whereas a system consists just one set of
info-/control-/obs-paths. Relatedly, in a submodel, downstream decisions may
be solved and replaced with a value node, whereas in a tree of systems, they
are not.
## 8 Discussion and Conclusion
This paper has described techniques for analyzing soluble influence diagrams.
In particular, we introduced ID homomorphisms, a method for transforming ID
graphs while preserving key properties, and showed how these can be used to
establish equivalent ID graphs with conveniently parameterizable “trees of
systems”. These techniques enabled us to derive the first completeness result
for a graphical criterion for value of information in the multi-decision
setting.
Given the promise of reinforcement learning methods, it is essential that we
obtain a formal understanding of how multi-decision behavior is shaped. The
graphical perspective taken in this paper has both advantages and
disadvantages. On the one hand, some properties cannot be distinguished from a
graphical perspective alone. On the other hand, it means our results are
applicable even when the precise relationships are unspecified or unknown.
There are a range of ways that this work could be beneficial. For example,
analogous results for the single-decision setting have contributed to safety
and fairness analyses (Armstrong et al. 2020; Cohen, Vellambi, and Hutter
2020; Everitt et al. 2021b, 2019; Langlois and Everitt 2021; Everitt et al.
2021a).
Future work could include applying the tools developed in this paper to other
incentive concepts such as value of control (Shachter 1986), instrumental
control incentives, and response incentives (Everitt et al. 2021a), to further
analyse the value of remembering past decisions (Shachter 2016; Lee and
Bareinboim 2020), and to generalize the analysis to multi-agent influence
diagrams (Hammond et al. 2021; Koller and Milch 2003).
## Acknowledgments
This work was supported in-part by the Leverhulme Centre for the Future of
Intelligence, Leverhulme Trust, under Grant RC2015-067.
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## Appendix A Background for the proofs
We review two properties of IDs — and all Bayesian networks — that we will use
throughout our proofs.
###### Lemma 22 (Active paths between ancestors contain only ancestors).
A path from an ancestor of node $N$ to another ancestor of $N$, that is active
conditional on only ancestors of $N$, contains only ancestors of $N$.
###### Proof.
Let $p\colon A\;\hbox{- - -}\;B$ be any active path where $A$ and $B$ are
ancestors of $N$ and let the colliders on $p$ be ${\bm{O}}$. Since $p$ is
active, any collider $O\in{\bm{O}}$ on $p$ is an ancestor of the conditioned
set, and is therefore an ancestor of $N$. For any chain or fork node $V$,
choose one of its outgoing edges along $p$ and follow $p$ until the next
collider or endpoint (${\bm{O}}$, $A$, or $B$). This path is directed, so $V$
is an ancestor of $O$, $A$, or $B$, and hence $M$. ∎
The standard definition of a walk is a sequence of consecutive edges. Unless a
node has an edge to itself, a walk is not alowed to visit the same node twice
in a row. Instead, we define a notion of a walk such that it is always allowed
to repeat the same node previously visited.
###### Definition 23 (Walk with node repetition).
A _walk with node repetition_ from $N^{1}$ to $N^{n}$ in a graph
$({\bm{V}},{E})$ is a sequence of nodes $N^{1},...,N^{n}$ such that for any
$i\in\\{1,...,n-1\\}$, either there is a link $N^{i}\to N^{i+1}$ or
$N^{i}\leftarrow N^{i+1}$ in ${E}$, or $N^{i}=N^{i+1}$.
We say that a node $N$ in a walk with node repetitions $p$ is a
collider/fork/chain node in a walk with node repetitions $w$ if it is a
collider/fork/chain node in the walk (without node repetitions) $w^{\prime}$
obtained by removing any nodes that are equal to their predecessor.
###### Lemma 24 (“Active” walk with node repetitions implies active path).
If there is a walk with node repetition from node $A$ to node $B$, such that
all fork and chain nodes are not in a set ${\bm{Z}}$, and all collider nodes
have a descendant in ${\bm{Z}}$, then there is a path between $A$ and $B$ that
is active given ${\bm{Z}}$.
###### Proof.
Assume there is a walk with node repetition $w\colon A\;\hbox{- - -}\;B$ such
that every collider in $w$ has a descendant in ${\bm{Z}}$ and every non-
collider in $w$ is not in ${\bm{Z}}$. Then let $p$ be the path obtained from
$w$ by replacing every segment $N\;\hbox{- - -}\;N$ with the node $N$.
Clearly, $p$ is a path, so we will proceed to show that it is active given
${\bm{Z}}$, by showing that it is active at each of its nodes.
Assume that $N$ is a collider in $p$. Then, $N$ was obtained from a segment in
$w$, $Y_{1}\to N\;\hbox{- - -}\;N\leftarrow Y_{2}$ where $N\;\hbox{- - -}\;N$
has length zero or greater. For this segment to be active in $w$, the first
collider in $N\;\hbox{- - -}\;N$ must have a descendant in ${\bm{Z}}$, and
thus so does $N$, and it is active in $p$. Assume instead that $N$ is a non-
collider. Then, $N$ was obtained from a segment in $w$, $Y_{1}\to N\;\hbox{- -
-}\;N\to Y_{2}$, $Y_{1}\leftarrow N\;\hbox{- - -}\;N\leftarrow Y_{2}$, or
$Y_{1}\leftarrow N\;\hbox{- - -}\;N\to Y_{2}$. In any case, for this segment
to be active in $w$, $N\not\in{\bm{Z}}$, so it is active in $p$, proving the
result. ∎
$O$$D_{1}$$D_{2}$$U$$\Pi_{1}$$\Pi_{2}$ (a) Insoluble ID graph
$O$$D_{1}$$D_{2}$$U$$\Pi_{1}$$\Pi_{2}$ (b) ID graph with perfect recall
$O_{2}$$D_{1}$$S$$O_{1}$$D_{2}$$U_{2}$$U_{1}$$\Pi_{1}$$\Pi_{2}$ (c) Soluble ID
graph
Figure 8: Multi-decision IDs.
For our analysis, we consider soluble ID graphs. This condition includes
graphs with perfect recall Fig. 8(b) but also includes some others, shown in
Fig. 8(c).
## Appendix B ID homomorphisms
### B.1 Properties preserved given an ID homomorphism
We now prove two properties that are preserved by any ID homomorphism:222An ID
homomorphism is analogous to the notion of graph homomorphism from graph
theory, which essentially requires that edges are preserved along the map. In
fact, if we would consider every node in an ID graph as a decision and as
having an edge to itself, then any ID homomorphism is also a graph
homomorphism when considering the two ID graphs as ordinary graphs (ignoring
node types). solubility, and VoI.
See 9
###### Proof.
Since ${\mathcal{G}}$ is soluble, there is a total ordering of decisions $<$
such that for all $D^{1}<D^{2}$, $\Pi^{D}\perp
U(D^{2})\mid\mathrm{\mathbf{{Fa}}}(D^{2})$. To show that
${\mathcal{G}}^{\prime}$ is also soluble, we use $<$ to construct an ordering
on the decisions of ${\mathcal{G}}^{\prime}$ that has the same property. We
define $<^{\prime}$ for decisions in ${\mathcal{G}}^{\prime}$: as
$D^{1}<^{\prime}D^{2}$ when:
* •
$h(D^{1})\neq h(D^{2})$ and $h(D^{1})<h({D^{2}})$; or
* •
$h(D^{2})=h(D^{2})$ and ${\mathcal{G}}^{\prime}$ contains $D^{1}\to D^{2}$.
This is a total order since whenever $h(D^{1})\neq h(D^{2})$ then either
$h(D^{1})<h(D^{2})$ or $h(D^{1})>h(D^{2})$ by the total order on
${\mathcal{G}}$, and whenever $h(D^{1})=h(D^{2})$ then
${\mathcal{G}}^{\prime}$ contains $D^{1}\to D^{2}$ or $D^{1}\leftarrow D^{2}$
by (Combines only linked decisions).
Now we show that for any two decisions $D^{1},D^{2}$ in
${\mathcal{G}}^{\prime}$, with $D^{1}<^{\prime}D^{2}$, that any path
$p\colon\Pi^{D^{1}}\to D^{1}\;\hbox{- - -}\;U$ for some
$U\in\textbf{Desc}(D^{2})$ cannot be active given
$\mathrm{\mathbf{{Fa}}}(D^{2})$. Consider two cases:
Case (1) : Assume $h(D^{2})\\!=\\!h(D^{1})$. Then $D^{1}\\!<\\!D^{2}$ so
${\mathcal{G}}^{\prime}$ contains $D^{1}\to D^{2}$ by definition of
$<^{\prime}$, and so any path $p\colon\Pi^{D^{1}}\to D^{1}\;\hbox{- - -}\;U$
that starts with the link $\Pi^{D^{1}}\to D^{1}\to Y$ is blocked at $D^{1}$
given $\mathrm{\mathbf{Pa}}({D^{2}})$. Any path that begins as $\Pi^{D^{1}}\to
D^{1}\leftarrow Y$ is blocked at the non-collider $Y$: the presence of
$D^{1}\leftarrow Y$ implies that ${\mathcal{G}}$ contains $h(D^{1})\leftarrow
h(Y)$ (Preserves links), so that ${\mathcal{G}}^{\prime}$ contains
$D^{2}\leftarrow Y$ (Covers all infolinks), and
$Y\in\mathrm{\mathbf{Pa}}(D^{2})$.
Case (2) : Assume $h(D^{2})\neq h(D^{1})$. We will prove the contrapositive:
if $\Pi^{D^{1}}\not\perp U(D^{2})\mid\mathrm{\mathbf{{Fa}}}(D^{2})$ in
${\mathcal{G}}^{\prime}$ then $\Pi^{h(D^{1})}\not\perp
U(h(D^{2}))\mid\mathrm{\mathbf{{Fa}}}(h(D^{2}))$ in ${\mathcal{G}}$ where
$h(D^{1})<h(D^{2})$. If $p\colon\Pi^{D^{1}}\to D^{1}\;\hbox{- - -}\;U(D^{2})$
is active given $\mathrm{\mathbf{{Fa}}}(D^{2})$, then consider the walk with
node repetition $w\colon\Pi^{h(D^{1})}\to h(D^{1})\;\hbox{- -
-}\;h(U(D^{\prime}))$ consisting of $f(V)$ for each node $V$ in $p$. We know
that each $V$ in $p$ is a (chain/fork/collider) if and only if $h(V)$ is a
(chain/fork/collider) in $w$, since if there is a link $N\to V$ or
$N\leftarrow V$ then there must be a link $h(N)\to h(V)$ or $h(N)\leftarrow
h(V)$ respectively by the (Preserves links) assumption of ID homomorphisms.
And that each node $V$ contains a descendant in
$\mathrm{\mathbf{{Fa}}}(D^{2})$ if and only if $h(V)$ contains a descendant in
$\mathrm{\mathbf{{Fa}}}(h(D^{2}))$. So every collider in $w$ has a descendant
in $\mathrm{\mathbf{{Fa}}}(h(D^{2}))$ while every non-collider does not. This
implies that $\Pi^{h(D^{1})}\not\perp
U(h(D^{2}))\mid\mathrm{\mathbf{{Fa}}}(h(D^{2}))$ by Lemma 24, and we know that
$h(D^{1})<h(D^{2})$ by the definition of $<^{\prime}$ so the result follows. ∎
We can now define how a homomorphism allows us to define a procedure for
transporting IDs between the two graphs, such that corresponding IDs and
policies lead to equivalent outcomes.
See 11
###### Proof.
We define the _ported ID_ ${\mathcal{M}}=({\mathcal{G}},\mathrm{dom},P)$ as
follows: Each non-utility node $N$ in ${\mathcal{G}}$ has
$\mathrm{dom}(N)=\prod_{N^{i}\in h^{-1}(N)}\mathrm{dom}(N^{i})$, and each
utility node $U$ in ${\mathcal{G}}$ has $\mathrm{dom}(U)=\mathbb{R}$. Each
non-decision node $N$ has as $P^{N}(n|\mathrm{\mathbf{pa}})$ the joint
conditional distribution of each
$\tilde{P}^{N^{i}}(n^{i}|\mathrm{\mathbf{pa}}^{i})$. We define the _ported
policy_ $\pi$ so that each decision $D$ has as
$\pi^{D}(d|\mathrm{\mathbf{pa}})$ the joint conditional distribution of each
$\pi^{D^{i}}(d^{i}|\mathrm{\mathbf{pa}})$. These in fact factor over
${\mathcal{G}}$ by property (b) of ID homomorphisms.
We show the result by induction on the graph of nodes $N^{i}$ in
${\mathcal{G}}^{4}$. Let $N=h(N^{i})$.
base step : Assume $N^{i}$ has no parents in ${\mathcal{G}}^{4}$. Then
$P(N\\!=\\!(n^{1},...,n^{k}))=P^{N}((n^{1},...,n^{k}))=\prod_{i=1}^{k}\tilde{P}^{N^{i}}(n^{i}|n^{1},...,n^{i-1})=\tilde{P}(N^{1}\\!=\\!n^{1},...,N^{k}=n^{k})$.
inductive step : Assume that for all parents $Y^{i}$ of $N^{i}$, letting
$Y=h(Y^{i})$, we have that
$P(Y\\!=\\!(y^{1},...,y^{k}))=\tilde{P}(Y^{1}\\!=\\!y^{1},...,Y^{k}=y^{k})$.
Then
$\displaystyle P(N\\!=\\!(n^{1},...,n^{k}))$
$\displaystyle=\sum_{y^{1},...,y^{k}}P(Y\\!=\\!(y^{1},...,y^{k}))\cdot
P^{N}((n^{1},...,n^{k})|y^{1},...,y^{k})\quad\quad$
$\displaystyle=\sum_{y^{1},...,y^{k}}\tilde{P}(Y^{1}\\!=\\!y^{1},...,Y^{k}\\!=\\!y^{k}))\cdot\prod_{i=1}^{k}\tilde{P}^{N^{i}}(n^{i}|n^{1},...,n^{i-1},y^{1},...,y^{k})\quad\quad$
$\displaystyle=\tilde{P}(N^{1}\\!=\\!n^{1},...,N^{k}=n^{k})\quad$
Which shows the result. ∎
We will write the “transported ID” and policy from Lemma 11 as $h(M)$ and
$h(\pi)$. This means that we also treat a homomorphism $h$ as a function
between IDs and policies. In fact, in order to show that ID homomorphisms
preserve VoI , we show that on policies, $h$ is a bijection, which relies on
the properties (c) and (d) of ID homomorphisms, and is the primary reason why
(c,d) are included:
###### Lemma 25.
Any ID homomorphism $h$ is a bijection from (optimal) policies on
$\tilde{\mathcal{M}}$ to (optimal) policies on
${\mathcal{M}}=h(\tilde{\mathcal{M}})$.
###### Proof.
We define an inverse for the map as follows: Take a policy $\pi_{D}$ on $M$.
This gives a joint distribution $\tilde{\pi}^{D^{i}}=(\pi^{D})^{i}$ over
$\mathrm{dom}(D^{i})$ for $D^{i}\in\mathcal{D}=h^{-1}(D)$. Moreover, for any
$X^{j}\in\mathrm{\mathbf{Pa}}(D)$ and for any $X^{j,k}\in h^{-1}(X^{j})$, each
decision $D^{i}$ has $X^{j,k}\in\mathrm{\mathbf{Pa}}(D^{i})$ by (Covers all
infolinks), and since these decisions $D^{i}$ form a complete graph (each
$D^{i}$ is linked to each $D^{j}$) by condition (Combines only linked
decisions), this distribution $\tilde{\pi}^{D^{i}}$ also factors over
$\tilde{\mathcal{G}}$ and hence is a policy $\tilde{M}$. But this is precisely
the definition of $\pi^{D}$ being the transported policy of
$\tilde{\pi}^{\mathcal{D}}$, so that $\pi^{D}\mapsto\pi^{\mathcal{D}}$ is
indeed the desired inverse. The optimal version of this lemma then follows
from Lemma 11. ∎
See 12
###### Proof.
Let $\mathcal{X}$ be the set of nodes $X^{j}$ in ${\mathcal{G}}^{\prime}$ such
that $h(X^{j})=X$, and $\mathcal{D}$ the set of nodes $D^{j}$ such that
$h(D^{j})=D$.
Firstly, note that $h$ is also a homomorphism from
${\mathcal{G}}^{\prime}_{\mathcal{X}\to\mathcal{D}}$ to ${\mathcal{G}}_{X\to
D}$ and from ${\mathcal{G}}^{\prime}_{\mathcal{X}\not\to\mathcal{D}}$ to
${\mathcal{G}}_{X\not\to D}$ (since in both cases, there is still an edge
$X^{i}\to D^{i}$ iff there is an edge $X\to D$). Hence, for any policy
$\pi^{\prime}$ on $M^{\prime}$ and letting $\pi=h(\pi^{\prime})$ be the
corresponding policy on $M$, apply Lemma 11 twice to conclude that
${\mathbb{E}}^{M_{X\to
D}}_{\pi}(\mathcal{U})={\mathbb{E}}^{M^{\prime}_{\mathcal{X}\to\mathcal{D}}}_{\pi^{\prime}}(\mathcal{U})$
and ${\mathbb{E}}^{M^{\prime}_{X\not\to
D}}_{\pi^{\prime}}(\mathcal{U})={\mathbb{E}}^{M_{\mathcal{X}\not\to\mathcal{D}}}_{\pi}(\mathcal{U})$.
Since the map that maps a policy on $M^{\prime}$ to the corresponding policy
on $M$ (see Lemma 11) is a bijection by Lemma 25, this implies that
$\max_{\pi}{\mathbb{E}}^{M_{X\to
D}}_{\pi}(\mathcal{U})=\max_{\pi^{\prime}}{\mathbb{E}}^{M^{\prime}_{\mathcal{X}\to\mathcal{D}}}_{\pi^{\prime}}(\mathcal{U})$
and $\max_{\pi}{\mathbb{E}}^{M_{X\not\to
D}}_{\pi}(\mathcal{U})=\max_{\pi^{\prime}}{\mathbb{E}}^{M^{\prime}_{\mathcal{X}\not\to\mathcal{D}}}_{\pi^{\prime}}(\mathcal{U})$.
These imply:
$\displaystyle\max\limits_{\pi}{\mathbb{E}}^{M_{X\to D}}_{\pi}(\mathcal{U})$
$\displaystyle=\max\limits_{\pi^{\prime}}{\mathbb{E}}^{M^{\prime}_{\mathcal{X}\to\mathcal{D}}}_{\pi^{\prime}}(\mathcal{U})\quad\quad$
$\displaystyle:\text{by the argument above}$
$\displaystyle\geq\max\limits_{\pi^{\prime}}{\mathbb{E}}^{M^{\prime}_{X^{i}\to
D^{i}}}_{\pi^{\prime}}(\mathcal{U})\quad\quad$ $\displaystyle:\text{more
infolinks cannot decrease max utility}$
$\displaystyle>\max\limits_{\pi^{\prime}}{\mathbb{E}}^{M^{\prime}_{X^{i}\not\to
D^{i}}}_{\pi^{\prime}}(\mathcal{U})\quad$ $\displaystyle:\text{by assumption:
$X^{i}$ is material for $D^{i}$}$
$\displaystyle\geq\max\limits_{\pi^{\prime}}{\mathbb{E}}^{M^{\prime}_{\mathcal{X}\not\to\mathcal{D}}}_{\pi^{\prime}}(\mathcal{U})\quad$
$\displaystyle:\text{more infolinks cannot decrease max utility}$
$\displaystyle=\max\limits_{\pi}{\mathbb{E}}^{M_{X\not\to
D}}_{\pi}(\mathcal{U})\quad$ $\displaystyle:\text{by the argument above}$
which shows the result. ∎
### B.2 Transformations that ensure a homomorphism
See 15
###### Proof.
We show that each of the four properties is preserved under composition:
(a) If $h$ and $h’$ preserve node types, then clearly so does $h\circ
h^{\prime}$.
(b) If ${\mathcal{G}}^{\prime\prime}$ contains $A\to B$ then by (b) for
$h^{\prime}$, ${\mathcal{G}}^{\prime}$ contains $h’(A)\to h^{\prime}(B)$ or
$h^{\prime}(A)=h^{\prime}(B)$. In either case, (b) for $h$ implies that
${\mathcal{G}}$ contains $h\circ h^{\prime}(A)\to h\circ h^{\prime}(B)$ or
$h\circ h^{\prime}(A)=h\circ h^{\prime}(B)$.
(c) If ${\mathcal{G}}$ contains $h\circ h^{\prime}(N)\to h\circ
h^{\prime}(D)$, then by (c) for $h$, ${\mathcal{G}}^{\prime}$ contains
$h’(N)\to h^{\prime}(D)$ and by the same argument ${\mathcal{G}}$ contains
$N\to D$.
(d) Assume $h\circ h^{\prime}(D^{1})=h\circ h^{\prime}(D^{2})$ and
$D^{1}\\!\neq\\!D^{2}$ in ${\mathcal{G}}^{\prime\prime}$. Then if
$h’(D^{1})\\!=\\!h^{\prime}(D^{2})$, by (d) for $h^{\prime}$,
${\mathcal{G}}^{\prime\prime}$ contains $D^{1}\\!\to\\!D^{2}$ or
$D^{2}\\!\to\\!D^{1}$ showing the result. If $h’(D^{1})\neq h^{\prime}(D^{2})$
then by (d) for $h$, ${\mathcal{G}}^{\prime}$ contains $h’(D^{1})\to
h^{\prime}(D^{2})$ or $h’(D^{2})\to h^{\prime}(D^{1})$, and hence by (c) for
$h^{\prime}$, ${\mathcal{G}}^{\prime\prime}$ contains $D^{1}\to D^{2}$ or
$D^{2}\to D^{1}$. ∎
See 13 color=blue!30]Ryan: I’ve simplified/shortened this a bit further. Feel
free to revert any changes is preferred.
###### Proof.
ID homomorphism condition (a) follows by definition. (b) follows from the
definition of $E^{\prime}$. (c,d) follow since they hold for all nodes $N$ by
definition, including the decisions. ∎
See 14
###### Proof.
The homomorphism properties follow: (a) by definition. (b) from
$E^{\prime}\\!\subseteq\\!E$, (c) from every $N\\!\to\\!D\in{\bm{D}}$ being in
$E$, (d) from $h$ being the identity map so every $D^{1}\\!\neq\\!D^{2}$ has
$h(D^{1})\\!\neq\\!h(D^{2})$. ∎
## Appendix C Systems and trees of systems in an ID graph
### C.1 Systems
Before detailing the properties of systems, we first recap the elements of a
system. We call $D^{s}$, $U^{s}$, $X^{s}$, and $\mathrm{infolink}^{s}\colon
X^{s}\to D^{s}$ the _decision node_ , _utility node_ , _info node_ , and
_infolink_ of $s$, respectively, and refer to $\mathrm{control}^{s}$,
$\mathrm{info}^{s}$ and $\mathrm{obs}^{s}(C)$ for each collider $C$ in
$\mathrm{info}^{s}$ as the _paths of $s$_.
###### Definition 26 (Elements of a system).
For a system $s$:
* •
An _obs node_ $O$ of $s$ is the penultimate node of each obs path
$\mathrm{obs}^{s}(C)$.333For “observation node”. But note that though $D^{i}$
does “observe” $X^{i}$, it is not an obsnode, since it is not the penultimate
node of an $\mathrm{obs}^{s}(C)$, but is the first node of
$\mathrm{info}^{s}$.
* •
The _question node_ $Q^{s}$, if $\mathrm{info}^{s}$ contains at least one fork
node, is the closest-to-$U^{s}$ fork node on $\mathrm{info}^{s}$.444This
implies that the segment of the info path from $Q^{s}$ to $U^{s}$ is a
directed path $Q^{s}\dashrightarrow U^{s}$, since there are no fork nodes on
that path, and it must begin and end with an arrow towards $U^{s}$.
* •
The _back section_ , if $\mathrm{info}^{s}$ contains a fork, is the set of
nodes in $X^{s}\;\hbox{- - -}\;Q^{s}$ in $\mathrm{info}^{s}$ (including
$X^{s}$ and $Q^{s}$) and in each $\mathrm{obs}(C^{i})$, except for $D^{s}$.
Otherwise, the back section is empty.
* •
The _front section_ consists of the nodes in any path in $s$ that are not in
the back section.
###### Definition 27 (Within-system links and paths).
A link $A\to B$ that is in $\mathrm{info}^{s}$, $\mathrm{control}^{s}$, or any
$\mathrm{obs}^{s}(C^{i})$, or the link $X^{s}\to D^{s}$ for some system $s$ is
called a _within-system- $s$_ or _within-system link_. A _within-system path_
is a path that contains only within-system links.
We will now prove a number of fundamental properties of systems.
###### Lemma 28 (Basic properties of a system in a soluble ID graph).
Any system $s$ in a soluble ID graph has the properties:
1. (a)
(No infolinks in the back-section) The back section of $s$ can only contain a
decision $D^{\prime}\in{\bm{D}}$ if $D^{\prime}=X^{s}$, and the infopath
$\mathrm{info}^{s}$ is front-door. Moreover then $D^{\prime}$ is not in any
$\mathrm{obs}^{s}(C^{i})$. color=green!30]Chris: Above can be phrased more
neatly. but not a priority.
2. (b)
(Infolinks in $s$ are descendants of $D^{s}$) An information link $N\to
D^{\prime}$ for $D^{\prime}\neq D^{s}$ can only be contained in a path in
system $s$ if the control path $\mathrm{control}^{s}$ contains a parent of
$D^{\prime}$, so that $D^{\prime}\in\textbf{Desc}(D^{s})$.
3. (c)
(Parents of ancestor decisions are parents of $D^{s}$) A node $N$ in system
$s$ can only be a parent of an ancestor decision $D^{\prime}$ of $D^{s}$ if
$N$ is also a parent of $D^{s}$.555Note that in a normal form tree (see
below), this link $N\to D^{\prime}$ is an out-of-tree link
###### Proof.
We prove each property in succession:
(a) (No infolinks in the back section) We will prove what restrictions are
implies by considering sequentially the cases where $D^{\prime}$ is in either
the infopath, or in the observation path. To begin with, let us state what we
know in both cases: $D^{\prime}$ must be an ancestor of $D^{s}$. As such,
$D^{\prime}<D^{s}$ in any topological ordering, so solubility requires that
$D^{\prime}\perp U^{s}\mid\mathrm{\mathbf{{Fa}}}(D^{s})$.
If $D^{\prime}\in\mathrm{info}^{s}$, then the path $p\colon\Pi^{\prime}\to
D^{\prime}\mathrel{\mathop{\;\hbox{- - -}\;}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{info}^{s}$}\vss}}}U^{s}$ may
be formed from by truncating the infopath $\mathrm{info}^{s}$. By solubility,
$p$ must be blocked given $\mathrm{\mathbf{{Fa}}}(D^{s})$. We know, however,
that $\mathrm{info}\colon X^{s}\mathrel{\mathop{\;\hbox{- -
-}\;}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{info}^{s}$}\vss}}}U^{s}$ is
active given $\mathrm{\mathbf{{Fa}}}({D^{s}})\setminus\\{X^{s}\\}$. If
$D^{\prime}\neq X^{s}$ then $p$ does not contain $X^{s}$, and so it is is
active given $\mathrm{\mathbf{{Fa}}}({D^{s}})$, violating solubility.
Moreover, if $D^{\prime}=X^{s}$ and $\mathrm{info}$ is a backdoor path, then
$p$ will have a collider at $D^{\prime}$, and solubility is violated once
again. So $\mathrm{info}^{s}$ can only contain a decision $D^{\prime}$ if
$D^{\prime}=X^{s}$ and $\mathrm{info}^{s}$ is frontdoor.
Now we will prove that $D^{\prime}\not\in\mathrm{obs}(C)$, by contradiction.
Suppose that $D^{\prime}\in\mathrm{obs}(C)$. Then, consider the path
$q\colon\Pi^{\prime}\to
D^{\prime}\mathrel{\mathop{\dashleftarrow}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{obs}^{s}(C)$}\vss}}}C\mathrel{\mathop{\;\hbox{-
- -}\;}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{info}^{s}$}\vss}}}U^{s}$,
constructed by truncating the observation path and infopath. By assumption,
the path $C\mathrel{\mathop{\dashrightarrow}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{obs}^{s}(C)$}\vss}}}Y\to D$
is minimal-length, so no node $W\neq Y$ on the path can be a parent of
$D^{s}$, and so $D^{s}\mathrel{\mathop{\dashleftarrow}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{obs}^{s}(C)$}\vss}}}C$ is
active given $\mathrm{\mathbf{{Fa}}}(D^{s})$. The segment
$C\mathrel{\mathop{\;\hbox{- - -}\;}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{info}^{s}$}\vss}}}U^{s}$ is
active given $\mathrm{\mathbf{{Fa}}}(D^{s})\setminus\\{X^{s}\\}$. Since
$\mathrm{info}^{s}$ is a path, the segment $C\mathrel{\mathop{\;\hbox{- -
-}\;}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{info}^{s}$}\vss}}}U^{s}$
cannot contain $X^{s}$, and thus is active given
$\mathrm{\mathbf{{Fa}}}(D^{s})$. So the path $q$ is active given
$\mathrm{\mathbf{{Fa}}}(D^{s})$, violating solubility.
Together, these two cases prove the result.
(b) (Infolinks in $s$ are descendants of $D^{s}$) We know from sublemma (a)
that the back section cannot contain any link $N\to D^{\prime}$. So
$D^{\prime}$ must lie in the front-section of $s$: either in
$Q^{s}\mathrel{\mathop{\;\hbox{- - -}\;}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{info}^{s}$}\vss}}}U^{s}$, or
in $\mathrm{control}^{s}$. In either case, we have
$U^{s}\in\textbf{Desc}(D^{s})$ and $U^{s}\in\textbf{Desc}(D^{\prime})$. So in
order for the ID graph to be soluble, we must have either
$\Pi^{D^{\prime}}\perp U^{s}\mid\mathrm{\mathbf{{Fa}}}(D^{s})$ or
$\Pi^{D^{s}}\perp U^{s}\mid\mathrm{\mathbf{{Fa}}}(D^{\prime})$.
We can show that the first case $\Pi^{D^{\prime}}\perp
U^{s}\mid\mathrm{\mathbf{{Fa}}}(D^{s})$ cannot hold. If $D^{\prime}$ is in
$\mathrm{control}^{s}$, note that $\mathrm{control}^{s}$ consists of only
descendants of $D^{s}$. If $D^{\prime}$ is in
$q:Q^{s}\mathrel{\mathop{\;\hbox{- - -}\;}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{info}^{s}$}\vss}}}U^{s}$
then note that $q$ is assumed to be active given
$\mathrm{\mathbf{{Fa}}}(D^{s})\setminus\\{X^{s}\\}$, and cannot contain
$X^{s}$. In either case, $\Pi^{D^{\prime}}\not\perp
U^{s}\mid\mathrm{\mathbf{{Fa}}}(D^{s})$. Hence we must have $\Pi^{D^{s}}\perp
U^{s}\mid\mathrm{\mathbf{{Fa}}}(D^{\prime})$, from which it follows that every
directed path from $D^{s}$ to $U^{s}$ (including $\mathrm{control}^{s}$) must
contain a parent of $D^{\prime}$.
(c) (Parents of ancestor decisions are parents of $D^{s}$) Assume $N$ is a
parent of $D^{\prime}$ in a path of $s$. It cannot be in
$\mathrm{control}^{s}$, because then $D^{\prime}$ would be a descendant of
$D^{s}$. So $N$ must be in $\mathrm{info}^{s}$ or one of
$\mathrm{obs}^{s}(C)$. If $N$ is in $\mathrm{info}^{s}$, consider the path
$p\colon\Pi^{D’}\to D’\leftarrow N\mathrel{\mathop{\;\hbox{- -
-}\;}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{info}^{s}$}\vss}}}U^{s}$. We
know $\mathrm{info}^{s}$ is active given
$\mathrm{\mathbf{{Fa}}}({D^{s}})\setminus\\{X^{s}\\}$. Hence if
$N\notin\mathrm{\mathbf{Pa}}(D^{s})$, then $p$ is active given
$\mathrm{\mathbf{Pa}}({D^{s}})$ and since it doesn’t contain $D^{s}$ also
active given $\mathrm{\mathbf{{Fa}}}(D^{s})$, violating solubility. Hence
$N\in\mathrm{\mathbf{Pa}}(D^{s})$.
Similarly, if $N$ is in $\mathrm{obs}^{s}(C)$, then consider the path
$q\colon\Pi^{D^{\prime}}\to D^{\prime}\leftarrow
N\mathrel{\mathop{\dashleftarrow}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{obs}^{s}(C)$}\vss}}}C\mathrel{\mathop{\;\hbox{-
- -}\;}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{info}^{s}$}\vss}}}U^{s}$.
Hence if $N\notin\mathrm{\mathbf{Pa}}(D^{s})$, then since
$\mathrm{obs}^{s}(C)$ is minimal-length, it holds that
$N\mathrel{\mathop{\dashleftarrow}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{obs}^{s}(C)$}\vss}}}C$ is
active given $\mathrm{\mathbf{{Fa}}}(D^{s})$, as in the proof of (a).
Moreover, the segment $C\mathrel{\mathop{\;\hbox{- - -}\;}\limits^{\vbox
to1.50694pt{\kern-2.0pt\hbox{$\scriptstyle\mathrm{info}^{s}$}\vss}}}U^{s}$ is
active given $\mathrm{\mathbf{{Fa}}}(D^{s})\setminus\\{X^{s}\\}$ by
assumption, and hence given $\mathrm{\mathbf{{Fa}}}(D^{s})$. Since
$D^{\prime}\in\textbf{Anc}(D^{s})$, $q$ is active given
$\mathrm{\mathbf{{Fa}}}(D^{s})$, again violating solubility. Hence again
$N\in\mathrm{\mathbf{Pa}}(D^{s})$. ∎
### C.2 Trees of systems
First, let us recap the definition of a tree of systems.
See 17 We define the _predecessor system_ and _predecessor path_ of system
$s^{i}$ as
$({\mathrm{predsys}}(s^{i}),{\mathrm{predpath}}(s^{i})):=\mathrm{pred}(s^{i})$.
Moreover, we will sometimes say “An ID graph with tree” to refer to an ID
graph, together with a tree on that ID graph.
###### Terminology.
If $s^{i}={\mathrm{predsys}}(s^{j})$ then we say that $s^{j}$ is a child
system of $s^{i}$. We will similarly apply the standard terminology of trees
and graphs: Ancestor system, descendant system.
###### Lemma 29 (A tree of systems has a tree structure).
Given a tree of systems $T=({\mathcal{S}},\mathrm{pred})$, the pair
$({\mathcal{S}},{\mathrm{predsys}})$ is a tree structure, i.e. it satisfies:
* •
There is a unique node $s^{\mathrm{root}}$ that has
${\mathrm{predsys}}(s^{\mathrm{root}})=s^{\mathrm{root}}$; and
* •
For any node $s$, there is some number $n\in\mathbb{N}$ such that
${\mathrm{predsys}}^{n}(s)=s^{\mathrm{root}}$.
###### Proof.
The first condition is satisfied directly by definition of
$s^{\mathrm{root}}$. For the second condition, we only need to show that for
any $s^{i}$, there is a sequence of systems $(s^{1},...,s^{n})$ such that
$s^{1}=s^{\mathrm{root}}$ and $s^{n}=s^{i}$, and
${\mathrm{predsys}}(s^{j})=s^{j-1}$ for all $1<j\leq n$. Assume by
contradiction that there is a system that doesn’t satisfy this, and let
${\mathcal{S}}^{*}$ be the set of all such systems. Then since the restrition
of ${\mathrm{predsys}}$ to ${\mathcal{S}}^{*}$ has no fixed points
($s^{\mathrm{root}}$ is the only fixed point and is not in ${\mathcal{S}}^{*}$
by definition), it must have some sequence $(\tilde{s}^{1},...,\tilde{s}^{k})$
with ${\mathrm{predsys}}(\tilde{s}^{1})=\tilde{s}^{k}$ and
${\mathrm{predsys}}(\tilde{s}^{j})=\tilde{s}^{j-1}$ for all $1<j\leq k$ (i.e.
a cycle). But this would imply that there is at least one pair of systems
$(\tilde{s}^{k},\tilde{s}^{m})$ with
${\mathrm{predsys}}(\tilde{s}^{k})=\tilde{s}^{m}$ but where
$D^{\tilde{s}^{k}}$ is a later decision than $D^{\tilde{s}^{m}}$,
contradicting Lemma 28(b). ∎
###### Lemma 30 (Basic properties of a tree in a soluble ID).
Let $T$ be a tree on a soluble ID graph. Then:
1. (a)
(Decisions in descendant systems are descendants) If $s^{\prime}$ is a
descendant system of $s$, then $D^{s^{\prime}}$ is a descendant node of
$D^{s}$.
2. (b)
(Info links to ancestor decisions only from obsnodes) Let $s^{\prime}$ be a
descendant system of $s$. If there is a link from a node $V$ in any path in
$s^{\prime}$ to any node in ${\bm{D}}\cap\textbf{Anc}(D^{s})$ (including
$D^{s}$), then either: i) $V$ is an obsnode in $s^{\prime}$, or ii)
$V=X^{s^{\prime}}=X^{s}$.
###### Proof.
We prove each property in succession:
Sublemma (a) : (Decisions in descendant systems are descendants) If
$s^{\prime}$ is a child system of $s$, then $D^{s^{\prime}}$ is a descendant
of $D^{s}$ by Lemma 28(b) (since it cannot lie in the back section by Lemma
28(a)). By induction the result follows: If any system $s^{\prime}$ with child
system $s^{\prime\prime}$ is a descendant system of $s$, then
$D^{s^{\prime\prime}}\in\textbf{Desc}(D^{s^{\prime}})$, and by the induction
assumption we know $D^{s^{\prime}}\in\textbf{Desc}(D^{s})$, so that
$D^{s^{\prime\prime}}\in\textbf{Desc}(D^{s})$.
Sublemma (b) : (Only info links from obsnodes to ancestor decisions) Since
$\mathrm{info}^{s}$ is active, and each $\mathrm{obs}^{s}(C)$ is a minimal
length path, the only parents of $D^{s^{\prime}}$ within system $s^{\prime}$
(i.e. the only nodes in
$\mathrm{\mathbf{Pa}}(D^{s^{\prime}})\cap{\bm{V}}^{s^{\prime}}$) are
$X^{s^{\prime}}$ and the obsnodes of $s^{\prime}$. Therefore, by Lemma 28(c)
and using Sublemma (a) that $D^{s}$ is an ancestor of $D^{s^{\prime}}$ (since
$s$ is an ancestor of $s^{\prime}$), these are the only nodes in $s’$ that can
be parents of $D^{s}$ or of ancestor decisions of $D^{s}$.
To show the result we show that $X^{s^{\prime}}$ cannot be such a parent when
$X^{s^{\prime}}\neq X^{s}$: Let $s^{*}$ be the closest-to-$s^{\prime}$
ancestor of $s^{\prime}$ in the tree such that $X^{s*}\neq X^{s^{\prime}}$.
Assume such $s^{*}$ exists and either equals $s$ or is a descendant of $s$
since otherwise $X^{s}$ would equal $X^{s^{\prime}}$, which would show the
result. We know that $X^{s^{\prime}}$ is in the system $s^{*}$, since it is
the closest-to-$s^{\prime}$ system such that $X^{s*}\neq X^{s^{\prime}}$, so
that there is a child system of $s^{*}$ whose info node equals
$X^{s^{\prime}}$ and hence must be part of an info-link in $s^{*}$. Hence
$X^{s^{\prime}}$ cannot be a parent of $D^{s^{*}}$ since the only parents
within a system of that system’s decision other than its info node are its
obsnodes, but $X^{s^{\prime}}$ cannot be one of the obsnodes since then
$D^{s^{\prime}}$ would have to be in the back section, which would violate
Lemma 28(a). But we assumed $s^{*}$ is a descendant system of $s$, and hence
$D^{s^{*}}$ is a descendant decision of $D^{s}$ (by Sublemma (a)), which
implies that $D^{s}$ and its ancestor decisions also don’t have
$X^{s^{\prime}}$ as a parent (due to the result shown in the previous
paragraph). ∎
### C.3 Normal form trees of systems
In this section, we will prove that in a _normal form tree_ , a system can
only get information from its own parents, and obsnodes of descendant systems.
See 18
We will also use the components of the definition of normal form tree
separately.
###### Lemma 31 (Concrete properties of position-in-tree-uniqueness).
A tree $T$ satisfied position-in-tree-uniqueness if and only if every node $N$
that is in some path of some system in $T$ lies in precisely one path $p$ of
one system $s$, with four exceptions:
* •
If $N$ is a collider node in path $p=\mathrm{info}^{s}$ then it is also the
first node in $\mathrm{obs}^{s}(N)$.
* •
If $N=U^{s}$ then it lies in both $\mathrm{info}^{s}$ and
$\mathrm{control}^{s}$.
* •
If $N$ is in an infolink $X^{s^{\prime}}\to D^{s^{\prime}}$ (with
$s^{\prime}\neq s$) on path $p$, then $N$ is also in
$\mathrm{info}^{s^{\prime}}$ (if $N=X^{s^{\prime}}$), or also in
$\mathrm{control}^{s^{\prime}}$ and in $\mathrm{obs}^{s^{\prime}}(C^{i})$ for
each collider $C^{i}$ in $\mathrm{info}^{s^{\prime}}$ (if $N=D^{s^{\prime}}$).
In both cases $N$ may also be the info node for exactly one of its child
systems $s^{1}$, of exactly one child system $s^{2}$ of $s^{1}$, and so on.
Formally, $N=X^{s^{1}}=...=X^{s^{n}}$ where each $s^{i}$ is a child system of
$s^{i-1}$.
###### Proof.
First we show that if a tree $T$ satisfies position-in-tree-uniqueness, then
the result is true. Assume that a tree $T$ does not satisfy the required
property, i.e. there is at least one node $N$ that is in multiple paths, but
without satisfying one of the exceptions. Then by Lemma 37 a different tree
$T^{\prime}$ can be obtained by applying graph transformation 1 (Definition
36), where $N$ is replaced with different nodes in those paths.
Now we show the other direction. Assume that the property holds. Assume $N$ is
part of two paths $p^{1}$ and $p^{2}$. Then one of the three exceptions must
apply. If the first exception applies, then $N$ is a collider, and
$p^{1}=\mathrm{info}^{s}$ and $p^{2}=\mathrm{obs}^{s}(N)$, so that replacing
$N$ with two separate nodes on $p^{1}$ and $p^{2}$ would make that the obspath
of $N$ no longer starts with a collider on $\mathrm{info}^{s}$. If the second
one applies, then replacing $N=U^{s}$ with two nodes would mean that the
control and info path no longer end at the same utility node. If the third
case applies and $N=X^{s^{\prime}}$ of a descendant system $s^{\prime}$ of
$s$, then $p^{1},p^{2}$ equal $\mathrm{info}^{s^{\prime}}$ and
${\mathrm{predpath}}^{s^{\prime}}$. Replacing $N$ with two nodes on the two
paths would break the required property on $\mathrm{pred}$ for $T$ to be a
tree. If the third case applies and $N=D^{s^{\prime}}$, then $p^{1},p^{2}$
equal two of: $\mathrm{control}^{s^{\prime}}$,
${\mathrm{predpath}}^{s^{\prime}}$ or one of $\mathrm{obs}^{s^{\prime}}$. If
one of them equals ${\mathrm{predpath}}^{s^{\prime}}$, then replacing $N$ with
two nodes would again break the required property on $\mathrm{pred}$ for $T$
to be a tree. Otherwise, it would mean that at least one of the
$\mathrm{obs}^{s^{\prime}}$ no longer ends at $D^{s^{\prime}}$, so that
$s^{\prime}$ would no longer be a system. ∎
###### Definition 32 (Base system and path of a node; chain of systems).
If $T$ is a normal form tree of systems, then we refer to the system $s$ and
the path $p$ from Lemma 31 (including in the exceptions) respectively as the
_base system_ and _base path_ of node $N$.
Note that this implies that a utility node $U^{s}$ has no base path. We refer
to the sequence of systems of which a node $N$ is the info node (in the third
exception) as the _chain of systems_ of $N$ (which is possibly empty).
###### Definition 33.
A within-tree-$T$ path for a normal form tree $T$ on an ID graph is a path
that contains only within-system links for the systems in $T$.
Note that we define the notion of within-tree path only for normal form trees,
since it is not sensible for trees that don’t satisfy position-in-tree-
uniqueness: If a node $N$ occurs in two unrelated systems, then a sequence of
within-tree links may jump between nodes in the tree that are not linked.
### C.4 Properties of normal form trees of systems
In this subsection, we will prove Lemma 20 — that the only information that
$D^{s}$ receives that is relevant within system $s$ is information that it
receives from its parents and obsnodes of descendant systems. To reach this
result, we first need to state some more fundamental properties of normal form
trees.
###### Lemma 34 (Properties of soluble ID graphs with trees that have
position-in-tree-uniqueness).
Any soluble ID graph ${\mathcal{G}}$ with a tree that has position-in-tree-
uniqueness has the following properties.
1. (a)
(A within-tree path corresponds to a walk with node repetition in the tree of
systems) For any within-tree path $p\colon N^{1}\;\hbox{- - -}\;N^{n}$, there
is a walk with node repetition in the tree of systems
$p_{\mathrm{systems}}\colon s^{1}\;\hbox{- - -}\;s^{m}$, with $m\geq n$,
together with a walk with node repetitions $p_{\mathrm{walk}}\colon
V^{1}\;\hbox{- - -}\;V^{m}$ in ${\mathcal{G}}$ such that each $V^{i}$ is in
some path in system $s^{i}$ and if we remove from $p_{\mathrm{walk}}$ every
node that equals its predecessor we obtain $p$.666Hence in particular, there
can only be a within-tree path between a node $N^{1}$ in system $s$ and node
$N^{n}$ in system $s^{\prime}$ if there is a path between $s$ and $s’$ in the
tree of systems.
2. (b)
(Within-tree links between systems only via $X^{s}$, $D^{s}$) If
$N-N^{\prime}$ is a within-tree link, where $N$ and $N^{\prime}$ are in nodes
in paths of systems $s$ and $s^{\prime}$ respectively, and $s\neq s^{\prime}$,
then $N-N^{\prime}$ must contain $X^{s}$ or $D^{s}$.
###### Proof.
We prove each sublemma in succession:
Sublemma (a) : (Within-tree paths correspond to walks with node repetition in
the tree of systems). We construct this walk with node repetition
$p_{\mathrm{systems}}$ recursively as follows, by iterating from $N^{1}$ to
$N^{n}$, using the fact that each link in $p$ is within-system for some system
(see definition of within-tree paths). For the base case, let $s^{1}$ equal
any of the systems that $N^{1}$ is a node in. Let $s^{k+1}$ and $V^{k+1}$ be
defined mutually based on $s^{k}$ and $V^{k}$: If the node $N’$ that is next
to $V^{k}$ on $p$ is also in system $s^{k}$, then let $s^{k+1}=s^{k}$ and let
$V^{k+1}=N’$, in which case the desired result follows that $s^{k}=s^{k+1}$
and that $V^{k}-V^{k+1}$ is a link in $p$. If it is not also in system
$s^{k}$, then by definition of within-tree path, $N’$ and $V^{k}$ are both in
some system $s’\neq s^{k}$, where $s’$ is part of the chain of systems of
$V^{k}$. Then let $s^{k+1}$ be the next system from $s^{k}$ in that chain, and
let $V^{k+1}=V^{k}$, from which the desired result follows that there is a
link $s^{k}-s^{k+1}$ and $V^{k+1}=V^{k}$. Together with the base case this
shows the result by induction.
Sublemma (b) : (Within-tree links between systems only via $X^{s}$, $D^{s}$).
Take any link $A-B$ with $A$ a part of $s$ and $B$ a part of some other system
$s’$. Then we must either have that $A$ is in both $s$ and in
${\mathrm{predsys}}(s)$, or that $B$ is in both $s$ and
${\mathrm{predsys}}(s)$. Whichever it is, by the position-in-tree-uniqueness
assumption, this can only be if that node equals $X^{s}$ or $D^{s}$, since any
node that is in multiple systems $s’$ must equal either $X^{s^{\prime}}$ or
$D^{s^{\prime}}$ for all systems $s^{\prime}$ except its base system. ∎
We now show graphically that in an ID graph with normal form tree a decision
$D^{s}$ cannot get relevant information about system $s$ from any paths via
nodes outside system $s$ and descendant systems. This will imply the
following:
See 20 color=blue!30]Ryan: graph knowledge lemma numbered as 43 here, but as
20 in the main paper
inline, color=green!30]Chris: I should try to simplify the proof below
further.
###### Proof.
Take any path from a node in $\bf{Back}^{s}$ to a node in
$\mathrm{\mathbf{Pa}}^{-s}\setminus\bf{{ObsDesc}}^{s}$. We will show that it
is inactive given $\mathrm{\mathbf{Pa}}^{s}\cup\bf{{ObsDesc}}^{s}$.
We first assume that the path starts from a node in the back section, so that
it is an ancestor of $D^{s}$. First note that since the decision of $s$ and
those of its descendant systems cannot be ancestors of $D^{s}$ (Lemma 30(a)),
this implies that if the path contains any of these it is necessarily inactive
given $\mathrm{\mathbf{Pa}}^{s}\cup\bf{{ObsDesc}}^{s}$, since active paths
between ancestors given ancestors contain only ancestors (Lemma 22).
So assume that the path does not contain the decision of $s$ (i.e. $D^{s}$)
nor those of its descendant systems. We will consider the initial within-tree
segment of the path.
By Lemma 34(a), this initial within-tree path corresponds to a walk with node
repetition in the tree of systems, and since the latter has a tree structure
by construction, this initial within-tree path either has to exit system $s$
via a node in its predecessor system, or stay within $s$ itself and its
descendant systems. The former can only happen via one of the links via
$X^{s}$ and $D^{s}$ by Lemma 34(b).
We first show that in this case, the path is blocked given
$\mathrm{\mathbf{Pa}}^{s}\cup\bf{{ObsDesc}}^{s}$. We already assumed that the
path doesn’t contain $D^{s}$, so assume that the link contains $X^{s}$. Since
$\mathrm{info}^{s}$ is front-door by the appropriateness assumption of normal
form tree, $X^{s}$ blocks the path, since
$X^{s}\in\mathrm{\mathbf{Pa}}^{s}\cup\bf{{ObsDesc}}^{s}$.
So we now assume that the initial within-tree segment does not exit into the
predecessor of $s$, and hence is contained within system $s$ and its
descendant systems. Consider the first link of the path that is out-of-tree.
At the start of this proof we assumed that the path doesn’t contain the
decisions $D^{s}$, nor $D^{s^{\prime}}$ of any of its ancestors $s^{\prime}$.
Hence by Lemma 28(a), the only decision that the initial within-tree segment
could contain is $X^{s}$ if that is a decision, but we just assumed that the
path doesn’t contain this.
So we assume now that the initial within-tree segment doesn’t contain any
decisions, so that the first out-of-tree link would have to be of the form
$N\to D$ for some decision $D$ (by the no redundant links assumption of normal
form trees, and using the fact that $N$ is inside the tree). $N$ can be either
in system $s$ or in a descendant system, and can be either an observation node
or some other node. Consider two exhaustive cases:
1. (a)
Assume $N$ is neither an obsnode in $s$ nor in a descendant system $s’$. Then
$D$ cannot be an ancestor of $D^{s}$, since the only info links from nodes in
$s$ or its descendant systems to $D^{s}$ or to ancestor decisions are from
obsnodes by Lemma 30(b), and hence the path cannot be active (active paths
between ancestors given ancestors contain only ancestors by Lemma 22).
2. (b)
Assume $N$ is an observation node of $s$ or of some descendant system of $s$.
Then $N$ blocks the path, since it is in
$\mathrm{\mathbf{Pa}}^{s}\cup\bf{{ObsDesc}}^{s}$.
This shows the result. ∎
### C.5 Obtaining a (homomorphically) transformed ID graph with a normal form
tree
We will prove that if an infolink is in the minimal $d$-reduction, then there
exists a transformed ID graph with a normal form tree and homomorphism to the
original. We will show that a series of three homomorphic transformations
yields a graph ${\mathcal{G}}^{3}$ with tree $T^{3}$ is in normal form, and
root infolink corresponding to that of $T$. Since each transformation is
homomorphic, their composition is a homomorphism from ${\mathcal{G}}^{3}$ to
${\mathcal{G}}$. The transformations are:
* •
First, we _obtain a full tree_ on ${\mathcal{G}}$.
* •
_Transformation 1_ obtains $({\mathcal{G}}^{1},T^{1})$, where $T^{1}$ has
property (a). This splits nodes other than $X^{s}$ and $D^{s}$, to ensure that
they do not appear in multiple positions in the tree.
* •
_Transformation 2_ obtains $({\mathcal{G}}^{2},T^{2})$, where $T^{2}$ has the
properties (a, b). This is done by modifying any backdoor infopath to be
front-door.
* •
_Transformation 3_ obtains $({\mathcal{G}}^{3},T^{3})$, where $T^{3}$ has the
properties (a, b, c). This consists of removing edges other than the within-
tree links.
We will not use the intermediate graphs ${\mathcal{G}}^{1},{\mathcal{G}}^{2}$,
except to define ${\mathcal{G}}^{3}$.
#### Obtain a full tree on ${\mathcal{G}}$
We will construct an arbitrary full tree using only infolinks in the minimal
$d$-reduction.
###### Lemma 35 (Existence of full tree).
Let ${\mathcal{G}}$ be a soluble ID graph whose minimal $d$-reduction
${\mathcal{G}}^{*}$ contains the link $X\to D$. Then there exists a full tree
of systems $T$ on ${\mathcal{G}}^{*}$ with root system on $X\to D$.
###### Proof.
We construct a tree iteratively. Since $X\to D$ is in ${\mathcal{G}}^{*}$,
there exists a directed path $p$ from $X$ to some $U\in\textbf{Desc}(D)$
active given $\mathrm{\mathbf{{Fa}}}(D^{i})\setminus\\{X\\}$. Let the infopath
be any such $p$, let the control path be any directed path from $D$ to $U$ and
let the obspaths be the shortest directed paths to $D$ from each collider in
$p$. Then, choose any infolink $X^{\prime}\to D^{\prime}$ in a path $q$ of any
system $s$ that lacks an associated system. Since $X^{\prime}\to D^{\prime}$
is in ${\mathcal{G}}^{*}$, we can choose paths in the same fashion and repeat
this procedure until every infolink that is traversed has its own system. This
process halts, because a path in a system $s$ (whose decision is $D^{s}$) can
only contain an infolink $X^{\prime}\to D^{\prime}$ if
$D^{\prime}\in\textbf{Desc}(D^{s})$. This is because: i)
$\mathrm{control}^{s}$ is directed, ii) $\mathrm{info}^{s}$ only contains
infolinks in $\textbf{Desc}(D^{s})$ (Lemma 28(b)), and iii) $\mathrm{obs}^{s}$
cannot contain any infolinks. So a full tree has been constructed. ∎
#### Transformation 1 (split): ensuring position-in-tree-uniqueness
For Transformation 1, a node $N$ is copied into a different node (of unchanged
type) for each position that $N$ occupies in the tree. More precisely, we
replace each node $N$ that is in path $p$ in system $s$, with the new node
$\mathrm{New}(N,s,p)$. This function is defined such that each node
$\mathrm{New}(N,s,p)$ has a unique position in the tree, which basically means
that it is only a part of path $p$ in system $s$, except that we need to make
sure that certain nodes are in multiple paths (e.g. a collider node $C$ in a
path $\mathrm{info}^{s}$ must be in both $\mathrm{info}^{s}$ and in
$\mathrm{obs}^{s}(C)$). We don’t delete the original occurrences of each node
$N$, so that the original graph is a subgraph of the transformed graph.
color=blue!30]Ryan: Seems we should be able to remove the name $Nsplit$ and
just substitute in New()
###### Definition 36 (Graph transformation 1).
Let $T^{0}=({\mathcal{S}}^{0},\mathrm{pred}^{0})$ be a tree on an ID graph
${\mathcal{G}}^{0}=({\bm{V}}^{0},E^{0})$. Then define
$\mathrm{transf}^{1}({\mathcal{G}}^{0},T^{0})=({\mathcal{G}}^{1},T^{1})$ where
${\mathcal{G}}^{1}=({\bm{V}}^{1},E^{1})$, together with homomorphism
$h^{0\leftarrow 1}\colon{\bm{V}}^{1}\to{\bm{V}}^{0}$ as
* •
Obtain any ${\mathcal{G}}^{1}$ and $h^{0\leftarrow 1}$ from Lemma 13, by
adding for each node $N$ a set of copies
$\mathrm{Copies}(N)=\\{N_{\mathrm{split}}\mid\textnormal{$\exists
s\in{\mathcal{S}}^{0},\exists$ path $p\in s$ such that $N\in p$, and
$N_{\mathrm{split}}=\mathrm{New}(N,s,p)$ }\\},$
where by tree recursion on $T^{0}$ (which has tree structure: Lemma 29) we
define $\mathrm{New}(N,s,p)$
$=\begin{cases}\mathrm{New}(N,{\mathrm{predsys}}(s),{\mathrm{predpath}}(s))&\textnormal{if
$s\neq\mathrm{root}(T^{0})$ and $N\in\\{X^{s},D^{s}\\}$}\\\
(N,s,\\{\mathrm{info}^{s},\mathrm{control}^{s}\\})&\textnormal{if
$N=U^{s}$}\\\ (N,s,\mathrm{info}^{s})&\textnormal{if
$p=\mathrm{obs}^{s}(N)$}\\\ N&\textnormal{if $s=\mathrm{root}(T^{0})$ and
$N\in\\{X^{s},D^{s}$\\}}\\\ (N,s,p)\quad&\textnormal{otherwise }\end{cases}.$
* •
$T^{1}$ is the tree $({\mathcal{S}}^{1},\mathrm{pred}^{1})$, where the system
$s_{\mathrm{split}}^{i}\in{\mathcal{S}}^{1}$ is defined as the system
$(\mathrm{split}(s^{i},\mathrm{info}^{T,s^{i}}),\mathrm{split}(s,\mathrm{control}^{T,s^{i}}),\mathrm{split}(s,\mathrm{obs}^{T,s}))$
for $s^{i}\in{\mathcal{S}}^{T}$, where
$\mathrm{split}(s^{i},p)^{j}=\mathrm{New}(p^{j},s^{i},p),$ and where $p^{j}$
denotes the $j$’th node of a path $p$. (this indeed gives a path, since there
is an edge between $\mathrm{split}(s^{i},p)^{j}=\mathrm{New}(p^{j},s^{i},p)$
and $\mathrm{split}(s^{i},p)^{j+1}=\mathrm{New}(p^{j+1},s^{i},p)$ because
there is an edge between $p^{j}$ and $p^{j+1}$ and by definition of $E^{1}$
using $p^{j}\neq p^{j+1}$). Moreover $\mathrm{pred}^{1}$ is the same as
$\mathrm{pred}^{0}$ except that each $s$ in $T^{0}$ is replaced with its
transformed $s_{\mathrm{split}}$.
###### Lemma 37 (Transformation 1 preserves tree).
Let $({\mathcal{G}}^{1},T^{1})=\mathrm{transf}^{1}({\mathcal{G}}^{0},T^{0})$.
If $T^{0}$ is a tree of systems on ${\mathcal{G}}^{0}$ with root link $X\to
D$, then $T^{1}$ is a tree of systems on ${\mathcal{G}}^{1}$ with root link
$X^{\prime}\to D^{\prime}$ with $h^{0\leftarrow 1}(X^{\prime})=X$ and
$h^{0\leftarrow 1}(D^{\prime})=D$.
###### Proof.
First we show that $T^{1}$ satisfies the three conditions of a tree of
systems: (1) We will show below that each indexed element of
${\mathcal{S}}^{1}$ is indeed a system. (2) since
$\mathrm{pred}^{0}=\mathrm{pred}^{1}$, and since $T^{0}$ is a tree of systems,
the required condition on $\mathrm{pred}^{1}$ is satisfied. (3) We show that
each system’s infolink is an infolink on its predecessor path: The nodes
$X^{s_{\mathrm{split}}}$ and $D^{s_{\mathrm{split}}}$ in ${\mathcal{G}}^{1}$
equal $\mathrm{New}(X^{s},s,\mathrm{info}^{s})$ and
$\mathrm{New}(D^{s},s,\mathrm{control}^{s})$ for $X^{s}$ and $D^{s}$ in
${\mathcal{G}}^{0}$. By definition of $\mathrm{New}(N,s,p)$, this is indeed an
infolink on ${\mathrm{predpath}}(s)$.
It remains to be shown that $s_{\mathrm{split}}$ indeed is a system for each
$s\in{\mathcal{S}}^{0}$:
Step (1) : We show that $\mathrm{control}^{s_{\mathrm{split}}}$ is a directed
path to a utility node. $\mathrm{control}^{s_{\mathrm{split}}}$ is a path from
$\mathrm{New}(D^{s},{s},\mathrm{control}^{s})$ to
$\mathrm{New}(U^{s},{s},\mathrm{control}^{s})$ and by definition of $E^{1}$
this is indeed a directed path (since $\mathrm{control}^{s}$ is directed in
$T$);
Step (2) : We show that $\mathrm{info}^{s_{\mathrm{split}}}$ is an active path
to the same utility node. Firstly, $\mathrm{info}^{s_{\mathrm{split}}}$ is a
path from $\mathrm{New}(X^{s},{s},\mathrm{info}^{s})$ to
$\mathrm{New}(U^{s},{s},\mathrm{info}^{s})$, and since
$\mathrm{New}(U^{s},{s},\mathrm{info}^{s})=\mathrm{New}(U^{s},{s},\mathrm{control}^{s})$
by definition of $\mathrm{New}$ for utility nodes, therefore the control and
info path indeed end at the same utility node. Secondly, it follows easily
from the definition of $E^{1}$, that a node
$\mathrm{New}(N,s,\mathrm{info}^{s})$ blocks the path if and only if $N$
blocks $\mathrm{info}^{s}$, and the latter is active by assumption, so that
$\mathrm{info}^{s_{\mathrm{split}}}$ is active as well;
Step (3) : Finally, we show that the $\mathrm{obs}^{s_{\mathrm{split}}}$ are
minimal length paths from collider nodes on
$\mathrm{info}^{s_{\mathrm{split}}}$ to $D^{s_{\mathrm{split}}}$. Firstly,
$\mathrm{obs}^{s_{\mathrm{split}}}(\mathrm{New}(C,s,\mathrm{info}^{s}))$ is a
path from $\mathrm{New}(C,s,\mathrm{obs}^{s}(C))$ to
$\mathrm{New}(D^{s},s,\mathrm{obs}^{s}(C))$. By definition of $L$, the former
equals $(C,(s,\mathrm{info}^{s}))$ and the latter equals
$\mathrm{New}(D^{s},{\mathrm{predsys}}(s),{\mathrm{predpath}}(s))$ if
$s\neq\mathrm{root}(T)$ and $(D^{s},(s,\mathrm{control}^{s}))$ if
$s=\mathrm{root}(T)$, which in both cases equals
$\mathrm{New}(D^{s},s,\mathrm{control}^{s})=D^{s_{\mathrm{split}}}$, so that
this is indeed a valid obspath. To show that it’s minimal length, assume by
contradiction that there is a shorter path and denote its $j$’th node by
$(N^{j},(s^{j},p^{j}))$, so that there are links
$(N^{j},s^{j},p^{j})\to(N^{j+1},s^{j+1},p^{j+1})$. Then by definition of
$E^{1}$, ${\mathcal{G}}^{0}$ contains an edge $N^{j}\to N^{j+1}$, and hence
this path in ${\mathcal{G}}^{0}$ must also be shorter than
$\mathrm{obs}^{s}(C)$, contradicting the assumption that $s$ is a system.
This shows that $T^{1}$ is a tree of systems. Finally, The root infolink of
$T^{1}$ is $X\to D$ by definition of $\mathrm{New}$, and $h^{0\leftarrow
1}(X)=X$ and $h^{0\leftarrow 1}(D)=D$, so it is mapped to the root info link
of $T$. ∎
###### Lemma 38 (Transformation 1 ensures position-in-tree-uniqueness).
Let $({\mathcal{G}}^{0},T^{0})$ be any soluble ID graph with complete tree.
Then $({\mathcal{G}}^{1},T^{1})=\mathrm{transf}^{1}({\mathcal{G}}^{0},T^{0})$
is an ID graph with complete tree that satisfies (a) position-in-tree-
uniqueness.
###### Proof.
We first show that the split preserves fullness, then that it ensures
position-in-tree-uniqueness.
(i) (full tree). color=green!30]Chris: my sense is this subproof can be
simplified, it seems very long for what it’s doing. Basically it’s a ”by
definition” kind of proof. We show that if ${T^{0}}$ is a full tree, then so
is $T^{1}$: Let $(X,s^{*,1},p^{*,1})\to(D,s^{*,2},p^{*,2})$ be an infolink in
${\mathcal{G}}^{1}$ on the path $p_{\mathrm{split}}$ in system
$s_{\mathrm{split}}$. We need to show that there is a system
$s_{\mathrm{split}}^{\prime}$ such that
$X^{s_{\mathrm{split}}^{\prime}}=(X,s^{*,1},p^{*,1})$ and
$D^{s_{\mathrm{split}}^{\prime}}=(D,s^{*,2},p^{*,2})$.
Note that this link in $p_{\mathrm{split}}$ implies that there is a
corresponding link in the original path $p$. By the definition of $T^{1}$, the
split path $p_{\mathrm{split}}$ was constructed from the original path $p$
(that has the same path type in system $s$ as $p_{\mathrm{split}}$ does in
$s_{\mathrm{split}}$) where if $(X,s^{*,1},p^{*,1})$ is the $i$’th node on
$p_{\mathrm{split}}$, it corresponds to the $i$’th node on $p$ by
$(X,s^{*,1},p^{*,1})=p_{\mathrm{split}}^{i}=\mathrm{New}(p^{i},s,p)$ and
similarly
$(D,s^{*,2},p^{*,2})=p_{\mathrm{split}}^{i+1}=\mathrm{New}(p^{i+1},s,p)$.
Hence $X=p^{i}$ and $D=p^{i+1}$, so that $X\to D$ is also an infolink on $p$
in $s$ in ${T^{0}}$. And since by assumption ${T^{0}}$ is full, there is a
system $s^{\prime}$ with $\mathrm{pred}^{0}(s^{\prime})=(s,p)$ such that
$X^{s^{\prime}}\to D^{s^{\prime}}$ equals $X\to D$. This implies also that the
desired system in $T^{1}$ exists: Since by definition $\mathrm{pred}^{1}$ is
equivalent to $\mathrm{pred}^{0}$ it implies that
$\mathrm{pred}^{1}(s_{\mathrm{split}}^{\prime})=(s_{\mathrm{split}},p_{\mathrm{split}})$,
and by construction of $s_{\mathrm{split}}$, $X^{s_{\mathrm{split}}^{\prime}}$
and $D^{s_{\mathrm{split}}^{\prime}}$ equal
$\mathrm{New}(X^{s^{\prime}},s^{\prime},\mathrm{info}^{s^{\prime}})=\mathrm{New}(X,s^{\prime},\mathrm{info}^{s^{\prime}})$
and
$\mathrm{New}(D^{s^{\prime}},s^{\prime},\mathrm{control}^{s^{\prime}})=\mathrm{New}(D,s^{\prime},\mathrm{control}^{s^{\prime}})$
respectively, which by definition of $\mathrm{New}(X,s,p)$ implies that they
equal $\mathrm{New}(X,s,p)$ and $\mathrm{New}(D,s,p)$ respectively, showing
the result.
(ii) (position-in-tree-uniqueness). Any node $N_{\mathrm{split}}$ in the tree
either equals one of $X,D$, or is a node of the form
$N_{\mathrm{split}}=(N,(s^{*},p^{*}))$. In the former case, let
$s^{*}=\mathrm{root}(T^{1})$ and let $p^{*}=\mathrm{info}^{\mathrm{root}}$ if
the node equals $X$ and $p^{*}=\mathrm{control}^{\mathrm{root}}$ if it equals
$D$. We will show that $s_{\mathrm{split}}^{*}$ and $p_{\mathrm{split}}^{*}$
are the node’s base system and base path respectively, by taking any path
$p_{\mathrm{split}}$ in any system $s_{\mathrm{split}}$ such that the node is
on $p_{\mathrm{split}}$, and showing that for the original path $p$ and system
$s$, either $p=p^{*}$ and $s=s^{*}$ or that one of the exceptions applies.
We will show this by induction on the tree: Assume that it holds for any
$p_{\mathrm{split}}^{\prime}$ in system $s_{\mathrm{split}}^{\prime}$ that is
an ancestor system of $s_{\mathrm{split}}$. Note that if
$N_{\mathrm{split}}=(N,(s^{*},p^{*}))$ is on path $p_{\mathrm{split}}$ in
system $s_{\mathrm{split}}$, then $N_{\mathrm{split}}=\mathrm{New}(N,s,p)$, so
consider two cases of the definition of $\mathrm{New}(N,s,p)$ separately:
Case (1) : (Assume $s\neq\mathrm{root}(T^{0})$ and $N\in\\{X^{s},D^{s}\\}$.)
Then
$N_{\mathrm{split}}=(N,s^{*},p^{*})=\mathrm{New}(N,s,p)=\mathrm{New}(N,{\mathrm{predsys}}(s),{\mathrm{predpath}}(s))$,
and we will use the induction assumption on
${\mathrm{predsys}}(s_{\mathrm{split}}),{\mathrm{predpath}}(s_{\mathrm{split}})$:
We know that $(N,s^{*},p^{*})$ lies on
${\mathrm{predpath}}(s_{\mathrm{split}})$ (since $T^{1}$ is a tree), and by
the induction assumption, either
${\mathrm{predsys}}(s_{\mathrm{split}})=s_{\mathrm{split}}^{*}$ and
${\mathrm{predpath}}(s_{\mathrm{split}})=p_{\mathrm{split}}^{*}$ (in which
case the third or fourth exception applies to $s_{\mathrm{split}}$ and
$p_{\mathrm{split}}$, showing the result), or one of the exceptions applies.
Since $X^{s}$ and $D^{s}$ aren’t utility nodes, and can’t be colliders on the
info path of ${\mathrm{predsys}}(s)$ (Lemma 28(a)), and ${\mathrm{predpath}}$
always is either an info or control path, only the third exception can apply
to ${\mathrm{predsys}}(s_{\mathrm{split}})$ and
${\mathrm{predpath}}(s_{\mathrm{split}})$, i.e. $(N,s^{*},p^{*})$ is the info
or decision node of ${\mathrm{predsys}}(s_{\mathrm{split}})$, where the latter
is a child system of $s_{\mathrm{split}}^{*}$ or it is the info node of an
unbroken chain of descendant systems of $s_{\mathrm{split}}^{*}$. In both
cases, $N$ cannot equal $D^{s}$, since the decision node of a system
(${\mathrm{predsys}}(s_{\mathrm{split}})$ in this case) is neither the
decision of an info link on its info path nor on its control path and hence
cannot equal the decision node of one of its child systems
($s_{\mathrm{split}}$ in this case), and hence $(N,s^{*},p^{*})$ must be the
info node of $s_{\mathrm{split}}$ and an unbroken chain of predecessor systems
between $s_{\mathrm{split}}$ and $s_{\mathrm{split}}^{*}$, so that it
satisfies the third exception.
Case (2) : (Assume $s=\mathrm{root}(T^{0})$ or $X\notin\\{X^{s},D^{s}\\}$.)
Then if $s=\mathrm{root}(T^{0})$ and $N\in\\{X^{s},D^{s}\\}$, then
$N_{\mathrm{split}}=N$, and the result follows easily (where
$p_{\mathrm{split}}$ may be an obs path in which case the final exception
applies). So assume otherwise, so that
$N_{\mathrm{split}}=(N,s^{*},p^{*})=\mathrm{New}(N,s,p)$, where $s^{*}=s$, and
we can easily match each of the cases of $\mathrm{New}(N,s,p)$ to the
exceptions, showing the result. ∎
#### Transformation 2 (split): ensuring no backdoor info-paths
In the second transformation, we turn any backdoor-info paths into frontdoor
infopaths.
###### Definition 39 (Transformation 2).
Let ${\mathcal{G}}^{1}=({\bm{V}}^{1},E^{1})$ be an ID graph with tree
$T^{1}=({\mathcal{S}}^{1},\mathrm{pred}^{1})$. Then
$\mathrm{transf}^{2}({\mathcal{G}}^{1},T^{1})=({\mathcal{G}}^{2},T^{2})$,
where ${\mathcal{G}}^{2}=({\bm{V}}^{2},E^{2})$ and $h^{1\leftarrow
2}\colon{\bm{V}}^{2}\to{\bm{V}}^{1}$ are defined as follows:
* •
Obtain any ${\mathcal{G}}^{2}$ and $h^{1\leftarrow 2}$ from Lemma 13, by
adding for each node $N$ a set of copies
$\mathrm{Copies}(N)=\begin{cases}\\{(N,``\mathrm{copy}",s)\\}&\textnormal{if
$N=X^{s}$ for some backdoor-info system $s$ in ${T^{1}}$}\\\
\emptyset\quad&\textnormal{otherwise }\end{cases}$
* •
$T^{2}$ is the tree $({\mathcal{S}}^{2},\mathrm{pred}^{2})$, where each system
$s_{\mathrm{split}}\in{\mathcal{S}}^{2}$ is obtained from $s$ by replacing the
first link $X^{s}\leftarrow N$ in $\mathrm{info}^{s}$ with the links
$X^{s}\to(X^{s},``\mathrm{copy}")\leftarrow N$, and extending
$\mathrm{obs}^{s}$ with $\mathrm{obs}^{s}((X^{s},``\mathrm{copy}"))$ to be the
path consisting of the single link $(X^{s},``\mathrm{copy}",s)\to D^{s}$.
Moreover $\mathrm{pred}^{2}$ is the same as $\mathrm{pred}^{1}$ except that
each $s$ in $T^{1}$ is replaced with its transformed $s_{\mathrm{split}}$.
###### Lemma 40 (Transformation 2 preserves tree).
Let $({\mathcal{G}}^{2},T^{2})=\mathrm{transf}^{2}({\mathcal{G}}^{1},T^{1})$.
If $T^{1}$ is a tree of systems on ${\mathcal{G}}^{1}$ with root link $X\to
D$, then $T^{2}$ is a tree of systems on ${\mathcal{G}}^{2}$ with root link
$X^{\prime}\to D^{\prime}$ with $h^{1\leftarrow 2}(X^{\prime})=X$ and
$h^{1\leftarrow 2}(D^{\prime})=D$.
###### Proof.
First we show that $T^{2}$ satisfies the three conditions of a tree of
systems: (1) We will show below that each indexed element of
${\mathcal{S}}^{2}$ is indeed a system. (2) since $\mathrm{pred}^{1}$ is
equivalent to $\mathrm{pred}^{2}$, and since ${T^{1}}$ is a tree of systems,
the required conditions on $\mathrm{pred}^{2}$ are satisfied. (3) For any
system $s$, $X^{s_{\mathrm{split}}}=X^{s}$ and $D^{s_{\mathrm{split}}}=D^{s}$
(i.e. they are unchanged under the split), and the front-section of
${\mathrm{predsys}}(s_{\mathrm{split}})$ is identical to that of
${\mathrm{predsys}}(s)$, and since infolinks can only be in the front section
(by Lemma 28(a)), so that the fact that ${T^{1}}$ is a tree of systems and
hence has $X^{s}\to D^{s}$ as an infolink on ${\mathrm{predpath}}(s)$, this
implies that $X^{s_{\mathrm{split}}}\to D^{s_{\mathrm{split}}}$ is an infolink
on ${\mathrm{predpath}}(s_{\mathrm{split}})$.
It remains to be shown that $s_{\mathrm{split}}\in{\mathcal{S}}^{2}$ indeed is
a system for each $s\in{\mathcal{S}}^{1}$:
Step (1) : $\mathrm{control}^{s_{\mathrm{split}}}$ is a directed path to a
utility node. $\mathrm{control}^{s_{\mathrm{split}}}$ is identical to
$\mathrm{control}^{s}$.
Step (2) : $\mathrm{info}^{s_{\mathrm{split}}}$ is an active path to the same
utility node. Since $\mathrm{info}^{s}$ is active given
$\mathrm{\mathbf{Pa}}(D^{s})$ by assumption, and
$\mathrm{info}^{s_{\mathrm{split}}}$ is identical to $\mathrm{info}^{s}$
except that if $\mathrm{info}^{s}$ is backdoor from $X^{s}$ then the first
link $X^{s}\leftarrow N$ is replaced by
$X^{s}\to(X^{s},``\mathrm{copy}")\leftarrow N$, hence by definition of
$E^{2}$, a node on $\mathrm{info}^{s_{\mathrm{split}}}$ is a parent of
$D^{s_{\mathrm{split}}}$ in ${\mathcal{G}}^{2}$ iff it is a parent of $D^{s}$
in ${\mathcal{G}}^{1}$ or if it equals $(X^{s},``\mathrm{copy}")$. Therefore,
$(X^{s},``\mathrm{copy}")$ doesn’t block because it’s a collider, and the
other nodes don’t block $\mathrm{info}^{s_{\mathrm{split}}}$ because by
assumption they didn’t block $\mathrm{info}^{s}$.
Step (3) : Finally, $\mathrm{obs}^{s_{\mathrm{split}}}$ are minimal length
paths from collider nodes on $\mathrm{info}^{s_{\mathrm{split}}}$ to
$D^{s_{\mathrm{split}}}$: For
$\mathrm{obs}^{s}_{\mathrm{split}}((X^{s},``\mathrm{copy}"))$, it is a single
link to $D^{s}$ and hence trivially minimal-length, so consider the other
colliders $C$ which are also on $\mathrm{info}^{s}$. Firstly, each
$\mathrm{obs}^{s_{\mathrm{split}}}(C)$ is identical to $\mathrm{obs}^{s}(C)$.
Secondly, the split doesn’t introduce shorter-length such paths, since any
path from $C$ to $D^{s_{\mathrm{split}}}$ via some newly added
$(X,``\mathrm{copy}")$ would correspond to a path via $X^{s}$ in
${\mathcal{G}}^{1}$ that is at least as short, using the fact that this ID
transformation is homomorphic and hence doesn’t introduce extra edges.
This shows that $T^{2}$ is a tree of systems. Finally, we show that the root
infolinks are equivalent: The if $s^{i}$ is the root system of ${T^{1}}$ then
$s_{\mathrm{split}}^{i}$ is the root system of $T^{2}$, and since the infolink
of each system in $T^{2}$ equals that of the corresponding system in ${T^{1}}$
by definition, we have $h^{1\leftarrow
2}(X^{s_{\mathrm{split}}^{\mathrm{root}}})=X^{s^{\mathrm{root}}}$ and
$h^{1\leftarrow
2}(D^{s_{\mathrm{split}}^{\mathrm{root}}})=D^{s^{\mathrm{root}}}$, showing the
result. ∎
###### Lemma 41 (Transformation 2 ensures appropriateness).
Let $({\mathcal{G}}^{1},T^{1})$ be any soluble ID graph with complete tree
satisfying property (a) position-in-tree-uniqueness. Then
$({\mathcal{G}}^{2},T^{2})=\mathrm{transf}^{2}({\mathcal{G}}^{1},T^{1})$ is an
ID graph with complete tree satisfying (a) and also (b) appropriateness.
###### Proof.
We first show that the split preserves fullness, position-in-tree-uniqueness
and no overlapping $X^{s}$, $Q^{\mathrm{pred}}$, then we show that it ensures
appropriateness:
(i) (full tree). We show that if ${T^{1}}$ is a full tree, so is $T^{2}$: For
any link $X\to D$ on a path in a system $s_{\mathrm{split}}$, that infolink
was also on the same path in system $s_{\mathrm{split}}$, since
$s_{\mathrm{split}}$ is identical to $s$ except for the first link on
$\mathrm{info}^{s_{\mathrm{split}}}$ but that link is in the back section and
hence cannot contain $X$ or $D$ by Lemma 28(a). Hence there is a system
$s^{\prime}$ in ${T^{1}}$ with $X\to D$ as its infolink. Hence since by
Definition 39 the infolink of $s_{\mathrm{split}}^{\prime}$ is the same as
that of $s^{\prime}$, there is a system in $T^{2}$ that has $X\to D$ as its
infolink, namely $s_{\mathrm{split}}^{\prime}$.
(ii) (a-position-in-tree-uniqueness). We show that if ${T^{1}}$ satisfies
position-in-tree-uniqueness, then so does $T^{2}$. We state the argument
informally: The nodes in $T^{2}$ are identical to those in ${T^{1}}$, except
for sometimes a split of $X^{s}$. In that case, $(X^{s},``\mathrm{copy}")$ is
a new node that only appears in system $s_{\mathrm{split}}$. Moreover, any
other nodes are precisely in system $s_{\mathrm{split}}$ if they were in
system $s$, so since the node satisfied the required property in ${T^{1}}$, it
also does so in $T^{2}$.
(iii) (b-no-backdoor-infopaths) Take any system
$s_{\mathrm{split}}\in{\mathcal{S}}^{2}$. If $s$ is frontdoor info, then
$s_{\mathrm{split}}$ is identical, so is also frontdoor info. If $s$ is
backdoor-info, then $s_{\mathrm{split}}$ is modified to be frontdoor-info. ∎
#### Transformation 3 (pruning): ensuring no-redundant-links
###### Definition 42 (Transformation 3).
Let ${\mathcal{G}}^{2}\\!=\\!({\bm{V}}^{2},E^{2})$ be an ID graph with tree
$T^{2}$. Then
$\mathrm{transf}^{3}({\mathcal{G}}^{2},T^{2})=({\mathcal{G}}^{3},T^{3})$ where
${\mathcal{G}}^{3}=({\bm{V}}^{3},E^{3})$ and the identity homomorphism
$h^{2\leftarrow 3}$ are obtained from ${\mathcal{G}}^{2}$ using Lemma 14 by
removing all Definition 18(c) (no-redundant-links) links are removed (which
are all into non-decision nodes), and where $T^{3}=T^{2}$.
###### Lemma 43 (Transformation 3 is homomorphic).
$h^{2\leftarrow 3}$ from Definition 42 is an ID homomorphism from
${\mathcal{G}}^{3}$ to ${\mathcal{G}}^{2}$.
###### Lemma 44 (Transformation 3 preserves tree).
Let $({\mathcal{G}}^{3},T^{3})=\mathrm{transf}^{3}({\mathcal{G}}^{2},T^{2})$.
If $T^{2}$ is a tree of systems on ${\mathcal{G}}^{2}$ with root link $X\to
D$, then $T^{3}$ is a tree of systems on ${\mathcal{G}}^{3}$ with root link
$X^{\prime}\to D^{\prime}$ with $h^{2\leftarrow 3}(X^{\prime})=X$ and
$h^{2\leftarrow 3}(D^{\prime})=D$.
###### Proof.
The tree $T^{3}$ is rooted at $X\to D$ such that $h^{2\leftarrow 3}(X)\to
h^{2\leftarrow 3}(D)$ because it is unchanged from $T^{2}$. $T^{3}$ is a tree
of systems because it is unchanged from $T^{2}$, while ${\mathcal{G}}^{3}$
retains every edge in any path of every system of $T^{2}$ — only redundant
links Definition 18(c) (no-redundant-links) are removed. ∎
###### Lemma 45 (Transformation 3 preserves (a,b) and ensures (c)).
Let $({\mathcal{G}}^{2},T^{2})$ be any soluble ID graph with complete tree
satisfying properties (a,b) of normal form trees. Then
$({\mathcal{G}}^{3},T^{3})=\mathrm{transf}^{3}({\mathcal{G}}^{2},T^{2})$ is an
ID graph with complete tree satisfying (a-c).
###### Proof.
The tree $T^{3}$ on ID graph ${\mathcal{G}}^{3}$ satisfies (b) because
$T^{3}\\!=\\!T^{2}$ and $T^{2}$ satisfies (b). It satisfies (a) because
$T^{3}\\!=\\!T^{2}$ and ${\mathcal{G}}^{3}$ has the same set of nodes, and a
subset of the edges of ${\mathcal{G}}^{2}$. color=blue!30]Ryan: Explain this.
It satisfies (c) by definition. ∎
#### Composing the transformations to obtain an ID graph with normal form tree
We will now perform these three transformations in order to obtain a normal
form tree.
See 19
###### Proof.
Given that the minimal $d$-reduction ${\mathcal{G}}^{*}$ of ${\mathcal{G}}$
contains $X\to D$, we can first pick an arbitrary full tree from Lemma 35 to
obtain a tree $T^{0}$ satisfying (a) position-in-tree-uniqueness.
Then, let
$({\mathcal{G}}^{\prime},T^{\prime})\\!=({\mathcal{G}}^{3},T^{3})=\\!\mathrm{transf}^{3}\circ\mathrm{transf}^{2}\circ\mathrm{transf}^{1}({\mathcal{G}}^{0},T^{0})$
and let $h=\\!h^{0\leftarrow 1}\circ h^{1\leftarrow 2}\circ h^{2\leftarrow 3}$
using Definition 36, Definition 39 and Definition 42. We show that these have
each of the desired properties.
Firstly, $T^{\prime}$ is normal form: Each transformation results in a tree
with one more property of normal form trees, and preserves the properties of
the previous transformations (Lemma 38, Lemma 41, Lemma 45).
Secondly, $h$ is a homomorphism from ${\mathcal{G}}^{\prime}$ to
${\mathcal{G}}^{0}$ since ID homomorphism is preserved under composition
(Lemma 15).
Thirdly, ${\mathcal{G}}^{\prime}$ is soluble since that is preserved under ID
homomorphisms (Lemma 9).
Fourthly, each transformation outputs a tree $T^{i}$ where $h^{(i-1)\leftarrow
i}$ maps nodes in the root infolink to nodes of infolink of $T^{i-1}$ (Lemma
37, Lemma 40 Lemma 44), so the composition has $h(X^{\prime})=X^{0}$ and
$h(D^{\prime})=D^{0}$.
Finally, Transformation 1 results in an ID graph with tree where the nodes in
the tree that are also in the original ID graph ${\mathcal{G}}^{0}$ are
precisely $X$ and $D$. And transformations 2-4 only remove and add nodes that
are not in ${\mathcal{G}}^{0}$, so the property also holds for $G^{\prime}$
and $T^{\prime}$, showing the result. ∎
## Appendix D Value of Information criterion completeness
In Appendix C we have shown that if an infolink $X\to D$ is present in the
minimal $d$-reduction of a soluble ID graph ${\mathcal{G}}$, then we can
choose a graph and tree ${\mathcal{G}}^{3},T^{3}$ so that ${\mathcal{G}}^{3}$
is homomorphic to ${\mathcal{G}}$, and $T^{3}$ is in normal form. In this
section, we will prove that we can use $T^{3}$ to parameterise
${\mathcal{G}}^{3}$ so that optimal performance can only be achieved by a
policy that has $\pi^{D}(\mathrm{\mathbf{pa}})=f(x)$ for a specific $f$ given
every $\mathrm{\mathbf{pa}}\in\mathrm{dom}(\mathrm{\mathbf{Pa}}(D))$ with
$P(\mathrm{\mathbf{pa}})>0$.
### D.1 Constructing an ID on nodes in a normal form tree
We will define an ID for only the nodes in the tree, excluding the root info
link, assuming that there is already an ID (possibly trivial) defined for all
the other nodes (including those in the root infolink). This result is more
general than is needed to prove positive VoI (wherein we will chose a trivial
ID) but this is done in order to help with generalizing to the
$\mathrm{Taskify}$ construction in the next section.
###### Definition 46 (Parameterization of a normal form tree).
Let ${\mathcal{G}}^{3}$ be a soluble ID graph together with a normal form tree
of systems $T^{3}$ with root info-link $X\to D$. Let ${\mathcal{G}}^{0}$ be
the subgraph consisting of $X,D$, and all nodes in ${\mathcal{G}}^{3}$ that
are not in $T^{3}$. Let $M^{0}=({\mathcal{G}}^{0},\mathrm{dom}^{0},P)$ be an
ID on on ${\mathcal{G}}^{0}$, and let $\pi_{\mathrm{Task}}^{D}$ (which we call
_the task for $D$_) be a deterministic decision rule for $D$ that depends only
on $X$. Then we define the ID $M^{3}=({\mathcal{G}}^{3},\mathrm{dom},P)$,
which are defined as follows:
* •
For each node $N$ in ${\mathcal{G}}^{0}$ except $D$, let
$\mathrm{dom}(N)=\mathrm{dom}^{0}(N)$, and let
$\mathrm{dom}(D)=\mathrm{dom}_{\mathrm{base}}(D)\times\mathbb{B}$, where
$\mathrm{dom}_{\mathrm{base}}(D)=\mathrm{dom}^{0}(D)$. For any other chance or
decision node $N$, we define the domain of a node by recursion on the tree
$T^{3}$. Let $s$ be the base system of $N$.777This uses the assumption that
$T^{3}$ is normal form, and hence satisfies Definition 18(a) (position-in-
tree-uniqueness), and using the properties that this implies by Lemma 31
Assume that the domains of the info node and decision node $X^{s},D^{s}$ of
$N$’s base system $s$ are already defined.888This is well-founded recursion,
and for the base case of $s=\mathrm{root}^{T}$, the domains of $X^{s}=X$ and
$D^{s}=D$ were already defined above. Then, if $s$ is of the non-directed-info
case then
$\mathrm{dom}_{\mathrm{base}}(N)=\begin{cases}\mathbb{B}&\textnormal{if $N$ is
on $\mathrm{obs}^{s}(C^{1})$, incl. $C^{1}$, the first obspath of $s$}\\\
\mathrm{dom}{(X^{s})}&\textnormal{if $N$ is in between $X^{s}\dashrightarrow
C^{1}$ on $\mathrm{info}^{s}$ }\\\
\mathbb{B}^{|\mathrm{dom}_{\mathrm{base}}{(D^{s})}|}&\textnormal{if $N$ in any
other part of $\mathrm{info}^{s}$, or any other $\mathrm{obs}^{s}$}\\\
\mathrm{dom}(D^{s})&\textnormal{if $N$ in $\mathrm{control}^{s}$}\end{cases},$
and if it is of the directed-info case then
$\mathrm{dom}_{\mathrm{base}}(N)=\mathrm{dom}(X^{s})$. Based on this:
$\mathrm{dom}(N)=\begin{cases}\mathrm{dom}_{\mathrm{base}}(N)\times\mathbb{B}&\textnormal{if
$N=D^{s^{\prime}}$ for a non-directed-info descendant $s^{\prime}$}\\\
\mathrm{dom}_{\mathrm{base}}(N)\quad&\textnormal{otherwise }\end{cases}.$
color=blue!30]Ryan: Relatedly, can we just refer to dom-base with $dom(Q^{s})$
or something, or do we need this term? color=green!30]Chris: We might be able
to get rid of it, but I’m not immediately sure how.
color=blue!30]Ryan: It seems like the domains are implied by the functions and
domains of the parents in all cases other than forks, and the root systems so
can we just define the functions directly? color=green!30]Chris: We maybe
should consider this, but it’s probably not a priority. It maybe has some
downsides in terms of understandability, but my guess is it’d be better if it
can be done clearly and rigorously. Probably the main problem is that decision
nodes need domains too.
* •
For each chance node $N$ in ${\mathcal{G}}^{0}$ (including
$X=X^{s^{\mathrm{root}}}$), let $P_{M^{3}}^{N}=P^{N}_{M^{0}}$. For any other
chance node $N$, let $s$ be the base system of $N$ and $p$ the base path of
$N$,999This uses the assumption that $T^{3}$ is normal form, and thus
satisfies (c) position-in-tree-uniqueness and let
$\pi_{\mathrm{Task}}^{D^{s}}$ be the task of the decision of system $s$,
defined for $s=\mathrm{root}$ as the task $\pi_{\mathrm{Task}}^{D}$ given
above, and as the identity operation _$\mathrm{id}^{p}$_ for every other
system. Then, writing $\pi_{\mathrm{Task}}^{D^{s}}(x)$ to refer to $d$ s.t.
$\pi_{\mathrm{Task}}^{D^{s}}(d|x)=1$, we let $P^{N}(n|\mathrm{\mathbf{pa}})=1$
iff $n=f^{N}(\mathrm{\mathbf{pa}})$, where
$f^{N}=\begin{cases}{\mathrm{id}}^{p}&\textnormal{if $\to N\to$ or $\leftarrow
N\leftarrow$ in $p$}\\\
N^{2}[\pi_{\mathrm{Task}}^{D^{s}}(N^{1})]&\textnormal{if $N^{1}\to N\leftarrow
N^{3}$ is the first collider on $\mathrm{info}^{s}$}\\\
{\mathrm{XOR}}^{p}&\textnormal{if $\to N\leftarrow$ is a collider on $p$ other
than the first}\\\ \mathrm{random}^{p}&\textnormal{if $\leftarrow N\to$ in
$p$}\end{cases}$
where $\mathrm{id}^{p}$ copies the output of the
$\mathrm{dom}_{\mathrm{base}}$ part of the previous node on $p$, |
[a]Sam Foreman
# HMC with Normalizing Flows
Taku Izubuchi Luchang Jin Xiao-Yong Jin James C. Osborn Akio Tomiya
###### Abstract
We propose using Normalizing Flows as a trainable kernel within the molecular
dynamics update of Hamiltonian Monte Carlo (HMC). By learning (invertible)
transformations that simplify our dynamics, we can outperform traditional
methods at generating independent configurations. We show that, using a
carefully constructed network architecture, our approach can be easily scaled
to large lattice volumes with minimal retraining effort. The source code for
our implementation is publicly available online at github.com/nftqcd/fthmc.
## 1 Introduction
### 1.1 2D $U(1)$ Gauge Theory
Figure 1: Plaquette $x_{P}$.
Let $U_{\mu}(n)=e^{ix_{\mu}(n)}\in U(1)$, with $x_{\mu}(n)\in[-\pi,\pi]$
denote the _link variables_ , where $x_{\mu}(n)$ is a link at the site $n$
oriented in the direction $\hat{\mu}$. Our goal is to generate an ensemble of
configurations, distributed according to
$p(x)\propto e^{-S(x)},\quad S(x)\equiv\sum_{P}1-\cos x_{P},$ (1)
where $S(x)$ is the Wilson action for the 2D $U(1)$ gauge theory111Explicitly,
on a square lattice with periodic boundary conditions., and
$x_{P}=x_{\mu}(n)+x_{\nu}(n+\hat{\mu})-x_{\mu}(n+\hat{\nu})-x_{\nu}(n)$ is the
sum of the links around the elementary plaquette as shown in Figure 1. For a
given lattice configuration, we can define the topological charge as
$Q=\frac{1}{2\pi}\sum_{P}\mathrm{arg}(x_{P})\in\mathbb{Z}$, where
$\mathrm{arg}(x_{P})\in[-\pi,\pi]$.
Traditional sampling techniques such as HMC are known to suffer from _critical
slowing down_ [1], a phenomenon characterized by the freezing of the
topological charge $Q$ as we approach physical lattice spacings. This effect
can be seen clearly in Figure 4(a), 4(b), where $Q$ typically remains stuck
for the duration of the HMC trajectories. In this work we describe a method
for training a normalizing flow model that is capable of sampling from
different topological charge sectors, thereby reducing the computational
effort required to generate independent configurations.
### 1.2 Field Transformations
For a random variable $z$ with a given distribution $z\sim r(z)$, and an
invertible function $x=f(z)$ with $z=f^{-1}(x)$, we can use the change of
variables formula to write
$p(x)=r(z)\left|\det\frac{\partial z}{\partial
x}\right|=r(f^{-1}(x))\left|\det\frac{\partial f^{-1}}{\partial x}\right|$ (2)
where $r(z)$ is the (simple) prior density, and our goal is to generate
independent samples from the (difficult) target distribution $p(x)$. This can
be done using _normalizing flows_ [2] to construct a model density $q(x)$ that
approximates the target distribution, i.e. $q(\cdot)\simeq p(\cdot)$ for a
suitably-chosen flow $f$.
Figure 2: Using a flow to generate data $x^{\prime}$. Image adapted from [3]
We can construct a normalizing flow by composing multiple invertible functions
$f_{i}$ so that $x\equiv\left[f_{k}\circ f_{k-1}\circ\cdots\circ f_{2}\circ
f_{1}\right](z)$. In practice, the functions $f_{i}$ are usually implemented
as _coupling layers_ , which update an “active” subset of the variables,
conditioned on the complimentary “frozen” variables [4, 5].
### 1.3 Affine Coupling Layers
A particularly useful template function for constructing our normalizing flows
is the affine coupling layer [6, 2],
$\displaystyle f(x_{1},x_{2})$
$\displaystyle=\left(e^{s(x_{2})}x_{1}+t(x_{2}),\,x_{2}\right),\quad\text{with}\quad\log
J(x)=\sum_{k}\left[s(x_{2})\right]_{k}$ $\displaystyle
f^{-1}(x^{\prime}_{1},x^{\prime}_{2})$
$\displaystyle=\left((x^{\prime}_{1}-t(x^{\prime}_{2}))e^{-s(x^{\prime}_{2})},\,x^{\prime}_{2}\right),\quad\text{with}\quad\log
J(x^{\prime})=\sum_{k}-\left[s(x^{\prime}_{2})\right]_{k}$
where $s(x_{2})$ and $t(x_{2})$ are of the same dimensionality as $x_{1}$ and
the functions act element-wise on the inputs.
In order to effectively draw samples from the correct target distribution
$p(\cdot)$, our goal is to minimize the error introduced by approximating
$q(\cdot)\simeq p(\cdot)$. To do so, we use the (reverse) Kullback-Leibler
(KL) divergence from Eq. 3, which is minimized when $p=q$.
$\displaystyle D_{\mathrm{KL}}(q\|p)$ $\displaystyle\equiv\int
dyq(y)\left[\log q(y)-\log p(y)\right]$ (3)
$\displaystyle\simeq\frac{1}{N}\sum_{i=1}^{N}\left[\log q(y_{i})-\log
p(y_{i})\right],\,\,\text{where}\,\,y_{i}\sim q$ (4)
## 2 Trivializing Map
Ultimately, our goal is to evaluate expectation values of the form
$\langle\mathcal{O}\rangle=\tfrac{1}{\mathcal{Z}}\int
dx\,\mathcal{O}(x)e^{-S(x)}.$ (5)
Using a normalizing flow, we can perform a change of variables $x=f(z)$, so
Eq. 5 becomes
$\displaystyle\langle\mathcal{O}\rangle$
$\displaystyle=\frac{1}{\mathcal{Z}}\int
dz\left|\det\left[J(z)\right]\right|\mathcal{O}(f(z))e^{-S(f(z))},\text{ where
}J(z)=\frac{\partial f(z)}{\partial z}$ (6)
$\displaystyle=\frac{1}{\mathcal{Z}}\int
dz\mathcal{O}(f(z))e^{-S(f(z))+\log|\det[J(z)]|}.$ (7)
We require the Jacobian matrix, $J(z)$, to be:
1. 1.
Injective (1-to-1) between domains of integration
2. 2.
Continuously differentiable (_or_ , differentiable with continuous inverse)
The function $f$ is a _trivializing map_ [7] when $S(f(z))-\log\left|\det
J(z)\right|=\text{const.}$, and our expectation value simplifies to
$\langle\mathcal{O}\rangle=\frac{1}{\mathcal{Z}^{\ast}}\int
dz\,\mathcal{O}(f(z)),\text{ where
}\frac{1}{\mathcal{Z}^{\ast}}=\frac{1}{\mathcal{Z}}\exp(-\text{const.}).$ (8)
## 3 Field Transformation HMC: fthmc
We can implement the trivializing map defined in Sec. 2 using a normalizing
flow model. For conjugate momenta $\pi$, we can write the Hamiltonian as
$H(z,\pi)=\frac{1}{2}\pi^{2}+S(f(z))-\log\left|\det J(f(z))\right|,$ (9)
and the associated equations of motion as
$\displaystyle\dot{z}$ $\displaystyle=\frac{\partial H}{\partial\pi}=\pi$ (10)
$\displaystyle\dot{\pi}$
$\displaystyle=-J(z)S^{\prime}(f(z))+\mathrm{tr}\left[J^{-1}\frac{d}{dz}J\right].$
(11)
If we introduce a change of variables, $\pi=J(z)\rho=J(f^{-1}(x))\rho$ and
$z=f^{-1}(x)$, the determinant of the Jacobian matrix reduces to $1$, and we
obtain the modified Hamiltonian
$\tilde{H}(x,\rho)=\frac{1}{2}\rho^{\dagger}\rho+S(x)-\log|\det J|.$ (12)
As shown in Figure 3, we can use a _field transformation_ ,
$f^{-1}:z\rightarrow x$ to perform HMC updates on the transformed variables
$x$, and $f:x\rightarrow z$ to recover the physical target distribution.
Figure 3: Normalizing flow with inner HMC block.
### 3.1 Hamiltonian Monte Carlo (HMC)
We describe the general procedure of the Hamiltonian Monte Carlo algorithm
[8].
1. 1.
Introduce $v\sim\mathcal{N}(0,\mathbb{I}_{n})\in\mathbb{R}^{n}$ and write the
joint distribution as
$p(x,v)=p(x)p(v)\propto e^{-S(x)}e^{-\frac{1}{2}v^{T}v}$ (13)
2. 2.
Evolve the joint system $(\dot{x},\dot{v})$ according to Hamilton’s equations
along $H=\text{const.}$ using the leapfrog integrator:
$\textbf{ (a.) }\tilde{v}\leftarrow
v-\frac{\varepsilon}{2}\partial_{x}S(x)\quad\textbf{ (b.)
}x^{\prime}\leftarrow x+\varepsilon\tilde{v}\quad\textbf{ (c.)
}v^{\prime}\leftarrow\tilde{v}-\frac{\varepsilon}{2}\partial_{x}S(x^{\prime})$
(14)
3. 3.
Accept or reject the proposal configuration using the Metropolis-Hastings
test,
$x_{i+1}=\begin{cases}x^{\prime},\text{ with probability
}A(x^{\prime}|x)\equiv\min\left\\{1,\frac{p(x^{\prime})}{p(x)}\left|\frac{\partial
x^{\prime}}{\partial x^{T}}\right|\right\\}\\\ x,\text{ with probability
}1-A(x^{\prime}|x)\end{cases}$ (15)
### 3.2 Volume Scaling
We use gauge equivariant coupling layers that act on plaquettes as the base
layer for our network architecture. As in [5], these layers are composed of
inner coupling layers which are implemented as stacks of convolutional layers.
One advantage of using convolutional layers is that we can re-use the trained
weights when scaling up to larger lattice volumes. Explicitly, when scaling up
the lattice volume we can initialize the weights of our new network with the
previously trained values. This approach has the advantage of requiring
minimal retraining effort while being able to efficiently generate models on
large lattice volumes.
## 4 Results
For traditional HMC, we see in Figure 4(a),4(b) that $Q\simeq 0$ for across
all trajectories for both $8\times 8$ and $16\times 16$ lattice volumes.
Conversely, we see in Figure 4(a),4(b) that the trained models are able to
sample from multiple values of $Q$ for both the $8\times 8$ and $16\times 16$
volumes.
The results in Figure 5 took $\sim 4$ hours to train using a single A100
Nvidia GPU. The performance of the trained sampler is limited by the
acceptance rate of the proposed configurations, which in turn, is ultimately
limited by the computational resources used to train the model. Because of
this, we would expect a continued improvement in performance with additional
training. For this relatively simple proof of concept, we were able to
demonstrate the usefuleness of our approach without requiring prohibitively
large upfront training costs.
(a) The average plaquette $x_{P}$ and topological charge $Q$ histories for the
trained model and HMC at $\beta=6$ with $V=8\times 8$.
(b) The same model from Figure 4(a) used to generate configurations on
$V=16\times 16$ lattice.
Figure 4: Comparison of lattice observables for both HMC and the trained model
at $V=8\times 8$, and $V=16\times 16$. Figure 5: Loss and Effective Sample
Size [9] (ESS) vs train epoch at $\beta=6$ on $V=8\times 8$ lattice.
## 5 Acknowledgments
This research was supported by the Exascale Computing Project (17-SC-20-SC), a
collaborative effort of the U.S. Department of Energy Office of Science and
the National Nuclear Security Administration. This research was performed
using resources of the Argonne Leadership Computing Facility (ALCF), which is
a DOE Office of Science User Facility supported under Contract
DE_AC02–06CH11357. This work describes objective technical results and
analysis. Any subjective views or opinions that might be expressed in the work
do not necessarily represent the views of the U.S. DOE or the United States
Government. Results presented in this research were obtained using the Python
[10], programming language and its many data science libraries [11, 12, 13]
## References
* [1] ALPHA collaboration, _Critical slowing down and error analysis in lattice QCD simulations_ , _Nucl. Phys. B_ 845 (2011) 93 [1009.5228].
* [2] D. Rezende and S. Mohamed, _Variational inference with normalizing flows_ , in _International conference on machine learning_ , pp. 1530–1538, PMLR, 2015.
* [3] L. Weng, _Flow-based deep generative models_ , _lilianweng.github.io/lil-log_ (2018) .
* [4] G. Kanwar, M.S. Albergo, D. Boyda, K. Cranmer, D.C. Hackett, S. Racanière et al., _Equivariant flow-based sampling for lattice gauge theory_ , _Phys. Rev. Lett._ 125 (2020) 121601 [2003.06413].
* [5] M.S. Albergo, D. Boyda, D.C. Hackett, G. Kanwar, K. Cranmer, S. Racanière et al., _Introduction to Normalizing Flows for Lattice Field Theory_ , _arXiv e-prints_ (2021) [2101.08176].
* [6] L. Dinh, J. Sohl-Dickstein and S. Bengio, _Density estimation using real NVP_ , _CoRR_ abs/1605.08803 (2016) [1605.08803].
* [7] M. Lüscher, _Trivializing Maps, the Wilson Flow and the HMC Algorithm_ , _Communications in Mathematical Physics_ 293 (2009) 899–919.
* [8] M. Betancourt, _A Conceptual Introduction to Hamiltonian Monte Carlo_ , _arXiv e-prints_ (2017) arXiv:1701.02434 [1701.02434].
* [9] V. Elvira, L. Martino and C.P. Robert, _Rethinking the Effective Sample Size_ , _arXiv e-prints_ (2018) arXiv:1809.04129 [1809.04129].
* [10] G. Van Rossum and F.L. Drake Jr, _Python Tutorial_ , Centrum voor Wiskunde en Informatica Amsterdam, The Netherlands (1995).
* [11] T.A. Caswell, M. Droettboom, J. Hunter, E. Firing, A. Lee, J. Klymak et al., _matplotlib/matplotlib v3.1.0_ , May, 2019. 10.5281/zenodo.2893252.
* [12] C.R. Harris, K.J. Millman, S.J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau et al., _Array programming with NumPy_ , _Nature_ 585 (2020) 357.
* [13] F. Perez and B.E. Granger, _IPython: A System for Interactive Scientific Computing_ , _Computing in Science and Engineering_ 9 (2007) 21.
|
#
Wenmeng Zhang School of Mathematical Sciences
Chongqing Normal University
Chongqing 401331, P. R. China<EMAIL_ADDRESS>, Kening Lu School of
Mathematics
Sichuan University
Chengdu, Sichuan 610064, P. R. China<EMAIL_ADDRESS>and Weinian Zhang
School of Mathematics
Sichuan University
Chengdu, Sichuan 610064, P. R. China<EMAIL_ADDRESS>
# Smooth invariant foliations without a bunching condition and Belitskii’s
$C^{1}$ linearization for random dynamical systems
Wenmeng Zhang School of Mathematical Sciences
Chongqing Normal University
Chongqing 401331, P. R. China<EMAIL_ADDRESS>, Kening Lu School of
Mathematics
Sichuan University
Chengdu, Sichuan 610064, P. R. China<EMAIL_ADDRESS>and Weinian Zhang
School of Mathematics
Sichuan University
Chengdu, Sichuan 610064, P. R. China<EMAIL_ADDRESS>
###### Abstract.
Smooth linearization is one of the central themes in the study of dynamical
systems. The classical Belitskii’s $C^{1}$ linearization theorem has been
widely used in the investigation of dynamical behaviors such as bifurcations,
mixing, and chaotic behaviors due to its minimal requirement of partial second
order non-resonances and low regularity of systems. In this article, we
revisit Belitskii’s $C^{1}$ linearization theorem by taking an approach based
on smooth invariant foliations and study this problem for a larger class of
dynamical systems (random dynamical systems). We assumed that the linearized
system satisfies the condition of Multiplicative Ergodic Theorem and the
associated Lyapunov exponents satisfy Belitskii’s partial second order non-
resonant conditions. We first establish the existence of $C^{1,\beta}$ stable
and unstable foliations without assuming the bunching condition for Lyapunov
exponents, then prove a $C^{1,\beta}$ linearization theorem of Belitskii type
for random dynamical systems. As a result, we show that the classical
Belitskii’s $C^{1}$ linearization theorem for a $C^{2}$ diffeomorphism $F$
indeed holds without assuming all eigenspaces of the linear system $DF(0)$ are
invariant under the nonlinear system $F$, a requirement previously imposed by
Belitskii in his proof.
###### Key words and phrases:
Invariant foliation; invariant distribution; bunching condition; random normal
form; cohomological equation.
This work was partially supported by from NSFC (11922105,
12090010,12090013,11971330,12171336,11831012) and National Key R&D Program of
China 2022YFA1005900.
###### 2020 Mathematics Subject Classification:
Primary: 37C15, 37H15; Secondary: 37C86, 37D25, 37G05
###### Contents
1. 1 Introduction
2. 2 Main result
3. 3 Random invariant distribution
4. 4 Random invariant foliation
5. 5 Decomposition of a cohomological equation
6. 6 Random normal form
7. 7 Solving the decomposed cohomological equation when $\alpha\in(0,1]$
8. 8 Solving the decomposed cohomological equation when $\alpha=0$
9. 9 Smoothness of the distribution
10. 10 Proof of the main result
## 1\. Introduction
Reducing a nonlinear system to the simplest form through a smooth conjugacy is
one of the fundamental themes in the study of dynamical systems. When the
simplest form is a linear system, finding such a conjugacy is called the
smooth linearization problem, which has widely been investigated under
different frameworks such as local diffeormorphisms near fixed points ([20,
33, 51, 59]), circle diffeomorphisms ([2, 14, 25, 35, 47, 62]), and toral
diffeomorphisms ([16, 31, 37, 45, 53]).
The study of smooth linearization goes back to Poincaré ([51]). He proved that
analytic linearization can be realized for an analytic diffeomorphism
$F(x)=\Lambda x+O(|x|^{2})$ in $\mathbb{C}^{d}$ when the eigenvalues
$r_{1},...,r_{d}$ of $\Lambda$ satisfy the contraction (or expansion)
condition:
$\max_{i=1,...,d}|r_{i}|<1\;(\text{or}\;\min_{i=1,...,d}|r_{i}|>1)$
and the non-resonant conditions of all orders:
$\displaystyle r_{j}\neq r_{1}^{m_{1}}\cdots r_{d}^{m_{d}}$ (1.1)
for each $j=1,...,d$, where $m_{i}\geq 0$ are all possible integers such that
$\sum_{i=1}^{d}m_{i}\geq 2$. Siegel ([57]) replaced Poincaré’s conditions by a
small-divisor (Diophantine) condition:
$|r_{j}-r_{1}^{m_{1}}\cdots r_{d}^{m_{d}}|\geq\frac{C}{|m|^{\mu}},\quad 1\leq
j\leq d,\;m\in\mathbb{Z}^{d}_{+},\;|m|=m_{1}+\cdots+m_{d}\geq 2,$
and proved that the analytic diffeomorphism is analytically conjugated to its
linear part $\Lambda$. Siegel’s condition allows $\Lambda$ to be hyperbolic
having both contraction and expansion. The simpler proof of Siegel’s theorem
was later given by V. Arnold ([3]), Moser ([46]) and Zehnder ([63]) by using
the KAM method.
$C^{k}$ smooth linearization in $\mathbb{R}^{d}$ was firstly obtained by
Sternberg ([58, 59]) in 1950s. He proved that for each $k\in\mathbb{N}$ there
is a sufficiently large integer $N>0$ such that a $C^{N}$ local hyperbolic
diffeomorphism $F(x)=\Lambda x+O(|x|^{2})$ is $C^{k}$ conjugate to its linear
part $\Lambda$ if the eigenvalues $r_{1},\cdots,r_{d}$ of $\Lambda$ satisfy
the non-resonant condition (1.1) up to order $N$, i.e., (1.1) holds for all
$m_{i}\geq 0$ with $2\leq\sum_{i=1}^{d}m_{i}\leq N$. Following this work,
there is an extensive literature on this subject, see for example Nagumo-Isé
([48]), Chen ([12]), Nelson ([49]), Sell ([55]), and Banyaga, de la Llave, and
Wayne ([5]). A crucial condition in these works is the high order non-resonant
condition. In the meantime, it also requires high order regularity of systems.
A fundamental question is whether a smooth linearization can be realized with
the lowest order non-resonance, or even a part of the lowest order non-
resonance (we call it the partial lowest order non-resonance), together with
low order regularity of the systems. Without assuming any non-resonant
conditions, Hartman ([19]) and Grobman ([17]) independently showed in 1960’s
that a $C^{1}$ local hyperbolic diffeomorphism can be $C^{0}$ linearized.
However, the $C^{0}$ conjugacy usually does not preserve dynamical properties
such as the characteristic directions, the derivatives of the Poincaré mapping
of global orbits, the iteration rates of systems and the differentiable
structure of smooth manifolds. In order to maintain such properties, the
$C^{1}$ conjugacy is essential.
In 1960, Hartman ([18]) proved that a $C^{1,1}$ contractive diffeomorphism in
$\mathbb{R}^{d}$ admits local $C^{1}$ linearization without assuming any non-
resonances. In the same paper, he also proved that a $C^{1,1}$ typically
hyperbolic diffeomorphism admits local $C^{1}$ linearization under the
following condition:
$\displaystyle|r_{1}|/|r_{t}|<|r_{{t}+1}|^{-1},\qquad|r_{t+1}|/|r_{d}|<|r_{{t}}|,$
(1.2)
where the eigenvalues $r_{1},\cdots,r_{d}$ are ordered as
$\displaystyle|r_{1}|\geq\cdots\geq|r_{t}|>1>|r_{{t}+1}|\geq\cdots\geq|r_{d}|$
(1.3)
for ${t}\in\\{1,...,d-1\\}$. Condition (1.2) means that the contractive
spectrum and the expansive spectrum both lie in a “narrow-band” area in
$\mathbb{C}$, which is referred to as the bunching condition (see for example
[33, pp.603-604]. Since the bunching condition (1.2) holds automatically in
$\mathbb{R}^{2}$, as a result, a $C^{1,1}$ typically hyperbolic diffeomorphism
in $\mathbb{R}^{2}$ admits local $C^{1}$ linearization. Samovol ([54]) in 1972
proved that a $C^{N}$ typically hyperbolic diffeomorphism with sufficiently
large $N\in\mathbb{N}$ admits local $C^{1}$ linearization if the eigenvalues
satisfy a partial second order non-resonant condition:
$\displaystyle r_{j}\neq r_{i}r_{\kappa}$ (1.4)
for $i=1,...,{t}$, $\kappa={t}+1,...,d$ and $j=1,...,d$. Note that condition
(1.4) is not only of the lowest order but also a part of the second order non-
resonance. However, it retains the requirement of $C^{N}$ smoothness of the
system for large $N$. Samovol’s strategy is to transform the $C^{N}$ system to
its polynomial normal form by Sternberg’s normal form theorem, which requires
$N$ to be sufficiently large, and then to give an explicit form of the
conjugacy, which can linearize the polynomial normal form and can be proved to
be $C^{1}$ under (1.4).
In 1973, this high order regularity requirement was removed by Belitskii
([7]). He stated that a $C^{1,1}$ typical hyperbolic diffeomorphism $F$ in
$\mathbb{R}^{d}$ admits local $C^{1}$ linearization if a condition
$\displaystyle|r_{j}|\neq|r_{i}|\,|r_{\kappa}|$ (1.5)
holds for $i=1,...,{t}$, $\kappa={t}+1,...,d$ and $j=1,...,d$. Clearly, this
result gives the above Hartman’s $C^{1}$ linearization results since (1.2)
implies (1.5). We notice that condition (1.5) allows some second order
resonances to appear. Condition (1.5), called the strong non-resonance of
order 2, is a little stronger than (1.4), but counter examples given in [8,
pp.139-142] show that such a condition is sharp, i.e., it cannot be removed
for $C^{1}$ linearization of a $C^{2}$ system in general.
Belitskii’s theorem is a truly remarkable and optimal result for $C^{1}$
linearization, and has many applications to homoclinic bifurcations ([4, 28,
29]), heteroclinic tangencies ([36]), mixing of hyperbolic systems ([15]),
invariant measures of partially hyperbolic systems ([27]), and solutions of
higher order semilinear equations ([11, 13]). However, Belitskii’s proof was
commented “In our opinion, the proofs of Theorems given by Belitskii should be
recognized as insufficient and incomprehensible” by Bronstein and Kopanskii
([10, p.191]). Actually, in his proof, Belitskii needs to establish the
surjectivity of nonlinear operators deduced from the conjugacy equation, where
all eigenspaces of $\Lambda$ are in fact assumed to be invariant under
$F(x)=\Lambda x+O(|x|^{2})$. This is a very restrictive assumption, which will
be elaborated just after the following Corollary.
In this article, we revisit Belitskii’s $C^{1}$ linearization theorem, taking
a different approach based on the smooth invariant foliations, which was not
involved in Samovol and Belitskii’s proofs, and study this problem for random
dynamical systems, a larger class of dynamical systems. We first establish the
existence of $C^{1,\beta}$ stable and unstable foliations without assuming the
bunching condition (1.2) but only the partial second order non-resonant
condition (1.5). Then we prove a Belitskii type of $C^{1,\beta}$ linearization
theorem for random dynamical systems under a partial second order non-resonant
condition in terms of Lyapunov exponents.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and
$(\theta^{n})_{n\in\mathbb{Z}}$ be a measurable $\mathbb{P}$-measure
preserving dynamical system on $\Omega$. A random dynamical system (or a
cocycle) on the space $\mathbb{R}^{d}$ over the dynamical system
${\theta}^{n}$ is a measurable mapping
$F:\mathbb{Z}\times\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d},\quad(n,\omega,x)\mapsto
F(n,\omega,x),$
such that the mapping $F(n,\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$
forms a cocycle over $\theta^{n}$:
$F(0,\omega,\cdot)={\rm id}~{}~{}\mbox{(the identity mapping)},\quad\hbox{ for
all }\;\omega\in\Omega,$ $F(n+m,\omega,\cdot)=F(n,\theta^{m}\omega,\cdot)\circ
F(m,\omega,\cdot),\quad\hbox{ for all
}\;m,n\in\mathbb{Z},\quad\omega\in\Omega.$
Replacing $\mathbb{Z}$ by $\mathbb{R}$ gives a continuous time random
dynamical system. A typical example is the solution operator for a stochastic
differential equation ([1]).
We consider a $C^{2,\alpha}$ ($\alpha\in[0.1]$) random dynamical system
$F(n,\omega,x)$ (that is, $F$ is $C^{2,\alpha}$ in $x$ for each $n$ and
$\omega$ and its derivatives are measurable with respect to $\omega$), and
write the time-one mapping $F(1,\omega,x)$ as $F(\omega,x):=F(1,\omega,x)$.
Then $F(\omega,\cdot)$ is a random diffeomorphism, whose $i$-th order
derivative is denoted by $D^{i}F(\omega,\cdot)$ for every $i\in\mathbb{N}$.
This random diffeomorphism generates the random dynamical system
$F(n,\omega,\cdot)=\begin{cases}F(\theta^{n-1}\omega,\cdot)\circ\cdots\circ
F(\omega,\cdot),&n>0,\\\ I,&n=0,\\\
F^{-1}(\theta^{n}\omega,\cdot)\circ\cdots\circ
F^{-1}(\theta^{-1}\omega,\cdot),&n<0.\end{cases}$
We assume that $F(\omega,\cdot)$ has a fixed point $x=0$ for all
$\omega\in\Omega$, to which a large class of random dynamical systems can be
converted (see [1, p.310]), and is a locally tempered $C^{2,\alpha}$ random
diffeomorphism, that is, there is a tempered ball
$V(\omega)=B_{\rho(\omega)}(0)=\\{x\in\mathbb{R}^{d}:\|x\|<\rho(\omega)\\}$,
where $\rho(\omega)$ is a random variable tempered from below (i.e.,
$\lim_{n\to\pm\infty}\frac{1}{n}\min\\{0,\log\rho(\theta^{n}\omega)\\}=0,\;\mathbb{P}-a.s.$),
such that
$\displaystyle\|D^{i}F(\omega,x)\|\leq
M(\omega),\quad\|D^{2}F(\omega,x)-D^{2}F(\omega,y)\|\leq
L(\omega)\|x-y\|^{\alpha},\quad\forall x,y\in V(\omega),$
where $M(\omega),L(\omega)>0$ are random variable tempered from above (i.e.,
$\lim_{n\to\pm\infty}\frac{1}{n}\max\\{0,\log
M(\theta^{n}\omega)\\}=0,\;\mathbb{P}-a.s.$). The size of a tempered ball may
decrease as $\omega$ varies, but these changes along each orbit
$\theta^{n}\omega$ are at a subexponential rate. The upper bounds of
$M(\omega)$ and $L(\omega)$ may grow to infinity as $\omega$ varies, but along
each orbit $\theta^{n}\omega$, it may increase only at a subexponential rate.
This nonuniform behavior is one of the intrinsic features of random dynamical
systems.
Two local tempered random diffeomporhisms $F$ and $G$ with fixed point $x=0$
are said to be $C^{1,\beta}$ locally conjugate for $\beta\in[0,1]$ if there
exists a $C^{1,\beta}$ random diffeomorphism $\Phi(\omega,x)$ defined on a
tempered ball $V(\omega)$ with $\Phi(\omega,0)=0$ such that
$\Phi(\theta\omega,F(\omega,x))=G(\omega,\Phi(\omega,x))\quad\hbox{for}\quad
x\in V(\omega),\quad\mathbb{P}-a.s.$
This conjugacy relation implies that $h$ carries orbits of $\varphi$ to orbits
of $\psi$ when the orbits stay in the corresponding tempered ball.
We write $F(\omega,x)$ as
$F(\omega,x)=\Lambda(\omega)x+f(\omega,x)=\Lambda(\omega)x+O(\|x\|^{2}),$
where $\Lambda(\omega)=DF(\omega,0)\in Gl(d,\mathbb{R})$. Assume that
$\Lambda(\omega)$ satisfies the conditions of the Multiplicative Ergodic
Theorem, that is,
$\max\\{0,\log\|\Lambda(\cdot)\|\\}\in
L^{1}(\Omega,\mathcal{F},\mathbb{P})\quad\hbox{and}\quad\max\\{0,\log\|\Lambda^{-1}(\cdot)\|\\}\in
L^{1}(\Omega,\mathcal{F},\mathbb{P}),$
and that $\mathbb{P}$ is an ergodic invariant measure for $\theta$. By the
Multiplicative Ergodic Theorem (see Section 2 for details), there exists a
$\theta$-invariant set $\tilde{\Omega}\subset\Omega$ of full measure such that
for each $\omega\in\tilde{\Omega}$, $\Lambda(n,\omega)$ (the linear random
dynamical system generated by $\Lambda(\omega)$) has $p$ Lyapunov exponents
$\lambda_{j}$, $j=1,\cdots,p$. We assume that $\Lambda(n,\omega)$ is
hyperbolic, namely, there is no zero Lyapunov exponent. We order them as
follows:
$\lambda_{1}>\cdots\lambda_{\tau}>0>\lambda_{\tau+1}\cdots>\lambda_{p}.$
For the remainder of this article, we use $\Omega$ to denote $\tilde{\Omega}$
and assume that these statements are true for all $\omega\in\Omega$.
Our main result is the following theorem on $C^{1,\beta}$ foliations and
linearization for random dynamical systems in the typically hyperbolic case,
in which there are both positive and negative Lyapunov exponents.
Theorem. Let $F:\Omega\times V(\omega)\to\mathbb{R}^{d}$ be a tempered
$C^{2,\alpha}$ $(\alpha\in[0,1])$ random diffeomorphism with an
$\epsilon$-slowly continuous second order derivative when $\alpha=0$. Assume
its linearization $\Lambda(\omega)$ at hyperbolic fixed point $x=0$ satisfies
the condition of the Multiplicative Ergodic Theorem. If the Lyapunov exponents
satisfy a partial second order non-resonant condition:
$\displaystyle\lambda_{j}\neq\lambda_{i}+\lambda_{\kappa}$ (1.6)
for all $i=1,...,{\tau}$, $\kappa={\tau}+1,...,{{p}}$ and $j=1,...,{{p}}$,
then the following assertions are true:
* (i)
Foliations: $F$ has $C^{1,\beta_{\alpha}}$ random stable and unstable
foliations with $C^{2,\alpha}$ leaves in a tempered ball $U(\omega)\subset
V(\omega)$, where $\beta_{\alpha}=0$ when $\alpha=0$ and
$\beta_{\alpha}\in(0,\alpha]$ when $\alpha\in(0,1]$.
* (ii)
Conjugacy: $F(\omega,\cdot)$ is conjugate to its linear part $\Lambda(\omega)$
in a tempered ball $U(\omega)\subset V(\omega)$ by a $C^{1,\beta_{\alpha}}$
random diffeomorphism $\Phi:\Omega\times U(\omega)\to\mathbb{R}^{d}$
satisfying $\Phi(\omega,0)=0$ and $D\Phi(\omega,0)={\rm id}.$
Remark. (1) The assumption of $\epsilon$-slowly continuity means that
$e^{-\epsilon|n|}D^{2}F(\theta^{n}\omega,\cdot)$ is equicontinuous with
respect $n\in\mathbb{Z}$ for each fixed $\omega\in\Omega$. It is only
necessary for random dynamical systems in the case of $\alpha=0$ since, when
$\alpha\in(0,1]$ or $F$ is deterministic, the second order derivative of
$F(\omega,\cdot)$ is automatically $\epsilon$-slowly continuous. (2) The non-
resonant condition (1.6) obviously does not imply the bunching condition:
$\lambda_{1}-\lambda_{\tau}<-\lambda_{\tau+1}\quad\text{and}\quad\lambda_{\tau+1}-\lambda_{p}<\lambda_{\tau}.$
(1.7)
(3) When $F$ is deterministic, the relationship between Lyapunov exponent
$\lambda$ and its corresponding eigenvalue $r$ is $|r|=e^{\lambda}$. Thus
condition (1.6) becomes (1.5).
As a result, we have the following $C^{1,\beta_{\alpha}}$ linearization
theorem.
Corollary. Let $F:V\to\mathbb{R}^{d}$ be a $C^{2,\alpha}$ ($\alpha\in[0,1]$)
diffeomorphism over a neighborhood $V\subset\mathbb{R}^{d}$ of $x=0$. Suppose
the eigenvalues of its linearization $\Lambda:=DF(0)$ at the hyperbolic fixed
point $x=0$ satisfy (1.5). Then $F$ is conjugate to its linear part $\Lambda$
in a small neighborhood $U\subset V$ by a $C^{1,\beta_{\alpha}}$
diffeomorphism $\Phi:U\to\mathbb{R}^{d}$ satisfying $\Phi(0)=0$ and
$D\Phi(0)={\rm id},$ where $\beta_{\alpha}=0$ when $\alpha=0$ and
$\beta_{\alpha}\in(0,\alpha]$ when $\alpha\in(0,1]$.
When $\alpha=0$, this corollary is exactly the same as the above-mentioned
Belitskii’s theorem [7] except the $C^{2}$ smoothness, which is stronger than
his $C^{1,1}$ smoothness. In [7], Belitskii proved the above result under an
assumption that all eigenspaces of $\Lambda$ are invariant under the nonlinear
system $F$. To be more precise, in [7, Lemma 3.1], he assumed that the
subspaces $L_{I}:=\bigoplus_{i\in I\subset Q}L_{i}$ of $\mathbb{R}^{d}$ are
invariant under $F$ for all subsets $I$ of $Q=\\{1,\cdots,q\\}$ with
$q\in\mathbb{N}$, where each $L_{i}$ is the eigenspace of $\Lambda$
corresponding to the eigenvalue lying on the circle
$\\{z\in\mathbb{C}:|z|=\rho_{i}\\}$ and those $\rho_{i}$, being the moduli of
the eigenvalues of $\Lambda$, are ordered as
$\rho_{1}>\rho_{2}>\cdots>\rho_{k}>1>\rho_{k+1}>\cdots>\rho_{q}.$
In other words, Belitskii assumed that the invariant manifolds of $F$
associated with any spectral splitting according to their moduli are all flat,
i.e., linear spaces. This can be achieved only if one can construct a
$C^{1,1}$ invariant manifold of $F$ for any spectral splitting:
$\\{\rho_{1},\cdots,\rho_{q}\\}=\\{\rho_{i}:i\in I\\}\cup\\{\rho_{i}:i\in
Q\setminus I\\}.$
In this case, the invariant manifold is given by the graph of a $C^{1,1}$
mapping
$\phi:L_{I}=\bigoplus_{i\in I}L_{i}\mapsto L_{I^{\prime}}=\bigoplus_{i\in
Q\setminus I}L_{i}$
Using these manifolds to change the coordinate system, one obtain flat
invariant manifolds for the new system.
By the theory of invariant manifolds, $F$ has $C^{1,1}$ stable manifold and
unstable manifold, i.e., according to the spectral slitting:
$Q=\\{\rho_{i}:\rho_{i}<1\\}\cup\\{\rho_{i}:\rho_{i}>1\\}.$
However, in order to have $C^{1,1}$ pseudo-stable and pseudo-unstable
manifolds for $F$, a spectral gap condition is needed. de la Llave and Wayne
[39, Example 5.1] gave an example of a 2-dimensional $C^{\infty}$
diffeomorphism with a small spectral gap, whose pseudo-stable manifold is not
$C^{(\ln 3/\ln 2)+\varepsilon}$ for any $\varepsilon>0$, so the pseudo-stable
manifold is not $C^{1,1}$ either. To better understand the issue here, we
borrow their Example 5.1 and add one trivial stable component to get the
diffeomorphism
$F(x)=\begin{pmatrix}1/2&0&0\\\ 0&2&0\\\
0&0&3\end{pmatrix}\begin{pmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{pmatrix}+\begin{pmatrix}0\\\ 0\\\ \varphi(x_{2})\end{pmatrix},$
(1.8)
where $\varphi$ is a $C^{\infty}$ function with a compact support. Note that
the eigenvalues satisfy the non-resonant condition (1.5), but the pseudo-
stable manifold associated with the splitting $\\{1/2,2\\}\cup\\{3\\}$ is not
$C^{1,1}$. Consequently, one can not transform equation (1.8) by a conjugacy
to a $C^{1,1}$ system with invariant manifolds all being linear spaces.
Furthermore, a spectral splitting such as
$Q=\\{\rho_{1},\rho_{q}\\}\cup\\{\rho_{i}:i\in Q,i\not=1,q\\}$
may not yield a $C^{1,1}$ invariant manifold (as a graph over $L_{1}\bigoplus
L_{q}$) either. The existence of such a $C^{1,1}$ invariant manifold may need
much stronger non-resonant conditions (see .e.g. [38]). In short, it is quite
restricted to assume that all the invariant manifolds of $F$ based on spectral
decompositions are linear spaces. We emphasize that the non-resonant condition
(1.5) implies neither the spectral gap condition needed for $C^{1,1}$
invariant manifolds as we see for example (1.8) nor the bunching condition
(1.2) required in Hartman’s theorem.
On the other hand, Bronstein and Kopanskii stated in [10, p.191] that “Theorem
9.4 coincides, in fact, with Theorem 5 (i.e., Belitskii’s theorem), but the
proof is new.” However, we find that there is no “Theorem 9.4” in their
monograph [10]. In fact, there are only Theorems 9.1 and 9.6 in [10, Chapter
II, Section 9] for the smooth linearization of mappings. The two theorems are
the same type as Sternberg’s theorem since $F$ was required to be sufficiently
smooth, and the derivatives of $F$ up to sufficiently high order were required
to be equal to $0$ at the origin, which is equivalent to a corresponding non-
resonant condition. These conditions on high order smoothness and vanishing
derivatives make it possible to get a contractive operator, whose fixed point
is used to construct the conjugacy, by the homotopy method (for Sternberg’s
theorem) or by the conjugacy relation (for Bronstein and Kopanskii’s results).
In this article, we do not make any assumption on the invariant manifolds of
$F$, except for the non-resonant condition (1.6) and the $C^{2,\alpha}$
regularity of $F$. We take a different approach from Belitskii’s. We construct
$C^{1,\beta_{\alpha}}$ random stable and unstable foliations with
$C^{2,\alpha}$ leaves and utilize them to decouple the system into a
contraction and an expansion. The $C^{2,\alpha}$ smoothness of the leaves can
be obtained by using the Hadamard graph transformation or the Lyapunov-Perron
method. However, without the bunching condition on the Lyapunov exponents, it
is more complicated to prove the $C^{1,\beta_{\alpha}}$ smoothness of the
foliations with respect to the base point only under the non-resonant
condition (1.6).
To obtain the $C^{1,\beta_{\alpha}}$ smoothness, it is sufficient to show that
the tangent spaces of leaves of the stable and unstable foliations are
$C^{1,\beta_{\alpha}}$ distributions thanks to Frobenius’ theorem. We
construct continuous bases for these stable and unstable distributions which
are uniformly $C^{1,\beta_{\alpha}}$ along each local leaf of the stable and
unstable foliations. This will be done by solving a random cohomological
equation along each leaf of all intermediate (pseudo-center) foliations, which
is the main step to overcome the difficulty of no bunching condition (see the
end of section 7). The non-resonant condition (1.6) plays a key role in
establishing the smoothness, which will be used to prove the result on random
normal form and to show the invertibility of cohomological operators (see
(7.36) and (8.10) below). To deal with the non-uniformity of hyperbolicity,
the Lyapunov norm is used in solving the random cohomology equation. Our proof
also involves the roughness of non-uniformly exponential dichotomy and
Journé’s lemma ([32]).
When the Lyaponov exponents satisfy the bunching condition (1.7) instead of
(1.6), the $C^{1,\beta_{\alpha}}$ smoothness of the random stable and unstable
foliations with respect to the base point can be obtained by using the
Lyapunov-Perron approach, see for example [56]. There is an extensive study of
regularity of invariant foliations. The optimal regularity of invariant
foliations for hyperbolic dynamical systems under bunching conditions was
obtained by Hasselblatt ([21, 22, 23]), and Hasselblatt and Wilkinson ([24]).
Moreover, Push, Shub and Wilkinson ([52]) established the regularity of
invariant foliations of partially hyperbolic diffeomorphisms under bunching
conditions.
Finally, we mention some existing results which are closely relevant to the
results derived in this article. The theorems of random version were
established for Poincaré’s analytic linearization, Siegel’s analytic
linearization, Sternberg’s smooth linearization and Hartman-Grobman’s $C^{0}$
linearization ([6, 40, 41, 42, 60]), but no result for random dynamical
systems on $C^{1}$ linearization until very recently. In [44], the authors of
the current paper proved a random version of Hartman’s $C^{1,\beta_{\alpha}}$
linearization for a $C^{1,\alpha}$ ($\alpha\in(\alpha_{0},1]$) random
contraction (or expansion) in $\mathbb{R}^{d}$, where $\alpha_{0}\in(0,1)$ is
a constant depending on the Lyapunov exponents.
## 2\. Main result
In this section, we first review some of the basic concepts and facts for
random dynamical systems taken from Arnold’s book [1]. We then state our main
results.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with a sample set
$\Omega$, a $\sigma$-algebra $\mathcal{F}\subset 2^{\Omega}$ and a probability
measure $\mathbb{P}$ on $\mathcal{F}$, and let
$(\Omega,\mathcal{F},\mathbb{P},(\theta^{n})_{n\in\mathbb{Z}})$ (denoted by
$\theta$ for short) be a metric dynamical system (see [1, p.536]). A mapping
${F}:\mathbb{Z}\times\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d},\quad(t,\omega,x)\mapsto{F}(t,\omega,x),$
where $\mathbb{Z}$ is endowed with its Borel $\sigma$-algebra
${\mathcal{B}}(\mathbb{Z})$, is called a $C^{N}$ ($N\geq 1$) random dynamical
system (abbreviated as RDS) on the measurable space ($\mathbb{R}^{d}$,
${\mathcal{B}}$) over $\theta$ if
* (1)
$F(n,\omega,\cdot)$ is
$\mathcal{B}(\mathbb{Z})\otimes\mathcal{F}\otimes\mathcal{B}(\mathbb{R}^{d})$-measurable
and $F(n,\omega,\cdot)$ is $C^{N}$ with measurable derivatives, and
* (2)
the mappings ${F}(n,\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$,
$n\in\mathbb{Z}$, form a cocycle over $\theta$, i.e.,
$\displaystyle{F}(0,\omega,\cdot)={\rm id}~{}\mbox{{\rm(}the identity
mapping{\rm)}},\quad\forall\omega\in\Omega,$
$\displaystyle{F}(n+m,\omega,\cdot)={F}(n,\theta^{m}\omega,\cdot)\circ{F}(m,\omega,\cdot),\quad\forall
n,m\in\mathbb{Z},\quad\forall\omega\in\Omega.$
Clearly, the time-one mapping
${F}(\omega,\cdot):={F}(1,\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a
$C^{N}$ random diffeomorphism. Conversely, a $C^{N}$ random diffeomorphism
${F}(\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$ generates a $C^{N}$ RDS by
$\displaystyle{F}(n,\omega,\cdot):=\begin{cases}{F}(\theta^{n-1}\omega,\cdot)\circ\cdots\circ{F}(\omega,\cdot),&n>0,\\\
{\rm id},&n=0,\\\
{F}^{-1}(\theta^{n}\omega,\cdot)\circ\cdots\circ{F}^{-1}(\theta^{-1}\omega,\cdot),&n<0.\end{cases}$
(2.1)
When $\theta$ is the identity mapping, the RDS becomes a usual deterministic
system.
A random variable $K:\Omega\to(0,\infty)$ is said to be tempered if it is
tempered from above, i.e.,
$\lim_{n\to\pm\infty}\frac{1}{n}\max\\{0,\log
K(\theta^{n}\omega)\\}=0,\qquad\mathbb{P}-a.s.,$
and is tempered from below, i.e.,
$\lim_{n\to\pm\infty}\frac{1}{n}\min\\{0,\log
K(\theta^{n}\omega)\\}=0,\qquad\mathbb{P}-a.s.$
Note that $K(\omega)$ is tempered from above (or below) if and only if
$1/K(\omega)$ is tempered from below (or above). When $K(\omega)$ is tempered
above, it follows from [1] that for any constant $\epsilon>0$ there is a
tempered random variable $K_{\epsilon}(\omega)$ such that
$K_{\epsilon}(\omega)\geq K(\omega)$ and
$\displaystyle e^{-\epsilon|n|}K_{\epsilon}(\omega)\leq
K_{\epsilon}(\theta^{n}\omega)\leq
e^{\epsilon|n|}K_{\epsilon}(\omega),\quad\forall n\in\mathbb{Z}.$ (2.2)
Such $K_{\epsilon}(\omega)$ is called an $\epsilon$-slowly varying random
variable. We call $V(\omega):=\\{x\in\mathbb{R}^{d}:\|x\|\leq\rho(\omega)\\}$
a tempered ball if $\rho(\omega)>0$ is a random variable tempered from below.
For an integer $N\in\mathbb{N}$ and a real number $\alpha\in[0,1]$, we call
${F}(\omega,x)$ a tempered $C^{N,\alpha}$ random diffeomorphism (see [42]) if
${F}(\omega,\cdot)$ is a $C^{N}$ diffeomorphism and the $i$-th derivative
$D^{i}{F}(\omega,x)$ are measurable with respect to $\omega$ such that
$\displaystyle\begin{split}&\|D^{i}{F}(\omega,x)\|\leq M(\omega),\quad\forall
i=0,1,...,N,\quad\forall\omega\in\Omega,\\\
&\|D^{N}{F}(\omega,x)-D^{N}{F}(\omega,y)\|\leq
L(\omega)\|x-y\|^{\alpha},\quad\forall x,y\in V(\omega),\end{split}$ (2.3)
where $M(\omega),L(\omega)>0$ are random variables tempered from above. In
particular, if $\alpha=0$ then it means that $F(\omega,x)$ is $C^{N}$.
Moreover, in the case of $\alpha=0$, we can further assume that
$D^{N}F(\omega,\cdot)$ is $\epsilon$-slowly continuous with respect to $x$ for
each fixed $\omega\in\Omega$, where $\epsilon>0$ is a small constant, i.e.,
$e^{-\epsilon|n|}D^{N}F(\theta^{n}\omega,\cdot)$ is equicontinuous with
respect $n\in\mathbb{Z}$ for each fixed $\omega\in\Omega$. When
$D^{N}F(\omega,x)$ is Hölder continuous with a tempered Hölder constant, then
$D^{N}F(\omega,\cdot)$ is $\epsilon$-slowly continuous; when the system is
deterministic, the uniform continuity of $D^{N}F$ implies that it is
$\epsilon$-slowly continuous.
We assume that ${F}(n,\omega,0)=0$ for any $n\in\mathbb{Z}$ and any
$\omega\in\Omega$, and we use the notation $DF(\omega,\cdot)$ (instead of
$D^{1}F(\omega,\cdot)$) for the first order derivative of $F$ with respect to
$x$.
Consider a linear random diffeomorphism
$\Lambda(\omega):=\Lambda(\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$. Then
it generates a linear random dynamical system
$\displaystyle{\Lambda}(n,\omega):=\begin{cases}\Lambda(\theta^{n-1}\omega)\cdots\Lambda(\omega),&n>0,\\\
{\rm id},&n=0,\\\
\Lambda^{-1}(\theta^{n}\omega)\cdots\Lambda^{-1}(\theta^{-1}\omega),&n<0.\end{cases}$
(2.4)
Assume that $\Lambda(\omega)$ satisfies the conditions of the Multiplicative
Ergodic Theorem, that is,
$\max\\{0,\log\|\Lambda(\cdot)\|\\}\in
L^{1}(\Omega,\mathcal{F},\mathbb{P})\quad\hbox{and}\quad\max\\{0,\log\|\Lambda^{-1}(\cdot)\|\\}\in
L^{1}(\Omega,\mathcal{F},\mathbb{P}).$ (2.5)
and $\mathbb{P}$ is an ergodic invariant measure for $\theta$. Then, by the
Multiplicative Ergodic Theorem ([1]), there exists a $\theta$-invariant set
$\tilde{\Omega}\subset\Omega$ of full measure such that for each
$\omega\in\tilde{\Omega}$, the linear random dynamical system
$\Lambda(n,\omega)$ has $p$ Lyapunov exponents: $\lambda_{j}$ with the
multiplicity $d_{j}$, $j=1,...,p$, and the corresponding Oseledets spaces
$E_{j}(\omega)$ with dimension $d_{j}$ form an invariant splitting of the
phase space $\mathbb{R}^{d}$:
$\mathbb{R}^{d}=E_{1}(\omega)\oplus\cdots\oplus E_{{p}}(\omega).$
Since $\mathbb{P}$ is ergodic, all $\lambda_{i}$, $d_{i}$, and $p$ are
constant. For the remainder of this article, we use $\Omega$ to denote
$\tilde{\Omega}$ and assume that these statements are true for all
$\omega\in\Omega$.
Assume that the linear random dynamical system $\Lambda(n,\omega)$ is
hyperbolic, i.e., there is no zero Lyapunov exponent. We arrange the Lyapunov
exponents as
$\displaystyle\lambda_{1}>\cdots>\lambda_{\tau}>0>\lambda_{\tau+1}>\cdots>\lambda_{{p}}$
(2.6)
for a certain $\tau\in\\{1,...,{p}-1\\}$. It follows from [42, Lemma 2.8] that
$\Lambda(n,\omega)$ is conjugate to a block diagonal linear random dynamical
system by a random isomorphism and the corresponding Oseledets spaces are
given by
$E_{j}(\omega)=X_{j}:=\\{0\\}\times\cdots\times\\{0\\}\times\mathbb{R}^{d_{j}}\times\\{0\\}\times\cdots\times\\{0\\}\subset\mathbb{R}^{d}.$
Moreover, for any small $\epsilon>0$ one has
$\displaystyle\begin{split}&\|\Lambda(n,\omega)|_{X_{j}}\|\leq
K(\omega)e^{(\lambda_{j}-\epsilon)n},\qquad\forall n<0,\\\
&\|\Lambda(n,\omega)|_{X_{j}}\|\leq
K(\omega)e^{(\lambda_{j}+\epsilon)n},\qquad\forall n\geq 0,\end{split}$ (2.7)
for each $j=1,...,p$, where $K(\omega)>0$ is a random variable tempered from
above. In particular, it satisfies the exponential dichotomy
$\displaystyle\begin{split}&\|\Lambda(n,\omega)|_{X_{u}}\|\leq
K(\omega)e^{(\lambda_{\tau}-\epsilon)n},\qquad\forall n<0,\\\
&\|\Lambda(n,\omega)|_{X_{s}}\|\leq
K(\omega)e^{(\lambda_{\tau+1}+\epsilon)n},\qquad\forall n\geq 0,\end{split}$
(2.8)
where $\lambda_{\tau}>0>\lambda_{\tau+1}$ and $X_{u}:=X_{1}\oplus\cdots\oplus
X_{\tau},$ $X_{s}:=X_{{\tau}+1}\oplus\cdots\oplus X_{{p}}.$ Notice that this
exponential dichotomy is nonuniform since $K(\omega)$ may be arbitrarily large
as $\omega$ varies. However, along each orbit
$(\theta^{n}\omega)_{n\in\mathbb{Z}}$, $K(\omega)$ can increase only at a
subexponential rate which is one of the intrinsic features of RDS. The
following definition on random conjugacy can be found in [1, 42, 44].
###### Definition 2.1.
Two tempered random diffeomporhisms $F(\omega,\cdot)$ and $G(\omega,\cdot)$
are said to be $C^{k,\beta}$ conjugate, where $k\geq 0$ is an integer and
$\beta\in[0,1]$ is a real, if there exists a $C^{k,\beta}$ random
diffeomorphism $\Phi(\omega,\cdot):V(\omega)\to\mathbb{R}^{d}$ such that
$\displaystyle\Phi(\theta\omega,F(\omega,x))=G(\omega,\Phi(\omega,x)),\qquad\forall
x\in V(\omega),$ (2.9)
where $V(\omega)$ is a tempered ball. In particular, as mentioned in the
introduction, $F$ is said to be $C^{k,\beta}$ linearized if $G$ is a linear
random diffeomporhism.
Now, we restate our main theorem of this paper for the sake of completeness of
this section.
###### Theorem 2.1.
Let $F:\Omega\times V(\omega)\to\mathbb{R}^{d}$ be a tempered $C^{2,\alpha}$
$(\alpha\in[0,1])$ random diffeomorphism with an $\epsilon$-slowly continuous
second order derivative when $\alpha=0$. Assume that its linearization
$\Lambda(\omega)=DF(\omega,0)$ at the hyperbolic fixed point $x=0$ satisfies
the condition (2.5). If the Lyapunov exponents satisfy a partial second order
non-resonant condition
$\displaystyle\lambda_{j}\neq\lambda_{i}+\lambda_{\kappa}$ (2.10)
for $i=1,...,{\tau}$, $\kappa={\tau}+1,...,{{p}}$ and $j=1,...,{{p}}$, then
the following assertions are true:
* (i)
Foliations: $F$ has $C^{1,\beta_{\alpha}}$ random stable and unstable
foliations with $C^{2,\alpha}$ leaves in a tempered ball $U(\omega)\subset
V(\omega)$, where $\beta_{\alpha}=0$ when $\alpha=0$ and
$\beta_{\alpha}\in(0,\alpha]$ when $\alpha\in(0,1]$.
* (ii)
Conjugacy: $F(\omega,\cdot)$ is conjugate to its linear part $\Lambda(\omega)$
in a tempered ball $U(\omega)\subset V(\omega)$ by a $C^{1,\beta_{\alpha}}$
random diffeomorphism $\Phi:\Omega\times U(\omega)\to\mathbb{R}^{d}$
satisfying $\Phi(\omega,0)=0$ and $D\Phi(\omega,0)={\rm id}.$
Remark that the assumption of $\epsilon$-slowly continuity is only necessary
for random dynamical systems in the case of $\alpha=0$ since, when
$\alpha\in(0,1]$ or $F$ is deterministic, the second order derivative of
$F(\omega,\cdot)$ is automatically $\epsilon$-slowly continuous. Moreover,
when $F$ is deterministic, the relationship between Lyapunov exponent
$\lambda$ and its corresponding eigenvalue $r$ is $|r|=e^{\lambda}$. Thus
condition (2.10) becomes (1.5). As a result, we have the following $C^{1}$
linearization theorem.
###### Corollary 2.1.
Let $F:V\to\mathbb{R}^{d}$ be a $C^{2,\alpha}$ ($\alpha\in[0,1]$)
diffeomorphism over a neighborhood $V\subset\mathbb{R}^{d}$ of $x=0$. Suppose
the eigenvalues of its linearization $\Lambda:=DF(0)$ at hyperbolic fixed
point $x=0$ satisfy (1.5). Then $F$ is conjugate to its linear part $\Lambda$
in a small neighborhood $U\subset V$ by a $C^{1,\beta_{\alpha}}$
diffeomorphism $\Phi:U\to\mathbb{R}^{d}$ satisfying $\Phi(0)=0$ and
$D\Phi(0)={\rm id},$ where $\beta_{\alpha}=0$ when $\alpha=0$ and
$\beta_{\alpha}\in(0,\alpha]$ when $\alpha\in(0,1]$.
This corollary extends Belitskii’s $C^{1}$ linearization theorem to
$C^{1,\beta_{\alpha}}$ linearization, where $\beta_{\alpha}>0$ whenever
$\alpha>0$. Remark that $C^{1,\beta_{\alpha}}$ linearization with
$\beta_{\alpha}>0$ has important applications in problems such as Lorenz
attractors ([26]) and topological entropy ([30]).
## 3\. Random invariant distribution
In this section, we study the Hölder continuity of random invariant
distributions. We first extend the local random diffeomorphism
$F(\omega,x)=\Lambda(\omega)x+f(\omega,x)$ defined on a tempered ball
$V(\omega)=\\{x\in\mathbb{R}^{d}:\|x\|\leq\rho(\omega)\\}$ to a global one. In
what follows, for a random variable (for example $K:\Omega\to(0,\infty)$), we
use $K(\omega)^{n}$ to denote $K(\omega)$ to the power of $n\in\mathbb{Z}$ and
let $U(\omega):=\\{x\in\mathbb{R}^{d}:\|x\|\leq\rho(\omega)/2\\}$, where
$\|x\|:=\max\\{\|x_{1}\|,...,\|x_{p}\|\\}\quad{\rm
for}~{}~{}x=x_{1}+\cdots+x_{p}\in X_{1}\oplus\cdots\oplus X_{p}.$
Consider a random smooth cut-off function
$u(\omega,\cdot):\mathbb{R}^{d}\to(0,1)$ such that
$\displaystyle u(\omega,x)=\left\\{\begin{array}[]{lll}1,&\forall x\in
U(\omega),\vspace{1ex}\\\ 0,&\forall x\in\mathbb{R}^{d}\backslash
V(\omega),\end{array}\right.\quad\rho(\omega)^{r}\|D^{r}u(\omega,x)\|\leq
C_{u},~{}\forall r=1,2,3,$ (3.3)
where $C_{u}$ is a positive constant. The construction of $u$ is given in
Appendix.
Let $\tilde{F}(\omega,x)$ be an extension of $F(\omega,x)$ given by
$\displaystyle\tilde{F}(\omega,x)=\begin{cases}\Lambda(\omega)x+u(\omega,x)f(\omega,x),&\quad\text{for
all}\;x\in V(\omega)\\\ \Lambda(\omega)x,&\quad\text{for
all}\;x\in\mathbb{R}^{d}\backslash V(\omega).\end{cases}$
Then, by elementary estimations, we have the following result for the
extension $\tilde{F}(\omega,x)$.
###### Lemma 3.1.
The following properties hold for $\tilde{F}(\omega,x)$:
* (i)
$\tilde{F}(\omega,x)=F(\omega,x)$ for all $x\in U(\omega)$.
* (ii)
The derivatives of $\tilde{F}$ satisfy the following estimates.
$\displaystyle\begin{split}&\|D\tilde{F}(\omega,x)-\Lambda(\omega)\|\leq\delta(\omega),\qquad\|D^{2}\tilde{F}(\omega,x)\|\leq
M_{\epsilon}(\omega),\\\
&\|D^{2}\tilde{F}(\omega,x)-D^{2}\tilde{F}(\omega,y)\|\leq
L_{\epsilon,\rho}(\omega)\|x-y\|^{\alpha},\quad\forall
x,y\in\mathbb{R}^{d},\end{split}$
where $\delta(\omega):=M_{\epsilon}(\omega)\rho(\omega)$,
$M_{\epsilon}(\omega)$ and $L_{\epsilon,\rho}(\omega)$ are $\epsilon$-slowly
varying random variables for any $\epsilon>0$ such that
$M_{\epsilon}(\omega)\geq(3C_{u}+1)M(\omega)$ and
$L_{\epsilon,\rho}(\omega)\geq 7C_{u}M(\omega)/\rho(\omega)+L(\omega)$. Here
$M(\omega)$ and $L(\omega)$ are given in (2.3) with $N=2$.
In what follows, we still let $F(\omega,\cdot)$ denote the global mapping
$\tilde{F}(\omega,\cdot)$, and therefore
$F(\omega,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$ satisfies that
$\displaystyle\begin{split}&\|DF(\omega,x)-\Lambda(\omega)\|\leq\delta(\omega),\qquad\|D^{2}F(\omega,x)\|\leq
M_{\epsilon}(\omega),\\\ &\|D^{2}F(\omega,x)-D^{2}F(\omega,y)\|\leq
L_{\epsilon,\rho}(\omega)\|x-y\|^{\alpha},\quad\forall
x,y\in\mathbb{R}^{d},\end{split}$ (3.4)
and $D^{2}F(\omega,\cdot)$ is $\epsilon$-slowly continuous when $\alpha=0$.
Note that the third inequality of (3.4) holds for every Hölder exponents
$\tilde{\alpha}\in[0,\alpha]$ because $F(\omega,x)-\Lambda(\omega)x$ is equal
to $0$ outside the small neighborhood $V(\omega)$.
Throughout this paper, we choose $\epsilon>0$ being sufficiently small such
that
$\displaystyle\epsilon<\frac{1}{100}\min\Big{\\{}$ $\displaystyle
1,~{}\min_{i=1,...,p-1}\frac{\lambda_{j}-\lambda_{j+1}}{2\lambda_{\max}}\min\\{\lambda_{\tau},-\lambda_{\tau+1}\\},\min_{\mbox{\tiny$\begin{array}[]{c}i\in\\{1,...,\tau\\},\kappa\in\\{\tau+1,...,{p}\\},\\\
j\in\\{1,...,{p}\\}\end{array}$}}|\lambda_{i}+\lambda_{\kappa}-\lambda_{j}|\Big{\\}},$
(3.7)
where $\lambda_{\rm max}:=\max\\{2\lambda_{1},-2\lambda_{{p}}\\}>0$. Note that
the choice of $\epsilon$ depends on the Lyapunov exponents only. Let the
diameter of $V(\omega)$ be defined by
$\rho(\omega):=1/(M_{\epsilon}(\omega)\mathfrak{M}_{\epsilon}(\omega))>0$
for a given $\epsilon$-slowly varying tempered random variable
$\mathfrak{M}_{\epsilon}(\omega)>0$ satisfying
$\displaystyle\begin{split}\mathfrak{M}_{\epsilon}(\omega)\geq\max\bigg{\\{}&4,~{}\delta_{\lambda}^{-1}K_{\epsilon}(\theta\omega),~{}8C_{\lambda}K_{\epsilon}(\omega)^{3},~{}(2M_{\epsilon}(\omega)K_{\epsilon}(\omega)^{5})^{2},~{}(2K_{\epsilon}(\omega)^{2})^{4},\\\
&\big{\\{}12K_{\epsilon}(\omega)^{2}e^{\lambda_{j}-\lambda_{j+1}-3\epsilon}\big{\\}}^{4(2\lambda_{\max}-\lambda_{j+1}-\epsilon)/(\lambda_{j}-\lambda_{j+1}-3\epsilon)},\\\
&2C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega),2e^{\lambda_{\max}}C^{3}_{\lambda}K_{\epsilon}(\omega)K_{\epsilon}(\theta\omega)\bigg{\\}}\end{split}$
(3.8)
where $K_{\epsilon}(\omega)\geq\max\\{1,K_{\lambda}K(\omega)\\}>0$ and
$C_{\epsilon}(\omega):=4(K_{\epsilon}(\omega))^{2}M_{\epsilon}(\omega)>0$ are
$\epsilon$-slowly varying random variables, $\delta_{\lambda}>0$ is a small
constant and $C_{\lambda},K_{\lambda}\geq 1$ are constants. Note that the
constants $\delta_{\lambda},C_{\lambda},K_{\lambda}$ are determined by the
Lyapunov exponents only, which will be given in Lemma 3.2 and (3.20). Then, as
seen in Lemma 3.1,
$\displaystyle\delta(\omega):=M_{\epsilon}(\omega)\rho(\omega)=1/\mathfrak{M}_{\epsilon}(\omega)>0$
(3.9)
is small and tempered.
Next, we obtain from (2.1) that
$\displaystyle
D{F}(n,\omega,x)=\begin{cases}D{F}(\theta^{n-1}\omega,{F}(n-1,\omega,x))\circ\cdots\circ
D{F}(\omega,{F}(0,\omega,x)),&n>0,\\\ {\rm id},&n=0,\\\
\big{\\{}D{F}(\theta^{n}\omega,D{F}(n,\omega,x))\big{\\}}^{-1}\circ\cdots\circ\big{\\{}D{F}(\theta^{-1}\omega,D{F}(-1,\omega,x))\big{\\}}^{-1},&n<0.\end{cases}$
(3.10)
For every $\omega\in\Omega$ and $x\in\mathbb{R}^{d}$, we set
$\varpi:=(\omega,x)\in{\bf\Omega}:=\Omega\times\mathbb{R}^{d},\qquad\vartheta\varpi:=(\theta\omega,F(\omega,x)),\qquad{\bf\Lambda}(\varpi):=DF(\omega,x).$
Then, $D{F}(n,\omega,x)$ is a cocycle generated by ${\bf\Lambda}(\varpi)$
driven by $\vartheta^{n}\varpi:=(\theta^{n}\omega,F(n,\omega,x))$. The first
inequality of (3.4) implies that ${\bf\Lambda}(\varpi)$ is a small
perturbation of $\Lambda(\omega)$ with the difference
$\|{\bf\Lambda}(\varpi)-\Lambda(\omega)\|\leq\delta(\omega).$
Note that $\Lambda(\omega)=\Lambda(\varpi)$ since $\Lambda(\omega)$ does not
depend on $x$. Using (2.7) and applying the roughness of the tempered
exponential dichotomy ([64, Theorem 1]) to the system
$e^{-(\lambda_{j}+\lambda_{j+1})n/2}\Lambda(n,\omega)$ for each
$j=1,\cdots,p-1$ with the splitting
$\mathbb{R}^{d}=\big{(}X_{1}\oplus\cdots\oplus
X_{j}\big{)}\oplus\big{(}X_{j+1}\oplus\cdots\oplus X_{p}\big{)},$
we have the following lemma.
###### Lemma 3.2.
There are positive constants $\delta_{\lambda}$ and $K_{\lambda}$ depending on
the Lyapunov exponents only such that if
$\delta(\omega)\leq\delta_{\lambda}/K_{\epsilon}(\theta\omega)$, then there is
an invariant splitting for the cocycle ${\bf\Lambda}(n,\varpi)$ generated by
${\bf\Lambda}(\varpi)$
$\mathbb{R}^{d}=E_{1}(\omega,x)\oplus\cdots\oplus E_{{p}}(\omega,x)$
such that for each $j=1,\cdots,p$,
$\displaystyle\begin{split}&\|{\bf\Lambda}(n,\varpi)|_{E_{j}}\|\leq
K_{\lambda}K(\omega)e^{(\lambda_{j}-2\epsilon)n},\qquad\forall n<0,\\\
&\|{\bf\Lambda}(n,\varpi)|_{E_{j}}\|\leq
K_{\lambda}K(\omega)e^{(\lambda_{j}+2\epsilon)n},\qquad\forall n\geq
0.\end{split}$
The invariance means that
${\bf\Lambda}(n,\varpi)E_{j}(\varpi)=E_{j}(\vartheta^{n}\varpi)$. Without loss
of generality, we still let $\epsilon$ denote $2\epsilon$. Note that
$D{F}(n,\omega,x)={\bf\Lambda}(n,\varpi)\;\text{ and
}F(\omega,x)=F(1,\omega,x).$
Thus, we have
$\displaystyle
DF(\omega,x)\\{E_{j}(\omega,x)\\}=E_{j}(\theta\omega,F(\omega,x)),\qquad\forall
j=1,...,{p},$ (3.11)
and
$\displaystyle\begin{split}&\|DF(n,\omega,x)|_{E_{j}(\omega,x)}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}-\epsilon)n},\qquad\forall n<0,\\\
&\|DF(n,\omega,x)|_{E_{j}(\omega,x)}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}+\epsilon)n},\qquad\forall n\geq
0,\end{split}$ (3.12)
where $K_{\epsilon}(\omega)\geq K_{\lambda}K(\omega)$ is an $\epsilon$-slowly
varying random variable. We point out that the measurability of
$E_{j}(\omega,x)$ with respect to $\omega$ follows from that $E_{j}(\omega,x)$
is the tangent spaces of an intermediate foliation whose partial derivative is
measurable with respect to $\omega$ (see (4.38) below in the next section). As
in the deterministic case (see [50, Sections 2 and 6]), for each $j=1,...,p$,
$E_{j}(\omega):=\\{E_{j}(\omega,x)\subset\mathbb{R}^{d}:x\in\mathbb{R}^{d}\\}$
is called a random intermediate distribution and $x$ is the base point of the
distribution. In particular,
$E_{u}(\omega):=\\{E_{u}(\omega,x)\subset\mathbb{R}^{d}:x\in\mathbb{R}^{d}\\}\mbox{
and
}E_{s}(\omega):=\\{E_{s}(\omega,x)\subset\mathbb{R}^{d}:x\in\mathbb{R}^{d}\\}$
are called the random unstable distribution and random stable distribution
respectively, with fibers
$\displaystyle E_{u}(\omega,x):=E_{1}(\omega,x)\oplus\cdots\oplus
E_{\tau}(\omega,x),$ $\displaystyle
E_{s}(\omega,x):=E_{\tau+1}(\omega,x)\oplus\cdots\oplus E_{{p}}(\omega,x)$
satisfying the following:
* (i)
$\mathbb{R}^{d}=E_{u}(\omega,x)\oplus E_{s}(\omega,x);$
* (ii)
The splitting is invariant, i.e.,
$\displaystyle\begin{split}&DF(\omega,x)E_{u}(\omega,x)=E_{u}(\theta\omega,F(\omega,x)),\\\
&DF(\omega,x)E_{s}(\omega,x)=E_{s}(\theta\omega,F(\omega,x));\end{split}$
(3.13)
* (ii)
For each $\varrho_{0}\in(0,\,\min\\{\lambda_{\tau},-\lambda_{\tau+1}\\})$,
there exists a random variable $K_{\epsilon}(\omega)>0$ tempered from above
such that
$\displaystyle\begin{split}\|DF(n,\omega,x)|_{E_{u}(\omega,x)}\|&\leq
K_{\epsilon}(\omega)e^{\varrho_{0}n},\qquad\forall n<0,\\\
\|DF(n,\omega,x)|_{E_{s}(\omega,x)}\|&\leq
K_{\epsilon}(\omega)e^{-\varrho_{0}n},\qquad\forall n\geq 0.\end{split}$
(3.14)
Finally, we show that each $E_{j}(\omega,x)$ is Hölder continuous with respect
to $x$. Recall that the distance between two subspaces $E$ and $\tilde{E}$ of
$\mathbb{R}^{d}$ is given by
$\displaystyle{\rm dist}(E,\tilde{E}):=\max\Big{(}\max_{v_{1}\in
E,\,\|v_{1}\|=1}{\rm
dist}(v_{1},\tilde{E}),\max_{v_{2}\in\tilde{E},\,\|v_{2}\|=1}{\rm
dist}(v_{2},E)\Big{)},$ (3.15)
where ${\rm dist}(v,E):=\min_{w\in E}\|v-w\|$.
###### Theorem 3.1.
For all $x,y\in\mathbb{R}^{d}$, $\omega\in\Omega$ and all $j=1,...,{p}$,
$\displaystyle\begin{split}&{\rm
dist}(E_{j}(\omega,x),E_{j}(\omega,y))\leq\|x-y\|^{\beta_{E}},\\\ &{\rm
dist}(E_{j}(\omega,x),E_{j}(\omega,y))\leq\delta_{E}(\omega),\end{split}$
(3.16)
where
$\beta_{E}:=\min_{\ell=1,...,p-1}(\lambda_{\ell}-\lambda_{\ell+1}-3\epsilon)/(6\lambda_{\max})\in(0,1)$
and $\delta_{E}(\omega):=\delta(\omega)^{\beta_{E}}.$
Proof. First of all, we define the subspaces
$\displaystyle E_{+}(\omega,x):=E_{1}(\omega,x)\oplus\cdots\oplus
E_{j}(\omega,x),$ $\displaystyle
E_{-}(\omega,x):=E_{j+1}(\omega,x)\oplus\cdots\oplus E_{{p}}(\omega,x)$
and claim that for all integers $m\geq 0$
$\displaystyle\begin{split}\|DF(m,\omega,x)\zeta_{-}\|&\leq
2K_{\epsilon}(\omega)e^{(\lambda_{j+1}+\epsilon)m}\|\zeta_{-}\|,\\\
\|DF(m,\omega,x)\zeta_{-}^{\bot}\|&\geq(2K_{\epsilon}(\omega))^{-1}e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{-}^{\bot}\|\end{split}$
(3.17)
for $\zeta_{-}\in E_{-}(\omega,x)$ and
$\zeta_{-}^{\bot}\in\\{E_{-}(\omega,x)\\}^{\bot}$, the orthogonal complement
of $E_{-}(\omega,x)$. In fact, we have from (3.12) that
$\displaystyle\begin{split}&\|DF(n,\omega,x)\zeta_{+}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}-\epsilon)n}\|\zeta_{+}\|,\qquad\forall
n<0,\\\ &\|DF(n,\omega,x)\zeta_{-}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j+1}+\epsilon)n}\|\zeta_{-}\|,\qquad\forall
n\geq 0,\end{split}$ (3.18)
for $\zeta_{+}\in E_{+}(\omega,x)$ and $\zeta_{-}\in E_{-}(\omega,x)$. Then
the first inequality of (3.17) follows from the second one of (3.18)
immediately.
Now, we show that the second inequality of (3.17) holds. We first notice that
for $m\geq 0$
$\displaystyle\big{(}DF(-m,\omega,x)\big{)}^{-1}=DF(m,\theta^{-m}\omega,F(-m,\omega,x))$
and therefore the first inequality of (3.18) yields
$\displaystyle\|DF(m,\theta^{-m}\omega,F(-m,\omega,x))\tilde{\zeta}_{+}\|\geq(1/K_{\epsilon}(\omega))e^{(\lambda_{j}-\epsilon)m}\|\tilde{\zeta}_{+}\|$
for $\tilde{\zeta}_{+}\in E_{+}(\theta^{-m}\omega,F(-m,\omega,x))$. Replacing
$F(-m,\omega,x)$ and $\theta^{-m}\omega$ with $x$ and $\omega$ respectively,
we obtain
$\displaystyle\|DF(m,\omega,x)\zeta_{+}\|\geq(1/K_{\epsilon}(\theta^{m}\omega))e^{(\lambda_{j}-\epsilon)m}\|\zeta_{+}\|\geq(1/K_{\epsilon}(\omega))e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{+}\|.$
(3.19)
Let $\Pi_{\pm}(\omega,x):\mathbb{R}^{d}\to E_{\pm}(\omega,x)$ be the
projections associated with $\mathbb{R}^{d}=E_{+}(\omega,x)\oplus
E_{-}(\omega,x)$ and let $\pi_{\pm}:\mathbb{R}^{d}\to X_{\pm}$ be the
projections associated with $\mathbb{R}^{d}=X_{+}\oplus X_{-}$, where
$\displaystyle X_{+}:=X_{1}\oplus\cdots\oplus X_{j},\quad
X_{-}:=X_{j+1}\oplus\cdots\oplus X_{{p}}.$
Then, using the arguments as in [64, (3.41)-(3.44) and (3.16)], we have
$\displaystyle\|\Pi_{+}(\omega,x)-\pi_{+}\|\leq
C_{\lambda}\delta(\omega)K_{\epsilon}(\omega),\quad\|\Pi_{-}(\omega,x)-\pi_{-}\|\leq
C_{\lambda}\delta(\omega)K_{\epsilon}(\omega),$ (3.20)
where $C_{\lambda}>0$ is a constant depending only on the Lyapunov exponents.
Using the second inequality of (3.20), when $\delta(\omega)$ is small enough,
there are $p-j$ vectors
$\displaystyle
v_{1}^{-}:=(v_{1,1},\cdots,v_{1,j},1,\cdots,0)^{T},~{}...,~{}v_{p-j}^{-}:=(v_{p-j,1},\cdots,v_{p-j,j},0,\cdots,1)^{T}\in
E_{-}(\omega,x)$
with $v_{i,j}\in\mathbb{R}$ ($T$ means the transpose of a vector) such that
$\displaystyle\max\\{|v_{i,1}|,\cdots,|v_{i,j}|\\}=\|v_{i}^{-}-\pi_{-}v_{i}^{-}\|=\|(\Pi_{-}(\omega,x)-\pi_{-})v_{i}^{-}\|\leq
C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\|v_{i}^{-}\|,$
which implies that $\|v_{i}^{-}\|=\max\\{|v_{i,1}|,\cdots,|v_{i,j}|,1\\}=1$
since $\delta(\omega)>0$ is sufficiently small. Therefore
$\displaystyle\max\\{|v_{i,1}|,\cdots,|v_{i,j}|\\}=\|v_{i}^{-}-\pi_{-}v_{i}^{-}\|\leq
C_{\lambda}\delta(\omega)K_{\epsilon}(\omega).$ (3.21)
For any vector
$\zeta_{-}^{\bot}:=(\zeta_{1},...\zeta_{j},\zeta_{j+1},...,\zeta_{p})\in(E_{-}(\omega,x))^{\bot},$
since $\zeta_{-}^{\bot}$ is perpendicular to $v_{1}^{-},\cdots,v_{p-j}^{-}$,
we have $B\zeta_{-}^{\bot}=0,$ where
$\displaystyle B:=\begin{pmatrix}v_{1,1}&\cdots&v_{1,j}&1&\cdots&0\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
v_{p-j,1}&\cdots&v_{p-j,j}&0&\cdots&1\end{pmatrix}.$
Thus, $B\zeta_{-}^{\bot}=0$ is equivalent to
$\displaystyle\begin{pmatrix}\zeta_{j+1}\\\ \vdots\\\
\zeta_{p}\end{pmatrix}=\begin{pmatrix}-v_{1,1}&\cdots&-v_{1,j}\\\
\vdots&\ddots&\vdots\\\
-v_{p-j,1}&\cdots&-v_{p-j,j}\end{pmatrix}\begin{pmatrix}\zeta_{1}\\\ \vdots\\\
\zeta_{j}\end{pmatrix},$
which together with (3.21) indicates that
$\displaystyle\|\pi_{-}\zeta_{-}^{\bot}\|\leq
jC_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\|\zeta_{-}^{\bot}\|\leq
C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\|\zeta_{-}^{\bot}\|.$
Here, we omit $j\leq p$ (the number of the Lyapunov exponents) since
$C_{\lambda}>0$ denotes a sufficiently large constant depending on the
Lyapunov exponents. It implies that
$\|\pi_{+}\zeta_{-}^{\bot}\|\geq\|\zeta_{-}^{\bot}\|-\|\pi_{-}\zeta_{-}^{\bot}\|\geq(1-C_{\lambda}\delta(\omega)K_{\epsilon}(\omega))\|\zeta_{-}^{\bot}\|.$
Therefore,
$\displaystyle\|\Pi_{+}(\omega,x)\zeta_{-}^{\bot}\|$
$\displaystyle\geq\|\pi_{+}\zeta_{-}^{\bot}\|-\|\Pi_{+}(\omega,x)\zeta_{-}^{\bot}-\pi_{+}\zeta_{-}^{\bot}\|$
$\displaystyle\geq\\{1-2C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\\}\|\zeta_{-}^{\bot}\|$
(3.22)
and
$\displaystyle\|\Pi_{-}(\omega,x)\zeta_{-}^{\bot}\|$
$\displaystyle\leq\|\pi_{-}\zeta_{-}^{\bot}\|+\|\Pi_{-}(\omega,x)\zeta_{-}^{\bot}-\pi_{-}\zeta_{-}^{\bot}\|$
$\displaystyle\leq
2C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\|\zeta_{-}^{\bot}\|.$ (3.23)
It follows from (3.18)-(3.23) that
$\displaystyle\|DF(m,\omega,x)\zeta_{-}^{\bot}\|$
$\displaystyle\geq\|DF(m,\omega,x)\Pi_{+}(\omega,x)\zeta_{-}^{\bot}\|-\|DF(m,\omega,x)\Pi_{-}(\omega,x)\zeta_{-}^{\bot}\|$
$\displaystyle\geq(1/K_{\epsilon}(\omega))e^{(\lambda_{j}-2\epsilon)m}\|\Pi_{+}(\omega,x)\zeta_{-}^{\bot}\|-K_{\epsilon}(\omega)e^{(\lambda_{j+1}+\epsilon)m}\|\Pi_{-}(\omega,x)\zeta_{-}^{\bot}\|$
$\displaystyle\geq(1/K_{\epsilon}(\omega))\\{1-2C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)\\}e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{-}^{\bot}\|$
$\displaystyle\quad-2C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)^{2}e^{(\lambda_{j+1}+\epsilon)m}\|\zeta_{-}^{\bot}\|$
$\displaystyle\geq(1/K_{\epsilon}(\omega))e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{-}^{\bot}\|-4C_{\lambda}\delta(\omega)K_{\epsilon}(\omega)^{2}e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{-}^{\bot}\|$
$\displaystyle\geq(2K_{\epsilon}(\omega))^{-1}e^{(\lambda_{j}-2\epsilon)m}\|\zeta_{-}^{\bot}\|$
(3.24)
since $\mathfrak{M}_{\epsilon}(\omega)\geq
8C_{\lambda}K_{\epsilon}(\omega)^{3}$ (i.e.,
$0<\delta(\omega)\leq(8C_{\lambda}K_{\epsilon}(\omega)^{3})^{-1}$) by (3.8).
This shows the second inequality of (3.17) and (3.17) is proved.
Next, we know from the first inequality of (3.4) that
$\displaystyle\|DF(\omega,x)-DF(\omega,y)\|\leq\|DF(\omega,x)-\Lambda(\omega)\|+\|DF(\omega,y)-\Lambda(\omega)\|\leq
2\delta(\omega),\quad\forall x,y\in\mathbb{R}^{d},$
and from (3.12) that
$\displaystyle\|DF(m,\omega,x)\|\leq\max_{j=1,...,p}\|DF(m,\omega,x)|_{E_{j}(\omega,x)}\|\leq
K_{\epsilon}(\omega)e^{\lambda_{\max}m},\qquad\forall m\geq 0,$ (3.25)
where $\lambda_{\rm max}:=\max\\{2\lambda_{1},-2\lambda_{{p}}\\}>0.$ Then, we
have
$\displaystyle\|DF(\theta^{m}\omega,F(m,\omega,x))-DF(\theta^{m}\omega,F(m,\omega,y))\|$
$\displaystyle=\|DF(\theta^{m}\omega,F(m,\omega,x))-DF(\theta^{m}\omega,F(m,\omega,y))\|^{1/2+1/2}$
$\displaystyle\leq(2\delta(\theta^{m}\omega))^{1/2}\\{M_{\epsilon}(\theta^{m}\omega)\|F(m,\omega,x)-F(m,\omega,x)\|\\}^{1/2}$
$\displaystyle\leq(2\delta(\theta^{m}\omega))^{1/2}\\{M_{\epsilon}(\theta^{m}\omega)K_{\epsilon}(\omega)e^{\lambda_{\max}m}\|x-y\|\\}^{1/2}$
$\displaystyle\leq\\{2\delta(\omega)M_{\epsilon}(\omega)K_{\epsilon}(\omega)\\}^{1/2}e^{(1/2)(\lambda_{\max}+2\epsilon)m}\|x-y\|^{1/2}$
(3.26)
by the second inequality of (3.4) and
$\displaystyle\|DF(\theta^{m}\omega,F(m,\omega,x))-DF(\theta^{m}\omega,F(m,\omega,y))\|\leq
2\delta(\theta^{m}\omega)\leq 2e^{\epsilon m}\delta(\omega).$ (3.27)
In the above (3.26)-(3.27), we know that
$e^{-\epsilon|k|}\delta(\theta^{k}\omega)\leq\delta(\omega),$ as known from
(3.9).
Notice the fact that
$\displaystyle a_{m-1}...a_{0}-b_{m-1}...b_{0}$
$\displaystyle=a_{m-1}...a_{1}\,(a_{0}-b_{0})+(a_{m-1}-b_{m-1})\,b_{m-2}...b_{0}$
$\displaystyle=\sum_{i=0}^{m-1}a_{m-1}...a_{i+1}\,(a_{i}-b_{i})\,b_{i-1}...b_{1}.$
It follows from (3.26) that
$\displaystyle\|DF(m,\omega,x)-DF(m,\omega,y)\|$
$\displaystyle=\|D{F}(\theta^{m-1}\omega,{F}(m-1,\omega,x))\circ\cdots\circ
D{F}(\omega,{F}(0,\omega,x))$
$\displaystyle\quad-D{F}(\theta^{m-1}\omega,{F}(m-1,\omega,y))\circ\cdots\circ
D{F}(\omega,{F}(0,\omega,y))\|$
$\displaystyle\leq\sum_{i=0}^{m-1}\|DF(m-i-1,\theta^{i+1}\omega,F(i+1,\omega,x))\|$
$\displaystyle\quad\cdot\|DF(\theta^{i}\omega,F(i,\omega,x))-DF(\theta^{i}\omega,F(i,\omega,y))\|\,\|DF(i,\omega,y)\|$
$\displaystyle\leq\sum_{i=0}^{m-1}K_{\epsilon}(\theta^{i+1}\omega)e^{\lambda_{\max}(m-i-1)}\\{2\delta(\omega)M_{\epsilon}(\omega)K_{\epsilon}(\omega)\\}^{1/2}e^{(1/2)(\lambda_{\max}+2\epsilon)i}\|x-y\|^{1/2}K_{\epsilon}(\omega)e^{\lambda_{\max}i}$
$\displaystyle\leq\sum_{i=0}^{m-1}e^{\epsilon(i+1)}e^{\lambda_{\max}(m-1)}e^{(1/2)(\lambda_{\max}+2\epsilon)i}\\{2\delta(\omega)M_{\epsilon}(\omega)K_{\epsilon}(\omega)^{5}\\}^{1/2}\|x-y\|^{1/2}$
$\displaystyle\leq e^{2\lambda_{\max}m}\delta(\omega)^{1/4}\|x-y\|^{1/2}$
(3.28)
since
$\mathfrak{M}_{\epsilon}(\omega)\geq(2M_{\epsilon}(\omega)K_{\epsilon}(\omega)^{5})^{2}$
(i.e.,
$\delta(\omega)\leq(2M_{\epsilon}(\omega)K_{\epsilon}(\omega)^{5})^{-2}$) by
(3.8), where
$me^{\epsilon(i+1)}e^{(1/2)(\lambda_{\max}+2\epsilon)i}\leq
e^{\lambda_{\max}m}$
as $0\leq i\leq m-1$ and $\epsilon>0$ is small. Similarly, it follows from
(3.27) that
$\displaystyle\|DF(m,\omega,x)-DF(m,\omega,y)\|$
$\displaystyle\leq\sum_{i=0}^{m-1}\|DF(m-i-1,\theta^{i+1}\omega,F(i+1,\omega,x))\|$
$\displaystyle\quad\cdot\|DF(\theta^{i}\omega,F(i,\omega,x))-DF(\theta^{i}\omega,F(i,\omega,y))\|\,\|DF(i,\omega,y)\|$
$\displaystyle\leq\sum_{i=0}^{m-1}K_{\epsilon}(\theta^{i+1}\omega)e^{\lambda_{\max}(m-i-1)}2e^{\epsilon
m}\delta(\omega)K_{\epsilon}(\omega)e^{\lambda_{\max}i}$
$\displaystyle\leq\sum_{i=0}^{m-1}e^{\epsilon(i+1)}e^{\lambda_{\max}(m-1)}2e^{\epsilon
m}\delta(\omega)K_{\epsilon}(\omega)^{2}\leq
e^{2\lambda_{\max}m}\delta(\omega)^{3/4}$ (3.29)
since $\mathfrak{M}_{\epsilon}(\omega)\geq(2K_{\epsilon}(\omega)^{2})^{4}$
(i.e., $\delta(\omega)\leq(2K_{\epsilon}(\omega)^{2})^{-4}$) by (3.8).
Hence, using (3.17) and (3.28)-(3.29), we apply [50, Lemma 3.10] for every
fixed $\omega\in\Omega$, which is given exactly as follows (in notations used
in [50]).
###### Lemma 3.3.
Let $(A_{n})_{n\in\mathbb{N}}$ and $(B_{n})_{n\in\mathbb{N}}$ be two sequences
of real $N\times N$ matrices such that for some $\Delta\in(0,1)$ and $a>1$,
$\|A_{n}-B_{n}\|\leq\Delta a^{n}$
for all positive integers $n$. Assume that there exist subspaces
$E_{A},E_{B}\in\mathbb{R}^{N}$ and numbers $0<\lambda<\mu$ and $C>1$ such that
$\lambda<a$ and for each $n\geq 0$,
$\displaystyle\|A_{n}v\|\leq C\lambda^{n}~{}if~{}v\in
E_{A};\qquad\|A_{n}w\|\geq C^{-1}\mu^{n}\|w\|~{}if~{}w\in{E_{A}}^{\bot};$
$\displaystyle\|B_{n}v\|\leq C\lambda^{n}~{}if~{}v\in
E_{B};\qquad\|B_{n}w\|\geq C^{-1}\mu^{n}\|w\|~{}if~{}w\in{E_{B}}^{\bot}.$
Then
${\rm dist}(E_{A},E_{B})\leq
3C^{2}\frac{\mu}{\lambda}\Delta^{\frac{\log\mu-\log\lambda}{\log
a-\log\lambda}}.$
Notice that if $\|x-y\|<1$ then the notations of [50, Lemma 3.10] and ours
have the correspondence:
$\displaystyle n:=m,\quad A_{m}:=DF(m,\omega,x),\quad B_{m}:=DF(m,\omega,y),$
$\displaystyle\Delta:=\delta(\omega)^{1/4}\|x-y\|^{1/2}~{}{\rm
or}~{}\delta(\omega)^{3/4},\quad a:=e^{2\lambda_{\max}},\quad\mbox{(see
\eqref{DFmm-2} or \eqref{DFmm-1})}$ $\displaystyle
E_{A}:=E_{-}(\omega,x),\quad E_{B}:=E_{-}(\omega,y),\quad\mbox{(see \eqref{ED-
md})}$ $\displaystyle
C:=2K_{\epsilon}(\omega),\quad\lambda:=e^{\lambda_{j+1}+\epsilon},\quad\mu:=e^{\lambda_{j}-2\epsilon}.\quad\mbox{(see
\eqref{ED-md})}$
Then, all conditions of Lemma 3.3 are verified and we conclude that for all
$x,y$ satisfying $\|x-y\|<1$
$\displaystyle{\rm dist}(E_{-}(\omega,x),E_{-}(\omega,y))$ $\displaystyle\leq
12K_{\epsilon}(\omega)^{2}e^{\lambda_{j}-\lambda_{j+1}-3\epsilon}\big{(}\delta(\omega)^{1/4}\|x-y\|^{1/2}\big{)}^{(\lambda_{j}-\lambda_{j+1}-3\epsilon)/(2\lambda_{\max}-\lambda_{j+1}-\epsilon)}$
$\displaystyle\leq\|x-y\|^{(1/2)(\lambda_{j}-\lambda_{j+1}-3\epsilon)/(2\lambda_{\max}-\lambda_{j+1}-\epsilon)}$
since
$\mathfrak{M}_{\epsilon}(\omega)\geq\big{\\{}12K_{\epsilon}(\omega)^{2}e^{\lambda_{j}-\lambda_{j+1}-3\epsilon}\big{\\}}^{4(2\lambda_{\max}-\lambda_{j+1}-\epsilon)/(\lambda_{j}-\lambda_{j+1}-3\epsilon)}$
by (3.8), i.e.,
$0<\delta(\omega)\leq\big{\\{}12K_{\epsilon}(\omega)^{2}e^{\lambda_{j}-\lambda_{j+1}-3\epsilon}\big{\\}}^{-4(2\lambda_{\max}-\lambda_{j+1}-\epsilon)/(\lambda_{j}-\lambda_{j+1}-3\epsilon)},$
and that
$\displaystyle{\rm dist}(E_{-}(\omega,x),E_{-}(\omega,y))$ $\displaystyle\leq
12K_{\epsilon}(\omega)^{2}e^{\lambda_{j}-\lambda_{j+1}-3\epsilon}\delta(\omega)^{(3/4)(\lambda_{j}-\lambda_{j+1}-3\epsilon)/(2\lambda_{\max}-\lambda_{j+1}-\epsilon)}$
$\displaystyle\leq\delta(\omega)^{(1/2)(\lambda_{j}-\lambda_{j+1}-3\epsilon)/(2\lambda_{\max}-\lambda_{j+1}-\epsilon)}.$
A similar conclusion holds in the subspaces $E_{+}(\omega,x)$. Then, by
choosing different $j\in\\{1,...,p\\}$ and intersecting appropriate subspaces,
when $\|x-y\|<1$ we prove (3.16) with
$\beta_{E}:=\min_{\ell=1,...,p-1}\frac{\lambda_{\ell}-\lambda_{\ell+1}-3\epsilon}{6\lambda_{\max}},\qquad\delta_{E}(\omega):=\delta(\omega)^{\beta_{E}},$
where $\beta_{E}\in(0,1)$, as known from (3.7). When $\|x-y\|\geq 1$, the
first inequality of (3.16) can be indicated by the second inequality of (3.16)
as $\delta_{E}(\omega)<1$. This completes the proof of Theorem 3.1. $\Box$
The original idea of the above proof comes from [9] for the invariant
distributions of Anosov diffeomorphims, i.e., hyperbolic diffeomorphims
defined on manifolds, from which we see that the classical method only gives
Hölder continuity of distributions in the base point $x$. In fact, it is
widely believed that the distributions are not $C^{1}$ (not even Lipschitzian)
smooth with respect to $x$ in general without the bunching condition (see e.g.
[21, 33, 50]). However, in the following Theorem 9.1, we will show that in the
local case the stable and unstable distributions are $C^{1,\beta_{\alpha}}$
under some mild assumptions (i.e., the assumptions of Theorem 2.1).
In order to prove Theorem 9.1, we need to study the intermediate foliations in
section 4 and then decompose a cohomological equation for the distribution in
every intermediate foliation in section 5, where two lemmas concerning random
normal form and the decomposed cohomological equation will be proved in
sections 6-8.
## 4\. Random invariant foliation
In this section, we present the results on various invariant foliations for
random dynamical systems according to the splits of Lyapunov exponents. Some
of results are borrowed from [1, 42, 43, 56].
First, from [1, pp.362-363] and [42, Theorem 3.8 and 3.9], there are the
random unstable foliation
$\\{{\mathcal{W}}_{u}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ and the random stable
foliation $\\{{\mathcal{W}}_{s}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ of
$\mathbb{R}^{d}$ with leaves ${\mathcal{W}}_{u}(\omega,x)$ and
${\mathcal{W}}_{s}(\omega,x)$, which are measurable with respect to $\omega$
and continuous with respect to $x$, such that
$\mathbb{R}^{d}=\cup_{x\in\mathbb{R}^{d}}{\mathcal{W}}_{u}(\omega,x)=\cup_{x\in\mathbb{R}^{d}}{\mathcal{W}}_{s}(\omega,x)$
and
$\displaystyle
F(\omega,{\mathcal{W}}_{u}(\omega,x))\subset{\mathcal{W}}_{u}(\theta\omega,F(\omega,x)),\quad
F(\omega,{\mathcal{W}}_{s}(\omega,x))\subset{\mathcal{W}}_{s}(\theta\omega,F(\omega,x)).$
(4.1)
Notice that (4.1) means the invariance of foliations under the action of
$F(\omega,\cdot)$. Let $\pi_{u}:\mathbb{R}^{d}\to X_{u}$ and
$\pi_{s}:\mathbb{R}^{d}\to X_{s}$ be the projections associated with
$\mathbb{R}^{d}=X_{u}\oplus X_{s}$ and let
$\displaystyle\Lambda_{*}(\omega):=\Lambda(\omega)|_{X_{*}},\quad\Lambda_{*}(n,\omega):=\Lambda(n,\omega)|_{X_{*}}\qquad{\rm
for}~{}~{}*=u,s.$
Consider the following random Lyapunov-Perron equations (abbreviated as the
L-P equations) for constructing the unstable and stable foliations
respectively.
$\displaystyle\begin{split}p_{n}(\omega,x,y_{u})&=\Lambda_{u}(n,\omega)(y_{u}-\pi_{u}x)-\sum_{k=n}^{-1}\Lambda_{u}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{u}(p_{k})(\omega,x,y_{u})\\\
&\quad+\sum_{k=-\infty}^{n-1}\Lambda_{s}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{s}(p_{k})(\omega,x,y_{u}),\qquad\forall
n<0,\\\
p_{0}(\omega,x,y_{u})&=y_{u}-\pi_{u}x+\sum_{k=-\infty}^{-1}\Lambda_{s}({-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{s}(p_{k})(\omega,x,y_{u}),\qquad
n=0,\end{split}$ (4.2)
where
$\Xi_{k}^{*}(p_{k})(\omega,x,y_{u}):=\pi_{*}f\big{(}\theta^{k}\omega,p_{k}(\omega,x,y_{u})+F(k,\omega,x)\big{)}-\pi_{*}f\big{(}\theta^{k}\omega,F(k,\omega,x)\big{)}$
for $*=u,s$, and
$\displaystyle\begin{split}q_{n}(\omega,x,y_{s})&=\Lambda_{s}(n,\omega)(y_{s}-\pi_{s}x)+\sum_{k=0}^{n-1}\Lambda_{s}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{s}(q_{k})(\omega,x,y_{s})\\\
&\quad-\sum_{k=n}^{\infty}\Lambda_{u}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{u}(q_{k})(\omega,x,y_{s}),\qquad\forall
n>0,\\\
q_{0}(\omega,x,y_{s})&=y_{s}-\pi_{s}x-\sum_{k=0}^{\infty}\Lambda_{u}({-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{u}(q_{k})(\omega,x,y_{s}),\qquad
n=0,\end{split}$ (4.3)
where
$\Xi_{k}^{*}(q_{k})(\omega,x,y_{s}):=\pi_{*}f\big{(}\theta^{k}\omega,q_{k}(\omega,x,y_{s})+F(k,\omega,x)\big{)}-\pi_{*}f\big{(}\theta^{k}\omega,F(k,\omega,x)\big{)}$
for $*=u,s$. It follows from [42, Theorem 3.8 and 3.9] that there is a
constant $C_{\lambda}>0$ depending only on the Lyapunov exponents such that if
$\delta(\omega)\leq C_{\lambda}/(K(\theta\omega)K(\omega)),$ (4.4)
then both equation (4.2) and equation (4.3) have continuous solutions
$(p_{n}(\omega,\cdot,\cdot))_{n\leq 0}$ and
$(q_{n}(\omega,\cdot,\cdot))_{n\geq 0}$ respectively. Notice that the
condition (4.4) holds since (3.8) and (3.9), which insures that the associated
Lyapunov-Perron operators are contraction. Furthermore, the leaves of random
unstable foliation and random stable foliation are given by
$\displaystyle\begin{split}{\mathcal{W}}_{u}(\omega,x):=\\{x+p_{0}(\omega,x,y_{u}):y_{u}\in
X_{u}\\},\quad{\mathcal{W}}_{s}(\omega,x):=\\{x+q_{0}(\omega,x,y_{s}):y_{s}\in
X_{s}\\},\end{split}$ (4.5)
respectively and pass through the point $x\in\mathbb{R}^{d}$. This point $x$
is called the base point of the leaves.
Next, we consider random pseudo-unstable foliations and pseudo-unstable
foliations. As we did in section 3, we decompose $\mathbb{R}^{d}=X_{+}\oplus
X_{-}$ for $1\leq j<d$, where
$\displaystyle X_{+}:=X_{1}\oplus\cdots\oplus X_{j},\quad
X_{-}:=X_{j+1}\oplus\cdots\oplus X_{{p}}.$
Let $\pi_{\pm}$ be the associated projections. Set
$\displaystyle\Lambda_{\pm}(\omega):=\Lambda(\omega)|_{X_{\pm}},\quad\Lambda_{\pm}(n,\omega):=\Lambda(n,\omega)|_{X_{\pm}},\quad
f_{\pm}:=\pi_{\pm}f.$
Consider the following Lyapunov-Perron equations associated with the pseudo-
unstable and pseudo-stable invariant foliations respectively,
$\displaystyle\begin{split}p_{n,j}(\omega,x,y_{+})&=\Lambda_{+}(n,\omega)(y_{+}-\pi_{+}x)-\sum_{k=n}^{-1}\Lambda_{+}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{+}(p_{k,j})(\omega,x,y_{+})\\\
&\quad+\sum_{k=-\infty}^{n-1}\Lambda_{-}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{-}(p_{k,j})(\omega,x,y_{+}),\qquad\forall
n<0,\\\
p_{0,j}(\omega,x,y_{+})&=y_{+}-\pi_{+}x+\sum_{k=-\infty}^{-1}\Lambda_{-}({-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{-}(p_{k,j})(\omega,x,y_{+}),\qquad
n=0,\end{split}$ (4.6)
where
$\Xi_{k}^{\pm}(p_{k,j})(\omega,x,y_{+}):=f_{\pm}\big{(}\theta^{k}\omega,p_{k,j}(\omega,x,y_{+})+F(k,\omega,x)\big{)}-f_{\pm}\big{(}\theta^{k}\omega,F(k,\omega,x)\big{)},$
and
$\displaystyle\begin{split}q_{n,j}(\omega,x,y_{-})&=\Lambda_{-}(n,\omega)(y_{-}-\pi_{-}x)+\sum_{k=0}^{n-1}\Lambda_{-}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{-}(q_{k,j})(\omega,x,y_{-})\\\
&\quad-\sum_{k=n}^{\infty}\Lambda_{+}({n-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{+}(q_{k,j})(\omega,x,y_{-}),\qquad\forall
n\geq 0,\\\
q_{0,j}(\omega,x,y_{-})&=y_{-}-\pi_{-}x-\sum_{k=0}^{\infty}\Lambda_{+}({-k-1},\theta^{k+1}\omega)\,\Xi_{k}^{+}(q_{k,j})(\omega,x,y_{-}),\qquad
n=0,\end{split}$ (4.7)
where
$\Xi_{k}^{\pm}(q_{k,j})(\omega,x,y_{-}):=f_{\pm}\big{(}\theta^{k}\omega,q_{k,j}(\omega,x,y_{-})+F(k,\omega,x)\big{)}-f_{\pm}\big{(}\theta^{k}\omega,F(k,\omega,x)\big{)}.$
Using the same arguments used in [56, Lemma 4.5], one has that equations (4.6)
and (4.7) have continuous solutions
$(p_{n,j}(\omega,\cdot,\cdot))_{n\leq 0}\in
C^{-}_{\varrho}:=\big{\\{}\\{p_{n}\\}_{n\leq
0}:\;p_{n}\in\mathbb{R}^{d},\;\;\sup_{n\leq 0}\\{e^{-\varrho
n}\|p_{n}\|\\}<\infty\big{\\}}$
and
$(q_{n,j}(\omega,\cdot,\cdot))_{n\geq 0}\in
C^{+}_{\varrho}:=\big{\\{}\\{q_{n}\\}_{n\geq
0}:\;q_{n}\in\mathbb{R}^{d},\;\;\sup_{n\geq 0}\\{e^{-\varrho
n}\|q_{n}\|\\}<\infty\big{\\}}$
respectively, where
$\varrho\in(\lambda_{j+1}+3\epsilon,\lambda_{j}-3\epsilon)$ and condition
(4.4) is used to show that the associated Lyapunov-Perron operators are
contraction in the Banach spaces $C^{-}_{\varrho}$ and $C^{+}_{\varrho}$
respectively.
Using these solutions, one obtains the following:
* $\bullet$
The strong-unstable (including only unstable directions) foliations
$\\{{\mathcal{W}}_{\leq j}^{uu}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ for
$j\leq\tau$.
* $\bullet$
The pseudo-unstable (including unstable directions and stable ones with larger
Lyapunov exponents) foliations $\\{{\mathcal{W}}_{\leq
j}^{pu}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ for $j>\tau$.
* $\bullet$
The pseudo-stable (including stable directions and unstable ones with smaller
Lyapunov exponents) foliations $\\{{\mathcal{W}}_{\geq
j+1}^{ps}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ for $j<\tau$.
* $\bullet$
The strong-stable (including only stable directions) foliations
$\\{{\mathcal{W}}_{\geq j+1}^{ss}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$ for
$j\geq\tau$.
The leaves of these foliations are given by
$\displaystyle\begin{split}&{\mathcal{W}}_{\leq
j}^{uu}(\omega,x):=\\{x+p_{0,j}(\omega,x,y_{+})\in\mathbb{R}^{d}:y_{+}\in
X_{+}\\},\quad\forall j\leq\tau,\\\ &{\mathcal{W}}_{\leq
j}^{pu}(\omega,x):=\\{x+p_{0,j}(\omega,x,y_{+})\in\mathbb{R}^{d}:y_{+}\in
X_{+}\\},\quad\forall j>\tau,\\\ &{\mathcal{W}}_{\geq
j+1}^{ps}(\omega,x):=\\{x+q_{0,j}(\omega,x,y_{-})\in\mathbb{R}^{d}:y_{-}\in
X_{-}\\},\quad\forall j<\tau,\\\ &{\mathcal{W}}_{\geq
j+1}^{ss}(\omega,x):=\\{x+q_{0,j}(\omega,x,y_{-})\in\mathbb{R}^{d}:y_{-}\in
X_{-}\\},\quad\forall j\geq\tau,\end{split}$ (4.8)
respectively. Note that the case $j=\tau$ are the random unstable and stable
foliations. The results on random strong unstable foliations and strong stable
foliations can also be found in [43].
The following theorem is on the regularity of the leaves and their continuity
with respect to the base point $x$.
###### Theorem 4.1.
Suppose that the random diffeomorphism
$F:\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ satisfies (3.4). Then, the
measurable mappings $p_{0,j}(\omega,x,\cdot)$ ($j\leq\tau$) and
$q_{0,j}(\omega,x,\cdot)$ ($j\geq\tau$) are tempered $C^{2,\alpha}$
${\rm(}\alpha\in[0,1]{\rm)}$ and $p_{0,j}(\omega,x,\cdot)$ ($j>\tau$) and
$q_{0,j}(\omega,x,\cdot)$ ($j<\tau$) are tempered $C^{1,\beta}$ with a
constant $\beta\in(0,1]$. All the above derivatives and Hölder coefficients
are uniformly bounded with respect to $x,y_{-},y_{+}$, and
$\displaystyle\begin{split}&\|\partial_{y_{+}}p_{0,j}(\omega,x,y_{+})-{\rm
id}_{+}\|\leq
C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega)\delta(\omega),\\\
&\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-{\rm id}_{-}\|\leq
C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega)\delta(\omega)\end{split}$
(4.9)
for all $j=1,...,p$ and all
$(\omega,x,y_{\pm})\in\Omega\times\mathbb{R}^{d}\times X_{\pm}$, where
$C_{\lambda}>0$ is a constant, ${\rm id}_{\pm}$ is the identity in $X_{\pm}$,
and $C_{\epsilon}(\omega):=4(K_{\epsilon}(\omega))^{2}M_{\epsilon}(\omega)$.
Moreover, the following results hold:
* (A1)
For every $\omega\in\Omega$, the mappings $x\mapsto p_{0,j}(\omega,x,\cdot)$
and $x\mapsto q_{0,j}(\omega,x,\cdot)$ are uniformly continuous in the
$C^{1,\beta}$-topology on $U_{1}$;
* (A2)
the mappings $x\mapsto\partial^{2}_{y_{+}^{2}}p_{0,j}(\omega,x,y_{+})$
($j\leq\tau$) and $x\mapsto\partial^{2}_{y_{-}^{2}}q_{0,j}(\omega,x,y_{-})$
($j\geq\tau$) are Hölder continuous with uniformly bounded Hölder coefficients
with respect to $x,y_{+},y_{-}$ on $U_{1}$,
where $U_{1}:=\\{x\in\mathbb{R}^{d}:\|x\|<1/2\\}$.
###### Remark 4.1.
(i) In the theorem above, we used the following simpler notations:
$\partial_{y_{+}}p_{0,j}(\omega,x,y_{+}):=\frac{\partial
p_{0,j}(\omega,x,y_{+})}{\partial{y_{+}}},\qquad\partial^{2}_{y_{+}^{2}}p_{0,j}(\omega,x,y_{+}):=\frac{\partial^{2}p_{0,j}(\omega,x,y_{+})}{\partial
y_{+}^{2}}.$
* (ii)
The most part of this theorem for pseudo-stable foliations was established in
[56] except estimate (4.9) and the uniform boundedness of derivatives with
respect to $x,y_{+},y_{-}$.
* (iii)
In both pseudo-unstable and pseudo-stable cases, this theorem gives that every
leaf of the foliations is $C^{1,\tilde{\beta}}$ for all
$\tilde{\beta}\in(0,\beta]$, where $\beta\in(0,1]$ is determined by the
spectral gap between $\lambda_{j}$ and $\lambda_{j+1}$. In fact, according to
[56, Theorem 3.1], the $\beta$ can be chosen as
$\displaystyle\beta=\min\Bigg{\\{}\min_{j=1,...,p-1}\frac{\lambda_{j}-\lambda_{j+1}}{\lambda_{\max}},~{}\min_{\mbox{\tiny$\begin{array}[]{c}i\in\\{1,...,\tau\\},\kappa\in\\{\tau+1,...,{p}\\},\\\
j\in\\{1,...,{p}\\}\end{array}$}}\frac{|\lambda_{i}+\lambda_{\kappa}-\lambda_{j}|}{2\lambda_{\max}}\Bigg{\\}}$
(4.12)
with $\lambda_{\max}:=\max\\{2\lambda_{1},-2\lambda_{p}\\}$.
* (iv)
Mapping $x\mapsto h(x,\cdot)$ is continuous in the $C^{1,\beta}$-topology
means that $h(x,y)$ is $C^{1}$ with respect to $y$ such that
$\displaystyle\|h(x,\cdot)-h(\tilde{x},\cdot)\|_{C^{1,\beta}}$
$\displaystyle:=\|h(x,\cdot)-h(\tilde{x},\cdot)\|_{C^{1}}+\sup_{y,\tilde{y}\in\mathbb{R}^{d}}\frac{\|\\{\partial_{y}h(x,y)-\partial_{y}h(\tilde{x},y)\\}-\\{\partial_{y}h(x,\tilde{y})-\partial_{y}h(\tilde{x},\tilde{y})\\}\|}{\|y-\tilde{y}\|^{\beta}}\to
0$
as $\|x-\tilde{x}\|\to 0$.
Proof of Theorem 4.1. We prove the theorem only for $q_{0,j}$ since the proof
for $p_{0,j}$ are almost the same. First, using (2.7), we have the following
exponential dichotomy:
$\displaystyle\begin{split}&\|\Lambda(n,\omega)|_{X_{+}}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}-\epsilon)n},~{}~{}\forall n<0,\\\
&\|\Lambda(n,\omega)|_{X_{-}}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j+1}+\epsilon)n},~{}~{}\forall n\geq
0,\end{split}$ (4.13)
where $K_{\epsilon}(\omega)\geq K(\omega)$. Then, using [56, Lemma 4.5], we
obtain that equation (4.7) has a unique solution
$(q_{n,j}(\omega,x,y_{-}))_{n\geq 0}$, measurable with respect to $\omega$ and
tempered $C^{2,\alpha}$ with respect to $y_{-}$ when $j\geq\tau$ (or tempered
$C^{1,\beta}$ when $j<\tau$), such that
$\displaystyle\sup_{n\geq 0}\\{e^{-\varrho
n}\|q_{n,j}(\omega,x,y_{-})\|\\}<\infty,\qquad\sup_{n\geq 0}\\{e^{-\varrho
n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})\|\\}<\infty,$
$\displaystyle\sup_{n\geq 0}\Bigg{\\{}e^{-\varrho
n}\frac{\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{n,j}(\omega,x,\tilde{y}_{-})\|}{\|y_{-}-\tilde{y}_{-}\|^{\beta}}\Bigg{\\}}<\infty.$
Furthermore, differentiating both sides of equation (4.7) with respect to
$y_{-}$, we get
$\displaystyle\begin{split}&\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})\\\
&=\Lambda_{-}(n,\omega)+\sum_{k=0}^{n-1}\Lambda_{-}(n-k-1,\theta^{k+1}\omega)Df_{-}(\theta^{k}\omega,q_{k,j}(\omega,x,y_{-})+F(k,\omega,x))\,\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\\\
&\quad-\sum_{k=n}^{\infty}\Lambda_{+}(n-k-1,\theta^{k+1}\omega)Df_{+}(\theta^{k}\omega,q_{k,j}(\omega,x,y_{-})+F(k,\omega,x))\,\partial_{y_{-}}q_{k,j}(\omega,x,y_{-}),\quad\forall
n>0,\\\ &\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})\\\ &={\rm
id}_{-}-\sum_{k=0}^{\infty}\Lambda_{+}(-k-1,\theta^{k+1}\omega)Df_{+}(\theta^{k}\omega,q_{k,j}(\omega,x,y_{-})+F(k,\omega,x))\,\partial_{y_{-}}q_{k,j}(\omega,x,y_{-}),\quad
n=0.\end{split}$ (4.14)
Then, for an arbitrarily given constant
$\varrho\in(\lambda_{j+1}+3\epsilon,\lambda_{j}-3\epsilon)$, we have
$\displaystyle e^{-\varrho n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})\|$
$\displaystyle\leq e^{-\varrho
n}\|\Lambda_{-}(n,\omega)\|+e^{-\varrho}\sum_{k=0}^{n-1}e^{-\varrho(n-k-1)}\|\Lambda_{-}(n-k-1,\theta^{k+1}\omega)\|\,\delta(\theta^{k}\omega)e^{-\varrho
k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|$
$\displaystyle\quad-e^{-\varrho}\sum_{k=n}^{\infty}e^{-\varrho(n-k-1)}\|\Lambda_{+}(n-k-1,\theta^{k+1}\omega)\|\delta(\theta^{k}\omega)e^{-\varrho
k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|$ $\displaystyle\leq
K_{\epsilon}(\omega)+C_{\lambda}K_{\epsilon}(\theta^{k+1}\omega)\delta(\theta^{k}\omega)\sup_{k\geq
0}\big{\\{}e^{-\varrho k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|\big{\\}}$
$\displaystyle\leq K_{\epsilon}(\omega)+\frac{1}{2}\sup_{k\geq
0}\big{\\{}e^{-\varrho
k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|\big{\\}},$
where (3.4) and (4.13), the fact that
$C_{\lambda}K_{\epsilon}(\theta^{k+1}\omega)\delta(\theta^{k}\omega)\leq 1/2$
(following from (3.8)) are used. Hence,
$\sup_{n\geq 0}\big{\\{}e^{-\varrho
n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})\|\big{\\}}\leq
2K_{\epsilon}(\omega).$
Similarly, we have that
$\sup_{n\geq 0}\big{\\{}e^{-\varrho n}\|q_{n,j}(\omega,x,y_{-})\|\big{\\}}\leq
2K_{\epsilon}(\omega)(\|x\|+\|y_{-}\|)$
and
$\sup_{n\geq
0}\big{\\{}e^{-(\varrho+2\epsilon)n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{n,j}(\omega,x,\tilde{y}_{-})\|\big{\\}}\leq
4(K_{\epsilon}(\omega))^{2}M_{\epsilon}(\omega)\|y_{-}-\tilde{y}_{-}\|^{\beta}.$
Summarizing the above discussion gives
$\displaystyle\begin{split}&q_{0,j}(\omega,x,\pi_{-}x)=0,\qquad\partial_{y_{-}}q_{0,j}(\omega,0,0)={\rm
id}_{-},\\\ &\sup_{n\geq 0}\big{\\{}e^{-\varrho
n}\|q_{n,j}(\omega,x,y_{-})\|\big{\\}}\leq
C_{\epsilon}(\omega)(\|x\|+\|y_{-}\|),\quad\sup_{n\geq 0}\big{\\{}e^{-\varrho
n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})\|\big{\\}}\leq
C_{\epsilon}(\omega),\\\
&\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,x,\tilde{y}_{-})\|\leq
C_{\epsilon}(\omega)\|y_{-}-\tilde{y}_{-}\|^{\beta},\end{split}$ (4.15)
where
$C_{\epsilon}(\omega):=4(K_{\epsilon}(\omega))^{2}M_{\epsilon}(\omega)>0$.
To see that (4.9) holds, we consider the case of $n=0$ in (4.14). It follows
from (4.13), (3.4) and (4.15) that
$\displaystyle\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-{\rm
id}_{-}\|=e^{(\varrho+2\epsilon)\,0}\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,{{0}},{{0}})\|$
$\displaystyle\leq
e^{-(\varrho+\epsilon)}\sum_{k=0}^{\infty}e^{-\epsilon(k+1)}e^{(\varrho+2\epsilon)(k+1)}\|\Lambda_{+}(-k-1,\theta^{k+1}\omega)\|e^{-\epsilon
k}\delta(\theta^{k}\omega)e^{-\varrho
k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|$ $\displaystyle\leq
e^{-(\varrho+\epsilon)}\sum_{k=0}^{\infty}e^{-\epsilon(k+1)}K_{\epsilon}(\theta^{k+1}\omega)e^{(\varrho-\lambda_{j}+3\epsilon)(k+1)}e^{-\epsilon
k}\delta(\theta^{k}\omega)e^{-\varrho
k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|$ $\displaystyle\leq
C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega)\delta(\omega),$
where the facts that $\varrho-\lambda_{j}+3\epsilon<0$ and
$e^{-\epsilon|k|}\delta(\theta^{k}\omega)\leq\delta(\omega)$ (by (3.9)) are
used. This proves the second inequality of (4.9).
Next, we prove (A1) for the mapping $x\mapsto q_{0,j}(\omega,x,\cdot)$. For
the purpose, we claim that
$\displaystyle\sup_{n\geq
0}\Big{\\{}e^{-\lambda_{\max}n}\|q_{n,j}(\omega,x,y_{-})-q_{n,j}(\omega,\tilde{x},y_{-})\|\Big{\\}}\leq
L_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}$ (4.16)
for all $x,\tilde{x},y_{-},\tilde{y}_{-}\in U_{1}$, where
$L_{\epsilon}(\omega)>0$ is an $\epsilon$-slowly varying random variable and
$\sigma\in(0,1)$ is a constant such that
$\sigma<(\lambda_{j}-\lambda_{j+1})/(4\lambda_{\max})$. In order to prove
(4.16), we notice from (3.25) that
$\displaystyle e^{-\lambda_{\max}k}\|F(k,\omega,x)-F(k,\omega,\tilde{x})\|$
$\displaystyle\leq
e^{-\lambda_{\max}k}\sup_{\xi\in\mathbb{R}^{d}}\|DF(k,\omega,\xi)\|\,\|x-\tilde{x}\|$
$\displaystyle\leq K_{\epsilon}(\omega)\|x-\tilde{x}\|,\qquad\forall k\geq 0.$
(4.17)
Using the notation $[h(\cdot)]_{\tilde{x}}^{x}:=h(x)-h(\tilde{x})$ for short,
we have
$\displaystyle\big{[}f(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))-f(\theta^{k}\omega,F(k,\omega,\cdot))\big{]}_{\tilde{x}}^{x}$
$\displaystyle=\big{[}f(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,x))\big{]}_{\tilde{x}}^{x}+\big{[}f(\theta^{k}\omega,q_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\cdot))\big{]}_{\tilde{x}}^{x}$
$\displaystyle\quad-\big{[}f(\theta^{k}\omega,F(k,\omega,\cdot))\big{]}_{\tilde{x}}^{x}$
$\displaystyle=\big{[}f(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,x))\big{]}_{\tilde{x}}^{x}$
$\displaystyle\quad+\int_{0}^{1}\big{[}Df(\theta^{k}\omega,tq_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\cdot))q_{k,j}(\omega,\tilde{x},y_{-})\big{]}_{\tilde{x}}^{x}dt.$
(4.18)
Then, setting
${\tilde{\lambda}}:=\lambda_{j+1}+(\lambda_{j}-\lambda_{j+1})/2,\qquad\|h(\cdot)\|_{\tilde{x}}^{x}:=\|h(x)-h(\tilde{x})\|$
for short, we obtain from (4.18), (3.4), (4.15) and (4.17) that
$\displaystyle
e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+\epsilon)k}\big{\|}f(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))-f(\theta^{k}\omega,F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}$
$\displaystyle\leq\sup_{\xi\in\mathbb{R}^{d}}\|Df(\theta^{k}\omega,\xi)\|\,e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}$
$\displaystyle\quad+e^{-(\lambda_{\max}\sigma+\epsilon)k}\Big{(}\big{\|}Df(\theta^{k}\omega,tq_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}\Big{)}^{\sigma}e^{-{\tilde{\lambda}}k}\|q_{k,j}(\omega,\tilde{x},y_{-})\|$
$\displaystyle\leq\delta(\theta^{k}\omega)e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}+e^{-\epsilon
k}M_{\epsilon}(\theta^{k}\omega)^{\sigma}\Big{(}e^{-\lambda_{\max}k}\|F(k,\omega,\cdot)\big{\|}_{\tilde{x}}^{x}\Big{)}^{\sigma}C_{\epsilon}(\omega)(\|\tilde{x}\|+\|y_{-}\|)$
$\displaystyle\leq\delta(\theta^{k}\omega)e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}\,+\tilde{M}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}$
(4.19)
for all $k\geq 0$ and all $\tilde{x},y_{-}\in U_{1}$, where
$\tilde{M}_{\epsilon}(\omega):=2M_{\epsilon}(\omega)^{\sigma}K_{\epsilon}(\omega)^{\sigma}C_{\epsilon}(\omega)>0$
and the following estimate is used:
$\big{\|}Df(\theta^{k}\omega,tq_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}\leq\Big{(}\big{\|}Df(\theta^{k}\omega,tq_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}\Big{)}^{\sigma}$
for $\sigma\in(0,1)$ since
$\|Df(\theta^{k}\omega,x)\|\leq\delta(\theta^{k}\omega)<1/2$ due to (3.4).
Notice that
${\tilde{\lambda}}+\lambda_{\max}\sigma+2\epsilon\in(\lambda_{j+1}+3\epsilon,\lambda_{j}-3\epsilon)$
with small $\epsilon$ due to the above choices of ${\tilde{\lambda}}$ and
$\sigma$, which implies that
$\sup_{k\geq
0}\Big{\\{}e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}\Big{\\}}<\infty$
by (4.15). Hence, we see from (4.7), (4.13) and (4.19) that for $n\geq 0$
$\displaystyle
e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}\|q_{n,j}(\omega,x,y_{-})-q_{n,j}(\omega,\tilde{x},y_{-})\|$
$\displaystyle\leq
e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}\|\Lambda_{-}(n,\omega)\|\|x-\tilde{x}\|$
$\displaystyle\quad+e^{-{\tilde{\lambda}}-(\lambda_{\max}\sigma+2\epsilon)}\sum_{k=0}^{n-1}e^{-{\tilde{\lambda}}(n-k-1)-(\lambda_{\max}\sigma+2\epsilon)(n-k-1)}\|\Lambda_{-}(n-k-1,\theta^{k+1}\omega)\|$
$\displaystyle\quad\cdot
e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}f_{-}(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))-f_{-}(\theta^{k}\omega,F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}$
$\displaystyle\quad+e^{-{\tilde{\lambda}}-(\lambda_{\max}\sigma+2\epsilon)}\sum_{k=n}^{\infty}e^{-{\tilde{\lambda}}(n-k-1)-(\lambda_{\max}\sigma+2\epsilon)(n-k-1)}\|\Lambda_{+}(n-k-1,\theta^{k+1}\omega)\|$
$\displaystyle\quad\cdot
e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}f_{+}(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))-f_{+}(\theta^{k}\omega,F(k,\omega,\cdot))\big{\|}_{\tilde{x}}^{x}$
$\displaystyle\leq
K_{\epsilon}(\omega)e^{(\lambda_{j+1}-{\tilde{\lambda}}-\lambda_{\max}\sigma-\epsilon)n}\|x-\tilde{x}\|^{\sigma}$
$\displaystyle\quad+e^{-{\tilde{\lambda}}-(\lambda_{\max}\sigma+2\epsilon)}\bigg{(}\sum_{k=0}^{n-1}K_{\epsilon}(\theta^{k+1}\omega)e^{(\lambda_{j+1}-{\tilde{\lambda}}-\lambda_{\max}\sigma-\epsilon)(n-k-1)}$
$\displaystyle\quad+\sum_{k=n}^{\infty}K_{\epsilon}(\theta^{k+1}\omega)e^{(\lambda_{j}-{\tilde{\lambda}}-\lambda_{\max}\sigma-3\epsilon)(n-k-1)}\bigg{)}$
$\displaystyle\quad\cdot\sup_{k\geq
0}\Big{\\{}\delta(\theta^{k}\omega)e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}\,+e^{-\epsilon
k}\tilde{M}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}\Big{\\}}$
$\displaystyle\leq K_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}$
$\displaystyle\quad+C_{\lambda}K_{\epsilon}(\theta^{k+1}\omega)\sup_{k\geq
0}\Big{\\{}\delta(\theta^{k}\omega)e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}\,+e^{-\epsilon
k}\tilde{M}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}\Big{\\}}$
$\displaystyle\leq
2C_{\lambda}K_{\epsilon}(\omega)\tilde{M}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma}+\frac{1}{2}\sup_{k\geq
0}\Big{\\{}e^{-{\tilde{\lambda}}k-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}_{\tilde{x}}^{x}\Big{\\}}$
(4.20)
when $\|x-\tilde{x}\|<1$. Here, the facts $\mathfrak{M}_{\epsilon}(\omega)\geq
2C_{\lambda}K_{\epsilon}(\theta\omega)$ (i.e.,
$\delta(\theta^{k}\omega)\leq(2C_{\lambda}K_{\epsilon}(\theta^{k+1}\omega))^{-1}$
by (3.8)) and
$\lambda_{j+1}-{\tilde{\lambda}}-\lambda_{\max}\sigma-\epsilon<0$,
$\lambda_{j}-{\tilde{\lambda}}-\lambda_{\max}\sigma-3\epsilon>0$ (see the
above choices of ${\tilde{\lambda}}$ and $\sigma$) are used. Hence, we have
$\displaystyle\sup_{n\geq
0}\Big{\\{}e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}\|q_{n,j}(\omega,x,y_{-})-q_{n,j}(\omega,\tilde{x},y_{-})\|\Big{\\}}\leq
4C_{\lambda}K_{\epsilon}(\omega)\tilde{M}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma},$
which proves (4.16) by setting an $\epsilon$-slowly varying random variable
$L_{\epsilon}(\omega)\geq
4C_{\lambda}K_{\epsilon}(\omega)\tilde{M}_{\epsilon}(\omega)>0$ since
$e^{-\lambda_{\max}n}\leq
e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}$.
Next, we prove that
$\displaystyle\sup_{n\geq
0}\Big{\\{}e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}\big{\|}\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{n,j}(\omega,\tilde{x},y_{-})\big{\|}\Big{\\}}\leq\tilde{L}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma^{2}},$
(4.21)
where
$\tilde{L}_{\epsilon}(\omega):=2C_{\lambda}K_{\epsilon}(\omega)M_{\epsilon}(\omega)(L_{\epsilon}(\omega)+K_{\epsilon}(\omega))^{\sigma}C_{\epsilon}(\omega)>0$.
In fact, notice that by (4.16) and (4.17) we have
$\displaystyle
e^{-(\lambda_{\max}\sigma+\epsilon)k}\big{\|}Df(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))\big{\|}^{x}_{\tilde{x}}$
$\displaystyle=\Big{(}\big{\|}Df(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))\big{\|}^{x}_{\tilde{x}}\Big{)}^{1-\sigma}$
$\displaystyle\quad\cdot e^{-\epsilon
k}\Big{(}e^{-\lambda_{\max}k}\big{\|}Df(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))\big{\|}^{x}_{\tilde{x}}\Big{)}^{\sigma}$
$\displaystyle\leq(2\delta(\theta^{k}\omega))^{1-\sigma}e^{-\epsilon
k}M_{\epsilon}(\theta^{k}\omega)^{\sigma}\Big{(}e^{-\lambda_{\rm
max}k}\big{\|}q_{k,j}(\omega,\cdot,y_{-})\big{\|}^{x}_{\tilde{x}}+e^{-\lambda_{\rm
max}k}\big{\|}F(k,\omega,\cdot)\big{\|}^{x}_{\tilde{x}}\Big{)}^{\sigma}$
$\displaystyle\leq
M_{\epsilon}(\omega)(L_{\epsilon}(\omega)+K_{\epsilon}(\omega))^{\sigma}\|x-\tilde{x}\|^{\sigma^{2}}$
(4.22)
when $\|x-\tilde{x}\|<1$. Then, using similar arguments to (4.20), we obtain
from (4.14) that
$\displaystyle
e^{-{\tilde{\lambda}}n-(\lambda_{\max}\sigma+2\epsilon)n}\|\partial_{y_{-}}q_{n,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{n,j}(\omega,\tilde{x},y_{-})\|$
$\displaystyle=C_{\lambda}K_{\epsilon}(\omega)\sup_{k\geq
0}\bigg{\\{}e^{-(\lambda_{\max}\sigma+2\epsilon)k}\big{\|}Df(\theta^{k}\omega,q_{k,j}(\omega,\cdot,y_{-})+F(k,\omega,\cdot))\big{\|}^{x}_{\tilde{x}}$
$\displaystyle\quad\cdot
e^{-{\tilde{\lambda}}k}\|\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})\|+\|Df(\theta^{k}\omega,q_{k,j}(\omega,\tilde{x},y_{-})+F(k,\omega,\tilde{x}))\|$
$\displaystyle\quad\cdot
e^{-{\tilde{\lambda}}k-(\lambda_{\max}+2\epsilon)k}\big{\|}\partial_{y_{-}}q_{k,j}(\omega,\cdot,y_{-})\big{\|}^{x}_{\tilde{x}}\bigg{\\}}$
$\displaystyle\leq
C_{\lambda}K_{\epsilon}(\omega)M_{\epsilon}(\omega)(L_{\epsilon}(\omega)+K_{\epsilon}(\omega))^{\sigma}C_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma^{2}}$
$\displaystyle\quad+\frac{1}{2}\sup_{k\geq
0}\Big{\\{}e^{-{\tilde{\lambda}}k-(\lambda_{\max}+2\epsilon)k}\big{\|}\partial_{y_{-}}q_{k,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{k,j}(\omega,\tilde{x},y_{-})\big{\|}\Big{\\}},$
(4.23)
which yields (4.21).
Setting $n=0$ in (4.21) we obtain that
$\displaystyle\big{\|}\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},y_{-})\big{\|}\leq\tilde{L}_{\epsilon}(\omega)\|x-\tilde{x}\|^{\sigma^{2}}.$
(4.24)
Therefore for any small constant $\varepsilon>0$, we obtain
$\displaystyle\frac{\big{\|}\\{\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},y_{-})\\}-\\{\partial_{y_{-}}q_{0,j}(\omega,x,\tilde{y}_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},\tilde{y}_{-})\\}\big{\|}}{\|y_{-}-\tilde{y}_{-}\|^{(1-\varepsilon)\beta}\|x-\tilde{x}\|^{\varepsilon\sigma^{2}}}$
$\displaystyle\leq\bigg{(}\frac{\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,x,\tilde{y}_{-})\|+\|\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},y_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},\tilde{y}_{-})\|}{\|y_{-}-\tilde{y}_{-}\|^{\beta}}\bigg{)}^{1-\varepsilon}$
$\displaystyle\quad\cdot\bigg{(}\frac{\|\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},y_{-})\|+\|\partial_{y_{-}}q_{0,j}(\omega,x,\tilde{y}_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},\tilde{y}_{-})\|}{\|x-\tilde{x}\|^{\sigma^{2}}}\bigg{)}^{\varepsilon}$
$\displaystyle\leq 2C_{\epsilon}(\omega)\tilde{L}_{\epsilon}(\omega),$ (4.25)
due to (4.15). Combining (4.16) (setting $n=0$), (4.24) with (4.25) together
gives that
$\displaystyle\|q_{0,j}(\omega,x,\cdot)-q_{0,j}(\omega,\tilde{x},\cdot)\|_{C^{1,(1-\varepsilon)\beta}}$
$\displaystyle=\|q_{0,j}(\omega,x,\cdot)-q_{0,j}(\omega,\tilde{x},\cdot)\|_{C^{1}}$
$\displaystyle\quad+\sup_{y_{-},\tilde{y}_{-}\in
X_{-}}\frac{\big{\|}\\{\partial_{y_{-}}q_{0,j}(\omega,x,y_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},y_{-})\\}-\\{\partial_{y_{-}}q_{0,j}(\omega,x,\tilde{y}_{-})-\partial_{y_{-}}q_{0,j}(\omega,\tilde{x},\tilde{y}_{-})\\}\big{\|}}{\|y_{-}-\tilde{y}_{-}\|^{(1-\varepsilon)\beta}}$
$\displaystyle\leq\\{L_{\epsilon}(\omega)+\tilde{L}_{\epsilon}(\omega)+2C_{\epsilon}(\omega)\tilde{L}_{\epsilon}(\omega)\\}\|x-\tilde{x}\|^{\varepsilon\sigma^{2}}$
when $\|x-\tilde{x}\|<1$. Since $\beta$ is in an open interval determined by
the spectral gaps (see (4.12)) and $\varepsilon>0$ can be sufficiently small,
it is reasonable to use $\beta$ to denote $(1-\varepsilon)\beta$ and therefore
the mapping $x\mapsto q_{0,j}(\omega,x,\cdot)$ is uniformly continuous in the
$C^{1,\beta}$-topology on $U_{1}$. This proves (A1) for the mapping $x\mapsto
q_{0,j}(\omega,x,\cdot)$.
Similarly to (4.24), in the case of $j\geq\tau$, we show that the mapping
$x\mapsto\partial^{2}_{y_{-}^{2}}q_{0,j}(\omega,x,y_{-})$ is Hölder continuous
with uniformly bounded Hölder coefficients with respect to $x,y_{-}$ on
$U_{1}$. This proves (A2) for the mapping $x\mapsto q_{0,j}(\omega,x,\cdot)$
by using a similar discussion to the one given below (4.24). The proof of this
theorem is completed. $\Box$
For the sake of convenient representations, we use notations
$\displaystyle{\mathcal{W}}_{\leq
j}^{u}(\omega,x):=\left\\{\begin{array}[]{lll}{\mathcal{W}}_{\leq
j}^{uu}(\omega,x),&\\!j\\!<\\!\tau,\vspace{1ex}\\\
{\mathcal{W}}_{u}(\omega,x),&\\!j\\!=\\!\tau,\vspace{1ex}\\\
{\mathcal{W}}_{\leq
j}^{pu}(\omega,x),&\\!j\\!>\\!\tau,\end{array}\right.\qquad{\mathcal{W}}_{\geq
j+1}^{s}(\omega,x):=\left\\{\begin{array}[]{lll}{\mathcal{W}}_{\geq
j+1}^{ps}(\omega,x),&\\!j\\!<\\!\tau,\vspace{1ex}\\\
{\mathcal{W}}_{s}(\omega,x),&\\!j\\!=\\!\tau,\vspace{1ex}\\\
{\mathcal{W}}_{\geq j+1}^{ss}(\omega,x),&\\!j\\!>\\!\tau.\end{array}\right.$
(4.32)
Notice that these leaves are graphs of the mappings
$y_{+}\mapsto p_{0,j}(\omega,x,y_{+})+\pi_{+}x-y_{+},\qquad y_{-}\mapsto
q_{0,j}(\omega,x,y_{-})+\pi_{-}x-y_{-}$
with Lipschitz constants $<1$ by (4.9). Then, corresponding to each $X_{j}$ we
introduce the leaf
$\displaystyle{\mathcal{W}}_{j}(\omega,x):=\left\\{\begin{array}[]{lll}{\mathcal{W}}_{\leq
1}^{u}(\omega,x),&\quad\mbox{if }j=1,\vspace{1ex}\\\ {\mathcal{W}}_{\leq
j}^{u}(\omega,x)\cap{\mathcal{W}}_{\geq j}^{s}(\omega,x),&\quad\mbox{if
}j=2,...,{{p}}-1,\vspace{1ex}\\\
{\mathcal{W}}_{\geq{{p}}}^{s}(\omega,x),&\quad\mbox{if
}j={{p}}.\end{array}\right.$ (4.36)
Clearly, for each $j\in\\{1,...,p\\}$ it defines a foliation
${\mathcal{W}}_{j}(\omega):=\\{{\mathcal{W}}_{j}(\omega,x)\\}_{x\in\mathbb{R}^{d}}$,
called the random intermediate foliation, and it is the graph of a mapping
$\check{q}_{j}(\omega,x,\cdot):X_{j}\to X_{1}\oplus\cdots\oplus X_{j-1}\oplus
X_{j+1}\oplus\cdots\oplus X_{p}$ such that
$\mathbb{R}^{d}=\cup_{x\in\mathbb{R}^{d}}{\mathcal{W}}_{j}(\omega,x)$ and the
following corollary holds.
###### Corollary 4.1.
Suppose that the random diffeomorphism
$F:\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ satisfies (3.4). Then for each
$j=1,...,p$ the measurable mapping $\check{q}_{j}(\omega,x,\cdot)$ is tempered
$C^{1,\beta}$ with a constant $\beta\in(0,1]$ such that
$\displaystyle\|\partial_{y_{j}}\check{q}_{j}(\omega,x,y_{j})-{\rm
id}_{j}\|\leq
C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega)\delta(\omega)$ (4.37)
for all $(\omega,x,y_{j})\in\Omega\times\mathbb{R}^{d}\times X_{j}$, where
${\rm id}_{j}$ is the identity in $X_{j}$. Moreover, the mapping
$x\mapsto\check{q}_{j}(\omega,x,\cdot)$ is uniformly continuous in the
$C^{1,\beta}$-topology on $U_{1}:=\\{x\in\mathbb{R}^{d}:\|x\|<1/2\\}$.
Concerning the relation between the intermediate distribution $E_{j}(\omega)$
and the intermediate foliation ${\mathcal{W}}_{j}(\omega)$, we further show
that
$\displaystyle E_{j}(\omega,x)$
$\displaystyle=\\{\partial_{y_{j}}\check{q}(\omega,x,\pi_{j}x)\,z_{j}\in\mathbb{R}^{d}:~{}z_{j}\in
X_{j}\\},\qquad\forall x\in\mathbb{R}^{d},$ (4.38)
(see (9.2) below in the stable case for more details), where
$\pi_{j}:\mathbb{R}^{d}\to X_{j}$ is defined by
$\pi_{j}(x_{1}+\cdots+x_{p}):=x_{j}$. It means that every fiber of
$E_{j}(\omega)$ at $x$ is the tangent space of the leaf of
${\mathcal{W}}_{j}(\omega)$ at $x$.
## 5\. Decomposition of a cohomological equation
In order to show that the stable and unstable distributions obtained in
section 3 are $C^{1,\beta_{\alpha}}$ with a constant
$\beta_{\alpha}\in[0,\alpha]$, we will prove that there are
$C^{1,\beta_{\alpha}}$ vectors fields $v^{s}_{j}(\omega,x)$ and
$v^{u}_{j}(\omega,x)$ that span $E_{s}(\omega,x)$ and $E_{u}(\omega,x)$ at
every point $x\in\mathbb{R}^{d}$ respectively (see [61, p.41]). This will be
done in sections 5-9 mainly for $E_{s}(\omega,x)$, as the proof for
$E_{u}(\omega,x)$ is similar.
In this section, we first derive a cohomological equation for the vector field
$v^{s}_{j}(\omega,x)$. Then, we solve the cohomological equation in 6 steps to
show that, for any $\bar{x}\in\mathbb{R}^{d}$ and $i=1,...,\tau$, there is a
basis (depending on $\bar{x}$ and $i$) of $E_{s}(\omega,x)$ which is
differentiable at $\bar{x}$ along every leaf
${\mathcal{W}}_{i}(\omega,\bar{x})$ of the intermediate foliation with a
(Hölder) continuous differential. Letting
$T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})$ be the tangent space of
${\mathcal{W}}_{i}(\omega,\bar{x})$ at $\bar{x}$, we have the following
theorem.
###### Theorem 5.1.
Suppose that the random diffeomorphism
$F:\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ satisfies (3.4) and that the
Lyapunov exponents satisfy (2.10). Then, for any point
$\bar{x}\in\mathbb{R}^{d}$ and $i=1,...,\tau$, the fibers $E_{s}(\omega,x)$
have bases
$\\{{v_{\bar{x},i}^{\kappa,\iota}}(\omega,x)\\}_{\kappa={\tau}+1,...,{p},~{}\iota=1,...,d_{\kappa}}$
for all $x\in{\mathcal{W}}_{i}(\omega,\bar{x})$ near $\bar{x}$, whose
differential (with respect to $x$)
$\displaystyle
dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x}):T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})\to\mathbb{R}^{d}~{}~{}(=T_{v_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})}\mathbb{R}^{d}\,)$
(5.1)
at the point $\bar{x}$ is $\beta_{v}$-Hölder continuous with respect to
$\bar{x}$ for every $\kappa={\tau}+1,...,{p}$ and $\iota=1,...,d_{\kappa}$,
where $\beta_{v}:=\min\\{\beta_{E},\epsilon/\lambda_{\max},\alpha\\}$.
Moreover, the differential and the Hölder coefficient are measurable with
respect to $\omega$ and are uniformly bounded with respect to $\bar{x}$ and
$i$.
###### Remark 5.1.
(i) The $\beta_{v}$-Hölder continuity of
$dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})$ with respect to $\bar{x}$
means that
$\displaystyle\|dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})\circ\hbar(\omega,\bar{x})-dv_{\bar{y},i}^{\kappa,\iota}(\omega,\bar{y})\circ\hbar(\omega,\bar{y})\|\leq
L(\omega)\|\bar{x}-\bar{y}\|^{\beta_{v}}$
when $\|\bar{x}-\bar{y}\|<1$, where $\hbar(\omega,\cdot)$ is a homeomorphism
such that $\hbar^{\pm 1}(\omega,\cdot)$ are $\beta_{v}$-Hölder continuous and
$\hbar^{-1}(\omega,\cdot)$ trivializes the bundle
${\mathcal{E}}:=\bigcup_{\bar{x}\in\mathbb{R}^{d}}T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})$0
with the base $\mathbb{R}^{d}$. If $\beta_{v}=0$ (when $\alpha=0$) then
$\beta_{v}$-Hölder continuity merely means continuity, and the differential is
uniformly bounded with respect to $\bar{x}$ and $i$.
(ii) The bases of $E_{s}(\omega,x)$ are not well defined for all
$x\in\mathbb{R}^{d}$ since
$\\{{v_{\bar{x},i}^{\kappa,\iota}}(\omega,x)\\}_{\kappa={\tau}+1,...,{p},~{}\iota=1,...,d_{\kappa}}$
depends on $\bar{x}$ and $i$. Therefore, it is possible that
$x\in{\mathcal{W}}_{i}(\omega,\bar{x})\cap{\mathcal{W}}_{j}(\omega,\bar{y})$
for certain $\bar{x},\bar{y}\in\mathbb{R}^{d}$ and $i,j\in\\{1,...\tau\\}$,
but $v_{\bar{x},i}^{\kappa,\iota}(\omega,x)\neq
v_{\bar{y},j}^{\kappa,\iota}(\omega,x)$. We will solve this problem in section
9 by finding a “canonical” basis of $E_{s}(\omega,x)$ not depending on
$\bar{x},i$ and giving the relation between the “canonical” basis and
$\\{{v_{\bar{x},i}^{\kappa,\iota}}(\omega,x)\\}_{\kappa={\tau}+1,...,{{p}},~{}\iota=1,...,d_{\kappa}}$.
Proof of Theorem 5.1. We consider the random cohomological equation for
$v(\omega,x)\in E_{s}(\omega,x)$, i.e.,
$\displaystyle
DF(\omega,x)v(\omega,x)=v(\theta\omega,F(\omega,x)),\quad\forall\omega\in\Omega,\quad\forall
x\in\mathbb{R}^{d}.$ (5.2)
In what follows, our goal is to find out the solutions $v(\omega,x)$ of
equation (5.2), which form the basis of $E_{s}(\omega,x)$ and are
differentiable at every point $x\in\mathbb{R}^{d}$ with (Hölder) continuous
derivatives. We will show this in 6 steps.
Step 1. Given any point $\bar{x}\in\mathbb{R}^{d}$, transform the orbit
$(F(n,\omega,\bar{x}))_{n\in\mathbb{Z}}$ to the origin $0$ to get a new system
(5.7), and then reduce to its normal form (5.10).
In order to study the differentiability of $v(\omega,\cdot)$ along the orbit
$(F(n,\omega,\bar{x}))_{n\in\mathbb{Z}}$, we let
$\varpi:=(\omega,\bar{x})\in\bar{\bf\Omega}:=\Omega\times\mathbb{R}^{d},\quad\vartheta\varpi:=(\theta\omega,F(\omega,\bar{x})),\quad{\bf\Lambda}(\varpi):=DF(\omega,\bar{x}).$
Then, we have
$\vartheta^{n}\varpi=(\theta^{n}\omega,F(n,\omega,\bar{x})),\quad{\bf\Lambda}(\vartheta^{n}\varpi)=DF(\theta^{n}\omega,F(n,\omega,\bar{x})),\quad\forall
n\in\mathbb{Z}.$
Note that the random matrix ${\bf\Lambda}(\varpi)$ generates the linear
cocycle
$\displaystyle{\bf\Lambda}(n,\varpi):=\begin{cases}{\bf\Lambda}(\vartheta^{n-1}\varpi)\cdots{\bf\Lambda}(\varpi),&n>0,\\\
{\rm id},&n=0,\\\
{\bf\Lambda}^{-1}(\vartheta^{n}\varpi)\cdots{\bf\Lambda}^{-1}(\vartheta^{-1}\varpi),&n<0.\end{cases}$
(5.3)
It is clear that ${\bf\Lambda}(n,\varpi)=DF(n,\omega,\bar{x})$. Therefore, by
(3.12) we have
$\displaystyle\begin{split}&\|{\bf\Lambda}(n,\varpi)|_{E_{j}(\varpi)}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}-\epsilon)n},\qquad\forall n<0,\\\
&\|{\bf\Lambda}(n,\varpi)|_{E_{j}(\varpi)}\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}+\epsilon)n},\qquad\forall n\geq
0.\end{split}$
Then, we use [1, Corollary 4.3.12] to block-diagonalize
${\bf\Lambda}(n,\varpi)$, namely, there is a random matrix $P(\varpi)\in
Gl(d,\mathbb{R})$ such that
$\displaystyle\bar{\bf\Lambda}(\varpi):=P(\vartheta\varpi){\bf\Lambda}(\varpi)P^{-1}(\varpi)={\rm
diag}(\bar{\bf\Lambda}_{1}(\varpi),...,\bar{\bf\Lambda}_{p}(\varpi)),$ (5.4)
and
$\displaystyle\begin{split}&\|\bar{\bf\Lambda}_{j}(n,\varpi)\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}-\epsilon)n},\qquad\forall n<0,\\\
&\|\bar{\bf\Lambda}_{j}(n,\varpi)\|\leq
K_{\epsilon}(\omega)e^{(\lambda_{j}+\epsilon)n},\qquad\forall n\geq
0.\end{split}$ (5.5)
Moreover precisely,
$P(\varpi):=(\vec{e}_{1,1}(\varpi),...,\vec{e}_{p,d_{p}}(\varpi))$ with
measurable unit vectors $\vec{e}_{j,\iota}(\varpi)$ for all $j=1,...,p$ and
$\iota=1,...,d_{j}$ is a transition matrix from the standard basis
$\mathfrak{C}$ of $\mathbb{R}^{d}$ to a basis
$\mathfrak{C}(\varpi):=\\{\vec{e}_{j,\iota}(\varpi)\\}_{j=1,...,p,\,\iota=1,...,d_{j}}$
associated with the measurable decompositions
$\mathbb{R}^{d}=E_{1}(\varpi)\oplus\cdots\oplus E_{p}(\varpi).$
By the second inequality of (3.16), we can choose
$\vec{e}_{j,\iota}(\varpi)\in\mathfrak{C}(\varpi)$ that is uniformly (with
respect to $\varpi$) close to the corresponding standard unit vector
$\vec{e}_{j,\iota}\in\mathfrak{C}$. Thus, for all $\varpi:=(\omega,\bar{x})$
and $\tilde{\varpi}:=(\omega,\bar{y})\in\bar{\bf\Omega}$,
$\displaystyle\|\vec{e}_{j,\iota}(\varpi)-\vec{e}_{j,\iota}(\tilde{\varpi})\|\leq
2{\rm dist}(E_{j}(\varpi),E_{j}(\tilde{\varpi}))$
by (3.15). Then, it follows from Theorem 3.1 that
$\displaystyle\begin{split}&\|P^{\pm 1}(\varpi)-P^{\pm
1}(\tilde{\varpi})\|\leq\|\bar{x}-\bar{y}\|^{\beta_{E}},\quad\forall\varpi:=(\omega,\bar{x}),~{}\tilde{\varpi}:=(\omega,\bar{y})\in\bar{\bf\Omega},\\\
&\|P^{\pm 1}(\varpi)-{\rm id}\|=\|P^{\pm 1}(\omega,\bar{x})-P^{\pm
1}(\omega,0)\|\leq\delta_{E}(\omega),\end{split}$ (5.6)
where the constant $2$ is omitted without loss of generality.
Now, for every $\varpi=(\omega,\bar{x})$ set
$\displaystyle\hat{\bf
F}(\varpi,x):=P(\vartheta\varpi)F(\omega,P^{-1}(\varpi)x+\bar{x})-P(\vartheta\varpi)F(\omega,\bar{x}),\quad
x\in\mathbb{R}^{d}.$ (5.7)
Then,
$\displaystyle\hat{\bf F}(\vartheta^{n}\varpi,x)$
$\displaystyle=P(\theta^{n+1}\omega,F(n+1,\omega,\bar{x}))F(\theta^{n}\omega,P^{-1}(\theta^{n}\omega,F(n,\omega,\bar{x}))x+F(n,\omega,\bar{x}))$
$\displaystyle\quad-P(\theta^{n+1}\omega,F(n+1,\omega,\bar{x}))F(\theta^{n}\omega,F(n,\omega,\bar{x})),$
which means that the problem of $F(\theta^{n}\omega,\cdot)$ near its orbit
$F(n,\omega,\bar{x})$ is converted to the problem of $\hat{\bf F}(\varpi,x)$
near the origin $0$. It is straightforward to compute that
$\displaystyle\hat{\bf F}(\varpi,{{0}})={{0}},\qquad D\hat{\bf
F}(\varpi,{{0}})=P(\vartheta\varpi){\bf\Lambda}(\varpi)P^{-1}(\varpi)=\bar{\bf\Lambda}(\varpi),$
and
$\displaystyle\begin{split}\|D^{2}\hat{\bf F}(\varpi,x)\|\leq
8M_{\epsilon}(\omega),\quad\forall x\in\mathbb{R}^{d},\end{split}$ (5.8)
due to (3.4). To show (5.8), we also need to note that
$\displaystyle\|P^{\pm 1}(\varpi)\|\leq
1+\delta_{E}(\omega)=1+\delta(\omega)^{\beta_{E}}=1+1/\mathfrak{M}_{\epsilon}(\omega)^{\beta_{E}}\leq
2$ (5.9)
by (5.6) and (3.8). For the sake of simplicity, we still use
$M_{\epsilon}(\omega)$ to denote $8M_{\epsilon}(\omega)$.
Next, we state a result on the normal form of $\hat{\bf F}(\varpi,x)$, which
will be proved in section 6.
###### Proposition 5.1.
There is a $C^{2,\alpha}$ random diffeomorphism
$N(\varpi,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$ transforming $\hat{\bf
F}(\varpi,\cdot)$ to $\bar{\bf F}(\varpi,\cdot)$, i.e.,
$\displaystyle\bar{\bf F}(\varpi,x):=N(\vartheta\varpi,\cdot)\circ\hat{\bf
F}(\varpi,\cdot)\circ N^{-1}(\varpi,x),$ (5.10)
such that
$\displaystyle\bar{\bf F}(\varpi,{{0}})={{0}},\quad D\bar{\bf
F}(\varpi,{{0}})=\bar{\bf\Lambda}(\varpi),\quad\partial^{2}_{x_{u}x_{s}}\bar{\bf
F}(\varpi,{{0}})={\bf 0}~{}~{}~{}\mbox{\rm(the zero mapping)},$ (5.11)
and
$\displaystyle\begin{split}&N^{\pm 1}(\varpi,0)=0,\qquad DN^{\pm
1}(\varpi,0)={\rm id},\quad\|DN^{\pm 1}(\varpi,x)-{\rm
id}\|\leq\delta(\omega)/K_{\epsilon}(\omega),\\\ &\|D^{2}N^{\pm
1}(\varpi,x)\|\leq\check{M}_{\epsilon}(\omega),\qquad\|D^{2}N^{\pm
1}(\varpi,x)-D^{2}N^{\pm
1}(\varpi,y)\|\leq\check{L}_{\epsilon,\delta}(\omega)\|x-y\|^{\alpha}\end{split}$
(5.12)
for all $x,y\in\mathbb{R}^{d}$, where $\check{M}_{\epsilon}(\omega)>0$ is an
$\epsilon$-slowly varying random variable depending on
$K_{\epsilon}(\omega),M_{\epsilon}(\omega)$ and
$\check{L}_{\epsilon,\delta}(\omega)>0$ is an $\epsilon$-slowly varying random
variable depending on $K_{\epsilon}(\omega)$, $M_{\epsilon}(\omega)$,
$\delta(\omega)$.
Moreover, $D^{2}N^{\pm 1}(\varpi,0)=D^{2}N^{\pm 1}(\omega,\bar{x},0)$ are
measurable with respect to $\omega$ and are continuous with respect to
$\bar{x}$ such that
$\displaystyle\|D^{2}N^{\pm 1}(\omega,\bar{x},0)-D^{2}N^{\pm
1}(\omega,\bar{y},0)\|\leq
C_{\lambda}K_{\epsilon}(\omega)^{3}\hat{L}_{\epsilon,\rho}(\omega)\|\bar{x}-\bar{y}\|^{\beta_{N}},$
(5.13)
when $\|\bar{x}-\bar{y}\|<1$, where
$\beta_{N}:={\min\\{\epsilon/\lambda_{\max},\alpha\\}}$ and
$\hat{L}_{\epsilon,\rho}(\omega)>0$ is an $\epsilon$-slowly varying random
variable depending on $K_{\epsilon}(\omega)$, $M_{\epsilon}(\omega)$ and
$L_{\epsilon,\rho}(\omega)$.
Remark that in the above definition of $\beta_{N}$, the choice of $\epsilon>0$
depends on the Lyapunov exponents only (see (3.7)). Moreover, a key part of
the above result is to obtain the third equality of (5.11), which will be used
to estimate the nonlinear part of a cohomological equation deduced from (5.2),
as will be seen in (7.17)-(7.20) below.
Using (5.7) and (5.10), we have that
$\displaystyle\bar{\bf F}(\varpi,x)={\Psi}(\vartheta\varpi,\cdot)\circ
F(\omega,\cdot)\circ{\Psi}^{-1}(\varpi,x),\quad\forall x\in\mathbb{R}^{d},$
(5.14)
where ${\Psi}(\varpi,x):=N(\varpi,\cdot)\circ P(\varpi)(x-\bar{x}).$ It
follows from (3.4) and (5.12) that
$\displaystyle\begin{split}&\|D\bar{\bf
F}(\varpi,x)-\bar{\bf\Lambda}(\varpi)\|\leq
C_{\lambda}\delta(\omega),\qquad\|D^{2}\bar{\bf
F}(\varpi,x)\|\leq\bar{M}_{\epsilon}(\omega),\\\ &\|D^{2}\bar{\bf
F}(\varpi,x)-D^{2}\bar{\bf
F}(\varpi,y)\|\leq\bar{L}_{\epsilon,\delta}(\omega)\|x-y\|^{\alpha},\quad\forall
x,y\in\mathbb{R}^{d},\end{split}$ (5.15)
and $D^{2}\bar{\bf F}(\varpi,\cdot)$ is $\epsilon$-slowly continuous when
$\alpha=0$, where $\bar{M}_{\epsilon}(\omega)>0$ is an $\epsilon$-slowly
varying random variable depending on
$K_{\epsilon}(\omega),M_{\epsilon}(\omega)$, and
$\bar{L}_{\epsilon,\delta}(\omega)>0$ is an $\epsilon$-slowly varying random
variable depending on
$K_{\epsilon}(\omega),M_{\epsilon}(\omega),\delta(\omega)$. As we saw in
section 3 (below (3.4)), the third inequality of (5.15) holds for all Hölder
exponents $\tilde{\alpha}\in[0,\alpha]$.
Step 2. Convert equation (5.2) to a new equation (5.17) with respect to
$\bar{\bf F}(\varpi,x)$, whose solution is represented by $\bar{\bf
v}(\varpi,x)$, and establish the relation between $v(\omega,x)$ and $\bar{\bf
v}(\varpi,x)$.
For any $v(\omega,x)$ satisfying equation (5.2), we define $\bar{\bf
v}(\varpi,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$ by
$\displaystyle\begin{split}\bar{\bf
v}(\varpi,x):=D{\Psi}(\varpi,{\Psi}^{-1}(\varpi,x))\,v(\omega,{\Psi}^{-1}(\varpi,x)).\end{split}$
(5.16)
Then, $\bar{\bf v}(\varpi,x)$ satisfies
$\displaystyle D\bar{\bf F}(\varpi,x)\bar{\bf v}(\varpi,x)=\bar{\bf
v}(\vartheta\varpi,\bar{\bf F}(\varpi,x)),\quad\forall x\in\mathbb{R}^{d}.$
(5.17)
In fact, using (5.2) and (5.14), we obtain that
$\displaystyle D\bar{\bf F}(\varpi,x)\bar{\bf v}(\varpi,x)$
$\displaystyle=D\bar{\bf
F}(\varpi,x)\,D{\Psi}(\varpi,{\Psi}^{-1}(\varpi,x))\,v(\omega,{\Psi}^{-1}(\varpi,x))$
$\displaystyle=D{\Psi}(\vartheta\varpi,F(\omega,\cdot)\circ{\Psi}^{-1}(\varpi,x))\,DF(\omega,{\Psi}^{-1}(\varpi,x))\,D{\Psi}^{-1}(\varpi,x)$
$\displaystyle\quad\cdot
D{\Psi}(\varpi,{\Psi}^{-1}(\varpi,x))\,v(\omega,{\Psi}^{-1}(\varpi,x))$
$\displaystyle=D{\Psi}(\vartheta\varpi,F(\omega,\cdot)\circ{\Psi}^{-1}(\varpi,x))\,DF(\omega,{\Psi}^{-1}(\varpi,x))\,v(\omega,{\Psi}^{-1}(\varpi,x))$
$\displaystyle=D{\Psi}(\vartheta\varpi,F(\omega,\cdot)\circ{\Psi}^{-1}(\varpi,x))\,v(\theta\omega,F(\omega,\cdot)\circ{\Psi}^{-1}(\varpi,x))$
$\displaystyle=D{\Psi}(\vartheta\varpi,{\Psi}^{-1}(\vartheta\varpi,\cdot)\circ\bar{\bf
F}(\varpi,x))\,v(\theta\omega,{\Psi}^{-1}(\vartheta\varpi,\cdot)\circ\bar{\bf
F}(\varpi,x))$ $\displaystyle=\bar{\bf v}(\vartheta\varpi,\bar{\bf
F}(\varpi,x)).$
The converse is also true. Thus, the problem of finding the solution
$v(\omega,x)$ of equation (5.2) is equivalent to finding the solution
$\bar{\bf v}(\varpi,x)$ of equation (5.17).
Let ${\mathcal{W}}_{i}(\varpi):={\mathcal{W}}_{i}(\omega,\bar{x})$ be the leaf
of an intermediate foliation of $F(\omega,x)$ (see (4.36)) passing through
$\bar{x}$ for every $i=1,...,\tau$. Then,
$\displaystyle\bar{\mathcal{N}}_{i}(\varpi):=\Psi(\varpi,{\mathcal{W}}_{i}(\varpi)),\quad\forall
n\in\mathbb{Z},$ (5.18)
is an invariant manifold (i.e., the leaf passing through $0$) of $\bar{\bf
F}(\varpi,x)$ corresponding to the subspace $X_{i}$. In fact, using (5.14), we
have that
$\displaystyle\bar{\bf F}(\varpi,\bar{\mathcal{N}}_{i}(\varpi))$
$\displaystyle=\bar{\bf
F}(\varpi,\cdot)\circ\Psi(\varpi,{\mathcal{W}}_{i}(\varpi))=\Psi(\vartheta\varpi,\cdot)\circ
F(\omega,{\mathcal{W}}_{i}(\varpi))$
$\displaystyle\subset\Psi(\vartheta\varpi,{\mathcal{W}}_{i}(\vartheta\varpi))=\bar{\mathcal{N}}_{i}(\vartheta\varpi).$
Step 3. Linearize $\bar{\bf F}(\varpi,x)$ along
$\bar{\mathcal{N}}_{i}(\varpi)$ for $i=1,...,\tau$.
By Corollary 4.1 (using (5.15) instead of (3.4)), we see that
$\bar{\mathcal{N}}_{i}(\varpi)$ is the graph of a mapping, i.e,
$\displaystyle\bar{\mathcal{N}}_{i}(\varpi)=\\{x_{i}+{\gamma}_{i}(\varpi,x_{i}):x_{i}\in
X_{i}\\},\quad i=1,...,\tau,$ (5.19)
where ${\gamma}_{i}(\varpi,\cdot):X_{i}\to X_{1}\oplus\cdots\oplus
X_{i-1}\oplus X_{i+1}\oplus\cdots\oplus X_{p}$ is a $C^{1,{\beta}}$ mapping
such that
$\displaystyle\begin{split}&{\gamma}_{i}(\varpi,{{0}})={{0}},\quad
D{\gamma}_{i}(\varpi,{{0}})={\bf 0},\quad\|D{\gamma}_{i}(\varpi,x_{i})\|\leq
C_{\lambda}K_{\epsilon}(\omega)C_{\epsilon}(\omega)\delta(\omega),\\\
&\|D{\gamma}_{i}(\varpi,x_{i})-D{\gamma}_{i}(\varpi,y_{i})\|\leq
L_{\epsilon,{\gamma}}(\omega)\|x_{i}-y_{i}\|^{\beta},\quad\forall
x_{i},y_{i}\in X_{i},\end{split}$ (5.20)
with an $\epsilon$-slowly varying tempered random variable
$L_{\epsilon,\gamma}(\omega)>0$. Using the transformation
$\varphi_{i}(\varpi,x):=x-{\gamma}_{i}(\varpi,x_{i}),$
we straighten up the invariant manifold $\bar{\mathcal{N}}_{i}({\varpi})$.
Then, we have a $C^{1,{\beta}}$ mapping $\bar{\bf
F}_{i}(\varpi,\cdot):X_{i}\to X_{i}$ by
$\displaystyle\bar{\bf
F}_{i}(\varpi,x_{i}):=\varphi_{i}(\vartheta\varpi,\cdot)\circ\bar{\bf
F}(\varpi,\cdot)\circ\varphi^{-1}_{i}(\varpi,x_{i}),$ (5.21)
which satisfies that
$\displaystyle\begin{split}&\bar{\bf F}_{i}(\varpi,{{0}})={{0}},\quad
D\bar{\bf F}_{i}(\varpi,{{0}})=\bar{\bf\Lambda}_{i}(\varpi),\quad\|D\bar{\bf
F}_{i}(\varpi,x_{i})-\bar{\bf\Lambda}_{i}(\varpi)\|\leq\delta^{*}(\omega),\\\
&\|D\bar{\bf F}_{i}^{\pm 1}(\varpi,x_{i})-D\bar{\bf F}_{i}^{\pm
1}(\varpi,y_{i})\|\leq
L_{\epsilon}^{*}(\omega)\|x_{i}-y_{i}\|^{\beta},\qquad\forall x_{i},y_{i}\in
X_{i},\end{split}$ (5.22)
by (5.15) and (5.20). Here, $\bar{\bf\Lambda}_{i}(\varpi)$ is given in (5.4),
$\delta^{*}(\omega):=C_{\lambda}K_{\epsilon}(\omega)^{2}C_{\epsilon}(\omega)\delta(\omega)>0$
and $L_{\epsilon}^{*}(\omega)>0$ depends on $L_{\epsilon,{\gamma}}(\omega)$
and $\bar{M}_{\epsilon}(\omega)$.
In order to solve equation (5.17), we will use the Lyapunov norms to overcome
the nonuniformity (see Lemma 7.1 below together with the remark given at the
end of section 7). Since the Lyapunov norms is applied only to the linear
system $\bar{\bf\Lambda}_{i}(\varpi)$, we need to linearize $\bar{\bf
F}_{i}(\varpi,x_{i})$.
The following lemma states that $\bar{\bf F}_{i}(\varpi,x_{i})$ is
$C^{1,\beta}$ with $\beta\in(0,1]$ given in (4.12) conjugate to its linear
part, which will be proved in section 7.
###### Lemma 5.1.
There is a $C^{1}$ diffeomorphism $\varphi_{i,*}(\varpi,\cdot):X_{i}\to X_{i}$
such that
$\displaystyle\bar{\bf\Lambda}_{i}(\varpi)\varphi_{i,*}(\varpi,x_{i})=\varphi_{i,*}(\vartheta\varpi,\cdot)\circ\bar{\bf
F}_{i}(\varpi,x_{i}),\quad\forall i=1,...,\tau,$ (5.23)
and
$\displaystyle\begin{split}D\varphi_{i,*}^{\pm 1}(\varpi,{{0}})={\rm
id}_{i},\quad\|D\varphi_{i,*}^{\pm 1}(\varpi,x_{i})-{\rm id}_{i}\|\leq
C^{*}_{4\epsilon}(\omega)\|x_{i}\|^{\beta},\quad\forall x_{i}\in
X_{i},\end{split}$ (5.24)
where $C^{*}_{4\epsilon}(\omega)>0$ is a $4\epsilon$-slowly varying tempered
random variable and ${\rm id}_{i}$ the identity mapping in $X_{i}$.
Step 4. Decompose equation (5.17), obtained in step 2, along every
$\bar{\mathcal{N}}_{i}(\varpi)$ for $i=1,...,\tau$ by using the conjugacy
obtained in step 3, and solve the decomposed equation (5.26).
Recall that $\varphi_{i}(\varpi,\cdot):\mathbb{R}^{d}\to\mathbb{R}^{d}$
straightens up $\bar{\mathcal{N}}_{i}({\varpi})$ which is tangent to $X_{i}$
at ${{0}}$. Let
$\displaystyle\bar{\varphi}_{i}(\varpi,\cdot):=\varphi_{i,*}(\varpi,\cdot)\circ\varphi_{i}(\varpi,\cdot)|_{\bar{\mathcal{N}}_{i}({\varpi})}:\bar{\mathcal{N}}_{i}({\varpi})\to
X_{i}.$ (5.25)
Then, we have
$\displaystyle\bar{\varphi}_{i}(\vartheta\varpi,\cdot)\circ\bar{\bf
F}(\varpi,\cdot)\circ\bar{\varphi}^{-1}_{i}(\varpi,x_{i})=\bar{\bf\Lambda}_{i}(\varpi)$
by (5.21) and (5.23). Replacing $x$ in (5.17) with
$\bar{\varphi}^{-1}_{i}(\varpi,x_{i})$, we obtain an equation
$\displaystyle D\bar{\bf
F}(\varpi,\bar{\varphi}_{i}^{-1}(\varpi,x_{i}))\,\bar{\bf
v}_{i}(\varpi,x_{i})=\bar{\bf
v}_{i}(\vartheta\varpi,\bar{\bf\Lambda}_{i}(\varpi)x_{i})$ (5.26)
for $i=1,...,\tau$, where $\bar{\bf
v}_{i}(\varpi,\cdot):X_{i}\to\mathbb{R}^{d}$ is defined by
$\displaystyle\bar{\bf v}_{i}(\varpi,x_{i}):=\bar{\bf
v}(\varpi,\bar{\varphi}_{i}^{-1}(\varpi,x_{i})).$ (5.27)
For $\kappa={\tau}+1,...,{p}$ and $\iota=1,...,d_{\kappa}$, where $d_{\kappa}$
is the dimension of $X_{\kappa}$, let
$\vec{e}_{\kappa,\iota}:=(0,...,1,...,0)^{T}$ be the standard unit vectors in
$X_{\kappa}$, i.e., the $(d_{1}+\cdots+d_{\kappa-1}+\iota)$-th component is 1
and others are 0.
The following proposition is on the existence of solutions of equation (5.26),
which will be proved in sections 7 and 8 for the cases of $\alpha\in(0,1]$ and
$\alpha=0$ respectively.
###### Proposition 5.2.
Given a $\kappa={\tau}+1,...,{p}$ and $d_{\kappa}$ standard unit vectors
$\vec{e}_{\kappa,\iota}\in X_{\kappa}$ ($\iota=1,...,d_{\kappa}$), for every
$\varpi\in\bar{\bf\Omega}$, there exist sets $\\{{\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,x_{i})\\}_{n\in\mathbb{Z}}$
($i=1,...,\tau$) of the solutions of equation (5.26) such that ${\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,{{0}})=\bar{\bf\Lambda}_{\kappa}(n,\varpi)\,\vec{e}_{\kappa,\iota}\in
X_{\kappa},$
$\displaystyle\frac{\|{\bar{\bf v}_{i}^{\kappa,\iota}}(\varpi,x_{i})-{\bar{\bf
v}_{i}^{\kappa,\iota}}(\varpi,{{0}})\|}{\|x_{i}\|}\to 0\qquad{\rm
as}~{}\|x_{i}\|\to 0,$ (5.28)
and
$\displaystyle\sup_{n\in\mathbb{Z}}\sup_{x_{i}\in
X_{i}\backslash\\{{{0}}\\}}\bigg{\\{}e^{-\lambda_{\kappa}n-12\epsilon|n|}\frac{\|{\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,x_{i})-{\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,{{0}})\|}{\|x_{i}\|^{\varsigma}}\bigg{\\}}\leq
M_{v}(\omega),$ (5.29)
where $M_{v}(\omega)>0$ is a tempered random variable and $\varsigma\in(0,1)$
is a constant satisfying
$\displaystyle\begin{split}0<\varsigma<\frac{1}{10}\min\Bigg{\\{}\frac{\lambda_{\tau}}{\lambda_{\max}},\frac{-\lambda_{\tau+1}}{\lambda_{\max}},\min_{\mbox{\tiny$\begin{array}[]{c}{\kappa\in\\{\tau+1,...,p\\},j\in\\{1,...,p\\}}\\\
{\rm such~{}that}~{}\lambda_{\kappa}-\lambda_{j}\neq
0\end{array}$}}\frac{|\lambda_{\kappa}-\lambda_{j}|}{\lambda_{1}}\Bigg{\\}}\end{split}$
(5.30)
with $\lambda_{\rm max}:=\max\\{2\lambda_{1},-2\lambda_{p}\\}>0$.
Step 5. Use $\\{\bar{\bf
v}_{i}^{\kappa,\iota}(\theta^{n}\varpi,x_{i})\\}_{n\in\mathbb{Z}}$ to get a
set $\\{v_{\bar{x},i}^{\kappa,\iota}(\theta^{n}\omega,x)\\}_{n\in\mathbb{Z}}$
of solutions of equation (5.2) for every $\omega\in\Omega$ on the leaves
$\\{{\mathcal{W}}_{i}(\theta^{n}\omega,F(n,\omega,\bar{x}))\\}_{n\in\mathbb{Z}}$
through (5.16). Then, show that $v_{\bar{x},i}^{\kappa,\iota}(\omega,x)$ is
differentiable at $\bar{x}$ along ${\mathcal{W}}_{i}(\omega,\bar{x})$ such
that the differential is (Hölder) continuous with respect to $\bar{x}$.
For each point $x\in{\mathcal{W}}_{i}(\varpi)$ for $i=1,...,\tau$, by the
invariance of foliation, we have that
$F(n,\omega,x)\in{\mathcal{W}}_{i}(\vartheta^{n}\varpi)$. Then, using (5.18)
and the fact that $\bar{\varphi}_{i}(\vartheta^{n}\varpi,\cdot)$ straightens
up $\bar{\mathcal{N}}_{i}(\vartheta^{n}\varpi)$, we get
$\displaystyle\bar{\varphi}_{i}(\vartheta^{n}\varpi,\cdot)\circ\Psi(\vartheta^{n}\varpi,F(n,\omega,x))\in
X_{i},\qquad\forall n\in\mathbb{Z},$ (5.31)
It implies that
${\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,\bar{\varphi}_{i}(\vartheta^{n}\varpi,\cdot)\circ{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))$
is well-defined. Then, by (5.16) (implying
$v(\omega,x)=\\{D{\Psi}(\varpi,x)\\}^{-1}\bar{\bf
v}(\varpi,{\Psi}(\varpi,x))$) and by (5.27), we obtain that
$\displaystyle{v_{\bar{x},i}^{\kappa,\iota}}(\theta^{n}\omega,F(n,\omega,x))$
$\displaystyle:=\\{D{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))\\}^{-1}\,{\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,\bar{\varphi}_{i}(\vartheta^{n}\varpi,\cdot)\circ{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))),\quad\forall
n\in\mathbb{Z},$ (5.32)
satisfies (5.2) at $F(n,\omega,x)$, i.e.,
$\displaystyle
DF(\theta^{n}\omega,F(n,\omega,x))\,{v_{\bar{x},i}^{\kappa,\iota}}(\theta^{n}\omega,F(n,\omega,x))={v_{\bar{x},i}^{\kappa,\iota}}(\theta^{n+1}\omega,F(n+1,\omega,x)).$
(5.33)
Since ${\Psi}^{-1}(\varpi,x)=P^{-1}(\varpi)\circ N^{-1}(\varpi,x)+\bar{x}$ due
to the definition of ${\Psi}$ given below (5.14), we have
${\Psi}^{-1}(\vartheta^{n}\varpi,x)=P^{-1}(\vartheta^{n}\varpi)\circ
N^{-1}(\vartheta^{n}\varpi,x)+F(n,\omega,\bar{x})$
by replacing $\varpi=(\omega,\bar{x})$ and $\bar{x}$ with
$\vartheta^{n}\varpi=(\theta^{n}\omega,F(n,\omega,\bar{x}))$ and
$F(n,\omega,\bar{x})$ respectively. Therefore,
$\displaystyle\\{D{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))\\}^{-1}$
$\displaystyle=D{\Psi}^{-1}(\vartheta^{n}\varpi,{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x)))$
$\displaystyle=P^{-1}(\vartheta^{n}\varpi)\,DN^{-1}(\vartheta^{n}\varpi,{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))),$
which together with (5.32) yields
$\displaystyle{v_{\bar{x},i}^{\kappa,\iota}}(\theta^{n}\omega,F(n,\omega,x))$
$\displaystyle=P^{-1}(\vartheta^{n}\varpi)\,DN^{-1}(\vartheta^{n}\varpi,{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x)))\,{\bar{\bf
v}_{i}^{\kappa,\iota}}(\vartheta^{n}\varpi,\bar{\varphi}_{i}(\vartheta^{n}\varpi,\cdot)\circ{\Psi}(\vartheta^{n}\varpi,F(n,\omega,x))).$
(5.34)
Next, we show the differentiability of
${v_{\bar{x},i}^{\kappa,\iota}}(\omega,x)$ at $x=\bar{x}$ along
${\mathcal{W}}_{i}(\omega,\bar{x})={\mathcal{W}}_{i}(\varpi)$. In view of
(5.31) with $n=0$, we let
$\displaystyle x_{i}:=\bar{\varphi}_{i}(\varpi,\cdot)\circ\Psi(\varpi,x)\in
X_{i}.$
Thus, $x\mapsto x_{i}(x)$ is a $C^{1}$ diffeomorphism from
${\mathcal{W}}_{i}(\varpi)$ to $X_{i}$ with the inverse $x_{i}\mapsto
x(x_{i}):=\Psi^{-1}(\varpi,\cdot)\circ\bar{\varphi}_{i}^{-1}(\varpi,x_{i})$
such that $x({{0}})=\bar{x}$. Then, in view of (5.34) with $n=0$ and with $x$
being replaced by
$x(x_{i}):=\Psi^{-1}(\varpi,\cdot)\circ\bar{\varphi}_{i}^{-1}(\varpi,x_{i})$,
we have that
$\displaystyle{v_{\bar{x},i}^{\kappa,\iota}}(\omega,x(x_{i}))=P^{-1}(\varpi)\,DN^{-1}(\varpi,\bar{\varphi}_{i}^{-1}(\varpi,x_{i}))\,{\bar{\bf
v}_{i}^{\kappa,\iota}}(\varpi,x_{i}),\quad\forall x_{i}\in X_{i}.$
Therefore,
$\displaystyle\frac{dv_{\bar{x},i}^{\kappa,\iota}(\omega,x(x_{i}))}{dx_{i}}\Big{|}_{x_{i}={{0}}}$
$\displaystyle=P^{-1}(\varpi)\\{D^{2}N^{-1}(\varpi,{{0}}){\rm
id}_{i}\\}\,\vec{e}_{\kappa,\iota}$
$\displaystyle=P^{-1}(\omega,\bar{x})\\{D^{2}N^{-1}(\omega,\bar{x},{{0}}){\rm
id}_{i}\\}\,\vec{e}_{\kappa,\iota}\in{\mathcal{L}}(X_{i},\mathbb{R}^{d})$
(5.35)
since $D\bar{\varphi}_{i}^{-1}(\varpi,{{0}})={\rm id}_{i}$ (see (5.20) and
(5.24)), ${\bar{\bf
v}_{i}^{\kappa,\iota}}(\varpi,{{0}})=\vec{e}_{\kappa,\iota}$ (see Proposition
5.2) and $D{\bar{\bf v}_{i}^{\kappa,\iota}}(\varpi,{{0}})=0$ (see (5.28)).
Then (5.35) determines the differential (5.1) of
$v_{\bar{x},i}^{\kappa,\iota}(\omega,\cdot):{\mathcal{W}}_{i}(\omega,\bar{x})\to\mathbb{R}^{d}$
along ${\mathcal{W}}_{i}(\omega,\bar{x})$ at $\bar{x}$ ([61, p.16]).
More precisely, since
$\frac{dx({{0}})}{dx_{i}}=\frac{\Psi^{-1}(\varpi,\cdot)\circ\bar{\varphi}_{i}^{-1}(\varpi,x_{i})}{dx_{i}}\Big{|}_{x_{i}=0}=P^{-1}(\varpi)\partial_{x_{i}}N^{-1}(\varpi,0)=P^{-1}(\omega,\bar{x}){\rm
id}_{i}\in{\mathcal{L}}(X_{i},\mathbb{R}^{d})$
by (5.12), we have that
$\\{P^{-1}(\omega,\bar{x})\vec{e}_{i,\hat{\iota}}\\}_{\hat{\iota}=1,...,d_{i}}$
is a basis of $T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})$, where
$\\{\vec{e}_{i,\hat{\iota}}\\}_{\hat{\iota}=1,...,d_{i}}$ is the standard
basis of $X_{i}$. Notice that every
$P^{-1}(\omega,\bar{x})\vec{e}_{i,\hat{\iota}}$ corresponds (one-to-one) to a
linear mapping
$\displaystyle
h_{\mathcal{W}_{i}}\mapsto\frac{\partial(h_{\mathcal{W}_{i}}\circ
x(x_{i}))}{\partial
x_{i,\hat{\iota}}}\Big{|}_{x_{i}=0}=Dh_{\mathcal{W}_{i}}(\bar{x})P^{-1}(\omega,\bar{x}){\rm
id}_{i,\hat{\iota}}\in{\mathcal{L}}(X_{i,\hat{\iota}},\mathbb{R})$ (5.36)
for all smooth functions
$h_{\mathcal{W}_{i}}:{\mathcal{W}}_{i}(\omega,\bar{x})\to\mathbb{R}$, where
${\rm id}_{i,\hat{\iota}}$ denotes the identity in the 1-dimensional space
$X_{i,\hat{\iota}}$. This observation enables us to use
$P^{-1}(\omega,\bar{x})\vec{e}_{i,\hat{\iota}}$ here to denote the tangent
vector for convenience, although (5.36) is exactly the definition of tangent
vector given in [61, Definition 1.19]. Then, we have that for any given ${\bf
a}_{\omega,\bar{x}}\in T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})~{}(\cong
X_{i})$, there are $c_{1},\cdots,c_{d_{i}}\in\mathbb{R}$ such that
$\displaystyle{\bf
a}_{\omega,\bar{x}}=\sum_{\hat{\iota}=1,...,d_{i}}c_{\hat{\iota}}P^{-1}(\omega,\bar{x})\vec{e}_{i,\hat{\iota}}.$
(5.37)
This enables us to compute the differential of
$v_{\bar{x},i}^{\kappa,\iota}(\omega,\cdot)$ along
${\mathcal{W}}_{i}(\omega,\bar{x})$ at $\bar{x}$ by (5.35), that is,
$\displaystyle dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})\,{\bf
a}_{\omega,\bar{x}}$
$\displaystyle=\sum_{\hat{\iota}=1,...,d_{i}}c_{\hat{\iota}}P^{-1}(\omega,\bar{x})\\{D^{2}N^{-1}(\omega,\bar{x},{{0}})\vec{e}_{i,\hat{\iota}}\\}\,\vec{e}_{\kappa,\iota}$
$\displaystyle=\sum_{\hat{\iota}=1,...,d_{i}}c_{\hat{\iota}}P^{-1}(\omega,\bar{x})D^{2}N^{-1}(\omega,\bar{x},{{0}})(\vec{e}_{i,\hat{\iota}},\vec{e}_{\kappa,\iota})$
(5.38)
(see [61, p.17, 1.23(a)]). Note that
$dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x}):T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x})\to\mathbb{R}^{d}~{}(=T_{v_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})}\mathbb{R}^{d})$.
Regarding the Hölder continuity of
$dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})$ with respect to $\bar{x}$, we
define the bundle
${\mathcal{E}}:=\bigcup_{\bar{x}\in\mathbb{R}^{d}}T_{\bar{x}}{\mathcal{W}}_{i}(\omega,\bar{x}),$
which is a subbundle of $T\mathbb{R}^{d}(=\mathbb{R}^{2d})$ (therefore, in
${\mathcal{E}}$, one can use the same norm as the one in $\mathbb{R}^{2d}$).
In order to trivialize ${\mathcal{E}}$, let ${\bf
a}:=\sum_{\hat{\iota}=1,...,d_{i}}c_{\hat{\iota}}\vec{e}_{i,\hat{\iota}}\in
X_{i}$ and define a mapping $\hbar:\Omega\times\mathbb{R}^{d}\times
X_{i}\to{\mathcal{E}}$ as
$\displaystyle\hbar(\omega,\bar{x},{\bf a})={\bf a}_{\omega,\bar{x}}.$
Since $\mathbb{R}^{d}\times X_{i}$ is a subspace of $\mathbb{R}^{2d}$, we
regard $\hbar(\omega,\cdot)$ as a mapping from a subset of $\mathbb{R}^{2d}$
into $\mathbb{R}^{2d}$, which is clearly one-to-one. Thus, in view of (5.37),
we see that $\hbar(\omega,\cdot)$ and $\hbar^{-1}(\omega,\cdot)$ are
measurable with respect to $\omega$, $\beta_{E}$-Hölder continuous with
respect to $\bar{x}$ with a uniformly bounded Hölder coefficient with respect
to $\bar{x}$ and $i$ by (5.6). Moreover, we have that
$\hbar(\omega,\bar{x},{\bf a})$ and $\hbar^{-1}(\omega,\bar{x},{\bf a})$ are
linear with respect to ${\bf a}$ and that $\hbar^{-1}(\omega,\cdot)$
trivializes $T_{\mathbb{R}^{d}}{\mathcal{W}}_{i}(\omega,\bar{x})$.
In what follows, we rewrite $\hbar(\omega,\bar{x},{\bf a})$ as
$\hbar(\omega,\bar{x}){\bf a}$ for convenience. Then, for any ${\bf a}\in
X_{i}$ such that $\|{\bf a}\|=1$, using (5.38), we have that
$\displaystyle\|dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})\circ\hbar(\omega,\bar{x}){\bf
a}-dv_{\bar{y},i}^{\kappa,\iota}(\omega,\bar{y})\circ\hbar(\omega,\bar{y}){\bf
a}\|=\|dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x}){\bf
a}_{\omega,\bar{x}}-dv_{\bar{y},i}^{\kappa,\iota}(\omega,\bar{y}){\bf
a}_{\omega,\bar{y}}\|$
$\displaystyle\leq\sum_{\hat{\iota}=1,...,d_{i}}\big{\|}c_{\hat{\iota}}P^{-1}(\omega,\bar{x})D^{2}N^{-1}(\omega,\bar{x},{{0}})(\vec{e}_{i,\hat{\iota}},\vec{e}_{\kappa,\iota})-c_{\hat{\iota}}P^{-1}(\omega,\bar{y})D^{2}N^{-1}(\omega,\bar{y},{{0}})(\vec{e}_{i,\hat{\iota}},\vec{e}_{\kappa,\iota})\big{\|}$
$\displaystyle\leq
4\check{M}_{\epsilon}(\omega)\|\bar{x}-\bar{y}\|^{\beta_{E}}+4C_{\lambda}K_{\epsilon}(\omega)^{3}\hat{L}_{\epsilon,\rho}(\omega)\|\bar{x}-\bar{y}\|^{\beta_{N}}$
$\displaystyle\leq
4\\{\check{M}_{\epsilon}(\omega)+C_{\lambda}K_{\epsilon}(\omega)^{3}\hat{L}_{\epsilon,\rho}(\omega)\\}\|\bar{x}-\bar{y}\|^{\beta_{v}}$
when $\|\bar{x}-\bar{y}\|<1$. Here, (5.6) and (5.12)-(5.13) are also used to
obtain the above estimates, and
$\beta_{v}:=\min\\{\beta_{E},\beta_{N}\\}=\min\\{\beta_{E},\epsilon/\lambda_{\max},\alpha\\}\geq
0.$
Hence,
$\displaystyle\|dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})\circ\hbar(\omega,\bar{x})-dv_{\bar{y},i}^{\kappa,\iota}(\omega,\bar{y})\circ\hbar(\omega,\bar{y})\|\leq
4\\{\check{M}_{\epsilon}(\omega)+C_{\lambda}K_{\epsilon}(\omega)^{3}\hat{L}_{\epsilon,\rho}(\omega)\\}\|\bar{x}-\bar{y}\|^{\beta_{v}},$
which means that the differential
$dv_{\bar{x},i}^{\kappa,\iota}(\omega,\bar{x})$ is $\beta_{v}$-Hölder
continuous with respect to $\bar{x}$ with a uniformly bounded Hölder
coefficient with respect to $\bar{x}$ and $i$.
Step 6. Show that when $x\in{\mathcal{W}}_{i}(\omega,\bar{x})$ is close to
$\bar{x}$, $v_{\bar{x},i}^{\kappa,\iota}(\omega,x)$ belong to
$E_{s}(\omega,x)$ for all $\kappa={\tau}+1,...,{p}$ and
$\iota=1,...,d_{\kappa}$, and form a basis of $E_{s}(\omega,x)$.
Next, we show that for every $x\in{\mathcal{W}}_{i}(\omega,\bar{x})$ near
$\bar{x}$ the set
$\\{{v_{\bar{x},i}^{\kappa,\iota}}(\omega,x)\\}_{\kappa={\tau}+1,...,{p},~{}\iota=1,...,d_{\kappa}}$
is a basis of $E_{s}(\omega,x)$. Notice that (5.34) gives
$\displaystyle{v_{\bar{x},i}^{\kappa,\iota}}(\theta^{n}\omega,F(n,\omega,\bar{x}))$ |
# Urban Spatiotemporal Data Synthesis via Neural Disaggregation
Bin Han<EMAIL_ADDRESS>University of WashingtonSeattleUSA and Bill Howe
<EMAIL_ADDRESS>University of WashingtonSeattleUSA
###### Abstract.
The level of granularity of open data often conflicts the benefits it can
provide. Less granular data can protect individual privacy, but to certain
degrees, sabotage the promise of open data to promote transparency and assist
research. Similar in the urban setting, aggregated urban data at high-level
geographic units can mask out the underline particularities of city dynamics
that may vary at lower areal levels. In this work, we aim to synthesize fine-
grained, high resolution urban data, by breaking down aggregated urban data at
coarse, low resolution geographic units. The goal is to increase the usability
and realize the values as much as possible of highly aggregated urban data. To
address the issue of simplicity of some traditional disaggregation methods –
1) we experimented with numerous neural-based models that are capable of
modeling intricate non-linear relationships among features. Neural methods can
also leverage both spatial and temporal information concurrently. We showed
that all neural methods perform better than traditional disaggregation
methods. Incorporating the temporal information further enhances the results.
2) We proposed a training approach for disaggregation task, Chain-of-Training
(COT), that can be incorporated into any of the training-based models. COT
adds transitional disaggregation steps by incorporating intermediate
geographic dimensions, which enhances the predictions at low geographic level
and boosts the results at higher levels. 3) We adapted the idea of
reconstruction (REC) from super-resolution domain in our disaggregation case —
after disaggregating from low to high geographic level, we then re-aggregate
back to the low level from our generated high level values. Both strategies
improved disaggregation results on three datasets and two cities we tested on.
urban computing, spatial disaggregation
††ccs: General and reference Empirical studies††ccs: Computing methodologies
Computer vision††ccs: Applied computing
## 1\. Introduction
High-quality, longitudinal, and freely available urban data, coupled with
advances in machine learning, improve our understanding and management of
urban environments (Han and Howe, 2023). Over the last two decades, cities
have increasingly released datasets publicly on the web, proactively, in
response to transparency regulation. For example, in the US, all 50 states and
the District of Columbia have passed some version of the federal Freedom of
Information (FOI) Act. While this first wave of open data was driven by FOI
laws and made national government data available primarily to journalists,
lawyers, and activists, a second wave of open data, enabled by the advent of
open source and web 2.0 technologies, was characterized by an attempt to make
data “open by default” to civic technologists, government agencies, and
corporations (Verhulst et al., 2020).
While making data publicly available possesses numerous benefits, such as
enabling extensive analyses of urban dynamics or assisting policymakers in
making urban development decisions (Vogel et al., 2011; Yu et al., 2017),
however, the open data might contain sensitive information about individuals.
Consequently, sharing such data at the individual level imposes potential
risks to individual privacy. For example, starting in 2017, NYC open data
portal no longer released longitudes and latitudes of taxi trips. Instead,
they released the central point of the geographic regions where the taxi trips
started and ended.
The level of granularity of open data often conflicts the benefits it can
provide. As stated in (Green et al., 2017), less granular data can protect
individual privacy, but to certain degrees, sabotage the promise of open data
to promote transparency and assist research. For instance, in population
disaggregation studies, highly aggregated population counts at large, coarse
areal units can hide critical local hotspots and trends, by smoothing out the
spatial variations (Monteiro et al., 2019a). Similar in the urban setting,
aggregated urban data at high-level geographic units can mask out the
underline particularities of city dynamics that may vary at lower areal
levels. Additionally, the predefined geographic units could often mismatch
with the units that users would like to work with for certain type of
analyses, not to mention that the boundaries of those pre-defined units could
also change overtime.
In this paper, we aim to synthesize fine-grained, high resolution urban data,
by breaking down aggregated urban data at coarse, low resolution geographic
units. The goal is to increase the usability and realize the values as much as
possible of highly aggregated urban data. Traditional spatial disaggregation
methods, such as areal weighting and dasymetric mapping, are frequently used
in the task. Areal weighting is a simple method that requires no auxiliary
information. It is operated under relatively simple assumptions, thus
hindering its capacities to achieve great disaggreagtion performances.
Dasymetric mapping methods are popular and leverage auxiliary information to
conduct weighted disaggregation. Machine learning and deep learning approaches
could be incorporated within to enhance the performances. However, dasymetric
mapping methods are highly reliant on auxiliary data, which are often not
available in urban setting. The quality of results are also highly dependent
on the relevance between auxiliary data and the source data.
To address the issue of simplicity of some traditional disaggregation methods,
we experimented with numerous neural-based models that are capable of modeling
intricate non-linear relationships among features. Neural methods can also
leverage both spatial and temporal information concurrently. We showed that
all neural methods perform better than traditional disaggregation methods. To
enhance the disaggregation results of neural methods, we proposed a training
strategies that can be adopted by any training-based methods — COT-REC: chain-
of-training with reconstruction. COT adds transitional disaggregation steps by
incorporating intermediate geographic dimensions, which enhances the
predictions at low geographic level and boosts the results at higher levels.
We adapted the idea of reconstruction (REC) from super-resolution domain in
our disaggregation case — after disaggregating from low to high geographic
level, we then re-aggregate back to the low level from our generated high
level values. Both strategies improved disaggregation results almost all
disaggregation tasks we tested.
In summary, we make the following contributions:
* •
We leveraged neural methods to break down aggregated urban data, which
outperformed traditional, simple disaggregation methods that are performed
under simple assumptions. We tested neural methods on four different datasets
covering different domains and grids.
* •
We proposed a training strategy — COT-REC. COT-REC adds transitional steps in
disaggregation steps, which enforces better results at lower geographic levels
thus enhancing higher lever disaggregation results. COT-REC provides better
results on almost all disaggregation tasks we experimented on.
## 2\. Related Work
Our work is in align with the domain of spatial disaggregation, which is a
process to transform values from a spatial scale at low resolution into higher
resolution. It is widely used in population studies (Qiu et al., 2019;
Wardropa et al., 2018), as well as in other areas, such as disease mapping
(Arambepola et al., 2020) and vaccination coverage (Utazi et al., 2018).
The most basic approach for spatial disaggregation is areal weighting (AW),
which breaks down the source value uniformly over the total number of target
regions. No auxiliary information is needed for this method, but the
assumption of homogeneity is implied — the values are evenly distributed among
the target regions (Goodchild et al., 1993). Even though this method is quite
simple, in the absence of auxiliary information to help with the
disaggregation task, it remains to be a valid and valuable approach.
Pycnophylactic interpolation (PI), which was first introduced by Tobler in
1979, works similar as areal weighting, but with slight refinement and
operates under different assumption. PI distributes the values from source
regions onto target regions in an iterative manner, aiming to avoid sharp
discontinuity between the target regions while to still preserve the total
counts in the source regions. Each iteration will adjust the value for each
target zone by considering the weighted average of its nearest neighbors,
which ensures the smooth boundaries.
Dasymetric mapping/interpolation/weighting (DM) approach is highly popular in
the domain of spatial disaggregation. Unlike areal weighting scheme,
dasymetric mapping uses auxiliary information to generate a weighted
distribution of source values onto the target regions. Recent methods
leveraged machine learning approaches to model the relationships between
auxiliary data and source values and determine the weights for disaggregation
(Stevens et al., 2015; Qiu et al., 2019; Monteiro et al., 2018, 2019b). Some
previously used auxiliary information include satellite images (Stevens et
al., 2015), 3D building information (Qiu et al., 2019), mobile phone usage
data (Deville et al., 2014), terrain elevation and human settlement (Monteiro
et al., 2019b) etc.
Despite the popularity of dasymetric mapping methods in spatial
disaggregation, there are several potential limitations of them (Comber and
Zeng, 2019). They are highly dependent on the availability of auxiliary
variables, which are not always available(e.g., remote sensing images or land
type data in population disaggregation tasks). Additionally, the quality of
the auxiliary information and the relevance between the auxiliary data and
source values have great impacts on the disaggregation performances.
## 3\. Datasets
We worked with four datasets: NYC taxi data, NYC bikeshare data, NYC 911 call
data, and Chicago taxi data. Each dataset contains records at the individual
level (e.g., individual taxi/bikeshare trip; individual 911-call location).
Those four datasets cover various domains and city grids.
* •
NYC Taxi Data: NYC taxi trip data were collected from NYC Open Data
portal111https://opendata.cityofnewyork.us/data/ from 01/01/2016 to
06/30/2016. The raw data are presented in tabular format. Each record
summarizes the information for one single taxi trip, which contains the
longitude and latitude of the location where the taxi took off.
* •
NYC Bikeshare Data: NYC bikeshare data were collected from NYC DOT
222https://citibikenyc.com/system-datafrom 01/01/2021 to 06/30/2021. Similar
to the taxi data, the raw data are presented in tabular format. Each data
point summarizes the information for one single bike trip, including the
longitude and latitude of the location where the bike was unlocked.
* •
NYC 911 Call Data: NYC 911 call data were collected from NYC Open Data
portal333https://data.cityofnewyork.us/Public-Safety/NYPD-Calls-for-Service-
Year-to-Date-/n2zq-pubd from 01/01/2021 to 06/29/2021. Similar to the taxi
data, the raw data are presented in tabular format. Each data point summarizes
the call information, including the longitude and latitude of the location
where the call was made.
* •
Chicago Taxi Data Data: Chicago taxi data were collected from Chicago Data
portal 444https://data.cityofchicago.org/Transportation/Taxi-
Trips-2022/npd7-ywjz from 01/01/2022 to 12/31/2022. Similar to NYC datasets,
the raw data are presented in tabular format. Each data point summarizes each
taxi trip information, including the longitude and latitude of the location
where the passenger was picked up.
The average counts for each geographic level and the corresponding standard
deviations are reported in Table 1. We noticed that the standard deviations
are rather large. This is because the uneven distribution of data points. Take
taxi data for example, upper Manhattan regions (e.g., Inwood) have way fewer
taxi rides than central Manhattan.
Data | PUMA | NTA | TRACT | BLOCK | EXTREME
---|---|---|---|---|---
Taxi | 1443.01 (1776.03) | 450.94 (689.15) | 50.99 (73.26) | 3.85 (8.02) | 1.27 (3.59)
Bikeshare | 189.53 (279.24) | 59.23 (97.86) | 6.7 (13.06) | 0.5 (2.53) | 0.17 (1.41)
911 Calls | 23.25 (14.22) | 7.27 (7.05) | 0.82 (1.34) | 0.06 (0.32) | 0.02 (0.19)
Data | Community | TRACT | BLOCK | EXTREME | -
Chicago Taxi | 27.71 (64.61) | 2.39 (10.91) | 0.06 (1.7) | 0.04 (1.35) | -
Table 1. Descriptive statistics of four datasets. The reported numbers are the
average counts per geographic region. The standard deviation is in the
parenthesis.
For NYC, we select four predefined geographic levels, covering lower to finer
geospatial resolutions — Public Use Microdata Area (PUMA), Neighborhood
Tabulation Areas (NTA), Census Tracts (Tract), and Census Blocks (Block).
Their geographic boundaries in 2010 are collected from NYC open data portal.
We generated an geographic resolution level — Extreme, which contains areal
units that are smaller than Census Block. This geographic resolution is to
prepare for the synthesis of individual data records. For Chicago, which has
different predefined geographic structures than NYC, we select geographic
levels as close as possible to those of NYC — Community Areas (COM), Census
Tracts (Tract), and Census Blocks (Block). We also generated an geographic
resolution level — Extreme, which contains areal units that are smaller than
Census Block. Figure 1 shows the resolutions of five geographic levels in NYC.
Figure 1. Visualizations of five different geographic levels. PUMA has the
most coarse resolution, while EXTREME has the finest resolution.
We construct two types of input data:
* •
Vector Inputs: We construct the aggregated hourly count data at the five
geographic levels, using the individual records from the datasets. For each
hour, we count how many records are located within each areal unit at each
geographic level. The vector data is represented as
$X_{vec}\in\mathbb{R}^{N\times d}$, where $N$ is the total number of hours,
and $d$ is the number of areal units in each geographic level (e.g. $d$ = 10
for PUMA; $d$=32 for NTA).
* •
Image Inputs: to disaggregate urban data, we need to model the spatial
correlations among the units between coarse and fine geographic level, as well
as within each level. Convolutional Neural Networks (CNN) are good at modeling
spatial connections. Therefore, we transform each vector input into an image.
We plot units at each geographic level. The pixels in each unit takes the
value of $\frac{Count\;Value}{\\#of\;pixels\;in\;unit}$
For all three NYC datasets, we use the month of June as the test split, the
month of May as the validation split, and the rest as training data. For
chicago taxi data, we use the month of December as the test split, November as
the validation split, and the rest as training data. The split information is
presented in Table 2.
Data | Train | Val | Test
---|---|---|---
Taxi | 2,904 | 744 | 720
Bikeshare | 2,904 | 744 | 720
911 Calls | 2,880 | 744 | 720
Chicago Taxi | 7,296 | 720 | 744
Table 2. Count of hourly data records in each split.
## 4\. Models
### 4.1. Baselines Models
As our baseline, we tested several neural models as well as simple,
traditional disaggregation methods:
* •
Constant Weighting (CW): a simple method that disaggregates data uniformly
over the total number of regions at the finer resolution.
* •
Areal Weighting (AW): a simple method that disaggregates data proportionally
to the areas of target regions at the finer resolution. The larger the region,
the larger the value it gets.
* •
Historical Ratios (HR): a simple method that disaggregates data based on
historical distribution ratios. Unlike AW that disaggregates uniformly, HR
calculates the average disaggregation ratios from the source regions to the
target regions based on historical data (e.g., training data). The ratios are
then applied on unseen data.
* •
Feedforward Neural Network (FNN): a multi-layer feedforward neural network,
which models the non-linear spatial connections among the areal units between
coarse and fine geographic levels. We name it spatial NN to distinguish it
from spatiotemporal NN, which incorporates both spatial and temporal
information.
* •
Convolutional Neural Network (CNN): a model with UNet architecture, which
takes in image inputs and models the spatial correlations among pixel values.
* •
LSTM: Long Short Term Memory network is a well-known network architecture
which is powerful in capturing sequential dependency. We chose T=5 as the
temporal sequence length based on the study (Han and Howe, 2023).
### 4.2. Proposed Methods
#### 4.2.1. Chain of Training (COT)
Based on preliminary experimental results, we noticed that as we disaggregate
into finer geographic regions (e.g., from puma to block compared with fromp
puma to nta), the task gets harder thus yielding worse results. We hypothesize
that if we can disaggregate in a hierarchical style, the results could be
boosted. Therefore, we propose a training approach for disaggregation task,
Chain-of-Training (COT), that can be incorporated into any of the training-
based models. The idea is simple — when we disaggregate from low to high
geographic resolution, we add transitional disaggregation steps by
incorporating intermediate geographic dimensions. See Figure 2 for detailed
illustration.
Figure 2. The visualizations show the model architectures to disaggregate from
PUMA level to BLOCK level. Left: architecture of a regular feedforward neural
network model with two hidden layers. The hidden dimensions could be of any
values. Right: The two hidden layers have the dimension sizes that are equal
to the count of NTA and TRACT regions.
The advantage of adding intermediate geographic dimensions is that it makes
the model layers interpretable. Additionally, when we have data at the high
geographic level, we automatically have data at the intermediate levels, which
can be utilized for training purposes. In regular training (e.g., FNN), the
objective function is the $\ell_{1}$ loss at the highest geographic level:
$\ell_{high}=\frac{1}{N}|\sum\hat{X_{high}}-X_{high}|$. With COT, we also
calculate the loses at the intermediate levels and back-propagate them
together with the loss at the last layer:
$\ell_{total}=\ell_{high}+\ell_{intermedite}$
$\ell_{intermedite}=\frac{1}{N}|\sum\hat{X_{intermedite}}-X_{intermedite}|$
We hypothesize that this would be particularly helpful when the disaggregation
range is wide (e.g. going from PUMA to BLOCK). Traing on those hidden layers
could potentially help with the disaggregation results by making improvements
step by step.
#### 4.2.2. Reconstruction (REC) Loss
Inspired by (Yuan et al., 2018) in the work of super-resolution, where they
generated high-resolution images from low-resolution, and restored the high-
resolution images backward to low-resolution. Then the models are jointly
trained on the loss calculated from both inputs and outputs. We adapted
similar idea in our disaggregation case — together with COT training
procedure, we first disaggregate from low to high geographic level, and then
re-aggregate back to the low level from our generated high level values. We
calculate disaggregation losses at both high and intermediate levels, as well
as reconstruction losses. We tested Full-reconstruction as shown in Figure 3.
For each high geographic level, we reconstruct all possible lower levels
beneath it, and take the average at each lower level. The objective function
is then:
$\ell_{total}=\alpha_{1}*\ell_{PUMA}\;+\;\alpha_{2}*\ell_{NTA}\;+\;\alpha_{3}\ell_{TRACT}\;+\;\alpha_{4}\ell_{BLOCK}\;+\;\alpha_{5}\ell_{EXTREME}$
Figure 3. Reconstruction regimes.
For NYC data, the models disaggregate from PUMA level data, into NTA, TRACT,
BLOCK, and EXTREME levels. Therefore, we have four disaggregation tasks.
Likewise, we have three disaggregation tasks for Chicago taxi data, which all
start from COM level. We evaluate the results using Mean Absolute Error (MAE).
For neural methods, the batch size is 8 and learning rate is 1e-4. We use
early stop setting to prevent over-fitting the training data.
## 5\. Experimental Results
In this section, we present the experimental results of all the models as well
as our proposed methods on four datasets.
### 5.1. Neural Methods Outperform Traditional Methods
For each dataset, the models are trained on the its own training set, and the
results are reported on the corresponding test set. The errors are normalized
by the area. Consequently, we could compare the results across different
disaggregation resolutions. Table 3 presents the quantitative disaggregation
results. For the convenience of comparison, we visualize the errors in Figure
4.
Figure 4. High-regime disaggregation results on three datasets. X-axis shows the disaggregation resolutions. Y-axis shows the error per squared meter. Different colors indicate different models. Dataset | Resolutions | CW | AW | HR | FNN | CNN | LSTM
---|---|---|---|---|---|---|---
Taxi | PUMA $\rightarrow$ NTA | 1.820705 | 1.400443 | 0.463015 | 0.364603 | 0.227565 | 0.209696
PUMA $\rightarrow$ TRACT | 1.850812 | 2.055297 | 0.960405 | 0.700117 | 0.592701 | 0.562141
PUMA $\rightarrow$ BLOCK | 3.088336 | 2.963414 | 1.757388 | 1.500080 | 1.509036 | 1.421459
| PUMA $\rightarrow$ EXTREME | 4.282211 | 3.907982 | 2.314584 | 2.661453 | 2.242518 | -
Bikeshare | PUMA $\rightarrow$ NTA | 0.296892 | 0.201911 | 0.107852 | 0.101145 | 0.086823 | 0.089066
PUMA $\rightarrow$ TRACT | 0.518658 | 0.490639 | 0.265565 | 0.255558 | 0.241655 | 0.241166
PUMA $\rightarrow$ BLOCK | 1.412735 | 1.354402 | 0.398009 | 0.390442 | 0.386120 | 0.357188
| PUMA $\rightarrow$ EXTREME | 1.559778 | 1.523974 | 0.406618 | 0.688281 | 0.445166 | -
911 Call | PUMA $\rightarrow$ NTA | 0.030705 | 0.023857 | 0.015683 | 0.016084 | 0.015639 | 0.016175
PUMA $\rightarrow$ TRACT | 0.067976 | 0.068798 | 0.054865 | 0.050578 | 0.050630 | 0.050546
PUMA $\rightarrow$ BLOCK | 0.123104 | 0.120838 | 0.101465 | 0.063181 | 0.064917 | 0.062418
| PUMA $\rightarrow$ EXTREME | 0.128736 | 0.127147 | 0.105850 | 0.064858 | 0.067793 | -
Table 3. Disaggregation results traditional and neural-based methods. The unit
of the error is count/block/hour.
We have several observations from the visualizations: 1) As the target
disaggregation resolution gets finer, the task gets harder, which is supported
by the observation that the errors are larger for finer high resolutions. 2)
Simple disaggregation methods, such CW and AW, do not perform well across all
resolutions in all three datasets. HR, though without any training, improves
the performances a lot over the simple methods. 3) Neural methods overall
outperform simple methods. However, models that utilize temporal information,
in addition to the spatial information, have better performances than those
models that only use spatial information. LSTM appears to be the best model
out of all.
### 5.2. COT+Reconstruction Improves Results
We tested the proposed COT-REC method on two models — FNN and LSTM, across all
three datasets. The results are presented in Table 4. We noticed that — 1)
when the plain models are trained together with COT or both COT and REC, the
performances are better than the plain results for most of the disaggregation
tasks. With Taxi data, training with both COT and REC improves the most, while
for the rest two datasets, COT alone generates better results. 2) For the
cases where the plain models outperform COT and COT-REC, three out of four are
disaggregation from PUMA to NTA, which is the easiest task. Additionally, the
differences are trivial and statistically insignificant.
Dataset | Model | Resolutions | Plain | +COT | +COT, +REC-Full
---|---|---|---|---|---
Taxi | FNN | PUMA $\rightarrow$ NTA | 0.2276 | 0.2302 | 0.2328
PUMA $\rightarrow$ TRACT | 0.5927 | 0.5752 | 0.5695
PUMA $\rightarrow$ BLOCK | 1.5090 | 1.5023 | 1.4890
PUMA $\rightarrow$ EXTREME | 2.2425 | 2.2328 | 2.2025
LSTM | PUMA $\rightarrow$ NTA | 0.2097 | 0.2101 | 0.2149
PUMA $\rightarrow$ TRACT | 0.5621 | 0.5448 | 0.5429
PUMA $\rightarrow$ BLOCK | 1.4215 | 1.4136 | 1.3926
Bike | FNN | PUMA $\rightarrow$ NTA | 0.0868 | 0.0806 | 0.0815
PUMA $\rightarrow$ TRACT | 0.2417 | 0.2314 | 0.2321
PUMA $\rightarrow$ BLOCK | 0.3861 | 0.3675 | 0.3913
PUMA $\rightarrow$ EXTREME | 0.4452 | 0.4224 | 0.4456
LSTM | PUMA $\rightarrow$ NTA | 0.0891 | 0.0830 | 0.0829
PUMA $\rightarrow$ TRACT | 0.2412 | 0.2335 | 0.2316
PUMA $\rightarrow$ BLOCK | 0.3572 | 0.3447 | 0.3771
911-Call | FNN | PUMA $\rightarrow$ NTA | 0.0156 | 0.0156 | 0.0157
PUMA $\rightarrow$ TRACT | 0.0506 | 0.0502 | 0.0531
PUMA $\rightarrow$ BLOCK | 0.0649 | 0.0638 | 0.0884
PUMA $\rightarrow$ EXTREME | 0.0678 | 0.0667 | 0.0952
LSTM | PUMA $\rightarrow$ NTA | 0.0162 | 0.0156 | 0.0156
PUMA $\rightarrow$ TRACT | 0.0505 | 0.0502 | 0.0527
PUMA $\rightarrow$ BLOCK | 0.0624 | 0.0625 | 0.0946
Table 4. Disaggregation results of neural-based methods and our proposed COT-
REC method. The unit of the error is count/block/hour.
## 6\. Ablation Study
In this section, we present two ablation studies about moddel weighting and
different reconstruction schemes.
### 6.1. Weighted v.s. Unweighted Models
Since we calculate errors at all geographic levels and train the models with
all the errors jointly, we need to weight each term in the objective function
properly. We observed that the disaggregating from PUMA to EXTREME is a much
harder task than from PUMA to NTA. Therefore, the loss term at the EXTREME
level would be larger than that at the NTA level. Consequently, the weights
for each term in the objective function is determined by the dimension of each
geographic level, which up-weight the NTA loss more. To show the effects of
proper weighting, we trained models without the weighting and report the
results in Table 5.
Dataset | Model | Resolutions | Plain | +COT | +COT, +REC-Full
---|---|---|---|---|---
Taxi | Unweighted FNN | PUMA $\rightarrow$ NTA | 0.2276 | 0.2313 | 0.2308
PUMA $\rightarrow$ TRACT | 0.5927 | 0.5773 | 0.5715
PUMA $\rightarrow$ BLOCK | 1.5090 | 1.5157 | 1.5884
PUMA $\rightarrow$ EXTREME | 2.2045 | 2.2328 | 2.3131
Weighted FNN | PUMA $\rightarrow$ NTA | 0.2276 | 0.2302 | 0.2328
PUMA $\rightarrow$ TRACT | 0.5927 | 0.5752 | 0.5695
PUMA $\rightarrow$ BLOCK | 1.5090 | 1.5023 | 1.4890
| PUMA $\rightarrow$ EXTREME | 2.2425 | 2.2328 | 2.2025
Table 5. Disaggregation results of weighted and unweighted models. The unit of
the error is count/block/hour.
We noticed that when disaggregating from PUMA to NTA, weighting does not
provide much benefits, though the differences are still insignificant. For the
other two disaggregation tasks, we observed larger improvement with weighted
models. The reason is that NTA loss is much smaller than EXTREME loss. Without
the weighting, EXTREME loss will dominate the learning process, which counters
the idea of enforcing better results at low levels.
### 6.2. Reconstruction Schemes
In the main text, we reported the results of Full reconstruction. Here we
explored two more reconstruction options, namely — Bridge reconstruction and
BottomUp reconstruction. Bridge reconstruction only reaggregates lower level
values from the adjacent higher level, such as from NTA to PUMA and from BLOCK
to TRACT. BottomUp reconstruction reaggregates lower lever values only from
the highest geographic level in the task. We trained the models in the same
was described above. The results are provided in Table 6. Full and BottomUp
reconstruction both provide promising results, though BottomUp reconstruction
sometimes generates way worse results than the plain model (e.g. PUMA to
TRACT). Therefore, Full reconstruction is relatively more stable. Bridge
reconstruction improves over the plain model results, but not as good as the
other two methods.
Dataset | Model | Resolutions | Plain | +COT | +COT, +REC-BottomUp | +COT, +REC-Bridge | +COT, +REC-Full
---|---|---|---|---|---|---|---
Taxi | FNN | PUMA $\rightarrow$ NTA | 0.2276 | 0.2302 | 0.2328 | 0.2328 | 0.2328
PUMA $\rightarrow$ TRACT | 0.5927 | 0.5752 | 0.6197 | 0.5780 | 0.5695
PUMA $\rightarrow$ BLOCK | 1.5090 | 1.5023 | 1.4908 | 1.5203 | 1.4890
PUMA $\rightarrow$ EXTREME | 2.2425 | 2.2328 | 2.1683 | 2.2911 | 2.2025
LSTM | PUMA $\rightarrow$ NTA | 0.2097 | 0.2101 | 0.2149 | 0.2149 | 0.2149
PUMA $\rightarrow$ TRACT | 0.5621 | 0.5448 | 0.5405 | 0.5412 | 0.5429
PUMA $\rightarrow$ BLOCK | 1.4215 | 1.4136 | 1.3926 | 1.4033 | 1.3926
Table 6. Disaggregation results of different reconstruction schemes. The unit
of the error is count/block/hour.
## 7\. Discussion
In this work, we aim to synthesize fine-grained, high resolution urban data,
by breaking down aggregated urban data at coarse, low resolution geographic
units. The goal is to increase the usability and realize the values as much as
possible of highly aggregated urban data. To address the issue of simplicity
of some traditional disaggregation methods – 1) we experimented with numerous
neural-based models that are capable of modeling intricate non-linear
relationships among features. Neural methods can also leverage both spatial
and temporal information concurrently. We showed that all neural methods
perform better than traditional disaggregation methods. Incorporating the
temporal information further enhances the results. 2) We proposed a training
approach for disaggregation task, Chain-of-Training (COT), that can be
incorporated into any of the training-based models. COT adds transitional
disaggregation steps by incorporating intermediate geographic dimensions,
which enhances the predictions at low geographic level and boosts the results
at higher levels. 3) We adapted the idea of reconstruction (REC) from super-
resolution domain in our disaggregation case — after disaggregating from low
to high geographic level, we then re-aggregate back to the low level from our
generated high level values. Both strategies improved disaggregation results
on three datasets and two cities we tested on.
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|
# Wireless earbuds for low-cost hearing screening
Justin Chan ⋄1 Antonio Glenn ⋄1 Malek Itani 1,2
Lisa R. Mancl3 Emily Gallagher 4 Randall Bly 4,5 Shwetak Patel 1,2 and
Shyamnath Gollakota1
⋄Co-primary student authors
1Paul G. Allen School of Computer Science and Engineering University of
Washington
2Department of Electrical and Computer Engineering University of Washington
3Department of Speech and Hearing Sciences University of Washington
4Seattle Children’s Hospital and Research Institute
5Department of Otolaryngology — Head and Neck Surgery University of
Washington
###### Abstract
We present the first wireless earbud hardware that can perform hearing
screening by detecting otoacoustic emissions. The conventional wisdom has been
that detecting otoacoustic emissions, which are the faint sounds generated by
the cochlea, requires sensitive and expensive acoustic hardware. Thus, medical
devices for hearing screening cost thousands of dollars and are inaccessible
in low and middle income countries. We show that by designing wireless earbuds
using low-cost acoustic hardware and combining them with wireless sensing
algorithms, we can reliably identify otoacoustic emissions and perform hearing
screening. Our algorithms combine frequency modulated chirps with wideband
pulses emitted from a low-cost speaker to reliably separate otoacoustic
emissions from in-ear reflections and echoes. We conducted a clinical study
with 50 ears across two healthcare sites. Our study shows that the low-cost
earbuds detect hearing loss with 100% sensitivity and 89.7% specificity, which
is comparable to the performance of a $8000 medical device. By developing low-
cost and open-source wearable technology, our work may help address global
health inequities in hearing screening by democratizing these medical devices.
## 1 Introduction
Figure 1: OAEbuds in use with an infant. Our low-cost wireless earbud can
perform hearing screening by detecting otoacoustic emissions (OAE) from the
cochlea.
The World Health Organization estimates that 5.3% of the world’s population
suffers from disabling hearing loss and 80% of people who need hearing care
live in low and middle-income countries [1, 2, 3]. Hearing loss is
particularly harmful for neuro-development if it is left undetected in early
childhood [4, 5, 6]. As a result, high-income countries have guidelines for
universal infant hearing screening — the Joint Committee on Infant Hearing,
the American Academy of Pediatrics and the Centers for Disease Control and
Prevention all recommend universal hearing screening [4, 5, 7, 8] that is now
implemented across almost all states, communities and hospitals in the United
States [9, 10].
Since the neonatal population cannot provide behavioural response to
conventional audiometry tests [11, 12, 13, 14], existing newborn hearing
screening technologies instead use the sounds generated by a healthy cochlea
called otoacoustic emissions (OAE) [15, 16]. While we think of the ear as a
biological organ that receives sounds like a microphone, a healthy cochlea,
the part of the inner ear responsible for converting sound waves into
electronic impulses for the brain, also generates sounds. These emissions are
created when the cochlea’s sensory hair cells vibrate in respond to external
sounds [15, 17, 18]. So, we could pick up these faint sounds and use their
absence to detect hearing loss.
The challenge is that detecting these faint sounds emitted from the cochlea
requires sensitive acoustic hardware and medical devices that are expensive
(5000-8000 dollars) [19, 20]. As a result, there is limited to no-hearing
screening in low and middle-income countries [21, 22]. Further, in rural and
resource-limited settings, getting access to hearing assessment may often
require travel to an urban setting and long wait times, significantly limiting
the accessibility of hearing care [2, 3, 6].
(a) (b)
Figure 2: OAEbuds hardware. (a) The 3D-printed enclosure with pediatric and
adult earbud tips. (b) OAEbud circuit board beside a penny for size
comparison.
We present OAEbuds, the first wireless earbud design for low-cost hearing
screening. Our hardware-software system reliably detects otoacoustic emissions
using low-cost acoustic hardware, while being in the form-factor of a wireless
earbud. The earbud hardware is designed to work across a wide demographic from
new-borns to adults.111OAE testing is not limited to just newborns but is also
used as part of clinical care in older kids and adults [23]. Our design
streams the digital acoustic data via Bluetooth to a nearby smartphone which
is then used for processing the signals and displaying the test results.
There are two key technical challenges in achieving this design with low-cost
acoustic components. First, since speaker hardware components are bulky, it is
challenging to incorporate the two-speaker design that has been proposed in
recent work [6] into the form-factor of a wireless earbud. Our experiments in
§2.1 show that transmitting the dual-tone signals used in [6] on a single low-
cost speaker hardware introduces nonlinearities that in turn creates inter-
modulation tones that can be confused for OAEs. Second, when an acoustic
signal is sent into an ear canal, it first gets reflected and creates echoes
not only inside the earbud case but also the ear drum and the walls of the ear
canal, before arriving at the cochlea (see Fig. 3). To accurately identify
OAEs, it is important to determine when the reflections and echoes of the
input stimuli end and the OAEs begin.
We design a two-step protocol that uses wireless sensing techniques to address
the above challenges using a single low-cost speaker in our wireless earbuds.
* •
Reflection time estimation. In the first step, we send frequency modulated
continuous wave (FMCW) signals as input stimuli. These signals get reflected
back from the earbud case, the ear drum and the ear canal which are captured
at the microphone. Since OAE signals are faint and non-linear, they do not
create linear FMCW reflections. So we can perform FMCW processing to estimate
the time-delays corresponding to the reflections and echoes and determine the
duration after which their power reduces below a preset threshold (see
§2.2.1).
* •
OAE signal extraction. In the second step, we transmit a train of wideband
pulses from the earbud speaker. Since the travelling sound wave traverses more
slowly in the cochlea [24], the otoacoustic emissions still arrive delayed in
time after the reflections and echoes of the input stimuli. To extract these
signals, we first reduce reflections by only considering the signals that
arrive after the duration estimated in the previous step. We then synchronize
the responses across multiple wideband pulses, combine them to improve the SNR
of OAE signals and detect them using our earbud system (§2.2.2).
Figure 3: In-ear signal propagation. During the measurement, the OAEbud plays
a broadband transmit (Tx) pulse to stimulate the cochlea to emit an OAE
signal. The signal received by microphone is a superposition of 1) unwanted
reflections from the ear canal, eardrum, and within the case and 2) the OAE
signal.
We designed an open-source wireless earbud hardware shown in Fig. 2 that is
capable of transmitting the above signals from its speaker and wirelessly
streaming the microphone audio. We designed our OAEbuds hardware using open
source eCAD software, outsourced fabrication and assembly ($28.3 per unit),
and 3D printed the enclosures in-house. The earbud is designed to support
multiple ear tip sizes that allows it to snugly fit for both new-born infants
and adults. The battery in the earbud can be recharged via a USB connection
within 3 hours. Our evaluation shows that on a single charge, the earbud can
be used to perform up to 91 hearing tests.
We perform a clinical study on 50 ears from 26 pediatric and adult patients
across two different healthcare sites. We perform testing with both our earbud
device as well as an FDA-cleared medical device that performs OAE detection.
For the tested patients, the attending clinicians determined the ground truth
for hearing loss using the patient’s hearing screen, audiograms, diagnostic
auditory brain response, and clinical history. Our evaluation shows that
OAEbuds achieves a sensitivity of 100% and specificity of 89.7% in screening
for hearing loss. In comparison, the FDA-cleared medical device achieves a
sensitivity of 83.3% and specificity of 92.1%. Our techniques also improve the
area under the curve (AUC) from 0.847 to 0.950 over existing OAE algorithms.
Finally, our system can output a ‘pass’ or ‘refer’ result for hearing
screening in under 70 seconds.
Contributions. We make the following contributions.
* •
We design the first wireless earbuds to achieve low-cost hearing screening by
detecting otoacoustic emissions.
* •
We introduce a two-step protocol that combines FMCW signals with wideband
pulses to separate reflections and echoes from OAEs while using a single low-
cost speaker.
* •
We perform a clinical study that shows our low-cost wireless earbud detecting
hearing loss with accuracies similar to a $8000 FDA-cleared medical device.
* •
Finally, we will make our code and hardware open source to help with adoption
across the target settings.
Comparison to prior work. The closest to our work is recent work [6] that uses
the two speakers in a wired earphone. It transmits a different frequency tone
from each speaker and uses an additional microphone that is placed next to the
ear to create a smartphone attachment. This work has multiple constraints that
limit its adoption in the target use-cases. 1) It uses a wired earphone and
external microphone that are connected to a smartphone. Since it uses the
smartphone’s ADC, DAC and AGC, it requires manual calibration for each
smartphone model, which is challenging to generalize. 2) It uses two frequency
tones that are transmitted from two different speakers and looks for inter-
modulation between the tones to detect OAE. The challenge is that it is
difficult to incorporate two speakers pointing into the ear-canal in a
wireless earbud form factor and hence the techniques used in this prior work
cannot be used for wireless earbuds (see §2.1). 3) The various hardware
components are attached using plastic tubing and glue which make it unreliable
and difficult to scale and introduces a DIY aspect to the system. In an
informal survey of clinicians in an African (anonymized) country conducted by
the authors, participants noted that this DIY-aspect could translate to
lowered patient confidence in the care received at the clinic. A low-cost yet
high-tech device would be required to achieve wider adoption by clinicians.
Our paper addresses the above limitations and designs the first wireless
earbuds for low-cost hearing screening. Compared to DIY devices, since our
wireless earbud is more integrated while being low-cost, it may help broaden
the adoption of our hearing screening tool.
## 2 System design
We first describe existing approaches to OAE sensing and their limits. We then
present our two-step protocol to estimate the reflection time and extract OAE
signals. Finally, we present our low-cost earbud hardware.
### 2.1 Existing OAE approaches
The challenge with reliably detecting OAEs is identifying them in the presence
of much stronger in-ear reflections. There are two key prior approaches.
Figure 4: Challenge of existing OAE approaches on a single-speaker earbud
design. Sending two stimulus tones $f_{1}$ and $f_{2}$ through a single
speaker setup to elicit OAEs creates a hardware non-linearity at
$2f_{1}-f_{2}$, which can be stronger than the OAE signal at that frequency.
* •
DPOAEs. Distortion-product otoacoustic emissions (DPOAEs) address the
reflection problem by using intermodulation. In particular, the cochlea is
stimulated by sending two tones $f_{1}$ and $f_{2}$. Given the nonlinear
response of the basilar membrane within the cochlea, it generates a nonlinear
intermodulation tone at the frequency $2f_{1}-f_{2}$ [6]. Since reflections
and echoes do not cause new frequency tones, this dual-tone approach can be
used to separate OAEs from in-ear reflections. The challenge with deploying
the DPOAE protocol on a single-speaker system is that hardware components
introduce nonlinear intermodulation distortion at frequencies which are linear
combinations of $f_{1}$ and $f_{2}$, $k_{1}f_{1}+k_{2}f_{2}$, where $k_{1}$
and $k_{2}$ are arbitrary integers. When $k_{1}=2$ and $k_{2}=-1$, this
matches the DPOAE signals produced by the cochlea. These non-linearities are
more prominent for low-cost speaker hardware. Fig. 4 shows the amplitude of
the intermodulation produced by a single low-cost speaker (Knowles
SR-32453-000, $4.42) when sending two tones 1640 and 2016 Hz at 65/55 dB SPL.
The figure shows that the unwanted intermodulation component has a sound level
of 28 dB SPL; in comparison the typical range of DPOAEs is 5–25 dB SPL [6]. As
a result, prior work [6] uses a two-speaker system to separately send the
$f_{1}$ and $f_{2}$ tones. Since wireless earbuds are generally constrained to
only a single speaker per bud due to size constraints, it is challenging to
use the DPOAE protocol on such a low-cost hardware.
* •
TEOAEs. The transient-evoked otoacoustic emission (TEOAE) protocol extracts
the OAEs in the presence of in-ear reflections and echoes using a single
speaker. Here a short biphasic click sequence is repeatedly sent, with a
polarity and amplitude pattern of $\\{1,1,1,-3\\}$ [25] . The key insight
behind this protocol is that the amplitude of the reflections are linearly
related to the amplitude of the transmitted clicks. So the responses of all
four clicks can be summed to cancel the reflections caused by the eardrum.
However since OAEs are non-linear in nature, they would not be canceled by
this addition operation. The challenge with deploying this protocol on a low-
cost system, is that 1) the clicks need to be perfectly synchronized and
phase-aligned, so that the reflections are cancelled out. Without exact
alignment, there will continue to be residual energy caused by imperfect
cancellation which will make it difficult to measure the OAEs. 2) Low-cost
speaker hardware also introduces non-linearities in polarity and amplitude
resulting in imperfect cancellation. Our evaluation in §4 shows that this
leads to degraded performance.
While other methods for eliciting OAEs have been proposed in the literature
[26, 27] they are not used in practice given uncertainty about their
reliability.
### 2.2 Our two-stage protocol
Instead of relying on the linearity of the acoustic hardware, we separate the
in-ear reflections from OAEs in the time domain. At a high level, we first
estimate the time delay at which reflections from both the ear and the
enclosure drop below a particular threshold. We then detect the OAEs over the
remaining time duration.
Figure 5: FMCW processing to calculate the time of arrival for reflections
from the ear canal. The OAEbud transmits a chirp into the ear canal, and
record the reflections from the ear and enclosure. It then performs an FFT
over the chirp duration to estimate the frequency shift $\Delta f_{i}$ and
time delay $\tau_{i}$ for the $i^{th}$ reflection from the ear. This estimate
is averaged across three chirps.
#### 2.2.1 Reflection time estimation
The time delay when reflections diminish will differ from one ear to another
due to differences in anatomical structures such as ear canal diameter which
increases with age. Further, hair and debris can change the reflection profile
significantly. At a high level, we send an FMCW signal to estimate when
reflections from the case and other parts of the ear diminish beyond a certain
threshold. Our algorithm then uses the remainder of the recording after this
time estimate to measure OAEs.
Although FMCW signals could be used to estimate the length of an individual
ear canal, and convert that to a time delay at which reflections diminish, we
find that in practice there is a significant amount of echos caused by
reflections from the case, ear drum and ear canal that result in a large delay
spread, much larger than the time of flight measurement for a typical ear
canal length of 2.5 cm. An analogy to this would be that if one shouts in an
empty cave, it can take several seconds for all the echos to diminish due to
the significant reflections that occur from the cave walls. Further we note
that the speed of sound is slower in the cochlea, meaning the OAE would take a
longer time to return compared to if the pulse were only sent into the air
medium, and this contributes further to a large delay spread [24].
We send a chirp with linearly increasing frequency from $f_{0}$ to $f_{1}$
where the frequency at a given time $t$ is denoted as
$f(t)=f_{0}+\frac{Bt}{T}$, where $B$ and $T$ are the bandwidth and duration of
the chirp. The phase is computed by integrating $f(t)$ over time, resulting in
the function: $\phi(t)=2\pi(f_{0}t+B\frac{t^{2}}{2T})$. The signal that is
then transmitted in the time domain is defined as $x(t)=cos(\phi(t))$, as
shown in Fig. 5.
Each of the echoes from the ear canal are delayed chirps that arrive at the
microphone as the received signal, $y(t)$, that is a combination of all
echoes. To estimate the multipath profile of the ear canal, we multiply the
receiver signal with the transmitted signal, $x(t)y(t)$. Using the
trigonometric identify, $2cos(A)cos(B)=cos(A-B)+cos(A+B)$ and filtering out
the high frequency term, $cos(A+B)$, we can translate the time delays of each
of the echoes into frequency shifts between the transmitted and received
chirps. Note that in contrast to radio signals that have both I and Q
components, acoustic signals operate in the real space. So instead of using a
downchirp, we multiply the received cosine signal with the transmitted signal
and apply a low-pass filter.
In Fig. 5, we show a plot of the transmitted FMCW signal, along with several
reflections in the frequency domain, each with its own time delay $\tau_{i}$
for the $i^{th}$ reflection. In order to determine individual time delays
$\tau_{i}$ when the reflections end, we compare the differences in frequencies
between the transmitted and reflected signals. Specifically, a time delay
$\tau_{i}$ will result in a frequency shift of $\Delta f_{i}$ for the
reflected signal from the transmitted signal and can be computed as follows:
$\tau_{i}=\frac{\Delta f_{i}T}{f_{1}-f_{0}}$
To obtain a precise resolution for our reflection time estimate, we send an
FMCW signal that is close to the maximum bandwidth allowable by the sampling
rate of our system. As the maximum sampling rate of our OAEbuds hardware is
31250 Hz, we send an FMCW signal with a bandwidth of 5 to 15 kHz so that the
upper frequency is close to the Nyquist frequency of 15625 Hz. The time
resolution of an FMCW system is, $\frac{1}{2B}$ , where $B$ is the bandwidth.
This corresponds to a time resolution of 0.05 ms when $B$=10 kHz. We set the
length of our signal based on the maximum number of samples that can be stored
in our hardware’s memory. In our system we use a 200 ms FMCW signal which
corresponds to 6250 samples.
Figure 6: Estimating when reflections diminish. By measuring the frequency
shifts from reflections of a FMCW signal transmitted into the ear canal, we
can estimate the time delay $t_{D}$ after which reflections diminish to a
predetermined power threshold.
Fig. 6 shows the result of this processing in a normal adult ear. We can
observe a peak in the zeroth bin that corresponds to the incident chirp, and
peaks at subsequent bins that correspond to reflections arriving at increasing
time delays. To minimize the interfering effects of reflections in our OAE
measurement, we select a time delay that corresponds to the frequency shift
where the power level diminishes below a preset power threshold. In our
implementation, if the power level of a frequency bin decreases below 55 dB
from the power of the incident signal, we use the time delay, $t_{D}$,
corresponding to that frequency bin. If such a bin cannot be found, a default
delay value of 12 ms is used.
We note two key points about our earbud system.
* •
Performing this estimation is important particularly given the tonotopic
geometry of the cochlea (Fig. 3) where the high frequency OAEs exit the
cochlea first, followed by the low frequencies. The cochlea has a coiled shape
where the beginning of the coil responds to high frequency sounds, while the
inner most curled part of the coil responds to low frequency sounds. As such,
when a stimulus pulse is sent into the cochlea, it is the high frequency OAEs
that exit first, and it is these frequencies that would also be most affected
by the reflections. By setting a time delay that is too low, there is a risk
that the reflections will be confused for the OAEs, whereas setting the delay
too high may only result in a measurement over the low frequency OAEs, but few
of the high frequency OAEs. Our algorithm allows us to minimize the power of
reflections while increasing the power of the OAE signals.
* •
Interestingly, the ear canal creates a closed enclosure that can create a
large number of strong echoes. As a result, while the length of the ear canal
is only around 2.5 cm [28], which translates to an acoustic round trip time of
0.15 ms, given the large number of echoes created within the ear canal, we can
have reflections as shown in Fig. 6 that arrive even at 5-10 ms. This
emphasizes the need for using a system that computes the time-delays for the
in-ear reflections which can be much longer than the time it takes to traverse
the ear canal. We also note that we compute the above time delay by averaging
the values across three continuous FMCW chirps.
#### 2.2.2 OAE signal extraction
After the time delay has been identified, our system needs to reliably measure
the faint otoacoustic emissions that are as low as -10 to 30 dB SPL using low-
cost microphones that would not have the same sensitivity of the high-end
expensive microphones in medical devices. To extract these faint OAE signals,
at a high level, we combine the OAE responses across multiple pulses to
increase the SNR of OAEs. This can be challenging especially given that the
target population of this test is young infants who may move, fidget or
otherwise cause noise throughout the measurement. Our measurement should also
be able to reliably distinguish between periods of noise caused by the patient
and legitimate OAE signals, as an incorrect classification would result in an
inaccurate measurement or an overly lengthy measurement that would result in
patient discomfort.
In the rest of this section, we first describe our transmission scheme and
then describe the various steps needed to extract the OAE signals.
Pulse transmission scheme. We transmit a sequence of short $500~{}\mu$s pulses
and apply a brick-wall filter with a bandwidth from $f_{start}=0~{}kHz$ to
$f_{end}=5~{}kHz$, to cover the full range of frequencies of clinical interest
in hearing screening (Fig. 7). We use a sampling rate of 15625 Hz, which
corresponds to 8 samples to represent the pulse. We multiply the pulse with a
hamming window to reduce the effect of ringing. Each of the pulses are
separated by a gap of 20 ms, to ensure that the OAEs have enough time to
arrive at the microphone. We perform these measurements over the course of
67.5 seconds, which corresponds to approximately 3300 pulses.
Figure 7: OAEbuds pulse transmission scheme. A pulse of $500~{}\mu$s with a
bandwidth from 0–5 kHz is transmitted every 20 ms into the ear to cover the
range of frequencies important for hearing screening. The recorded signal
consists of reflections of the input stimuli from the ear canal which overlap
with the OAE signal.
Decoding algorithms. We describe the various steps to combine the OAE
responses across pulses and extract higher SNR OAE signals.
Step 1. Pulse synchronization. The first step of our algorithm is to establish
synchronization with the start of the pulse. To do this, we perform cross
correlation of the first one second window with the transmitted pulse, and
look for peaks with a minimum peak prominence set to 0.3e8. If we do not find
such a peak in this window, we proceed to the next one second window. We note
that we may not find peaks within the first one second window if the user
initially begins the measurement outside the ear where the amplitude of the
reflections will be lower compared to in the ear. Once the start of a pulse
can be found, we can add a fixed offset of 20.5 ms to find the start of the
next pulse. We also use this windowed approach instead of performing cross
correlation over the entire one minute recording as it allows for real-time
computation of the OAEs over the course of the measurement. In other words,
our algorithm is able to incrementally compute the OAE result over windows of
one second intervals, and provide continuous feedback to the user about the
OAE results, and whether the environment is too noisy. This will allow the
user to be able to recognize possible problems in probe fit or environmental
noise in real-time instead of having to wait until the entire one minute
measurement is complete.
Step 2: Noise detection. After we have a set of peaks corresponding to the
start of all the pulses in a 1-second window, our next step is to determine
which pulses are usable for subsequent processing and identify pulses that
have been affected by noise in the environment. To do this, we apply a sliding
correlation window over batches of four pulses and calculate the correlation
between each of the adjacent pulses. We sum the calculated correlation values
and divide it by the sum of the received power within that batch. This
produces a normalized value that is invariant to pulse amplitude differences
across batches. If this normalized value is above 0.95, we consider that batch
to be usable for subsequent measurement, else we discard that batch. This
allows us to discard specific OAE signals within the overall measurement that
have been corrupted by noise.
Step 3: Combining OAE responses across pulses. For all usable batches in a
given window, we look for OAE responses using the time delay $t_{D}$ computed
in the previous section. We use the window size of $t_{P}-t_{D}-t_{guard}$
where $t_{P}$ is the gap between consecutive pulses and $t_{guard}$ is a guard
period which we set to 1 ms. In other words, we look for OAEs starting from
the time when the reflections have diminished up till the start of the next
pulse, minus a small guard period. We then average the power of all odd
numbered pulses to compute, $P^{OAE}_{odd}$ and all even numbered pulses for
$P^{OAE}_{even}$. We then compute the signal and noise power, $P_{signal}$ and
$P_{noise}$ in the time domain as follows:
$P_{signal}=\frac{P^{OAE}_{odd}+P^{OAE}_{even}}{2},P_{noise}=\frac{\lvert
P^{OAE}_{odd}-P^{OAE}_{even}\rvert}{2}$
To obtain the SNRs across 1 to 5 kHz, we convert the above signals to the
frequency domain by performing an FFT. We repeat the above process for each
frequency band by taking an average across adjacent frequency bins.
Specifically we perform an average over the following bands: 750–1250 Hz,
1250–1750 Hz, 1750–2500 Hz, 2500–3500 Hz, and 3500–4500 Hz. The result of this
step is a set of SNR measurements for each frequency band.
Hearing testing algorithms. Finally, we describe the algorithms we run while
performing the hearing test.
a) Computing the pass/refer result. To calculate the ‘pass’ or ‘refer’
screening result, as with existing medical devices, we determine if at least
2/3 of the 5 frequency bands are above a preset threshold (8 dB in our case).
These parameters were determined by performing a parameter sweep over
different number of frequency bands and SNR thresholds to determine values
that optimize our clinical performance. Further, we check that the absolute
sound level of the signal component is above a preset value of -10 dB SPL,
which is typically regarded as the minimum sound level of an OAE. This ensures
that spurious reflections or noise that were not discarded during previous
filtering steps are not mistaken as OAEs. Additionally, if the average noise
level across the frequency bands exceeds 6 dB SPL, we mark the measurement as
noisy.
Figure 8: Checking if the probe is in the ear. By measuring the amplitude of a
chirp at the 200 Hz frequency during the beginning of a measurement, we can
detect whether the ear probe is inside or outside the ear.
b) Determining if probe is in the ear. To determine when the measurement can
begin, our system performs a check for whether the probe is probably placed in
the ear. To do this, we send a sequence of 20 ms chirps from 100 to 5500 Hz
and measure the amplitude of the frequency response to determine if the probe
has formed a snug fit with the ear. We find that the frequency at 200 Hz is
representative of whether the probe is outside or inside the ear (Fig. 8). If
the average sound level in this frequency range exceeds a predefined threshold
for 50 chirps (1 s), we mark the probe tip as being in the ear and begin the
measurement.
### 2.3 Hardware design
We design a custom hardware solution based on the ISP1807 Bluetooth Low Energy
(BLE) module, which combines a Nordic nRF52840 microcontroller with a variety
of other components such as capacitors, oscillators and an antenna. The device
is equipped with a pair of pulse-density modulated (PDM) microphones (TDK
Invensense T3903) and a speaker (PUI Audio AS01008MR-3) driven by a digital
pulse-code modulation (PCM) input Class D amplifier (Maxim Integrated
MAX98357A). The system is powered by a 3.7 V, 100 mAh Lithium Polymer Battery,
and a buck converter (Texas Instruments LM3671) is used to bring the system
voltage down to 3.3V. A Micro-USB connector is used to program the device over
SWD and charge the battery via a charger IC (Analog Devices LTC4124). Battery
information, such as cell voltage and state of charge (SOC), is probed using a
fuel gauge (Maxim Integrated MAX17048). A high level overview of the system is
shown in Fig. 9.
Figure 9: OAEbuds hardware design. We include an additional microphone for
future research.
The speaker amplifier is interfaced using the controller’s Inter-IC Sound
(I2S) module. The device is preloaded with a fixed array in the controller’s
memory that holds the PCM representation of a signal (e.g., pulse). The
preloaded waveform must be generated at a sampling frequency matching that of
the I2S module clock signals. The sampling frequency is set to 15.625 kHz, the
smallest frequency compatible with our amplifier that is larger than twice the
pulse bandwidth. When the device starts emitting pulses, the controller
supplies a copy of the signal waveform to the I2S module, which transfers the
waveform to the amplifier using its direct memory access (DMA). The I2S module
is internally double buffered, and it triggers a callback for the controller
to supply a fresh audio buffer to output once a previous transfer finishes.
The microphones are interfaced using the controller’s PDM module.
Specifically, the module’s DMA is used to asynchronously convert microphone
PDM measurements to PCM values and load the results into memory. The two PDM
microphones are connected to the same serial clock line, running at 1 MHz,
which the PDM module internally decimates by a factor of 64. This yields an
overall sampling frequency of 15.625 kHz. The recording process also uses
double buffering and produces 312 channel-interleaved pairs of 16-bit samples
(equivalent to 20 ms at 15.625 kHz) at a time. The captured samples are then
divided into packets and transmitted over Bluetooth. To maximize throughput,
each packet contains 80 two-channel samples (240 bytes total), which, when
including the sequence number and overhead bytes, is the largest number of
samples we can transmit in a single packet. Additionally, we also use the
maximum possible data rate of 2 Mbps. In our design, the earbuds stream the
recorded acoustic signals via Bluetooth to a nearby smartphone and the
computation to extract the OAEs is performed on the smartphone.
Component | Cost (USD)
---|---
BLE Module | 10.67
Microphones | 2 $\times$ 0.80
Speaker | 1.06
Amplifier | 1.62
Charger | 5.12
Fuel Gauge | 1.89
MicroUSB Connector | 0.29
Switch | 1.52
Battery | 1.50
PCB Fabrication & Assembly | 1.63
3D Printed Case | 1.40
Total | 28.30
Table 1: Itemized hardware cost. Component prices are estimated for a
production lot of 1,000 devices.
(a)
(b)
(c)
Figure 10: Clinical study performance.(a) Audiograms for ears tested in
clinical study with normal hearing and different degrees of hearing loss. (b)
Performance of OAEbuds in comparison with commercial OAE medical device. (c)
Effect of measurement time on clinical performance.
The circuit schematic and PCB layout for the device was designed using KiCAD
and was fabricated and assembled by PCBWay. The enclosure was designed in
Fusion360 and 3D-printed using a Formlabs Form 3 resin printer. Our enclosure
is designed to have a tip diameter and length of 5.4 mm and 7.1 mm
respectively, which allows the rubber ear tips to have a snug fit with the
enclosure. We note that for all the ear tips in our study, the base diameter
is the same, and is able to fit easily on the enclosure. The enclosure also
has openings for a switch to power on and off the device, as well as for a
micro-USB charging port. The interior of the case is also lined with foam to
reduce the effect of acoustic reflections from within the case. Table. 1 shows
the cost of the individual components in our earbud device estimated using
Digikey, Mouser, Alibaba and PCBWay. The above numbers provide a ballpark cost
which can be further reduced at higher volumes.
## 3 Clinical study
Our study was approved by the Institutional Review Board and we obtained
informed consent for all adults and parental consent was obtained for
pediatric patients and patients aged 7 to 17 provided verbal or written
assent. We recruited patients from otolaryngology, craniofacial and hearing
loss clinics across two clinical sites. We also recruited adults without any
known concern for hearing loss ($n=28$ ears). We tested our device on 50 ears
from 26 newborns and adults up to 32 years (mean age: $18\pm 9$). We performed
testing on 10 ears between 1 week and 1 year of age. Measurements on adult
patients were performed in duplicates whenever possible, and a total of 75
measurements are used for subsequent analysis. The female-to-male ratio was
3.2.
To determine the ground truth hearing status of each patient, the best
available clinical information was interpreted by the attending physician or
clinician. This information includes the patient’s newborn hearing screen,
audiogram, diagnostic auditory brain response, and clinical and examination
history. Our patient population included sensorineural ($n=5$) and conductive
($n=1$) hearing loss, as well as auditory neuropathy ($n=1$) which is a form
of hearing loss that affects the auditory nerve’s ability to transmit sound to
the brain, but which does not affect the cochlea’s ability to produce OAEs. We
recruited patients with different degrees of hearing loss spanning the full
range of degrees from slight to profound (Fig. 10(a)). The degree of hearing
loss for a given ear is computed by taking the mean hearing level measured
from a patient’s audiogram in dB HL, and mapping it to the thresholds as
defined by the American Speech-Language-Hearing Association [29]. Our dataset
also had ears with middle ear infections due to fluid buildup ($n=3$), as well
as ears that recently had ear tubes ($n=2$). In total, 6 ears were classified
as having hearing loss, the remaining 44 ears were classified as having normal
hearing or having healthy outer hair cells in the cochlea. For our study, we
mark the patient with auditory neuropathy as having healthy outer hair cells
in the cochlea, as OAEs are expected in this patient.
During testing, all participants $>$ 6 months were instructed to sit upright
for the test. Younger patients were tested in the position that was most
comfortable for them and their parents, and included being asleep in a supine
position, or being cradled over the parent’s shoulder. All patients were first
tested with the commercial OAE device in each ear. Patients recruited from one
of the sites were tested with a commercial TEOAE device (Otoport Screener,
Otodynamics) that was used regularly at the clinic. This test was performed
across the 1, 1.5, 2, 3, 4 kHz bands. We set the device to continue measuring
for this full duration even if the test passed early. The remaining patients
were measured using a DPOAE device (OAE Hearing Screener, Welch Allyn) that
was available for us to use at other locations. This device tested at
frequency bands of 2, 3, 4, and 5 kHz. During this portion of the test, we
would select a disposable rubber ear tip (Grason & Associates LLC) size based
on visual examination of the patient’s ear canal. In our study, ear tip sizes
8, 9, 10, 11 an 12 were used 9, 4, 23, 8 and 2 times respectively. For the
pediatric population, we used three different ear tip sizes from 8 to 10 mm,
while for the adult population four different ear tip sizes from 8 to 12 mm.
This suggests that a relatively small number of ear tip sizes can accommodate
a large range of ear canal sizes. We note that commercial earbuds such as
AirPods contain four different ear tip sizes [30].
(a)
(b)
(c)
Figure 11: Subgroup analysis and benchmark results. (a) Subgroup analysis
comparing the mean SNR of OAEs measured in patients with hearing loss and
normal hearing during the clinical study. (b) Effect of background noise on
system performance for different sound levels. (c) Probe integrity check in a
close-ended tube is used to ensure that the system produces SNRs below the
cutoff for healthy hearing when measured outside the ear.
After completing the test with the commercial device, we proceeded to test the
patient with our wireless earbud device using the same rubber ear tip. For
pediatric patients, each ear was tested effectively for 45 to 68 seconds per
ear, depending on the compliance of each patient. Adult participants were
tested for 68 seconds in each ear, twice. All testing with both children and
adults was performed by two computer science graduate students. During
testing, we transmitted clicks with a duration of 500 $\mu$s and gaps of 20 ms
between clicks. The clicks were set to have a bandwidth of 0 to 5 kHz. The
clicks were sent at a sound level of 84 dB peSPL (pe = peak-equivalent).
The sampling rate of the speaker and microphone was set to 15625 Hz to allow
for streaming the data over Bluetooth. We measure for OAEs at the 1, 1.5, 2,
3, and 4 kHz bands. Specifically, we average the signal and noise responses in
the ranges of 750–1250 Hz, 1250–1750 Hz, 1750–2500 Hz, 2500–3500 Hz, and
3500–4500 Hz. On our device, we consider a measurement a passing screen if the
SNR of at least two frequency bands exceeds an SNR threshold of 8 dB, and the
absolute sound level of those passing frequency bands is greater than -10 dB
SPL, which prior work regarded as the minimum power for these OAEs [31].
### 3.1 Performance evaluation
To determine our SNR threshold on each frequency band, we generate a receiver-
operating curve (Fig. 10) to compute the sensitivity and specificity values
for SNR thresholds ranging from -20 to 40 dB in increments of 1 dB. We find
that the SNR threshold of 8 dB maximizes the sum of sensitivity and
specificity, yielding a sensitivity of 100.0% (95% CI, 64.6–100.0%) and
specificity of 89.7% (95% CI, 80.2–94.9%). We find that using two frequency
bands as the pass criteria yields an AUC of 0.958. Using three or four
frequency bands as the pass criteria results in AUCs of 0.950 and 0.884
respectively.
Comparison with medical device. In comparison to our earbuds, Fig. 10(b) shows
that the medical device had a lower AUC of 0.822 yielding a sensitivity of
83.3% (95% CI, 43.6–97.0%) and specificity of 92.1% (95% CI, 79.2–97.3%). The
ground truth for both our device and the medical device is the clinical
information that is interpreted by the physician including their clinical and
examination history as well as auditory brain response tests. We note that for
one ear with hearing loss, the medical device was unable to pass the probe
check despite numerous attempts to fit the ear with different sized ear tips,
even though the ear tip appeared to fit well visually. For this instance, we
marked the medical device as having failed the measurement.
Our device misclassified the hearing loss status for seven ears, three of
these ears had middle ear fluid and infection. In these ears, OAEs were not
detected, as the fluid acts as a barrier that blocks the OAEs from reaching
the outer ear, and it is expected that OAEs do not appear in these ears [32].
The commercial device similarly did not detect OAEs in ears with middle ear
fluid or infection. One of these ears had hearing loss, but OAEs were
detected. We suspect that this is due to reflections or ear tip fit issues, as
this was the only ear where the commercial device could not begin a test due
to a failure of the initial probe check despite repeated attempts to pass the
check. One of these ears recently had ear tubes removed which would have
resulted in a hole in the eardrum which would have begun healing, and may have
affected the ability to detect OAEs.
Effect of measurement time. To determine the effect of measurement time, we
set a limit on the maximum number of clicks used by our algorithm and compute
clinical performance when the measurement time is reduced to 15 or 30 seconds.
Fig. 10(c) shows that we are able to achieve an AUC of 0.892 and 0.915, which
is higher than that achieved by the commercial device of 0.822. At these
measurement times, our system obtained an optimal sensitivity of 100.0% (95%
CI, 61.0–100.0%) and 83.3% (95% CI, 43.6–97.0%) respectively and specificity
of 71.4% (95% CI, 59.3–81.1%) and 90.5% (95% CI, 80.7–95.6%) respectively.
When reducing the measurement duration to 5 seconds, our sensitivity and
specificity, as expected, reduces to 83.3% (95% CI, 43.6–97.0%) and 68.3% (95%
CI, 56.0–78.4%) respectively.
Subgroup analysis of hearing status. In Fig. 11(a), we show the average SNR
obtained in ears with hearing loss as well as with normal hearing. The plot
shows that for the ears with hearing loss the average SNR across all
frequencies is 3 dB, while it is 11 dB for ears with normal hearing, showing
that there is large separation in SNRs between the two classes of ears. We
note that in the hearing loss ears, although the average SNRs are positive,
none of the SNRs at any of the frequencies exceeded the 8 dB SNR cutoff. We
suspect the SNRs are positive due to residual reflections from within the
plastic case of our wireless earbuds. We also note that the SNRs for the OAEs
detected by our system are smaller at the lower frequencies. This is in line
with existing literature [33] which confirms that transient-evoked OAEs are
better at mid-range frequencies than the lower frequencies.
Test-retest evaluation. For the 25 ears where duplicate testing was performed,
the screening result for our earbuds matched in all but one ear. In our study,
we also tested the commercial device several times if we were not confident in
the probe fit in the ear during a given measurement. There were two ears where
the commercial device had differing results between tests. Both of these ears
were normal hearing ears and it took two and three attempts respectively for
these ears in order to obtain a passing screen result.
## 4 Micro-benchmarks
We provide various micro-benchmark evaluations including the effect of
background noise and a comparison of our wireless sensing techniques with
existing OAE algorithms. We also present an evaluation of system level issues
including power and run-time analyses. Effect of background noise. To measure
the effect of background noise on our device, we played noise of road traffic
from a laptop such that the sound level at a healthy ear varied from 50 to 70
dB SPL and measured the OAEs measured by our earbuds in an ear with normal
hearing. These sound levels reflect the typical range of ambient background
noise that would occur in a clinical testing facility that is not well
insulated from noise, and which might be situated close to a road. Fig. 11(b)
shows that at a noise of 50 dB SPL, the OAEs can be detected in the ear, and
are above the 8 dB threshold at all frequencies. This sound level is the
typical ambient noise level in an urban residence [34]. At 60 and 70 dB SPL,
the SNRs at all frequencies drop below the 8 dB threshold. These sound levels
are similar to conversational speech held at 1 m and a vacuum cleaner at 1 m
[35]. These results suggest that testing should be performed in a relatively
quiet environment to obtain reliable results.
Probe integrity check. Medical OAE devices use a closed-ended tube between
0.5–2 cc in volume as a probe integrity check [31]. This range of volumes is
selected to mimic the volume of the ear canal for the pediatric and adult
population. Similarly, we have our earbuds perform a measurement in a closed
ended plastic tube with a volume of 1 cc, and in a healthy ear. Fig. 11(c)
shows the SNRs obtained in both these scenarios when repeated three times. The
plot shows that the SNRs in the tube are below the SNR cutoff of 8 dB for all
frequencies across all measurements, and can be used as a probe integrity
check to ensure that the device is not incorrectly identifying OAEs.
(a)
(b)
Figure 12: Comparison of OAEbuds with prior TEOAE algorithms. Our OAEbuds
system achieves better performance compared to prior TEOAE algorithms when (a)
2 and (b) 3 frequency bands are required to be above the SNR threshold for the
hearing test to pass.
Comparison with prior OAE algorithms. We evaluate the performance of our
OAEbuds system using alternative variations to implementation. We test three
different protocols as follows 1) OAEbuds protocol with a fixed delay of 2.5
ms where reflections after a 2.5 ms duration from the pulse are removed, 2)
conventional TEOAE protocol with a delay of 2.5 ms and 3) conventional TEOAE
protocol with a higher delay of 12 ms. Conventional TEOAE systems transmit a
train of pulses with a polarity pattern of $\\{1,1,1,-3\\}$ [36]. The receiver
then adds up the response across the four pulses to generate the OAE signals.
We select 2.5 ms as the delay as this is what commercial TEOAE devices
typically use [37]. Fig. 12 shows the ROC curves for these protocol
implementations when using either 2 or 3 frequency bands to pass. We note that
our implementation of OAEbuds yields a better AUC compared to the conventional
protocol regardless of whether 2 or 3 frequency bands are needed to pass. We
note that when using a delay of 2.5 ms, a significant amount of the signal
will be reflections and not OAEs. Because of this, the optimal SNR threshold
for these modified protocol is significantly higher at 51 and 48 dB for the 2
and 3 frequency band scenarios respectively. The most likely reason why the
TEOAE protocol performs worse than OAEbuds is that it relies on the clicks
canceling out each other well. In practice, the cancellation is not perfect
and there will be a residual error in the cancellation process, in particular
on our low-cost acoustic hardware that has non-linearities. Although the goal
of this protocol is to cancel out reflections which are linear, it will also
cancel out the linear components of the OAEs themselves, which may also
contribute to lowered performance. We note that with TEOAE, using the lower
delay of 2.5 ms performs better than the 12 ms delay. With this lower delay,
the optimal SNR threshold is 21–22 dB, while it is 4–6 dB when running the
TEOAE protocol with a 12 ms delay.
(a)
(b)
Figure 13: Power analysis. (a) Number of tests that can be performed on a
single charge for tests of different durations. (b) Time required to charge an
OAEbud via a micro-USB connection.
Power analysis. To evaluate how long our earbuds last on a single charge, we
first charge the battery of the earbuds to its maximum level and evaluate its
performance over multiple tests. Fig. 13(a) shows the state of charge on the
battery as a function of the number of tests. Each of the lines represents a
different duration for a hearing test. To measure the voltages and state of
charge, we use an on-device fuel gauge. The fuel gauge uses the proprietary
ModelGauge algorithm to continuously track the battery’s state of charge
(SOC). It simulates the internal nonlinear dynamics of the battery model. By
analyzing voltage measurements over time, it can determine the state of charge
much more accurately. The fuel gauge data is interfaced over an I2C bus and it
allows us to read out the battery voltage and SOC.
The plots show that on a single charge, even when each test lasts around 60 s,
the earbud can support 91 hearing tests. For context, we note that on a
typical day, the hearing loss clinic in our institution sees around 20-30
patients. We measure how long it takes to charge the earbuds through a micro-
USB cable connected to a wall outlet. Fig. 13(b) shows that it takes around 3
hours to charge the earbud which in a practical setting can be done overnight
in a clinic.
Runtime analysis. To evaluate the feasibility of running our algorithm on a
mobile device like a smartphone, we convert our algorithm to C++ code that can
run on the Android smartphone platform and time how long it takes to execute
the operations in our system. On a Samsung Galaxy S9, we are able to process
windows of 1 s containing 48 clicks in less than a millisecond. This means
that we are able to provide a real-time update of the SNR values throughout a
OAE measurement. This real-time feedback can help a user determine if a probe
fit is snug or if there is too much noise at different points in a
measurement, and allows them to make any changes to how the measurement is
being performed.
## 5 Related work
Prior work can be broadly divided into three classes.
Health tracking using earphones. Prior work has explored the use of earphones
for monitoring physiological signals for cardiovascular sensing [38, 39],
blood pressure measurements [40] and respiration [41]. Earphones have also
been used for sensing jaw clenching [42], teeth motion and voice detection
[43, 44]. Prior work has also explored the use of smartphone attachments for
diagnosing middle ear conditions. [45] designed a paper cone that is attached
to a smartphone and used a machine learning classifier to detect middle ear
fluid behind the ear drum. [46] used a smartphone-connected wired earphone
attached to an external microphone to differentiate between middle ear fluid,
ear drum ruptures and wax blockage. [47] presented a smartphone attachment to
perform tympanometry where the pressure within the ear canal is changed to
assess the mobility of the ear drum. All these wired systems however are
designed for assessing the state of the middle ear and the ear drum. Hearing
screening, in contrast, is primarily focused on the state of the cochlea. The
cochlea is part of the inner-ear that is behind the ear drum and is primarily
responsible for converting sound waves into electrical impulses which are then
interpreted by the brain.
OAE devices. Prior work [48, 49] created a smartphone interface for the probes
from an existing commercial OAE device [50]. In addition to not being
wireless, this approach is still constrained by the cost of commercial OAE
probes that are expensive. Further, since it is directly connected to a
smartphone it requires calibration for each smartphone model which is
difficult to generalize. Recent commercial approaches [51, 52] use bone
conduction to monitor OAEs through a headband. In addition to not being in the
wireless earbud form factor, these have not been demonstrated to be low-cost.
[53] proposes to use a single transducer to measure OAEs. However it is
primarily focused on the transducer characterization and does not build an
end-to-end wireless earbud system. Further, none of these prior efforts have
performed clinical studies to evaluate efficacy. Finally, high-end
personalized earphones from companies like Nura Sound use OAE measurements to
customize music for an adult wearer [54, 55]. Our goal in this work is
complementary in that we create a low-cost and open-source earbud system that
is designed to achieve hearing loss screening with accuracies similar to
medical devices.
Earable platforms. Recent years have seen the introduction of earbud platforms
like eSense platform [56, 57], Clearbuds [58] and OpenEarable [59]. Like
eSense, the Clearbuds platform does not have speakers or microphones facing
the ear canal and is not designed for hearing loss screening. OpenEarable has
a rigid over the ear design that is hard to operate across different age
groups. We instead design an open-source wireless earbud platform to reliably
measure otoacoustic emissions for hearing loss screening using low-cost
hardware.
## 6 Limitations and discussion
We describe various limitations of our current system and discuss the
regulatory pathway.
Field studies. In practical deployments, OAEbuds will potentially be used by a
range of stakeholders including nurses, technicians and volunteers. Our
clinical study does show that graduate students with no formal training in
audiology were able to select the ear tips and snugly place the earbuds for
both infants and adults. However, field studies in low and middle-income
countries might be required to ensure that our design can be used as
advertised; this is however not in the scope of this paper.
Followup care and regulatory costs. Detecting hearing loss is an important
first step in addressing this complex public health problem. Other factors
include human resources for performing the tests, followup care and regulatory
costs. We however note that prior FDA clearances for OAE devices did not
require human testing [60], which significantly reduces the cost of clearance.
Software update to commercial earbuds. We develop a custom earbud instead of
using existing earbuds for two key reasons: 1) we wanted to achieve a lower
cost than existing wireless earbuds and demonstrate that OAEs can be detected
using low-cost acoustic components, and 2) commercial wireless earbuds do not
provide access to data from the in-ear microphone. We however note that the
microphones and speaker used in Apple AirPods and Pixel buds have a higher
quality. Given that all the hardware including an in-ear microphone are
already present in commercial earbuds, our paper shows that there is an
exciting possibility that using the algorithms presented here, commercial
earbuds can potentially enable OAE detection and hearing loss screening using
only a software update.
## 7 Conclusion
Over the next decade, the mobile systems community is uniquely positioned to
develop wearable and mobile technologies that help alleviate global health
inequity. We developed the first wireless earbuds that can detect otoacoustic
emissions and perform hearing screening using low-cost acoustic hardware. Our
work introduces two components, 1) a low-cost wireless earbud hardware for
hearing screening that works across infants and adults, and 2) wireless
sensing algorithms to reliably identify otoacoustic emissions in the presence
of in-ear reflections and echoes. Our clinical study demonstrates similar
sensitivity and specificity to commercial medical devices and shows the
potential of our design to enable hearing loss screening in low and middle
income countries.
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|
# Exploring phonon-like interactions in one-dimensional Bose-Fermi mixtures
Axel Gagge Department of Physics, Stockholm University, SE-106 91 Stockholm,
Sweden Th. K. Mavrogordatos Department of Physics, Stockholm University,
SE-106 91 Stockholm, Sweden ICFO – Institut de Ciències Fotòniques, The
Barcelona Institute of Science and Technology, 08860 Castelldefels
(Barcelona), Spain Jonas Larson<EMAIL_ADDRESS>Department of
Physics, Stockholm University, SE-106 91 Stockholm, Sweden
###### Abstract
With the objective of simulating the physical behavior of electrons in a
dynamic background, we investigate a cold atomic Bose-Fermi mixture confined
in an optical lattice potential solely affecting the bosons. The bosons,
residing in the deep superfluid regime, inherit the periodicity of the optical
lattice, subsequently serving as a dynamic potential for the polarized
fermions. Owing to the atom-phonon interaction between the fermions and the
condensate, the coupled system exhibits a Berezinskii-Kosterlitz-Thouless
transition from a Luttinger liquid to a Peierls phase. However, under
sufficiently strong Bose-Fermi interaction, the Peierls phase loses stability,
leading to either a collapsed or a separated phase. We find that the primary
function of the optical lattice is to stabilize the Peierls phase.
Furthermore, the presence of a confining harmonic trap induces a diverse
physical behavior, surpassing what is observed for either bosons or fermions
individually trapped. Notably, under attractive Bose-Fermi interaction, the
insulating phase may adopt a fermionic wedding-cake-like configuration,
reflecting the dynamic nature of the underlying lattice potential. Conversely,
for repulsive interaction, the trap destabilizes the Peierls phase, causing
the two species to separate.
Bose-Fermi mixture, Peierls instability, harmonic trap, lattice potential
###### pacs:
05.30.Rt, 42.50.Ct, 75.10.Kt
## I Introduction
Over the past decades, we have witnessed a rampant growth of experimental
methods devised to cool and control dilute gases. The attainment of Bose-
Einstein condensates (BECs) Anderson et al. (1995) was soon to be followed by
in-depth explorations of BEC dynamics in light-induced periodic potentials
Morsch et al. (2001); Denschlag et al. (2002), paving the way for the
groundbreaking demonstration of the Mott-superfluid phase transition. This
exemplary quantum phase transition was predicted by the Bose-Hubbard model
Fisher et al. (1989); Jaksch et al. (1998) for an atomic gas loaded into an
optical lattice Greiner et al. (2002). Since then, trapped and cooled dilute
atomic gases have developed into a versatile laboratory where quantum matter
can be studied in a controlled and detailed fashion, constituting one of the
most promising platforms for the realization of analog quantum simulators
Lewenstein et al. (2007); Bloch et al. (2008); Daley et al. (2022).
Many of the open questions in the field of quantum phase transitions,
especially those revolving around the formation and characterization of exotic
quantum phases of matter, cannot, however, be directly addressed by the Bose-
Hubbard model per se. Although the classical laser field forming the optical
lattice is, in principle, dynamic, the back-action between the trapped
particles (modeled as beam splitters) and the lattice is typically very weak
Asbóth et al. (2007, 2008). One may then, to a very good approximation, treat
the lattice as the outcome of applying a periodic static potential. Certainly,
this approximation would be sufficient to emulate many paradigmatic lattice
models. However, such classical potentials do not take into account any back-
action between the lattice and the conducting matter, which we know is
essential to explain several phenomena like the Peierls distortion Peierls and
Peierls (1955) [see Figs. 1 (a) and (b)] and superconductivity Mahan (2013).
In solids, effects of that kind result from electron-phonon interactions
emerging from the very nature of a dynamical lattice.
A central theme in our analysis is that assigning tractable degrees of freedom
to the lattice renders its description dynamic and enables the simulation of
some analog of phonon-like interactions. One possibility is to couple the
atoms to the light field of an optical resonator Deng et al. (2014); Pan et
al. (2015), where substantial Stark shifts are observed even in the presence
of a few photons. However, in current experiments, only a few modes of the
resonator actively participate in the light-matter interaction, and it is,
therefore, not possible to locally modify the dynamical lattice. Instead, one
must turn to multi-mode cavities Ballantine et al. (2017), which still pose
difficulties in reaching configurations similar to those encountered in real
solids Lewenstein et al. (2006). A viable alternative arises when considering
atoms directly interacting with an ionic crystal Bissbort et al. (2013). Here,
harmonically trapped ions form a Wigner crystal, while additional neutral
atoms move within this lattice. This ion-atom system bears obvious
similarities to a real solid, albeit being experimentally challenging. In
fact, the crystalline structure is not necessary for exploring phonon-like
interactions. In mixtures of different atomic species — either different atoms
or different internal states of the same atomic species — the interplay
between subsystems can lead to intriguing effects.
Coming now to single atomic species, the use of Feshbach resonances allows
experimenters to control the strength, and even the sign, of all involved
interactions Stan et al. (2004); Best et al. (2009); Kawaguchi and Ueda
(2012); Park et al. (2012); Ferrier-Barbut et al. (2014); DeSalvo et al.
(2019), aiming to probe the resulting phase diagram. In mixtures of bosons and
spin-polarized fermions (hereinafter referred to as BF mixtures), it is well
known that an attractive BF interaction leads to a so-called pairing phase in
the strongly correlated regime. This phase has been studied for weak BF
interactions, $g_{bf}$, in one dimension (1D) Cazalilla and Ho (2003);
Miyakawa et al. (2004); Rizzi and Imambekov (2008), as well as in twoBüchler
and Blatter (2003); Klironomos and Tsai (2007) and three dimensions Titvinidze
et al. (2008). The phase in question collapses if the interaction becomes too
strong, resulting in clumping of the atoms and breaking of translational
invariance. The effect of optical lattices on BF mixtures has also been
investigated in Refs. Albus et al. (2003); Mathey et al. (2004); Roth and
Burnett (2004); Salerno (2005); Pazy and Vardi (2005); Bruderer et al. (2007);
Lan and Lobo (2014). For deep lattices and/or very strong interaction, such
systems can be described by a BF Hubbard model as they enter an insulating
phase of composite fermions Lewenstein et al. (2004); Pollet et al. (2006).
The physical behavior is typically described within a Wannier-basis expansion
for both species, where the bosons can be construed as agents of effective
onsite energy shifts for the lighter fermions. The imposed approximations in
such a scheme omit certain back-action between the two subsystems in
comparison to the self-consistent analysis we employ here. Nevertheless, it is
still possible to encounter Peierls phases, supersolids, and charge density
waves Büchler and Blatter (2003); Lewenstein et al. (2004). Furthermore, if
the repulsive boson-boson interaction is weak, the system can enter a regime
of phase separation where the bosons and fermions completely avoid each other
Büchler and Blatter (2004).
Reporting from the experimental front, an early study focused on how the
coherence of an atomic condensate –- held in place by a cubic optical lattice
–- is affected by the presence of fermionic atoms Günter et al. (2006).
Quantum degeneracy for both bosons and fermions was attained in Ref. Ferrier-
Barbut et al. (2014), where the two species were treated on equal footing, a
trend followed in a series of subsequent papers Delehaye et al. (2015); Roy et
al. (2017); DeSalvo et al. (2017); Yao et al. (2016). Further experimental
investigations have also considered a particular regime emerging for a
condensate in weak contact with much lighter fermionic atoms DeSalvo et al.
(2019); Edri et al. (2020); DeSalvo et al. (2017). Here, the fermions
effectively induce a so-called Ruderman–Kittel–Kasuya–Yosida (RKKY) long-range
boson-boson interaction Ruderman and Kittel (1954). For attractive Bose-Fermi
interaction, it was demonstrated that a self-sustained trap may emerge for
those fermions located inside the bosonic condensate DeSalvo et al. (2017), an
effect which may as well lead to the formation of soliton trains DeSalvo et
al. (2019). The fermion-mediated spin-spin interaction in a spinor condensate
has also been recently observed by means of microwave spectroscopy Edri et al.
(2020).
Figure 1: (color online) Schematic description of the Peierls distortion. At a
fermion filling corresponding to a wavenumber $k_{F}$, opening up a gap of the
dispersion at $k_{F}$ will lower the kinetic energy of the fermions, as
depicted in (a). The gap opening results from a lattice distortion of the
dynamical lattice, as shown in (b). Here, the period has been doubled through
shifting every second atom by a distance $\delta$, and consequently, the size
of the Brillouin zone has been halved. Thus, for half filling, we encounter a
realization of the Su-Schrieffer–Heeger (SSH) model, while for other fillings,
the lattice distortion will generate another periodicity. In the normal phase,
the condensate will share the same periodicity as the lattice and
predominantly populate the lattice minima (c). In the Peierls phase, for half
filling, the bosonic density will alternate between every second site (d).
Upon comparison of (b) and (d), we note that the periodicity has been broken
in two different ways: in (b), the densities are held fixed, but the locations
of the sites have been shifted, while in (d), the locations are fixed and the
densities have been altered. The former can be seen as a ‘phase modulation,’
and the latter as an ‘amplitude modulation’ of the densities. Finally, in (e),
we depict the collapsed phase where all atoms, bosons, and fermions have been
compressed to a small region of the lattice.
Carrying on with the thread of recent investigations on collective phenomena
brought about by a coherent interaction between Bose-Fermi atomic clouds, in
this report, we study a 1D atomic BF mixture where only the bosons are subject
to an optical lattice. The motivation for considering such a configuration is
drawn from the resemblance to an actual solid – the optical lattice orders the
bosons in a crystalline structure. Our spin-polarized fermions experience only
the periodic potential arising from the boson atomic density but no externally
imposed static potential, with the occasional exception of a harmonic trap. A
similar idea, although experimentally more challenging, was proposed in Ref.
Johnson et al. (2016), where the crystalline order for the bosons was
established from a rapidly rotating condensate that created a triangular
Abrikosov vortex lattice. We focus on the limit of a weak boson-boson (BB)
interaction and an optical lattice of amplitude $\leq 20E_{R}$ ($E_{R}$ is the
recoil energy), where the bosonic gas is expected to form a superfluid, see
Fig. 1(c). In this regime, we can work within a mean-field approximation for
the condensed bosons, which act as a classical dynamical lattice felt by the
fermions. The coupled system is solved self-consistently such that every back-
action, at the mean-field level, is taken into account (without imposing, for
instance, any single-band nor tight-binding approximations). At this hybrid
mean-field level, the mixture displays a rich phase diagram. For strong BF
interaction, with a coupling rate $g_{bf}$, the system either experiences a
collapse ($g_{bf}\gg 0$), in which the two species overlap and populate only a
small fraction of the lattice, or a separation ($g_{bf}\gg 0$) where instead
the two species avoid each other and populate different parts of the lattice.
These are first-order transitions, even though translational invariance is
spontaneously broken in both cases. For a non-zero interaction beyond a
critical coupling $g_{bf}=g_{bf}^{c}$, an instability occurs such that the
period of the state is different from that of the underlying lattice; for
example, for half-filling, a periodic doubling is found. The system then
transitions from a Luttinger liquid (LL) phase into a Peierls phase via a
Berezinskii-Kosterlitz-Thouless (BKT) transition. The Peierls phase manifests
itself through a non-zero gap $\Delta_{P}$ at the Fermi wavenumber
$k_{\mathrm{F}}$, i.e. $\Delta_{P}\neq 0$ for $g_{bf}$ beyond the critical
interaction strength $g_{bf}^{c}$.
We further present an account of the effects arising due to the presence of a
confining harmonic trap. For a sufficiently strong repulsive boson-fermion
interaction, it has already been demonstrated that, at the mean-field level,
the trap induces a separation of the two species Nygaard and Mølmer (1999).
This instance suggests ruling out a Peierls-like phase when a trap breaks the
translational invariance. Inside a trap, the atomic densities vary in space,
which, in a local density approximation, translates to a local Fermi
wavenumber. A spatially varying wavenumber has a direct impact on the phases,
such as by smoothing the discontinuous transition between the (possible)
Peierls and collapsed/separated phases. Naturally, the presence of a trap
entails a larger number of involved wavenumbers and thereby a broadening of
the Bragg peaks in a time-of-flight detection. Additionally, we find that the
interplay between the trap, the dynamical boson field, and the fermions
results in an insulator bearing a clear fermionic “wedding cake structure”,
which differs from the density profile obtained for a static periodic
potential.
Our narrative is structured as follows. Section II introduces the Bose-Fermi
system under study and the model we employ for its description. There we also
discuss some basic concepts and known results pertaining to the properties of
the involved phases. The numerical methods are developed in Sec. III, while
the associated results are presented in Sec. IV, first for the translationally
invariant case IV.1, and subsequently in the presence of a harmonic trap IV.2,
where we focus on the quantum coherence registered in the one-particle density
matrix and on the decay of spatial correlations. Concluding remarks in Sec. V
close our paper out, while the two appendixes present two possible paths to
the derivation of effective model Hamiltonians taking quantum fluctuations
into account for both species, alongside some details underlying the employed
hybrid mean-field method.
## II Background: model Hamiltonian and phases
### II.1 The Hamiltonian
The physical system we study is a mixture of two atomic gases, one bosonic and
one fermionic. Owing to tight transverse confinement (in the sense of
$\omega_{\perp}\gg\omega_{\parallel}$), excited states of the transverse modes
can be neglected and the low-energy physical description is quasi-one
dimensional Das (2003); Miyakawa et al. (2004). To begin with, we will take
the gas to be of infinite extent in the longitudinal direction, and later
consider the effect of a trap. We work with dimensionless quantities, where
energies are scaled by the boson recoil energy, $E_{R}=\hbar^{2}k^{2}/2m_{b}$
with $k$ the lattice wavenumber and $m_{b}$ the atomic boson mass, such that
half the optical lattice wavelength sets the characteristic length scale of
the problem. The behavior of such a system is modeled by the Hamiltonian
$\begin{array}[]{lll}\hat{\mathcal{H}}&=&\displaystyle{\int
dx\;\hat{\psi}_{b}^{\dagger}\left(-\frac{\partial^{2}}{\partial
x^{2}}+V_{b}(x)+\frac{g_{b}}{2}\hat{\psi}_{b}^{\dagger}\hat{\psi}_{b}\right)\hat{\psi}_{b}}\\\
\\\ &&\displaystyle{+\int
dx\;\hat{\psi}_{f}^{\dagger}\left(-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+V_{f}(x)+g_{bf}\hat{\psi}_{b}^{\dagger}\hat{\psi}_{b}\right)\hat{\psi}_{f}},\end{array}$
(1)
where $r=m_{f}/m_{b}$ is the mass ratio between the two species, $g_{b}$ is
the effective 1D boson-boson (BB) interaction strength, and $g_{bf}$ is the
effective 1D BF interaction strength. Later on, we will assume the bosons to
be harmonically confined and subjected to an optical lattice, while the
fermions will be harmonically confined but assumed to have negligible
interaction with the optical lattice,
$\displaystyle V_{b}(x)$
$\displaystyle=\tfrac{1}{2}\omega_{b}^{2}x^{2}+V_{0}\sin^{2}(x),$ (2)
$\displaystyle V_{f}(x)$ $\displaystyle=r\tfrac{1}{2}\omega_{f}^{2}x^{2}.$ (3)
The last term on the RHS of Eq. (2) is the optical-lattice potential, also
denoted as $V_{OL}=V_{0}\sin^{2}(x)$, where the distance between potential
minima is $a=\pi$ in dimensionless units. At first, we remove the harmonic
confinement and keep only the periodic optical lattice. To ensure that the
lattice interacts only with the bosonic atoms, the light frequency should be
such that it is not quasi-resonant with any of the dipole transitions –
dictated by selection rules – of the fermionic atoms. The atomic field
operators $\hat{\psi}_{b,f}(x)$ and $\hat{\psi}_{b,f}^{\dagger}(x)$ obey the
standard commutation relations for bosonic and fermionic creation and
annihilation operators. Note that the Hamiltonian commutes with both particle
number operators ($\alpha=b,f$),
$\hat{N}_{\alpha}=\int
dx\;\hat{\psi}_{\alpha}^{\dagger}(x)\hat{\psi}_{\alpha}(x).$ (4)
For further use, we define the average particle number of bosons/fermions as
$\bar{n}_{\alpha}=\lim_{L\to\infty}\frac{1}{L}\int_{-L/2}^{L/2}dx\;\langle\hat{\psi}_{\alpha}^{\dagger}(x)\hat{\psi}_{\alpha}(x)\rangle,$
(5)
and the density $\nu_{\alpha}=a\bar{n}_{\alpha}=\pi\bar{n}_{\alpha}$ as the
filling of bosons/fermions per lattice site.
Hereinafter, we assume that the bosonic gas is dense enough to form a (quasi)
condensed state, but not so dense that the interaction energy becomes
dominant. In terms of the 1D and 3D scattering lengths $a_{s,1D}$ and
$a_{s,3D}$ this implies
$a_{s,3D}\ll\xi\ll a_{s,1D},$ (6)
where the healing length of the condensate is defined as $\xi=(8\pi
a_{s,3D}\rho_{b})^{-1/2}$, with $\rho_{b}$ the condensate density.
Furthermore, our approximations are consistent with limiting our attention to
optical lattice strengths satisfying $V_{0}\lesssim 20E_{R}$. We take the
fermions to be lighter than the bosons, i.e. $r=m_{f}/m_{b}<1$, which is
typically the case for the relevant experiments, e.g. for a Li6-Li7 mixture
Ferrier-Barbut et al. (2014) one has $r=0.86$ and for a Li6-Cs133 mixture
DeSalvo et al. (2017); Repp et al. (2013) one instead gets $r\approx 0.04$. We
will assume $g_{b}>0$ but allow any sign and magnitude of $g_{bf}$. Both the
BB and BF interactions should be possible to vary in the experiment by
deploying Feshbach resonances Stan et al. (2004); Best et al. (2009);
Kawaguchi and Ueda (2012); Park et al. (2012); Ferrier-Barbut et al. (2014).
### II.2 Pairing phase and Peierls instability
For low bosonic densities, the occurrence of a so-called pairing phase for BF
mixtures with attractive BF interaction is well known. For weak interaction
strength and light fermions, the development of modulated densities can be
understood from the fact that the fermion-mediated long-range interaction
between two bosons is an ultracold atomic analog to the RKKY interaction
Ruderman and Kittel (1954); Mering and Fleischhauer (2010); De and Spielman
(2014). These spatial modulations of the boson density alter the potential
landscape for the fermions, and subsequently instigate back-action between the
two species. The spatial modulations will occur at the wavenumber $2k_{F}$ if
we assume an RKKY interaction from a “flat” background of fermions with a
Fermi wavevector $k_{F}$. In Cazalilla and Ho (2003), a 1D BF mixture without
optical lattice was investigated using bosonization and a transition was
identified between a two-component gapless LL phase and a gapped pairing phase
with periodic density modulations. The transition was found to “flow” towards
the BKT fixed point with the gap opening as
$\Delta\sim\exp(-1/\sqrt{|g_{bf}-g_{bf}^{c}|})\Theta(g_{bf}-g_{bf}^{c}),$ (7)
where $\Theta(x)$ is the Heaviside step function, and $g_{bf}^{c}$ is the
critical Bose-Fermi coupling strength. A study using discretization of the
spatial coordinate and the numerical DMRG method also predicted a stable LL
phase for small negative $g_{bf}$ Rizzi and Imambekov (2008). For repulsive
interaction ($g_{bf}>0$), the aforementioned works found no pairing phase.
However, in Ref. Miyakawa et al. (2004), a pairing phase was indeed reported
upon employing the method of a random phase approximation.
A gap opening at $2k_{F}$ and the corresponding periodic modulation in the
density are reminiscent of the Peierls distortion. As first demonstrated by
Peierls Peierls and Peierls (1955), a metal is unstable towards lattice
distortions, i.e. a small displacement of the atoms. This phenomenon can be
understood by treating the atomic displacements on the mean-field level in the
well-known SSH model. Opening up a gap $\Delta_{P}$ in the fermion dispersion
at the Fermi wavevector $k_{F}$, as depicted in Fig. 1 (a), lowers the total
energy and corresponds to a lattice modulation of wavenumber $k_{P}=2k_{F}$,
as sketched in Fig. 1 (b). There are however also some discrepancies. The gap
$\Delta_{P}$ was derived from a mean-field treatment of the ions, while the
pairing gap $\Delta$ is the gap of excitations in the full quantum system.
Still, it is interesting to observe that the gap in the SSH model also has an
exponential form similar to (7). Moreover, in the original work of Peierls,
the instability was demonstrated for any non-zero interaction Peierls and
Peierls (1955), while the pairing effect arises as a consequence of a phase
transition occurring at finite interaction strength.
An optical lattice is, on intuitive grounds, expected to stabilize the pairing
phase against a collapse, since it renders the bosons less mobile by
generating a larger effective mass. On the other hand, using the same
argument, the lattice may also extend a possible metallic phase to larger
regions in the phase diagram. We may note that for BF mixtures where both
species are subject to an optical lattice, it has been reported that the
pairing phase appears for both repulsive as well as attractive BF interactions
Pazy and Vardi (2005). Furthermore, one could in principle expect a beating
between the involved length scales, the Fermi wavenumber $k_{F}$ and the
optical lattice wavenumber $k$, that could, in principle, give rise to so-
called Devil’s staircase structures Chanda et al. (2022).
A last remark on the periodically modulated Peierls phase is in order. We have
envisioned the boson superfluid as the agent creating a dynamical lattice for
the fermions. We may consider the opposite viewpoint of a condensate living in
a partially dynamic background. Within this perspective, the Peierls phase is
reminiscent of a supersolid Büchler and Blatter (2003); Titvinidze et al.
(2008), i.e., a superfluid state that has spontaneously broken the periodicity
of the Hamiltonian. Somewhat similar scenarios have been studied in atomic
condensates confined within optical resonators Baumann et al. (2010); Léonard
et al. (2017).
### II.3 Stability of the Bose-Fermi mixture
For sufficiently strong BF interaction, the system is unstable towards long-
wavelength fluctuations. The stability of BF mixtures without an optical
lattice has already been investigated in, e.g., Refs. Das (2003); Miyakawa et
al. (2004). Applying a hydrodynamic (mean-field) description of both fermions
and bosons, a linear stability analysis yields the condition
$\bar{n}_{f}\geq\frac{g_{bf}^{2}}{2g_{f}g_{b}},$ (8)
where the constant $g_{f}=\pi^{2}/r$ appears as an effective fermion
interaction strength. Note that the different definition compared to Ref. Das
(2003) is due to our use of rescaled dimensionless units. The fact that a high
fermion density stabilizes the mixture is particular to the 1D case.
Attractive interaction leads to a collapse of both species, while a repulsive
interaction induces a phase separation where the bosons and fermions either
avoid each other or lump together to form BF soliton mixtures Adhikari (2005);
Tylutki et al. (2016); Salerno (2005). More precisely, for repulsive boson-
fermion interaction, the bosons can form a dark soliton (density dip), filled
by a bright fermionic soliton. For attractive interaction, both species form a
bright soliton. This is reminiscent of the Townes solitons predicted for
electromagnetic waves Chiao et al. (1964), and recently demonstrated in
ultracold atomic Bose-Bose mixtures Chen and Hung (2021); Bakkali-Hassani et
al. (2021). Soliton solutions are known to be unstable beyond the mean-field
approach Krutitsky et al. (2010); Rubbo et al. (2012) but BF mixtures can form
stable self-bound systems (e.g., in Ref. Salasnich et al. (2007),) which in 1D
are stable within the mean-field approximation as well as to higher order
Rakshit et al. (2019). In the course of our analysis, we will find out that an
optical lattice increases the range of $g_{bf}$ values for which the mixture
is stable.
## III Methods
### III.1 A hybrid mean-field approximation
To investigate the pairing phase, we employ a “hybrid” approach that
intertwines quantum and classical dynamics, a method which has already been
followed in Refs.Wang et al. (2012); Karpiuk et al. (2004). We note that this
path bears similarities to the mean-field approximation employed to derive the
SSH model. It is also akin to the DFT method Wang et al. (2012). Due to its
affinity with the SSH model, we will refer to the gapped phase in the hybrid
approximation as the Peierls phase. For the ground state, we adopt the ansatz
$|\Psi\rangle=|\psi_{b}\rangle\otimes|\Psi_{f}\rangle,$ (9)
where $|\Psi_{f}\rangle$ is a general state of $N_{f}$ fermions, and
$|\psi_{b}\rangle$ is a (generalized) coherent state of the bosons satisfying
Solomon et al. (1999)
$\hat{\psi}_{b}(x)|\psi_{b}\rangle=\psi_{b}(x)|\psi_{b}\rangle,$ (10)
with the complex variable $\psi_{b}(x)$ called the condensate wavefunction or
order parameter. We impose the normalization
$\int_{0}^{L}dx\,|\psi_{b}(x)|^{2}=1$ for a finite system and thereby factor
out the total boson number $N_{b}$. If $m_{f}/m_{b}=r\ll 1$, it is justified
to assume an adiabatic evolution, similar to the Born-Oppenheimer
approximation in molecular physics Worth and Cederbaum (2004), where the
lighter fermions adjust approximately instantaneously according to the heavier
bosons. Under such conditions, any gauge potential Larson et al. (2020),
characterizing non-adiabatic corrections and arising due to back-action
between the condensate and the fermions, can be neglected. The fermionic part
of the Hamiltonian can then be diagonalized while keeping the bosonic degrees
of freedom fixed, thus acting as an “adiabatic potential”. The approximation
is equivalent to replacing the bosonic field operators with the condensate
wavefunction in the Hamiltonian, yielding the hybrid operator
$\hat{\mathcal{H}}[\psi_{b}]=\mathcal{E}_{b}[\psi_{b}]+\sum_{n}\epsilon_{f,n}\hat{\psi}_{f,n}^{\dagger}\hat{\psi}_{f,n},$
(11)
where the mean-field energy functional of the bosonic part is
$\mathcal{E}_{b}[\psi_{b}]=N_{b}\int_{0}^{L}dx\;\psi_{b}^{*}\left(-\frac{\partial^{2}}{\partial
x^{2}}+V_{b}+\frac{g_{b}N_{b}}{2}|\psi_{b}|^{2}\right)\psi_{b},$ (12)
and the second part is just the fermionic part of the Hamiltonian written in
diagonal form, $\epsilon_{f,n}$ being the eigenvalues of the Hartree equation
for the fermion orbitals,
$\left(-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+g_{bf}N_{b}|\psi_{b}|^{2}\right)\phi_{f,n}=\epsilon_{f,n}\phi_{f,n}.$
(13)
Note that the solution of this Hartree equation exhibits a functional
dependence on the boson density, hence from now on we will be writing
$\epsilon_{f,n}=\epsilon_{f,n}[\psi_{b}]$. From Eq. (11), a nonlinear
Schrödinger equation for $\psi_{b}$ can be derived,
$i\dot{\psi}_{b}=\left(-\frac{\partial^{2}}{\partial
x^{2}}+V_{b}+g_{b}N_{b}|\psi_{b}|^{2}+g_{bf}n_{f}\right)\psi_{b},$ (14)
where the expectation value of the fermion density can be calculated as
$n_{f}(x)=\langle\hat{\psi}_{f}^{\dagger}(x)\hat{\psi}_{f}(x)\rangle=\sum_{n=1}^{N_{f}}|\phi_{f,n}(x)|^{2}.$
(15)
To determine the ground state, we resort to Eq. (14) and numerically propagate
an initial condensate wavefunction in imaginary time, employing the split-
operator method Feit et al. (1982). This procedure amounts to solving the
eigenvalue problem (13) at each time step.
To assess the Peierls phase for a system of infinite extent, we will use the
following prescription: Assume a filling $\nu_{f}=1/2$, so that we expect
density modulations with a period of two sites. Since the modulations are
commensurate with the optical lattice, the entire problem is periodic and the
solutions of the Hartree equation are Bloch waves, with a reduced unit cell in
reciprocal space due to the doubled periodicity in the Peierls phase. Again,
for the system evolution, we propagate in imaginary time, solving the Hartree
equation at each time instant. The case of incommensurate fermion filling is
interesting in its own right, as it may lead to coexisting spatially separated
regions of commensurable and incommensurate phases Molina et al. (2007).
However, our method does not allow for an incommensurate filling at present,
whence we leave the problem aside for later investigation.
For the Peierls phase, we appeal to the same order parameter as in the SSH
model, namely the energy gap of fermionic excitations above the Fermi surface,
$\Delta_{P}:=\lim_{\epsilon\to
0^{+}}\Big{(}E_{f}(k_{F}+\epsilon)-E_{f}(k_{F}-\epsilon)\Big{)},$ (16)
which in turn is identical to the gap of the hybrid Hamiltonian (11). As we
discuss in Sec. II.2, this quantity should approximate the excitation gap of
the entire system.
Figure 2: (color online) Phase properties in the absence of a trap. (a) Phase
diagram in the $(g_{bf},g_{b})$ plane, marking the different phases: LL,
Peierls, collapsed, and separated. The solid black lines mark the boundaries
of first-order transitions, while the solid red lines represent the BKT
transitions. The mean-field prediction (8) is plotted with dotted black lines.
The critical coupling $g_{bf}^{c}$ for the LL-to-Peierls transition is
calculated from a least square fit of the Peierls gap to the expression (7),
using the hybrid mean-field method. (b) Same as for (a) but in the
$(g_{bf},V_{0})$ plane. Here it becomes clear that the presence of the optical
lattice alters the phase diagram quantitatively, in particular through
stabilizing the Peierls phase. (c) The Peierls gap as a function of $g_{bf}$,
once more calculated using the hybrid code, for $V_{0}=0$ (yellow), $V_{0}=4$
(green), and $V_{0}=16$ (blue), fitted to the BKT formula (7) (dotted lines).
(d) Boson and fermion densities in the Peierls phase for the parameters
$V_{0}=1$, $g_{b}=0.4$, and $g_{bf}=-1$. In all plots, we consider half
filling, $\nu_{f}=1/2$. Figure 3: (Color online) Density profiles in the
presence of a harmonic trap. The top row shows the boson density (magenta) and
the fermion density (green), while the bottom row shows the fermion density
per site. Only $x>0$ is shown since the densities are symmetric around $x=0$.
Both plots pertain to the ground state in a BF mixture in a trap, obtained
using the hybrid method. (a): “Wedding-cake” structured fermionic insulator,
with parameters $V_{0}=4$, $g_{bf}=-48$, $\omega_{b}=0.03$ and
$\omega_{f}=0.3$. (b): Peierls phase, with parameters $V_{0}=1$, $g_{bf}=-4$,
$\omega_{b}=0.072$ and $\omega_{f}=0.0072$. The inset in the upper plot zooms
on the density profile in the region $0\leq x/a\leq 8$, evincing density
modulations in the bulk, with their origin in the Peierls instability. (c):
Separated phase with parameters $V_{0}=4$, $g_{bf}=4$, $\omega_{b}=0.08$ and
$\omega_{f}=0.8$. In all simulations, we have set the remaining parameters to
$g_{b}=0.4$, $r=0.04$, $N_{f}=32$, and $N_{b}=400$.
### III.2 Thomas-Fermi model for the fermions
To ascertain the transition to the collapsed or separated phases, we resort to
a Thomas-Fermi model for the fermions (not to be confused with the Thomas-
Fermi approximation); for an introduction, see Refs. Parr and Yang (1989);
Spruch (1991) and for applications, see Refs. March et al. (2000); Salasnich
et al. (2002, 2007). We define a classical field
$\psi_{f}(x)=\sqrt{n_{f}(x)/N_{f}}$, called the Thomas-Fermi wavefunction,
where $n_{f}(x)$ has been defined in Eq. (15) under the normalization $\int
dx\,|\psi_{f}|^{2}=1$. The kinetic energy is approximated as a functional
$\mathcal{T}[\psi_{f}]=\int dx\;\frac{\pi^{2}N_{f}^{3}}{3r}|\psi_{f}|^{6}.$
(17)
This form can be motivated by means of a dimensional analysis, and is derived
in detail in Parr and Yang (1989); Spruch (1991). We obtain the Thomas-Fermi
energy functional as
$\mathcal{E}[\psi_{b},\psi_{f}]=\mathcal{E}_{b}[\psi_{b}]+\mathcal{E}_{f}[\psi_{b},\psi_{f}],$
(18)
where the first term on the right is given by (12) and the second by
$\begin{array}[]{lll}\mathcal{E}_{f}[\psi_{b},\psi_{f}]&=&\displaystyle{N_{f}\int
dx\Bigg{(}\frac{\pi^{2}N_{f}^{2}}{3r}|\psi_{f}|^{6}+\frac{1}{r}\left|\frac{\partial\psi_{f}}{\partial
x}\right|^{2}}\\\ \\\
&&+g_{bf}N_{b}|\psi_{b}|^{2}|\psi_{f}|^{2}\Bigg{)}.\end{array}$ (19)
Taking the functional derivative, we find an equation of a two-component
Gross-Pitaevskii type,
$\begin{array}[]{l}\displaystyle{i\partial_{t}\psi_{b}=\left(-\frac{\partial^{2}}{\partial
x^{2}}+V_{\text{OL}}+g_{b}N_{b}|\psi_{b}|^{2}+g_{bf}N_{f}|\psi_{f}|^{2}\right)\psi_{b}},\\\
\\\
\displaystyle{i\partial_{t}\psi_{f}=\left(-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+\frac{\pi^{2}N_{f}^{2}}{r}|\psi_{f}|^{4}+g_{bf}N_{b}|\psi_{b}|^{2}\right)\psi_{f}.}\end{array}$
(20)
Employing once more the split-operator method Feit et al. (1982), we then
evolve an initial state in imaginary time to determine the ground state. Due
to the mass separation between the two species, one may argue that an
adiabatic elimination of the fermion field should be justified. This would
result in an effective bosonic model where the fermion-mediated RKKY-like
boson-boson interactions would appear. Such an approximation, however, has a
negligible benefit to the numerical solution, whence we solve the full mean-
field model formulated by Eqs. (20).
## IV Results
### IV.1 Translation-invariant BF mixture
We first focus on the periodic case in the absence of a trap, where we set
$V_{\rm{trap}}=0$. This idealization targets the thermodynamic limit in the
system response, which is meaningful when assessing the universal properties
of the phase transitions reported herein. A Bose-Fermi mixture with no lattice
was studied in Ref. Cazalilla and Ho (2003), where it was found that, for
negative $g_{bf}$, a BKT transition separates a LL from a Peierls phase; that
gap had the form dictated by Eq. (7).
In the upper two frames of Fig. 2, we depict the numerically extracted phase
diagrams; frame (a) shows a region in the $(g_{bf},g_{b})$ plane when keeping
$V_{0}=0$ and $r=0.04$ constant, while frame (b) displays a region in the
$(g_{bf},V_{0})$ plane for $g_{b}=0.4$ and $r=0.04$ constant. Both (a) and (b)
show the region of stability of the Peierls phase towards collapse or
separation (black solid lines), located via a calculation using the Thomas-
Fermi approximation and compared to the mean-field prediction (8) (black
dotted lines). The transition is identified by detecting the discontinuity in
the overlap between boson and fermion densities. The LL-to-Peierls transition
(red solid lines) is calculated via the hybrid method, and the critical
coupling $g_{bf}^{c}$ is obtained from a least square fit of the expression
(7) to the numerical data. From frame (a) we conclude that the region of
stability has an extent which abides by the inequality (8), but more
importantly that our method predicts a Peierls (i.e. pairing) phase for
$g_{bf}>0$, in disagreement with Ref. Cazalilla and Ho (2003). The reason for
such a disparity could be simply attributed to the fact that the hybrid method
entails the unphysical assumption that the solution is periodic, which hinders
the detection of long-wavelength fluctuations. From the literature on Bose-
Fermi-Hubbard systems, where both gases are subject to an optical lattice, we
expect however an optical lattice to stabilize the Peierls phase for repulsive
interaction Mathey (2007). From frame (b) we infer that this is indeed the
case: the optical lattice stabilizes the Peierls phase beyond the boundary set
by condition (8) for a homogeneous configuration. More precisely, the extent
of this phase is growing with the optical lattice potential while the LL,
collapsed, and separated phases are shrinking for the shown parameter ranges.
Furthermore, the Peierls gap also grows with increasing lattice amplitude,
meaning that it should be easier to observe in an experiment with a stronger
optical lattice, provided the full system does not enter an insulating phase
due to the inclusion of an optical lattice.
To illustrate the type of transition occurring between the LL and Peierls
phases, Fig. 2(c) shows three examples of the Peierls gaps as a function of
$g_{bf}$, for $V_{0}=0$, $4$, and $8$, keeping $g_{b}=0.4$ constant. As
demonstrated by the dotted curves, close to the critical value, the
numerically extracted gaps fit very well the analytical expression (7) for the
gap of a BKT transition. Away from the critical point, there is a notable
“knee” feature, which occurs when the bosons only populate every other site,
and solely for $V_{0}\neq 0$.
Figure 2(d) displays the density of bosons and fermions within the unit cell,
as a typical example for a particular point picked in the phase diagram:
$V_{0}=1$, $g_{b}=0.4$, and $g_{bf}=-1$. The doubling of the periodicity is in
evidence. Contrary to most earlier studies, our method is capable of capturing
both amplitude and phase modulations, however, we only find modulations in
amplitude. For the repulsive case, $g_{bf}>0$, the situation is similar to
that shown in (d), but with the two densities being “out-of-phase” instead.
In order to experimentally probe the phase diagram, one must detect the order
parameter or any other quantity capable of telling the different phases apart.
The gap could, in principle, be extracted via pump-probe experiments Ernst et
al. (2010); Heinze et al. (2011), while one could also imagine transport
experiments to differentiate insulating and conduction phases, i.e. Peierls
vs. LL. As will be evident in the next section, the trap induces a varying
Fermi wave number $k_{F}$ which makes the Peierls gap opening not that
pronounced. In a time-of-flight detection, for instance, the Bragg peaks are
smeared out, and the particular one corresponding to the gap is almost
invisible. Hence, in a realistic setup, including the trap, time-of-flight
measurements might not turn out to be the most optimal scheme. Instead, the
onsite densities, which display alternating variations, could be accessed via
single-site resolution detection Bakr et al. (2009). This technique has been
successfully implemented for cold fermionic gases, including even detection of
higher order correlators Haller et al. (2015); Boll et al. (2016); Scherg et
al. (2021).
### IV.2 BF mixture in a harmonic trap
Current experiments rely on the application of harmonic potentials for the
attainment of sufficiently long trapping times. However, the trap may have a
relatively small frequency yet still confine the atoms, such that the system
locally experiences a periodic potential. To study these more realistic
situations, we include a harmonic trap in our analysis but still consider a
very tight transverse confinement and consequently a quasi-1D configuration.
Figure 4: From a Mott to a Peierls insulator with extended eigenstates in a
trap. One-particle density matrix,
$|\langle\hat{\psi}_{f}(x)\hat{\psi}_{f}^{\dagger}(x^{\prime})\rangle|^{2}$,
where only the top-right quadrant is plotted owing to the symmetry about
$x=0$. Two cases are shown: (a) a trapped mixture in the wedding cake phase,
for the same state as the one presented in the left column of Fig. 3; (b)
Peierls phase for the same state as in the central column of Fig. 3. The non-
diagonal part of the one-particle density matrix is significantly suppressed
in the wedding cake phase, where the decay is Gaussian and on a length scale
similar to the characteristic length of the optical lattice. In the Peierls
phase, there are instead periodic oscillations signifying coherence. Figure 5:
(Color online) Scaling of the two-point density matrix with the width of the
harmonic trap. The one-point density matrix $n(x,x^{\prime})$ is calculated
for $x^{\prime}=-x$ and plotted with $x$ scaled with $a\sqrt{k}$. The
oscillations of the one-point density matrix are fitted to the expression
$n(x,-x)=C\text{sinc}(bx)$ in line with Eq. (46). Left column: normal phase
($g_{bf}=0,\omega_{0,b}=0.005,\omega_{0,f}=0.07$). Right column: Peierls phase
($g_{bf}=-1.5,\omega_{0,b}=\omega_{0,f}=0.0036$). The harmonic confinements
have been set as $\omega_{b/f}=k\omega_{0,b/f}$, with $k=1,2,4$ for the top,
middle and bottom rows. In all simulations, we have fixed the remaining
parameters to $g_{b}=0.1$ and $V_{0}=0.25$.
We have used the hybrid method to investigate the different phases in the
aforementioned setup. Figure 3 presents the numerical results obtained for the
densities, along the main diagonal of the one-particle density matrix. In the
upper row, we present the real-space densities, and in the lower row, the
fermion density per site is depicted, which could, in principle, be accessed
in an experiment, as mentioned in the last section. Illustrated in the left
column is a novel insulating state not present in the translationally
invariant system. We refer to this profile as a “wedding cake”, drawing an
analogy to the wedding cake-like structure found in the harmonically confined
Bose-Hubbard model Jaksch et al. (1998) and also in fermionic systems Jördens
et al. (2008); Greif et al. (2016). In this state, the fermions form plateaus
(or “layers of the cake”) with an integer number of fermions per site. In the
figure, a plateau of three fermions per site is observed in the center of the
trap, transitioning in a step-like fashion to two and finally one fermion per
site as we move further out from the center. The wedding cake pattern results
from the interplay between kinetic, potential, and interaction energies and
only appears when an optical lattice is present. It cannot be accurately
modeled using a local density approximation due to the abrupt changes in
density.
In the central column of Fig. 3, we elaborate on the correspondence to the
Peierls phase occurring in the absence of a trap. Generally, the boson and
fermion clouds form a region of a size determined by the trapping strength
alongside the BB and BF interactions. The optical lattice induces a spatial
variation in the bosonic profile, imparted to the fermions via the BF
interaction. The Friedel oscillations of the fermions in this potential are
then imprinted on the bosons via the RKKY interaction. Because the Peierls
phase is a compressible supersolid, the wavelength varies continuously over
the width of the trap. Therefore, in the trapped system, the Peierls
modulations are generally not commensurate with the lattice. These modulations
may be detected using single-site addressing, leading to a spatial variation
in density that cannot be attributed to the profile of the trap or that of the
optical lattice. We note here that the beating between the fermion and the
optical lattice wavenumbers may give rise to local commensurate-incommensurate
transitions in which the fermionic wavenumber of an atomic density wave
spatially adjusts to the optical lattice period Molina et al. (2007).
For any positive value of $g_{bf}$ and any strength of the optical lattice, we
have observed that the system phases separate. The rightmost column of Fig. 3
displays the separated state, where the heavier atoms occupy the center of the
trap, surrounded by regions of the lighter species. Perhaps the most
surprising feature of Fig. 3 is the emergence of the wedding cake phase where
we would expect a collapsed phase. The integer filling per site clearly
indicates a fermionic Mott insulator state Rigol et al. (2003); Rigol and
Muramatsu (2004); Schneider et al. (2008). One way to ascertain whether this
instance is something more than a coincidence is to examine the off-diagonal
elements of the one-particle density matrix, given by
$\langle\hat{\psi}_{f}^{\dagger}(x)\hat{\psi}_{f}(x^{\prime})\rangle=\sum_{n=0}^{N_{f}-1}\phi_{f,n}^{*}(x)\phi_{f,n}(x^{\prime}).$
(21)
To derive the above expression, we have written
$\hat{\psi}_{f}(x)=\sum_{n}\phi_{f,n}(x)\hat{\psi}_{f,n},$ (22)
where $\hat{\psi}_{f,n}$ are fermionic operators and the orbitals
$\phi_{f,n}(x)$ are solutions of the Hartree equation (see Appendix B)
$\left(-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+g_{bf}N_{b}|\psi_{b}|^{2}\right)\phi_{f,n}=\epsilon_{f,n}\phi_{f,n}.$
(23)
An insulating state is expected to have an exponentially decaying one-particle
density matrix Cloizeaux (1964); He and Vanderbilt (2001). However, it turns
out that this is not the case for the BF mixture in a harmonic trap.
In Fig. 4, we focus into a more detailed comparison between the wedding cake
[frame (a)] and Peierls [frame (b)] phases, showcasing the corresponding one-
body density matrix. As seen in Fig. 3(a), the wedding cake is distinguished
by having an integer number of fermions per site. The one-particle density
matrix provides an additional way of distinguishing the phases. In the
wedding-cake phase, it displays a Gaussian decay on a length scale similar to
the characteristic length of the optical lattice. This feature, in turn,
signals the opening of a more pronounced gap in the spectrum and the presence
of an insulating phase. On the other hand, the spatial decay of the matrix
elements in the Peierls phase reveals a sustained coherence originating from
the individual trapped-particle wavefunctions, which we will attempt to
elucidate.
A characteristic example of the spatially decaying one-point density matrix
$n(x,x^{\prime})$ is plotted in Fig. 5 for $x^{\prime}=-x$. It is evident that
in the normal phase, the “wavelength” of the fitted sinc function scales with
$\sqrt{k}$, while the relation is not so direct in the Peierls phase. To
motivate the mechanism underlying the periodically decaying oscillations, we
start by setting $g_{bf}=0$, in which case the fermions are free from their
interaction with the bosons in the same trap. In this case, we find (see
Appendix C for further details)
$n(x,-x)=\frac{1}{\pi}\frac{\sin(\sqrt{2N}\,2x)}{2x}.$ (24)
Equation (24) explicitly reveals an algebraically decaying envelope and a
$\sqrt{N_{f}}$-dependence of the spatial frequency, with $N_{f}$ being the
number of fermions confined in the trap, both attributes evidenced in the left
plots of Fig. 5. While gapped phases might suggest an exponential decay of the
single-particle density Sachdev (1999), the spatial confinement imposed by the
trap implies that the extent of excitations is typically of the same order as
the characteristic trap length (unless very weak trap frequencies are
considered). Thus, we observe an algebraic decay instead of an exponential
one. Turning on an attractive Bose-Fermi interaction $g_{bf}$ to access the
regime of Peierls instability in the right frames of Fig. 5 marks the
departure from the sinc profile and the $\sqrt{N_{f}}$ scaling, yet the
periodicity is still evident, and the frequency carries on increasing with the
extent of the trap. The counterdiagonal now defines a particular cut through
ripples visible in Fig. 4(d) – remnants of quantum interference developing
symmetrically to the main diagonal and signifying a Peierls phase with a
larger spatial extent than the normal. The overall amplitude of the density
matrix along the counterdiagonal, depicted in the right frames of the figure,
also increases with $N_{f}$, in line with the prediction of Eq. (46). Taking
$N_{f}\to\infty$, we obtain the delta function as the familiar limit of the
sinc function, writing $n(x,-x)\propto\delta(x)$.
## V Concluding remarks
In this work, we have studied the response of a dilute mixture of two atomic
clouds of disparate masses, one fermionic and one bosonic, to an external
lattice potential alongside an atomic trap. The gases are confined in a
cylindrical trap so that the low-energy physical behavior of the system is
effectively one-dimensional, while the optical lattice imprints a period
structure onto the bosonic gas. We have assumed a weak optical lattice and a
high density of bosons, such that the bosons form a condensate, which amounts
to an effective interaction of the fermions with a coherent field.
We confirmed that a Peierls instability, manifested via a BKT-type phase
transition to a Peierls phase, persists when the optical lattice potential is
applied solely to the bosons. Moreover, the presence of this phase was found
to stabilize the system against collapse and separation, as well as to enhance
the significance of the Peierls effect, a property that is highly desirable
for experimental explorations of the associated phonon-like physics. The
presence of the Peierls phase was confirmed upon developing and applying a
hybrid mean-field method, which retained an amount of coherence able to
sustain sinc-type oscillations along the counterdiagonal of the one-particle
density matrix. It is interesting to note that such a method can reveal the
BKT nature of the phase transition, as well as the similarity to the SSH model
of the Peierls instability. The crucial scaling is captured by the method we
have developed owing to the fact that we take the deformation of the Fermi
surface into account, while we also retain the quantum degrees of freedom for
the fermions albeit within an adiabatic approximation. An algebraic decay of
correlations is found for all phases occurring in the trapped BF mixture, in
accordance with a BKT phase transition.
The natural step forwards would be to take quantum fluctuations of the boson
degrees of freedom into account. This entails the development of a formulation
and related numerical methods manifestly beyond any mean-field reduction, and
to that aim we outline possible directions for deriving the corresponding
many-body Hamiltonian in Appendix A below. Another promising direction for
future research concerns the investigation of fermion fillings incommensurate
with the optical lattice as well as studying the quenched time-evolution
problem of a single fermion in a dynamical bosonic potential. Furthermore, an
interesting extension replaces the optical lattice with a BB mixture, using
either trapped ions or dipole bosons. Finally, systems of the kind can be
analyzed in greater detail using matrix product states or bosonization
techniques.
## Appendix A Routes towards known quantum many-body models
In Sec. III.1, we reported on the presence of a Peierls phase at a hybrid
mean-field level. We argue that this description accounts for the pairing
effect qualitatively. However, there are clear limitations to such an
approach. Effectively, the model is a free fermion theory, and as long as
there is a finite Peierls gap, $\Delta\neq 0$, however we find a nontrivial
insulator with algebraically decaying order for the trapped BF mixture. This
is in line with what one expects from a BKT transition, where one phase should
exhibit algebraic decay. Furthermore, the classical field stemming from the
boson condensation cannot build up quantum correlations with the fermions,
while a quantum field can. To quantitatively analyze such situations one must
go beyond mean-field. In this appendix we outline, without going into details,
two possible approaches in order to derive effective Hamiltonians for which
both species are treated quantum mechanically.
In Ref. Bruderer et al. (2007), the condensate wavefunction was calculated for
zero coupling between the two species, and subsequently, Bogoliubov
excitations around the mean-field response were taken into account, arising
from a weak interaction. Due to the presence of a deep external lattice
potential in their model, a Hubbard-Holstein model is obtained which can be
understood in terms of polarons. In our case, we instead find the fermion-
phonon Hamiltonian
$\displaystyle\hat{\mathcal{H}}=$
$\displaystyle\hat{\mathcal{H}}_{f}+\sum_{\mu}\omega_{\mu}\int
dx\;\left(M_{\mu}\hat{b}_{\mu}+M_{\mu}^{*}\hat{b}_{\mu}^{\dagger}\right)\hat{\rho}_{f}$
$\displaystyle+g_{bf}\int
dx|\phi_{b}|^{2}\hat{\rho}_{f}+\sum_{\mu}\omega_{\mu}\hat{b}_{\mu}^{\dagger}\hat{b}_{\mu},$
(25)
where
$\displaystyle M_{\mu}(x)=u_{\mu}(x)-v_{\mu}(x)$ (26)
are given in terms of the $u_{\mu}(x),v_{\mu}(x)$ of the Bogoliubov
transformation and $\omega_{\mu}$ are the energies of the Bogoliubov modes.
The third term will confine the fermions to their lowest Bloch band if the
interaction is strong enough, and in this limit, the model approaches the
Hubbard-Holstein model of Ref. Bruderer et al. (2007). However, the approach
above would have to be modified since it rests on the assumption that $g_{bf}$
is weak.
For a deep optical lattice one could come up with another method: expand the
boson field in the corresponding Wannier functions, $w_{j}(x)$ localized at
site $j$, as Lewenstein et al. (2007); Bloch et al. (2008)
$\hat{\psi}_{b}(x)=\sum_{j}\hat{a}_{j}w_{j}(x).$ (27)
If we expand the fermion field as
$\hat{\psi}_{f}(x)=\sum_{k}\hat{c}_{k}e^{ikx}$ and impose the single-band and
tight-binding approximations, we arrive at the many-body Hamiltonian
$\displaystyle\hat{\mathcal{H}}_{mb}$
$\displaystyle=\hat{\mathcal{H}}_{BH}+\sum_{k}\\!\frac{k^{2}}{r}\hat{c}_{k}^{\dagger}\hat{c}_{k}$
$\displaystyle+\hat{N}_{b}\sum_{k,k^{\prime}}\\!\left[D(k-k^{\prime})\hat{c}_{k}^{\dagger}\hat{c}_{k^{\prime}}+\text{H.c.}\right]\\!,$
(28)
where
$\displaystyle\hat{\mathcal{H}}_{BH}$
$\displaystyle=-J\sum_{j}\left(\hat{a}_{j}^{\dagger}\hat{a}_{j+1}+\text{H.c.}\right)$
$\displaystyle+\frac{U}{2}\sum_{j}\hat{n}_{j}\left(\hat{n}_{j}-1\right)$ (29)
is the Bose-Hubbard Hamiltonian, with $J$ and $U$ the tunneling rate and
onsite interaction strengths, respectively,
$\hat{n}_{j}=\hat{a}_{j}^{\dagger}\hat{a}_{j}$, and $D(k-k^{\prime})$ is the
overlap integral (which is Gaussian in the harmonic approximation). Since
$\hat{N}_{b}$ is preserved, the quadratic fermionic Hamiltonian can be readily
diagonalized, and we find no back-action on the bosons due to the fermions. To
incorporate such effects one would need to go beyond the single-band or tight-
binding approximations. Alternatively, the approximations may be kept but
other additional degrees of freedom should be introduced corresponding to the
phonon modes. For heavy bosons, one can follow the idea of Ref. Maluckov et
al. (2013) to allow a variation in the position of the Wannier-function
centers $j\pi$, but keep the shape of the functions intact. Thus, we associate
a quantized shift $\hat{\delta}_{j}$ with each Wannier function,
$w_{j}(x)\rightarrow w_{j}(x-\hat{\delta}_{j})$. Assuming small shifts
$\hat{\delta x}_{j}\ll 1$ one may then expand to linear order in them and
derive an effective Fröhlich-like Hamiltonian Mahan (2013)
$\hat{\mathcal{H}}_{Fr}=\hat{\mathcal{H}}_{mb}+\hat{\mathcal{H}}_{BP}+\hat{\mathcal{H}}_{FBP},$
(30)
where the boson-phonon interaction is given by
$\begin{array}[]{lll}\hat{\mathcal{H}}_{BP}&=&\displaystyle{-J_{1}\sum_{j}\left(\hat{\delta}_{j+1}-\hat{\delta}_{j}\right)\left(\hat{a}_{j}^{\dagger}\hat{a}_{j+1}+\text{H.c.}\right)}\\\
\\\
&&\displaystyle{+\frac{U_{1}}{2}\sum_{j}\hat{\delta}_{j}\hat{n}_{j}\left(\hat{n}_{j}-1\right)}.\end{array}$
(31)
If we introduce the local phonon annihilation/creation operators
$\hat{d}_{j}/\hat{d}_{j}^{\dagger}$, the phonon displacement is expressed as
$\hat{\delta}_{j}=\left(\hat{d}_{j}+\hat{d}_{j}^{\dagger}\right)/2,$ (32)
while the fermion-boson-phonon interaction term takes the form
$\hat{\mathcal{H}}_{FBP}=\mu\hat{N}_{d}+\sum_{j}\sum_{k,k^{\prime}}\hat{n}_{j}D_{1}(k-k^{\prime})\hat{\delta}_{j}\hat{c}_{k}^{\dagger}\hat{c}_{k^{\prime}},$
(33)
where the first term is the bare phonon energy ($\mu$ is the characteristic
frequency and $\hat{N}_{d}=\sum_{j}\hat{d}_{h}^{\dagger}\hat{d}_{j}$). In
principle, within the harmonic approximation, the coefficients $J_{1}$,
$U_{1}$ and $D_{1}$ can be analytically determined. Both
$\hat{\mathcal{H}}_{BP}$ and $\hat{\mathcal{H}}_{FBP}$ describe phonon-
assisted tunneling, either between neighboring lattice sites (bosons) or
between different momentum modes (fermions). It may be noted that if
$g_{bf}=0$, then $D(k-k^{\prime})=D_{1}(k-k^{\prime})=0$ and $\hat{N}_{d}=0$
such that $\hat{\mathcal{H}}_{BP}=0$.
## Appendix B The hybrid mean-field–quantum method
In this appendix, we provide some further details about the employed hybrid
mean-field ansatz. As already mentioned in Sec. III.1, a gap of the form (7)
opens up in the pairing phase. A superfluid of bosons is gapless and, to the
lowest order, is described by the condensate wavefunction $\psi_{b}(x)$ – the
mean-field order parameter. We define the generalized coherent state
$\ket{\psi_{b}}=\exp\left(\int
dx\;\psi_{b}(x)\hat{\psi_{b}}^{\dagger}(x)-\text{H.c.}\right)\ket{0},$ (34)
where $\ket{0}$ is the bosonic vacuum. Taking the expectation value of the
time-dependent Schrödinger equation, we obtain an effective hybrid Hamiltonian
$\displaystyle\hat{\mathcal{H}}_{\text{eff}}=\mathcal{E}_{b}[\psi_{b}]+\hat{\mathcal{H}}_{f}[\psi_{b}]-i\matrixelement{\psi_{b}}{\partial_{t}}{\psi_{b}},$
(35)
where the first term is given by (12), the second term is
$\displaystyle\hat{\mathcal{H}}_{f}[\psi_{b}]=\int
dx\;\hat{\psi}_{f}^{\dagger}\Big{(}-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+V_{f}+g_{bf}|\psi_{b}|^{2}\Big{)}\hat{\psi}_{f},$ (36)
and the third term is the Berry connection Larson et al. (2020). We will
assume that the state of the condensate is slowly modified, so that this term
can be neglected.
The above is equivalent to a product-state ansatz for the ground state
$\ket{\Psi_{0}}=\ket{\psi_{b}}\otimes\ket{\Psi_{f}},$ (37)
where $\ket{\Psi_{f}}$ is a general fermion state of $N_{f}$ fermions. Such an
ansatz neglects any entanglement built between the bosons and fermion
subsystems; correlations may actually arise within the non-condensed fraction
of bosons and, more importantly, different configurations of the condensate
may get entangled with the fermions. In general, such mixed quantum-classical
dynamics is interesting in a much wider context in condensed matter physics,
high-energy physics, and quantum gravity, and has been investigated in detail
in Refs. Prezhdo and Kisil (1997); Caro and Salcedo (1999); Oppenheim et al.
(2020). In Ref. Prezhdo and Kisil (1997), a multi-configurational mean-field
approximation based on hybrid quantum-classical theory was developed, with the
central object being the quantum-classical distribution function – a map from
the classical phase space to the set of quantum density operators, defined as
$\hat{\rho}(\mathbf{q},\mathbf{p})=\sum_{ij}\varrho_{ij}|\Psi_{i}\rangle\langle\Psi_{j}|\delta(\mathbf{q}-\mathbf{q}_{ij})\delta(\mathbf{p}-\mathbf{p}_{ij}),$
(38)
where $\mathbf{q},\mathbf{p}$ are the classical generalized coordinates and
momenta, and $|\Psi_{i}\rangle$ is a basis state of the quantum subsystem. In
this formalism, “non-diagonal” contributions correspond to the quantum
subsystem generating a coupling between different trajectories of the
classical subsystems. When considering only a single trajectory, the quantum-
classical distribution function can be written as a single delta-function
term,
$\hat{\rho}[\psi_{b}^{\prime}]=|\Psi_{f}\rangle\langle\Psi_{f}|\delta[\psi_{b}-\psi_{b}^{\prime}],$
(39)
where the brackets indicate a functional dependence. In other words, the
system is represented by a single point in classical phase space, evolving
with a quantum density matrix of a pure state.
We find the ground state of $\hat{H}_{\text{eff}}$ self-consistently first
observing that it can be readily diagonalized by selecting
$\hat{\psi}_{f}(x)=\sum_{n}\phi_{f,n}(x)\hat{\psi}_{f,n},$ (40)
where the orbitals are solutions of the Hartree (single-particle) equation
$\left(-\frac{1}{r}\frac{\partial^{2}}{\partial
x^{2}}+g_{bf}N_{b}|\psi_{b}|^{2}\right)\phi_{f,n}=\epsilon_{f,n}\phi_{f,n}.$
(41)
Second, we define a non-linear Schrödinger equation for the condensate,
$\begin{array}[]{lll}i\dot{\psi}_{b}&=&\displaystyle{\frac{\delta}{\delta\psi_{b}^{*}}\matrixelement{\Psi_{f}}{\hat{H}_{\text{eff}}}{\Psi_{f}}}\\\
\\\ &=&\displaystyle{\left(-\frac{\partial^{2}}{\partial
x^{2}}+V_{b}+g_{b}N_{b}|\psi_{b}|^{2}+g_{bf}n_{f}\right)\psi_{b}},\end{array}$
(42)
where
$n_{f}(x)=\matrixelement{\Psi_{f}}{\hat{\psi}_{f}^{\dagger}(x)\hat{\psi}_{f}(x)}{\Psi_{f}}=\sum_{n}|\phi_{f,n}(x)|^{2}$
(43)
depends explicitly on the bosons through (41). The ground state of (42) is
found from imaginary time propagation utilizing the split-operator method Feit
et al. (1982).
As mentioned in Sec. III.1, the above method can also be easily adapted to
study a translationally invariant system of infinite extent for fermion
filling $\nu_{f}=1/2$. In this case, we assume a periodic solution
$\psi_{b}(x+4\pi)=\psi_{b}(x)$. The solutions of the Hartree equations are
then Bloch waves $e^{iqx/2}u_{f,n}(q,x)$. Due to the assumed double
periodicity of $4\pi$, the first Brillouin zone is halved and the fermionic
filling is in direct correspondence to the edges of the Brillouin zone. It is
straightforward to find the fermion density as the integral
$n_{f}(x)=\int_{1BZ}dq\;|u_{f,0}(q,x)|^{2}.$ (44)
Another important difference here is that we have to diagonalize the
Bloch/Hartree equation for each quasi-momentum – in practice for a large
number of sampled values. Apart from these discrepancies between the two
methods, the solution to Eq. (42) is also found using imaginary time
propagation.
## Appendix C Decay of correlations in the absence of Bose-Fermi interaction
For $g_{bf}=0$ (normal phase), the one-point density matrix can be calculated
analytically via the Cristoffel-Darboux formula, involving summed products of
individual orthogonal fermionic wavefunctions in terms of the Hermite
polynomials $H_{n}(x)$ (with $n=0,1,\ldots N_{f}$) derived from the quantum
harmonic oscillator eigenstates. Letting $N_{f}=N$ for convenience, the result
reads
$\displaystyle n(x,x^{\prime})$
$\displaystyle=\pi^{-1/2}\exp\left(-\frac{x^{2}+(x^{\prime})^{2}}{2}\right)$
$\displaystyle\times\frac{1}{N!2^{N+1}}\frac{H_{N}(x^{\prime})H_{N+1}(x)-H_{N}(x)H_{N+1}(x^{\prime})}{x-x^{\prime}}.$
(45)
For $x^{\prime}=-x$, along the counterdiagonal of the density matrix that we
select for Fig. 5, the above expression can be recast into the following form:
$\displaystyle n(x,-x)=\pi^{-1/2}\exp(-x^{2})\frac{1}{N!2^{N+1}}$ (46)
$\displaystyle\times\frac{2(-1)^{N}H_{N}(x)H_{N+1}(x)}{2x}\sim\frac{2^{2N}\Gamma((N+1)/2)\Gamma(N/2+1)}{\pi^{3/2}\,\,2^{N}\,N!}$
$\displaystyle\times\frac{\sin(\sqrt{2N}\,2x)}{2x}=\frac{1}{\pi}\frac{\sin(\sqrt{2N}\,2x)}{2x},$
which follows from the familiar asymptotic $N\gg 1$ expansion of quantum
harmonic oscillator eigenstates M. and I. (1983)
$e^{-x^{2}/2}H_{N}(x)\sim\frac{2^{N}}{\sqrt{\pi}}\Gamma\left(\frac{N+1}{2}\right)\cos(x\sqrt{2N}-N\pi/2)$
(47)
and the Legendre duplication formula for the gamma function. In fact, the
number of fermions confined in the trap need not be large. Even for
$N_{f}\gtrsim 10$ we note a reasonable agreement between the exact formula of
Eq. (C), and the asymptotic sinc profile of (46).
###### Acknowledgements.
We thank Maciej Lewenstein, Hannes Conners, and Christophe Salomon for helpful
discussions and comments. We acknowledge financial support from VR-
Vetenskapsrådet (The Swedish Research Council), and KAW (The Knut and Alice
Wallenberg foundation). Th. K. M. acknowledges support from: ERC AdG NOQIA;
Ministerio de Ciencia y Innovation Agencia Estatal de Investigaciones
(PGC2018-097027-B-I00/10.13039/501100011033,
EX2019-000910-S/10.13039/501100011033, QUANTERA DYNAMITE PCI2022-132919,
Proyectos de I+D+I “Retos Colaboración” QUSPIN RTC2019-007196-7); MICIIN with
funding from European Union NextGenerationEU(PRTR-C17.I1) and by Generalitat
de Catalunya; Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya
(European Social Fund FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452,
QuantumCAT U16-011424, co-funded by ERDF Operational Program of Catalonia
2014-2020); Barcelona Supercomputing Center MareNostrum (FI-2022-1-0042); EU
Horizon 2020 FET-OPEN OPTOlogic (Grant No 899794); EU Horizon Europe Program
(Grant Agreement 101080086 — NeQST); ICFO Internal “QuantumGaudi” project;
European Union’s Horizon 2020 research and innovation program under the Marie-
Skłodowska-Curie grant agreement No 101029393 (STREDCH) and No 847648 (“La
Caixa” Junior Leaders fellowships ID100010434: LCF/BQ/PI19/11690013,
LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013). Views and
opinions expressed in this work are, however, those of the author(s) only and
do not necessarily reflect those of the European Union, European Climate,
Infrastructure and Environment Executive Agency (CINEA), nor any other
granting authority. Neither the European Union nor any granting authority can
be held responsible for them.
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|
# Monte Carlo simulations in anomalous radiative transfer: a tutorial
Tiziano Binzoni Department of Basic Neurosciences, University of Geneva,
Geneva, Switzerland Department of Radiology and Medical Informatics,
University Hospital, Geneva, Switzerland Corresponding author:
<EMAIL_ADDRESS>Fabrizio Martelli Dipartimento di Fisica e
Astronomia dell’Università degli Studi di Firenze, Sesto Fiorentino, Firenze,
Italy
###### Abstract
Anomalous radiative transfer (ART) theory is a generalization of classical
radiative transfer theory. The present tutorial wants to show how Monte Carlo
(MC) codes describing photons transport in anomalous media can be implemented.
It is shown that the heart of the method consists in suitably describing, in a
“non-classical” manner, photons steps starting from fixed light sources or
from boundaries separating regions of the medium with different optical
properties. To give a better intuition of the importance of these particular
photon step lengths, it is also shown numerically that the described approach
is essential to preserve the invariance property for light propagation. An
interesting byproduct of the MC method for ART, is that it allow us to
simplify the structure of “classical” MC codes, utilized e.g. in biomedical
optics.
## 1 Introduction
Monte Carlo (MC) simulations represent the tool of choice when describing
“classical” photons migration in propagating media [1]. This is specially true
when exact solutions (except for statistical noise) of complex media are
needed. In fact, when dealing with the simultaneous presence of different
media types, complex geometries and varying optical parameters, it becomes
extremely difficult to use alternative approaches such us analytical solutions
or finite differences based methods. Moreover, the continuous increase of the
computational power, even for simple desktops, renders in general more and
more attractive and easy to use the MC methods. These are some of the reasons
why data generated by MC simulations have become nowadays a “gold standard”,
allowing to test alternative methods.
In recent years, a generalization of the classical theory allowing also to
describe anomalous radiative transfer (ART), has been introduced [2, 3]. ART
theory permits to modelize photons migration in anomalous media, i.e., media
where the classical Beer–Lambert–Bouguer’s law is not valid.
The release of the Beer–Lambert–Bouguer’s law hardly complexify the definition
of the analytical models describing ART and their related solutions. This
increase in complexity may be generated, e.g., by the need of the introduction
of models based on fractional (integro)-differential equations, where the
solutions are often difficult to obtain and manipulate. Moreover, when the
investigated optical medium is not infinite, and thus necessitates the
introduction of boundary conditions, the definition of the analytical models
and the obtainment of their solutions may represent a real challenge.
This is why in communities such as the biomedical optics, where the main aim
is not specifically to understand the mathematical theory but above all to
obtain solutions related to complex practical problems in photon migration,
more efficient and general purpose tools would be particularly appreciated. It
is precisely in the context of ART that the MC methods show their extreme
power, i.e., thanks to their relative simplicity in describing and solving the
considered problems.
In general, the MC methods describe how photons propagate step-by-step in the
media, through scattering, absorption, reflection and refraction events. The
main difference between the classical and the ART MC-approaches is given by
the specific choice of the function describing the probability $p(s)ds$ for a
photon to reach a distance $s\in[s-ds,s+ds]$ in the medium, without
interactions; where $p(s)$ is a probability density function (pdf).
As it is well known, in the classical case
$p(s)=\frac{1}{\ell}e^{-s/\ell},$ (1)
where $\ell$ is the mean free path for the investigated medium (in biomedical
optics $\ell$ is often reported as $1/\ell=\mu_{t}$; where $\mu_{t}$ is the
extinction coefficient). Historically, Eq. (1) has been determined by
considering the fact that a classical medium must satisfy
Beer–Lambert–Bouguer’s law.
In ART, it is not mandatory for $p(s)$ to be an exponential function and a
panoply of choices, depending on the specific optical properties of the
medium, are possible.
At this point, the fundamental remark is that to simulate ART models, by means
of the MC method, it is not sufficient to substitute in the MC code the
classical $p(s)$ [Eq. (1)] with the relative anomalous one. In fact, if we
proceed in such a simple way the fundamental reciprocity law (RL) at the basis
of optics will not be correctly reproduced by the simulations. As a short
reminder, in the present context the RL tells us that if we exchange light
sources and detectors (by inverting the direction of the detected photons; see
e.g. red line in Fig. 1), we must always measure (detect) the same photons
flux. For a more indepth approach to the RL see e.g. Ref. [4].
Thus, based on the findings derived from ART theory, the aim of the present
tutorial is to describe the suitable way to treat this problem. This will be
done by directly giving the necessary practical “recipes” allowing to build MC
codes for ART. We will see that the core of the solution is given by the
particular choice of the pdf allowing to generate the length $s$ of the first
step of photons that are starting from a medium boundary (defined by adjacent
regions with different optical parameters) or from a fixed light source. The
remaining steps, and the photons interactions with absorbers/scatterers, being
treated as in a classical MC code.
In this tutorial, it is considered that the reader already has a basic
knowledge on MC simulations applied to classical problems in photons
migration, typical e.g. for biomedical optics. Very simplified numerical
examples will be given, allowing to intuitively understand the influence of a
correct/incorrect approach.
To simplify the explanations, even if ART theory clearly include the classical
case, in the following sections we will sometimes adopt the convention that
the expression ART does not include the classical case. The context easily
clarifies this use.
## 2 Basic MC tools in ART
### 2.1 Basic probabilty density functions for photons steps
ART theory is a generalization of the classical radiative transfer theory. We
will recall here how in ART photons steps are generated and, in particular,
how the interactions with the boundaries and light sources are treated [5, 6].
In ART, photons steps, $s_{c}$, occurring inside the medium are generated by
giving a pdf
$p_{c}(s_{c};a,b,\dots),$ (2)
defining the probability $p_{c}(s_{c};a,b,\dots)ds_{c}$ to reach a distance
$s_{c}\in[s_{c}-\frac{ds_{c}}{2},s_{c}+\frac{ds_{c}}{2}]$ in the investigated
anomalous medium. Thus, the generalization of the classical radiative tranfer
theory consists in the fact that $p_{c}(s_{c};a,b,\dots)$ must not necessarily
have an exponential behavior [Eq. (1)]. In other words, ART can treat photons
propagation in media that are not memory-less. The parameters $a,b,\dots$
represent the constants defining a specific pdf, depending on the optical
characteristics of the medium. The index $c$ is to remember that the positions
of the sites where the scattering or absorbing events occur in the medium are
statistically correlated. In fact, these positions can statistically be
described by the use of a probability distribution function
$v_{k}(s_{c};a,b,\dots)$ which, in principle, can directly be derived from
$p_{c}(s_{c};a,b,\dots)$ (see Sec. 2.2.2). The specific choice of the model
for $p_{c}(s_{c};a,b,\dots)$ comes from the actual optical properties of the
physical system (medium) we want to describe. Thus, $p_{c}(s_{c};a,b,\dots)$
can in principle be assessed theoretically or experimentally.
Until this point, the ART approach appears to be similar to the classical,
i.e., once the suitable $p_{c}(s_{c};a,b,\dots)$ is defined, we can generate
the photons step lengths, and decide if the photons are scattered, absorbed,
reflected or refracted at the different interaction sites. However, the
fundamental difference clearly appears when a photon hits a boundary. In this
case, if the boundary is encountered when traveling along the step $s_{c}$,
then in ART the photon simply stops at the boundary. From this reached
position a new step is immediately generated, without the need to terminate
the previous path by applying the well known procedure usually utilized in the
classical case (see e.g. Ref. [7]).
When we generate a new step $s_{u}$ starting from a fixed boundary or a fixed
light source position (i.e. these positions do obviously not result from the
probability law $v_{k}(s_{c};a,b,\dots)$), the pdf can no more be $p_{c}(.)$.
In this case, a different pdf must be considered, i.e. [5],
$p_{u}(s_{u};a,b,\dots)=\frac{1-\int_{0}^{s_{u}}p_{c}(s_{c};a,b,\dots)ds_{c}}{\int_{0}^{+\infty}s_{c}p_{c}(s_{c};a,b,\dots)ds_{c}},$
(3)
where the index $u$ is to remember that the photon starts from an uncorrelated
origin (i.e. the position is fixed). Thus, $p_{u}(s_{u};a,b,\dots)ds_{c}$
represents the probability to reach a distance
$s_{u}\in[s_{u}-\frac{ds_{u}}{2},s_{u}+\frac{ds_{u}}{2}]$, if starting from a
fixed position (boundary or light source). All remaining steps are generated
with the law $p_{c}(s_{c};a,b,\dots)$.
As we mentioned in Sec. 1, Eq. (3) derives from the fact that, similarly to
the classical case, ART must also satisfy the fundamental RL. In other words,
if we do not take into account Eq. (3), e.g. by simply putting
$p_{u}(.)=p_{s}(.)$, the RL is no more satisfied (except for the classical
case, see below) and photons propagation in the medium will be described in a
wrong manner. Note that, Eq. (3) permits the RL to be satisfied for any choice
of optical parameters (scattering coefficients, phase functions, refractive
indexes, etc.) or geometries.
The random steps, $s_{c}$ and $s_{u}$, based on laws $p_{c}(s_{c};a,b,\dots)$
and $p_{u}(s_{u};a,b,\dots)$, can numerically be generated as usual by
analytically solving
$\xi=\int_{0}^{s_{i}}p_{i}(s_{i}^{\prime};a,b,\dots)ds_{i}^{\prime};\quad
i\in\\{c,u\\},$ (4)
as a function of $s_{i}$, where $\xi\in(0,1)$ is a uniformly distributed
random variable. If an analytical solution of Eq. (4) does not exist, a
numerical solution can be found, and a look-up table relating $\xi$ to $s_{i}$
can be derived.
This is all we need to know to generate an ART MC simulation, the remaining
part of the MC code being the same as in the classical case.
### 2.2 Extracting general medium properties from $p_{c}(s_{c};a,b,\dots)$
Once $p_{c}(s_{c};a,b,\dots)$, the geometry, the optical parameters and the
phase functions are given, it becomes possible to perform MC simulations in
the ART domain. However, before to run a specific simulation, it is maybe
interesting to know that from $p_{c}(s_{c};a,b,\dots)$ we can derive some very
general properties of the investigated anomalous medium. Strictly speaking,
the properties presented here hold for an infinite medium with pdf
$p_{c}(s_{c};a,b,\dots)$, but are of fundamental importance in understanding
the physics described by $p_{c}(s_{c};a,b,\dots)$. Thus, for the sake of
completeness, in the following paragraphs we will report four functions,
directly derived from $p_{c}(s_{c};a,b,\dots)$, describing some of these
properties. The reported equations are derived from well known results in
renewal theory and the interested reader can refer e.g. to Refs. [8, 9]. The
equations hold for a photon propagation through any homogeneous part of the
medium.
1) The first interesting function is the survival probability
$P_{c}(s_{c};a,b,\dots)$, i.e., in the language of optics, the probability
that a photon survive after a path of length $s_{c}$. This probability is
expressed as
$P_{c}(s_{c};a,b,\dots)=\int_{s_{c}}^{+\infty}p_{c}(s_{c}^{\prime};a,b,\dots)ds_{c}^{\prime}.$
(5)
In the classical case, $P_{c}(s_{c};a,b,\dots)$ corresponds to the
Beer–Lambert–Bouguer’s law (within a multiplicative factor).
2) Now, let be
$\tilde{p}_{c}(w_{c};a,b,\dots)=\mathcal{L}\\{p_{c}(s_{c};a,b,\dots);w_{c}\\},$
(6)
where $\mathcal{L}\\{.;.\\}$ represents the Laplace transform and $w_{c}$ the
obtained variable in the transformed domain. Then, we can express the average
number of photon interactions along a path of length $s_{c}$ as
$\langle
m(s_{c};a,b,\dots)\rangle=\mathcal{L}^{-1}\left\\{\frac{\tilde{p}_{c}(w_{c};a,b,\dots)}{w_{c}[1-\tilde{p}_{c}(w_{c};a,b,\dots)]};s_{c}\right\\};$
(7)
3) The probability that $k$ photon interactions occur along a path of length
$s_{c}$ is
$v_{k}(s_{c};a,b,\dots)=\mathcal{L}^{-1}\left\\{\frac{1-\tilde{p}_{c}(w_{c};a,b,\dots)}{w_{c}}\right.\\\
\left.\times[\tilde{p}_{c}(w_{c};a,b,\dots)]^{k};s_{c}\right\\};$ (8)
4) The probability that the sum of the first $k$ photon steps does not exceed
$s_{c}$ is
$F_{k}(s_{c};a,b,\dots)=\mathcal{L}^{-1}\left\\{\frac{[\tilde{p}_{c}(w_{c};a,b,\dots)]^{k}}{w_{c}};s_{c}\right\\}.$
(9)
The above equations allow us to better describe the physical system under
study, determined by a given choice of $p_{c}(s_{c};a,b,\dots)$.
## 3 ART MC simulations: explanatory examples
The scope of the following examples is to show the importance of Eq. (3) (the
core of the ART MC simulations). This will be done by demonstrating, with easy
to understand numerical examples, that if we neglect Eq. (3), a fundamental
physical property of the system cannot be reproduced, i.e., the invariance
property (IP) will not be satisfied (see below). Note that, the IP represents
a very powerful test allowing us to check the reliability of any MC code [10].
### 3.1 Two general analytical models
We will define here two possible models for $p_{c}(s_{c};a,b,\dots)$, i.e.,
the power law and the constant step models.
#### 3.1.1 Power law
The first model we will consider is called a “power law” and is expressed as
$p_{c}(s_{c};\ell,a)=\frac{a(a+1)\ell(a\ell)^{a}}{(a\ell+s_{c})^{a+2}};\quad
a>0,$ (10)
where $\int_{0}^{\infty}p_{c}(s_{c};\ell,a)ds_{c}=1$ and
$\int_{0}^{\infty}s_{c}p_{c}(s_{c};\ell,a)ds_{c}=\ell$. The asymptotic limit
($s_{c}\rightarrow+\infty$) of Eq. (10) is
$p_{c}(s_{c};\ell,a)\sim\frac{a(a+1)\ell(a\ell)^{a}}{s_{c}^{a+2}};\quad
s_{c}\gg 1.$ (11)
The specific $s_{c}$ dependence of Eq. (11) is particularly interesting
because it implies that it may exist an asymptotic diffusion limit of this
anomalous model [11] (related to a “classical” or a “fractional” diffusion
equation, depending on the choice of $a$; see below). Thus, this simple model
[Eq. (10)] represents a kind unified summary for the classical and anomalous
approach.
Then, the pdf $p_{u}(s_{u},\ell,a)$ can be derived by applying Eq. (3), to
obtain
$p_{u}(s_{u},\ell,a)=\frac{1}{\ell}\left(\frac{a\ell}{a\ell+s_{u}}\right)^{a+1}.$
(12)
Finally, the algorithms allowing to generate random $s_{c}$ and $s_{u}$ values
in the MC codes can be obtained by introducing Eqs. (10) and (12) into Eq.
(4), and by solving as a function of $s_{c}$ and $s_{u}$, respectively. This
gives
$s_{c}=a\ell\left[(1-\xi)^{-\frac{1}{a+1}}-1\right],$ (13)
and
$s_{u}=a\ell\left[(1-\xi)^{-\frac{1}{a}}-1\right].$ (14)
#### 3.1.2 Constant step
The second model we will consider in this tutorial is called “constant step”,
because it describes photons that “jump” always with steps of the same length
$\ell$ , i.e.,
$p_{c}(s_{c};\ell)=\delta(s_{c}-\ell),$ (15)
where $\int_{0}^{\infty}p_{c}(s_{c};\ell)ds_{c}=1$ and
$\int_{0}^{\infty}s_{c}p_{c}(s_{c};\ell)ds_{c}=\ell$.
Equation (3) is then written as
$p_{u}(s_{u};\ell)=\frac{1-\Theta(s_{u}-\ell)}{\ell},$ (16)
where $\Theta(.)$ is the Heaviside function.
The generating functions for $s_{c}$ and $s_{u}$ are then expressed as [Eq.
(4)]
$s_{c}=\ell;\quad\forall\xi,$ (17) $s_{u}=\ell\xi.$ (18)
### 3.2 Numerical MC examples: IP test
Let’s now define four different examples of MC simulations based on Eqs. (10)
and (15). In the following sub-sections we will give the geometry of the
problem, all the necessary optical quantities, together with the light sources
and detectors positions. The general properties (geometry independent) of the
four chosen tutorial examples are treated, and a short summary concerning the
IP is also reported.
#### 3.2.1 Monte Carlo simulations
Geometry: The common geometrical model chosen in this tutorial consists of
two spheres centered at the axis origin, with radious $r_{in}<r_{out}$ and
matched refractive indexes at all the boundaries (Fig. 1).
Figure 1: Schematic of the geometrical model utilized for the Monte Carlo
simulations, representing two 3D spheres centered at the axis origin.
The mean free paths inside the spheres are $l_{in}$ and $l_{out}$. The
absorption coefficients and the anisotropy parameters of the spheres are set
to zero (mandatory conditions to test the IP in ART; see Sec. 3.3.2.6).
Forward measurements: An isotropic and uniform radiance illumination
impinging on the surface of the external sphere (continuous uniform
distribution of Lambertian sources) is utilized as a light source. Note that,
in Fig. 1 only one representative photon is reported (black path; forward
propagation). Photons are detected on the whole surface of the external
sphere. The pathlength of each single photon, $s$, is computed and the mean
pathlength $\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC}}$ is assessed. Each
simulation consists of 100 independent simulations (each of them has $10^{7}$
photons trajectories) in order to evaluate the standard error of the mean.
Isotropic scattering is used. Random photons steps for the three examples of
power law are generated using Eqs. (13) and (14), for three representative $a$
values, i.e., $a\in\\{1,1/2,+\infty\\}$ with
$\ell\in\\{\ell_{in},\ell_{out}\\}$ (the choice of $\ell_{in}$ or $\ell_{out}$
depends on where the photon is situated along the path). For the constant step
example, photons steps are obtained using Eqs. (17) and (18) with
$\ell\in\\{\ell_{in},\ell_{out}\\}$.
Adjoint measurements: Adjoint measurements are assessed by inverting the
direction of each photon reaching the sphere surface (detector). Then, these
photons, with their new inverted directions (red-dashed path in Fig. 1), are
utilized as light source to generate the adjoint measurements, with the same
conditions utilized for the forward measurements. Even in this case, photons
are detected on the whole surface of the external sphere. At the end of the
simulation, the mean pathlength, $\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC}}$, is then computed.
The importance of $p_{u}(s_{u},\ell,a)$: Forward and adjoint MC simulations
are repeated two times: 1) By using the rules for the MC ART presented in Sec.
2.2.1, giving $\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC}}$ and $\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC}}$ and; 2) By setting $p_{u}(.)=p_{c}(.)$,
as it is usually done in classical MC, resulting in this case in two mean
paths that we will note $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC}_{\textrm{no-reciprocity}}}$ and $\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC}_{\textrm{no-reciprocity}}}$.
#### 3.2.2 Power law: $a=1$
In this case random steps $s_{c}$ and $s_{u}$ are generated as [Eqs. (13) and
(14)]
$s_{c}=\ell\left[(1-\xi)^{-\frac{1}{2}}-1\right],$ (19)
$s_{u}=\frac{\ell\xi}{1-\xi}.$ (20)
To better understand the physical properties of this model, we will compare it
with the classical one (see below), through the functions presented in Sec.
2.2.2. In this case, Eq. (5) is expressed as
$P_{c}(s_{c};\ell,a=1)=\left(\frac{\ell}{\ell+s_{c}}\right)^{2}.$ (21)
To obtain the remaining Eqs. (7), (8) and (9) let first compute Eq. (6), i.e.,
$\tilde{p}_{c}(w_{c};\ell,a=1)=2(w_{c}\ell)^{2}\Gamma(-2,w_{c}\ell)e^{w_{c}\ell}.$
(22)
where $\Gamma(.,.)$ is the incomplete gamma function. Then, insert Eq. (22) in
Eqs. (7), (8) and (9). Being the obtained expressions too complex to be
derived analytically, numerical solutions can be assessed by applying the
Talbolt’s inverse Laplace transform [12] (see Fig. 2).
Note that, the asymptotic limit [Eq. (11)] of this power law with $a=1$ can be
described by a classical diffusion equation [11].
#### 3.2.3 Power law: $a=1/2$
The random steps $s_{c}$ and $s_{u}$ for $a=1/2$ are generated as [Eqs. (13)
and (14)]
$s_{c}=\frac{\ell}{2}\left[(1-\xi)^{-\frac{2}{3}}-1\right],$ (23)
$s_{u}=\frac{\ell}{2}\left[(1-\xi)^{-2}-1\right].$ (24)
This is an interesting case, because the asymptotic limit of this model is
related to a fractional diffusion equation with derivative of order $3/2$
[11]. In this case [Eq. (5)],
$P_{c}(s_{c};\ell,a=1/2)=\left(\frac{\ell}{\ell+2s_{c}}\right)^{\frac{3}{2}}$
(25)
and Eq. (6) can be expressed as
$\tilde{p}_{c}(w_{c};\ell,a=1/2)=\frac{\sqrt{2}}{2}(\ell
w_{c})^{\frac{3}{2}}\Gamma(\frac{1}{2},\frac{\ell w_{c}}{2})e^{\frac{\ell
w_{c}}{2}}-\ell w_{c}+1$ (26)
Then, Eqs. (7), (8) and (9) can be obtained numerically with the same
procedure presented in Sec. 3.3.2.2 (see Fig. 2).
#### 3.2.4 Power law: $a=+\infty$
The random step $s_{c}$ and $s_{u}$ are generated as
$s_{c}=s_{u}=-\ell\ln(1-\xi).$ (27)
In this case, $s_{c}=s_{u}=s$, because the correlated and uncorrelated pdfs
are the same, i.e.,
$p_{c}(s;\ell,a)=p_{u}(s;\ell,a)=\frac{1}{\ell}e^{-s/\ell}.$ (28)
We immediately recognize here the pdf of the classical model, related to the
underlying Beer–Lambert–Bouguer’s law. In fact [Eq. (5)],
$P_{c}(s_{c};\ell,a=+\infty)=e^{-s_{c}/\ell}$ (29)
It is in this sense, i.e., when $a=+\infty$, that the classical model is
naturally included in the tutorial ART power model of Sec. 3.3.1.
In this special case, by deriving before Eq. (6), i.e.,
$\tilde{p}_{c}(w_{c};\ell,a=+\infty)=\frac{1}{\ell w_{c}+1},$ (30)
Eqs. (7), (8) and (9) can be completely obtained analytically, i.e.,
$\langle m(s_{c};\ell,a=+\infty)\rangle=s_{c}/\ell$ (31)
$v_{k}(s_{c};\ell,a=+\infty)=\frac{(s_{c}/\ell)^{k}}{k!}e^{-s_{c}/\ell}$ (32)
$F_{k}(s_{c};\ell,a=+\infty)=\sum_{n=k}^{+\infty}\frac{(s_{c}/\ell)^{n}}{n!}e^{-s_{c}/\ell}$
(33)
#### 3.2.5 Constant step
By following the same procedure utilized for the power law, we obtain for the
constant step model
$P_{c}(s_{c};\ell)=1-\Theta(s_{c}-\ell),$ (34)
$\tilde{p}_{c}(w_{c};\ell)=e^{-w_{c}\ell},$ (35) $\langle
m(s_{c};\ell)\rangle=\left\lfloor\frac{s_{c}}{\ell}\right\rfloor,$ (36)
where $\left\lfloor.\right\rfloor$ is the floor function and,
$v_{k}(s_{c};\ell)=\Theta(s_{c}-k\ell)-\Theta(s_{c}-(k+1)\ell),$ (37)
$F_{k}(s_{c};\ell)=\Theta(s_{c}-k\ell).$ (38)
#### 3.2.6 Invariance property (IP)
As it was mentioned in Sec. 2.2.1, the introduction of Eq. (3) allows us to
suitably taking into account the RL in the MC ART simulations. Neglecting this
rule, the RL may not be satified. In this context, the examples chosen for
this tutorial represent a special instructive case. In fact, it is easy to see
that, due to the absence of absorption and the particular geometry of the
light source and detector, the RL will always be satisfied, independently of
the remaining optical parameters or other functions utilized for the MC
simulation. The reason of this special behavior is due to the fact that all
the lunched photons always reach the detector (i.e., same photon flux through
the detector in forward or adjoint direction). This means that testing the RL
is not always a good test to check a MC code, and that a more universal method
is needed. This is why the IP is introduced in the following paragraph and is
then exploited to show the importance of the right choice of $p_{u}(.)$ [Eq.
(3)].
The IP [13, 14] is a very powerful property that in the case of anomalous
transport states the following: let be a uniform and isotropic radiance
illumination applied at the external boundary, $S$, of a finite, scattering
and non-absorbing medium of any shape, volume V, matched refractive indexes
and isotropic phase function. Then, if we apply the rules reported in Sec.
2.2.1 (to suitably describe ART), the mean pathlength, $\langle
s\rangle_{\textrm{theory}}$, spent inside V by photons outgoing from $S$, is
an invariant quantity independent of the scattering strength. Note that the
volume $V$ can also be the union of any number of sub-volumes with different
scattering coefficients but matched refractive indexes (external medium also
matched). The quantity $\langle s\rangle_{\textrm{theory}}$ is expressed as
$\langle s\rangle_{\textrm{theory}}=4\frac{V}{S}.$ (39)
This property holds for anomalous and classical photon propagation and can be
utilized to test the validity of the MC codes. It is worth to note that in the
case of classical photon transport (Beer–Lambert–Bouguer’s law), the
hypothesis of an isotropic phase function is not necessary.[13, 14, 15] A
generalization of Eq. (39) for unmatched refractive indexes also exists [16,
15], but this goes far from the aim of the present tutorial. However, in this
context we need a general formulation valid for ART, implying the use of an
isotropic phase function.
## 4 Results
### 4.1 General comparisons of the classical and the ART models
Before to proceed with the MC simulations, let us compare (semi)-analytically
the power law and the constant step models with the classical one [Eq. (28)].
In Fig. 2
Figure 2: Probabilistic functions (see Sec. 2.2.2) characterizing the models
chosen as a tutorial cases (see Secs. 3.3.2.2, 3.3.2.3, 3.3.2.4) and 3.3.2.5).
the power law [Eqs. (21) and (25) ], for $a=1$ and $a=1/2$, and the constant
step [Eq. (34)] models, are compared with the special case $a=+\infty$ [Eq.
(29)], related to the classical Beer–Lambert–Bouguer’s law. For simplicity we
have chosen the case $\ell=1$, but the general considerations remain valid for
other $\ell$ values.
In Fig. 2a the classical model for $P_{c}(.)$ appears as a straight line
(logarithmic scale). It is known from renewal theory that this model is
representative of a Markovian or memoryless process. This means that a future
photon step length only depends on the present state and not on the past
history of the photon propagation. The remaining models appearing in Fig. 2a,
that deviate from this behavior, are not memoryless. This explains in part why
it may be more complex to express these models through an
(integro)-differential equation formalism. In general, $P_{c}(.)$ may be
assessed experimentally by the measurement of the so called ballistic photons,
while the remaining functions in Fig. 2 may only be indirectly derived by
means of a mathematical procedure. Figure 2a clearly shows that the power low
curves for ART go slower to zero compared to the classical case. This means
that there is a higher probability for a photon to suddenly undergo very long
steps. This may be e.g. the case of photons that pass through a non-
homogeneous distribution of clouds.
Figure 2b represents the average number of scattering events (or steps),
$\langle m_{c}(.)\rangle$, along a path of length $s_{c}$. The figure
highlights the fact that even if in general the average step is always $\ell$
for the power law [Eq. (10)] and the constant step [Eq. (15)] models, for the
specific ART condition it is not possible to simply divide the path length
$s_{c}$ by $\ell$ to find the $\langle m_{c}(.)\rangle$ value, as it can be
done in the classical case [Eq. (31)] (black color; straight line). In fact,
Fig. 2b shows that ART curves are not linear and thus a specific calculation
must be performed [Eq. (7)]. This particular behavior is generated by the
“memory” of the ART models. Function $\langle m_{c}(.)\rangle$ is extremely
important because renewal theory tells us that it uniquely determines the
process generating the photons’ steps.
Figure 2c reports the probability $v_{k}(.,.)$ that k scattering events occur
on a path of length $s_{c}=5$ mm. As expected, in the classical case
$v_{k}(.,.)$ is a Poisson distribution [Eq. (32)], due to the exponentially
distributed photons steps. The function $v_{k}(.,.)$ can be utilized to
reproduce the statistical spatial distribution of the scattering sites inside
the medium, and gives us an idea on how the medium is structured.
Figure 2d reports the probability $F_{k}(.)$ that the sum of the first $k$
photon steps does not exceed $s_{c}=5$ mm. It can be seen that the classical
case has smaller values than the ART power law (apart for the cases $k=1,2$).
The exceptions for $k=1,2$ can be intuitively explained by looking e.g. Fig.
2a from which we can deduce e.g. that the probability to have a long step
larger than 5 mm is higher for the classical case.
Thus, Fig. 2 gives us the main features of a given ART model compared to the
classical one. The above (semi)-analytical approach may be sometimes useful
for the interpretation of the MC simulations.
### 4.2 MC results
The results of the MC simulations, for the three cases and the geometrical
model appearing in Fig. 1, are summarized in Table 1.
Table 1: Explanatory tests of the IP for the ART theory. Parameters of the simulations: $r_{out}=5$ mm, $r_{in}=2$ mm, $\ell_{out}=1$ mm, $\ell_{in}=3$ mm, $\langle s\rangle_{\textrm{theory}}=6.\bar{6}$ mm (for the remaining parameters see text). PL: power law; CS: constant step; BLB: Beer–Lambert–Bouguer’s. Values in parenthesis are the standard error of the mean. pdf model | $\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC}}$ | $\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC}_{\textrm{no-reciprocity}}}$ | $\langle s\rangle^{\textrm{Adj}}_{\textrm{MC}}$ | $\langle s\rangle^{\textrm{Adj}}_{\textrm{MC}_{\textrm{no-reciprocity}}}$
---|---|---|---|---
PL $(a=1)$ | $6.6666\;(0.0004)$ | $5.4777\;(0.0002)$ | $6.6668\;(0.0002)$ | $5.4242\;(0.0002)$
PL $(a=1/2)$ | $6.6667\;(0.0002)$ | $4.9731\;(0.0002)$ | $6.6667\;(0.0002)$ | $4.9030\ (0.0002)$
CS | $6.6666\;(0.0004)$ | $9.3866\;(0.0004)$ | $6.6668\;(0.0004)$ | $9.6856\ (0.0004)$
BLB $(a=+\infty)$ | $6.6668\;(0.0003)$ | $6.6664\;(0.0002)$ | $6.6666\;(0.0003)$ | $6.6668\;(0.0003)$
The theoretical mean pathlength that we must reproduce with the MC simulations
is $\langle s\rangle_{\textrm{theory}}=\frac{4}{3}r_{out}$ mm [Eq. (39)].
From Table 1 it clearly appears that, $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC}}$ perfectly reproduces the theoretical
$\langle s\rangle_{\textrm{theory}}$ IP-value (within the statistical error)
for all cases, i.e., the power law, the constant step and the classical model.
Moreover, $\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC}}=\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC}}$ (within the statistical error).
However, if the condition for the RL is not satisfied, i.e., if we neglect the
special behavior for uncorrelated scattering events and set $p_{u}(.)$ equal
to $p_{s}(.)$, then the MC simulations generates results in disagreement with
the laws of optics, i.e., $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}\neq\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}$, $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}\neq\langle
s\rangle_{\textrm{theory}}$ and $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}\neq\langle
s\rangle_{\textrm{theory}}$, when $a\in\\{1,1/2\\}$. The only exception being
the classical case ($a=+\infty$), where all results remain valid (last line of
Table 1). This is obviously due to the fact that in this case
$p_{u}(.)=p_{s}(.)$ [Eq. (28)].
Considering the geometry of the problem, the inequality $\langle
s\rangle^{\textrm{Fwd}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}\neq\langle
s\rangle^{\textrm{Adj}}_{\textrm{MC${}_{\textrm{no-reciprocity}}$}}$, for the
ART, immediately tells us that the light reaching the detector in the forward
direction do not represent an isotropic and uniformly distributed radiance. In
other words, one of the conditions for the IP is not satisfied for the adjoint
case. As expected, in the classical case this problem does not subsist, i.e.,
$\langle s\rangle^{\textrm{Fwd}}_{\textrm{MC${}_{\textrm{no-
reciprocity}}$}}=\langle s\rangle^{\textrm{Adj}}_{\textrm{MC${}_{\textrm{no-
reciprocity}}$}}$.
In Table 1, the data are exact up to the third decimal but, obviously, if we
increase the number of photons trajectories per simulation, we can also
increase the precision of the results at any desired level.
## 5 Discussion and conclusions
In Sec. 3 and 4 we have seen, through some tutorial example, the importance to
distinguish between correlated and uncorrelated scattering events, through the
pdfs $p_{c}(.)$ and $p_{u}(.)$. In fact, this approach not only satisfies (by
construction of the theory) the RL for photon propagation, but it is also
mandatory if one want to satisfy the universal IP.
In Table 1 we have also seen that there is an exception for the case
$a=+\infty$, corresponding to the classical case where photons propagate under
the Beer–Lambert–Bouguer’s law. In fact, in this special case (the only one)
$p_{u}(.)=p_{s}(.)$, and as a consequence the results remain exact in all
cases (Table 1; last line).
In practice, the ART theory is a generalization of the classical one, the
latter appearing as a particular case. Thus, the ART MC method can be applied
to solve classical problems in photon transport, e.g., in biomedical optics.
This may greatly simplify the writing of MC codes (the photon step just stops
at the boundary, before the next step), because it does not necessitate to
treat the photons reaching the boundaries with more complex algorithms as it
is classically done [7].
But then, do we have two different algorithms to treat the photons
interactions with the boundaries in classical MC ? Which is the right one?
Actually, the classical and the ART based MC methods give the same results.
Intuitively, it is probably hard to see that the two approaches are
equivalent. For this reason, another simplified example might maybe help the
reader to have a more concrete idea. Let be a medium with only one internal
boundary and mean free paths $\ell_{1}$ and $\ell_{2}$. Let generate a photon
step, $s=s_{1}+s_{2}$, as in Fig. 3.
Figure 3: Schematic of a photon that crosses a boundary separating two regions
with different mean free path.
It is well known that in classical MC the random step $s$ is obtained as
$s=\begin{cases}-\ell_{1}\ln\xi;&\mbox{if }s\leq s_{1}\\\
\frac{\ell_{2}}{\ell_{1}}(-\ell_{1}\ln\xi-s_{1})+s_{1};&\mbox{if }s\mbox{
computed above}>s_{1},\end{cases}$ (40)
where $\xi\in(0,1)$ is a uniform distributed random variable.
In the same way, as it was explained in this tutorial, in ART the MC random
step $s$ is generated as
$s=\begin{cases}-\ell_{1}\ln\xi_{1};&\mbox{if }s\leq s_{1}\\\
-\ell_{2}\ln\xi_{2}+s_{1};&\mbox{if }s\mbox{ computed
above}>s_{1},\end{cases}$ (41)
where $\xi_{1},\xi_{2}\in(0,1)$ are independent uniform distributed random
variables. It easy to see (e.g. numerically) that Eqs. (40) and (41) generate
random $s$ with the same pdf. This equivalence is valid in general for any
choice of complex geometries, number of boundaries, optical properties, etc.
This is the reason why, in the case of classical simulations where Eq. (1)
holds, ART and the classical MC approaches are equivalent.
At this point, without entering to much in too technical details, it is maybe
worth making a general consideration concerning the RL and the IP. In fact,
historically, the pdf $p_{u}(.)$ has been derived independently for the RL and
for the IP theory. Each theory having its own independent (but compatible)
physical assumptions allows us to assess $p_{u}(.)$. In this didactical
contribution, we have chosen a point of view that is not historical but that
should better highlight the relationship existing between RL and IP. In fact,
the RL is a more “general” law than the IP. This, because RL holds e.g. also
for any light source, detector configuration, optical properties values, or
phase function choice. Thus, the pdf for a step starting from a fixed position
must be $p_{u}(.)$, if we want RL to be satisfied.
On the other hand, historically, the pdf $p_{u}(.)$ necessary to satisfy the
IP has also been derived independently [14] (the obtained $p_{u}(.)$ equals
the one satisfying the RL), but by using different hypothesis (actually, a
subset of conditions of the RL). We realize here that this derivation for the
IP is not strictly necessary, because $p_{u}(.)$ is already given by the RL.
In practice, it is possible to demonstrate the IP by considering $p_{u}(.)$
already known. We invite the interested reader to revisit the original papers
on IP (e.g. Ref. [14]) with this point of view. This might help to have a more
coherent vision on the strong relationship existing between RL and IP.
The few examples presented in this tutorial had the aim to show how to
generate MC simulations for any $p_{c}(.)$ describing a desired medium.
However, we must be aware of the fact that to describe the exact physics of
the investigated medium, the first moment of $p_{c}(.)$ must be finite [Eq.
(3)]. From a practical point of view this constraint on $p_{c}(.)$ is not a
real problem, because when simulating media representing actual physical
systems, it is unlikely that photons goes to infinity without any interaction.
This keeps the average step length finite.
In conclusion, it may be worth to note that an MC code compatible with the ART
theory may also be of a more general interest, than a simple technicality, for
tissue optics. In fact, it has been shown that some biological tissues, such
as bone and lung, have a particular fractal-like structure [17, 18, 19]. As
reported in Ref. [20] by Davis and Mineev-Weinstein, (multi-)fractal
structures may produce anomalous photon propagation. This observation may open
a new domain of exploration in biomedical optics, and adapted MC codes may
represent one of the important tools to address this interesting topic.
## References
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|
# The creation of a massive UCD by tidal threshing from NGC 936
Sanjaya Paudel,1,2 Pierre-Alain Duc,3 Sungsoon Lim,1 Mélina Poulain,4 Francine
R. Marleau,5 Oliver Müller,6 Rubén Sánchez-Janssen,7 Rebecca Habas,3,8 Patrick
R. Durrell,9 Nick Heesters,6Daya Nidhi Chhatkuli,10 Suk-Jin Yoon,1,2
1Department of Astronomy, Yonsei University, Seoul, 03722, Republic of Korea
2Center for Galaxy Evolution Research, Yonsei University, Seoul, 03722,
Republic of Korea
3Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR
7550, F-67000 Strasbourg, France
4Space Physics and Astronomy Research Unit, University of Oulu, P.O. Box 3000,
FI-90014, Oulu, Finland
5Institut für Astro- und Teilchenphysik, Universität Innsbruck,
Technikerstraße 25/8, Innsbruck, A-6020, Austria
6Institute of Physics, Laboratory of Astrophysics, École Polytechnique
Fédérale de Lausanne (EPFL), 1290 Sauverny, Switzerland
7UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh,
EH9 3HJ, UK
8INAF Osservatorio Astronomico di Teramo, via Maggini, I-64100, Teramo, Italy
9Youngstown State University, One University Plaza, Youngstown, OH 44555 USA
10Central Department of Physics, Tribhuvan University, Kirtipur 44618,
Kathmandu, Nepal E-mail<EMAIL_ADDRESS>
(Accepted 2023 August 30. Received 2023 August 28; in original form 2023
August 03.)
###### Abstract
We study a compact nucleus embedded in an early-type dwarf galaxy, MATLAS-167,
which is in the process of disruption by the tidal force of the neighboring
giant S0 galaxy, NGC 936, in a group environment. Using the imaging data of
the MATLAS survey, we analyze the stellar tidal tail of MATLAS-167 and its
central compact nucleus, designated as NGC 936_UCD. We find that NGC 936_UCD
has a luminosity of Mg = $-$11.43$\pm$0.01 mag and a size of 66.5$\pm$17 pc,
sharing the global properties of Ultra Compact Dwarf galaxies (UCDs) but
significantly larger and brighter compared to the typical UCD populations
observed in the Virgo cluster. By integrating the total luminosity of both the
tidal stream and MATLAS-167, we estimate that the disrupted dwarf progenitor
possesses a luminosity of Mg = $-$15.92$\pm$0.06 mag, a typical bright dE
luminosity. With the help of the optical spectrum observed by the SDSS survey,
we derive the simple stellar population properties of NGC 936_UCD: a light-
weighted age of 5.6$\pm$0.7 Gyr and metallicity of [Z/H] = $-$0.83$\pm$0.3
dex. Our findings suggest that tidal threshing is a possible formation
mechanism of bright UCD populations in close proximity to giant galaxies.
###### keywords:
galaxies: dwarf — galaxies: evolution — galaxies: groups: general — galaxies:
interactions — galaxies: nuclei
††pubyear: 2022††pagerange: The creation of a massive UCD by tidal threshing
from NGC 936–References
## 1 INTRODUCTION
Ultra-compact dwarf galaxies (UCDs) bridge the gap between galaxies and star
clusters in terms of mass, size, and luminosity, making it difficult to
clearly distinguish between the two classes of stellar systems (Hilker et al.,
1999; Drinkwater et al., 2000; Phillipps et al., 2001; Evstigneeva et al.,
2008; Norris et al., 2014). The question at the heart of this discussion is
whether UCDs are the largest star clusters or the smallest galaxies (Mieske et
al., 2002; Kissler-Patig et al., 2006). UCDs are larger, brighter, and more
massive than the typical globular clusters (GCs) with typical half-light radii
of 10 $\lesssim$ Rh $\lesssim$ 100 pc, and luminosities Li $\gtrsim$ 105 L☉
(Haşegan et al., 2005; Mieske et al., 2008; Misgeld & Hilker, 2011; Norris et
al., 2014; Voggel et al., 2016). Their stellar population is old ($\gtrsim$5
Gyr), with a wide range of metal content, mostly sub-solar (Firth et al.,
2009; Paudel et al., 2010; Chilingarian et al., 2011; Janz et al., 2016; Zhang
et al., 2018; Forbes et al., 2020; Fahrion et al., 2019). The central velocity
dispersions ($\sigma_{v}$) of UCDs are similar to dwarf galaxies, with a
typical value of 20 $\lesssim$ $\sigma_{v}$ $\lesssim$ 50 $km\,s^{-1}$. Their
dynamical mass estimates show that they have mass-to-light ratios, which are,
on average, about twice as large as those of GCs (Hilker et al., 2007;
Baumgardt & Mieske, 2008; Frank et al., 2011; Mieske et al., 2013; Janz et
al., 2015). Recent high spatial resolution spectroscopic observations show
that a fraction of UCDs also hosts a central intermediate-mass black hole
(Seth et al., 2014; Ahn et al., 2017, 2018; Afanasiev et al., 2018; Voggel et
al., 2019).
Figure 1: Comparison between the SDSS and the MATLAS image. The left panel
shows a tri-color image of NGC 936 from SDSS, created by combining $g$-, $r$-,
and $i$-band images. The right panel shows a deep $g$-band image from the
MATLAS, which clearly reveals the filament and low surface brightness plumes
around NGC 936. Both images have a field of view of
5.5$\arcmin$$\times$5.5$\arcmin$. The position of the disrupted dwarf,
MATLAS-167, is highlighted by a green circle in both images. While only the
star cluster is visible in the SDSS image, the underlying low surface
brightness host is revealed in the MATLAS image.
Since the discovery of UCDs, there has been a significant amount of research
focused on understanding their origins. It has become clear that UCDs are not
a uniform population and can be formed through a variety of different
processes (Hilker, 2011). Two main formation pathways are frequently discussed
in the literature (e.g., Fellhauer & Kroupa, 2002). The first involves tidal
disruption, with UCDs proposed as the remnant nuclei of tidally disrupted
galaxies (Drinkwater et al., 2003; Gregg et al., 2003; Goerdt et al., 2008;
Pfeffer & Baumgardt, 2013; Pfeffer et al., 2014). In this scenario, a
nucleated dwarf galaxy in a cluster or group environment may undergo complete
tidal disruption, leaving behind a naked dense stellar core (known as a
nuclear star cluster). The remnant dense nuclear star cluster is
gravitationally strong enough to retain its stars against tidal disruption
(Bekki et al., 2003). Evidence in support of the tidal disruption origin of
UCDs includes the presence of features such as tidal tails, extended haloes,
SMBH, and asymmetries around these objects (Voggel et al., 2016; Wittmann et
al., 2016; Schweizer et al., 2018; Evstigneeva et al., 2008; Liu et al.,
2020). Other nucleated dwarf galaxies undergoing disruption have been
discovered. They include the Sagittarius dwarf galaxy around the Milky Way, a
so-called dog-leg tidal stream around NGC 1407 (Galianni et al., 2010;
Amorisco et al., 2015) and extremely diffuse nucleated dwarf galaxies at the
Virgo cluster (Mihos et al., 2015).
The second scenario suggests that UCDs are the high-mass end of the GC mass
function (Kroupa, 1998; Fellhauer & Kroupa, 2002; Mieske et al., 2002; Brüns
et al., 2011) and bright UCDs might have formed through the merger of GCs
(Kissler-Patig et al., 2006). It is also argued that UCDs can be primordial
objects formed in an intense burst of star formation (Murray, 2009).
There is a wide range of properties among known UCDs, and they share
characteristics with both GCs and the nuclei of dwarf galaxies. This suggests
multiple formation processes contribute to their creation (Francis et al.,
2012). However, it is likely that stripped nuclei account for at least some
percentage of the UCD population due to various similarities to compact galaxy
nuclei (Drinkwater et al., 2003; Paudel et al., 2010). These include
overlapping luminosity distributions and similar size$-$luminosity
relationships (Evstigneeva et al., 2008), internal velocity dispersions
(Drinkwater et al., 2003), positions on the color$-$magnitude diagram, and
stellar population properties (Côté et al., 2006; Evstigneeva et al., 2008;
Paudel et al., 2010; Brodie et al., 2011; Chilingarian et al., 2011; Spengler
et al., 2017; Zhang et al., 2018).
In this work, we identify a star cluster located at the end of a tidal stream
that is likely to have originated from the disruption of an early-type dwarf
galaxy (dE), MATLAS-167. The star cluster is bright, $M_{g}$ =
$-$11.43$\pm$0.01 mag, and compact, likely a surviving nucleus of MATLAS-167
disrupted by the tidal force of the nearby giant galaxy NGC 936 located at
23.0 Mpc111The distance is measured using the surface brightness fluctuation
method by Tonry et al. (2001). away from us. We propose that the nuclear star
cluster is in the process of forming a UCD through tidal stripping.
## 2 Data and Analysis
The aim of the Mass Assembly of early Type gaLAxies with their fine Structures
(MATLAS) project is to conduct a comprehensive imaging survey of local
elliptical galaxies that were selected from the ATLAS3D legacy survey
(Cappellari et al., 2011; Duc et al., 2015). Its primary objective is
identifying and documenting low surface brightness features such as stellar
streams, filaments, and shells surrounding giant early-type and dwarf galaxies
(Bílek et al., 2020; Habas et al., 2020; Marleau et al., 2021). This project
has a magnitude limit of 29 mag arcsec2 for extended low surface brightness
objects. Through a thorough visual examination of all the galaxies in the
survey, we have identified a system of ongoing disruption of a dwarf galaxy
around the giant S0 galaxy, NGC 936. The disrupting dwarf galaxy is
MATLAS-167, and it is cataloged as a dE galaxy in the dwarf galaxy catalog,
which has a prominent bright nucleus at the center (Poulain et al., 2021).
In Figure 1, we compare the SDSS color image and the MATLAS $g$-band image. As
expected, the SDSS image does not reveal any stream, and only a compact source
is visible (see the green circle). On the other hand, the deeper MATLAS
$g$-band image displays a spectacular view of the tidal stream around NGC 936.
The compact star cluster is embedded in a stellar stream, which we have marked
by a green circle. We consider it a putative UCD (hereafter NGC 936_UCD). It
is located at the end of the stream, which forms an almost semi-circular
trajectory around NGC 936. The focus of this study is the nature of the
interaction between NGC 936 and MATLAS-167 and the evolution of NGC 936_UCD.
Figure 2: On-sky position of member galaxies in the NGC 936 group. Green
symbols represent giant galaxies, while dEs and star-forming dwarf galaxies
are represented by red and blue symbols, respectively. Black dots indicate
nucleated dEs. Additionally, the diagram includes two large symbols to
indicate the position of NGC 936 itself and its disrupted satellite,
MATLAS-167.
NGC 936 is a barred S0 galaxy classified as S0Bb in the RC3 catalog (de
Vaucouleurs et al., 1991). It is the most dominant galaxy in the group, which
includes three other massive galaxies. It has a face-on orientation with an
inclination of $<$10 degrees as shown in Figure 1, and it has a prominent
central bar. The MATLAS search for dwarf galaxies identified 27 dwarf galaxies
around NGC 936, and their distribution around NGC 936 is shown in Figure 2.
Only 7 out of 27 dwarf galaxies are star-forming dwarf galaxies (Habas et al.,
2020; Poulain et al., 2021). Among 20 dEs, the nucleated fraction is nearly
50%.
Table 1: Physical properties of NGC 936_UCD
Properties | Values | Unit | Note
---|---|---|---
R.A. | 02:27:32.88 | h:m:s | 1
Decl. | $-$01:13:49.31 | d:m:s | 2
$M_{g}$ | $-$11.43$\pm$0.01 | mag | 3
$z$ | 0.0039 | | 4
$g-r$ | 0.60$\pm$0.01 | mag | 5
Re | 66.5$\pm$17 | pc | 6
$M_{g}$ | $-$15.92$\pm$0.06 | mag | 7
$\Delta$d | 23 | kpc | 8
$\Delta$vr | 260 | $km\,s^{-1}$ | 9
1) R.A. of NGC 936_UCD
2) Decl. of NGC 936_UCD
3) The absolute $g$-band magnitude of NGC 936_UCD
4) Redshift measured from the SDSS spectrum of NGC 936_UCD
5) $g-r$ color of NGC 936_UCD
6) Effective radius of NGC 936_UCD
7) The $g$-band integrated magnitude of putative dwarf galaxies along the
streams
8) Sky-projected separation between NGC 936_UCD and NGC 936
9) Relative line-of-sight velocity between NGC 936_UCD and NGC 936
### 2.1 Imaging and Photometry
#### 2.1.1 The nucleus
In this work, we used Megacam CHFT images obtained by the MATLAS survey (Duc
et al., 2015; Bílek et al., 2020). The MATLAS survey consisted of $g$, $r$,
and $i$-band images, where the $g$-band is the deepest and $i$-band is the
best in image quality. We, therefore, used the $g$-band images for the
photometric measurement and surface photometry of the host galaxy. The
$i$-band, particularly, was used for size measurement of compact nucleus,
which can provide a better spatial resolution than others. All $g$, $r$, and
$i$-band images are observed in 0.19$\arcsec$ pixel-1 spatial resolution, and
the $i$-band has a median PSF of 0.89$\arcsec$ which corresponds to 99 pc at
the distance of NGC 936 (23 Mpc).
Figure 3: The $g$-band surface brightness profile of MATLAS-167 along its
major axis. The best-fit Sérsic function is shown by the red line. We also
show a 45$\arcsec$$\times$45$\arcsec$ $g$-band image of MATLAS-167 and the
residual after subtracting the best-fit model image in the inset. The vertical
dash line represents the size of NGC 936_UCD.
To accurately measure the flux of the nucleus, ensuring that the surrounding
galaxy light does not contaminate it, we employed a method that involves
subtracting the host galaxy light. To accomplish this, we utilized the IRAF
$ellipse$ task, which outputs an azimuthally averaged value along an elliptic
path with a function of galactocentric radii. Figure 3 depicts the $g$-band
light profile of MATLAS-167 along the major axis, with the black dots
representing the observed data points and the red line representing the best-
fitted Sérsic function. To avoid any interference from the central nucleus, we
excluded the inner (r $\leq$ 4$\arcsec$) data points during the fit.
The best-fitted parameters derived from the best-fitted Sérsic function are an
effective radius (Re) of 12.23$\arcsec$ and a Sérsic index (n) of 1.4. To
construct a two-dimensional representation of the observed galaxy, we
incorporated the one-dimensional best-fitted flux into the output of the IRAF
ellipse fit and employed the $bmodel$ task. Considering this best-fit Sérsic
model represents the bound component of MATLAS-167, it has a luminosity of
$M_{g}$ = $-$12.76 mag.
Subsequently, we performed aperture photometry of the compact nucleus in the
resulting model-subtracted residual images. For the measurement of the total
flux and its magnitude, we used an aperture that is roughly twice the size of
the Full Width at Half Maximum (FWHM). To determine the FWHM, we utilized
multiple bright, unsaturated stars in the field as references. To eliminate
background contributions, we selected an annulus with inner and outer radii of
twice and thrice the FWHM, respectively. Total brightness is $M_{g}$ =
$-$11.43$\pm$0.01 mag, and $g-r$ color is 0.6$\pm$0.01 mag.
Figure 4: The $g$-band surface brightness map of the field around NGC 936. The
unrelated foreground and background objects are masked out manually. The green
box in the left panel represents the zoom-in area shown in the left panel,
which is prepared after subtracting the model of NGC 936\. The green polygon
in the left panel delineates the aperture used to carry out the photometry.
Figure 5: The SDSS fiber spectrum (black), together with its best-fit SSP
model spectrum (red). The residuals are shown in the lower panel. The fit is
generally consistent within 5 percent of the observed flux (the horizontal
lines).
The size of NGC 936_UCD was determined by analyzing the galaxy-subtracted
$i$-band image, where it was partially resolved. To perform the measurement,
we utilized the publicly available software $ishape$, and explored both MOFFAT
and KING profiles (Larsen, 1999). The software convolves a model light profile
with a provided PSF and fits it to the source. The analysis resulted in a size
estimate of 0.56$\arcsec$ for the KING15 profile and 0.64$\arcsec$ for the
MOFFAT15 profile, both exhibiting a similar uncertainty of 0.16$\arcsec$. When
translated into physical units, these values correspond to sizes of 62 pc and
71 pc for the KING15 and MOFFAT15 profiles, respectively. The discrepancies in
residuals between the two models were not statistically significant.
Consequently, we opted to adopt the average of these two measurements,
yielding a final size estimate of 66.5$\pm$17 pc. This is relatively large for
a typical NSC of stellar mass $<$107 M☉ and NSC of the stellar mass of
$\approx$107 M☉ or $M_{g}$$\approx$$-$12 mag typically have effective radius
of $\approx$50 pc (Böker et al., 2004; Georgiev et al., 2016)
#### 2.1.2 Surface photometry of the tidal stream
A ring filter was utilized to remove foreground stars and compact background
galaxies from the images, and any residual artifacts were manually subtracted
using the IRAF task $imedit$. In Figure 4, we show the $g$-band surface
brightness map of the field around NGC 936 after cleaning and masking
unrelated foreground and background objects. The background gradient of halo
light from the nearby giant galaxy NGC 936 is subtracted. First, a constant
sky-background level is subtracted across the entire image. The constant sky
background level is derived using 10 independent sky regions of size
10$\times$10 pixel boxes from which we sampled the sky background and
calculated an overall median. Subsequently, we masked MATLAS-167 and its tidal
tail region and ran $ellipse$ task to model NGC 936, and then subtracted this
model of NGC NGC 936 from the image.
To measure the total flux of filamentary structure, we conducted aperture
photometry using a polygonal aperture (see the green polygon in Figure 4).
Since the surface brightness of the faint filaments was too low for automatic
detection, an aperture is defined visually. We excluded pixels below the S/N
threshold from the measurement, and the resulting values are presented in
Table 1. The total brightness in $g$-band we measured was Mg =
$-$15.92$\pm$0.04 mag. However, we want to emphasize that this estimate may
not account for additional starlight below our detection threshold or behind
NGC 936, and it may also include contamination from faint point sources.
Therefore, caution should be exercised when interpreting these measurements as
the accreted galaxy luminosity. We followed a similar procedure in the
$r$-band image, measuring the flux within the identical polygonal aperture,
and found that the color of the full stream is $g-r$ = 0.72$\pm$0.06 mag.
### 2.2 Spectroscopy
The SDSS targeted NGC 936_UCD for spectroscopic observation, which we
retrieved from the SDSS archive server, and it proved to be of sufficient
quality and high signal-to-noise ratio to perform a detailed stellar
population study. The SDSS spectrum is observed with a fiber of radius
1.5$\arcsec$, which is nearly three times NGC 936_UCD size. However, the light
contribution of NGC 936_UCD in the fiber is dominant, i.e., $>$90%.
To extract the maximum amount of information from the spectrum, we employed a
full-spectrum fitting method, which exploits the extensive wavelength coverage
of SDSS optical spectroscopy. This fitting method involves modeling the
spectrum using a combination of simple stellar populations (SSPs) defined by
their age and metallicity. We utilized the publicly available code UlySS by
Koleva et al. (2008) for this purpose. We used an SSP model provided by
Vazdekis et al. (2010), based on MILES stellar library (Sánchez-Blázquez et
al., 2006). This model considers the effects of different stellar evolutionary
phases, such as the main sequence, red giant branch, and asymptotic giant
branch. We fitted the observed spectrum of wavelength range 4100 to 7000 Å
after smoothing the SDSS spectrum by a three-pixel Gaussian kernel. The
quality of the model comparison with the SDSS spectrum is shown in Figure 5,
where the observed spectrum typically matches within 5 percent of the modeled
flux. The analysis yielded a light-weighted SSP age of 5.6$\pm$0.8 Gyr and
[Z/H] of $-$0.83$\pm$0.3 dex.
## 3 Discussion
Figure 6: Relation between the luminosity of dEs and their nuclei. NGC 936_UCD
is shown in red and the comparison sample of the Virgo cluster dEs are shown
in blue, which we obtained from Sánchez-Janssen et al. (2019). The arrow in
NGC 936_UCD represents our measurement of MATLAS-167 flux is a lower limit.
the In the right panel, we show the magnitude distribution of the Virgo
cluster dE nuclei and UCDs. The Virgo cluster UCDs magnitudes are obtained
from Liu et al. (2020).
### 3.1 Comparison of UCDs and dE Nuclei Properties
UCDs and dE nuclei are compact and dense stellar systems of high mass. They
often contain predominantly old stellar populations, indicating their
formation in the early stages of galaxy evolution. In this section, we make a
comparative analysis between UCDs and dEs nuclei and find the position of NGC
936_UCD. For this purpose, we use the Virgo cluster UCDs and dE nuclei as
reference samples.
The relationship between the luminosity of dEs and their nuclei is depicted in
Figure 6. The Virgo cluster dE sample is obtained from Sánchez-Janssen et al.
(2019), shown in blue. NGC 936_UCD is represented by a red dot. As
anticipated, a well-established correlation emerges between the luminosity of
dEs and that of their nuclei, placing NGC 936_UCD among the brightest objects
situated in the upper-right corner. It is important to note, however, that the
estimated luminosity of the NGC 936_UCD host galaxy represents a lower limit,
implying that its actual position on the plot may have been even further to
the right.
Figure 7 illustrates the relationship between the derived SSP properties and
local projected density. In this analysis, we utilized UCD and dE nuclei
samples from the Virgo cluster, as studied by Paudel et al. (2010). The local
density was determined by calculating the circular projected area enclosing
the 10th neighbor. The results indicate a weak correlation between the local
projected density and the ages of the nuclei. More importantly, an age break
is observed at approximately $\sim$ 4 (100 kpc)2. Almost all UCDs are located
in the high-density region as defined above, and their age distribution
overlaps with that of dE nuclei situated in high-density environments. A
similar trend is identified in the metallicity distribution of dE nuclei,
where those in high-density environments exhibit lower metallicity compared to
nuclei of dE located in low-density environments. Notably, the SSP properties
of NGC 936_UCD, being situated in a relatively dense region, $>$4 (100 kpc)2,
resemble those of Virgo UCDs or dE nuclei located in dense regions.
Figure 7: Comparison of age and metallicity of the Virgo cluster UCDs (black)
and dEs nuclei (blue) with respect to the local projected density. The data
are from Paudel et al. (2010). NGC 936_UCD is shown in red. Figure 8: Relation
between the distance of UCDs from their nearest bright galaxy and their
luminosities. The blue symbol represents the median distance in the magnitude
bin, accompanied by an error bar that indicates the normalized standard
deviation. NGC 936_UCD is shown in red.
### 3.2 Tidal Interaction and Formation of UCDs
Observations have shown that UCDs have a size$-$luminosity distribution and
internal velocity dispersion similar to compact nuclei (Drinkwater et al.,
2003; Evstigneeva et al., 2008; Pfeffer & Baumgardt, 2013). The high dynamical
mass-to-light ratios of UCDs suggest that they may contain a significant
amount of dark matter, which may have been inherited from the parent dwarf
galaxies during the tidal disruption (Baumgardt & Mieske, 2008; Mieske et al.,
2008). Several UCDs display indications of asymmetrical or tidal features,
while others reveal the presence of stellar envelopes or the status of
transitional objects from dwarf galaxies to UCDs (Wittmann et al., 2016).
State-of-the-art high-resolution imaging and spectroscopic observations of
these compact objects have allowed us to search for the presence of
supermassive black holes (SMBH). Particularly, recent observations have
revealed that all the top three most massive UCDs of the Virgo cluster possess
SMBH (Ahn et al., 2017, 2018; Seth et al., 2014) and these SMBHs account for a
substantial portion of their overall mass. These trends provide compelling
evidence of their tidal stripping origin (Voggel et al., 2019).
The tidal threshing scenario has been proposed to account for the origin of
intra-cluster GCs, which is quite faint compared to the UCDs (West et al.,
1995). Our analysis suggests that massive UCDs are likely to form through
tidal stripping. Based on Figure 6, it is evident that the disrupted nucleated
dE, MATLAS-167, stands out as one of the brighter dEs, and its nucleus
luminosity is comparable to the brighter UCDs observed in the Virgo cluster.
In fact, considering the combined luminosity of MATLAS-167 and its tidal
stream, it surpasses the luminosity of all other dEs identified by the MATLAS
dwarf galaxy survey around NGC 936 (Habas et al., 2020).
The close proximity of NGC 936_UCD to a giant galaxy raises questions about
whether the special environment plays a role in the formation and evolution of
bright UCDs. To shed light on this issue, we show a relation between UCD
brightness and distance to the nearest bright galaxy (Mr $<$$-$19 mag) of the
Virgo cluster UCD sample studied by Liu et al. (2020) in Figure 8. The figure
reveals that bright UCDs tend to be closer to bright galaxies than faint UCDs,
indicating a potential link between bright UCD formation and proximity to a
bright galaxy. We find that almost all UCDs of $M_{g}$ < $-$12 mag are located
within 20 kpc sky-projected distance from their nearest giant neighbor galaxy.
NGC 936_UCD, located at a sky-projected distance of 19 kpc away from a giant
galaxy, NGC 936, is consistent with the observed trend in the Virgo cluster.
To quantify the observed trend, we sub-sample the UCD sample into faint (Mg
$>$ $-$11 mag) and bright categories and compute the two-point correlation
coefficient between these subsets and massive galaxies. We find a significant
disparity in the correlation coefficients. Specifically, the correlation
coefficient between bright galaxies and bright UCDs is almost double (2.05)
compared to that of bright galaxies and faint UCDs (0.96).
Indeed, the destruction of a bright dwarf necessitates a strong tidal force,
which can typically be attained in the vicinity of a massive galaxy or within
a densely populated cluster core. Consequently, the substantial tidal force
exerted by giant galaxies appears to be advantageous in destroying bright
dwarf galaxies, thereby resulting in exposed luminous nuclei commonly referred
to as UCDs. This line of reasoning strongly supports the hypothesis that the
tidal stripping mechanism is not only accountable for the formation of low-
mass intra-cluster GC but also massive UCDs. Remarkably, these objects
represent the two extremes of the mass function of compact stellar systems.
## Acknowledgements
S.-J.Y. and S.P. acknowledge support from the Basic Science Research Program
(2022R1A6A1A03053472) through the National Research Foundation (NRF) of Korea.
S.P. and S.-J.Y., respectively, acknowledge support from the Mid-career
Researcher Program (No. RS-2023-00208957) and the Mid-career Researcher
Program (No. 2019R1A2C3006242) through the NRF of Korea. O.M. is grateful to
the Swiss National Science Foundation for financial support under the grant
number PZ00P2_202104. Mélina Poulain is supported by the Academy of Finland
grant n:o 347089.
## DATA AVAILABILITY
Most of the data underlying this article are publicly available. The derived
data generated in this research will also be shared on reasonable request to
the corresponding author.
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|
Polarization parameters of the quasi–elastic $(p,2p)$ reaction with nuclei at
1 GeV
O.V. Miklukho, G.M. Amalsky, V.A. Andreev, S.V. Evstiukhin, O.Ya. Fedorov,
G.E. Gavrilov, A.A. Izotov, A.Yu. Kisselev, L.M. Kochenda, M.P. Levchenko,
V.A. Murzin, D.V. Novinsky, A.N. Prokofiev, A.V. Shvedchikov, S.I. Trush, A.A.
Zhdanov
B.P. Konstantinov Petersburg Nuclear Physics Institute, Gatchina, RUSSIA
New experimental data on the polarization and spin-correlation parameters in
the (p,2p) reaction with nuclei at 1 GeV are presented. The experiment was
aimed to study a modification of the proton–proton scattering matrix.
Comments: 11 pages, 5 figures. Presented at XV International Workshop on
High–Energy Spin Physics, Dubna, October 2013. Submitted to the Workshop
proceedings.
Category: Nuclear Experiment (nucl–ex)
## 1 Introduction
There were some speculations on modifications of nucleon and meson masses and
sizes, and of meson–nucleon coupling constants, and, as a consequence, of
nucleon–nucleon scattering matrix in nuclear medium [1–3]. These speculations
were motivated by a variety of theoretical points of view, including the
renormalization effects due to strong relativistic nuclear fields,
deconfinement of quarks, and partial chiral symmetry restoration.
This work is a part of the experimental program in the framework of which the
medium–induced modifications of the nucleon–nucleon scattering amplitudes are
studied at the PNPI synchrocyclotron with the 1 GeV proton beam [4–8]. The
intermediate–energy quasi–free $(p,2p)$ reaction is a good experimental tool
to study such effects, since in the first approximation, this reaction can be
considered as a proton–proton scattering in the nuclear matter. Usage of
S–shell protons (with zero orbital momentum) is preferred because
interpretation of obtained data in this case is essentially simplified since
the effective polarization is not involved [9]. The polarization observables
in the reaction are compared with those in the elastic $pp$ scattering. In our
exclusive experiment, a two–arm magnetic spectrometer is used, the shell
structure of the nuclei being evidently distinguished. To measure polarization
characteristics of the reaction, each arm of the spectrometer was equipped
with a multi-wire–proportional chamber polarimeter.
In the early PNPI–RCNP experiment [5], the polarizations $P_{1}$ and $P_{2}$
of both secondary protons from the $(p,2p)$ reactions at 1 GeV with the
1$S$–shell protons of the nuclei 6Li, 12C and with the 2$S$–shell protons of
the 40Ca nucleus has been measured at nuclear proton momenta close to zero.
The polarization observed in the experiment, as well as the analyzing power
$A_{y}$ in the RCNP experiment at the 392 MeV polarized proton beam [10, 11],
drastically differed from those calculated in the framework of
non–relativistic Plane Wave Impulse Approximation (PWIA) and of spin–dependent
Distorted Wave Impulse Approximation (DWIA) [12], based on free space
proton–proton interaction. This difference was found to have a negative value
and to increase monotonously with the effective mean nuclear density
$\bar{\rho}$ [10]. The latter is determined by the absorption of initial and
secondary protons in nucleus matter. The observed inessential difference
between the non–relativistic PWIA and DWIA calculations pointed out only to a
small depolarization of the secondary protons because of proton–nucleon re-
scattering inside a nucleus. All these facts strongly indicated a modification
of the proton–proton scattering amplitudes due to the modification of the main
properties of hadrons in the nuclear matter.
Later, the results of the experiment with a 4He target broke the
above–mentioned dependence of the difference between the experimental
polarization values and those calculated in the framework of the PWIA on the
effective mean nuclear density [6]. The difference for the 4He nucleus proved
to be smaller than that for the 12C nucleus. This evidently contradicts the
elastic proton–nucleus scattering experiment. According to the experiment, the
4He nucleus has the largest mean nuclear density. The important feature of the
experiment with the 4He nucleus was a possibility to see the medium effect
without any contribution from multi–step processes (for instance, from the
$(p,2pN)$ reactions). These processes could take place when there were
nucleons of outer shells as in other nuclei. Therefore, they could not cause
the systematic difference between the polarizations $P_{1}$ and $P_{2}$
clearly obtained for the first time in the experiment [6].
Here we present the polarization data for the reaction with the nuclei 4He,
6Li, 12C (1$S$–shell), and 40Ca (2$S$–shell) obtained with a much better
statistical accuracy in our last experiments. New data on the polarization in
the reaction with the 1$S$–shell protons of the 28Si nucleus are presented.
The 1$S$–state of the 28Si nucleus has a larger value of the mean proton
binding energy $E_{s}$ (50 MeV) than that of the 12C nucleus (35 MeV). We also
present the polarization measured in the reaction with the $P$–shell and
$D$–shell protons of the 12C and 28Si nuclei, respectively.
In recent experiments, the research program was extended to measure the spin
correlation parameters $C_{ij}$ in the $(p,2p)$ reaction with the 4He and 12C
nuclei. Measurements of the parameters in the reaction with nuclei were for
the first time performed. The main attention was concentrated on the spin
correlation parameter $C_{nn}$ since its value is the same in the
center–of–mass and laboratory systems. Besides, this parameter is not
distorted by the magnetic fields of the two–arm spectrometer because of the
proton anomalous magnetic moment [13]. Since the polarization and the spin
correlation parameter $C_{nn}$ are expressed differently through the
scattering matrix elements [3], the measurement of both these polarization
observables can provide a more comprehensive information about a modification
of the hadron properties in the nuclear medium.
## 2 Experimental method
The general layout of the experimental setup is shown in Fig. 1 [14]. The
experiment is performed at non–symmetric scattering angles of the final state
protons in the coplanar quasi–free scattering geometry with a complete
reconstruction of the reaction kinematics. The measured secondary proton
momenta $K_{1}$, $K_{2}$ (kinetic energies $T_{1}$, $T_{2}$) and the
scattering angles $\Theta_{1}$, $\Theta_{2}$ are used together with the proton
beam energy $T_{0}$ to calculate nuclear proton separation energy $\Delta E$ =
$T_{0}$-$T_{1}$-$T_{2}$ and the residual nucleus momentum $K_{r}$ for each
$(p,2p)$ event. In the impulse approximation, the $K_{r}$ is equal to the
momentum $K$ of the nuclear proton before the interaction (${\bf K_{r}}$ =
-${\bf K}$).
External proton beam of the PNPI synchrocyclotron was focused onto the target
TS of a two–arm spectrometer consisting of the magnetic spectrometers MAP and
NES. The beam intensity was monitored by the scintillation telescope M1, M2,
M3 and was at the level of about 5$\cdot$1010 protons/(s$\cdot$cm2).
Solid nuclear targets TS made of CH2 (for the setup calibration), 6Li, 12C,
28Si, and 40Ca, as well as a cryogenic target made of liquid helium 4He (or
liquid hydrogen for calibration) were used in the experiment [6, 14].
The spectrometers were used for registration of the secondary protons from the
$(p,2p)$ reaction in coincidence and for measurement of their momenta and
outgoing angles. The polarization of these protons $P_{1}$ and $P_{2}$,
Figure 1: The experimental setup. TS is the target of the two–arm
spectrometer; Q1$\div$Q4 are magnetic quadrupoles; D1, D2 are dipole magnets;
C1, C2 are collimators; S1$\div$S4 and M1$\div$M3 are scintillation counters;
PC1$\div$PC4, PC1’, PC4’ (PC5$\div$PC8, PC5’, PC8’) and A1 (A2) are the
proportional chambers and the carbon analyzer of the high–momentum
(low–momentum) polarimeter, respectively.
and the spin correlation parameters $C_{ij}$ were measured by the polarimeters
located in the region of focal planes of the spectrometers MAP and NES (Fig.
1). The first index of the $C_{ij}$, $i$ (where $i$ is $n$ or $s^{,}$), and
the second index $j$ (where $j$ is $n$ or $s^{,,}$) correspond to the forward
scattered proton analyzed by the MAP polarimeter and the recoil proton
analyzed by the NES polarimeter, respectively. The unit vector ${\bf n}$ is
perpendicular to the scattering plane of the reaction. Unit vectors
${\bf{s^{,}}}$ and ${\bf s^{,,}}$ are perpendicular to the vector ${\bf n}$
and to the coordinate axes z, and z,, (Fig. 1) of the polarimeters.
The overall energy resolution (on $\Delta E$) of the spectrometer estimated
from the elastic proton–proton scattering with the 22 mm thick cylindrical CH2
target was about 5 MeV (FWHM). The spectra which was analysed is presented in
Fig.2 [14].
Figure 2: Proton separation energy spectra for elastic $pp$–scattering (left
panel) and for the $(p,2p)$ reaction with 12C nuclei (right panel). In the 12C
spectrum the accidental background contribution is subtracted.
The track information from the proportional chambers of both polarimeters was
used in the off-line analysis to find the azimuthal $\phi_{1}$, $\phi_{2}$ and
polar $\theta_{1}$, $\theta_{2}$ angles of the proton scattering from the
analyzers A1, A2 for each $(p,2p)$ event.
The polarization parameters were estimated by folding the theoretical
functional shape of the azimuthal angular distribution into experimental one
[8], using the CERNLIB MINUIT package and a $\chi^{2}$ likelihood estimator.
This method permits to realize the control over $\chi^{2}$ in the case the
experimentally measured azimuthal distribution is distorted due to the
instrumental problems.
The time difference (TOF) between the signals from the scintillation counters
S2 and S4 was measured. It served to control the accidental coincidence
background. The events from four neighboring proton beam bunches were
recorded. Three of them contained the background events only and were used in
the off-line analysis to estimate the background polarization parameters and
the background contribution at the main bunch containing the correlated events
[14].
The recoil proton spectrometer NES was installed at a fixed angle
$\Theta_{2}\simeq$ 53.2∘. At a given value of the $S$–shell mean binding
energy of the nucleus under investigation, the angular and momentum settings
of the MAP spectrometer and the momentum setting of the NES spectrometer were
chosen to get a kinematics of the $(p,2p)$ reaction close to that of the
elastic proton–proton scattering. In this kinematics, the momentum $K$ of the
nuclear $S$–proton before the interaction is close to zero. At this condition,
the counting rate of the $S$-shell proton knockout reaction should be maximal.
## 3 Results and discussion
In Fig. 3, the polarizations $P_{1}$, $P_{2}$ in the $(p,2p)$ reaction with
the $S$–shell protons of the nuclei 4He, 6Li, 12C, 28Si, 40Ca are plotted
versus the $S$–shell proton binding energy $E_{s}$ [14]. For all nuclei
(excluding 4He), the effective mean nuclear density $\bar{\rho}$, normalized
on the saturation nuclear density $\rho_{0}\approx$ 0.18 fm-3, is given. The
actual calculation of the effective mean nuclear density $\bar{\rho}$, which
is determined by absorption of the incident and both outgoing protons, was
carried out following a procedure [10] using the computer code THREEDEE [12].
The potential model of a nucleus employed by the code is not correct for the
4He nucleus. The calculated value of the $\bar{\rho}$ in this case is strongly
unreliable [6]. The 4He data should be excluded in comparison with theoretical
models which differ from the PWIA.
The points ($\circ$) and ($\bullet$) in the figure correspond to the
polarizations $P_{1}$
Figure 3: Polarizations $P_{1}$ and $P_{2}$ of the protons scattered at the
angles $\Theta_{1}$ ($\circ$) and $\Theta_{2}$ ($\bullet$) in the $(p,2p)$
reaction with the $S$–shell protons of nuclei at 1 GeV. The points at $E_{s}$
= 0 correspond to the elastic proton-proton scattering. The curves correspond
the theoretical calculations described in the text.
and $P_{2}$ of the forward scattered protons at the angle $\Theta_{1}$ =
21${}^{\circ}\div$25∘ (with energy $T_{1}$ = 745$\div$735 MeV) and of the
recoil protons scattered at the angle $\Theta_{2}\simeq$ 53.2∘ (with energy
$T_{2}$ = 205$\div$255 MeV). The points at the $E_{s}$=0 are the polarizations
$P_{1}$ and $P_{2}$ in the elastic proton-proton scattering at the angles
$\Theta_{1}$ = 26.0∘ and $\Theta_{2}$ = 53.2∘ ($\Theta_{cm}$ = 62.25∘).
In Fig. 3, the experimental data are compared with the results of the non-
relativistic PWIA, DWIA calculations (the dashed and solid curves,
respectively) [14] and the DWIA* calculation with the relativistic effect, the
distortion of the nucleon Dirac spinor in nuclear medium, taken into account
(the dotted, $M^{*}_{N}$, curve) [2,14]. For the calculations, the computer
code THREEDEE was used [12] using an on–shell factorized approximation and the
final energy prescription. A global optical potential, parametrized in the
relativistic framework and converted to the Shrödinger–equivalent form, was
used to calculate the distorted wave functions of incident and outgoing
protons in the case of DWIA and DWIA*. A conventional well–depth method was
used to construct the bound–state wave function. The DWIA* calculations were
carried out in the Shrödinger–equivalent form [5]. In this approach, a
modified $NN$ interaction in medium is assumed due to the effective nucleon
mass (smaller than the free mass), which affects the Dirac spinors used in the
calculations of the $NN$ scattering matrix. A linear dependence of the
effective mass of nucleons on the nuclear density was assumed in the
calculations.
The results of the polarization studies:
1\. The difference of the final proton polarizations $P_{1}$ and $P_{2}$ found
in the PWIA, DWIA and DWIA* is quite small (less than 0.005) for all nuclei
under investigation.
2\. The difference between the PWIA and DWIA results is small. This indicates
that the distortion in the conventional non-relativistic framework does not
play any essential role in the polarization for the kinematic conditions under
consideration (the transferred momenta $q$ = 3.2$\div$3.7 fm-1).
3\. Predictions of the DWIA* with relativistic corrections (distortion of the
proton Dirac spinor in nuclear medium) are close to experimental data for the
forward scattered proton polarization $P_{1}$.
4\. A significant difference is observed between the measured polarization of
the scattered proton $P_{1}$ and that of the recoil proton $P_{2}$.
Note that the difference between the measured polarizations $P_{1}$ and
$P_{2}$ was also observed in the reaction with the $D$–shell protons of the
28Si nucleus and was not seen in the reaction with the $P$–shell protons of
the 12C (Fig. 4).
Figure 4: Polarization in the $(p,2p)$ reaction with the external shell
protons of the 12C and 28Si.
The experimental data on the spin correlation parameters $C_{ij}$ in the
reactions with the 4He and 12C are given in Fig. 5. The dashed and dotted
curves in the figure correspond to the PWIA calculations of the $C_{nn}$ and
$C_{s^{,}{s^{,,}}}$ parameters using the current Arndt’s group phase-shift
analysis (SP07). The mixed $C_{s^{,}{s^{,,}}}$ parameter was found by taking
into account its distortion in the magnetic field of the spectrometers. The
points at the $E_{s}$ = 0 correspond to the elastic proton-proton scattering.
As seen in Fig. 5, the $C_{nn}$ data (as well as the $C_{s^{,}{s^{,,}}}$ data)
are described in the framework of the PWIA. The question arises, there is no
the nuclear medium modification of the $C_{nn}$ parameter as it is for the
polarization of the final protons (Fig. 3)? Whether this fact is connected to
the strong
Figure 5: Spin correlation parameters $C_{ij}$ in the $(p,2p)$ reaction at 1
GeV with the $S$–shell protons of the 4He and 12C nuclei at the secondary
proton scattering angles $\Theta_{2}$ = 53.22∘, $\Theta_{1}$ = 24.21∘ and
$\Theta_{2}$ = 53.22∘, $\Theta_{1}$ = 22.71∘, respectively. The points at
$E_{s}$ = 0 correspond to the elastic proton-proton scattering ($\Theta_{1}$ =
26.0∘, $\Theta_{cm}$ = 62.25∘). The curves are the results of calculations
described in the text.
polarization dropping for the recoil proton? It is possible that some spin-
flip mechanism compensates the nuclear medium effect in the $C_{nn}$.
Due to the parity conservation in the elastic proton-proton scattering, the
spin correlation parameters $C_{ns^{\prime\prime}}$ and $C_{s^{\prime}n}$
should be equal to 0. This is confirmed by the experimental data at the
$E_{s}$ = 0 in the Fig. 5. For the $(p,2p)$ reaction, we see some deviation of
the parameters from zero. It may be related to the spin-flip mechanism
mentioned above. Note that test calculations of all spin correlation
parameters for the accidental coincidence background give zero values as
should be expected.
To find an explanation of the observed effects, let us assume that there is a
spin-flip interaction of the recoil (nuclear) proton with the residual
nucleus, which is not taken into account by the theoretical approaches. This
additional interaction mechanism, governed by the Pauli exclusion principle in
a nucleus, reverses the proton spin direction and, as a consequence, changes
the signs of the polarization and the spin correlation parameter $C_{nn}$.
The relative contribution ($\alpha$) of the spin–flip mechanism in the
interaction with a residual nucleus, which is mainly determined by the proton-
nucleon re-scattering at small angles, can be found from experiment via the
relative polarization dropping ($g_{p}$) for the recoil proton. First define
the averaged polarization of the recoil proton:
$<{P}_{2}>~{}=~{}\frac{P_{2}+\alpha(-P_{2})}{1+\alpha}~{}=~{}\frac{P_{1}+\alpha(-P_{1})}{1+\alpha}~{}=~{}\frac{(1-\alpha)P_{1}}{1+\alpha}.$
(1)
In the equation we used the fact that all employed theories give equal values
of the polarizations $P_{1}$ and $P_{2}$. The averaged value of the $C_{nn}$
can also be calculated using the equation:
$<{C}_{nn}>~{}=~{}\frac{C_{nn}+\alpha(-C_{nn})}{1+\alpha}~{}=~{}\frac{(1-\alpha)C_{nn}}{1+\alpha}.$
(2)
The relative polarization dropping $g_{p}$ is determined as:
$g_{p}=\frac{P_{1}-<{P}_{2}>}{P_{1}}=g_{C_{nn}}=\frac{C_{nn}-<{C}_{nn}>}{C_{nn}}=\frac{2\alpha}{1+\alpha}.$
(3)
It can be seen that the proposed spin–flip interaction couples in simple form
the relative dropping of the polarization and the $C_{nn}$ parameter $g_{p}$ =
$g_{C_{nn}}$. From experimental data we find $g_{p}$(4He) = 0.153$\pm$0.018,
$g_{p}$(12C) = 0.325$\pm$0.031 and make corrections to the PWIA calculations
using the formula $C_{nn}$-cor = (1-$g_{p}$)$C_{nn}$ (the solid curve, PWIA-C,
in Fig. 5). One can see from the figure that the the experimental $C_{nn}$
points lie above the curve. So it can be expected that the nuclear medium
modification enhances the $C_{nn}$ parameter, while the polarization is
reduced.
From the experimental $g_{p}$ data, the probability of the spin–flip
interaction can be defined for the corresponding residual nuclei: $\alpha$(3H)
= 0.083$\pm$0.010, $\alpha$(11B) = 0.194$\pm$0.022.
What could be the nature of the considered spin–flip interaction? It was first
time proposed by D.I. Blokhintsev that there are the fluctuations of nuclear
density in nuclei, or the dense nucleon associations [15]. The reflection of
the recoil proton off the objects is similar to the spin–flip interaction
considered above. As a result, a proton belonging to a correlation, with
opposite spin direction (due to the Pauli principle) is detected.
Nucleon correlations are intensively studied in the JLAB using electron beam.
The CLAS collaboration gives the probability for a given nucleon to belong to
a two-nucleon correlation in nucleus with A nucleons $a_{2N}$(3He) =
0.080$\pm$0.016, $a_{2N}$(12C) = 0.193$\pm$0.041 [16].
We can see that there is a coincidence between the PNPI $\alpha$ and the JLAB
$a_{2N}$ for the corresponding residual nuclei. The model of the spin–flip
interaction for explanation of the PNPI polarization data is currently being
developed. Preliminary results suggest that the ratio of the $\alpha$ and
$a_{2N}$ is very close to unity.
References
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|
# Elasticizing Linux via Joint Disaggregation of Memory and Computation
Ehab Ababneh, Zaid Al-Ali, Sangtae Ha, Richard Han, Eric Keller
Department of Computer Science, University of Colorado Boulder
### Abstract
In this paper, we propose a set of operating system primitives which provides
a scaling abstraction to cloud applications in which they can transparently be
enabled to support scaled execution across multiple physical nodes as resource
needs go beyond that available on a single machine. These primitives include
_stretch_ , to extend the address space of an application to a new node,
_push_ and _pull_ , to move pages between nodes as needed for execution and
optimization, and _jump_ , to transfer execution in a very lightweight manner
between nodes. This joint disaggregation of memory and computing allows for
transparent elasticity, improving an application’s performance by capitalizing
on the underlying dynamic infrastructure without needing an application re-
write. We have implemented these primitives in a Linux 2.6 kernel,
collectively calling the extended operating system, ElasticOS. Our evaluation
across a variety of algorithms shows up to 10x improvement in performance over
standard network swap.
## 1 Introduction
We are in the midst of a significant transition in computing, where we are
consuming infrastructure rather than building it. This means that applications
have the power of a dynamic infrastructure underlying them, but many
applications struggle to leverage that flexibility. In this paper, we propose
supporting this at the operating system level with new primitives to support
scaling.
To gain some context on the challenge with scaling, we first discuss how it is
predominantly handled today. The most straight forward option, which required
no changes to applications, is to simply get a bigger (virtual) machine as
load for an application increases. Cloud providers, such as Amazon[2], offer a
wide range of machine sizes which cost anywhere from less than a penny per
hour to a few dollars per hour. For cost efficiency, companies wish to use the
“right” size machine, which might change over time. But, transitioning from
one VM size to another can pose challenges. In some cases, we can take
snapshots (e.g., with CRIU [8]) and migrate the application to a
bigger/smaller VM, but this can be disruptive, and the management of the
application needs scripts and other infrastructure to trigger scaling.
An alternative is to re-write the applications with scaling in mind. To
leverage the scaling, commonly applications are built around frameworks such
as Hadoop[12], Apache Spark[26], MPI[7] or PGAS [1]. These frameworks are
designed with the flexibility of being able to execute tasks on a varying
amount of distributed resources available. The problem here is two fold.
First, to leverage this, the application needs to be built for this – a
significant challenge (requiring a re-write) for any existing application, and
forcing application developers to evaluate and become fluent in the latest
frameworks and potentially adapt the application as the frameworks change.
Second, and perhaps more challenging, is that not every application fits into
one of these frameworks.
Another approach to scaling is to replicate VMs/containers when an application
becomes popular and requires more resources. This too introduces burdens on
the programmer in order to synchronize shared data and state across multiple
replicas, as well as to script their applications to spawn/delete replicas
depending on load.
In short, in each case, _the burden of scaling is placed on programmers_. We
argue that developers shouldn’t need to be experts in cloud management and
other frameworks, in addition to also needing to be fluent in programming and
their application domain. Instead, the operating system should provide more
support. Broadly speaking, the job of an operating system is to make the life
of an application developer easier (through abstraction). A modern OS provides
virtual memory abstractions, so developers do not have to coordinate memory
use among applications, network socket abstractions, so developers can send
messages without needing to be intimately familiar with the underlying network
protocols, and many other abstractions (file system, device, multi-tasking)
all to support developers._We propose that scaling should be an OS
abstraction_.
Related Work: We are not the first to propose that operating systems should
support scaling. Scaling of memory approaches are popular and include efforts
such as RAMCloud [22], which requires refactoring in user space to utilize its
memory scaling capabilities. An early approach to sharing memory called DSM
[21, 19, 17, 16, 14, 6] suffered from scaling issues, but more recently
disaggregation-based approaches towards memory have emerged that are centered
around transparent scaling of memory behind the swap interface, such as NSwap,
Infiniswap, X-Swap and Memx [20, 10, 29, 4]. Scaling of computation approaches
include process migration to machines with more resources [8, 13, 3, 25, 18,
27], in addition to the scaling frameworks and replication methods mentioned
previously. Approaches to accelerate process migration [24, 15] have been
proposed to hide the latency of migration by copying most of the process state
in the background and only copying a small delta to the new machine after
halting the process. Single system image (SSI) OSs such as Kerrighed, MOSIX,
Sprite and Amoeba [15, 3, 5, 28] have been created to support operation across
a distributed cluster of machines. These approaches typically employ a process
migration model to move computation around cluster nodes and require
applications to be recompiled for these specialized OSs.
These prior efforts in OS scaling suffer from a variety of limitations.
Network swap-based approaches, while being a step in the right direction of
disaggregation in the data center, miss the opportunity to exploit _joint
disaggregation_ of computation and memory for improved performance. Execution
is typically assumed to be pinned on one machine, while memory pages are
swapped back and forth remotely across the network. This can result in
excessive swapping of pages over the network. In these cases, movement of
computation to a remote machine towards a cluster of locality stored in the
remote machine’s memory would result in substantially faster execution and
lower network overhead, as we will show later.
Figure 1: ElasticOS Vision for Cloud Data Centers.
Combining current network swap approaches with existing process migration
techniques to alleviate excessive network swapping overhead would suffer two
major limitations. First, each decision to move computation would incur the
overhead of copying the entire address space. This is a significant amount of
overhead to impose on the network. Second, even with accelerated process
migration, there is a substantial delay between the time the decision is made
to migrate and when that is completed, at which time the conditions that
triggered the original migration decision may be obsolete due to the length of
time needed to copy all the state.
Figure 2: Illustration of ElasticOS abstractions. Each box labeled with a
number above is a compute node, with the shaded boxes within represent
individual pages. Starting with execution on a single machine in (0), when
memory nears being filled, we stretch to two nodes in (1) and balance the
pages in (2). We then push and pull pages in (3), with the red shaded pages
going from node 1 to 2 (push) and from node 2 to 1 (pull). Finally, in (4) and
(6) we are seeing too many page faults (resulting in pull), so decide to jump
from node 1 to 2 in (5) and from node 2 to 1 in (7), respectively.
Introducing ElasticOS: In response to these shortcomings, we introduce four
primitives to realize the scaling OS abstraction – _stretch_ , _jump_ , _push_
, and _pull_. These scaling abstractions are designed to be transparent,
efficient, and practically useful. Our approach is inspired by an early work
that hypothesized elasticizing operating systems as a hot research topic, but
did not build a working implementation of the proposed concept [11]. _Stretch_
is used when an application becomes overloaded (e.g., a lot of thrashing to
disk is occurring), so the operating system _stretches_ the application’s
address space to another machine – extending the amount of memory available to
the application. Push and pull allow memory pages to be transferred between
machines which the application has been stretched to, whether proactively to
optimize placement, or reactively to make it so the data is available where it
is needed. _Jump_ allows program execution to transfer to a machine which the
application has been stretched to. Unlike heavyweight process migration, our
jump primitive is a lightweight transfer of execution that only copies the
small amount of state needed to begin execution immediately on the remote
machine, such as register state and the top of the stack. Any additional state
that is needed is faulted in using pulls from the rest of the distributed
address space. Having both jumping and push/pull allows for the OS to choose
between moving the data to be where the execution needs it, and moving the
execution to be where the data is. This supports the natural, but not
necessarily perfect locality that exists in applications.
To demonstrate the feasibility of this scaling approach, we extended the Linux
kernel with these four primitives, and call the extended Linux, ElasticOS.
Figure 1 provides a high level view of ElasticOS. We see that an instance of
ElasticOS is capable of spanning a number of nodes in the data center, and
that the number of spanned nodes can elastically scale up or down depending
upon application demand. The application is executed within ElasticOS, and the
scaling primitives are used to support this execution across a distributed
collection of resources.
To demonstrate the desirability of these four primitives, we evaluated a set
of applications with large memory footprints and compared against network
swap, which supports the pull and push primitives, and itself has shown
performance improvements of being able to scale memory resources transparently
across multiple machines. We illustrate the additional benefit of also
transparently scaling computing resources across multiple machines, forming a
system with joint disaggregation of memory and computation. Our evaluation
shows up to 10x speedup over network swap, as well as a reduction of network
transfer between 2x and 5x.
In summary, we make the following contributions.
* •
Introduce scaling as a new OS abstraction, specifically with four primitives:
stretch, push, pull, and jump.
* •
Provide an architecture and implementation of these abstractions in Linux.
* •
Demonstrate through an evaluation on Emulab servers that ElasticOS achieves up
to 10x speed up over network swap across a range of applications, and up to 5X
reduction in network overhead.
## 2 ElasticOS Primitives in Action
In this section, we describe the four primitives through an illustration of a
running program. Figure 2 graphically presents each of the primitives. In this
figure, we can see nodes 1 and 2, with pages inside of each node – this
represents the physical memory and whether a given page is used (shaded) or
unused (unshaded) in physical memory. As a starting point, an application is
running on a single machine. Over time, this application grows in memory use
to nearly the size of the amount of memory in the entire node (label 0 in the
figure). This is when ElasticOS decides to stretch the process, that is to
scale out by using memory on a second node (label 1). At this point, the
memory available to the application has grown (doubled in the figure, since it
is now on two nodes with equal memory, which is not required in ElasticOS).
ElasticOS can choose to balance the pages at this point, to transfer pages to
the (new) remote node (label 2). These can be chosen by a means, such as least
recently used.
Once the process is stretched, this means that the process is effectively
running on multiple machines, but each node only hosts some of the pages. At
this point, execution continues on the original machine. As not all of the
pages are on this machine (which would have naturally happened over time, even
if we didn’t balance pages), when the process tries to access a page, it might
trigger a page fault. In ElasticOS, the page fault handler is modified to
handle this situation. At this point, we perform a pull, where a page from a
remote machine (that caused the fault), is transferred to the local machine
and the process is resumed. The process will be able to make progress, as the
page that is being accessed (and caused a fault) is now local.
If space is needed to perform a pull, we can perform a push to free up memory
for the incoming page by transferring a page to a remote node (that we have
stretched the application to). Push (and pull) is more versatile, as they can
be performed proactively as well – moving pages around, in the background, to
optimize the placement for locality (label 3).
The idea of locality is important, especially in regards to our final
primitive, jump. Assuming that programs have locality, there is a certain
point at which, when we transition into a new pocket of locality, that the
amount of data that forms that locality is high. It is therefore advantageous
to jump execution to the data, rather than pull it all into the local node (as
is done in network swap). In the figure, in steps (4 and 6), the area
highlighted in red represents an island of locality that would be more
advantageous to jump to rather than pulling the entire group of pages to the
local machine. When to jump is an important decision – jumping too much can
hurt performance (constantly transferring execution, without making progress),
but not jumping enough can also hurt performance (transferring lots of data
back and forth between machines). As such, we created an initial algorithm,
and implemented it as a flexible module within which new decision making
algorithms can be integrated seamlessly.
## 3 ElasticOS Architecture
Figure 3: EOS Architecture.
In this section, we describe the main components of the ElasticOS
architecture. ElasticOS can be built as a service integrated into existing and
commercially-available operating systems. Figure 3 illustrates the main
functional elements that enable a process (e.g., a.out) to be stretched for
distributed execution over two ElasticOS nodes. For clarity purposes, we
depict the perspective of pushing and pulling from the perspective of node 1,
but in reality all nodes have symmetric capabilities to enable pushing,
pulling, and jumping in all directions.
In the remainder of this section, we will provide a more detailed
architectural overview focusing on mechanisms that are roughly OS-independent
in order to achieve stretching (3.1), pushing (3.2), pulling (3.3), and
jumping (3.4). The discussion of OS-dependent elements specific to the Linux
implementation is reserved for Section 4.
### 3.1 Stretching
Figure 4: Stretching.
Stretching is responsible for enabling a process to span multiple nodes. This
consists of an initial stretch operation, as well as on going synchronization.
Initial stretch operation: In order for a process to span multiple nodes, it
needs a process shell on each machine. In this way, stretching resembles
previous Checkpoint/Restore (C/R) works [8, 3], except that less information
needs to be written into the checkpoint. Here we will need to create a process
shell that will remain in a suspended state rather than wholly-independent
runnable replica. This makes stretching faster than standard C/R. It requires
kernel-space process meta-data. These include virtual memory mappings (mmaps),
the file descriptor table, scheduling class, and any other meta-data which is
not updated frequently. Other information that is typically modified at a high
rate such as pending signals, register state, and stack frames need not be in
the checkpoint and will be carried over from the running process whenever it
jumps (3.4).
As shown in Figure 4, stretching is triggered by the EOS manager, which
continuously monitors process’ memory usage and issues a newly-created signal
(SIGSTRETCH) whenever it detects a process that is too big to fit into the
node where it is running. Our special kernel-space handler (eos_sig_handler)
intercepts the signal and instructs the process-export module (p_export) to
send the checkpoint using a pre-created TCP socket to a process-import module
(p_import) waiting in the other node. The latter will, then, create a shell
process by allocating the necessary kernel-space structures and filling them
in with checkpoint data.
State Synchronization: After the process has been stretched, and its replica
has been created on another machine, additional changes in process state on
the first machine will need to be propagated to the replica. This is handled
in two ways. Rapid changes in state are handled using the jumping mechanism,
as explained later. Changes in state at a more intermediate time scale such as
mapping new memory regions and opening or closing files are handled using
multicast sockets to listeners on each participating node.
One of the pitfalls to avoid here is that the operating system scheduler may
delay flushing all such synchronization messages until after a jump is
performed. If this happens, the system may arrive at an incorrect state or
even crash. So, it is crucial to flush all synchronization message before a
jump is performed.
### 3.2 Pushing
Now that the process has presence on more than one machine, its memory pages
are pushed between nodes in order to balance the load among participating
nodes. Our page pusher piggybacks on existing OS’s swap management (See Figure
5).
Typically, the swap daemon scans least-recently used (LRU) lists to select
least recently used page frames for swapping. Our page balancer modifies this
page scanner in order to identify pages mapped by elasticized processes
(shaded pages in Figure 5) using reverse mapping information associated with
the page. These are then sent to a virtual block device client (VBD), similar
to the one described in [10], after updating the respective page table entries
(PTEs) in the elastic page table. The VBD then forwards the page along with
relevant information such as process ID, and the page’s virtual starting
address to the page injection module (pg_inject) on the node, which will then
allocate a new page, fill it with the proper content, and update the replicas
elastic page table.
Figure 5: Pushing.
Maintaining accurate information in the elastic page tables when pushing pages
is very crucial to correct execution. As we will see later, jumping depends on
this information for locating pages in the system.
### 3.3 Pulling
Partitioning the process’s memory footprint will, inevitably, result in
references to remote pages. These are handled by our modified page fault
handler (Figure 6). On a page fault, the handler will consult the elastic page
table to identify the page’s location. If it happened to be on a remote node,
the page’s starting virtual address and process ID is forwarded to the VBD,
which will then contact the page extraction module (pg_extract) on the
respective node to pull the page. Once it receives the page’s content, the VBD
client, then restores the process’s access to the page.
Whenever a remote page fault is handled as described above, page fault
counters are updated. This is required by ElasticOS’s jumping policy (Section
3.4), which will always try to co-locate execution with its most-referenced
memory.
Figure 6: Pulling.
### 3.4 Jumping
Figure 7: Jumping.
Jumping is the act of transferring execution from one node to another. For
this, there is both a jumping mechanism that performs a lightweight process
migration, and the jumping policy to determine when to jump.
Jumping mechanism: Jumping is an lightweight mechanism similar to
checkpoint/restore systems. In contrast to stretching, with jumping, the
process does actually transfer execution, and only carries in the checkpoint
the information that changes at a high rate. This includes CPU state, the top
stack frames, pending signals, auditing information, and I/O context. The
overall size of jumping checkpoint data is dominated by the stack frames, so
it is very important to include only the topmost stack memory pages that are
necessary for correct execution.
As shown in Figure 7, whenever a jump is deemed necessary by the jumping
policy in the EOS Manager, it sends a special signal (SIGJUMP) to the process,
which is then routed to the eos_sig_handler which will then instruct the
p_export module to checkpoint the process and send the information to the
other node’s p_import module. The latter will fill in the appropriate kernel-
space structures and set the process’s state to runnable. Notice here that
when jumping, no new structures need to be allocated since the process has
been already stretched to the target node. Also, notice that the process at
the source node will remain in a suspended state. In essence, jumping
resembles rescheduling a process from one CPU to another across the boundaries
of a single machine.
Jumping Policy Algorithm: Maximizing locality is crucially important to the
application’s performance. A naive approach to moving execution and memory
pages around in the system will, inevitably, increase the rate of remote page
faults leading to poor performance. Thus, a good policy for moving processes
close to their most frequently used memory is of critical importance.
ElasticOS can achieve this goal by overcoming two challenges, namely having a
good sense of how to group inter-dependent memory pages together on the same
node, and detecting which of those groups is the most frequently accessed one.
The first challenge can be overcome by taking advantage of the natural
groupings memory pages belonging to an application tend to form due to recency
of reference. This property is already evident in the wide adoption of the LRU
algorithm for page replacement in most modern OSs. Thus, we can extend LRU
algorithms to work in a multi-node system, where pages evicted from one node’s
RAM are immediately shipped to another node via our pushing mechanism.
The second challenge can be addressed by implementing a jumping policy that:
1) monitors the process’s page accesses to find the ”preferred” node, and 2)
reschedules the process to the preferred node if it is running on any of the
other ones.
Bear in mind that accurately tracking memory references for a particular
process can be a challenging task since CPUs do not report every memory access
to the OS for performance reasons. This leaves us with options that provide
the ”next best thing”, such as counting the number of time the CPU sets
PG_ACCESSED flag for a particular page frame when it is accessed in the X86_64
architecture or tracking handled page faults.
## 4 ElasticOS Implementation
We implemented ElasticOS as a fork of the Linux kernel v2.6.38.8. We chose the
2.6 kernel because it contains the key features that we needed to demonstrate
elasticity, e.g. support for 64-bit x86 architectures and a reasonably
featured virtual memory manager, while avoiding unnecessary complexity and
instability in later kernels.
System Startup: Whenever a machine starts, it sends a message on a pre-
configured port announcing its readiness to share its resources. The message
includes two groups of information. First, connectivity parameters such as IP
addresses and port numbers. Second, information noting the machine ’s
available resources, which includes total and free RAM. Next, each
participating node records the information received about the newly-available
node and initiates network connections for the various clients. Finally, EOS
manager is started, which will periodically scan processes and examines their
memory usage searching for opportunities for elasticity.
Identifying such opportunities can be achieved by examining the per-process
counters Linux maintains to keep track of memory usage. They include: 1)
`task_size` inside each process’s memory descriptor (i.e., `struct mm_struct`)
which keeps track of the size of mapped virtual memory, 2) `total_vm` inside
the same structure to track the process’s mapped RAM pages, 3) `rss_stat` of
type `struct mm_rss_stat` which contains an array of counters that further
breaks down `task_size` into different categories (i.e., anonymous and file-
mapped RAM pages used, and swap entries), and 4) `maj_flt` variable inside the
`struct task_struct` which counts the number of swap-ins triggered by the
process.
Linux also maintains memory utilization indicators called watermarks. There
are three levels of watermarks: min, low, and high. These levels drive the
kernel swap daemon’s (kswapd) activity. When memory usage reaches the high
watermark, page reclaim starts, and when it goes down to low watermark, page
reclaim stops.
ElasticOS leverages these watermarks and the level of kswapd’s activity to
detect periods of memory pressure. Further, it identifies specific memory-
intensive processes using the counters mentioned above and marks them for
elasticity.
Stretching Implementation: The Linux kernel forces each process to handle
pending signals upon entering the CPU. This is when our in-kernel signal
handler, the p_export module, checks for pending ElasticOS-specific signals.
This design choice of checkpoint creation logic placement gives us access to
register state, while preventing the process from updating its own memory
while a checkpoint is in progress.
The handler, then, accesses the process information while writing them to a
socket initialized during system startup. At the other end of the socket, the
p_import module collects the information and uses it to create the new shell
process.
The key items that are included in this checkpoint consist of: contents of the
process descriptor (struct task_struct), memory descriptor and (struct
mm_struct) virtual memory mappings (struct vm_area_struct), open files
information (struct files_struct), scheduling class information (struct
sched_class), signal handling information (struct sighand_struct), and few
others. The overall size of the this checkpoint in our experiments averages
around nine kilobytes, which are dominated by the size of the process’s data
segment which is also included in the checkpoint.
Note, that we do not need to copy memory pages containing the code, since our
implementation assumes that the same file system is available on all
participating nodes. Instead, we carry over with the checkpoint data the
mapped file names. Our p_import module will locate and map the same files at
the appropriate starting addresses.
P_import handle the process creation the same way as if it were forked locally
while substituting missing values with others from the local machine. For
example, it assigns the newly created process a ”baby sitter” to replace the
real parent from the home node.
Finally, the p_import module leaves the newly created process in a suspended
state and informs the p_export module that it can allow the original process
in the source node to resume execution.
Pushing and Pulling Implementation: We extend Linux’s second-chance LRU page
replacement algorithm by adding multi-node page distribution awareness to it.
In this version, pages selected for swapping out belong to elasticized
processes and are pushed to another node and injected into the address space
of the process’ duplicate there. Second-chance LRU groups pages in reference-
based chronological order within the pages list. So, it is most likely that
pages at the rear of the queue, which are typically considered for eviction,
are related in terms of locality of reference.
One challenge that needed to be solved to implement page balancing is
identifying pages belonging to an elasticized process and what virtual address
they are mapped to. Luckily, Linux maintains a functionality called reverse
mapping, which links anonymous pages to their respective virtual area map. By
walking this chain of pointers and then finding which process owns that map,
we can tell them apart from other pages owned by other processes in the
system. Then, with simple calculations we can find the starting virtual
address of that page. As for moving pages from one machine to another, we
created a virtual block device (VBD) that sends page contents using a socket
connected to a page server on the other machine (VBD Server) rather than
storing it to a storage medium. This was shown in Figure 6. This virtual block
device is added to the system as a swap device. All pages belonging to an
elasticized process sent to the other machine are allocated swap entries from
this device. This swap entry is inserted into the page table of the
elasticized process where the page is mapped. As a result, if that page needs
to be faulted in later on, the swap entry will route the page fault to our
VBD. This design choice allows us to reuse Linux’s page eviction and faulting
code.
Jumping: Whenever a remote page fault occurs, a remote page counter is
incremented. We keep track of the number of remote page faults to use it later
on for jumping. As the page remote fault counter builds up, it will show the
tendency of where page faults are ”going”. If the remote faults count value
hits a predetermined threshold, then the system could determine that the
process would better exploit locality of reference if it jumps to the remote
node. Jumping starts by sending a special signal to the target process, which
is handled by an in-kernel checkpoint module. This module will, then, copy
only the necessary information for the process to resume on the other node.
This information includes: 1) the thread context, which contains the register
state and other flags, 2) pending signals (i.e., struct sigpending contents
inside struct task_struct), 3) auditing counters, and 4) the stack page frames
(i.e., RAM pages mapped by the vm_area_struct with the flag VM_GROWSDOWN set).
In our tests, the checkpoint size was roughly 9KBs and was dominated by the
two stack page frames (4KBs each). Other information about the process will be
synchronized using a special module described next. These pieces of
information are sent to the restart module in the remote node via a pre-
established TCP connection. In its turn, the restart module updates the
process information with the checkpoint data, and sends a (SIGCONT) to the
process. This will inform the scheduler that it is ready to run again. The
process on the source machine will remain in an interruptible wait state
(i.e., suspended). This will guarantee that only one clone of the process is
running at any given instance.
State Synchronization: The state synchronization component is built as a
collection of user-space programs and a kernel module. The user space portion
simply sets up the network connections and then passes their socket
descriptors to the kernel module, which exposes hook functions to the main
kernel.
When an elasticized process issues a system call that modifies its in-kernel
data structures (e.g., mmap), the appropriate kernel module hook function is
called (e.g., sync_new_mmap), which will then multi-cast a message to all
participating nodes. The message will contain all necessary information (e.g.,
region’s starting address, its length, mapping flags, and file name) to apply
the same operation on all process replicas. Multi-cast listeners, then, relay
the message to the appropriate hook functions, who will apply the change
(i.e., call mmap on the process replica).
## 5 Performance Evaluation
In this section, we focus on evaluation of the performance of ElasticOS.
Specifically, we look to quantify the benefit of joint disaggregation (memory
and computation) by comparing against network swap, which is a one dimensional
(scaling memory), which has previously been shown to have performance benefits
over not scaling memory [20, 10]. We note that we do not explicitly compare
against just process migration, as the use cases are different, where
process/VM migration is commonly used to move execution permanently and
triggered by contention for resources or for other operational reasons (e.g.,
planned maintenance) – making it heavier weight and not well suited for
comparison.
### 5.1 Experimental Setup
We evaluated ElasticOS on the Emulab testbed [9]. We used Emulab D710 nodes
with 64-bit Quad Core Xeon processor, 12 gigabytes RAM, and a gigabit NIC. We
choose Emulab D710 nodes because they support Linux kernel 2.6. Our
experimental setup for each experiment consists of two nodes connected via
gigabit Ethernet ports, transported through a network switch.
To evaluate, we ran tests on a variety of algorithms representing the type of
processing that would be a target use case for ElasticOS – large graphs or
lists to be processed. Shown in Table 1 is a summary of these applications,
and the footprint of each application – note that the footprint goes beyond
the limits of a single node in Emulab. Specifically, these algorithms
typically use 11GB of memory on the first machine, and stretch to a remote
machine for the additional memory.
Table 1: Tested algorithms and their memory footprints. Algorithm | Memory Footprint
---|---
Depth First Search | 330 million nodes (15 GB)
Linear Search | 2 billion long int (15 GB)
Dijkstra | 3.5 billion int weights (14 GB)
Block Sort | 1.8 billion long int (13 GB)
Heap Sort | 1.8 billion long int (14 GB)
Count Sort | 1.8 billion long int (14 GB)
In our experimental setup, we employed a basic jumping algorithm to trigger
transfer of execution. A simple remote page fault counter is updated for each
remote pull, and whenever a counter threshold value is reached, then a process
will jump its execution to the remote machine. In addition, the counter is
then reset. We tested the algorithms with different counter threshold values
(32 up to 4M).
For each algorithm, we measure its execution time as well as network traffic
generated, and compare results of ElasticOS and network swap. To provide a
comparison with network swap, hereafter termed Nswap, in a manner which
isolated the gains to simply the benefit of jumping and not any implementation
differences, we use ElasticOS code, but disable jumping. In this way, Nswap
tests pin a process on one machine, but use the memory of a remote machine as
a swap space. In our experiments, both ElasticOS and Nswap spanned two
machines. Emulab provides isolation of networking and execution in the testbed
from external disturbances.
Table 2: Micro-benchmarks of ElasticOS primitives. Primitive | Latency | Network Transfer
---|---|---
Stretch | 2.2ms | 9KB
Push | 30-35us | 4KB
Pull | 30-35us | 4KB
Jump | 45-55us | 9KB
### 5.2 Micro-benchmarks
An important metric when evaluating ElasticOS is the performance of each
individual primitive. These are summarized in Table 2, based on our
measurements on Emulab D710 nodes. We’ll note that jumping is very fast,
taking only 45-55 microseconds. This is substantially lower than reported
numbers for process or VM migration, which are measured in seconds (e.g., one
benchmark states CRIU’s downtime is roughly 3 seconds [23]). Stretching is
also only performed once – when a decision is made that this process would
benefit from scaling in the future.
We measured pushing and pulling to be between 30-35 microseconds – roughly the
time to submit the request and transfer a page (4KB) of data across a network.
For jumping to be effective in speeding up execution in ElasticOS, there must
be locality. That is, the time for a single jump must be less than the time
for the number of remote page pulls that would be saved by jumping. For our
performance microbenchmarks, for jumping to be efficient, the process should
save at least two remote page pulls. As we show next, the locality is much
greater than this, resulting in substantial speedups.
Figure 8: Execution Time Comparison.
### 5.3 Execution Time and Network Traffic
There are two key metrics to consider when comparing ElasticOS (with jumping,
pulling and pushing), to network swap (just pulling and pushing). The first is
overall execution time. Here, the key premise behind jumping is that to
exploit locality, we should transfer execution to where the data is, rather
than pull in the data to where the execution is. The second is the amount of
network traffic – jumping needs to transfer context (e.g., the current stack),
and pulling/pushing transfers pages.
In Figure 8, we show our measured average execution time for both Nswap and
ElasticOS for each of the algorithms we have evaluated. These execution times
are averaged over four runs using the threshold that achieves the most
improvement. We observe that in the best case, ElasticOS shows substantial
performance benefits for most algorithms. For example, Linear Search
experienced about an order of magnitude speedup in execution performance,
Depth First Search (DFS) achieved about 1.5X delay improvement, while
Dijkstra’s algorithm achieved no speedup.
Table 3 describes the specific threshold values where best performance was
achieved in ElasticOS for each algorithm. It also lists the total number of
jumps at that threshold as well as the frequency of jumping for each algorithm
at that threshold. The jumping rate ranges from less than once per second to
hundreds of times per second.
While Figure 8 represents the best case, we were also interested in
understanding whether we could find universal threshold values that achieves
performance improvements - perhaps not the best - regardless of the algorithm.
Our analysis found that, regardless of the algorithm, using any threshold
value above 128, Elastic OS performs better than Nswap for any algorithm,
either in delay, network overhead or both.
The use of jumping to exploit locality improves the execution time by enabling
more local pages to be accessed, rather than needing to go across a network
(which is orders of magnitude slower). This also reduces the amount of network
traffic, even taking into account the data transfer needed to perform a jump.
Shown in Figure 9 are our measured results for each of the algorithms tested.
We can see that ElasticOS reduces the amount of traffic on the network for all
algorithms tested by a significant amount – from a 5x reduction for Linear
Search to about 2x reduction for DFS. By avoiding the process of swapping in
and out to remote machines through lightweight jumping, we save a large amount
of data and control traffic associated with avoidable remote page faults.
Also, even if we did not achieve any delay improvements running ElasticOS, we
still can obtain network traffic reduction. For example, Dijkstra’s algorithm
did not achieve any delay improvement, even though Table 3 shows that Dijkstra
had 520 jumps, but these jumps helped reducing its network overhead by 70%. In
examining the behavior of Dijsktra’s, its initial set of jumps before settling
down to execution on one machine resulted in substantial overhead savings.
Table 3: Jumping Thresholds. Algorithm | Threshold | Number | Jumping
---|---|---|---
| | of jumps | frequency
| | | (jumps/sec)
DFS | 8K | 180 | 0.6
Block Sort | 512 | 1032 | 12.3
Heap Sort | 512 | 3454 | 12.4
Linear Search | 32 | 3054 | 157.4
Count Sort | 4096 | 198 | 0.6
Dijkstra | 512 | 520 | 1.4
Figure 9: Network Traffic Comparison.
### 5.4 Understanding Application Specific Behavior
We previously showed that each algorithm has a varying degree of improvements.
While the simple reasoning is that it is due to locality, here we examine
three of the algorithms in detail to really understand this behavior.
#### 5.4.1 Linear Search
For Linear Search, the memory access pattern is simple and predictable, namely
the memory address space is accessed in a linear fashion. As a result,
consecutive memory pages tend to age in LRU lists together, and end up being
swapped to the remote machine together. When a process jumps towards a remote
page, it is very likely for the process to find a chunk of consecutive pages
to access, exploiting locality of these pages, which saves the process a
significant amount of time by avoiding swap overhead. Figure 10 shows delay
improvements on Linear Search with respect to jumping threshold. Linear Search
tends to perform better when the counter threshold is smaller, hence jumping
early is better when accessing the address space in a linear fashion. Table 3
shows the highest frequency of jumping for linear search, as well as the
lowest threshold value used. We also observe that as the threshold for jumping
increases, jumping will occur less often and eventually not at all, hence
explaining why the delay curve for ElasticOS converges to Nswap.
Figure 10: Linear Search Execution Time.
#### 5.4.2 Depth First Search
On the other hand, Depth First Search has a non linear memory access pattern.
When the algorithm starts a depth first search, the search starts at the root
node, and traverses the graph branch by branch, from root to the end (depth)
of the branch. While the graph nodes are laid out in a certain order in the
memory space, the access pattern of DFS does not match this layout. This
increased randomness of access to pages means that there is less locality to
exploit on each jump than occurred for Linear Search, and hence less gain
versus Nswap compared to Linear Search. Figure 11 shows different execution
times of DFS for various counter threshold sizes. ElasticOS achieves at best
about a 1.5x improvement in delay over Nswap across a wide range of counter
thresholds, namely larger than 64. However, for very small values of threshold
less than or equal to 64, DFS performs worse. Figure 12 shows that when the
threshold value is very small, DFS experiences a large number of jumps. Also,
our tests showed that DFS’s best performance happens when the threshold value
is large compared to other algorithms as shown in Table 3.
Figure 11: Depth First Search Execution Time. Figure 12: Depth First Search
Number of Jumps.
The shape of the graph in DFS can also impact the memory access pattern. For
example increasing the depth of the graph would make branches longer,
resulting in a longer branch that occupies more memory pages, increasing the
chance of a single branch having pages located both on local and remote
machines. This would increase the chances of jumping more and performing
poorly. Figure 13 shows DFS performance on ElasticOS for different graph
depths with a fixed jumping counter size of 512. Increasing the graph depth
eventually results in poorer performance. Figure 14 shows that this poorer
performance occurs when there is excessive jumping for deep graphs. To make
ElasticOS perform better on such graph depth we need to increase the jumping
counter size to values larger than 512, to avoid jumping too much.
Figure 13: Depth First Search Performance on Different Depths. Figure 14:
Depth First Search Jumps on Different Depths.
#### 5.4.3 Dijkstra’s Algorithm
ElasticOS achieved very little gain when executing Dijkstra’s algorithm when
compared to Nswap. Dijkstra’s algorithm scans through an adjacency matrix,
then learns and stores information about the shortest path in a separate
array. However, Dijkstra does not necessarily access all nodes in the
adjacency matrix, because some nodes are not connected, or one of the paths
was excluded for being too long. Since Dijkstra’s algorithm keeps track of the
shortest path nodes in a separate array, it only accesses the adjacency matrix
nodes once, and keeps useful information in the shortest path array. Based on
how Dijkstra’s algorithm works, it does not access memory frequently, and only
accesses part of the allocated memory. Therefore, most of Dijkstra’s execution
time does not involve many remote page faults. Since jumping saves time wasted
on remote page faults, Dijkstra does not gain much delay improvement, because
it does not jump due to very small number of remote page faults. Figure 15
confirms that Dijkstra’s algorithm spends most of its execution time on one
machine without jumping. Our experiments showed that only a relatively small
set of jumps happened at the beginning, and the rest of the time execution
stayed on one machine.
Figure 15: Maximum Time Spent on a Machine without Jumping.
## 6 Discussion and Future Work
We intend to upgrade ElasticOS to a newer version of Linux. We plan to
investigate improved jumping algorithms that better exploit locality by
actively learning about elasticized process’ memory access patterns during run
time and employing adaptive jumping thresholds. Probabilistic models will be
investigated. In addition, we will explore whether incorporating into the
jumping decision the burstiness of remote page faulting brings any benefit.
Also, we are considering a more proactive approach to controlling the swap out
operation for elasticized processes by modifying kswapd. If we selectively
swap out pages to remote machines, we might be able to create islands of
locality on remote machines, thus, making jumping more efficient. We also can
pin memory pages, and prevent them from being swapped, which would allow us to
control how the memory address space is distributed across participating
machines. We plan to test a wider variety of algorithms, including SQL-like
database operations. We intend to expand testing to more than two nodes.
## 7 Conclusion
In this paper, we have implemented within Linux four new primitives, namely
stretch, push, pull, and jump, to support scaling as an OS abstraction. This
extended Linux system is called ElasticOS. These primitives transparently
achieve joint disaggegration of computation and memory, enabling both data to
move towards execution, as well as execution to move towards data within a
stretched address space spanning multiple nodes in a data center. Our
evaluation results were obtained from Emulab deployments of ElasticOS testing
a variety of different application algorithms, and indicate that such joint
disaggregation achieves up to 10X speedup in execution time over network swap,
as well as 2-5X reductions in network overhead.
## 8 Acknowledgments
This research was supported by NSF CCF grant # 1337399, funded under the NSF
program Exploiting Parallelism and Scalability ”XPS: SDA: Elasticizing the
Linux Operating System for the Cloud”. We also wish to thank Sepideh Goodarzy
and Ethan Hanner.
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|
# Pathological limits in statistical mechanics
C. Y. Chen Beihang University, PRC, Beijing 100191<EMAIL_ADDRESS>
###### Abstract
It is unveiled that some of the multi-variable limits used in statistical
mechanics have been ill-defined in the mathematical sense. Along the line, it
is suggested that significant progresses in non-equilibrium gas dynamics can
be made by redefining, reinterpreting, and reformulating those limits.
Keywords: Boltzmann’s equation, Phase space, Liouville’s theorem,
Discontinuity.
## I Introduction
In mathematics, the term “pathological object”, or “monster”, refers to
something whose behavior is unexpectedly bad, counter-intuitive, and
inexplicable in terms of supposedly relevant theories. As a matter of fact,
there are popular bookscounter1 ; counter2 ; counter3 titled ”Counterexamples
in analysis”, ”Counterexamples in topology” and ”Counterexamples in
probability”, in which many of such counterexamples can be identified as
typical pathological objects.
One of the reasons behind those pathological objects is that basic concepts in
analysis, such as infinitely small, infinitely large, limit and continuity,
are simple but complex. If explained with help of examples, they appear to be
intuitive and rudimentary; however if explained by the textbook concepts,
which is intended to be theoretically rigorous and generally applicable, they
possess certain amount of abstraction and intricacy. After they are
incorporated with other concepts to build up more complex objects, the
abstraction and intricacy possessed by them may turn themselves into more
tricky ones. Professional mathematicians are fully aware of the risk. When
they propose a new theory in mathematics, they spend, quite often, a lot of
time and effort to recheck every building block of the theory.
Understandably, nonmathematicians pay less or no attention to things like
that. According to some prominent physicists, mathematics could be described
as a tool unreasonably effective in physical scienceswigner . This description
of mathematics has inspired and continue to inspire physicists to apply
mathematics more diligently and more creatively. But one question may still
remain in the mind of some physicists: In what sense and to what extent should
we, as physicists, also pursue the mathematical rigor? Considering that a
simple but ambiguous answer to this question would help no one, this paper is
devoted to an example exemplifying that pathological objects in mathematics
and half-baked ideas in physics may somehow find each other and result in
long-lasting mistakes (reminding us of Murphy’s law: anything that can go
wrong will go wrong).
The concrete topic of this paper is related to a number of multi-variable
limits used in statistical mechanics. Unlike the papers on multi-variable
limits in mathematics, the language, scope, perspective and objective of this
paper are mostly physics-oriented. Physicists and applied mathematician should
be able to find things fundamentally interesting and important.
## II Partial limits of a multi-variable limit
In the conceptual sense the multi-variable limit is not much different from
the single-variable limit, but dealing with multi-variable limits is much more
complex. As far as mathematics is concerned, the main problem is that a
considerable number of multi-variable “limits”, though looking like and
seeming like a limit, are actually ill-defined and must be forbidden from
mathematical use. (This paper will show that the problem in physics becomes
how to define and use variants of such ill-defined “limits”.) In this section,
we look at this issue from the mathematical perspective.
To get some taste of it, let’s first examine the expression below:
$\lim\limits_{(x,y,z)\to(0,0,0)}\frac{x^{2}+y^{2}+\sin(3z^{2})}{x^{2}+z^{2}}.$
(1)
Its limiting behavior around the limit point $(x,y,z)=(0,0,0)$ can be
investigated through the following if-then (hypothesis-conclusion) analysis.
If we let $x$ and $y$ tend to zero much faster than $z$ does, then the
expression goes to $3$. If $x$ is kept relatively stationary while letting $y$
and $z$ go to the infinitesimal at a regular pace, independently or in any
combined way (e.g., $y^{2}=z^{2}\to 0$ or $y^{2}+z^{2}\leq\rho^{2}\to 0$),
then the expression approaches the value of $1$. If $x$ goes to zero at the
fastest pace, then the expression takes on a value from $3$ to $\infty$,
depending on in what way the two remaining variables, $y$ and $z$, go to their
own zeros. This analysis shows that a multi-variable “limit” can indeed have
multi behavior and the multi behavior can be revealed by performing several
if-then cycles (as long as they are designed properly).
In today’s mathematics, there is a special rule governing such multi-variable
“limits”courant ; thomas , and the rule may be described as follows: the limit
of a function can be considered as legitimately defined if and only if the
function approaches a unique and definite value no matter in what way the
function’s variables proceed with their limiting processes. Different people
may describe the rule in different manners. For instance, it can be
interpreted as saying that a well-defined limit must be path-independent, in
which the term path refers to a path in the variable space that leads to the
limit point. Or, it can be stated like this: a well-defined limit must be
hypothesis-free, by which it is meant that the unique and definite value of
the limit must not be associated with an explicit or implicit hypothesis
(presupposition).
As said already, revealing multi behavior of a multi-variable “limit” can be
accomplished by performing several well-designed if-then cycles. To make such
analysis be a symbolically concise approach, we introduce the following
concepts and notations.
Consider, for instance, a three-variable function $f(x,y,z)$ whose limiting
behavior at a limit point $(a,b,c)$ is of concern. The expression
$f([\hskip 1.0ptx\rangle_{a},[\hskip 1.0pty\rangle_{b},[\hskip
1.0ptz\rangle_{c})\equiv\lim\limits_{(x,y,z)\to(a,b,c)}f(x,y,z)$ (2)
will, hereafter, be called an intended limit (rather than a limit). By the
term “intended limit”, it is meant that there is a large chance that this
“limit” is ill-defined.
Each of the following expressions will be regarded as a partial limit produced
from $f([\hskip 1.0ptx\rangle_{a},[\hskip 1.0pty\rangle_{b},[\hskip
1.0ptz\rangle_{c})$:
$f([\hskip 1.0ptx\rangle_{a},y,z),\,f([\hskip 1.0ptx\rangle_{a},[\hskip
1.0pty\rangle_{b},z),\,f([\hskip 1.0ptx{\&}y\rangle_{(a,b)},z),\,\cdots,$ (3)
in which the variables enclosed in $[\hskip 1.0pt..\rangle$ are called the
active ones approaching their limits promptly, the variables not enclosed in
$[\hskip 1.0pt..\rangle$ are called the inactive ones whose limiting processes
are suspended until the active variables finish their limiting processes, and
the variables connected by “&” represent the ones behaving themselves in a
combined way. (When considering partial limits in physics, inactive variables
will be allowed to be inactive permanently.)
Take expression (1) as an example. It has quite a few partial limits, of which
the following three can be evaluated like single-variable limits:
$\displaystyle f(x^{2},[\hskip 1.0pty^{2}\rangle_{0},[\hskip
1.0ptz^{2}\rangle_{0})\to{x^{2}}/{x^{2}}\to 1,$ (4) $\displaystyle f([\hskip
1.0ptx^{2}\rangle_{0},y^{2}=z^{2})\to[z^{2}+\sin(3z^{2})]/z^{2}\to 4$ (5)
and
$f([\hskip 1.0ptx^{2}=z^{2}\rangle_{0},y^{2})\to{y^{2}}/{0}\to\infty.$ (6)
A simple observation of these expressions tells us that a partial limit’s
evaluation corresponds to an if-then cycle.
A conclusion is now evident: an intended limit can be identified as ill-
defined if an infinity or two different finite values emerge from the
evaluation of its partial limits. (In this paper, $\infty$ is not considered
as a legitimate value of a limit due to the two facts: i. $\infty$ is not a
definite value. ii. No measurable quantity in physics is $\infty$-valued.) As
an application of the conclusion, we can, by virtue of Eqs. (4) and (5) or by
virtue of Eq. (6) alone, deduce that expression (1) is ill-defined.
In mathematics (and in physics) there are quantities whose behavior is limit-
like from one perspective and function-like from another perspective.
Interestingly, whether they can be regarded as well-defined (or whether they
have multi behavior) can be investigated with help of the concept of partial
limit.
As an example, let’s suppose our task is to reveal the behavior of a two-
variable function defined as
$g(x,y)=\frac{1}{x^{2}y^{2}+1}$ (7)
around and along the $y$-axis.
It is obvious that
$g([\hskip 1.0ptx\rangle_{0},y)\equiv\lim_{x\to 0}g(x,y)=1\quad({\rm
for\;any\;definite}\;y).$ (8)
This result is impressive and may be of use for some purposes (such example in
physics will be given in the next section). But, can we thus claim that
$g(x,y)$ is an invariant around and along the $y$-axis? At first sight, this
claim can be proven with ease. Denoting $g([\hskip 1.0ptx\rangle_{0},y)$ as
$f(y)$, we obtain, from Eq. (8), $f(y_{0})=1$ and $f(y=y_{0}+\epsilon)=1$.
According to the usual conception, this can be rewritten as
$df/dy=0,$ (9)
which is synonymous to saying that $f(y)$ is an invariant along the $y$-axis.
From Eq. (9), one may further assert that $g(x,y)$ is an invariant around and
along the $y$-axis.
Figure 1: (a) $g=1/(x^{2}y^{2}+1)$ versus $y$ for different $x$; (b), (c) and
(d) $g=1/(x^{2}y^{2}+1)$ versus $x$ for different $y$.
Can this proof be taken for granted? In fact, we have two ways to refute it.
Firstly, let’s examine how the proof treats its hypothesis. Notice that
although $f(y)$ looks like a single-variable function of $y$, it is not. The
expression $f(y)$, or $g([\hskip 1.0ptx\rangle_{0},y)$, is and only is a
partial limit associated with the hypothesis that $[\hskip 1.0ptx\rangle_{0}$
prevails over $[\hskip 1.0pty\rangle_{\infty}$. When the above proof finally
concludes that $g$ is an invariant around and along the $y$-axis, this
hypothesis has been illogically disregarded and forgotten. Secondly, let’s
check out whether $g$ has multi behavior around and along the $y$-axis.
Obviously, the hypothesis that $[\hskip 1.0ptx\rangle_{0}$ prevails over
$[\hskip 1.0pty\rangle_{\infty}$ is not a unique hypothesis that we can
possibly have. If we let $x$ stay rather stationary and let $y$ go bigger and
bigger at a relatively fast pace, the value of $g$, denoted as $g(x,[\hskip
1.0pty\rangle_{\infty})$, will keep decreasing; namely, we have
$g(x,[\hskip 1.0pty\rangle_{\infty})\to 0\quad({\rm
for\;any\;definite\;}x\not=0).$ (10)
This behavior can be schematically seen from Fig. 1(a). The multi behavior
given by Eqs. (8) and (10) informs us that
$g([\hskip 1.0ptx\rangle_{0},[\hskip 1.0pty\rangle_{\infty})\to{\rm undef},$
(11)
in which undef means undefinable.
In summary, “the behavior of $g$ around and along the $y$-axis” involves two
competing processes: the first is related to “around” and the second to
“along”. If we choose either one of the two processes and let it prevail over
the other, the behavior of $g$ is well-defined. However, if we forget to make
the choice or allow the choice to be unclear, questing for “the behavior of
$g$ around and along the $y$-axis” is a self-misleading task.
It is also instructive to view the issue from the continuity and discontinuity
perspective. Figs. 1(b), 1(c) and 1(d) show that $g(x,y)$ behaves itself ever-
increasingly like a discontinuous function along the $y$-axis, which reminds
us of a plain fact that in a region where a certain type of discontinuity gets
involved defining a limit thereat is not really proper.
Several conclusive and/or speculative remarks are the following. Firstly, the
concept of partial limit is helpful in terms of identifying ill-defined multi-
variable limits. Secondly, if an “invariant” is established with help of
limiting processes, it is quite possible that the “invariant” is a partial
limit associated with an ad hoc hypothesis. Thirdly, considering a function’s
limiting behavior in a domain where a certain type of discontinuity gets
involved is in principle a risky business. Finally, if an intended limit in
physics proves to be ill-defined, we might be compelled to use some of its
partial limits as substitutes (with certain ad hoc hypotheses accepted).
## III The distribution function for collision-free and force-free gas
When used to describe a gas in equilibrium, the concept of distribution
function is definitely a well-defined limit. But, can it be qualified as well-
defined when applied in gas dynamics? Since no paper has addressed the
question, we may reasonably believe there is no need to be concerned. However,
the issue is not that whether the distribution function is well-defined in the
academic sense but that if the distribution function involves multi behavior
in terms of describing gas dynamics we will have no choice but to deal with it
properly. We shall here investigate the issue with a collision-free and force-
free gas being used as the testbed.
One of the stadard ways of dealing with the dynamics of an ideal gas can be
called the Liouville-theorem approachliou ; nolte ; lerner , and the
Liouville-theorem approach has the following four-level structure:
i. Liouville’s theorem (path-constancy of phase volume).
ii. Path-invariance theorem of distribution function.
iii. The collisionless Boltzmann equation.
iv. The collisional Boltzmann equation.
(Note that level i listed above should be considered as the “ground level”,
level ii as the level immediately upon the ground level, and so on so forth…)
Historically, there were a few people who doubted the validity of the
collisional Boltzmann equation, but almost no serious scientists have openly
challenged this four-level structure.
Another way to derive the collisional Boltzmann equation is through the so-
called fluid-mechanics approachreif ; dorf , in which the involved phase space
is divided into many many small phase volume elements, and, then, how
particles get in, and out of, these volume elements is investigated and
formulated.
A common consensus in the community is that the Liouville-theorem approach and
the fluid-mechanics approach are essentially equivalent. Another common
consensus is that if a computational work of this theory needs to be
performed, the algorithm should be based on the fluid-mechanics approach.
Oddly, it is relatively easy to find out weak points of this conventional
wisdom. Suppose that our task is to examine the dynamic evolution of a
Boltzmann-type gas that suffers rather scarce particle-particle collisions.
According to the four-level structure listed above, two approaches can be
taken. The first will herein be called the path-approach and the second the
equation-approach; and the two approaches are respectively related to the
first half and second half of the four-level structure. In the path-approach,
the path-invariance theorem (level ii) plays the central role, and it works
like this: with help of two inputs, the path-information given by Newtonian
equations and the initial distribution function, the theorem yields the
distribution function $f^{(0)}(t)$. Since particle-particle collisions are
ignored, $f^{(0)}(t)$ may be regarded as the zeroth-order solution. (What
happens on the boundary in this approach is determined after $f^{(0)}(t)$ is
obtained.) However, in the equation-approach, the collisional Boltzmann
equation (level iv) plays the central role, and it works like the following:
with help of two inputs, the boundary condition and the initial condition
given in advance, the equation yields its solution $f(t)$. Then, interesting
questions arise: Are these two solutions, $f^{(0)}(t)$ and $f(t)$, basically
consistent? If not, can we reconcile them in a systematic way? Why the path-
information gets direct involvement only in the path-approach not in the
equation-approach? How come the two approaches treat the boundary condition so
differently?
Figure 2: Demonstration of how a phase volume element of a collisionless and
force-free gas moves and evolves.
Leaving these questions aside for time being, let’s start our long exploration
with the first half of the Liouville-theorem approach. Consider the phase
space shown by Fig. 2 (given only in the $x-v_{x}$ plane for technical
reasons), in which a material phase volume element, denoted as
$\delta\beta\equiv\delta x\delta v_{x}$, is moving and evolving. Suppose that
the initial distribution function in the concerned region is well defined.
[The term “well-defined” is not truly simple. For the discussion here, let’s
assume that the particles in and around $(\delta\beta)_{0}$ initially belong
to a gas in equilibrium.] Since the volume of $\delta\beta$ remains unchanged
in the process (due to Liouville’s theorem) and no particles move out of, or
move in, $\delta\beta$ (due to the fact that the gas is collision-free), the
particle density at the center of $(\delta\beta)_{t}$ will remain unchanged.
On the basis of this reasoning, almost all textbooks concludelandau ; stocker
; huang ; harris that the distribution function in phase space behaves like
an incompressible fluid and the path-invariance theorem of distribution
function is a corollary of Liouville’s theorem.
Figure 3: An alternative view on how the distribution function evolves in the
$x$-$v_{x}$ phase space.
Interestingly, there is an alternative view from which we can get a different
conclusion. In Fig. 3, the initial distribution function $f_{0}(x,v_{x})$ has
been sliced into many thin slices according to their different $v_{x}$.
Obviously, as time proceeds each slice will move rightward with its own speed,
along its own path. Those slices, as a whole, will spread wider and wider in
the position dimension while each of them becomes thinner and thinner in the
velocity dimension. (We shall regard it as the velocity tearing effect
although the distribution function will not be truly torn apart.) If we are a
number of identical observers located one by one along a slice’s path, we will
definitely see that the number of the particles around a slice’s peak keeps
decreasing.
To comprehend these two figures, let’s do a bit of analytic work. When
thinking about the path-invariance theorem of distribution function, we must
concern ourselves with the following quantity:
$\hat{f}(\delta\beta,l)=\frac{\delta N(\delta\beta,l)}{|\delta\beta|},$ (12)
in which $l$ represents the length of a particle’s path and $\delta\beta$
stands for a small phase volume element whose “center” (defined by whatever
definition) is moving along the particle’s path, $|\delta\beta|$ is the volume
of $\delta\beta$ and $\delta N$ is the particle number in $\delta\beta$.
To most physicists who have been accustomed to the textbook treatment, the $l$
in Eq. (12) must be a definite (finite) one, and the $\delta\beta$ must be
infinitesimal (approaching zero rather freely). Namely, the distribution
function in their mind is nothing but the partial limit $\hat{f}([\hskip
1.0pt\delta\beta\rangle_{0},l)$. For convenience and clarity, we shall name
such distribution function the zero-volume distribution function; and denote
it as $f_{t}({\bf r},{\bf v})$ where $({\bf r},{\bf v})$ represents the
endpoint of the path $0\to l$. We then have, by virtue of Fig. 2,
$\hat{f}([\hskip 1.0pt\delta\beta\rangle_{0},l)=\hat{f}([\hskip
1.0pt\delta\beta\rangle_{0},0)$ (13)
or
$f_{t}({\bf r},{\bf v})=f_{0}({\bf r}_{0},{\bf v}_{0})$ (14)
in which $({\bf r}_{0},{\bf v}_{0})$ is the starting phase point of the path
$0\to l$. For historical reasons, we still consider Eq. (13) or (14) as the
path-invariance theorem of distribution function although the distribution
function here has been defined as the partial limit associated with the
specific hypothesis that $[\hskip 1.0pt\delta\beta\rangle_{0}$ prevails.
For the following four reasons, we realize that defining the zero-volume
distribution function is not sufficient. i. According to the last section, in
order to know the complete behavior of Eq. (12), we need to evaluate it under
different hypotheses. One of such different hypotheses is that $l$ becomes
larger and larger at a pace much faster than that of $\delta\beta\to 0$. ii.
In terms of measuring the distribution function along a path by physical
means, the involved $\delta\beta$ must be finite. iii. In any numerical
treatment, setting $\delta\beta$ to be infinitesimal is pointless. iv. It
turns out (in this paper and the other publications of the authorchen1 ) that
the gas dynamics of real gases possesses a number of mechanisms that can, and
will, produce particles distributing discontinuously so that defining the
zero-volume distribution function becomes very difficult or impossible. Thus,
we define
$f_{\delta\beta}(l)\equiv\hat{f}(\delta\beta,l),$ (15)
in which $\delta\beta$ stands for a phase volume element with a definite size
and a definite shape and $f_{\delta\beta}(l)$ represents a function of $l$
defined by Eq. (12). We shall regard such distribution function as a finite-
volume distribution function (or the average distribution function over the
volume element $\delta\beta$). As has been shown by Fig. 3, if such
$\delta\beta$ initially encloses a peak of the initial distribution function,
$f_{\delta\beta}(l)$ will decrease continuously along the path; and the end
behavior can be expressed by
$f_{\delta\beta}(l\to\infty)\equiv\hat{f}(\delta\beta,[\hskip
1.0ptl\rangle_{\infty})\to 0.$ (16)
(If one wishes, one can use the fluid-mechanics approach to confirm this path-
behavior.)
We now realize that neither the zero-volume distribution function nor the
finite-volume distribution function is sufficient in terms of describing the
path-behavior of a distribution function. While the zero-volume distribution
function can reveal more details of the local structure (if definable), the
finite-volume distribution function is more useful and more meaningful in many
other senses. [It turns out that in usual gases the velocity distribution will
vary more violently than the position distribution. Under this realization, we
may sometimes define $f([\hskip 1.0ptd^{3}{\bf r}\rangle_{0},d^{3}{\bf v})$
instead.]
As a matter of fact, we can use the zero-volume distribution function to
further uncover the necessity of using the finite-volume distribution
function. With reference to Fig. 3, let’s suppose that $f_{0}$ is well-defined
and that $\partial f_{0}/\partial v_{x0}$ is quite small
($\equiv\epsilon\approx 0$) in the region concerned. Then, we have
$\left.\frac{\partial f_{t}}{\partial
x}\right|_{x_{0}=x-v_{x}t,v_{x0}=v_{x}}=\left.\frac{\partial f_{0}}{\partial
x_{0}}\right|_{x_{0}}{\hskip 59.75095pt}$ (17)
and
$\left.\frac{\partial f_{t}}{\partial
v_{x}}\right|_{x_{0}=x-v_{x}t,v_{x0}=v_{x}}=-t\left.\frac{\partial
f_{0}}{\partial x_{0}}\right|_{x_{0}}+\epsilon.\hskip 30.0pt$ (18)
These two expressions are quite informative. In reference to Fig. 3 and Fig.
4, Eq. (17) shows that if the initial distribution function (deemed as the
source distribution function) has a definite profile in the position space,
the distribution function at a later time (deemed as the image distribution
function) will have infinitely many such profiles in the position space due to
the fact that each $\bf v$ can carry the source profile to a new place; and
Eq. (18) shows that when velocities carry the source distribution function to
a new place $\bf r$, a new profile in the velocity space will be produced, and
the new velocity profile is roughly the same as the source profile but the
size of it is expanded or compressed by the factor $t$.
Figure 4: Illustration of how the source distribution function creates many
new profiles (a) in the position space and (b) in the velocity space.
Armed with the notion that $\partial f/\partial({\bf r},{\bf v})\sim\infty$
can be used to characterize the discontinuity of distribution function, we
should be able to view the above gas dynamics from a new perspective. If
$\partial f_{0}/\partial{\bf r}_{0}$ is fairly large somewhere, the
distribution functions along paths will be increasingly like discontinuous
ones in the velocity space (due to the fact that $t|\partial
f_{0}/\partial{\bf r}_{0}|\to\infty$). We shall call this phenomenon the
dynamic quasi-discontinuity. Furthermore, if $f_{0}$ is already discontinuous
somewhere in the position space (caused by the initial condition or by the
boundary condition), the whole phase space will be full of discontinuity,
which can also be shown by Fig. 4.
From Fig. 4(b), it should be noted that the velocity distribution function at
any position $\bf r$ is a compressed image of the entire position space
(different position points have different affecting times though); and thus a
large or overwhelming fraction of a gas’s particles must involve quasi-
discontinuity and ordinary discontinuity. In this sense, the gas dynamics of
real gas must be based on the finite-volume distribution function which can
treat continuous and discontinuous particles in a unifying way.
Figure 5: Schematic of how to determine the finite-volume distribution
functions in the phase elements $(\delta\beta)_{t}^{i}$,
$(\delta\beta)_{t}^{ii}$, $(\delta\beta)_{t}^{iii}$ and
$(\delta\beta)_{t}^{iv}$ from the given initial distribution function.
Now, we give an example of how to evaluate the finite-volume distribution
function under the condition that the initial distribution function is of
piecewise continuity. In reference to Fig. 5, $(\delta x)_{1}$ and $(\delta
x)_{2}$ are two regions in each of which there exists a gas in its own
equilibrium, and the other regions are in the vaccum. Suppose that after $t=0$
all the walls confining the gases disappear suddenly and completely. Our task
is to determine the finite-volume distribution functions at a later time $t>0$
in the phase elements $(\delta\beta)_{t}^{i}$, $(\delta\beta)_{t}^{ii}$,
$(\delta\beta)_{t}^{iii}$ and $(\delta\beta)_{t}^{iv}$. The discussion given
in this section suggests that we should first locate the source regions of
those phase elements. By performing the time reversal, we find that the source
regions are $(\delta\beta)_{0}^{i}$, $(\delta\beta)_{0}^{ii}$,
$(\delta\beta)_{0}^{iii}$ and $(\delta\beta)_{0}^{iv}$ respectively. Since the
initial distribution function is well-defined in the piecewise sense, the
finite-volume distribution functions in $(\delta\beta)_{t}^{i}$,
$(\delta\beta)_{t}^{ii}$, $(\delta\beta)_{t}^{iii}$ and
$(\delta\beta)_{t}^{iv}$ can be easily evaluated with help of
$\displaystyle f_{\delta\beta}(t)$
$\displaystyle=\int_{(\delta\beta)_{t}}f_{t}({\bf r},{\bf v})d^{3}{\bf
r}d^{3}{\bf v}\Big{/}|(\delta\beta)_{t}|$
$\displaystyle=\int_{(\delta\beta)_{0}}f_{0}({\bf r}_{0},{\bf v}_{0})d^{3}{\bf
r}_{0}d^{3}{\bf v}_{0}\Big{/}|(\delta\beta)_{0}|.$ (19)
If the initial distribution function is not well-defined (for instance if the
particles come from the boundary-particle interaction), we can determine
$f_{\delta\beta}(t)$ by regarding each patch of the boundary surface as a
point-wise source of particleschen1 .
In some sense, the investigation done in this section has clearly shown the
advantages of the path-approach. i. Instead of giving an unsolved equation, It
gives directly treatable formulas. ii. The involved cause-effect relationship
is very clear and very straightforward and no unnecessary mistakes can be
made. iii. Dealing with continuity and discontinuity with almost the same
accuracy and efficiency.
In next several sections, we shall step by step take particle-particle
collisions into account.
## IV Scattering cross sections in different reference frames
To deal with particle-particle collisions between particle beams, several
types of scattering cross sections are employed in textbooks of statistical
mechanicsreif ; kubo . It turns out that one of the cross sections therin is
also ill-defined.
For simplicity, all particles considered in this paper are distinguishable and
subject only to short-range elastic interaction (in terms of classical
mechanics), but have the same mass, size and shape. Furthermore, throughout
this paper, the assumption of molecular chaos is taken for granted whenever a
gas system is in the consideration.
Although the usual examination in textbooks starts with a beam of particles
falling upon a resting target particle, we shall, for consistence of this
paper, directly discuss what will happen when two particle beams meet in
space.
Imagine the setup illustrated in Fig. 6a where two narrow particle beams move
towards each other. The particles in the first beam, with the velocity ${\bf
v}$, will be referred to as incident particles; and the particles in the
second beam, with the velocity $\bf w$, as target particles. Inside the two
beams, the positions of these particles are completely randomized. If a
collision between an incident particle and a target particle occurs, their
incoming velocities $\bf v$ and ${\bf w}$ will be changed to the outgoing
velocities ${{\bf v}^{\prime}}$ and ${\bf w}^{\prime}$ respectively. The small
region where those collisions may take place is denoted as $dU$.
Figure 6: (a) Particles of two beams colliding in a small region $dU$; and (b)
a pair of the outgoing particles shown in the velocity space.
We then define the velocity of the center-of-mass and the velocity of the
incident particles relative to the center-of-mass as
${\bf c}\equiv({\bf v}+{\bf w})/2\quad{\rm and}\quad{\bf u}\equiv({\bf v}-{\bf
w})/2$ (20)
respectively; after a collision, these two velocities become
${\bf c}^{\prime}\equiv({{\bf v}^{\prime}}+{\bf w}^{\prime})/2\quad{\rm
and}\quad{\bf u}^{\prime}\equiv({{\bf v}^{\prime}}-{\bf w}^{\prime})/2$ (21)
respectively. Since elastic collisions are our concern, the energy-momentum
conservation law reads
${\bf c}^{\prime}={\bf c}\quad{\rm and}\quad|{\bf u}^{\prime}|=|{\bf u}|\equiv
u.$ (22)
It should be mentioned that for a velocity pair $({\bf v},{\bf w})$ there
exist infinitely many spreading velocity pairs $({{\bf v}^{\prime}},{\bf
w}^{\prime})$, and the paired velocities ${{\bf v}^{\prime}}$ and ${\bf
w}^{\prime}$ will symmetrically fall on two sides of the spherical shell
labeled as the energy-momentum shell (EMS) shown in Fig. 6b.
Armed with these notions, we have the differential scattering cross section in
the center-of-mass reference frame taking the form
$\sigma({\bf u},{\bf
u}^{\prime})=\sigma(\Omega)=\frac{dN}{d\Omega}=u^{2}\frac{dN}{dS},$ (23)
in which the solid angle $\Omega$ is defined by the direction of ${\bf
u}^{\prime}$ with respect to ${\bf u}$, $d\Omega$ is an infinitesimal solid-
angle element about $\Omega$, $dS$ is the infinitesimal surface element on the
EMS subtending $d\Omega$ and $dN$ represents the average number of the
incident particles emerging after collisions within $d\Omega$, or on $dS$, per
unit incident flux (in terms of the relative speed $2u$), unit target
($n_{{\bf w}}|dU|=1$) and unit time.
Eq. (23) shows that $\sigma$ is a two-variable $0/0$ limit. It is easy to see
that the incident particles will, after collisions, distribute on the EMS
continuously (in the statistical sense); and thus the magnitude of $\sigma$
will be independent of how $d\Omega$, or $dS$, shrinks to an infinitesimal
one. Based on this realization, we admit that $\sigma$ is well-defined.
Then, let’s look at another scattering cross section in textbooksreif ; kubo ,
which is in this paper referred to as the cross section in the laboratory
frame. The cross section, denoted as $\sigma^{l}$ instead of $\sigma$, is
defined in the form
$\sigma^{l}({\bf v},{\bf w}\to{{\bf v}^{\prime}},{\bf
w}^{\prime})=\frac{dN}{d^{3}{{\bf v}^{\prime}}d^{3}{\bf w}^{\prime}}$ (24)
where $d^{3}{{\bf v}^{\prime}}\equiv
dv_{x}^{\prime}dv_{y}^{\prime}dv_{z}^{\prime}$ is a small velocity element
enclosing ${{\bf v}^{\prime}}$, $d^{3}{\bf w}^{\prime}\equiv
dw_{x}^{\prime}dw_{y}^{\prime}dw_{z}^{\prime}$ is a small velocity element
enclosing ${\bf w}^{\prime}$, and $dN$ is the number of the incident particles
scattered into $d^{3}{{\bf v}^{\prime}}$ per unit incident flux, unit target
and unit time (while the involved target particles fall into $d^{3}{\bf
w}^{\prime}$ presumably). The fact that $\sigma^{l}$ has been defined and
applied without any specification about $d^{3}{\bf v^{\prime}}$ and $d^{3}{\bf
w^{\prime}}$ tacitly implies that this $\sigma^{l}$ is supposed to be a
legitimate $0/0$ limit.
Now, we inspect whether or not $\sigma^{l}$ is indeed well-defined.
A simple comparison betwee these two defining equations gives us rather useful
information. Eq. (23) says that the collision-produced particles have a
2-dimensional distribution in the velocity space (which is related to the fact
that there are 6 unknown variables concerning ${\bf v^{\prime}}$ and $\bf
w^{\prime}$ and there are only 4 energy-momentum equations to constrain them).
But, Eq. (24) seemingly suggests that ${\bf v}^{\prime}$ and ${\bf
w}^{\prime}$ have six free space to move in. For convenience, we shall call it
the problem of “dimension mismatch”.
With help of the concepts presented in the last section, the dimension
mismatch can be analyzed easily. It is obvious that $\sigma^{l}$ defined by
Eq. (24) is just an intended limit. One of its partial limits takes the form
$\sigma^{l}([\hskip 1.0ptd^{3}{{\bf v}^{\prime}}\rangle_{0},d^{3}{\bf
w}^{\prime})=\lim\limits_{d^{3}{\bf v^{\prime}\to 0}}\frac{dN}{d^{3}{\bf
v^{\prime}}d^{3}{\bf w^{\prime}}}.$ (25)
If we let $d^{3}{\bf w^{\prime}}$ be a spherical ball with a definite radius
and let $[\hskip 1.0ptd^{3}{{\bf v}^{\prime}}\rangle_{0}$ be an ever-shrinking
spherical ball in the velocity space with the radius $\rho\to 0$. Assuming the
centers of the two balls to lie on the EMS symmetrically as shown in Fig. 6b,
Eq. (25) becomes
$\sigma^{l}([\hskip 1.0ptd^{3}{{\bf v}^{\prime}}\rangle_{0},d^{3}{\bf
w}^{\prime})=\frac{\sigma\pi\rho^{2}/u^{2}}{(4\pi\rho^{3}/3)d^{3}{\bf
w^{\prime}}}=\frac{3\sigma}{4u^{2}d^{3}{\bf w^{\prime}}}\cdot\frac{1}{\rho},$
(26)
in which we have used $dN=\sigma\cdot dS/u^{2}=\sigma\pi\rho^{2}/u^{2}$ given
by Eq. (23).
Alternatively, if we let $[\hskip 1.0ptd^{3}{{\bf v}^{\prime}}\rangle_{0}$
stand for an ever-shrinking cube whose side length is $b\to 0$, whose center
lies on the EMS, and whose top and bottom are parallel with the EMS locally,
we get, in, otherwise, the same context,
$\sigma^{l}([\hskip 1.0ptd^{3}{{\bf v}^{\prime}}\rangle_{0},d^{3}{\bf
w}^{\prime})=\frac{\sigma b^{2}/u^{2}}{b^{3}\cdot d^{3}{\bf
w}^{\prime}}=\frac{\sigma}{u^{2}d^{3}{\bf w^{\prime}}}\cdot\frac{1}{b}.$ (27)
These results show that $\sigma^{l}$ is multi-valued; furthermore, if $\rho\to
0$ or $b\to 0$, $\sigma^{l}$ is $\infty$-valued. In fact, by changing the
shape of $d^{3}{\bf v^{\prime}}$, the value of $\sigma^{l}$ can be any value
from $0$ to $\infty$chen2 . Therefore, $\sigma^{l}$ cannot, and should not, be
used in any mathematical and physical formalism.
In the usual textbook treatment,
$\sigma^{l}({\bf v},{\bf w}\to{\bf v^{\prime}},{\bf
w^{\prime}})=\sigma^{l}({\bf v^{\prime}},{\bf w^{\prime}}\to{\bf v},{\bf w})$
(28)
is given as the expression for the time-reversibility of particle-particle
collision. The discussion given above has shown that this expression is
misleading. To express the time-reversibility in a correct way, we need at
least to avoid the problem of dimension mismatch. An inspection of Eq. (23)
tells us that the equation can be rewritten as
$\sigma(\Omega_{\bf uu^{\prime}})=u^{2}\frac{dN}{dS_{\bf
v^{\prime}}}\equiv\sigma({\bf v},{\bf w}\to dS_{\bf v^{\prime}},dS_{\bf
w^{\prime}}),$ (29)
in which $dS_{\bf v^{\prime}}$ and $dS_{\bf w^{\prime}}$ stand for two
infinitesimal symmetric patches located symmetrically on the two side of the
EMS (allowing $\bf v^{\prime}$ and $\bf w^{\prime}$ to fall on them
respectively). This new definition of the cross section gives a good picture
about what happens with the velocities $\bf v$, $\bf w$, $\bf v^{\prime}$ and
$\bf w^{\prime}$ in the laboratory reference frame. With help of this new
definition of $\sigma$, we have
$\sigma({\bf v},{\bf w}\to dS_{\bf v^{\prime}},dS_{\bf
w^{\prime}})=\sigma({\bf v^{\prime}},{\bf w^{\prime}}\to dS_{\bf v},dS_{\bf
w}),$ (30)
whose validity can be understood from $\sigma(\Omega_{\bf
uu^{\prime}})=\sigma(\Omega_{\bf u^{\prime}u})$.
We have seen the usefulness and powerfulness of $\sigma$, we shall in the next
two sections use $\sigma$ alone to solve various problems involving particle-
particle collisions.
## V the scattering-out rate from a particle beam
The most fundamental collision problem has been considered in the last section
with reference to Fig. 6. Apart from the ill-defined cross section in the
laboratory fram, there seems nothing really new to physicists. Strangely, when
we include some of the errorproof formulas into the standard kinetic theory,
we shall still meet with unexpected difficultieschen2 . In this section, we
reinvestigate the issue and give a better solution for it.
According to textbooks of statistical mechanicslerner ; reif , the time
evolution of the distribution function of an ideal dilute gas should be
described by the Boltzmann equation
$R_{t}=R^{\bf v}+R^{\bf F}+R^{\rm s.in}-R^{\rm s.out},$ (31)
in which $R_{t}\equiv(\partial f/\partial t)_{{\bf r},{\bf v}}$ is the change
rate of the local distribution function at $({\bf r},{\bf
v})\equiv(x,y,z,v_{x},v_{y},v_{z})$, $R^{\bf v}=-{\bf v}\cdot(\partial
f/\partial{\bf r})$ represents the particles driven by the velocity $\bf v$,
$R^{\bf F}=-{\bf F}/m\cdot(\partial f/\partial{\bf v})$ stands for the
influence of the macroscopic force ${\bf F}\equiv(F_{x},F_{y},F_{z})$, and
$R^{\rm s.in}-R^{\rm s.out}$ is called the Boltzmann scattering operator.
Although different approaches deal with the Boltzmann equation differently,
the Boltzmann collision operator $R^{\rm s.in}-R^{\rm s.out}$ is formulated
unanimously by examining what happens in a small phase volume element
$d\beta\equiv d^{3}{\bf r}d^{3}{\bf v}$ fixed in phase space. This type of
approach may be referred to as the Eulerian approach; and every term in Eq.
(31) should be understood likewise.
On the understanding, the change rate of the local distribution function
$R_{t}$ is defined by
$R_{t}=\lim\limits_{dt\to 0,d\beta\to
0}\frac{(dN)_{\delta\beta,0-dt}}{dtd\beta}=\left.\frac{\partial f}{\partial
t}\right|_{{\bf r},{\bf v}},$ (32)
in which $(dN)_{\delta\beta,0-dt}$ is the net increment of the particle number
in $d\beta$ from time $0$ to time $dt$, which shows us that $R_{t}$ is
supposed to be a multi-variable limit with $dt\to 0$ and $\delta\beta\to 0$.
Actually, every term in Eq. (31) can be expressed similarly.
To be less burdensome in this section, we take on the following
simplifications: i. ${\bf F}=0$. ii. The direction of $\bf v$ is consistent
with the direction of the $x$-axis. And, iii. $R^{\rm s.in}$ needs not to be
taken into account (which means in the consideration of this section there
will be no particle-particle collisions that can produce particles with $\bf
v$). Thus, Eq. (31) becomes $R_{t}=R^{v_{x}}-R^{\rm s.out}$, or
$R_{t}=\frac{[{(dN)^{v_{x}\rm.in}-(dN)^{v_{x}\rm.out}}]}{dtd\beta}-\frac{(dN)^{\rm
s.out}}{dtd\beta},$ (33)
in which the first term $R^{v_{x}}$ caused by $v_{x}$ will be called the fluid
term and the second term $R^{\rm s.out}$ caused by particle-particle
collisions will be called the kinetic term. In what follows, our task is to
inspect whether each of the two terms is well-defined and inspect whether the
events related to the two terms are mutually exclusive (in order to obey the
addition rule for probabilities).
Figure 7: Dynamics of particle number: (a) Particles moving in and out of
$d\beta$ due to $v_{x}dt$. (b) Particles scattered out of $d\beta$ by $f_{\bf
w}$. (c) The combination of these two effects.
The fluid term $R^{v_{x}}$ can indeed be analyzed by fluid mechanics. From
Fig. 7a, we obtain:
$\displaystyle(dN)^{v_{x}\rm.in}-(dN)^{v_{x}\rm.out}$
$\displaystyle=v_{x}dtdydzd^{3}{\bf v}(f^{l}-f^{r})$
$\displaystyle=-v_{x}\frac{\partial f}{\partial x}dtd\beta,$ (34)
where $f^{l}$ and $f^{r}$ are the local distribution functions of $f$ on the
left and right ends of $dx$ respectively. Hence, we have
$R^{v_{x}}=\frac{(dN)^{v_{x}\rm.in}-(dN)^{v_{x}\rm.out}}{dtd\beta}=-v_{x}({\partial
f}/{\partial x}).$ (35)
It is noted that this result is independent of how $dt$ and $d\beta$ shrink to
their zeros (by choice or by chance), and thus $R^{v_{x}}$ is indeed well-
defined in the mathematical sense.
The kinetic term $R^{\rm s.out}$ can be analyzed with help of Fig. 7b, which
is essentially the same as Fig. 6a except that the possible collisions are
marked with letters x in Fig. 7b. In reference to Fig. 6, we find, from Eq.
(23), that the number of the particles initially belonging to $n_{\bf v}$ and
eventually scattered into the patch $dS$ on the EMS is
${(dN)^{dS}}=\sigma{(d\Omega)\cdot(2un_{\bf v})\cdot(n_{{\bf w}}d^{3}{\bf
r})\cdot(dt)},$ (36)
where $2un_{\bf v}$, $n_{\bf w}d^{3}{\bf r}$ and $dt$ are no longer unit ones,
and $d\Omega=dS/u^{2}$. The total number of the scattering-out particles is
$(dN)^{\rm s.out}=\int 2u\sigma n_{\bf v}n_{{\bf w}}d\Omega d^{3}{\bf r}dt.$
(37)
By replacing $n_{\bf v}$ with $f_{\bf v}d^{3}{\bf v}$ and replacing $n_{{\bf
w}}$ with $\int d^{3}{\bf w}f_{{\bf w}}$ (assuming there are more beams
entering $\delta\beta$ and involving the collisions), we arrive at
$R^{\rm s.out}=\lim\frac{(dN)^{\rm s.out}}{dtd^{3}{\bf r}d^{3}{\bf v}}=\int
2u\sigma f_{\bf v}f_{{\bf w}}d\Omega d^{3}{\bf w}.$ (38)
To get it, no hypothesis about how $d^{3}{\bf r}$, $d^{3}{\bf v}$ and $dt$
shrink to their zeros has been introduced, which shows that the $R^{\rm
s.out}$ here is a legitimate $0/0$ limit.
So far the discussion given in this section is completely consistent with the
standard textbook treatment.
Let’s now check out whether or not the above two terms are “mutually
exclusive”. In Fig. 7c, the particles related to the fluid term and the
kinetic term are shown in a combined way. The figure shows that if we pay
attention to the particles in the shaded region we find that there is double
counting concerning $(dN)^{v_{x}\rm.out}$ and $(dN)^{\rm s.out}$.
The direct consequence of this double counting is that $R_{t}$, cannot be
recognized as a well-defined limit, although the standard theory, represented
by expression (32), seems to suggest otherwise.
To reveal the problem more explicitly, let’s do a thought experiment. Consider
that the experiment shown in Fig. 7c is done in a computer simulation. By
letting the incoming particle beam represented by $f_{\bf v}$ be a stationary
one during the experiment, and letting the incoming colliding particle beam
represented by $f_{\bf w}$ be applied to $\delta\beta$ after $t=0$, a direct
evaluation of Eq. (32) gives us that
$R_{t}([\hskip 1.0ptvdt<dx\rangle_{0},dydzd^{3}{\bf v})=-R^{\rm s.out}+{\kappa
R^{\rm s.out}}/2,$ (39)
where $\kappa\equiv v_{x}dt/(dx)$ is a real number between 0 and 1. In
physics, this expression means that the value of $R_{t}$ strongly depends on
the experiment parameters $dt$ and $dx$. In mathematics, it means that $R_{t}$
is an ill-defined limit. All these are not expected from the viewpoint of the
standard theory.
As a matter of fact, there is a simpler way to find out what the problem is.
According to the Eulerian approach, equation (31) is constructed on the basic
assumption that the particles in $\delta\beta$ can be partitioned into several
groups, such as the particles driven into $\delta\beta$ by $v_{x}$, the
particles driven out of $\delta\beta$ by $v_{x}$, the particles driven out of
$\delta\beta$ by collisions, the particles driven into $\delta\beta$ by
collisions and so on so forth. However, when $\delta\beta\to 0$ and $dt$ is
slightly long, dividing the particles therein into such groups will become a
mission impossible (those particles are no longer “mutually exclusive”).
Interestingly, although the problem indicated above is a fundamental one, it
is still surmountable. If we look at $R^{\rm s.out}$ from the perspective of
the path-approach (or look at it in the Lagrangian reference frame), the
difficulties associated with $R^{\rm s.out}$ disappear almost entirely. In
fact, there are two pieces of good news. The first is that, for the particles
in a moving beam, the only leaving mechanism is the one that has been
formulated by $R^{\rm s.out}$. The second is that Eq. (38) can be integrated
meaningfully and easily. To see it, let’s rewrite Eq. (38) as
$\left.\frac{df_{\bf v}}{dt}\right|_{l}=v\frac{df_{\bf v}}{dl}=-f_{\bf v}\int
d^{3}{\bf w}d\Omega 2u\sigma f_{{\bf w}},$ (40)
where $l$ represents the particles’ path (a curve in the position space,
generally speaking) and $v\equiv|{\bf v}|$ the particles’ speed. This formula
is essentially the same as another one in the textbook treatmentreif :
$\frac{dp}{pdt}=v\frac{dp}{pdl}=-\tau^{-1}=-\int d^{3}{\bf w}d\Omega 2u\sigma
f_{\bf w},$ (41)
in which $p$ is the survival probability of a test particle and $\tau$ is the
average collision time (or relaxation time).
These two formulas inform us that for a test particle or for a small group of
similar particles represented by $f$ the path-dynamics is exactly the same.
The integration of Eq. (41) over a finite path, rather than over a period of
time at a fixed phase point, gives us the following path-survival probability
$p(\Delta l)=\exp\left(-\int_{\Delta l}\frac{dl}{|\bf v|}\int d^{3}{\bf
w}d\Omega 2u\sigma f_{\bf w}\right).$ (42)
This path-survival probability formula sheds a lot of light on the gas
dynamics that we wish to explore. It greatly highlights that all the
inferences associated with Fig. 4a and 4b in Sect. III make sense even in
presence of particle-particle collisions as long as certain path-survival
probabilities are taken into account. In particular, it tells us that if a
local distribution involves discontinuity (for whatever reasons), the
discontinuity will affect almost everywhere of the gas system.
## VI The particles scattered into a phase volume element
In Sect. IV, we have shown that the standard expression for time-reversibility
of particle-particle collision is a misunderstanding. In this section, we
shall try to formulate the particles scattered into a phase volume element
without taking any help from the time-reversibility.
Let’s first follow the textbook treatment and try to define the scattering-in
rate in an explicit way. In reference to Fig. 6, we accept the notation in
which the two colliding particle beams are represented by $n_{\bf v^{\prime}}$
and $n_{\bf w^{\prime}}$ and the phase volume element used to collect the
scattered particles is $d^{3}{\bf v}$ (which means that the collected
particles initially belong $n_{\bf v^{\prime}}$). Under the notation, the
scattering-in rate is supposed to be defined by
$R^{\rm s.in}=\lim\frac{(dN)^{\rm s.in}}{dtd^{3}{\bf r}d^{3}{{\bf v}}},$ (43)
where $(dN)^{\rm s.in}$ is the number of the particles collected by $d^{3}{\bf
v}$ during $dt$ and $d^{3}{\bf r}$ is a position volume element in the
collision region.
Evidently, like $\sigma^{l}$, this definition suffers the problem of dimension
mismatch: while $d^{3}{\bf v}$ is a three-dimensional volume element, the
scattered particles will distribute only on a two-dimensional surface in the
velocity space. Without any difficulty, we find that if $d^{3}{\bf v}$ is a
spherical ball with radius $\rho$ whose center lies on the EMS determined by
$\bf v^{\prime}$ and $\bf w^{\prime}$, then we obtain, with help of Eq. (36),
$R^{\rm s.in}([\hskip 1.0ptdt\rangle_{0},[\hskip 1.0ptd^{3}{\bf
r}\rangle_{0},d^{3}{\bf v})=\frac{3\sigma n_{\bf v^{\prime}}n_{\bf
w^{\prime}}}{2u}\frac{1}{\rho}.$ (44)
By letting $\rho\to 0$, we know that this scattering-in rate is ill-defined,
if the defining equation (43) is insisted upon.
Figure 8: (a) Two gases meeting in a collision region. (b) The collision-
produced distribution function on the half-line OA.
To find a way out of the difficulty, let’s consider the situation shown by
Fig. 8, where $f_{{\bf v}^{\prime}}$ and $f_{{\bf w}^{\prime}}$ represent the
distribution functions of two “colliding” gases. Our goal is to determine the
distribution function $f_{\bf v}$ which is employed to stand for the
collision-produced particles initially belong to $f_{\bf v^{\prime}}$.
For the following two reasons, we shall first use the collisional Boltzmann
equation to deal with the situation: i. In view of that many of us still
belive that the collisional Boltzmann equation can give a qualitatively and
quantitatively correct answer to the problem, we should take it as the first
step. ii. The solution given by the standard theory must give us a number of
helpful hints.
According to the Boltzmann equation in the textbook treatmentreif , $f_{\bf
v}$ on the half-line OA satisfies (with the half-line OA chosen to be the
positive $x$-axis)
$v_{x}\frac{\partial f_{\bf v}(x)}{\partial x}=\int 2u\sigma f_{\bf
v^{\prime}}f_{\bf w^{\prime}}d^{3}{\bf w}d\Omega,$ (45)
in which we have set $v_{y}=v_{z}=0$, $(\partial f_{\bf v}/\partial t)=0$ and
${\bf F}=0$, and we have disregarded all higher-order collisions. This leads
to
$f_{\bf v}(x)=\frac{1}{v_{x}}\int{dx}\int 2u\sigma f_{{\bf v}^{\prime}}f_{{\bf
w}^{\prime}}d^{3}{\bf w}d\Omega.$ (46)
This expression can be numerically integrated, and the final result is
schematically shown by the dotted curve in Fig. 8b.
But, our physical sense would tell us a different story. If we, as an
observer, move further and further away along the OA-axis, what shall we
observe? Considering that the collision region will gradually behave like a
point-like particle source, the inverse square law must eventually take
control. In other words, the solid curve shown in Fig. 8b is supposed to
describe the practical behavior of $f_{\bf v}(x)$.
Although Eq. (46) does not give us a correct result, it still hints us that
the particle-particle collisions along the segment OA can serve as a particle
source giving contribution to the distribution function somewhere else (named
as the contributed distribution function hereafter). A careful observation of
Eq. (46) unveils two shortcomings of it: i. The source region has been
confined on a line segment, which has zero volume and cannot be considered as
a reasonable source region. And ii. The velocity volume element used to
collect the collision-produced particles has been implicitly assumed to be an
infinitesimal three-dimensional one, which is not proper according to the
discussion that has been just given.
Figure 9: (a) A detector with an inlet area $d\alpha$ is placed outside the
collision region shown in Fig. 8a. (b) Only the particles whose velocities are
within $v^{2}dvd\Omega_{0}$, where $d\Omega_{0}=d\alpha/|{\bf r}-{\bf
r}_{0}|^{2}$, can be detected by the detector.
Based on these understandings, we arrive at several new guidelines concerning
the would-be treatment of Fig. 8a: i. Each spot of the collision region will
be deemed as a point-like “particle source”. And, ii. The contributed
distribution function will be defined as the finite-volume distribution
function in the velocity space to avoid the infinity difficulty caused by
dimension mismatch. In relation to Fig. 9a, to make the source-contribution
relationship as clear as possible, it is also assumed that the position point
$\bf r$, at which the contributed distribution function is of interest, is
located outside the collision region (for clarity of thinking), and that an
“idealized” detector, as a substitute for a human observer, is placed at $\bf
r$ to collect the collision-produced particles. With respect to the detector,
we investigate the following “distribution function”:
$\bar{f}([\hskip 1.0ptd^{3}{\bf r}\rangle_{0},\delta
v,\delta\Omega)\equiv\frac{1}{{\bar{v}}^{2}\delta
v\delta\Omega}\lim\limits_{dtd\alpha\to 0}\int_{\delta
v\delta\Omega}\frac{dN}{vdtd\alpha},$ (47)
where $[\hskip 1.0ptd^{3}{\bf r}\rangle_{0}$ represents an infinitesimal
position volume element just at the detector’s inlet, $\delta v$ is a small
but finite speed increment, $\delta\Omega$ is a small but finite solid-angle
element in the velocity space,
$\bar{v}^{2}\equiv(v_{l}^{2}+v_{l}v_{h}+v_{h}^{2})/3$ with $v_{l}$ and $v_{h}$
being the lowest and highest bounds of $\delta v$ respectively, $dN$ is the
number of the particles registered during $dt$ by the detector whose
velocities are confined in $\bar{v}^{2}\delta
v\delta\Omega=(v_{h}^{3}-v_{l}^{3})\cdot\delta\Omega/3$ and $d\alpha$ stands
for the infinitely small opening area of the detector’s inlet.
Explanation is needed about this definition. As implied by the form of
$\bar{f}$, $\bar{f}$ here is nothing but the partial limit associated with the
assumption that $d^{3}{\bf r}$ is infinitesimal and $\bar{v}^{2}\delta
v\delta\Omega$ is finite. In other words, it is an ordinary distribution
function in the position space, but an average distribution function in the
velocity space. While its differentiation in the velocity space makes no
sense, its integration can be used to obtain macroscopic quantities as if it
is a regular distribution function.
By examining the upstream paths of the particles collected by the detector in
Fig. 9a, it is found that only the collisions taking place in the spatial cone
$\\{-\delta\Omega\\}_{\bf r}$ are relevant to $\bar{f}$; and thus the region
included in $\\{-\delta\Omega\\}_{\bf r}$ will be called the effective zone.
Supposing ${\bf r}_{0}$ is a spatial point in the effective zone, how do the
collisions taking place around ${\bf r}_{0}$ contribute to $\bar{f}$? Due to
the infinite smallness of $d\Omega_{0}$ which is defined as
$d\Omega_{0}=d\alpha/|{\bf r}-{\bf r}_{0}|^{2},$ (48)
and the finiteness of $\delta\Omega$, any collision-produced particle that can
enter the detector at a speed within $\delta v$ is qualified as a one
belonging to $dN$ in Eq. (47).
Thus, from Eq. (36), the contribution to $dN$ due to the collisions in
$d^{3}{\bf r}_{0}$ can be written as
$dt\int_{\delta v}\int_{v^{2}dvd\Omega_{0}}2uf_{{\bf v}^{\prime}}d^{3}{\bf
v}^{\prime}d^{3}{\bf r}_{0}f_{{\bf w}^{\prime}}d^{3}{\bf
w}^{\prime}\sigma\cdot(dS/u^{2}),$ (49)
where $dS$ is the surface element lying on the EMS defined by ${\bf
v}^{\prime}$ and ${\bf w}^{\prime}$ and enclosed by $v^{2}dvd\Omega_{0}$ as
shown in Fig. 9b.
To truly integrate Eq. (49), we invoke the variable transformation $d^{3}{\bf
v}^{\prime}d^{3}{\bf w}^{\prime}\to d^{3}{\bf c}^{\prime}d^{3}{\bf
u}^{\prime}$, where ${\bf c}^{\prime}=({\bf v^{\prime}}+{\bf w^{\prime}})/2$
and ${\bf u}^{\prime}=({\bf v^{\prime}}-{\bf w^{\prime}})/2$, and obtain
$dtd^{3}{\bf r}_{0}\int_{\delta v}\int_{v^{2}dvd\Omega_{0}}d^{3}{\bf
c}^{\prime}d^{3}{\bf u}^{\prime}2uJf_{{\bf v}^{\prime}}f_{{\bf
w}^{\prime}}\sigma\cdot(dS/u^{2}),$ (50)
in which $J$ is the Jacobian
$J=\left\|\frac{\partial({\bf v}^{\prime},{\bf w}^{\prime})}{\partial({\bf
c}^{\prime},{\bf u}^{\prime})}\right\|=8.$ (51)
Noticing that
$\int d^{3}{\bf u}^{\prime}=\int u^{\prime 2}du^{\prime}d\Omega^{\prime}=\int
d\Omega^{\prime}\int u^{2}du,$ (52)
where $\Omega^{\prime}$ is the direction of ${\bf u}^{\prime}$, and that
$\int_{v^{2}dvd\Omega_{0}}dudS=v^{2}dvd\Omega_{0}=\frac{v^{2}dvd\alpha}{|{\bf
r}-{\bf r}_{0}|^{2}},$ (53)
we arrive at
$\bar{f}=\frac{1}{{\bar{v}}^{2}\delta v\delta\Omega}\int d^{3}{\bf
r}_{0}dvd^{3}{\bf c}^{\prime}d\Omega^{\prime}\frac{16uv\sigma}{|{\bf r}-{\bf
r}_{0}|^{2}}f_{\bf v^{\prime}}f_{{\bf w}^{\prime}},$ (54)
where $\int d^{3}{\bf r}_{0}$ is over the effective zone given by
$\\{-\delta\Omega\\}_{\bf r}$, $\int dv$ is over $\delta v$, $\int d^{3}{\bf
c}^{\prime}$ is over the entire velocity space, and $\int d\Omega^{\prime}$ is
over $0-4\pi$. It is noted that several quantities in this formula, such as
$\sigma$, $f_{{\bf v}^{\prime}}$ and $f_{{\bf w}^{\prime}}$, are defined with
help of ${\bf u}$ and ${\bf u}^{\prime}$, in which ${\bf u}={\bf v}-{\bf
c}={\bf v}-{\bf c}^{\prime}$ with ${\bf v}=v({\bf r}-{\bf r}_{0})/|{\bf
r}-{\bf r}_{0}|$, and ${\bf u}^{\prime}$ is defined by the magnitude of ${\bf
u}$ and the direction of $\Omega^{\prime}$.
If the collision region is relatively small, the result given by Eq. (54)
evidently satisfies the inverse square law. If interested, readers may use
computational means to verify (or deny) this approach. With slight
modifications, the formulation can be applied to many other situationschen3 .
## VII Summary
In conclusion, we have shown that
* •
A number of multi-variable limits used in the existing kinetic theory are
actually ill-defined.
* •
By redefining, reinterpreting and reformulating these ill-defined limits, we
can make significant progresses in gas dynamics.
* •
To formulate the quantities introduced in this pepr, we need to give up the
equation-type approach and accept the path-integral approach.
* •
In real dynamic gases there are a considerable number of particles that are
discontinuously distributed. The newly introduced path-approach can treat them
with almost the same accuracy and effectiveness.
The concepts, methodologies and conclusions given by this paper are
fundamentally different from those in the conventional theory. Possible impact
on fluid mechanics, turbulence studies, plasma physics or quantum statistical
mechanics remains to be seen.
The author is very grateful to Drs. M. Berry, O. Penrose, Robert G.
Littlejohn, Guo Han-ying, Zhang Tian-rong, Guan Ke-ying, Ying Xing-ren, Wen
Duanzhi, V. Travkin, W. Hoover and many others for their direct or indirect
encouragement. Discussions with them have been pleasant and stimulating.
## References
* (1) Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis, Courier Corporation, 2003.
* (2) Lynn Arthur Steen and J. Arthur Seebach Jr. Counterexamples in Topology, Springer-Verlag New York Inc. 1978.
* (3) Jordan M. Stoyanov, Counterexamples in Probability, Dover Books on Mathematics, 2013, Third Edition.
* (4) Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences”. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959.
* (5) R. Courant and F. John, Introduction to Calculus and Analysis, Volume II/1, Springer-Verlag New York, inc., 1989.
* (6) E. T. Whittaker and G. N. Watson, Thomas’ Calculus, Cambridge University Press, 1996.
* (7) L. Liouville, J. Math. Pures Appl. 3, 342 (1838).
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* (11) J.R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical mechanics, Cambridge Lecture Notes in Physics, 1999.
* (12) L.D.Landau and E.M.Lifshitz, Statistical Physics, 3rd edition Part 1, Butterworth-Heinemann, 2013.
* (13) Walter Greiner, Ludwig Neise and Horst Stöcker, Thermodynamics and Statistical Mechanics, Springer-Verlag New York, 1995.
* (14) K. Huang Statistical Mechanics, Wiley, 2nd edition, 1987\.
* (15) F. Schwabl, Statistical Mechanics, Springer-Verlag Berlin Heidelberg, 2nd Edition, 2006.
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* (17) R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II, second edition, Springer-Verlag, 1991.
* (18) C. Y. Chen, arXiv: 0812.4343 [physics.gen-ph]; cond-mat/0608712, 0504497, 0412396; physics/0312043, 0311120, 0305006, 0010015, 0006033, 0006009, 9908062.
* (19) C. Y. Chen, Il Nuovo Cimento B, V117B, p177 (2002).
* (20) C. Y. Chen, (2015) An Alternative Approach to Particle-Particle Collisions, Journal of Modern Physics, 6, 772-779. doi: 10.4236/jmp.2015.66082.
|
# Electric field effect on electron gas spins in two-dimensional magnets with
strong spin-orbit coupling
K.S. Denisov<EMAIL_ADDRESS>Ioffe Institute, 194021
St.Petersburg, Russia
###### Abstract
The recent rise of material platforms combining magnetism and two-
dimensionality of mobile carriers reveals a diverse spectrum of spin-orbit
phenomena and stimulates its ongoing theoretical discussions. In this work we
use the density matrix approach to provide a unified description of subtle
microscopic effects governing the electron gas spin behavior in the clean
limit upon electric perturbations in two-dimensional magnets with strong spin-
orbit coupling. We discuss that an inhomogeneity of electrostatic potential
generally leads to the electron gas spin tilting with the subsequent formation
of equilibrium skyrmion-like spin textures and demonstrate that several
microscopic mechanisms of 2DEG spin response are equally important for this
effect. We analyze the dynamics of 2DEG spin upon an oscillating electric
field with a specific focus on the emergent electric dipole spin resonance. We
address the resonant enhancement of magneto-optical phenomena from the spin
precession equation perspective and discuss it in terms of the resonant spin
generation. We also clarify the connection of both static and dynamic spin
phenomena arising in response to a scalar perturbation with the electronic
band Berry curvature.
## I Introduction
The recent advances in the development of spintronics devices extensively use
relativistic spin-orbit properties of free carriers interacting with magnetic
layers. The spin-orbit coupling (SOC) of charge carriers generally opens up
the possibility to deal with the magnetization purely by electrical means; the
magnetization orientation can be detected electrically by virtue of the
anisotropic magnetoresistance effect [1, 2, 3], while electric current-induced
spin-orbit torque occurs to be a highly effective tool for switching its
direction [4, 5, 6, 7]. Nonstationary dynamics of carriers in presence of SOC
can result in stimulated photon emission, as in case of terahertz spintronic
light emitter [8, 9, 10, 11] and spin Hall nano-oscillators [12, 13]. Apart
from kinetic phenomena spin-orbit effects can modify equilibrium spin
configurations via indirect RKKY exchange interaction [14, 15, 16] and lead to
the formation of magnetic skyrmions [17, 18] due to Dzyaloshinskii–Moriya
terms [19, 20]. An efficient charge-to-spin conversion wanted for modern
spintronics needs is often realized when turning to a two-dimensional electron
gas, as the reduction of the dimensionality tends to be accompanied by the
lowering of symmetry and by the subsequent increase in SOC [19, 21]. There are
an increasing number of different material platforms that allow one to combine
systematically stronger SOC magnitudes of 2D electrons directly with a
magnetic component, the examples include van der Waals heterostructures [22]
either proximitized by magnetic layer [23, 24, 25, 26, 27] or being intrinsic
ferromagnets [28, 29, 30, 31], semiconductor nanostructures doped by magnetic
dopants [32, 33], surface states of magnetic topological insulators [34, 35],
or layered magnetic heterostructures [36, 19]. Moreover, combining magnetism
with 2D conductive channels additionally offers new functionalities, such as
spin tunnel field-effect transistors [37], spin inversion effect [24] or novel
class of spinterfaces [38].
In order to fully benefit from two-dimensional magnetic systems it is of key
importance to have a comprehensive understanding of how the spin density of
electron gas in a 2D channel responds to an applied electric field, that is
the understanding of free electron gas magnetoelectric properties. However, a
complete microscopic treatment of the related phenomena appears to be
extremely challenging, even despite there is a few theoretical approaches
effectively dealing with multiband systems (e.g. wave-packet dynamics theory
[39, 40, 41, 42, 43], diagrammatic and ab-initio calculations [44, 45, 46,
47]). The difficulty lies in the fact that in spin-orbital systems multiple
microscopic mechanisms of quite a subtle character often contribute on the
equal footing, which hinders a simplified consideration. In particular, an
exchange interaction induced spin splitting in combination with strong spin-
orbit coupling generally lead to a geometrical structure of electronic band
states featured by nonzero Berry curvature in k-space. Treating different
spin-related phenomena with account for the electronic band geometry remains
an ongoing discussion. It covers, for instance, the issues of the Liouville’s
theorem with account for the Berry phase [48, 49, 50], the Hall conductivity
modifications in presence of real-space magnetic textures [51], or, concerning
the anomalous and spin Hall effects, the interplay between Karplus-Luttinger
anomalous velocity and disorder-induced mechanisms [52, 53, 54, 55]; the
latters have recently been enriched by the electron scattering on a pair of
impurities [55, 56]. Moreover, when calculating spin-related quantities a
specific class of coarse graining effects should be taken into account, as is
clearly demonstrated in [40, 42].
In this paper we respond to an ever-growing role that two-dimensional magnetic
systems plays for spintronics and consider in detail a complex pattern of
microscopic effects relevant for the magnetoelectric behavior of 2DEG in the
clean limit. Based on the density matrix approach we describe the most
significant spin-response mechanisms of two-dimensional spin-orbital systems
within the unified framework, reveal the interconnection between different
microscopic effects and clarify its relation to an electronic band geometry.
The theoretical model and the density matrix description are formulated in
Sec. II. In Sec. III we analyze a magnetoelectric effect in thermal
equilibrium, namely we consider the formation of equilibrium spin textures and
local persistent electric currents arising due to an inhomogeneous
electrostatic potential. We discuss in detail semiclassical electron dynamics
with account for a spin-to-momentum locking and identify microscopic
mechanisms responsible for the magnetoelectric response. Namely, we attribute
the generation of an extra-spin density directed within 2DEG plane both to the
non-adiabatic correction to the electron spin precession and to the correlated
change of charge and spin electron densities, the latter scenario is sometimes
referred as spin-dipole effect [40]. We provide a unified treatment of these
mechanisms using the density matrix, derive general equations governing the
contribution due to each mechanism independently and reveal the role that the
Berry curvature plays for the emergent phenomena.
In Sec. IV we turn to the dynamical regime and investigate the 2DEG spin
dynamics upon an oscillating electric field. We focus specifically on spin
resonance phenomena due to electric dipole transitions, also referred as the
electric dipole spin resonance (EDSR). We derive the precession equation for
2DEG spin density capturing the spin resonance scenario, and clarify the
relation of the band states Berry curvature with the spin response
susceptibility. We also discuss the spin resonance in terms of optical
conductivity and describe the associated magneto-optical properties of 2DEG.
In particular, we describe how the EDSR induced generation of the in-plane
spin density is accompanied by the resonant enhancement of the Hall
conductivity, the latter is responsible for magneto-optical Kerr and Faraday
effects. We classify different spin polarizations emerging in the dynamical
regime and present analytic expressions for the spin resonance related optical
conductivity.
## II Theoretical framework
### II.1 Model band structure
We consider a two-dimensional electron gas with parabolic bands affected both
by the Rashba effect and by an exchange interaction with a magnetic host. We
assume that the magnetization responsible for the spin splitting is directed
along $z$-axis perpendicular to the electron motion plane. The so-called
Rashba ferromagnet model covers all the physics relevant for our consideration
and allows one address the related spin phenomena in the most transparent way.
The effective Hamiltonian describing this model is given by
$\mathcal{H}=\frac{\bm{k}^{2}}{2m}+\bm{\Omega}_{k}\cdot\hat{\bm{S}},$ (1)
here the first term describes the parabolic dispersion with an effective mass
$m$, and $\bm{\Omega}_{k}$ is an effective $k$-space magnetic field acting on
the electron spin $\hat{\bm{S}}=\hat{\bm{\sigma}}/2$; $\hat{\bm{\sigma}}$ is
the vector of Pauli matrices. The field $\bm{\Omega}_{k}$ leads to a spin
splitting of the electronic subbands, in our model $\bm{\Omega}_{k}$ consists
of two parts
$\displaystyle\bm{\Omega}_{k}=\bm{\Omega}^{so}(\bm{k})-\Omega_{0}\bm{e}_{z},\qquad\bm{\Omega}^{so}(\bm{k})=2\lambda_{so}\left[\bm{e}_{z}\times\bm{k}\right],$
(2)
where $\bm{\Omega}^{so}(\bm{k})$ describes the spin-orbit Rashba interaction
with the coupling constant $\lambda_{so}$, and the second term is due to an
exchange interaction with a magnetic background, the parameter $\Omega_{0}$
describes the corresponding splitting of spin subbands at zero momentum. The
eigenstates of Eq. 1 Hamiltonian can be written in the following form
$\psi_{k}^{\pm}=e^{i\bm{kr}}|u_{k}^{\pm}\rangle,\qquad|u_{k}^{+}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}b_{k}\\\
-ie^{i\varphi}a_{k}\end{pmatrix},\quad|u_{k}^{-}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}-ie^{-i\varphi}a_{k}\\\
b_{k}\end{pmatrix},$ (3)
where $(a_{k},b_{k})=(1\pm\Omega_{0}/\Omega_{k})^{1/2}$. We use the notation
$\eta=(\pm)$ for two electron spin subbands. The states $\psi_{k}^{\eta}$ are
characterized by the electron spin $\bm{s}_{k}^{\eta}=\langle
u_{k}^{\eta}|\hat{\bm{S}}|u_{k}^{\eta}\rangle$ directed either parallel or
antiparallel to $\bm{\Omega}_{k}$
$\displaystyle\bm{s}_{k}^{\pm}=\pm\frac{1}{2}\bm{n}_{k},\qquad\bm{n}_{k}=\frac{\bm{\Omega}_{k}}{{\Omega}_{k}},\qquad\Omega_{k}=\sqrt{\Omega_{0}^{2}+(2\lambda_{so}k)^{2}},$
(4)
where the unit vector $\bm{n}_{k}$ points along the direction of
$\bm{\Omega}_{k}$.
The energy dispersion corresponding to $\eta$-subband is
$\varepsilon_{k}^{\eta}=k^{2}/2m+\eta\Omega_{k}/2$. The presence of
$k$-dependent spin splitting leads to the renormalization of effective masses
nearby $k\approx 0$, namely $m^{\pm}=m/(1\pm\xi)$, where the parameter
$\xi\equiv 2m\lambda_{so}^{2}/\Omega_{0}$. We focus on systems with
sufficiently strong exchange interaction, when $\Omega_{0}$ greatly exceeds
the spin-orbital coupling. We thus take the parameter $\xi<1$, at that the
effective mass $m^{-}>0$ is positive and the lower energy branch is a
monotonic function of the momentum, see Fig. 4b.
Let us discuss the role of the spin splitting terms. The presence of the
Rashba effect induced spin-momentum locking directly manifests itself in the
velocity operator
$\hat{\bm{v}}=\frac{i}{\hbar}\left[\mathcal{H},{\bm{r}}\right]=\frac{\hat{\bm{k}}}{m}+2\lambda_{so}[\bm{e}_{z}\times\hat{\bm{S}}],$
(5)
where the second term is sensitive to the instantaneous direction of the
electron spin. While the average velocity for the eigen spin states is
determined by the unperturbed spin vector $\bm{s}_{k}^{\eta}$
$\bm{v}_{k}^{\eta}\equiv\langle\psi_{k}^{\eta}|\hat{\bm{v}}|\psi_{k}^{\eta}\rangle=\frac{\bm{k}}{m}+2\lambda_{so}[\bm{e}_{z}\times\bm{s}_{k}^{\eta}],$
(6)
the changes in the direction of an electron spin caused by external fields can
directly affect the average of the velocity operator and, correspondingly,
influence the orbital motion.
The presence of a magnetic gap due to the magnetization directed perpendicular
to 2DEG plane leads additionally to the fact that electron band states acquire
a geometric structure. Indeed, the electron spin direction in $\bm{k}$-space
forms a hedgehog pattern which underlies the appearance of the Berry curvature
$\mathcal{F}_{k}^{\eta}=i\langle\nabla_{\bm{k}}u_{k}^{\eta}|\times|\nabla_{\bm{k}}u_{k}^{\eta}\rangle$.
For a spin-$1/2$ Hamiltonian this Berry curvature can be expressed as follows
$\mathcal{F}_{k}^{\eta}=\eta\frac{1}{4\pi}\bm{n}_{k}\cdot\left[\frac{\partial\bm{n}_{k}}{\partial{k_{x}}}\times\frac{\partial\bm{n}_{k}}{\partial{k_{y}}}\right]=\eta~{}2\lambda_{so}^{2}\frac{\Omega_{0}}{\Omega_{k}^{3}},$
(7)
and we keep the notation $\mathcal{F}_{k}=|\mathcal{F}_{k}^{\eta}|$ for its
absolute value. The total Berry flux $Q_{F}^{\eta}$ accumulated by electrons
from $\eta$ subband up to the Fermi energy $\mu$ is given by
$Q_{F}^{\eta}=\sum_{k<k_{F}^{\eta}}\mathcal{F}_{k}^{\eta}=\eta~{}\frac{1}{4\pi}\left(1-\frac{\Omega_{0}}{\Omega_{F}^{\eta}}\right),\qquad\Omega_{F}^{\eta}=\sqrt{\Omega_{0}^{2}+(2\lambda_{so}k_{F}^{\eta})^{2}},$
(8)
where $\Omega_{F}^{\pm}$ is the spin splitting energy for $\eta=(\pm)$
subbands at the Fermi energy, see Fig. 3b. The strong spin-orbit coupling
considered in our work means that we do not account for the disorder-induced
smearing of SOC features of electronic bands.
### II.2 Density matrix approach
Let us firstly discuss the structure of the density matrix ${f}^{0}$ for 2DEG
in thermal equilibrium without external perturbations. The general form is
${f}^{0}=(e^{\beta(\hat{\mathcal{H}}-\mu)}+1)^{-1}$, where $\hat{\mathcal{H}}$
is given by Eq. 1, $\beta$ is the inverse temperature and $\mu$ is the Fermi
energy. In this work we focus on zero temparature limit $\beta\to\infty$. The
density matrix $\hat{f}_{k}^{0}$ in the momentum representation is a $2\times
2$ matrix which can be presented as follows (we keep hats for spin indices
only)
$\hat{f}_{k}^{0}=\frac{1}{2}n_{k}^{0}+\bm{S}_{k}^{0}\cdot\hat{\bm{\sigma}}.$
(9)
We note that $\hat{f}_{k}^{0}$ is diagonal in the basis of eigen states
$\psi_{k}^{\pm}$, so we can present it as a sum of $\eta=(\pm)$ spin subband
contributions $\hat{f}_{k}^{\eta}$
$\displaystyle\hat{f}_{k}^{0}=\hat{f}_{k}^{+}+\hat{f}_{k}^{-},\qquad\hat{f}_{k}^{\eta}=n_{k}^{\eta}\left(\frac{1}{2}+\bm{{s}}_{k}^{\eta}\cdot\hat{\bm{\sigma}}\right),$
(10)
where $n_{k}^{\eta}=(e^{\beta(\varepsilon_{k}^{\eta}-\mu)}+1)^{-1}$ is the
Fermi-Dirac distribution function of electrons in the spin subband with energy
$\varepsilon_{k}^{\eta}$. The terms in Eq. 9 are given
$n_{k}^{0}=n_{k}^{+}+n_{k}^{-}$, and
$\bm{S}_{k}^{0}=n_{k}^{+}\bm{s}_{k}^{+}+n_{k}^{-}\bm{s}_{k}^{-}$, here
$\bm{s}_{k}^{\eta}$ corresponds to the eigen spin states from Eq. 4. The
equilibrium spin density $\bm{S}_{0}$ is directed perpendicular to the 2DEG
plane
$\bm{S}_{0}=\frac{1}{2}\sum_{k}{\rm
Sp}\left(\hat{f}_{k}^{0}\cdot\hat{\bm{\sigma}}\right)=\sum_{k}\left(n_{k}^{+}\bm{s}_{k}^{+}+n_{k}^{-}\bm{s}_{k}^{-}\right)=\bm{e}_{z}~{}\frac{\Omega_{F}^{-}-\Omega_{F}^{+}}{16\pi\lambda_{so}^{2}}.$
(11)
We note that when both spin subbands are populated ($\mu>\Omega_{0}/2$) the
equilibrium spin density takes value ${S}_{0}=m\Omega_{0}/4\pi$ independent of
the Fermi energy, this is specific for Hamiltonian from Eq. 1.
The application of a scalar potential $U(\bm{r},t)$ deviates the electron
distribution from Eq.10. In this paper we focus on spatially smooth
perturbations ($k_{F}\cdot\nabla_{k}\ll 1$ and $\lambda_{F}\cdot\nabla_{r}\ll
1$) and study the electron gas response in the classical limit. For this
purpose we introduce the Wigner density matrix $\hat{f}_{k}(\bm{r},t)$ in the
following form
$\displaystyle\hat{f}_{k}(\bm{r},t)=\frac{1}{2}n_{k}(\bm{r},t)+\bm{S}_{k}(\bm{r},t)\cdot\hat{\bm{\sigma}},$
(12)
where $n_{k}(\bm{r},t),\bm{S}_{k}(\bm{r},t)$ can be treated as particle and
spin distribution functions locally in real space. In particular, the 2DEG
spin density perturbation emerging in the real space at point $\bm{r}$ can be
found from
$\delta\bm{S}(\bm{r},t)=\frac{1}{2}\sum_{k}{\rm
Sp}\left(\hat{f}_{k}(\bm{r},t)\cdot\hat{\bm{\sigma}}\right)-\bm{S}_{0}=\sum_{k}\bm{S}_{k}(\bm{r},t)-\bm{S}_{0}.$
(13)
In the clean limit $\hat{f}_{k}(\bm{r},t)$ satisfies the kinetic equation [57]
$\displaystyle\frac{\partial\hat{f}_{k}}{\partial
t}+\frac{1}{2}\Bigl{\\{}\left(\hat{\bm{v}}\cdot\nabla_{\bm{r}}\right);\hat{f}_{k}\Bigr{\\}}-\left[\bm{\Omega}_{k}\times\bm{S}_{k}\right]\cdot\hat{\bm{\sigma}}+\left(\bm{F}\cdot\nabla_{\bm{k}}\right)\hat{f}_{k}=0,$
(14)
where $\\{;\\}$ stands for the anticommutator, $\nabla_{\bm{r},\bm{k}}$ are
the nabla operators, $\bm{F}(\bm{r},t)=-\nabla_{\bm{r}}U(\bm{r},t)$ describes
the dynamical force acting on electrons, and the third term takes into account
the precession of the electron spin in the effective magnetic field
$\bm{\Omega}_{k}$. Let us draw the attention to the anticommutator type of
ordering between $\hat{\bm{v}}$ and $\hat{f}_{k}$ that appears in the second
term. This ordering directly stems from the Wigner transformation procedure
[58] and it is especially important to describe accurately the response in the
inhomogeneous regime.
## III Static spin textures
We start our analysis by inspecting the redistribution of the 2DEG charge and
spin densities nearby smooth electrostatic defects, such as Coulomb centres or
gating potential perturbations. The geometric character of electronic band
states and the associated nonzero Berry curvature underline the appearance of
chiral spin textures and adjoint persistent electric currents that surround
electrostatic potential inhomogeneity, see Fig. 1. In [59] we used the Kubo
formalism to address the nonlocal regime of the spin density response due to
short-range impurities. In this section, instead, we provide a detailed
semiclassical description of this phenomenon and accompany it by the
comprehensive physical analysis.
Figure 1: Formation of skyrmion-like spin textures and the distribution of the
persistent electric currents nearby electrostatic defects.
### III.1 General mechanisms of the intrinsic spin generation
Let us qualitatively discuss the effect of the electron spin non-adiabatic
rotation upon the precession in a slowly varying magnetic field [60, 61, 62].
We start by considering the precession equation for an electron spin $\bm{s}$
rotating upon a time-dependent frequency $\bm{\Omega}(t)$
$\frac{d\bm{s}}{dt}=\left[\bm{\Omega}(t)\times\bm{s}\right].$ (15)
Assuming the adiabatically slow rotation of $\bm{\Omega}(t)$, i.e. that the
characteristic time $\tau$ of its variation satisfies $\Omega\tau\gg 1$, the
zero-order solution of the precession equation simply describes the electron
spin $\bm{s}^{0}(t)=\bm{\Omega}(t)/2|\bm{\Omega}(t)|$ remaining co-aligned
with the instant direction of $\bm{\Omega}(t)$. However, the adiabatic
rotation of $\bm{s}^{0}(t)$ can be maintained only due to the appearance of
the non-adiabatic correction $\delta\bm{s}(t)$ directed perpendicular to the
instant vector $\bm{\Omega}(t)$. Naturally, this correction exists in the
first order in $(\Omega\tau)^{-1}$ and it can be found from the precession
equation keeping only the leading term due to $\bm{s}^{0}(t)$ in the time
derivative
$\frac{d\bm{s}^{0}}{dt}=\left[\bm{\Omega}(t)\times\delta\bm{s}(t)\right]\quad\rightarrow\quad\delta{\bm{s}}(t)=\frac{1}{2\Omega^{3}}\left[\bm{\Omega}\times\frac{d\bm{\Omega}}{dt}\right].$
(16)
The appearance of $\delta{s}\propto(\Omega\tau)^{-1}{s}^{0}$ is a general
property of the precession equation. Naturally, this is also valid when a
Larmor frequency stems from an effective magnetic field in k-space due to a
spin-orbit coupling. In this case, however, the vector $\bm{\Omega}_{k}$ that
governs the spin dynamics of an electron with momentum $\bm{k}$ varies in time
only provided that the electron momentum does not remain constant along its
trajectory $\dot{\bm{k}}\neq 0$, which is the case if $\bm{F}\neq 0$. The non-
adiabatic spin component acquired by an electron can be estimated from Eq. 16
by replacing the time derivative by $d/dt\to\dot{\bm{k}}\cdot\nabla_{\bm{k}}$
$(\dot{\bm{k}}\cdot\nabla_{\bm{k}})\bm{s}_{k}^{0}=\left[\bm{\Omega}_{k}\times\delta\bm{s}_{k}\right]\quad\rightarrow\quad\delta\bm{s}_{k}=\frac{1}{2\Omega_{k}^{3}}\left[\bm{\Omega}_{k}\times(\dot{\bm{k}}\cdot\nabla_{\bm{k}})\bm{\Omega}_{k}\right].$
(17)
We conclude that an electron moving along its classical trajectory with finite
acceleration has its spin always slightly tilted compared to the instantaneous
direction of $\bm{\Omega}_{k}$. Moreover, in view of the spin-momentum locking
such an intrinsically generated extra-spin leads to the change in the electron
velocity $\delta\bm{v}_{k}=2\lambda_{so}(\bm{e}_{z}\times\delta\bm{s}_{k})$.
The second spin-related phenomenon being important for the collective response
of 2DEG concerns the spin-dipole effect [40]. This mechanism is relevant when
the single electron density $|\psi(\bm{r})|^{2}$ deviates from the homogeneous
distribution and acquires some finite $\bm{r}$-dependence nearby an
inhomogeneity. Let us consider an electron at the unperturbed plane-wave state
$\psi_{k}^{\pm}$ from Eq. 3 with the momentum $\bm{k}$, its spin
$\bm{s}_{k}^{\pm}$ is determined by $\bm{\Omega}_{k}$. The corresponding
density $|\psi_{k}^{\pm}|^{2}$ is spatially homogeneous. In fact, the smooth
spatial variation of the density for such electron is possible only provided
that its wave-function gets an admixture of other plane-wave band states
$\psi_{k^{\prime}}^{\pm}$ with momenta $\bm{k}^{\prime}$ slighty differing
from $\bm{k}$. Essentially, the added states have different spin orientation
$\bm{s}_{k^{\prime}}^{\pm}\neq\bm{s}_{k}^{\pm}$, so the resulting average spin
density appears to be slightly tilted. In terms of the wave-packet dynamics
[63, 40] the mixing of spin-orbital states leads to the fact that the charge
and spin centers of the electron wave-packet do not coincide, which creates an
additional spin polarization. This scenario is specifically important for
localized electron states [64, 65]. We emphasize that the spin-dipole effect
is essentially connected with the spatial variation of the electron density.
In particular, if a given external field keeps an electron gas in the
homogeneous state, the spin-dipole contribution will be absent. The appearance
of the non-adiabatic correction from Eq. 17, on the contrary, is not connected
with the change of an electron density, it simply tracks the exact electron
spin dynamics along quasiclassical trajectories.
### III.2 Density matrix in a static inhomogeneous setting
We proceed with giving a rigorous description of the outlined phenomena based
on the kinetic equation for the density matrix. Let us consider an electron
gas subjected to an electrostatic potential $U(\bm{r})$ smoothly varying in
space. Since the unperturbed density matrix
$\hat{f}_{k}^{0}=\hat{f}_{k}^{+}+\hat{f}_{k}^{-}$ given by Eq. 10 has two
parts corresponding to $\eta=(\pm)$ subband states, the linear response
correction $\delta\hat{f}_{k}(\bm{r})=\hat{f}_{k}(\bm{r})-\hat{f}_{k}^{0}$
will be determined independently by two subband terms
$\delta\hat{f}_{k}(\bm{r})=\delta\hat{f}_{k}^{+}(\bm{r})+\delta\hat{f}_{k}^{-}(\bm{r})$.
We present the corresponding correction $\delta\hat{f}_{k}^{\eta}$ as follows
$\displaystyle\delta\hat{f}_{k}^{\eta}(\bm{r})=\frac{1}{2}\delta
n_{k}^{\eta}(\bm{r})+\delta\bm{S}_{k}^{\eta}(\bm{r})\cdot\hat{\bm{\sigma}},$
(18)
where $\delta n_{k}^{\eta}(\bm{r}),\delta\bm{S}_{k}^{\eta}(\bm{r})$ are the
perturbations of the electron density and spin distribution functions,
respectively.
The key suggestion implemented in this paper is to use the following ansats
for the linear response spin density
$\delta\bm{S}_{k}^{\eta}(\bm{r})=\delta
n_{k}^{\eta}(\bm{r})\bm{s}_{k}^{\eta}+n_{k}^{\eta}\delta\bm{s}_{k}^{\eta}(\bm{r})+\delta\bm{\mathcal{S}}_{k}^{\eta}(\bm{r}),$
(19)
where we took into account all possible types of
$\delta\bm{S}_{k}^{\eta}(\bm{r})$ variation. Indeed, the first term describes
the change of the electron spin distribution due to the change in the density
$\delta n_{k}^{\eta}$. The second term corresponds to the change of the spin
vector $\delta\bm{s}_{k}^{\eta}$ for each individual electron independently of
the electron number distribution. The third term is the remaining linear-order
variation, which is essentially neither due to $\delta n_{k}^{\eta}(\bm{r})$
or $\delta\bm{s}_{k}^{\eta}(\bm{r})$ separately; thus
$\delta\bm{\mathcal{S}}_{k}^{\eta}$ describes the correlated change of both
the electron spin and charge densities. Naturally, the second and the third
terms in this expansion turn out to describe the non-adiabatic spin tilting
and the spin-dipole effects, respectively.
We proceed with calculating $\delta\bm{S}_{k}^{\eta}(\bm{r})$ from the kinetic
equation 14. In what follows we keep in Eq. 14 only the terms linear in $U$
and $\bm{F}=-\nabla_{\bm{r}}U$. In this limit the change of the electron
density $\delta n_{k}^{\eta}$ can be determined independently from the scalar
part of Eq. 14. Taking the trace over Eq. 14 we get
$\left(\bm{v}_{k}^{\eta}\cdot\nabla_{r}+\bm{F}(\bm{r})\cdot\nabla_{k}\right)n_{k}^{\eta}(\bm{r})=0.$
(20)
Here $\bm{v}_{k}^{\eta}$ is the electron group velocity given by Eq. 6. In the
linear response regime the correction $\delta n_{k}^{\eta}$ is given by:
$\delta n_{k}^{\eta}(\bm{r})=U(\bm{r})(\partial
n_{k}^{\eta}/\partial\varepsilon)$, where $\varepsilon$ is the electron
energy. The change in the overall 2DEG density is $\delta n(\bm{r})=\delta
n^{+}(\bm{r})+\delta n^{-}(\bm{r})$, where $\delta
n^{\eta}(\bm{r})=-\nu_{F}^{\eta}~{}U(\bm{r})$ and $\nu_{F}^{\eta}$ is the
density of states in $\eta$ subbands taken at the Fermi energy.
Correspondingly, the perturbation of the spin density Eq. 13 due to the first
term in Eq. 19 is given by
$\delta\bm{S}^{(1)}(\bm{r})=\sum_{k,\eta}\bm{s}_{k}^{\eta}\cdot\delta
n_{k}^{\eta}(\bm{r})=\bm{e}_{z}\Omega_{0}\left(\frac{\nu_{F}^{+}}{\Omega_{F}^{+}}-\frac{\nu_{F}^{-}}{\Omega_{F}^{-}}\right)U(\bm{r}).$
(21)
The term $\delta\bm{S}^{(1)}(\bm{r})$ is responsible for the change in the
out-of-plane spin density component and it appears even if there is no spin-
orbit interaction. A complex spin-orbital electron dynamics is responsible for
an extra spin response described by $\delta{\bm{s}}_{k}^{\eta}$ and
$\delta\bm{\mathcal{S}}_{k}^{\eta}$. We notice that
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ are absent in a
homogeneous setting, thus the expansion of
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ starts with the
linear term $\nabla_{r}U$. Taking the trace over Eq. 14 multiplied by
$\hat{\bm{\sigma}}$ and keeping only the terms linear in $\nabla_{\bm{r}}$
gradient we get
$\displaystyle\left[\bm{\Omega}_{k}\times\delta\bm{s}_{k}^{\eta}(\bm{r})\right]-\left(\bm{F}(\bm{r})\cdot\nabla_{k}\right)\bm{s}_{k}^{\eta}=0,$
(22)
$\displaystyle\left[\bm{\Omega}_{k}\times\delta\bm{\mathcal{S}}_{k}^{\eta}(\bm{r})\right]+\left[\bm{s}_{k}^{\eta}\times\left(\bm{s}_{k}^{\eta}\times{\bm{\Omega}^{so}({\nabla}_{\bm{r}}n_{k}^{\eta})}\right)\right]=0,$
(23)
where $\bm{\Omega}^{so}(\nabla_{r}n_{k}^{\eta})$ is obtained from Eq. 2 by
replacing $\bm{k}\to\nabla_{\bm{r}}n_{k}^{\eta}(\bm{r})$.
Let us comment on the relation between
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ and the previously
described kinematic effects. The first equation Eq. 22 can be satisfied by
changing the electron spin vector $\delta\bm{s}_{k}^{\eta}$ independently of a
particular density distribution $n_{k}^{\eta}$, it thus indeed describes the
spin rotation of individual electrons due to the precession in the effective
magnetic field $\bm{\Omega}_{k}$. Naturally, the nonzero term
$\delta\bm{s}_{k}^{\eta}$ is exactly the non-adiabatic correction to the
instant spin vector $\bm{s}_{k}^{\eta}$ which follows adiabatically the local
direction of $\bm{\Omega}_{k}$. The solution of the equation 22 replicates the
result from Eq.17
$\delta\bm{s}_{k}^{\eta}(\bm{r})=\eta\frac{1}{2\Omega_{k}^{3}}\Bigl{[}\bm{\Omega}_{k}\times(\bm{F}(\bm{r})\cdot\nabla_{\bm{k}})\bm{\Omega}_{k}\Bigr{]}.$
(24)
It is worth noting that $\delta\bm{s}_{k}^{\eta}$ is nonlinear with respect to
$\bm{\Omega}_{k}$. The second equation Eq. 23 describes the appearance of
$\delta\bm{\mathcal{S}}_{k}^{\eta}$, the general form of the solution is given
by
$\delta\bm{\mathcal{S}}_{k}^{\eta}(\bm{r})=-\frac{1}{4\Omega_{k}^{2}}\left[\bm{\Omega}_{k}\times\bm{\Omega}^{so}({\nabla}_{r}n_{k}^{\eta})\right].$
(25)
Importantly, the additional spin density $\delta\bm{\mathcal{S}}_{k}^{\eta}$
responds directly to the spatial gradient of the electron density
$\nabla_{r}n_{k}^{\eta}(\bm{r})$ entering in $\bm{\Omega}^{so}$. In fact, this
allows us to refer $\delta\bm{\mathcal{S}}_{k}^{\eta}$ as the correlational
term: it is neither due to the independent change in the number of electrons
or due to the individual electron spin rotation. Instead,
$\delta\bm{\mathcal{S}}_{k}^{\eta}$ describes the simultaneous change in the
electron spin due to the variation in its spatial density, it is indeed
relevant to the spin-dipole effect.
### III.3 Interplay between microscopic mechanisms and the role of Berry
curvature
The explicit evaluation of extra-spin density terms from Eq. 24,25 for the
Rashba ferromagnet model gives the following expressions
$\displaystyle\delta\bm{s}_{k}^{\eta}=\eta\frac{e\mathcal{F}_{k}}{2\lambda_{so}}\cdot\bm{E}(\bm{r})-\eta\frac{2e\lambda_{so}^{2}}{\Omega_{k}^{3}}\left[\bm{k}\times\bm{E}(\bm{r})\right],$
(26)
$\displaystyle\delta\bm{\mathcal{S}}_{k}^{\eta}=-\mathcal{F}_{k}\cdot\frac{\Omega_{k}}{4\lambda_{so}}\nabla_{\bm{r}}n_{k}^{\eta}(\bm{r})+\eta\frac{\lambda_{so}}{2\Omega_{k}^{2}}\bm{e}_{z}\left(\bm{\Omega}_{k}\cdot\nabla_{\bm{r}}\right)n_{k}^{\eta}(\bm{r}),$
(27)
where $\mathcal{F}_{k}$ is the magnitude of the Berry curvature from Eq. 7,
and the density gradient
$\nabla_{r}n_{k}^{\eta}(\bm{r})=-e\bm{E}(\bm{r})(\partial
n_{k}^{\eta}/\partial\varepsilon)$ is due to the redistribution of electrons
in the vicinity of an electrostatic potential inhomogeneity.
We note that various terms from Eqs. 26, 27 give rise to quite different spin
phenomena. For instance, the second terms in
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ depend on the
electron momentum direction and they are particularly important for the
generation of spin currents in nonmagnetic systems (they survive at
$\Omega_{0}\to 0$); the second term in $\delta\bm{s}_{k}^{\eta}$ is
responsible for the universal spin Hall conductivity [62]. Alternatively, it
keeps significance for spin dynamics, see the details in Sec. IV. Below we
focus on the local magnetoelectric effect, that is the appearance of an
equilibrium spin density in response to the local electric field. This
phenomenon stems from the first terms in
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$; they can directly
generate an additional spin density at a given point in a space as they
survive averaging over the electron momentum direction. Moreover, these terms
can be explicitly expressed in terms of the Berry curvature, thus they are
specific for topological systems.
The equilibrium spin density perturbations coupled with the Berry curvature of
electronic states have only in-plane components; substituting Eqs. 26, 27 to
the spin density perturbation from Eq. 13 we get
$\displaystyle\delta\bm{S}_{\parallel}(\bm{r})=\sum_{k,\eta}n_{k}^{\eta}\delta\bm{s}_{k}^{\eta}(\bm{r})+\delta\bm{\mathcal{S}}_{k}^{\eta}(\bm{r})\equiv\left(\chi_{t}+\chi_{d}\right)\cdot\bm{E}(\bm{r}),$
(28)
where the magnetoelectric susceptibilities $\chi_{t,d}$ correspond to the non-
adiabatic spin tilting and spin-dipole effects, respectively. The evaluated
expressions for $\chi_{t},\chi_{d}$ are given by
$\chi_{t}=\frac{e}{2\lambda_{so}}\left(Q_{F}^{+}+Q_{F}^{-}\right),\qquad\chi_{d}=-e\frac{\lambda_{so}\Omega_{0}}{2}\left(\frac{\nu_{F}^{+}}{\Omega_{F+}^{2}}+\frac{\nu_{F}^{-}}{\Omega_{F-}^{2}}\right),$
(29)
where $Q_{F}^{\pm}$ is the total Berry flux from Eq. 8. It is important to
emphasize that both the non-adiabatic spin tilting and the spin-dipole effects
are equally important to describe correctly the emergent spin patterns in
2DEG. In Fig. 2 we plot the dependence of the overall spin-response
coefficient $\chi\equiv\chi_{t}+\chi_{d}$ (solid lines) along with the partial
contributions from $\chi_{t}$ and $\chi_{d}$ (dotted lines) on the electron
gas Fermi energy $\mu$. We note that the terms $\chi_{t}$ and $\chi_{d}$ are
generally of the same order of magnitude. Moreover, in case when the electron
gas populates both spin subbands $\mu>\Omega_{0}/2$ the overall response
entirely disappears $\chi_{t}+\chi_{d}=0$ (this feature was previously noted
by [59, 66]). In the opposite case when electrons fill only the lowest spin-
subband $\mu<\Omega_{0}/2$ the terms $\chi_{t},\chi_{d}$ have opposite signs,
which results in the sign-altering dependence of $\chi$ on the Fermi energy.
We finally note that when either the spin-orbit coupling or the exchange
interaction is absent, the coefficients $\chi_{t}=\chi_{d}=0$ turn to zero and
the corresponding equilibrium spin patterns disappear.
Figure 2: The dependence of the susceptibility $\chi=\chi_{t}+\chi_{d}$ on
the Fermi energy for two values of $\xi$ parameter: (a) $\xi=0.6$ and (b)
$\xi=0.2$.
### III.4 Discussion
Let us discuss the physical significance of the described phenomena. We
firstly comment on the role that intrinsic mechanisms described by Eqs. 26, 27
play for the charge and spin transport on distances that greatly exceed the
mean free path. The non-adiabatic spin precession lies in the basis of the
Karplus-Luttinger mechanism of the anomalous Hall effect (AHE) [67, 68, 69],
of the so-called intrinsic mechanisms of the spin Hall (SHE) [62] and spin-
galvanic effects [70]. However, in order to estimate correctly the overall
electron gas response one has to additionally examine the disorder effects. In
particular, the intrinsic contribution to AHE, which is due to the anomalous
velocity term $\delta\bm{v}_{k}^{\eta}\propto
e\mathcal{F}_{k}^{\eta}\cdot\left[\bm{e}_{z}\times\bm{E}\right]$, is generally
cancelled out by the contributions due to side-jump scattering processes [57,
53, 56]. Alternatively, considering the generation of spin currents upon the
applied homogeneous electric field one has to carefully account for the
emergent nonequilibrium phenomena [71, 70, 72, 73]; e.g. the spin Hall current
due to the intrinsic mechanism is often compensated by the nonequilibrium spin
current arising nearby the sample boundaries [58, 74, 75].
However, the contributions
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ preserve the
importance in the nondissipative regime, when the underlying electrostatic
perturbation varies at the distances much smaller than the mean free path. In
particular, this matters for 2DEG charge and spin distribution around an
ionized impurity, at that the typical spatial scale under consideration is the
Thomas-Fermi screening length. The distribution of an excessive 2DEG spin
density emerging around an axially symmetric perturbation forms a skyrmion-
like vortex pattern which is schematically shown in Fig. 1. One concludes that
a smooth electrostatic potential disorder in topological spin polarized 2DEG
inevitably generates chiral spin textures, which can be particularly important
for the transport properties of the corresponding system; the formation of
non-collinear spin order generally leads to the topological Hall effect [76,
77, 78]. Moreover, in view of the spin-velocity coupling the formation of a
mesoscopic in-plane spin density is accompanied by the generation of the
persistent electrical current density
$\bm{j}(\bm{r})=e~{}2\lambda_{so}\left[\bm{e}_{z}\times\delta{\bm{S}(\bm{r})}\right]$.
In this regard an axially symmetric perturbation from Fig. 1 is additionally
featured by radially propagating electric currents. The presence of local
equilibrium currents also maintains the orbital magnetization, this effect has
been considered in [66].
It is worth mentioning that the considered magnetoelectric susceptibility of
free electrons generally opens up a possibility to directly affect the host
magnetization by a mesoscopic electric perturbation. The electric field-
induced 2DEG spin density lies in 2D channel plane and it is perpendicular to
the orientation of host magnetization, thus it is able to produce torque-like
effects. However, these issues remain poorly investigated, even despite its
importance for the magnetization control at nanoscales.
The microscopic mechanisms under consideration are general for multiband
systems. In the appendix A we present the connection of our method with the
wave-packet quasiclassical technique used in [39, 40, 41]. In the appendix B
we relate $\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ to the
Kubo formula method for the charge-spin correlation functions used in [59]. In
particular, we show that the non-adiabatic spin precession is described by the
interband correlation functions, while the spin-dipole effect stems from the
intraband ones.
## IV Spin dynamics and magneto-optical effects
### IV.1 Electric dipole spin resonance
In this section we focus on the electron gas spin dynamics in presence of an
oscillating electric field and describe the corresponding optical properties
of a magnetic two-dimensional system. The optical response of a 2D conductive
channel is generally encoded in the optical conductivity $\sigma(\omega)$. In
particular, the absorption coefficient $\alpha(\omega)=(4\pi/c){\rm
Re}[\sigma_{xx}(\omega)]$ is connected with the longitudinal part of
conductivity $\sigma_{xx}$. Also, since the time-reversal symmetry is broken
in presence of magnetism, different magneto-optical effects are possible, e.g.
the magneto-optical Kerr effect (MOKE), that is the rotation of the reflected
light polarization by the complex Kerr angle $\phi_{K}$. MOKE generally
appears in a conductive media due to nonzero optical Hall conductivity
$\sigma_{H}(\omega)$, for a 2D layer and normal incidence [79] one can expess
$\phi_{K}=\sigma_{H}/\sigma_{xx}\sqrt{1+(4\pi i/\omega)\sigma_{xx}}$.
Importantly, the considered geometry opens up the possibility to realize the
resonant enhancement of the Hall conductivity and, thus, of the related
magneto-optical effects.
Commonly, MOKE is seen to acquire a resonance structure due to interband
transitions affected by the combined effect of the spin-orbit coupling and the
electron spin polarization; the corresponding intrinsic contributions to the
Hall conductivity at finite frequencies have been investigated in a number of
papers [80, 81, 44, 82]. The general idea that we are going to explore in this
paper and which stands in the basis for the enhancement of magneto-optical
phenomena is that the optical properties of magnetic 2D systems can be
understood in terms of the electric dipole spin resonance (EDSR).
Correspondingly, the part of the optical conductivity responsible for the
resonant features can be directly related to the resonantly generated spin
density of 2DEG.
Figure 3: The electric dipole spin resonance scheme and the appearance of MOKE
due to the resonant Hall current generation
$\bm{j}_{\omega}\propto\left[\bm{e}_{z}\times\delta\bm{S}_{\omega}\right]$ .
Let us illustrate this process in more detail, see Fig. 3. The exchange
interaction field gives rise to a momentum-independent Zeeman splitting of the
electron spin subbands, for the considered geometry it is directed
perpendicular to the 2DEG plane. In fact, the spin-orbit interaction can be
viewed as $\bm{k}$-dependent effective magnetic field
$\bm{\Omega}^{so}(\bm{k})$ acting on electron spins. The applied in-plane ac-
electric field $\bm{E}_{\omega}e^{-i\omega t}$ causes the electron’s momentum
oscillations $\delta\bm{k}\propto\bm{E}_{\omega}e^{-i\omega t}$, so the
associated spin-orbital field also oscillates with frequency $\omega$. We note
that $\bm{\Omega}^{so}(\bm{k})$ is perpendicular to the out-of-plane exchange
interaction component $\bm{\Omega}_{0}$. Naturally, this makes it possible to
induce spin transitions when the electric field frequency coincides with the
magnitude of the Zeeman spin splitting $\hbar\omega=\Omega_{0}$, which is
exactly the EDSR scheme [83]. This spin resonance causes the equilibrium
electron spin density $\bm{S}_{0}\parallel\bm{e}_{z}$ from Eq. 11 to rotate
onto 2DEG plane, thus resonantly generating an excessive in-plane spin density
$\delta\bm{S}_{\omega}$. In view of the spin-orbit coupling Eq. 5 between the
velocity and spin operators, the accumulation of $\delta\bm{S}_{\omega}$
immediately leads to a resonant enhancement of the associated electric current
density
$\delta\bm{j}_{\omega}=2e~{}\lambda_{so}\left[\bm{e}_{z}\times\delta\bm{S}_{\omega}\right]$
and of the corresponding contribution to the optical conductivity.
Importantly, the in-plane spin density appears in tilted polarization with
respect to the vector of the electric field, see Fig 3. In particular, the
manifestation of the nonzero Berry curvature lies in the fact, that there
exists the ”perpendicular” polarization of the spin density, which gives rise
to the anomalous velocity $\delta\bm{v}_{k}^{\eta}\propto
e\mathcal{F}_{k}^{\eta}\cdot\left[\bm{e}_{z}\times\bm{E}\right]$ directed
perpendicular to $\bm{E}_{\omega}$ and responsible for the the magneto-optical
effects. The resonant generation of the spin density in this polarization
leads to the enhancement of $\sigma_{H}(\omega)$.
### IV.2 Density matrix in the dynamical regime
Let us consider an oscillating electric field $\bm{E}_{\omega}e^{-i\omega t}$
applied in plane of the electron gas. We assume that the system remains
homogeneous and present $\hat{f}_{k}$ in the following form
$\hat{f}_{k}(t)=\frac{1}{2}n_{k}(t)+\bm{S}_{k}(t)\cdot\hat{\bm{\sigma}}.$ (30)
We keep to the high-frequency regime when $\omega$ greatly exceeds the typical
inverse relaxation time $\tau_{sc}^{-1}$ due to the scattering processes. The
distribution function $n_{k}(t)=n_{k}+\delta n_{k}(\omega)e^{-i\omega t}$
satisfies the scalar part of the kinetic equation Eq. 14
$\frac{\partial n_{k}(t)}{\partial
t}-e\left(\bm{E}(t)\cdot\nabla_{k}\right)n_{k}(t)=0,$ (31)
Since the equilibrium part contains terms from both spin subbands
$n_{k}=n_{k}^{+}+n_{k}^{-}$, the linear response perturbation $\delta
n_{k}(\omega)=\delta n_{k}^{+}(\omega)+\delta n_{k}^{-}(\omega)$ generally
contains two contributions
$\delta
n_{k}^{\eta}(\omega)=-\frac{e\bm{E}\cdot\bm{v}_{k}^{\eta}}{i\omega}\left(-\frac{\partial
n_{k}^{\eta}}{\partial\varepsilon}\right).$ (32)
The equation governing 2DEG spin dynamics is obtained similarly to Eq. 14 and
reads as
$\displaystyle\frac{\partial\bm{S}_{k}(t)}{\partial
t}-\left[\bm{\Omega}_{k}\times\bm{S}_{k}(t)\right]+e\left(\bm{E}(t)\cdot\nabla_{k}\right)\bm{S}_{k}(t)=0.$
(33)
At zero electric field this equation describes the electron spin precession
around $\bm{\Omega}_{k}$. The static regime solution in this case corresponds
to the equilibrium spin distribution
$\bm{S}_{k}^{\pm}\parallel\bm{\Omega}_{k}$ directed parallel or antiparallel
to the spin splitting field, while the non-stationary solution describes the
electron spin precession around $\bm{\Omega}_{k}$ with an eigenfrequency
$\Omega_{k}$. The nonzero $\bm{E}$, in its turn, drives the spin dynamics due
to the spin transfer in the momentum space. Naturally, when the frequency of
an external field $\omega$ coincides with the precession frequency of the
$\bm{k}$-electrons, the EDSR conditions are fulfilled leading to the resonant
rotation. This rotation occurs with the Rabi frequency
$\omega_{R}\propto\lambda_{so}E$, which goes to zero at small electric fields.
Naturally, in case of vanishing $\omega_{R}$ we can consider the linear
response regime with
$\bm{S}_{k}(t)=\bm{S}_{k}^{0}+\delta\bm{S}_{k}(\omega)e^{-i\omega t}$
differing from the equilibrium value
$\bm{S}_{k}^{0}=n_{k}^{+}\bm{s}_{k}^{+}+n_{k}^{-}\bm{s}_{k}^{-}$ by the
linear-order correction $\delta\bm{S}_{k}(\omega)$. This is justified when the
ongoing evolution of $\bm{S}_{k}(t)$ due to the Rabi oscillations is
interrupted by the spin relaxation processes. We thus introduce the
phenomenological spin relaxation rate $\Gamma$ and assume
$\omega_{R}\ll\Gamma\ll\Omega_{0}$.
In the linear response regime we can consider the spin response
$\delta\bm{S}_{k}(\omega)=\delta\bm{S}_{k}^{+}(\omega)+\delta\bm{S}_{k}^{-}(\omega)$
independently for each spin subband (recall that
$\bm{S}_{k}^{0}=n_{k}^{+}\bm{s}_{k}^{+}+n_{k}^{-}\bm{s}_{k}^{-}$). It is
convenient to present the linearized part in the following way
$\delta\bm{S}_{k}^{\eta}(\omega)=\delta
n_{k}^{\eta}(\omega)\bm{s}_{k}^{\eta}+n_{k}^{\eta}\delta\bm{s}_{k}^{\eta}(\omega)$,
where $\delta n_{k}^{\eta}(\omega)$ is determined by Eq. 32 and the equation
for $\delta\bm{s}_{k}^{\eta}(\omega)$ is given by
$\left(-i\omega+\Gamma\right)\delta\bm{s}_{k}^{\eta}(\omega)-\left[\bm{\Omega}_{k}\times\delta\bm{s}_{k}^{\eta}(\omega)\right]+e\left(\bm{E}_{\omega}\cdot\nabla_{k}\right)\bm{s}_{k}^{\eta}=0.$
(34)
Let us introduce the notation
$\delta\bm{s}_{k0}^{\eta}\equiv\delta\bm{s}_{k}^{\eta}(\omega\to 0)$ for the
additional electron spin density from Eq. 24 emerging in the static limit, we
note that $(\delta\bm{s}_{k0}^{\eta}\cdot\bm{\Omega}_{k})=0$. The third term
in this equation can be presented as follows
$e\left(\bm{E}_{\omega}\cdot\nabla_{k}\right)\bm{s}_{k}^{\eta}=\left[\bm{\Omega}_{k}\times\delta\bm{s}_{k0}^{\eta}\right]$.
The spin density perturbation $\delta\bm{s}_{k}^{\eta}(\omega)$ lies in the
plane perpendicular to $\bm{\Omega}_{k}$, the two independent polarizations
for $\delta\bm{s}_{k}^{\eta}(\omega)$ are given by $\delta\bm{s}_{k0}^{\eta}$
and $\left[\bm{n}_{k}\times\delta\bm{s}_{k0}^{\eta}\right]$, where
$\bm{n}_{k}=\bm{\Omega}_{k}/\Omega_{k}$ from Eq. 4. The solution of the
precession equation can be written in terms of these two vectors as follows
$\delta\bm{s}_{k}^{\eta}(\omega)=-\frac{\Omega_{k}^{2}}{\left(\omega-\Omega_{k}+i\Gamma\right)\left(\omega+\Omega_{k}+i\Gamma\right)}\left(\delta\bm{s}_{k0}^{\eta}+\frac{-i\omega+\Gamma}{\Omega_{k}}\left[\bm{n}_{k}\times\delta\bm{s}_{k0}^{\eta}\right]\right).$
(35)
The first term is directly due to the finite-frequency evolution of the non-
adiabatic spin tilt mechanism. The second term exists only at finite
frequencies and it arises from the electron spin retardation in the momentum
space. The denominator has a pole structure which reflects the EDSR with the
multiple resonances determined by $\omega=\Omega_{k}$.
The resulting correction to the density matrix can be presented as a sum of
two terms $\delta\hat{f}_{k}=e^{-i\omega t}(\delta\hat{f}_{k}^{\rm
den}+\delta\hat{f}_{k}^{\rm spin})$, where $\delta\hat{f}_{k}^{\rm den,spin}$
take the following form
$\displaystyle\delta\hat{f}_{k}^{\rm den}=\frac{1}{2}\left(\delta
n_{k}^{+}(\omega)+\delta n_{k}^{-}(\omega)\right)+\left(\delta
n_{k}^{+}(\omega)\bm{s}_{k}^{+}+\delta
n_{k}^{-}(\omega)\bm{s}_{k}^{-}\right)\cdot\hat{\bm{\sigma}},$ (36)
$\displaystyle\delta\hat{f}_{k}^{\rm
spin}=\left(n_{k}^{+}\delta\bm{s}_{k}^{+}(\omega)+n_{k}^{-}\delta\bm{s}_{k}^{-}(\omega)\right)\cdot\hat{\bm{\sigma}}.$
(37)
### IV.3 Resonant spin response and optical conductivity
We start the discussion of the optical conductivity. The contribution
$\delta\hat{f}_{k}^{\rm den}$ is related specifically to the perturbation of
the electron density and it gives rise to the dominant part of the
longitudinal conductivity
$\bm{j}_{\omega}=e\sum_{k,\eta}\delta
n_{k}^{\eta}(\omega)\cdot\bm{v}_{k}^{\eta}=\sigma_{xx}^{0}(\omega)\bm{E}_{\omega},\qquad\sigma_{xx}^{0}(\omega)=\frac{ie^{2}}{\omega}~{}\frac{{v_{F+}^{2}\nu_{F}^{+}}+{v_{F-}^{2}\nu_{F}^{-}}}{2}.$
(38)
This is simply the Drude conductivity at finite frequency and it describes
nondissipative retardation of the 2DEG density in ac-electric field. On the
contrary, the term $\delta\hat{f}_{k}^{\rm spin}$ is due to the spin rotation
only. This contribution is responsible for the spin resonance related
phenomena and below we consider its role in more detail.
The density of an electric current $\delta\bm{j}_{\omega}$ emerging due to the
spin part of the density matrix $\delta\hat{f}_{k}^{\rm spin}$ is coupled with
an induced in-plane spin density $\delta\bm{S}_{\omega}$ of 2DEG
$\displaystyle\delta\bm{j}_{\omega}=2e\cdot\lambda_{so}\left[\bm{e}_{z}\times\delta\bm{S}_{\omega}\right],$
(39) $\displaystyle\delta\bm{S}_{\omega}=\frac{1}{2}\sum_{k}{\rm
Sp}\left(\delta\hat{f}_{k}^{\rm
spin}\cdot\hat{\bm{\sigma}}\right)=\sum_{k,\eta}n_{k}^{\eta}~{}\delta\bm{s}_{k}^{\eta}(\omega).$
(40)
Since $\delta\bm{s}_{k}^{\eta}(\omega)$ generally has two polarizations, see
Eq. 35, the overall spin $\delta\bm{S}_{\omega}$ and correspondingly the
associated current $\delta\bm{j}_{\omega}$ are also featured by two
independent polarizations
$\displaystyle\delta\bm{S}_{\omega}=\chi_{l}(\omega)\left[\bm{e}_{z}\times\bm{E}_{\omega}\right]+\chi_{H}(\omega)\bm{E}_{\omega},$
(41)
$\displaystyle\delta\bm{j}_{\omega}=\sigma_{l}(\omega)\bm{E}_{\omega}+\sigma_{H}(\omega)\left[\bm{e}_{z}\times\bm{E}_{\omega}\right],$
(42)
where $\sigma_{l,H}(\omega)=2e\lambda_{so}\chi_{l,H}(\omega)$. By this we
identified the contributions to the optical conductivity related to the
magnetoelectric spin susceptibility.
The correction to the longitudinal conductivity $\sigma_{l}(\omega)$ is
related to the retardation term
$\left[\bm{n}_{k}\times\delta\bm{s}_{k0}^{\eta}\right]$ in Eq. 35. Using the
formula Eq. 26 for $\delta\bm{s}_{k0}^{\eta}$ and averaging over momentum
directions we get (below we restore the Planck constant $\hbar$)
$\sigma_{l}(\omega)=-ie^{2}\cdot\sum_{k}\frac{\left(n_{k}^{-}-n_{k}^{+}\right)\hbar\omega}{\left(\hbar\omega-\Omega_{k}+i\Gamma\right)\left(\hbar\omega+\Omega_{k}+i\Gamma\right)}\frac{\lambda_{so}^{2}}{\Omega_{k}}\left(1+\frac{\Omega_{0}^{2}}{\Omega_{k}^{2}}\right).$
(43)
The straightforward calculation of this integral gives
$\sigma_{l}(\omega)=-\frac{ie^{2}}{16\pi\hbar}\left[\frac{2\Omega_{0}^{2}}{\hbar\omega}\left(\frac{1}{\Omega_{\rm
min}}-\frac{1}{\Omega_{F}^{-}}\right)+\left(1+\frac{\Omega_{0}^{2}}{(\hbar\omega)^{2}}\right)\ln{\left(\frac{\hbar\omega+\Omega_{F}^{-}}{\hbar\omega-\Omega_{F}^{-}}\cdot\frac{\hbar\omega-\Omega_{\rm
min}}{\hbar\omega+\Omega_{\rm min}}\right)}\right],$ (44)
where $\Omega_{\rm min}=\Omega_{0}$ for $\mu<\Omega_{0}/2$ and $\Omega_{\rm
min}=\Omega_{F}^{+}$ for $\mu>\Omega_{0}/2$. The expression from above remains
well-defined at $\Gamma\to 0$. In fact, the poles $\hbar\omega=\Omega_{k}$ in
the denominator of $\delta\bm{s}_{k}^{\eta}(\omega)$ lie in the continuum
spectrum, so the overall response of closely lying resonances merges onto the
$\omega$-regular curve featured by the Van Hove singularities at the edges of
the spin splittins $\hbar\omega=(\Omega_{0},\Omega_{F}^{\pm})$.
The real part of the longitudinal conductivity describes the energy
dissipation. The presence of the resonant poles in Eq. 35 reflects the
appearance of a finite absorption. Indeed, the absorption coefficient is
nonzero in the frequency range $\Omega_{\rm min}<\hbar\omega<\Omega_{F}^{-}$
(see Fig. 4b) corresponding to EDSR, the expression is given by
$\alpha(\omega)=\frac{4\pi}{c}{\rm
Re}\left[\sigma_{l}(\omega)\right]=\frac{\pi e^{2}}{4\hbar
c}\left[1+\left(\frac{\Omega_{0}}{\hbar\omega}\right)^{2}\right],\qquad\Omega_{\rm
min}<\hbar\omega<\Omega_{F}^{-}.$ (45)
Figure 4: (a) The dependence of optical conductivities $\sigma_{l,H}$ on
frequency exhibits a resonant structure due to EDSR. (b) Electron band
structure and the transition energies $\Omega_{F}^{\pm}$ at the Fermi level.
The Hall conductivity $\sigma_{H}(\omega)$ stems from the Berry curvature
related term in $\delta\bm{s}_{k0}^{\eta}$. Taking into account Eqs. 26, 35
and averaging over the momentum direction we express $\sigma_{H}(\omega)$
$\sigma_{H}(\omega)=-\frac{e^{2}}{\hbar}\sum_{k}\frac{(n_{k}^{-}-n_{k}^{+})~{}\Omega_{k}^{2}}{\left(\hbar\omega-\Omega_{k}+i\Gamma\right)\left(\hbar\omega+\Omega_{k}+i\Gamma\right)}\cdot\mathcal{F}_{k}.$
(46)
The evaluation of this expression gives the following result
$\displaystyle\sigma_{H}(\omega)=-\frac{e^{2}}{4\pi\hbar}\frac{\Omega_{0}}{\hbar\omega}\ln{\left(\frac{\hbar\omega+\Omega_{F}^{-}}{\hbar\omega-\Omega_{F}^{-}}\cdot\frac{\hbar\omega-\Omega_{\rm
min}}{\hbar\omega+\Omega_{\rm min}}\right)}.$ (47)
Importantly, the Hall conductivity has the same resonance-aware logarithmic
term as $\sigma_{l}(\omega)$. Fig. 4 demonstrates the resonant enhancement of
the Hall conductivity in the EDSR absorption frequency range. Namely, we plot
the dependence of ${\rm Re}[\sigma_{l}(\omega)]$ and the absolute value
$|\sigma_{H}(\omega)|$ on the electric field frequency. It is clearly seen
from Fig. 4 that the increase in $|\sigma_{H}(\omega)|$ magnitude occurs
exactly in the same frequency range where ${\rm Re}[\sigma_{l}(\omega)]\neq 0$
is nonzero. In Fig. 5 we plot the dependences of real and imaginary parts of
the spin-resonance related optical conductivities $\sigma_{l,H}(\omega)$ on
frequency. The parameters are the same as in Fig. 4. The Van Hove
singularities give rise to the pronounced peaks in $|\sigma_{l,H}(\omega)|$ at
the boundary of the absorption band
$\hbar\omega=\Omega_{F}^{+},\Omega_{F}^{-}$. For the parameters taken in this
plot ($\mu=1.3\Omega_{0}$) the lower boundary is determined by
$\Omega_{F}^{+}$, see Fig. 4, as the electrons populate both spin subbands. We
also note that the behavior of $\sigma_{l,H}(\omega)$ when approaching the
static limit $\omega\to 0$ is different, see Fig. 5. While the longitudinal
part goes to zero $\sigma_{l}\to 0$, the Hall conductivity has a finite
nonzero limit $\sigma_{H}\to(e^{2}/\hbar)(Q_{F}^{+}+Q_{F}^{-})$ determined by
the total Berry flux $Q_{F}^{\pm}$ from Eq. 8 and reflecting the appearance of
persistent electric currents associated with the magnetoelectric
susceptibility. In the static limit, however, the accurate calculation of
$\sigma_{H}$ for a macroscopic sample requires one to take into account the
disorder effect [84].
Figure 5: The dependence of the optical conductivities $\sigma_{l}(\omega)$
(frame a) and $\sigma_{H}(\omega)$ (frame b) on the frequency $\omega$, the
parameters $\xi=0.5,\mu=1.3~{}\Omega_{0}$.
### IV.4 Discussion
The calculations of the optical conductivity of multiband systems is typically
performed using the Kubo formula [80, 81, 44, 82]. In the Appendix C we relate
the spin polarization and the density contributions from the density matrix
approach with different terms from the Kubo formalism. In Table 1 we summarize
the correspondence between these approaches; naturally the spin resonance
related terms are connected with the interband contributions $\sigma^{\rm
inter}$ to the conductivity.
Let us comment on the role of spin relaxation and electron scattering. The
multiple-peak structure of $\sigma_{l,H}(\omega)$ visible in Fig. 4 can be
well resolved only provided that the spin-orbit interaction splitting
($|\Omega_{F}^{\pm}-\Omega_{0}|\gg\tau_{sc}^{-1}$) exceeds the energy
broadening due to scattering processes. This requires rather strong spin-orbit
coupling. In the opposite case, the resonance profile of
$\sigma_{l,H}(\omega)$ will merge onto the single resonant-peak structure
centered at $\Omega_{0}$ with the line-shape sensitive to particular
scattering and spin relaxation processes, in analogy with EDSR due to an
electron gas in nonmagnetic semiconductors [85]. Interestingly, the Hall
conductivity can possess an additional information on spin relaxation times.
Kubo formula | $\sigma_{xx}^{\rm intra}$ | $\sigma_{xx}^{\rm inter}$ | $\sigma_{H}^{\rm inter}$
---|---|---|---
Density matrix | $\displaystyle\delta\hat{f}_{k}^{\rm den}$ | $\displaystyle n_{k}^{\eta}\left[\bm{n}_{p}\times\delta\bm{s}_{k0}^{\eta}\right]\cdot\hat{\bm{\sigma}}$ | $\displaystyle n_{k}^{\eta}\delta\bm{s}_{k0}^{\eta}\cdot\hat{\bm{\sigma}}$
Table 1: Density matrix and Kubo formula correspondence
We note that the finite absorption due to the electric dipole spin resonance
in 2DEG is not strictly limited to the case when the Zeeman field has an out-
of-plane component. In fact, most of the EDSR experiments with 2DEG in
nonmagnetic semiconductors [86, 87, 88] were carried out for the in-plane
magnetic field geometry. This is particularly useful when one aims to suppress
the orbital quantization effects and to focus on the spin-related response
only. On the contrary, combining spin-orbital electronic channels with
magnetism allows one to orient the Zeeman field perpendicular to the 2DEG
plane without breaking the spectrum onto Landau levels. Moreover, in this
setting the electron band states are featured by the appearance of a
topological structure. Studying experimentally the electronic spin resonance
phenomena in these systems seems of high interest as EDSR has an extra degree
of freedom that is the strong enhancement of the adjoint magneto-optical
effects.
Finally, the presented interpretation of the magneto-optical effects
enhancement in terms of spin resonance is equally relevant for other two-
dimensional models beyond Rashba ferromagnets. For instance, e.g. massive
Dirac metals [89], honeycomb lattices [90] or Haldane model [91] demonstrate
similar resonant features of the Hall conductivity.
## Summary
In summary, we have considered various spin-orbital phenomena leading to a
nontrivial behavior of an electron gas spin density upon application of the
electric field in two-dimensional magnets. Based on the density matrix
formalism we identified different microscopic mechanisms responsible for the
2DEG spin tilting in presence of an inhomogeneous electrostatic potential, and
described microscopic features of spin resonance upon oscillating electric
field with specific focus on optical conductivity and magneto-optical
phenomena. We traced the connection of the considered spin phenomena with the
Berry curvature of electronic band states thereby specifying the role of
electrons band topology. The presented analysis clarifies the basics of the
electron gas magnetoelectric response in two-dimensional magnets and
contributes to the ongoing discussion of its spintronics applications.
## Acknoledgments
The Author thanks I.V. Rozhansky, M.M. Glazov, P.S. Alekseev and N.S. Averkiev
from the Ioffe Institute for a very fruitful discussion of the results and for
giving useful advices. The work has been carried out with the financial
support of the Russian Science Foundation (project 18-72-10111). K.S.D. also
thanks the Foundation for the Advancement of Theoretical Physics and
Mathematics “BASIS”.
## Appendix A Wave-packet dynamics semiclassical approach
The semiclassical theory of band electrons moving in a spatially varying
adiabatic perturbation $U(\bm{r})$ can be built by considering the wave-packet
dynamics [42]. Let us introduce the wave packet $|W_{k}^{n}\rangle$ consisting
of the $n$-th band Bloch states $|u_{k}^{n}\rangle$, its centre of mass
coordinates in real and momentum spaces are located at $(\bm{r}_{c},\bm{k})$.
The average of the physical quantity ${Q}$ described by the operator $\hat{Q}$
can be expressed in the following way [40]
${Q}=\sum_{k,n}f_{n}(\bm{k},\bm{r})\cdot\langle
W_{k}^{n}|\hat{Q}|W_{k}^{n}\rangle|_{\bm{r}=\bm{r}_{c}}-\nabla_{\bm{r}}\cdot\sum_{k,n}f_{n}(\bm{k},\bm{r})\cdot\langle
W_{k}^{n}|\hat{Q}\cdot\left(\hat{\bm{r}}-\bm{r}\right)|W_{k}^{n}\rangle|_{\bm{r}=\bm{r}_{c}},$
(48)
where the first term treats the wave packet as a point particle with the
distribution function $f_{n}(\bm{k},\bm{r})$, and the second term is the
first-order correction due to the wave-packet finite size effects. The great
advantage of this consideration is that it allows one to describe the electron
dynamics in terms of semiclassical equations. For instance, in the
nondissipative regime $f_{n}(\bm{k},\bm{r})$ satisfies the Liouville’s
equation
$\frac{df_{n}}{dt}=\frac{\partial f_{n}}{\partial
t}+\left\\{f_{n};\mathcal{H}\right\\}=0,$ (49)
where $\mathcal{H}=\varepsilon_{k}^{n}+U(\bm{r})$ is the classical Hamiltonian
function in $n$-th electron band with energy $\varepsilon_{k}^{n}$. The
Poisson bracket $\\{A;B\\}$ for $A,B$ physical quantities depending on
($\bm{r},\bm{k}$) takes into account the kinematic Berry phase [43, 49, 50]
$\displaystyle\left\\{A;B\right\\}=\omega_{\alpha\beta}\cdot(\partial_{\alpha}A)(\partial_{\beta}B),\qquad\omega_{\alpha\beta}=\begin{pmatrix}\varepsilon_{\alpha\beta\gamma}\Omega_{\gamma}^{n}&\delta_{\alpha\beta}\\\
-\delta_{\alpha\beta}&0\end{pmatrix},\qquad(\alpha,\beta)=({\bm{r},\bm{k}}),$
(50)
where $\omega_{\alpha\beta}$ is the antisymmetric Poisson matrix,
$\varepsilon_{\alpha\beta\gamma}$ is the Levi-Civita tensor, and
$\bm{\Omega}^{n}$ is the Berry curvature in $n$-th Bloch band defined as
follows
$\bm{\Omega}^{n}=\nabla_{\bm{k}}\times\mathcal{A}_{k}^{n}=i\langle\nabla_{\bm{k}}u_{k}^{n}|\times|\nabla_{\bm{k}}u_{k}^{n}\rangle,$
where $\mathcal{A}_{k}^{n}$ is the Berry connection. The expression for the
Liouville’s equation with account for the explicit form of
$\omega_{\alpha\beta}$ is given by:
$\displaystyle\frac{\partial f_{n}}{\partial
t}+\left(\frac{\partial\varepsilon_{k}^{n}}{\partial\bm{k}}+\left[\dot{\bm{k}}\times\bm{\Omega}_{n}\right]\right)\cdot\frac{\partial
f_{n}}{\partial\bm{r}}+\dot{\bm{k}}\cdot\frac{\partial
f_{n}}{\partial\bm{k}}=0$ (51)
where $\dot{\bm{k}}=-\nabla_{\bm{r}}\mathcal{H}=-\nabla_{\bm{r}}U(\bm{r})$.
The second term in brackets describes a full electron velocity
${\bm{v}}=\\{\mathcal{H};\bm{r}\\}=\bm{v}_{k}^{n}-[\nabla_{\bm{r}}U,\bm{\Omega}^{n}]$,
here $\bm{v}_{k}^{n}=\nabla_{k}\varepsilon_{k}^{n}$.
Let us apply this technique to calculate the emerging spin density nearby the
electrostatic inhomogeneity. We focus on the linear response regime. Following
Eq. 48 we present the spin density $\bm{S}(\bm{r})$ as follows
$\displaystyle\bm{S}(\bm{r})=\sum_{k,n}f_{n}(\bm{k},\bm{r})\cdot\langle
W_{k}^{n}|\hat{\bm{S}}|W_{k}^{n}\rangle-\nabla_{\bm{r}}\cdot\sum_{k,n}f_{n}(\bm{k},\bm{r})\langle
u_{k}^{n}|\hat{\bm{S}}\left(i\nabla_{\bm{k}}-\mathcal{A}_{k}^{n}\right)|u_{k}^{n}\rangle.$
(52)
In the second term we took into account that the wave packet
$|W_{k}^{n}\rangle$ is strongly localized nearby $\bm{k}$ in the momentum
space and we can approximate it as follows $|W_{k}^{n}\rangle\approx
e^{i\bm{kr}}|u_{k}^{n}\rangle$, which leads us directly to the expression in
Eq. 52. The unperturbed spin density $\bm{S}_{0}$ corresponds to
$U(\bm{r})=0$, at that $f_{n}(\bm{k},\bm{r})=f_{n}^{0}(\bm{k})$ and
$\bm{S}_{0}$ is given by
$\bm{S}_{0}=\sum_{k,n}f_{n}^{0}(\bm{k})\langle
u_{k}^{n}|\hat{\bm{S}}|u_{k}^{n}\rangle.$ (53)
The linear order deviations from $\bm{S}_{0}$ arise from three different
origins. Firstly, the distribution function
$f_{n}(\bm{k},\bm{r})=f_{n}^{0}(\bm{k})+\delta f_{n}(\bm{k},\bm{r})$ in
presence of $U$ is modified according to Eq. 51
$\left(\bm{v}_{k}^{n}\cdot\nabla_{\bm{r}}\right)\delta
f_{n}(\bm{k},\bm{r})+\bm{F}(\bm{r})\cdot\frac{\partial
f_{n}^{0}}{\partial\bm{k}}=0,\qquad\delta
f_{n}(\bm{k},\bm{r})=-U(\bm{r})\left(-\frac{\partial
f_{n}^{0}}{\partial\varepsilon}\right).$ (54)
Taking into account the redistribution of the electron density in the first
term in Eq. 52 and approximating $\langle
W_{k}^{n}|\hat{\bm{S}}|W_{k}^{n}\rangle\approx\langle
u_{k}^{n}|\hat{\bm{S}}|u_{k}^{n}\rangle$ we obtain the contribution identical
with Eq. 21 in the density matrix approach
$\displaystyle\delta\bm{S}^{(1)}(\bm{r})=\sum_{k,n}\delta
f_{n}(\bm{k},\bm{r})\langle u_{k}^{n}|\hat{\bm{S}}|u_{k}^{n}\rangle.$ (55)
Also, the inhomogeneous structure of $f_{n}$ gives rise to the spin-dipole
contribution, that is the second term in Eq. 52
$\delta\bm{\mathcal{S}}(\bm{r})=-\bm{F}(\bm{r})\cdot\sum_{k,n}\left(-\frac{\partial
f_{n}^{0}}{\partial\varepsilon}\right)\langle
u_{k}^{n}|\hat{\bm{S}}\left(i\nabla_{\bm{k}}-\mathcal{A}_{k}^{n}\right)|u_{k}^{n}\rangle.$
(56)
The straightforward evaluation of this expression for the Rashba ferromagnet
model leads to the susceptibility $\chi_{d}$ given by Eq. 29. Finally, there
is also the linear order perturbation which is not associated with the change
in the electron distribution. In fact, the first term in Eq. 52 is determined
by the average spin of an electron wave packet $\bm{s}_{k}^{n}(t)=\langle
W_{k}^{n}|\hat{\bm{S}}|W_{k}^{n}\rangle$, which satisfies the precession
equation
$\frac{d\bm{s}_{k}^{n}}{dt}=\left[\bm{\Omega}_{k}\times\bm{s}_{k}^{n}\right].$
(57)
According to our discussion from III.1, the wave-packet spin acquires a non-
adiabatic correction $\delta\bm{s}_{k}^{n}$ linear in $\bm{F}$ and given by
Eq. 17. This term gives rise to the spin perturbation
$\delta\bm{S}=\sum_{(k,n)}f_{n}^{0}\delta\bm{s}_{k}^{n}$ identical to
$\chi_{t}$ contribution to the spin susceptibility from Eq. 29.
## Appendix B Kubo formula in the static limit
In this appendix we relate the semiclassical description of magnetoelectric
susceptibility in terms of the density matrix with the Kubo formula for the
charge-spin correlation functions, considered in detail in [59]. The spin
density induced in 2DEG by the change in the potential energy $U(\bm{r})$ is
given in linear response by
$\delta\bm{S}(\bm{r})=\int\frac{d\bm{q}}{(2\pi)^{2}}e^{i\bm{qr}}\bm{\mathcal{Q}}(\bm{q})U(\bm{q}),$
(58)
where $U(\bm{q})$ is the Fourier component of $U(\bm{r})$ and the static
charge-spin correlation function $\bm{\mathcal{Q}}(\bm{q})$ can be computed
from the Kubo formula
$\displaystyle\bm{\mathcal{Q}}(\bm{q})=\sum_{m,n}\bm{\mathcal{Q}}^{mn}(\bm{q}),$
(59)
$\displaystyle\bm{\mathcal{Q}}^{mn}(\bm{q})=\sum_{k}f_{k}^{m}\frac{\langle
u_{k}^{m}|\bm{\hat{S}}|u_{k+q}^{n}\rangle\langle
u_{k+q}^{n}|u_{k}^{m}\rangle}{\varepsilon_{k}^{m}-\varepsilon_{k+q}^{n}+i0}-f_{k+q}^{m}\frac{\langle
u_{k}^{n}|\bm{\hat{S}}|u_{k+q}^{m}\rangle\langle
u_{k+q}^{m}|u_{k}^{n}\rangle}{\varepsilon_{k}^{n}-\varepsilon_{k+q}^{m}+i0}.$
The terms $\bm{\mathcal{Q}}^{nn}$ with $m=n$ describe the intraband
contributions, while $\bm{\mathcal{Q}}^{mn}$ with $m\neq n$ correspond to the
interband ones.
The Kubo formula 59 has been explicitly evaluated for an arbitrary wavevector
$\bm{q}$ in [59] for Rashba ferromagnet and Dirac models. Here we focus on the
semiclassical regime when the potential $U$ changes smoothly on the Fermi
wavelength $\lambda_{F}$ scale, so the following relation is fulfilled
$\lambda_{F}\cdot\nabla_{r}U\ll U$. In this case the spin response becomes
local and the correlation function for the Rashba ferromagnet model takes the
following form $\bm{\mathcal{Q}}=i\bm{q}\cdot\chi$, where $\chi$ is the
$q$-independent coefficient describing the susceptibility
$\delta\bm{S}(\bm{r})=\chi\cdot\bm{E}(\bm{r})$.
We now proceed with considering the role of intra- and interband terms. In the
intraband contribution $\bm{\mathcal{Q}}^{nn}$ we replace
$(f_{k}^{n}-f_{k+q}^{n})/(\varepsilon_{k}^{n}-\varepsilon_{k+q}^{n}+i0)\approx\partial
f_{k}^{n}/\partial\varepsilon$ and keep only the ${q}$-linear terms in the
matrix elements. At that the expression takes the following form
$\bm{\mathcal{Q}}^{nn}=-i\bm{q}\cdot\sum_{k}\left(-\frac{\partial
f_{n}^{0}}{\partial\varepsilon}\right)\langle
u_{k}^{n}|\hat{\bm{S}}\left(i\nabla_{\bm{k}}-\mathcal{A}_{k}^{n}\right)|u_{k}^{n}\rangle,$
(60)
where $\mathcal{A}_{k}^{n}=i\langle u_{k}^{n}|\nabla_{{k}}u_{k}^{n}\rangle$ is
the Berry connection. When taking the Fourier transform Eq. 58
$\bm{\mathcal{Q}}^{nn}$ gives exactly the spin perturbation
$\delta\bm{\mathcal{S}}$ in form of Eq. 56 corresponding to the spin-dipole
term within the semiclassical wave-packet approach. We thus conclude that the
spin-dipole effect from Eq. 29 is related to the intraband terms in the Kubo
formula.
In the interband contributions $\bm{\mathcal{Q}}^{mn}$ we also keep only the
linear terms with respect to $\bm{q}$, which brings us to the following
expression
$\displaystyle\bm{\mathcal{Q}}^{mn}(\bm{q})=i\bm{q}\cdot\sum_{k}f_{k}^{m}~{}{\rm
Re}\left(\frac{\langle
u_{k}^{n}|\hat{\bm{\sigma}}|u_{k}^{m}\rangle\cdot{\mathcal{A}}_{k}^{mn}}{\varepsilon_{k}^{m}-\varepsilon_{k}^{n}}\right),$
(61)
where $\mathcal{A}_{k}^{mn}=i\langle u_{k}^{m}|\nabla_{k}u_{k}^{n}\rangle$.
The straightforward calculations for the Rashba ferromagnet model gives
$\bm{\mathcal{Q}}^{mn}(\bm{q})=i\bm{q}\sum_{k}f_{k}^{m}\cdot\frac{\mathcal{F}_{k}^{mn}}{2\lambda_{so}},$
(62)
where $\mathcal{F}_{k}^{mn}=\nabla_{{k}}\times\mathcal{A}_{k}^{mn}$ is the
Berry curvature. The interband terms are related exactly to the non-adiabatic
spin tilt effect described by $\delta\bm{s}_{k}^{\eta}$ in the density matrix
formalism and given by $\chi_{t}$ susceptibility from Eq. 29.
## Appendix C Kubo formula in the dynamical regime
In this appendix we relate the Kubo formula calculations of the optical
conductivity with the spin resonance related terms emerging in the density
matrix approach. Kubo formula for the conductivity is given by
$\sigma_{\alpha\beta}(\omega)=\frac{ie^{2}}{S}\sum_{k,m,n}\frac{f_{k}^{m}-f_{k+q}^{n}}{\varepsilon_{k}^{m}-\varepsilon_{k+q}^{n}}\cdot\frac{v_{(k,m),(k+q,n)}^{\alpha}v_{(k+q,n),(k,m)}^{\beta}}{\varepsilon_{k}^{m}-\varepsilon_{k+q}^{n}+\hbar\omega+i0},$
(63)
where $\bm{q}\to 0$ and ${\bm{v}}_{ij}$ is the proper matrix element of the
velocity operator between $i,j$ states. We consider firstly the longitudinal
conductivity $\sigma_{xx}(\omega)$. The contribution to $\sigma_{xx}(\omega)$
due to intraband terms has the form
$\sigma_{xx}^{\rm intra}(\omega)=\frac{ie^{2}}{\hbar\omega}\sum_{m}\int
d\varepsilon~{}\nu_{m}(\varepsilon)\left(-\frac{\partial
f_{k}^{m}}{\partial\varepsilon}\right)\langle|v_{{k},m}^{x}|^{2}\rangle,$ (64)
where $\nu_{m}$ is the density of states in the corresponding band $m$ and
$\langle|v_{{k},m}^{x}|^{2}\rangle$ is the angular averaged square of the
matrix element modulus. This part describes the Drude conductivity at
$\omega\tau_{sc}\gg 1$ due to the perturbation of the electron density and it
corresponds to Eq. 38 from the main text. For the Rashba ferromagnet model the
evaluation of the integral gives
$\sigma_{xx}^{\rm
intra}(\omega)=i\frac{e^{2}}{\omega}\cdot\frac{{v_{F+}^{2}\nu_{F}^{+}}+{v_{F-}^{2}\nu_{F}^{-}}}{2}.$
(65)
The contribution to $\sigma_{xx}(\omega)$ due to interband terms in case of
the Rashba ferromagnet model has the following form
$\sigma_{xx}^{\rm
inter}(\omega)=\frac{ie^{2}}{S}\sum_{k}\frac{f_{k}^{-}-f_{k}^{+}}{-\Omega_{k}}\cdot\frac{\langle|v_{(k,-),(k,+)}^{x}|^{2}\rangle}{\hbar\omega-\Omega_{k}+i0}+\frac{f_{k}^{+}-f_{k}^{-}}{\Omega_{k}}\cdot\frac{\langle|v_{(k,-),(k,+)}^{x}|^{2}\rangle}{\hbar\omega+\Omega_{k}+i0}.$
(66)
The angular averaged term is
$\langle|v_{(k,-),(k,+)}^{x}|^{2}\rangle=(\lambda_{so}^{2}/2)(1+\Omega_{0}^{2}/\Omega_{k}^{2})$.
Using this formula and combining the denominators in $\sigma_{xx}^{\rm inter}$
we get the following expression
$\sigma_{xx}^{\rm
inter}(\omega)=-ie^{2}\cdot\sum_{k}\frac{\left(f_{k}^{-}-f_{k}^{+}\right)\hbar\omega}{\left(\hbar\omega-\Omega_{k}+i\Gamma\right)\left(\hbar\omega+\Omega_{k}+i\Gamma\right)}\frac{\lambda_{so}^{2}}{\Omega_{k}}\left(1+\frac{\Omega_{0}^{2}}{\Omega_{k}^{2}}\right),$
(67)
which repeats Eq. 43 for $\sigma_{l}(\omega)$. We thus conclude that
$\sigma_{xx}^{\rm inter}(\omega)$ is related to
$\left[\bm{n}_{k}\times\delta\bm{s}_{k0}^{\eta}\right]$ polarization in terms
of the in-plane spin density (see Eqs. 35, 26 from the main text). It is
instructive to analyze the energy absorption due to the spin resonance. For
this purpose we write down explicitly the expression for the real part of the
longitudinal conductivity due to the interband terms
${\rm Re}[\sigma_{xx}^{\rm inter}(\omega)]=\frac{\pi e^{2}}{\hbar\omega
S}\sum_{k}\left(f_{k}^{-}-f_{k}^{+}\right)\left|v_{(k,-),(k,+)}^{x}\right|^{2}\cdot\delta\left(\varepsilon_{k}^{-}-\varepsilon_{k}^{+}+\hbar\omega\right).$
(68)
The expression has the form of the Fermi golden rule, its straigthforward
calculation leads to the Eq. 45.
We now turn to the transversal component of the conductivity. The interband
contribution can be expressed as:
$\sigma_{yx}^{\rm
inter}(\omega)=\frac{ie^{2}}{S}\sum_{k}\frac{f_{k}^{-}-f_{k}^{+}}{-\Omega_{k}}\frac{v_{-+}^{y}v_{+-}^{x}}{\hbar\omega-\Omega_{k}+i0}+\frac{f_{k}^{+}-f_{k}^{-}}{\Omega_{k}}\frac{\left(v_{-+}^{y}v_{+-}^{x}\right)^{\ast}}{\hbar\omega+\Omega_{k}+i0}$
(69)
The angular averaged combination of matrix elements $\langle
v_{-+}^{y}v_{+-}^{x}\rangle=-i\lambda_{so}\Omega_{0}/\Omega_{k}$ is purely
imaginary. Combining both terms we obtain
$\sigma_{yx}^{\rm
inter}(\omega)=-\frac{e^{2}}{S}\sum_{k<k_{F}^{-}}\frac{\Omega_{k}^{2}}{(\hbar\omega-\Omega_{k}+i0)(\hbar\omega+\Omega_{k}+i0)}\cdot\mathcal{F}_{k},$
(70)
which is same expression Eq. 46 that we get via the density matrix formalism
considering $\bm{s}_{\bm{k}0}$ contribution to the spin density (see Eqs. 35,
26 from the main text).
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|
J0537.5+0959 | J053745.9+095759 | J053745.9+095759 | -11.35 | 2.64 | 0.02 | 1.47 | -0.89 | -5.39 | 0.38 | 2.49
J0539.2$-$6333 | J054002.9-633216 | J054002.9-633216 | -11.79 | 2.09 | -0.03 | 1.04 | -1.04 | -4.03 | 1.19 | 2.55
J0544.8+5209 | J054424.5+521513 | J054424.5+521513 | -11.6 | 2.64 | -0.7 | 0.94 | -0.69 | -5.19 | 1.12 | 2.56
J0610.8$-$4911 | J061100.0-491034 | J061100.0-491034 | -11.98 | 1.93 | -0.65 | 1.21 | 0.31 | -4.77 | 0.31 | 2.15
J0610.8$-$4911 | J061031.8-491222 | J061031.8-491222 | -11.98 | 1.93 | -0.65 | 1.21 | -0.75 | -1.92 | -0.02 | 0.01
J0620.7$-$5034 | J062045.7-503350 | J062045.7-503350 | -11.98 | 1.88 | 0.16 | 1.32 | -0.28 | -4.43 | 0.45 | 2.06
J0633.9+5840 | J063400.1+584036 | J063400.1+584036 | -11.9 | 2.1 | -0.6 | 0.92 | -1.03 | -5.33 | 0.53 | 2.11
J0650.6+2055 | J065035.4+205557 | J065035.4+205557 | -11.25 | 1.79 | 0.3 | 0.98 | 0.05 | -4.01 | 0.64 | 1.66
J0704.3$-$4829 | J070421.8-482645 | J070421.8-482648 | -11.61 | 2.07 | 0.08 | 1.24 | -0.39 | -4.64 | 0.64 | 1.73
J0738.6+1311 | J073843.4+131330 | J073843.4+131330 | -11.57 | 2.6 | -0.36 | 1.23 | -0.97 | -2.22 | -0.03 | 0.08
J0800.1$-$5531 | J080013.1-553408 | J080013.1-553408 | -11.64 | 2.47 | -0.49 | 1.02 | -0.89 | -4.74 | 0.2 | 1.21
J0800.1$-$5531 | J075949.3-553254 | J075949.3-553254 | -11.64 | 2.47 | -0.49 | 1.02 | -0.94 | -4.15 | 1.12 | 2.36
J0800.9+0733 | J080056.5+073235 | J080056.5+073235 | -11.94 | 1.83 | 0.55 | 0.61 | -0.34 | -3.92 | 0.75 | 2.19
J0827.0$-$4111 | J082705.4-411159 | J082705.4-411159 | -11.54 | 2.1 | -0.14 | 0.99 | -0.4 | -5.37 | 0.63 | 1.74
J0838.5+4013 | J083902.8+401548 | J083903.0+401546 | -11.99 | 1.69 | -1.68 | 0.5 | -0.4 | -4.6 | 0.39 | 1.54
J0903.5+4057 | J090342.8+405503 | J090342.8+405503 | -12.02 | 1.96 | 0.11 | 1.03 | -0.89 | -4.14 | 1.02 | 2.76
J0906.1$-$1011 | J090616.2-101430 | J090616.1-101426 | -11.76 | 2.07 | -0.3 | 0.48 | -0.69 | -4.73 | 1.12 | 2.56
J0910.1$-$1816 | J091003.9-181613 | J091003.9-181613 | -11.8 | 1.95 | 0.36 | 1.1 | -0.21 | -4.77 | 0.35 | 2.2
J0914.5+6845 | J091430.0+684509 | J091429.7+684509 | -11.9 | 1.88 | 0.24 | 1.02 | -0.31 | -4.56 | 0.47 | 2.27
J0928.4$-$5256 | J092818.7-525701 | J092818.7-525701 | -11.38 | 2.0 | -0.17 | 0.89 | -0.21 | -5.13 | 0.4 | 2.41
J0930.9$-$3030 | J093058.0-303118 | J093057.9-303118 | -11.67 | 1.96 | -0.36 | 0.74 | -0.67 | -4.75 | 0.38 | 1.91
J0934.5+7223 | J093334.0+722101 | J093333.7+722101 | -11.64 | 2.91 | -4.0 | 1.03 | -0.74 | -4.46 | 0.73 | 3.05
J0938.8+5155 | J093835.0+515455 | J093834.8+515453 | -11.87 | 2.06 | 0.01 | 1.08 | -0.67 | -4.69 | 0.88 | 2.83
J1008.2$-$1000 | J100749.3-094910 | J100749.4-094912 | -11.39 | 3.08 | -0.03 | 1.05 | -1.11 | -4.87 | 1.05 | 2.67
J1008.2$-$1000 | J100802.5-095918 | J100802.5-095918 | -11.39 | 3.08 | -0.03 | 1.05 | -0.39 | -4.77 | 1.15 | 3.31
J1008.2$-$1000 | J100848.6-095450 | J100848.6-095450 | -11.39 | 3.08 | -0.03 | 1.05 | 0.61 | -3.12 | 0.68 | 2.21
J1011.1$-$4420 | J101132.0-442255 | J101132.0-442255 | -11.81 | 2.02 | -0.16 | 0.96 | 0.86 | -3.91 | 0.4 | 2.13
J1016.1$-$4247 | J101620.8-424723 | J101620.7-424723 | -11.42 | 1.87 | 0.43 | 1.04 | -0.56 | -4.45 | 0.68 | 1.86
J1016.2$-$5729 | J101625.7-572807 | J101625.7-572807 | -10.9 | 2.54 | 0.07 | 1.08 | -0.34 | -4.9 | 0.49 | 0.68
J1018.1$-$2705 | J101750.2-270550 | J101750.2-270550 | -11.47 | 2.54 | 0.23 | 1.04 | -0.29 | -4.49 | 0.33 | 1.8
J1018.1$-$4051 | J101807.6-404408 | J101807.6-404408 | -11.57 | 2.52 | 0.27 | 1.09 | -1.09 | -5.28 | 0.92 | 2.64
J1018.1$-$4051 | J101801.5-405520 | J101801.4-405519 | -11.57 | 2.52 | 0.27 | 1.09 | -1.03 | -4.63 | 1.08 | 1.83
J1024.5$-$4543 | J102432.6-454428 | J102432.5-454428 | -11.6 | 1.92 | -0.03 | 0.89 | 0.1 | -4.94 | 0.32 | 1.75
J1034.7$-$4645 | J103438.7-464405 | J103438.7-464405 | -11.7 | 2.17 | -0.13 | 1.13 | 0.21 | -5.27 | 0.54 | 2.23
J1048.4$-$5030 | J104824.2-502941 | J104824.2-502941 | -11.74 | 1.69 | 0.31 | 0.94 | -0.78 | -4.27 | 0.68 | 2.23
J1049.8+2741 | J104938.8+274217 | J104938.8+274213 | -11.79 | 2.13 | -0.41 | 1.09 | -0.68 | -4.11 | 0.29 | 1.14
J1106.7+3623 | J110636.7+362648 | J110636.5+362650 | -11.5 | 2.51 | -0.07 | 1.86 | -1.0 | -4.66 | 1.08 | 2.6
J1111.4+0137 | J111114.2+013431 | J111114.2+013431 | -11.77 | 2.42 | -0.48 | 0.7 | 0.11 | -4.88 | 0.48 | 1.47
J1119.9$-$1007 | J111948.2-100704 | J111948.4-100707 | -11.69 | 2.15 | 0.29 | 1.38 | -0.69 | -4.56 | 0.81 | 2.52
J1122.0$-$0231 | J112213.8-022916 | J112213.7-022914 | -11.71 | 2.37 | 0.16 | 0.92 | -0.29 | -4.22 | 0.51 | 2.18
J1146.0$-$0638 | J114600.8-063851 | J114600.8-063851 | -11.46 | 1.75 | 0.27 | 0.82 | -0.55 | -4.86 | 0.34 | 1.98
J1155.2$-$1111 | J115514.7-111125 | J115514.7-111125 | -11.74 | 1.93 | 0.28 | 1.05 | -0.55 | -5.16 | 0.19 | 2.73
J1220.1$-$2458 | J122014.5-245949 | J122014.5-245949 | -11.56 | 2.03 | -0.34 | 1.05 | 0.01 | -4.83 | 0.51 | 2.07
J1243.7+1727 | J124351.6+172643 | J124351.8+172645 | -11.95 | 2.01 | -0.17 | 0.69 | -1.0 | -4.38 | 0.51 | 2.58
J1256.8+5329 | J125630.4+533203 | J125630.5+533205 | -11.4 | 2.64 | 0.07 | 1.02 | -1.15 | -4.71 | 1.16 | 2.53
J1320.3$-$6410 | J132016.9-641353 | J132015.9-641349 | -11.33 | 2.63 | 0.56 | 1.15 | 2.32 | -4.29 | 0.01 | -0.17
J1326.0+3507 | J132544.4+350450 | J132544.4+350450 | -11.67 | 2.15 | 0.24 | 1.75 | -0.84 | -4.68 | 0.66 | 1.83
J1326.0+3507 | J132622.3+350628 | J132622.2+350625 | -11.67 | 2.15 | 0.24 | 1.75 | -1.19 | -4.57 | 1.06 | 2.41
J1415.9$-$1504 | J141546.1-150229 | J141546.1-150229 | -11.68 | 2.18 | 0.19 | 0.98 | -0.55 | -4.56 | 0.67 | 1.61
J1429.8$-$0739 | J142949.7-073302 | J142949.5-073305 | -11.9 | 2.34 | 0.59 | 0.96 | -0.24 | -3.57 | 0.5 | 3.25
J1513.0$-$3118 | J151244.8-311647 | J151244.8-311647 | -11.42 | 2.43 | 0.42 | 1.0 | -0.46 | -3.02 | -0.02 | 0.11
J1514.8+4448 | J151451.0+444957 | J151451.0+444957 | -11.18 | 2.36 | 0.62 | 1.85 | -1.66 | -4.95 | 0.72 | 2.7
J1528.4+2004 | J152836.0+200424 | J152835.8+200421 | -11.91 | 2.14 | -0.36 | 0.98 | -0.17 | -4.83 | 0.36 | 2.36
J1545.0$-$6642 | J154459.0-664148 | J154458.9-664147 | -11.42 | 1.71 | -0.1 | 0.81 | 0.41 | -4.68 | 0.6 | 1.51
J1557.2+3822 | J155711.9+382032 | J155711.9+382032 | -12.14 | 1.98 | 0.09 | 0.98 | -0.4 | -4.88 | 0.35 | 1.89
J1623.7$-$2315 | J162334.1-231750 | J162334.1-231750 | -11.18 | 2.62 | -0.07 | 0.85 | -0.55 | -4.69 | 0.93 | 2.94
J1631.8+4144 | J163146.7+414634 | J163146.7+414633 | -11.93 | 1.8 | 0.01 | 1.16 | -0.44 | -4.23 | 0.65 | 2.39
J1637.5+3005 | J163739.3+301015 | J163739.2+301013 | -11.68 | 2.82 | -0.3 | 0.91 | -0.23 | -5.12 | 1.01 | 3.13
J1637.5+3005 | J163728.2+300958 | J163728.1+300953 | -11.68 | 2.82 | -0.3 | 0.91 | -1.27 | -4.94 | 1.2 | 2.53
J1644.8+1850 | J164457.3+185149 | J164457.2+185150 | -11.61 | 2.06 | 0.38 | 0.97 | -0.89 | -4.56 | 0.81 | 2.84
J1645.0+1654 | J164500.0+165510 | J164459.8+165513 | -11.73 | 2.26 | -0.11 | 0.85 | -0.41 | -4.54 | 0.41 | 1.44
J1650.9$-$4420 | J165124.2-442142 | J165124.2-442142 | -10.57 | 2.64 | 0.64 | 0.88 | 3.18 | -5.96 | 0.18 | 0.2
J1651.7$-$7241 | J165151.5-724310 | J165151.5-724310 | -11.98 | 1.99 | -1.11 | 0.49 | 0.0 | -4.71 | 0.35 | 1.57
J1720.6$-$5144 | J172032.7-514413 | J172032.7-514413 | -11.57 | 2.06 | 0.29 | 0.87 | 0.44 | -4.75 | 0.37 | 1.72
J1818.5+2533 | J181830.9+253707 | J181831.2+253707 | -11.4 | 2.6 | 0.38 | 1.17 | -1.23 | -3.84 | 0.2 | 1.67
J1820.4$-$1609 | J182029.6-161044 | J182029.6-161044 | -10.19 | 2.67 | -0.48 | 1.08 | -0.89 | -4.91 | 0.54 |
J1846.9$-$0227 | J184650.7-022904 | J184650.7-022904 | -10.47 | 2.5 | 0.73 | 0.98 | 1.5 | -3.56 | -0.09 | -0.16
J1910.8+2856 | J191059.4+285635 | J191059.4+285635 | -11.45 | 1.81 | 0.32 | 1.34 | -0.96 | -4.84 | 0.18 | 2.56
J1910.8+2856 | J191052.2+285624 | J191052.2+285624 | -11.45 | 1.81 | 0.32 | 1.34 | -0.02 | -5.23 | 0.63 | 2.3
J1918.0+0331 | J191803.6+033030 | J191803.6+033030 | -11.44 | 1.77 | 0.31 | 1.25 | -0.47 | -5.44 | 0.5 | 1.26
J1927.5+0154 | J192731.3+015357 | J192731.3+015357 | -11.56 | 1.79 | 0.29 | 1.16 | -0.3 | -4.79 | 0.52 | 2.34
J1955.3$-$5032 | J195512.5-503012 | J195512.5-503012 | -11.62 | 2.43 | 0.22 | 1.21 | -0.36 | -4.95 | 0.42 | 2.08
J2008.4+1619 | J200827.6+161844 | J200827.6+161844 | -11.65 | 2.12 | 0.31 | 1.12 | -0.19 | -4.99 | 0.55 | 2.16
J2041.1$-$6138 | J204112.0-613952 | J204112.0-613949 | -11.59 | 2.18 | -0.0 | 0.63 | -0.75 | -4.44 | 0.54 | 2.09
J2046.9$-$5409 | J204700.5-541246 | J204700.7-541245 | -11.8 | 1.9 | 0.33 | 0.95 | -0.58 | -4.67 | 0.34 | 2.17
J2109.6+3954 | J210936.4+395513 | J210936.4+395513 | -11.48 | 1.69 | 0.47 | 1.11 | -0.38 | -5.43 | 0.68 | 2.37
J2114.9$-$3326 | J211452.0-332532 | J211452.1-332533 | -11.44 | 2.07 | -0.52 | 1.07 | -0.25 | -4.72 | 0.55 | 1.8
J2159.6$-$4620 | J215935.9-461954 | J215936.1-461953 | -11.55 | 1.72 | 0.04 | 1.15 | -0.79 | -4.42 | 0.52 | 2.6
J2207.1+2222 | J220704.3+222234 | J220704.1+222232 | -11.84 | 1.85 | -0.12 | 1.15 | -0.6 | -4.58 | 0.48 | 3.06
J2222.9+1507 | J222253.9+151052 | J222253.9+151055 | -11.88 | 2.11 | -0.07 | 0.87 | -0.69 | -4.11 | 0.77 | 3.15
J2225.8$-$0804 | J222552.9-080416 | J222552.9-080416 | -11.65 | 1.97 | 0.12 | 1.08 | -0.77 | -5.01 | 0.25 | 2.95
J2237.2$-$6726 | J223709.3-672614 | J223709.3-672614 | -11.76 | 2.01 | -0.12 | 1.25 | -0.26 | -4.81 | 0.4 | 1.8
J2240.3$-$5241 | J224017.5-524117 | J224017.7-524113 | -11.2 | 2.16 | 0.04 | 1.27 | -1.51 | -4.66 | 0.58 | 2.08
J2247.7$-$5857 | J224745.0-585501 | J224745.0-585501 | -11.65 | 2.55 | -0.5 | 1.12 | -0.66 | -5.03 | 0.34 | 2.69
J2303.9+5554 | J230351.7+555618 | J230351.7+555618 | -11.63 | 1.83 | 0.11 | 0.89 | 0.12 | -5.12 | 0.42 | 1.57
J2311.6$-$4427 | J231145.6-443221 | J231145.6-443221 | -11.84 | 2.27 | 0.05 | 0.98 | -0.89 | -4.34 | 0.79 | 2.53
J2317.7+2839 | J231740.0+283954 | J231740.0+283954 | -11.51 | 2.0 | -0.47 | 1.43 | -1.43 | -4.71 | 0.48 | 2.53
J2326.9$-$4130 | J232653.2-412713 | J232653.1-412711 | -11.47 | 2.78 | -1.19 | 1.34 | -0.97 | -3.87 | 1.1 | 2.8
J2336.9$-$8427 | J233627.1-842648 | J233627.1-842648 | -11.72 | 2.15 | -0.5 | 0.95 | -0.39 | -4.75 | 0.51 | 2.26
J2337.7$-$2903 | J233730.2-290241 | J233730.2-290241 | -11.87 | 2.16 | 0.24 | 0.61 | -0.19 | -4.56 | 0.66 | 2.41
J2351.4$-$2818 | J235136.5-282154 | J235136.5-282154 | -11.79 | 2.35 | -0.01 | 0.65 | -0.73 | -3.31 | -0.04 | 1.26
Note. — All the parameters are described in Table 1.
Table 3: The association results based on the neural network probabilities for all the 4FGL unassociated sources 4FGL Name | UVOT | P_fsrq | SIMBAD/SDSS | redshift
---|---|---|---|---
| | | FSRQ candidates from NN results |
J1008.2$-$1000 | J100802.5$-$095918 | 0.993 | QSO | 1.66(Krogager et al., 2018)
J1008.2$-$1000 | J100749.4$-$094912 | 0.993 | $\cdots$ | $\cdots$
J1637.5+3005 | J163728.1+300953 | 0.992 | QSO | 1.2; Fig 10a
J1637.5+3005 | J163739.2+301013 | 0.991 | Seyfert I | 0.155; Fig 9f
| | | BL Lac candidates from NN results |
J0026.1$-$0732 | J002611.6$-$073115 | 0.007 | blazar candidate | 0.5phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J0031.5$-$5648 | J003135.1$-$564640 | 0.008 | $\cdots$ | $\cdots$
J0057.9+6326 | J005758.1+632642 | 0.006 | blazar candidate | 0.18phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J0156.3$-$2420 | J015624.6$-$242003 | 0.007 | blazar candidate | Chang et al. (2019)
J0159.0+3313 | J015905.0+331255 | 0.008 | $\cdots$ | $\cdots$
J0231.0+3505 | J023112.2+350445 | 0.006 | $\cdots$ | $\cdots$
J0301.6$-$5617 | J030115.1$-$561644 | 0.008 | $\cdots$ | $\cdots$
J0302.5+3354 | J030226.7+335448 | 0.007 | $\cdots$ | $\cdots$
J0327.6+2620 | J032737.2+262008 | 0.007 | $\cdots$ | $\cdots$
J0409.2+2542 | J040921.6+254440 | 0.008 | $\cdots$ | $\cdots$
J0610.8$-$4911 | J061031.8$-$491222 | 0.007 | $\cdots$ | $\cdots$
J0610.8$-$4911 | J061100.0$-$491034 | 0.007 | QSO | Warwick et al. (2012)
J0620.7$-$5034 | J062045.7$-$503350 | 0.007 | BL Lac | 0.25phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J0633.9+5840 | J063400.1+584036 | 0.009 | blazar candidate | 0.29phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J0650.6+2055 | J065035.4+205557 | 0.006 | BL Lac | 0.3phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J0704.3$-$4829 | J070421.8$-$482648 | 0.009 | $\cdots$ | $\cdots$
J0800.9+0733 | J080056.5+073235 | 0.007 | blazar candidate | 0.44phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J0827.0$-$4111 | J082705.4$-$411159 | 0.009 | $\cdots$ | $\cdots$
J0838.5+4013 | J083903.0+401546 | 0.006 | BL Lac | 0.19; Fig 8b
J0903.5+4057 | J090342.8+405503 | 0.009 | QSO | 0.89; Fig 8c
J0910.1$-$1816 | J091003.9$-$181613 | 0.007 | blazar candidate | 0.45phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J0914.5+6845 | J091429.7+684509 | 0.007 | blazar candidate | 0.45phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J0928.4$-$5256 | J092818.7$-$525701 | 0.007 | $\cdots$ | $\cdots$
J0930.9$-$3030 | J093057.9$-$303118 | 0.007 | $\cdots$ | $\cdots$
J1011.1$-$4420 | J101132.0$-$442255 | 0.007 | $\cdots$ | $\cdots$
J1016.1$-$4247 | J101620.7$-$424723 | 0.007 | $\cdots$ | $\cdots$
J1024.5$-$4543 | J102432.5$-$454428 | 0.007 | BL Lac | 0.37phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J1048.4$-$5030 | J104824.2$-$502941 | 0.006 | $\cdots$ | $\cdots$
J1146.0$-$0638 | J114600.8$-$063851 | 0.006 | BL Lac | 0.64phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J1155.2$-$1111 | J115514.7$-$111125 | 0.007 | BL Lac | 0.47phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J1220.1$-$2458 | J122014.5$-$245949 | 0.007 | blazar candidate | 0.48phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J1243.7+1727 | J124351.8+172645 | 0.007 | QSO | 0.94** The redshift values provided by the SDSS$-$DR16 and SIMBAD catalogs are different; 0.94 and 2.14 respectively. In addition, the redshift estimate has varied from 0.5 to 2.14 in different data releases for SDSS. The value provided in this table corresponds to the latest data release, DR16.; Fig 9b
J1545.0$-$6642 | J154458.9$-$664147 | 0.006 | BL Lac | 0.23phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J1557.2+3822 | J155711.9+382032 | 0.008 | $\cdots$ | $\cdots$
J1631.8+4144 | J163146.7+414633 | 0.006 | galaxy | 0.71; Fig 9e
J1651.7$-$7241 | J165151.5$-$724310 | 0.007 | $\cdots$ | $\cdots$
J1720.6$-$5144 | J172032.7$-$514413 | 0.008 | $\cdots$ | $\cdots$
J1910.8+2856 | J191052.2+285624 | 0.007 | BL Lac | 0.3phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J1910.8+2856 | J191059.4+285635 | 0.006 | CV | Pourbaix et al. (2004)
J1918.0+0331 | J191803.6+033030 | 0.006 | blazar candidate | 0.23phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J1927.5+0154 | J192731.3+015357 | 0.006 | $\cdots$ | $\cdots$
J2046.9$-$5409 | J204700.7$-$541245 | 0.007 | $\cdots$ | $\cdots$
J2109.6+3954 | J210936.4+395513 | 0.007 | $\cdots$ | $\cdots$
J2114.9$-$3326 | J211452.1$-$332533 | 0.007 | $\cdots$ | $\cdots$
J2159.6$-$4620 | J215936.1$-$461953 | 0.006 | blazar candidate | 0.52phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J2207.1+2222 | J220704.1+222232 | 0.007 | blazar candidate | 0.55; Fig 10b
J2225.8$-$0804 | J222552.9$-$080416 | 0.008 | $\cdots$ | $\cdots$
J2237.2$-$6726 | J223709.3$-$672614 | 0.008 | $\cdots$ | $\cdots$
J2303.9+5554 | J230351.7+555618 | 0.006 | $\cdots$ | $\cdots$
J2317.7+2839 | J231740.0+283954 | 0.007 | galaxy | 0.57 ; Fig 10c
| | | ambiguous candidates from NN results |
J0004.4$-$4001 | J000434.2$-$400035 | 0.018 | BL Lac | 0.10phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J0025.4$-$4838 | J002536.9$-$483810 | 0.014 | $\cdots$ | $\cdots$
J0037.2$-$2653 | J003729.5$-$265045 | 0.597 | $\cdots$ | $\cdots$
J0058.3$-$4603 | J005806.3$-$460419 | 0.011 | $\cdots$ | $\cdots$
J0118.3$-$6008 | J011824.0$-$600753 | 0.035 | $\cdots$ | $\cdots$
J0120.2$-$7944 | J011914.7$-$794510 | 0.947 | $\cdots$ | $\cdots$
J0125.9$-$6303 | J012548.1$-$630245 | 0.022 | $\cdots$ | $\cdots$
J0209.8+2626 | J020946.5+262528 | 0.101 | blazar candidate | 0.68; Fig. 8a
J0240.2$-$0248 | J024004.6$-$024505 | 0.406 | $\cdots$ | $\cdots$
J0259.0+0552 | J025857.6+055244 | 0.01 | $\cdots$ | $\cdots$
J0406.2+0639 | J040607.7+063919 | 0.244 | $\cdots$ | $\cdots$
J0427.8$-$6704binbin this source is classified as binary source in the most recent third data release 4FGL catalog (4FGL$-$DR3) (Abdollahi et al., 2022) | J042749.6$-$670435 | 0.141 | $\cdots$ | $\cdots$
J0537.5+0959 | J053745.9+095759 | 0.938 | $\cdots$ | $\cdots$
J0539.2$-$6333 | J054002.9$-$633216 | 0.015 | $\cdots$ | $\cdots$
J0544.8+5209 | J054424.5+521513 | 0.959 | $\cdots$ | $\cdots$
J0738.6+1311 | J073843.4+131330 | 0.695 | $\cdots$ | $\cdots$
J0800.1$-$5531 | J075949.3$-$553254 | 0.52 | $\cdots$ | $\cdots$
J0800.1$-$5531 | J080013.1$-$553408 | 0.29 | $\cdots$ | $\cdots$
J0906.1$-$1011 | J090616.1$-$101426 | 0.015 | $\cdots$ | $\cdots$
J0934.5+7223 | J093333.7+722101 | 0.3 | $\cdots$ | $\cdots$
J0938.8+5155 | J093834.8+515453 | 0.014 | QSO | 0.41; Fig 8d
J1008.2$-$1000 | J100848.6$-$095450 | 0.972 | Seyfert I | 0.05(Jones et al., 2009)
J1016.2$-$5729 | J101625.7$-$572807 | 0.461 | $\cdots$ | $\cdots$
J1018.1$-$2705 | J101750.2$-$270550 | 0.647 | $\cdots$ | $\cdots$
J1018.1$-$4051 | J101801.4$-$405519 | 0.951 | $\cdots$ | $\cdots$
J1018.1$-$4051 | J101807.6$-$404408 | 0.966 | $\cdots$ | $\cdots$
J1034.7$-$4645 | J103438.7$-$464405 | 0.017 | blazar candidate | 0.33phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J1049.8+2741 | J104938.8+274213 | 0.012 | radio galaxy | 0.14; Fig 8e
J1106.7+3623 | J110636.5+362650 | 0.963 | QSO | 1.55; Fig 8f
J1111.4+0137 | J111114.2+013431 | 0.125 | radio galaxy | 0.26; Fig 9a
J1119.9$-$1007 | J111948.4$-$100707 | 0.029 | $\cdots$ | $\cdots$
J1122.0$-$0231 | J112213.7$-$022914 | 0.191 | BL Lac | $\cdots$
J1256.8+5329 | J125630.5+533205 | 0.984 | QSO | 0.99; Fig 9c
J1320.3$-$6410 | J132015.9$-$641349 | 0.71 | $\cdots$ | $\cdots$
J1326.0+3507 | J132622.2+350625 | 0.055 | QSO | 1.04; Fig 9d
J1326.0+3507 | J132544.4+350450 | 0.028 | $\cdots$ | $\cdots$
J1415.9$-$1504 | J141546.1$-$150229 | 0.015 | $\cdots$ | $\cdots$
J1429.8$-$0739 | J142949.5$-$073305 | 0.36 | $\cdots$ | $\cdots$
J1513.0$-$3118 | J151244.8$-$311647 | 0.496 | Rot Variable Star | Kiraga (2012)
J1514.8+4448 | J151451.0+444957 | 0.831 | $\cdots$ | $\cdots$
J1528.4+2004 | J152835.8+200421 | 0.013 | blazar candidate | 0.52phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J1623.7$-$2315 | J162334.1$-$231750 | 0.958 | $\cdots$ | $\cdots$
J1644.8+1850 | J164457.2+185150 | 0.015 | $\cdots$ | $\cdots$
J1645.0+1654 | J164459.8+165513 | 0.031 | $\cdots$ | $\cdots$
J1650.9$-$4420 | J165124.2$-$442142 | 0.692 | $\cdots$ | $\cdots$
J1818.5+2533 | J181831.2+253707 | 0.721 | $\cdots$ | $\cdots$
J1846.9$-$0227 | J184650.7$-$022904 | 0.469 | $\cdots$ | $\cdots$
J1955.3$-$5032 | J195512.5$-$503012 | 0.487 | $\cdots$ | $\cdots$
J2008.4+1619 | J200827.6+161844 | 0.014 | $\cdots$ | $\cdots$
J2041.1$-$6138 | J204112.0$-$613949 | 0.012 | $\cdots$ | $\cdots$
J2222.9+1507 | J222253.9+151055 | 0.013 | $\cdots$ | $\cdots$
J2240.3$-$5241 | J224017.7$-$524113 | 0.013 | blazar candidate | 0.25phph photometric redshift from the High$-$Synchrotron Peaked blazars catalog by Chang et al. (2019)
J2247.7$-$5857 | J224745.0$-$585501 | 0.552 | $\cdots$ | $\cdots$
J2311.6$-$4427 | J231145.6$-$443221 | 0.073 | $\cdots$ | $\cdots$
J2326.9$-$4130 | J232653.1$-$412711 | 0.938 | $\cdots$ | $\cdots$
J2336.9$-$8427 | J233627.1$-$842648 | 0.01 | $\cdots$ | $\cdots$
J2337.7$-$2903 | J233730.2$-$290241 | 0.017 | $\cdots$ | $\cdots$
J2351.4$-$2818 | J235136.5$-$282154 | 0.159 | Brightest Galaxy in the Cluster | Lauer et al. (2014) |
# On the Relationship between CH3OD Abundance and Temperature in the Orion KL
Nebula
Olivia H. Wilkins Division of Chemistry and Chemical Engineering, California
Institute of Technology, Pasadena, CA 91125 USA<EMAIL_ADDRESS>Geoffrey A. Blake Division of Chemistry and Chemical Engineering, California
Institute of Technology, Pasadena, CA 91125 USA
###### Abstract
The relative abundances of singly-deuterated methanol isotopologues,
[CH2DOH]/[CH3OD], in star-forming regions deviate from the statistically
expected ratio of 3. In Orion KL, the nearest high-mass star-forming region to
Earth, the singly-deuterated methanol ratio is about 1, and the cause for this
observation has been explored through theory for nearly three decades. We
present high-angular resolution observations of Orion KL using the Atacama
Large Millimeter/submillimeter Array to map small-scale changes in CH3OD
column density across the nebula, which provide a new avenue to examine the
deuterium chemistry during star and planet formation. By considering how CH3OD
column densities vary with temperature, we find evidence of chemical processes
that can significantly alter the observed column densities. The astronomical
data are compared with existing theoretical work and support D-H exchange
between CH3OH and heavy water (i.e., HDO and D2O) at methanol’s hydroxyl site
in the icy mantles of dust grains. The enhanced CH3OD column densities are
localized to the Hot Core-SW region, a pattern that may be linked to the
coupled evolution of ice mantel chemistry and star formation in giant
molecular clouds. This work provides new perspectives on deuterated methanol
chemistry in Orion KL and informs considerations that may guide future
theoretical, experimental, and observational work.
Current address: NASA Postdoctoral Program Fellow, NASA Goddard Space Flight
Center, Greenbelt, MD, 20771 USA Division of Geological and Planetary
Sciences, California Institute of Technology, Pasadena, CA 91125 USA
## 1 Introduction
The relative abundances of site-specific stable isotopologues, particularly
those involving deuterated compounds, are powerful tools that can be used to
trace chemical evolution in the interstellar medium, and during star and
planet formation. For example, the relative abundances of heavy water (i.e.,
HDO and D2O) in comets and meteorites can provide insights into the processing
of water between the primordial molecular cloud and present-day Earth.1, 2
Interstellar compounds such as N2H+ and CH3OH have D/H ratios that are higher
than the cosmic value of ${\sim}10^{-5}$,3 and that ratio is a function of
temperature, with higher D/H ratios signalling formation in colder, typically
denser, environments.4
In many high-mass and low-mass protostars alike, the deuterium chemistry of
methanol—namely, the relative abundances of the singly-deuterated isotopomers
CH2DOH and CH3OD—has presented itself as a mystery. Methanol is one of the
simplest complex (having $\geq$6 atoms) organic molecules, and it is found at
every stage of star formation, from cold cloud cores and hot cores/corinos to
outflows and circumstellar disks. 5, 6 As such, it is commonly used as a
tracer of other complex organics.
In prestellar and protostellar cores, methanol forms primarily via successive
hydrogenation of frozen CO on grain mantles (eq 1).7, 8
$\ch{CO->[H]HCO->[H]H2CO->[H]H3CO->[H]CH3OH}$ (1)
Statistically, we would expect the [CH2DOH]/[CH3OD] ratio to be 3 since there
are three methyl hydrogen sites compared to a single hydroxyl site. This
statistical ratio has been observed in the massive star-forming region NGC
7538-IRS1 but not toward many other star-forming regions.9 Low-mass cores
generally exhibit ratios ${>}3$ and as much as $\utilde{>}$10,10, 11 while
high-mass protostars tend to have [CH2DOH]/[CH3OD] ratios of ${<}3$.12, 13, 14
Astrochemical models have predicted that the [CH2DOH]/[CH3OD] ratio should be
$\geq$10 in prestellar cores and that CH3OD is only efficiently formed on icy
grains at later evolutionary stages when the ices are warmed due to the
presence of young (proto)stars.15 Deviations from the statistical ratio in
high-mass star-forming regions have also been attributed to grain surface
chemistry,16, 17 but investigations into the intricacies of such processes—and
the potential role of gas processing—are ongoing.
The Orion Kleinmann-Low (Orion KL) nebula is a high-mass star-forming region
notable for its peculiar methanol deuteration. At a distance of $\sim$388
pc,18 Orion KL is uniquely situated to explore the relationship between
relative deuterated methanol abundances and environmental conditions because
sub-environments within the nebula can be resolved, even with modest imaging
capabilities. The two most well-studied regions within Orion KL are the Hot
Core and Compact Ridge. The Hot Core region contains denser and warmer gas
($n_{\rm H_{2}}\sim 10^{7}$ cm-3, $T_{\rm kin}\sim 200$ K) whereas the Compact
Ridge, to the southwest,***In astronomical maps, north is up, east is left,
and west is right. is cooler and less dense ($n_{\rm H_{2}}\utilde{<}10^{6}$
cm-3, $T_{\rm kin}\sim 100{-}150$ K).19, 20 These regions are also the
prominent sites of nitrogen-bearing and oxygen-bearing compounds,
respectively.19, 21 Extending southwest from the Hot Core toward the Compact
Ridge is the Hot Core-SW, which is physically and chemically heterogeneous,
with possible sources of internal heating22 and both oxygen- and nitrogen-
bearing compounds.23, 24 Flanking these regions are compact sources, such as
Source I—an edge-on disk thought to be internally heated25, 26—to the west
within the Hot Core region and the (sub)millimeter sources SMA1 and C22—a
protostar and possible hot core, respectively27, 21—to the southwest of Source
I and along the northwestern edge of the Hot Core-SW.
Jacq et al. 28 reported the first definitive detection of CH2DOH toward Orion
KL on angular scales between 12′′ and 26′′ (centered on the Hot Core and
Source I region). They combined their measurements with past CH3OD
measurements to report a [CH2DOH]/[CH3OD] ratio in the range of 1.1-1.5. Neill
et al. 29 similarly reported a ratio of $1.2\pm 0.3$ based on local
thermodynamic equilibrium models of ${\sim}30{-}44^{\prime\prime}$
observations of the Hot Core and the Compact Ridge, while Peng et al. 30
reported an even lower ratio of $0.7\pm 0.3$ toward Orion KL, using an angular
resolution of $3.\\!\\!^{\prime\prime}6\times 2.\\!\\!^{\prime\prime}3$.
There has been extensive debate about whether the apparent CH3OD enhancements
are the result of grain-surface or gas-phase processes. Early on, Jacq et al.
28 concluded that their observed ratio was evidence of grain-surface
processing followed by injection into the gas phase, perhaps by thermal
desorption. Shortly after, chemical models of gas-phase exchange rejected the
grain-surface hypothesis on the premise that such chemistry would require
unrealistically high [HDO]/[H2O] ratios.31 Rodgers and Charnley 32 criticized
the assumed statistical [CH2DOH]/[CH3OD] ratio of 3 since D and H react with
species other than CO and H2CO, which could affect the relative abundances of
the singly-deuterated methanol isotopologues. Osamura et al. 33 used models to
suggest that ion-molecule reactions in the gas phase lead to the loss of
CH3OD, which they conclude accounts for high [CH2DOH]/[CH3OD] ratios in low-
mass star-forming regions, provided methanol is efficiently regenerated in the
dissociative recombination of protonated methanol with electrons. This work
also finds that D-H exchange on the methyl site is inefficient.
Nevertheless, D-H exchange at the hydroxyl group of methanol on icy grain
mantles has emerged as a favored explanation for the [CH2DOH]/[CH3OD] ratios
observed in massive star-forming regions. 34, 17, 30 In this mechanism,
deuterated water in the ice reacts with CH3OH to produce CH3OD:
$\ch{CH3OH+HDO<=>CH3OD+H2O}$ (2) $\ch{CH3OH+D2O<=>CH3OD+HDO}$ (3)
However, the intricacies of this exchange are still being investigated.
This work provides a new observational perspective on the possibility of D-H
exchange at the methanol hydroxyl site by mapping gas-phase CH3OD abundances
in Orion KL at sub-arcsecond ($\sim$0$.\\!\\!^{\prime\prime}$7) angular
resolution using the Atacama Large Millimeter/submillimeter Array (ALMA),
which corresponds to linear scales of $\sim$270 au at the nebula’s distance.
This allows us to plot gas-phase CH3OD as a function of the local line-of-site
temperature across relatively small scales within the nebula and explore
temperature-dependent chemical processes that may affect the observed CH3OD
chemistry.
## 2 Methods
### 2.1 Observations
Observations of Orion KL were taken in ALMA Band 4 during Cycle 5 (project
code: ADS/JAO.ALMA#2017.1.01149, PI: Wilkins) on 2017 December 14, completely
on the main 12-m array. The pointing center was set to
$\alpha_{\mbox{\scriptsize J2000}}=05^{\mbox{\scriptsize
h}}35^{\mbox{\scriptsize m}}14.\\!\\!^{\mbox{\scriptsize s}}50$,
$\delta_{\mbox{\scriptsize
J2000}}=-05^{\circ}22^{\prime}30.\\!\\!^{\prime\prime}9$. These observations
employed 49 antennas during one execution block. All spectra were obtained in
a single local oscillator set-up consisting of 10 spectral windows; as such,
the uncertainties for quantities derived from these spectra are dominated by
thermal noise and mostly unaffected by calibration uncertainty. Of these
spectral windows, the targeted CH3OD lines were contained in three spectral
windows with a spectral resolution of 244 kHz ($\sim$0.5 km s-1) covering
143.51-143.97, 153.16-153.40, and 154.84-155.07 GHz. Projected baselines were
between 15.1 m and 3.3 km (7.6 and 1650 k$\lambda$, where $\lambda\sim 2$ mm
is the wavelength), and the primary beam was 39.1′′. The on-source integration
time was 2062 s. Precipitable water vapor was 3.7 mm, and typical system
temperatures were around 75-125 K.
Calibration was completed using standard CASA (version 5.1.1-5) calibration
pipeline scripts. The source J0423$-$0120 was used as a calibrator for
amplitude, atmosphere, bandpass, pointing, and WVR (Water Vapor Radiometer)
variations, and J0541$-$0211 was used as the phase and WVR calibrator.
The CH3OD data introduced here were prepared in the same way as the ^13CH3OH
images presented by Wilkins et al. 22 In brief, the data cubes were created
from measurement sets split to include only baselines of $\leq$500 m,
resulting in a synthesized beam of $0.\\!\\!^{\prime\prime}74\times
0.\\!\\!^{\prime\prime}63$. Cubes were reduced with continuum emission
estimated from line-free channels subtracted using the uvcontsub function
followed by imaging using the tclean algorithm with robust weighting, a Briggs
parameter of 1.5 (i.e., semi-natural weighting) for deconvolution, and the
‘auto-multithresh’ masking algorithm 35 in conjunction with interactive
tclean. The images have a noise-level of $\sigma_{\mbox{\scriptsize RMS}}\sim
1.3$ mJy beam-1.
### 2.2 Deriving CH3OD Parameters
The CH3OD column density as a function of position was derived via pixel-by-
pixel fits of the CH3OD transitions shown listed in Table 1 assuming optically
thin lines (see Table S1 of the Supporting Information) in local thermodynamic
equilibrium (LTE).†††Python script available at
https://github.com/oliviaharperwilkins/LTE-fit. Integrated intensity maps of
each transition are providing in the Supporting Information (Figure S2). For
each coordinate-space pixel in the data cubes, a spectrum within a single
synthesized beam centered on that pixel was extracted. The CH3OD rotational
temperature ($T_{\rm rot}$) profile was assumed to be the same as that
previously derived22 from ^13CH3OH since the transitions of both isotopologues
have similar upper energy states $E_{u}$. Column density, line width
(${\sim}0.8{-}3.5$ km s-1), and local standard of rest (LSR) velocity
(${\sim}7{-}9$ km s-1) were determined by simultaneous fits using LMFIT, a
least-squares fitting software package.‡‡‡https://doi.org/10.5281/zenodo.11813
Because all lines used in the fit were observed simultaneously, the
uncertainties in excitation, which are derived from relative fluxes, should be
dominated by thermal noise rather than by multiple sources of calibration
uncertainty.
Table 1: Transitions of CH3OD used for line fits.⋆
Transition | $\nu$ | $E_{u}$ | $S_{ij}\mu^{2}$ | $g_{u}$
---|---|---|---|---
(GHz) | (K) | (Debye2)
$5_{(1,4)}-5_{(0,5)}$ A | 143.7417 | 39.48 | 11.2 | 11
$7_{(1,6)}-7_{(0,7)}$ A | 153.3240 | 68.05 | 14.7 | 15
$3_{(-1,2)}-2_{(0,1)}$ E | 154.9628 | 17.71 | 2.2 | 7
⋆From Anderson et al. 36 Column (2): rest frequency $\nu$ of the transitions;
(3): upper state energy $E_{u}$; (4): product of the transition line strength
and the square of the electric dipole moment $S_{ij}\mu^{2}$; (5): upper-state
degeneracy $g_{u}$
## 3 Results and Discussion
### 3.1 CH3OD Column Density
Figure 1: Derived CH3OD column density and percent propagated uncertainty
shown by the color maps in the upper and lower panels, respectively. The 2 mm
($\sim$150 GHz) continuum emission is shown by the grey contours at
$2\sigma_{RMS},4\sigma_{RMS},8\sigma_{RMS},16\sigma_{RMS},32\sigma_{RMS},64\sigma_{RMS}$.
The Hot Core (HC), Source I (I), IRc7, and IRc4 are shown in red; SMA1 and C22
are shown by the teal diamonds; the methyl formate emission peak (MF1)24 is
labeled in yellow, and the Compact Ridge (CR) is labeled in blue. The
0$.\\!\\!^{\prime\prime}$7 synthesized beam is shown by the black ellipse in
the bottom left corner.
The column density profile derived from a pixel-by-pixel fit of the ALMA data
image cubes was used to show small-scale variations in CH3OD column densities
across Orion KL. As shown in Figure 1, the derived CH3OD column density
($N_{\rm tot}$) is generally on the order of $10^{17}$ cm-2 and peaks south of
SMA1 and C22. In general, the uncertainties (standard errors calculated using
LMFIT) for these values are $<$10% throughout the region southwest of the Hot
Core (Hot Core-SW), which is the region of interest for most of the discussion
in this work; higher uncertainties, up to $\sim$25%, characterize IRc4 and the
western (left) edge of the Compact Ridge. The relationship between the CH3OD
and ^13CH3OH column densities (Figure S3 in the Supporting Information) and
rotational temperature ($T_{\rm rot}$, Figure S4 in the Supporting
Information) is illustrated by Figures 2a and 2b. In general,
[CH3OD]/[^13CH3OH] falls between 1.5 and 5.3. Assuming a local interstellar
^12C/^13C ratio of $68\pm 15$,37 this suggests [CH3OD]/[CH3OH] $\approx
0.015$-$0.104$, which encompasses the ratio of 0.01-0.06 in Orion KL reported
by Mauersberger et al. 38 on much larger angular scales of 15′′-23′′. The
discrepancy on the higher end of these ranges may be the result of CH3OD
abundance enhancements on smaller spatial scales being diluted in the single-
dish data, but otherwise, the agreement suggests that the flux recovered in
our observations is representative of that collected by single-dish
observations. A comparison of [CH3OD]/[^12CH3OH] ratios derived from different
assumed ^12C/^13C values is presented in Table S2 of the Supporting
Information.
(a)
(b)
Figure 2: (a) Column densities $N_{\rm tot}$ of ^13CH3OH (horizontal axis,
from Wilkins et al. 22) and CH3OD (vertical axis, this work) with each point
representing a single pixel in Figure 1. Each pixel is colored by its
rotational temperature. The grey lines are labeled by the [CH3OD]/[^13CH3OH]
ratios (1 to 5) they represent. (b) Two-dimensional histogram (50 points per
bin) showing the methanol [CH3OD]/[^13CH3OH] ratios plotted as a function of
temperature.
(a)
(b)
Figure 3: (a) Two-dimensional histogram (50 points per bin) showing the
derived CH3OD column densities $N_{\rm tot}$ against rotational temperature
$T_{\rm rot}$ of ^13CH3OH. The blue solid line shows the power-law fit to the
data if there were no D-H exchange (eq 4). The black dashed line shows the
modeled D-H exchange (eq 5) followed by desorption based on ice experiments by
Souda et al. 39 and assumptions by Faure et al. 40 The blue dotted curve is
the power-law fit after the CH3OD enhanced by D-H exchange thermally desorbs
off the grains. (b) The solid curve shows the density profile with the
underlying power-law (blue curve in a, eq 4) subtracted. The dotted curve
shows the sputtered CH3OD2+ profile reported by Souda et al. 39 but scaled for
comparison to the CH3OD column density profile in this work.
Figure 3a, in which each point represents 50 binned pixels for which a column
density and rotational temperature pair were derived, shows that the CH3OD
column density increases with rotational temperature, which is characteristic
of thermal desorption in which material is sublimed from the grains as the
environment warms. 41 However, the profile also contains a “shark-tooth”
feature where the column density starts to rise more steeply at $\sim$110 K
before peaking close to 185 K. At temperatures higher than 185 K, there is a
sharp decrease in the CH3OD column density, and the relationship between
column density and rotational temperature returns to the underlying trend.
Power-law distributions are commonly used to characterize the temperature and
density profiles of star-forming regions and young stellar objects (YSOs).42,
43 In Figure 3a, the fitted underlying power-law relationship between $T_{\rm
rot}$ and $N_{\rm tot}$, described by eq 4, is shown by the solid blue line.
$N_{\rm tot}=1.5\times 10^{14}T_{\rm rot}^{1.4148}$ (4)
The solid black line in Figure 3b shows the “shark-tooth” from Figure 3a with
the underlying power-law between $T_{\rm rot}$ and $N_{\rm tot}$ subtracted.
### 3.2 Grain-Surface Processes
The rapid rise in gas-phase CH3OD column density between $\sim$110 K and
$\sim$120 K is consistent with D-H exchange between methanol and heavy water
(HDO, D2O) on the ices at $\sim$100 K. Souda et al. 39 experimentally
investigated hydrogen bonding between water and methanol in low-temperature
ices warmed from 15 K to 200 K under ultrahigh vacuum conditions. They
observed that when CH3OH was adsorbed onto D2O ice, secondary CH3OD2+
ions—evidence of D-H exchange at the hydroxyl site—sputtered off the ice
analogue surfaces predominantly between 140 and 175 K. Follow-up analyses by
Kawanowa et al. 44 describes this as a “rapid and almost complete H/D
exchange” to yield the sputtered CH3OD2+ species. The fact that we see a
similar sudden increase in CH3OD column densities at similar temperatures
(Figure 3b, dotted line), with discrepancies owing to the differences in
pressure between ultrahigh vacuum and even the densest regions of interstellar
medium, supports a similar rapid exchange in Orion KL.
Models of D-H exchange between water and methanol in ices by Faure et al. 40
successfully reproduced gas-phase CH3OH deuterium fractionation in Orion KL
using initial ice abundances of $n_{S}(\ch{CH3OH})=2.0\times
10^{-6}n_{\ch{H}}$, $n_{S}(\ch{HDO})=3.0\times 10^{-7}n_{\ch{H}}$, and
$n_{S}(\ch{CH3OD})=6.0\times 10^{-9}n_{\ch{H}}$. Taking these initial ice
abundances, we modeled the change in gas-phase CH3OD column density following
rapid D-H exchange on the ices and subsequent desorption. Specifically, we
assumed an initial ice column density of $N_{S}(\ch{CH3OD})=6.0\times
10^{-9}N_{\ch{H}}=6.0\times 10^{14}\mbox{ cm}^{-2}$, since $N_{\ch{H}}\sim
10^{23}$ cm-2 across Orion KL (including the Hot Core, Compact Ridge, and
Extended Ridge),45, 46 and an initial gas-phase column density of
$N(\ch{CH3OD})=1\times 10^{16}\mbox{ cm}^{-2}$, based on the column densities
measured at 100 K in this work after subtracting the underlying power-law in
eq 4.
The enhancement of gas-phase CH3OD column density from D-H exchange with water
was then modeled. In the absence of directly analogous temperature-programmed
desorption measurements of CH3OH and CH3OD themselves, we fit the CH3OD2+
curve of Souda et al. 39 (Figure 3b, dotted line) between 110 K and 145 K. The
relative intensity amplitude was normalized to the column densities observed
for CH3OD in our Orion KL observations, resulting in a relationship of the
form
$N_{S}^{\prime}(\ch{CH3OD})=6.2\times 10^{14}T-6.0\times 10^{16}$ (5)
where $N_{S}^{\prime}$ is the additional (solid) CH3OD available [cm-2] for
desorption at a given temperature $T$. The desorption rate coefficient is
expressed as
$k_{des}=\nu_{des}e^{(-E_{d}/T)}$ (6)
where the pre-exponential factor $\nu_{des}$ and the binding energy $E_{d}$
are taken to be approximately the values for annealed amorphous solid
water—$2.0\times 10^{12}$ s-1 and 5200 K, respectively—under the assumption
that the methanol desorbs with water, which is in excess.47, 48, 49, 50, 40,
51
It follows that the change in gas-phase $N(\ch{CH3OD})$ is approximated by
multiplying the rate coefficient (eq 6) by the total solid CH3OD column
density, which includes the additional solid CH3OD available following D-H
exchange (eq 5) at temperature $T$. Thus, the rate at which $N(\ch{CH3OD})$
changes in the gas phase between 100 and 150 K is approximated by
$\frac{d}{dt}N(\ch{CH3OD})=k_{des}\left[N_{S}(\ch{CH3OD})+N_{S}^{\prime}(\ch{CH3OD})\right]$
(7)
using temperature steps of 1 K, the corresponding time steps for which were
determined by
$\Delta t=\left(\frac{T-T_{0}}{T_{max}-T_{0}}\right)^{1/n}t_{h}$ (8)
where $\Delta t$ is the time elapsed since $t=0$; $T_{0}$ and $T_{max}$ are
the initial (10 K) and maximum (300 K) temperatures, respectively; $t_{h}$ is
the heating timescale; and $n$ is the order of heating, which is assumed to be
2 following the previous work.52 The initial gas and solid CH3OD column
densities were assumed, respectively, to be $N(\ch{CH3OD})=9.0\times 10^{15}$
cm-2 (approximated from the CH3OD density profile with the underlying power-
law subtracted) and $N_{S}(\ch{CH3OD})=6.0\times 10^{-9}N_{\ch{H}}$ cm-2
(based on assumptions used by Faure et al. 40 in their D-H exchange models).
The resulting desorption model from eq 7 with $t_{h}=10^{3}$ yr is shown by
the black dashed line in Figure 3a. Longer timescales (i.e., $t_{h}\geq
10^{4}$ yr) characteristic of massive YSOs do not follow the increasing CH3OD
profile as closely. Although potential internal heating sources have been
suggested within the Hot Core-SW,22 Li et al. 53 conclude that this region,
part of the “elongated ridge” comprising the Hot Core and Source I, is
predominantly heated externally by shocks induced by the Orion KL explosion,
which took place about 500 years ago. Furthermore, the models by Faure et al.
40 suggest the D-H exchange in Orion KL reaches steady-state in ${<}10^{3}$
yr. After D-H exchange, the power-law relationship from eq 4 applies, but with
a coefficient of $3.2\times 10^{14}$ (dotted blue curve in Fig. 3a).
The hypothesis that the CH3OD column density profile in Figure 3 is the result
of rapid D-H exchange on the grains relies on two assumptions regarding
temperature. First, we assume that the rotational temperature $T_{\rm rot}$ is
an appropriate estimate for the kinetic temperature $T_{kin}$. This assumption
is based on the fact that the Compact Ridge, a spatial component toward the
southwestern region of Orion KL that is characterized as being rich in oxygen-
bearing molecules, has a fairly high density of ${\sim}10^{6}$ cm-3, 19, 20
implying that LTE is a reasonable assumption. Second, we assume that the dust
and gas are thermally coupled. Li et al. 54 found that other quiescent regions
(no infrared sources, no evident outflows) in the Orion Molecular Cloud are
thermally coupled. Models by Bruderer et al. 55 also support coupling between
dust and gas temperature at the relevant densities. This gas-grain thermal
coupling was also demonstrated in models of several massive star-forming
regions, including the Orion KL Compact Ridge, by Garrod and Herbst.56 As
such, it is reasonable to assume that the gas temperatures shown in Figure 3
are also representative of the temperatures of the dust, on which methanol
forms, and that any decoupling between the dust and gas is negligible for the
purposes of this discussion.
### 3.3 Enhanced Deuteration in the Hot Core-SW
Above $\sim$180 K, the profile in Figure 3a drops sharply. We attribute this
to the enhanced deuteration being localized to the Hot Core-SW and that there
simply is little gas above 180 K in this region. Figure 4a shows the
distribution of CH3OD column densities minus the underlying power-law
relationship (eq 4). Figure 4b maps [CH3OD]/[^13CH3OH], confirming that the
profile in Figure 4a is indeed the result of excess CH3OD in the Hot Core-SW
and not enhanced abundances of methanol in general.
(a)
(b)
Figure 4: (a)Map of gas-phase CH3OD excess column density across Orion KL
after subtracting the underlying power-law relationship (eq 4) between
rotational temperature and column density. Darker shading indicates a larger
deviation from the underlying power-law. (b) Map of [CH3OD]/[^13CH3OH].
We considered two chemical explanations for this trend, neither of which
adequately explain the observed patterns. One avenue that has been proposed
for CH3OD depletion is gas-phase D-H exchange via protonation of the hydroxyl
group (eqs 9 and 10) followed by dissociative recombination (eqs. 11 and
12).33.
$\ch{CH3OD+H3O+->CH3ODH++H2O}$ (9)
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ch{CH2DOH+H3O+->CH2DOH2++H2O}}$
(10) $\ch{CH3ODH++e-->CH3OH+D},\ch{CH3OD+H}$ (11) $\ch{CH2DOH2++e-->CH2DOH+H}$
(12)
The methyl H/D site is not exchangeable; therefore, in this model, only CH3OD
can be depleted while CH2DOH cannot, which has been suggested as an
explanation for the low relative CH3OD abundances in low-mass star-forming
regions.33 However, this is unlikely to account for the decrease in gas-phase
CH3OD at warmer temperatures in the Orion KL Hot Core-SW following enrichment
on grain surfaces because dissociative recombination reactions tend to destroy
gas-phase methanol (and its isotopologues), with methanol production
comprising the smallest branching ratio (3%) listed in KIDA57. A drop in the
^13CH3OH column density at temperatures above 180 K is not seen in Orion KL
(see Figure S5 in the Supporting Information), suggesting that the observed
decrease of the [CH3OD]/[^13CH3OH] ratio (Figure 2b) and of the CH3OD column
density (Figure 3a) above 180 K cannot simply be explained by protonation of
the methanol (e.g., by H3O+) followed by dissociative recombination.
Another mechanism considered was the neutral-radical reaction with the
hydroxyl radical (OH); however, this reaction is too slow at $\sim$200 K to
account for the observed patterns (see Section S4 of the Supporting
Information).
Instead, we propose that the drop in enhanced CH3OD column density is the
result of environment rather than chemistry. That is, the enhanced CH3OD
column density profile is limited to warm gas in the Hot Core-SW with
temperatures that max out around 200 K. The enhanced deuteration in this
region, brought about by temperature-dependent surface D-H exchange, may be
the result of evolutionary state. However, this region is not directly
associated with any known YSOs, including SMA1 (a young, high-mass protostar)
and C22 (a possible hot core).27, 21 Although there is no known self-
illuminated source (e.g., embedded protostar) in the Hot Core-SW to drive
grain warming and associated D-H exchange in ices, it has been suggested that
there is a potential (hidden) source of internal heating there.22 And, the
complex history and star formation patterns in Orion KL will lead to quite
varied thermal histories of various sub-regions of the giant molecular cloud
complex. Thus, dedicated work to elucidate the nature of the Hot Core-SW
region is needed to better understand how this environment could affect the
observed chemistry.
### 3.4 Methyl Group Chemistry
If the hump observed in the CH3OD column density versus temperature profile is
indeed evidence of surface D-H exchange at the hydroxyl site, then we would
expect a smooth profile (i.e., without a similar hump) in the profile of
CH2DOH. Unfortunately, we do not have sufficient CH2DOH transitions in our
data to test this hypothesis directly. However, Carroll 58 mapped the physical
parameters of CH2DCN toward Orion KL using data from ALMA (project:
ADS/JAO.ALMA#2013.1.01034, PI: Crockett). Figure 5 shows a histogram of the
CH2DCN column density and rotational temperature using these data where they
overlap with CH3OD emission in the current observations. In this plot, we see
a consistent power-law relationship between temperature and column density,
which supports the conclusion by Osamura et al. 33 that the methyl groups of
complex organics would not undergo grain-surface or gas-phase D-H exchange. In
other words, the hump visible in Figure 3a is indeed likely the result of
chemistry specific to the hydroxyl site of methanol. Future dedicated
observations of CH2DOH lines with ALMA in this region are necessary to confirm
this hypothesis.
Figure 5: Two-dimensional histogram with 20 points per bin showing the derived
CH2DCN column density against rotational temperature, using data presented by
Carroll 58.
### 3.5 Comparison to Past Studies
The evidence presented here adds to the growing list of observational and
theoretical evidence in favor of a grain-surface mechanism for CH3OD
enrichment in massive star-forming regions. A key difference between this work
and that of past observations is that, here, we map CH3OD column density
across much of Orion KL, including the Compact Ridge, whereas past work
derives one value for the Compact Ridge as a whole. Furthermore, resolving the
small-scale structure of CH3OD column densities (and temperature) allows us to
look at how column density is related to the line-of-site temperature,
something that has not yet been extensively investigated through observations
but is now possible with the sensitivity and spatial grasp of ALMA.
Computational models to assess D-H exchange generally have investigated
temporal variations in relative CH3OD abundances at a single temperature or
have looked at singly-deuterated methanol chemistry across a range of
temperatures but less than 140 K.33, 40, 14 Previous models of grain-surface
D-H exchange in Orion KL have assumed methanol is completely sublimated at
temperatures $>$110 K, whereas our observations suggest D-H exchange on the
ice may be important up to 125 K. Thus, the observations presented here probe
a temperature regime beyond that of existing astrochemical models and call for
revised models to investigate D/H chemistry at higher temperatures.
Specifically, temperature-dependent models of grain-surface D-H exchange in
which methanol sublimates completely at higher temperatures (e.g., 125 K) are
needed to more robustly explain the patterns observed in this work.
Perhaps the leading criticism of proposed grain-surface chemistry prompting
the enhancement of CH3OD abundances relative to CH2DOH is that such processes
would require a large initial [HDO]/[H2O] ratio. For example, models by
Charnley et al. 31 suggest that the initial [HDO]/[H2O] ratio in the ice
mantles would need to be $\sim$0.1,31 which is significantly larger than the
ratio of $\sim$0.003 reported by Neill et al. 45 for compact regions of Orion
KL. However, Thi et al. 59 suggest that the [HDO]/[H2O] ratio can exceed 0.01
in dense (${\geq}10^{6}$ cm-3), warm ($T>100$ K) regions (such as those
observed here) via neutral-neutral reactions—such as the formation of HDO from
OH + HD, OD + H2, and OD + OH—which may be promising for the hypothesis of a
grain-surface CH3OD enhancement. Even more promising is a model presented by
Faure et al. 40, who reproduced observed [CH2DOH]/[CH3OD] ratios in the
Compact Ridge assuming a primitive [HDO]/[H2O] fractionation of 0.006, only a
factor of 2 larger than the observed ratio reported by Neill et al. 45. That
is, the key initial condition is that of the deuteration state of methanol and
water in the icy grain mantles, which are exceedingly difficult to measure
directly with previous observational capabilities. JWST will offer greatly
improved capabilities to attempt such measurements, going forward.
A remaining question, then, is what makes methanol deuteration in low-mass
star-forming regions so different from that in high-mass star-forming regions?
Ratajczak et al. 12 suggest observational biases, namely that, since high-mass
objects tend to be further away than those low-mass objects where deuterium
chemistry has been studied, measurements of the [CH2DOH]/[CH3OD] ratio may be
affected, particularly if the spatial distributions of the two isotopomers are
different. High angular resolution mapping of high-mass star-forming regions,
like that presented in Figure 1, would address this by comparing the CH2DOH
and CH3OD column densities only where their emission overlapped, as was done
for different spectral components of the high-mass star-forming region NGC
6334I by Bøgelund et al. 14 As stated previously, the observations presented
here do not have sufficient CH2DOH lines available to test the spatial
correlation of site-specific deuterated isotopologues, and dedicated high
angular resolution observations targeting low-energy CH2DOH lines are
necessary to further address this issue.
Another conjecture for the different deuterium fractionation patterns in
massive YSOs compared to low-mass star-forming regions is that there is simply
less deuteration in massive protostars because of the warmer environments.14
Faure et al. 40 reproduced relative singly-deuterated methanol abundances for
Orion KL (a high-mass source) and IRAS 16293-2422 (a low-mass object) using
kinetic models that were identical except for the initial deuterium
fractionation ratios. They reported that the Orion KL Compact Ridge’s gas-
phase deuterium chemistry could be modeled assuming similar primitive
deuteration of water and methanol ices ($\sim$0.2-0.3%), whereas IRAS 16293’s
gas-phase deuterated methanol chemistry required a significantly higher
deuterium fractionation in methanol (12%) than water (1%). Their model shows
complete methanol desorption by $\sim$110 K. Such conditions of extreme
deuteration only occur in very cold, dense environments where extensive
molecular depletion occurs, including that of CO and N2. Under such
conditions, D${}_{3}^{+}$ becomes the dominant molecular ion, whose
dissociative recombination results in the arrival of hydrogen atoms onto grain
mantles with a D/H ratio of $>$1.
As seen in Figure 3a, the desorption model based on work by Faure et al. 40
(dashed line) matches nicely the CH3OD column density rise between 100 and 110
K; however, the CH3OD in our data increases at temperatures up to $\sim$125 K.
This slight discrepancy might be addressed by temperature-programmed
desorption experiments, for example, studying the release of CH3OD directly
rather than via sputtered CH3OD2+ detected by Souda et al. 39 Furthermore,
thermal desorption strongly depends on the composition of the underlying
surface. Such questions require more robust chemical networks for deuterium
chemistry as well as a better understanding of the initial chemical conditions
of both high-mass and low-mass star-forming regions.
## 4 Conclusion
We provide observational evidence in support of rapid D-H exchange in
methanol-containing ices, specifically at the hydroxyl site, between $\sim$100
and 125 K in Orion KL, using high angular resolution ALMA Band 4 observations
of CH3OD to map the small-scale variations in CH3OD column density for the
first time, and to compare the observed column densities to the line-of-site
rotational temperatures mapped at the same angular resolution (and derived
previously from ^13CH3OH).22
We fit power-law relationships and toy models of D-H exchange at methanol’s
hydroxyl (-OH) site followed by CH3OD thermal desorption to the observed CH3OD
column density profile in Orion KL. These analyses suggest that D-H exchange
and rapid CH3OD desorption increase the gas-phase CH3OD column density between
100 and 125 K. Enhanced CH3OD column densities are limited to the Hot Core-SW,
which has been previously suggested to harbor a potential source of internal
heating, which could explain the enhanced CH3OD column densities between 125
and 185 K. In this interpretation, there is simply little gas in this region
at higher temperatures that would display CH3OD enhancements.
Future investigations—through observations, experiments, and computational
models—are needed to further constrain the peculiar D-methanol chemistry in
Orion KL and other star-forming regions. The work presented here would be
aided by dedicated high-resolution observations of CH2DOH, spectroscopic
experiments measuring the kinetics of CH3OD formation via D-H exchange in
heavy water ice (HDO and D2O) and subsequent desorption, and temperature-
dependent astrochemical models of possible CH3OD loss at higher temperatures
(185-225 K).
The Supporting Information includes an explanation of optical depth
approximations (Section S1); supplemental maps (Section S2), specifically
CH3OD integrated intensity (Figure S2), ^13CH3OH column density (Figure S3),
and ^13CH3OH rotational temperature (Figure S4) maps; a comparison of derived
[CH3OD]/[^12CH3OH] ratios (Section S3); and loss calculations of CH3OD by gas-
phase reactions with OH (Section S4).
## Acknowledgements
This work makes use of the following ALMA data: ADS/JAO.ALMA#2017.1.01149 and
ADS/JAO.ALMA#2013.1.01034.ALMA is a partnership of ESO (representing its
member states), NSF (USA), and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Tawain), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO,
and NAOJ. The National Radio Astronomy Observatory (NRAO) is a facility of the
National Science Foundation (NSF) operated under Associated Universities, Inc.
(AUI). This research made use of APLpy, an open-source plotting package for
Python,60 and the KInetic Database for Astrochemistry (KIDA), an online
database of kinetic data.57
This work has been supported by the NSF Graduate Research Fellowship Program
under grant No. DGE-1144469 and NRAO Student Observing Support under Award No.
SOSPA6-014. OHW was additionally supported by an ARCS Los Angeles Founder
Chapter scholarship. GAB gratefully acknowledges support from the NSF AAG
(AST-1514918) and NASA Astrobiology (NNX15AT33A) and Exoplanet Research (XRP,
NNX16AB48G) programs. This work benefited from discussions with Brandon
Carroll, Dana Anderson, Aida Behmard, Cam Buzard, Steve Charnley, and
Catherine Walsh. Many thanks to the anonymous referees for their thoughtful
comments and help in improving the manuscript. OHW thanks Erica Keller, Sarah
Wood, and the NRAO North American ALMA Science Center (NAASC) for their
assistance with the data reduction.
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# Fourier decay of equilibrium states on hyperbolic surfaces
Gaétan Leclerc
###### Abstract
Let $\Gamma$ be a (convex-)cocompact group of isometries of the hyperbolic
space $\mathbb{H}^{d}$, let $M:=\mathbb{H}^{d}/\Gamma$ be the associated
hyperbolic manifold, and consider a real valued potential $F$ on its unit
tangent bundle $T^{1}M$. Under a natural regularity condition on $F$, we prove
that the associated $(\Gamma,F)$-Patterson-Sullivan densities are stationary
measures with exponential moment for some random walk on $\Gamma$. As a
consequence, when $M$ is a surface, the associated equilibrium state for the
geodesic flow on $T^{1}M$ exhibit Fourier decay, in the sense that a large
class of oscillatory integrals involving it satisfies power decay. It follows
that the non-wandering set of the geodesic flow on convex-cocompact hyperbolic
surfaces has positive Fourier dimension, in a sense made precise in the
appendix.
## 1 Introduction
### 1.1 State of the art
Since the early work of Dolgopyat on the decay of correlation of Anosov flows
[Do98], we know that the rate of mixing of hyperbolic flows (with respect to
some equilibrium states) may be linked with the spectral properties of twisted
transfer operators. This idea has been widely used and generalized: see, for
example, [St01], [BV05], [St11], and [DV21], to only name a few related work.
In concrete terms, exponential mixing can be reduced to exhibiting enough
cancellations in sums of exponentials. It turns out that this (rather
complicated) process can be sometimes simplified if one knows that the Fourier
transform of those equilibrium states exhibit Fourier decay. This idea has
been explored and discussed in Li’s work on stationary measures [Li20], as
well as in [MN20], and more recently in a preprint by Khalil [Kh23]. This
connection with the exponential mixing of dynamical systems has sparked recent
interest in studying the behavior of the Fourier transform of measures.
Historically, this was motivated by understanding sets of unicity for Fourier
series [KS64], which lead us to discover that the Fourier properties of a
measure may be used to study the arithmetic properties of its support. This
idea is encoded in the notion of Fourier dimension: see for exemple a recent
preprint of Fraser [Fr22] (introducing a new notion of Fourier dimension
spectrum) and the references therein. Let us introduce this notion.
The Fourier dimension is better understood if we first recall a well known
formula for the Hausdorff dimension. If $E\subset\mathbb{R}^{d}$ is a compact
subset of some euclidean space, a corollary (see for example [Ma15]) of a
lemma by Frostmann [Fro35] yields the following identity:
$\dim_{H}E=\sup\left\\{\alpha\in[0,d]\ |\ \exists\mu\in\mathcal{P}(E),\
\int_{\mathbb{R}^{d}}|\widehat{\mu}(\xi)|^{2}|\xi|^{\alpha-d}d\xi<\infty\right\\},$
where $\dim_{H}E$ is the Hausdorff fractal dimension of $E$, $\mathcal{P}(E)$
is the set of all (borel) probability measures supported on $E$, and where
$\widehat{\mu}:\mathbb{R}^{d}\rightarrow\mathbb{C}$, given by
$\widehat{\mu}(\xi):=\int_{\mathbb{R}^{d}}e^{-2i\pi x\cdot\xi}d\mu(x),$
is the Fourier transform of the measure $\mu\in\mathcal{P}(E)$. The condition
on the measure in the supremum can be though as a decay condition on average.
In particular, the inner integral is finite if $\widehat{\mu}(\xi)$ decays
like $|\xi|^{-\alpha/2-\varepsilon}$ for large $\xi$. With this in mind, the
following notion is quite natural to introduce: we define the Fourier
dimension of $E\subset\mathbb{R}^{d}$ by the formula
$\dim_{F}E:=\sup\left\\{\alpha\in[0,d]\ |\ \exists\mu\in\mathcal{P}(E),\exists
C\geq 1,\forall\xi\in\mathbb{R}^{d}\setminus\\{0\\},\ |\widehat{\mu}(\xi)|\leq
C|\xi|^{-\alpha/2}\right\\}.$
While it is clear that $0\leq\dim_{F}E\leq\dim_{H}E\leq d$, we do not always
have equality between the two notions. For example, it is well known that the
triadic Cantor set has Hausdorff dimension $\ln 2/\ln 3$, but has Fourier
dimension $0$. In fact, exhibiting deterministic (fractal) sets with positive
Fourier dimension is quite a challenge. One of the earliest examples of
deterministic (fractal) set with positive Fourier dimension was discovered by
Kaufmann in 1980 [Ka80], involving sets related to continued fractions.
Kaufmann’s method was optimised by Queffelec and Ramare in [QR03], and was
more recently generalized by Jordan and Sahlsten [JS16]. A year later,
Kaufmann [Ka81] found deterministic examples of (fractal) _Salem sets_ in
$\mathbb{R}$, that is, sets $E$ satisfying $\dim_{H}E=\dim_{F}E$. This example
is related to diophantine approximations. The construction was generalized in
$\mathbb{R}^{2}$ in [Ha17], and then in $\mathbb{R}^{d}$ in [FH23]. Let us
mention that there is a whole bibliography on random constructions of Salem
sets: the interested reader may look at the references in [FH23]. Returning to
the study of sets with positive Fourier dimension and of sets of unicity, let
us also mention that a there has been a lot of interest for Cantor sets
appearing from linear IFS. Related work includes [LS19a], [LS19b], [So19],
[Br19], [VY20], and [Ra21].
In 2017, Bourgain and Dyalov [BD17] introduced a new method to prove
positivity of the Fourier dimension for some sets. The paper takes place in a
dynamical context. If we fix some Schottky group
$\Gamma<\text{PSL}(2,\mathbb{R})$, then $\Gamma$ acts naturally on
$\mathbb{R}$. One can prove that there exists a cantor set
$\Lambda_{\Gamma}\subset\mathbb{R}$, called a limit set, which is invariant by
$\Gamma$. On this limit set, there is a family of natural probability measures
associated with the dynamics that are called Patterson-Sullivan measures.
Using results from additive combinatorics, more specifically a sum-product
phenomenon, Bourgain and Dyatlov managed to prove power decay for the Fourier
transform of those probability measures. In particular, the limit set
$\Lambda_{\Gamma}\subset\mathbb{R}$ has positive Fourier dimension. An
essential feature of $\Gamma$ was the _nonlinearity_ of the dynamics.
The method introduced in this paper inspired numerous generalizations,
beginning with a paper of Li, Naud and Pan [LNP19] proving power decay for
Patterson-Sullivan measures over (Zariski dense) Fuschian Schottky groups
$\Gamma<\text{PSL}(2,\mathbb{C})$. In this paper, Li proves that such measures
may be seen as stationary measures with finite exponential moment, which
allows them to use several results from the topic of random walks on groups.
From this, they obtain positivity of the Fourier dimension of the associated
limit set $\Lambda_{\Gamma}\subset\mathbb{C}$.
From there, at least two different directions exists for generalization. The
first one is to notice that Patterson-Sullivan measures are equilibrium states
for some hyperbolic dynamical system. A natural generalization is then given
by Sahlsten and Steven in [SS20]: for one dimensional and totally nonlinear
IFS, one can show that any equilibrium state exhibit Fourier decay. In
particular, Sahslten and Steven obtain positive Fourier dimension for a large
class of nonlinear Cantor sets. This paper use the method introduced by
Bourgain-Dyatlov, and is also inspired by previous techniques appearing in
[JS16]. See also [ARW20] for some related work on nonlinear IFS and pointwise
normality. Some complementary remarks on the work of Sahlsten and Steven may
be found in [Le22]. Past the one-dimensionnal setting, it was proved by the
author in [Le21] that the same results hold true in the context of hyperbolic
Julia sets in the complex plane. Some decay results are also true, in the
unstable direction, for equilibrium states of (sufficiently bunched) nonlinear
solenoids [Le23].
A second natural direction to look at is for result concerning stationary
measures with exponential moment for random walks on groups. Li proved in
[Li18] and [Li20] several Fourier decay results in the context of random walks
over $SL_{n}(\mathbb{R})$ (a crucial property of this group in the proofs is
its splitness). Past the split setting, further results seems difficult to
achieve.
### 1.2 Our setting of interest
In this paper, we are interested in studying the Fourier properties of
equilibrium states for the geodesic flow on convex-cocompact surfaces of
constant negative curvature. More details on our setting will be explained
during the paper, but let’s quickly introduce the main objects at play. A
usefull reference is [PPS15].
We work on hyperbolic manifolds, that is, a riemannian manifold $M$ that may
be written as $M=\mathbb{H}^{d}/\Gamma$, where $\mathbb{H}^{d}$ is the
hyperbolic space of dimension $d$, and where $\Gamma$ is a (non-elementary,
discrete, without torsion, orientation preserving) group of isometries of
$\mathbb{H}^{d}$. The geodesic flow $\phi=(\phi_{t})_{t\in\mathbb{R}}$ acts on
the unit tangent bundle of $M$, denoted by $T^{1}M$. We say that a point $v\in
T^{1}M$ is wandering for the flow if there exists an open neighborhood
$U\subset T^{1}M$ of $v$, and a positive number $T>0$ such that:
$\forall t>T,\ \phi_{t}(U)\cap U=\emptyset.$
The set of non-wandering points for $\phi$, denoted by $NW(\phi)\subset
T^{1}M$, is typically fractal and is invariant by the geodesic flow. We will
work under the hypothesis that the group $\Gamma$ is convex-cocompact, which
exactly means that $\text{NW}(\phi)$ is supposed compact. In particular, _the
case where $M$ is itself compact is authorized_. Under this condition, the
flow $\phi$ restricted to $NW(\phi)$ is Axiom A.
In this context, for any choice of Hölder regular potential
$F:T^{1}M\rightarrow\mathbb{R}$, and for any probability measure
$m\in\mathcal{P}(T^{1}M)$ (the set of borel probability measures on $T^{1}M$)
invariant by the geodesic flow, one can consider the _metric pressure_
associated to $m$, defined by:
$P_{\Gamma,F}(m)=h_{m}(\phi)+\int_{\text{NW}(\phi)}Fdm,$
where $h_{m}(\phi)$ denotes the entropy of the time-1 map of the geodesic flow
with respect to the measure $m$. Notice that any probability measure invariant
by the geodesic flow must have support included in the non-wandering set of
$\phi$. The _topological pressure_ is then defined by
$P(\Gamma,F):=\sup_{m}P_{\Gamma,F}(m),$
where the sup is taken over all the $\phi$-invariant probability measures $m$.
Those quantities generalize the variationnal principle for the topological and
metric entropy (that we recover when $F=0$). It is well known that this
supremum is, in fact, a maximum: see for example [BR75] or [PPS15].
###### Theorem 1.1.
Let $\Gamma$ be convex-cocompact, $M:=\mathbb{H}^{d}/\Gamma$, and
$F:T^{1}M\rightarrow\mathbb{R}$ be a Hölder regular potential. Then there
exists a unique probability measure $m_{F}$ invariant by $\phi$ such that
$P_{\Gamma,F}(m_{F})=P({\Gamma,F})$. This measure is called the equilibrium
state associated to $F$ and its support is the non-wandering set of the
geodesic flow. When $F=0$, $m_{F}$ is the measure of maximal entropy.
Theorem 6.1 in [PPS15] also gives us a description of equilibrium states. To
explain it, recall that the _Hopf coordinates_ allows us to identify
$T^{1}\mathbb{H}^{d}$ with
$\partial_{\infty}\mathbb{H}^{d}\times\partial_{\infty}\mathbb{H}^{d}\times\mathbb{R}$,
where $\partial_{\infty}\mathbb{H}^{d}$ denotes the _ideal boundary_ of the
hyperbolic space (diffeomorphic to a sphere in our context). The measure
$m_{F}$ lift into a $\Gamma$-invariant measure $\tilde{m}_{F}$ on
$T^{1}\mathbb{H}^{d}$, which can then be studied in these coordinates. The
interesting remark is that $\tilde{m}_{F}$ may be seen as a product measure,
involving what we call $(\Gamma,F)$-Patterson-Sullivan densities, which are
generalization of the usual Patterson-Sullivan probability measures. More
precisely, there exists $\mu_{F}$ and $\mu_{F}^{\iota}$, two Patterson-
Sullivan densities supported on the ideal boundary
$\partial_{\infty}\mathbb{H}^{d}$, such that one may write (in these Hopf
coordinates):
$d\tilde{m}_{F}(\xi,\eta,t)=\frac{d\mu_{F}(\xi)\otimes
d\mu_{F}^{\iota}(\eta)\otimes dt}{D_{F}(\xi,\eta)^{2}},$
where $D_{F}$ is the potential gap (or gap map), that we will define later.
More details on Patterson-Sullivan densities can be found in section 2 (which
will be devoted to recalling various preliminary results). Since the Hopf
coordinates are smooth on $\mathbb{H}^{d}$, we see that one may reduce Fourier
decay for $m_{F}$ to proving Fourier decay for Patterson-Sullivan densities.
This reduction is the content of section 4. Then, to prove Fourier decay for
those measures, several possibilities exists. With our current techniques,
this may only be achieved when $d=2$, so that Patterson-Sullivan densities are
supported on the circle.
The first possibility would be to use the fact that, in this low dimensionnal
context, there exists a coding of the dynamics of the group $\Gamma$ on the
ideal boundary: see for exemple [BS79] or [AF91]. Using these, one should be
able to get Fourier decay for Patterson-Sullivan densities by adapting the
proof of Bougain and Dyatlov in [BD17]. The second possibility would be to
adapt the argument found in Li’s appendix [LNP19] to prove that Patterson-
Sullivan densities are actually stationary measures with exponential moment
(for a random walk on $\Gamma$). Since in dimension 2, isometries of
$\mathbb{H}^{2}$ may be seen as elements of $\text{SL}_{2}(\mathbb{R})$, one
could then apply Li’s work [Li20] to get Fourier decay. This is the strategy
that we choose to follow in section 3. Finally, let us enhance the fact that
we are only able to work under a regularity condition (R) (see definition
2.13) that ensure Hölder regularity for our measures of interest. We now state
our main results.
###### Theorem 1.2 (Compare Theorem 3.2).
Let $\Gamma$ be a convex-cocompact group of isometries of $\mathbb{H}^{d}$,
and let $F:T^{1}(\mathbb{H}^{d}/\Gamma)\rightarrow\mathbb{R}$ be a Hölder
potential satisfying (R). Let
$\mu\in\mathcal{P}(\partial_{\infty}\mathbb{H}^{d})$ be a $(\Gamma,F)$
Patterson-Sullivan density. Then there exists $\nu\in\mathcal{P}(\Gamma)$ with
exponential moment such that $\mu$ is $\nu$-stationary and such that the
support of $\nu$ generates $\Gamma$.
Theorem 1.2 is our main technical result. The strategy is inspired by the
appendix of [LNP19], but in our setting, some additionnal difficulties appear
since the potential may be non-zero. For example, the proof of Lemma A.12 in
[LNP19] fails to work in our context. Our main idea to replace this lemma is
to do a carefull study of the action of $\Gamma$ on the sphere at infinity: we
will be particularly interested in understanding its contractions properties.
This is the content of section 2. The proof of Theorem 1.2 is in section 3.
Once this main technical result is proved, one can directly use the work of Li
[Li20] and get:
###### Corollary 1.3 ([Li20], Theorem 1.5).
Let $\Gamma$ be a convex-cocompact group of isometries of $\mathbb{H}^{2}$,
and let $F:T^{1}(\mathbb{H}^{2}/\Gamma)\rightarrow\mathbb{R}$ be a Hölder
potential satisfying (R). Let $\mu\in\mathcal{P}(\Lambda_{\Gamma})$ be a
$(\Gamma,F)$ Patterson-Sullivan density. There exists $\varepsilon>0$ such
that the following hold. Let $R\geq 1$ and let
$\chi:\partial_{\infty}\mathbb{H}^{2}\simeq\mathbb{S}^{1}\rightarrow\mathbb{R}$
be a $\alpha$-Hölder map supported on some compact $K$. Then there exists
$C\geq 1$ such that, for any $C^{2}$ function
$\varphi:\partial_{\infty}\mathbb{H}^{2}\rightarrow\mathbb{R}$ such that
$\|\varphi\|_{C^{2}}+(\inf_{K}|\varphi^{\prime}|)^{-1}\leq R$, we have:
$\forall s\in\mathbb{R}^{*},\
\left|\int_{\partial_{\infty}\mathbb{H}^{2}}e^{is\varphi}\chi
d\mu\right|\leq\frac{C}{|s|^{\varepsilon}}.$
Using the previous Corollary 1.3 and using the Hopf coordinates, we can
conclude Fourier decay for equilibrium states on convex-cocompact hyperbolic
surfaces. The proof is done in section 4.
###### Theorem 1.4 (Compare Theorem 4.5).
Let $\Gamma$ be a convex-cocompact group of isometries of $\mathbb{H}^{2}$,
and let $F:T^{1}(\mathbb{H}^{2}/\Gamma)\rightarrow\mathbb{R}$ be a Hölder
potential satisfying (R). Let $m_{F}$ be the associated equilibrium state.
There exists $\varepsilon>0$ such that the following holds. Let
$\chi:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}$ be a Hölder map supported on a
compact neighborhood of some point $v_{o}\in T^{1}\mathbb{H}^{2}$, and let
$\varphi:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}^{3}$ be a $C^{2}$ local
chart containing the support of $\chi$. There exists $C\geq 1$ such that:
$\forall\zeta\in\mathbb{R}^{3}\setminus\\{0\\},\
\left|\int_{NW(\phi)}e^{i\zeta\cdot\varphi(v)}\chi(v)dm_{F}(v)\right|\leq\frac{C}{|\zeta|^{\varepsilon}},$
where $\zeta\cdot\zeta^{\prime}$ and $|\zeta|$ denotes the euclidean scalar
product and the euclidean norm on $\mathbb{R}^{3}$. In other word, the
pushforward measure $\varphi_{*}(\chi dm_{F})\in\mathcal{P}(\mathbb{R}^{3})$
exhibit power Fourier decay.
###### Remark 1.5.
We will see in section 4 that the argument to prove Theorem 1.4 from Corollary
1.3 is fairly general. In particular, if one is able to prove Fourier decay
for $(\Gamma,F)$-Patterson-Sullivan densities in some higher dimensionnal
context, this would prove Fourier decay for equilibrium states in higher
dimensions. For example, [LNP19] precisely proves Fourier decay for Patterson-
Sullivan densities with the potential $F=0$ when
$\Gamma<\text{PSL}(2,\mathbb{C})$ is a Zariski-dense Kleinian Schottky group.
This yields power decay for the measure of maximal entropy on
$M:=\mathbb{H}^{3}/\Gamma$ in this context.
###### Remark 1.6.
With our result in mind, it is natural to try to give some sense to the
sentence $\dim_{F}NW(\phi)>0$. The problem is that the notion of Fourier
dimension is not well defined on manifolds. In the appendix, we suggest some
natural notions of Fourier dimensions for sets living in a manifold, in
particular a notion of _lower Fourier dimension_ , that measure persistence of
the positivity of the Fourier dimension under deformations. The sentence is
then made rigorous in Remark A.7 and Example A.23.
### 1.3 Acknowledgments
I would like to thank my PhD advisor, Frederic Naud, for pointing out to me
the existing bibliography on Patterson-Sullivan densities and for encouraging
me to work in the context of hyperbolic surfaces. I would also like to thank
Jialun Li for explaining to me his work on Fourier decay for stationary
measures with exponential moment in the context of split groups. Finally, I
would like to thank Paul Laubie for some very stimulating discussions.
## 2 Preliminaries
### 2.1 Moebius transformations preserving the unit ball
In this first paragraph we recall well known properties of Moebius
transformations. Usefull references for the study of such maps are [Be83],
[Ra06] and [BP92]. The group of all Moebius transformations of
$\mathbb{R}^{d}\cup\\{\infty\\}$ is the group generated by the inversion of
spheres and by the reflexions. This group contains dilations and rotations.
Denote by $\text{Mob}(B^{b})$ the group of all the Moebius transformations
$\gamma$ such that $\gamma$ preserves the orientation of $\mathbb{R}^{d}$, and
such that $\gamma(B^{d})=B^{d}$, where $B^{d}$ denotes the open unit ball in
$\mathbb{R}^{d}$. These maps also acts on the unit sphere $\mathbb{S}^{d-1}$.
These transformations can be put in a normal form as follows.
###### Lemma 2.1 ([Ra06], page 124).
Define, for $b\in B^{d}$, the associated hyperbolic translation by:
$\tau_{b}(x)=\frac{(1-|b|^{2})x+(|x|^{2}+2x\cdot b+1)b}{|b|^{2}|x|^{2}+2x\cdot
b+1}.$
Then $\tau_{b}\in\text{Mob}(B^{d})$. Moreover, for every
$\gamma\in\text{Mob}(B^{d})$, $\tau_{\gamma(0)}^{-1}\gamma\in
SO(d,\mathbb{R})$.
It follows that the distortions of any Moebius transformation
$\gamma\in\text{Mob}(B^{d})$ can be understood by studying the distortions of
hyperbolic translations. The main idea is the following: if $\gamma(o)$ is
close to the unit sphere, then $\gamma$ contracts strongly on a large part of
the sphere. Let us state a quantitative statement:
###### Lemma 2.2 (First contraction lemma).
Let $\gamma\in\text{Mob}(B^{d})$. Suppose that $|\gamma(o)|\geq c_{0}>0$.
Denote by $x_{\gamma}^{m}:=\gamma(o)/|\gamma(o)|$, and let
$\varepsilon_{\gamma}:=1-|\gamma(o)|$. Then:
1. 1.
There exists $c_{1},c_{2}>0$ that only depends on $c_{0}$ such that
$\forall x\in\mathbb{S}^{d-1},\ |x-x^{m}_{\gamma}|\geq
c_{1}\varepsilon_{\gamma}^{2}\Longrightarrow|\gamma^{-1}x-\gamma^{-1}x^{m}_{\gamma}|\geq
c_{2}.$
2. 2.
For all $c\in(0,1)$, there exists $C\geq 1$ and a set
$A_{\gamma}\subset\mathbb{S}^{d-1}$ such that $\text{diam}(A_{\gamma})\leq
C\varepsilon_{\gamma}$ and such that:
$\forall x\in\mathbb{S}^{d-1}\setminus A_{\gamma},\
|\gamma(x)-x_{\gamma}^{m}|\leq c\varepsilon_{\gamma}.$
###### Proof.
Let $\gamma\in\text{Mob}(B^{d})$. Since $\gamma=\tau_{\gamma(o)}\Omega$ for
some $\Omega\in SO(n,\mathbb{R})$, we see that we may suppose
$\gamma=\tau_{b}$ for some $b\in B^{d}$. Without loss of generality, we may
even choose $b$ of the form $\beta e_{d}$, where $e_{d}$ is the d-th vector of
the canonical basis of $\mathbb{R}^{d}$, and where
$\beta=|\gamma(o)|\in[c_{0},1[$. Denote by $\pi_{d}$ the projection on the
$d-th$ coordinate. We find:
$\forall x\in\mathbb{S}^{d-1},\
\pi_{d}\tau_{b}(x)=\frac{(1+\beta^{2})x_{d}+2\beta}{2\beta
x_{d}+(1+\beta^{2})}=:\varphi(x_{d}).$
The function $\varphi$ is continuous and increasing on $[-1,1]$, and fixes
$\pm 1$. Computing its value at zero gives
$\varphi(0)=\frac{2\beta}{1+\beta^{2}}\geq 1-\varepsilon_{\gamma}^{2}$, which
proves the first point. Computing its value at $-\beta$ gives
$\varphi(-\beta)=\beta$, which (almost) proves the second point. The second
point is proved rigorously by a direct computation, noticing that
$1-\varphi(-1+C\varepsilon_{\gamma})=\frac{\varepsilon_{\gamma}}{C}\frac{1-C\varepsilon_{\gamma}/2}{1-(1-1/(2C))\varepsilon_{\gamma}}\leq\varepsilon_{\gamma}/C.$
∎
Finally, let us recall a well known way to see $\text{Mob}(B^{d})$ as a group
of matrices.
###### Lemma 2.3.
Let $q:\mathbb{R}^{d+1}\rightarrow\mathbb{R}^{d+1}$ be the quadratic form
$q(t,\omega):=-t^{2}+\sum_{i}\omega_{i}^{2}$ on $\mathbb{R}^{d+1}$. We denote
by $SO(d,1)$ the set linear maps with determinant one that preserves $q$. Let
$H:=\\{(t,\omega)\in\mathbb{R}\times\mathbb{R}^{d}\ ,\ q(t,\omega)=-1\
,t>0\\}.$ Define the stereographic projection $\zeta:B^{d}\rightarrow H$ by
$\zeta(x):=\left(\frac{1+|x|^{2}}{1-|x|^{2}},\frac{2x}{1-|x|^{2}}\right)$.
Then, for any $\gamma\in\text{Mob}(B^{d})$, the map
$\zeta\gamma\zeta^{-1}:H\rightarrow H$ is the restriction to $H$ of an element
of $SO(d,1)$.
###### Proof.
It suffices to check the lemma when $\gamma$ is a rotation or a hyperbolic
translation. A direct computation shows that, when $\Omega\in
SO(d,\mathbb{R})$, then $\zeta\Omega\zeta^{-1}$ is a rotation leaving
invariant the $t$ coordinate, and so it is trivially an element of $SO(d,1)$.
We now do the case where $\gamma=\tau_{\beta e_{d}}$ is a hyperbolic
translation. We denote by $x$ the variable in $B^{d}$ and $(t,\omega)$ the
variables in $\mathbb{R}^{d+1}$. The expression $\zeta(x)=(t,\omega)$ gives
$\frac{1+|x|^{2}}{1-|x|^{2}}=t,\quad\frac{2x}{1-|x|^{2}}=\omega,\
\text{and}\quad\zeta^{-1}(t,\omega)=\frac{\omega}{1+t}.$
For $\alpha\in\mathbb{R}$, denote $s_{\alpha}:=\sinh(\alpha)$ and
$c_{\alpha}:=\cosh(\alpha)$. There exists $\alpha$ such that
$\beta=s_{\alpha}/(c_{\alpha}+1)=(c_{\alpha}-1)/s_{\alpha}$. For this
$\alpha$, we also have $\beta^{2}=(c_{\alpha}-1)/(c_{\alpha}+1)$, and
$1-\beta^{2}=2/(c_{\alpha}+1)$. Now, we see that
$\tau_{\beta e_{d}}(x)=\frac{(1-\beta^{2})x+(|x|^{2}+2x_{d}\beta+1)\beta
e_{d}}{\beta^{2}|x|^{2}+2x_{d}\beta+1}=\frac{\frac{2}{c_{\alpha}+1}x+(|x|^{2}+2x_{d}\frac{c_{\alpha}-1}{s_{\alpha}}+1)\frac{s_{\alpha}}{c_{\alpha}+1}e_{d}}{\frac{c_{\alpha}-1}{c_{\alpha}+1}|x|^{2}+2x_{d}\frac{s_{\alpha}}{c_{\alpha}+1}+1}$
$=\frac{2x+(s_{\alpha}(1+|x|^{2})+2x_{d}(c_{\alpha}-1))e_{d}}{1-|x|^{2}+c_{\alpha}(1+|x|^{2})+2s_{\alpha}x_{d}}=\frac{\omega+\left(s_{\alpha}t+(c_{\alpha}-1)\omega_{d}\right)e_{d}}{1+c_{\alpha}t+s_{\alpha}\omega_{d}}$
$=\zeta^{-1}\left(c_{\alpha}t+s_{\alpha}\omega_{d},\omega+\left(s_{\alpha}t+(c_{\alpha}-1)\omega_{d}\right)e_{d}\right),$
and so $\zeta\tau_{\beta
e_{d}}\zeta^{-1}(t,\omega)=\left(c_{\alpha}t+s_{\alpha}\omega_{d}\ ,\
\omega+\left(s_{\alpha}t+(c_{\alpha}-1)\omega_{d}\right)e_{d}\right)$ is
indeed linear in $(t,\omega)$. In this form, checking that $\zeta\tau_{\beta
e_{d}}\zeta^{-1}\in SO(d,1)$ is immediate. ∎
###### Remark 2.4.
From now on, we will allow to directly identify elements of
$\text{Mob}(B^{d})$ with matrices in $SO(d,1)$. (By continuity of
$\gamma\mapsto\zeta\gamma\zeta^{-1}$, we even know that thoses matrices lies
in $SO_{0}(d,1)$, the connected component of the identity in $SO(d,1)$). It
follows from the previous explicit computations that, for any matrix norm on
$SO(d,1)$, and for any $\gamma\in\text{Mob}(B^{d})$ such that $|\gamma(o)|\geq
c_{0}$, there exists $C_{0}$ only depending on $c_{0}$ such that
$\|\gamma\|\leq C_{0}\varepsilon_{\gamma}^{-1}.$
### 2.2 The Gibbs cocycle
In this paragraph, we introduce our geometric setting. For an introduction to
geometry in negative (non-constant) curvature, the interested reader may refer
to [BGS85], or to the first chapters of [PPS15].
Let $M=\mathbb{H}^{d}/\Gamma$ be a hyperbolic manifold of dimension $d$, where
$\Gamma\subset Iso^{+}(\mathbb{H}^{d})$ denotes a non-elementary and discrete
group of isometries of the hyperbolic space without torsion that preserves the
orientation. Let $T^{1}M$ denotes the unit tangent bundle of $M$, and denote
by $p:T^{1}M\rightarrow M$ the usual projection. The projection lift to a
$\Gamma$-invariant map
$\tilde{p}:T^{1}\mathbb{H}^{d}\rightarrow\mathbb{H}^{d}$. Fix
$F:T^{1}M\rightarrow\mathbb{R}$ a Hölder map: we will call it a potential. The
potential $F$ lift to a $\Gamma$-invariant map
$\tilde{F}:T^{1}\mathbb{H}^{d}\rightarrow\mathbb{R}$. All future constants
appearing in the paper will implicitely depend on $\Gamma$ and $F$.
To be able to do use our previous results about Moebius transformations, we
will work in the conformal ball model for a bit. In this model, we can think
of $\mathbb{H}^{d}$ as being the unit ball $B^{d}$ equipped with the metric
$ds^{2}:=\frac{4\|dx\|^{2}}{(1-\|x\|^{2})^{2}}$. The ideal boundary
$\partial_{\infty}\mathbb{H}^{d}$ (see [BGS85] for a definition) of
$\mathbb{H}^{d}$ is then naturally identified with $\mathbb{S}^{d-1}$, and its
group of orientation-preserving isometries with $\text{Mob}(B^{d})$. On the
ideal boundary, there is a natural family of distances
$(d_{x})_{x\in\mathbb{H}^{d}}$ called visual distances (seen from $x$),
defined as follow:
$d_{x}(\xi,\eta):=\lim_{t\rightarrow+\infty}\exp\left(-\frac{1}{2}\left(d(x,\xi_{t})+d(x,\eta_{t})-d(\xi_{t},\eta_{t})\right)\right)\in[0,1],$
where $\xi_{t}$ and $\eta_{t}$ are any geodesic rays ending at $\xi$ and
$\eta$. To get the intuition behind this quantity, picture a finite tree with
root $x$ and think of $\xi$ and $\eta$ as leaves in this tree.
###### Lemma 2.5 ([PPS15] page 15 and [LNP19] lemma A.5).
The visual distances are all equivalent and induces the usual euclidean
topology on $\mathbb{S}^{d-1}\simeq\partial_{\infty}\mathbb{H}^{d}$. More
precisely:
$\forall
x,y\in\mathbb{H}^{d},\forall\xi,\eta\in\partial_{\infty}\mathbb{H}^{d},\ \
e^{-d(x,y)}\leq\frac{d_{x}(\xi,\eta)}{d_{y}(\xi,\eta)}\leq e^{d(x,y)}.$
In the ball model, the visual distance from the center of the ball is the sine
of (half of) the angle.
The sphere at infinity $\partial_{\infty}\mathbb{H}^{d}$ takes an important
role in the study of $\Gamma$. For any $x\in\mathbb{H}^{d}$, the orbit $\Gamma
x$ accumulates on $\mathbb{S}^{d-1}$ (for the euclidean topology) into a
(fractal) _limit set_ denoted $\Lambda_{\Gamma}$. This limit set is
independant of $x$. We will denote by $\text{Hull}(\Lambda_{\Gamma})$ the
convex hull of the limit set: that is, the set of points $x\in\mathbb{H}^{d}$
such that $x$ is in a geodesic starting and finishing in $\Lambda_{\Gamma}$.
Since $\Gamma$ acts naturally on $\Lambda_{\Gamma}$, $\Gamma$ acts on
$\text{Hull}(\Lambda_{\Gamma})$. Without loss of generality, we can assume
that $o\in\text{Hull}(\Lambda_{\Gamma})$, and we will do from now on. We will
say that $\Gamma$ is convex-cocompact if $\Gamma$ is discrete, without torsion
and if $\text{Hull}(\Lambda_{\Gamma})/\Gamma$ is compact. In particular, in
this paper, we allow $M$ to be compact.
We will suppose througout the paper that $\Gamma$ is convex cocompact. In this
context, the set
$\Omega\Gamma=p^{-1}(\text{Hull}\Lambda_{\Gamma}/\Gamma)\subset T^{1}M$ is
compact, and it follows that $\sup_{\Omega\Gamma}|F|<\infty$. In particular,
$\tilde{F}$ is bounded on $\tilde{p}^{-1}(\text{Hull}(\Lambda_{\Gamma}))$,
which is going to allow us to get some control over line integrals involving
$F$. Recall the notion of line integral in this context: if
$x,y\in\mathbb{H}^{d}$ are distinct points, then there exists a unique unit
speed geodesic joining $x$ to $y$, call it $c_{x,y}$. We then define:
$\int_{x}^{y}\tilde{F}:=\int_{0}^{d(x,y)}\tilde{F}(\dot{c}_{x,y}(s))ds.$
Beware that if $\tilde{F}(-v)\neq\tilde{F}(v)$ for some $v\in T^{1}M$, then
$\int_{x}^{y}\tilde{F}$ and $\int_{y}^{x}\tilde{F}$ might not be equal.
We are ready to introduce the _Gibbs cocycle_ and recall some of its
properties.
###### Definition 2.6 ([PPS15], page 39).
The following Gibbs cocycle
$C_{F}:\partial_{\infty}\mathbb{H}^{d}\times\mathbb{H}^{d}\times\mathbb{H}^{d}\rightarrow\mathbb{R}$
is well defined and continuous:
$C_{F,\xi}(x,y):=\underset{t\rightarrow+\infty}{\lim}\left(\int_{y}^{\xi_{t}}\tilde{F}-\int_{x}^{\xi_{t}}\tilde{F}\right)$
where $\xi_{t}$ denotes any geodesic converging to $\xi$.
###### Remark 2.7.
Notice that if $\xi$ is the endpoint of the ray joining $x$ to $y$, then
$C_{F,\xi}(x,y)=-\int_{x}^{y}\tilde{F}.$
For $A\subset\mathbb{H}^{d}$, we call shadow of $A$ seen from $x$ the set
$\mathcal{O}_{x}A$ of all $\xi\in\partial_{\infty}\mathbb{H}^{d}$ such that
the geodesic joining $x$ to $\xi$ intersects $A$. (The letter $\mathcal{O}$
stands for Ombre in french.)
###### Proposition 2.8 ([PPS15], prop 3.4 and 3.5).
We have the following estimates on the Gibbs cocycle.
1. 1.
For all $R>0$, there exists $C_{0}>0$ such that for all $\gamma\in\Gamma$ and
for all $\xi\in\mathcal{O}_{o}B(\gamma(o),R)$ in the shadow of the
(hyperbolic) ball $B(\gamma(o),R)$ seen from $o$, we have:
$\left|C_{F,\xi}(o,\gamma(o))+\int_{o}^{\gamma(o)}\tilde{F}\right|\leq C_{0}$
2. 2.
There exists $\alpha\in(0,1)$ and $C_{0}>0$ such that, for all
$\gamma\in\Gamma$ and for all $\xi,\eta\in\Lambda_{\Gamma}$ such that
$d_{o}(\xi,\eta)\leq e^{-d(o,\gamma(o))-2}$,
$\left|C_{F,\xi}(o,\gamma(o))-C_{F,\eta}(o,\gamma(o))\right|\leq
C_{0}e^{\alpha d(o,\gamma(o))}d_{o}(\xi,\eta)^{\alpha}.$
(The hypothesis asking $\xi,\eta$ to be very close can be understood as an
hypothesis asking for the rays $[o,\xi[,[o,\eta[$ and
$[\gamma(o),\xi[,[\gamma(o),\eta[$ to be close. This way, we can use the
Hölder regularity of $\tilde{F}$ to get some control.)
### 2.3 Patterson-Sullivan densities
In this paragraph, we recall the definition of $(\Gamma,F)$ Patterson-Sullivan
densities, and we introduce a regularity condition. To begin with, recall the
definition and some properties of the critical exponent of $(\Gamma,F)$.
###### Definition 2.9 ([PPS15], Lemma 3.3).
Recall that $F$ is supposed Hölder, and that $\Gamma$ is convex-cocompact. The
critical exponent of $(\Gamma,F)$ is the quantity
$\delta_{\Gamma,F}\in\mathbb{R}$ defined by:
$\delta_{\Gamma,F}:=\underset{n\rightarrow\infty}{\limsup}\
\underset{n-c<d(x,\gamma y)\leq n}{\frac{1}{n}\ln\ {\sum_{\gamma\in\Gamma}}\
e^{\int_{x}^{\gamma y}\tilde{F}}},$
for any $x,y\in\mathbb{H}^{d}$ and any $c>0$. The critical exponent dosen’t
depend on the choice of $x,y$ and $c$.
###### Theorem 2.10 ([PPS15], section 3.6 and section 5.3).
Let $\Gamma\subset Iso^{+}(\mathbb{H}^{d})$ be convex-cocompact, and note
$M:=\mathbb{H}^{d}/\Gamma$. Let $F:T^{1}M\rightarrow\mathbb{R}$ be a Hölder
regular potential. Then there exists a unique (up to a scalar multiple) family
of finite nonzero measures $(\mu_{x})_{x\in\mathbb{H}^{d}}$ on
$\partial_{\infty}\mathbb{H}^{d}$ such that, for all $\gamma\in\Gamma$, for
all $x,y\in\mathbb{H}^{d}$ and for all
$\xi\in\partial_{\infty}\mathbb{H}^{d}$:
* •
$\gamma_{*}\mu_{x}=\mu_{\gamma x}$
* •
$d\mu_{x}(\xi)=e^{-C_{F-\delta_{\Gamma,F},\xi}(x,y)}d\mu_{y}(\xi)$
Moreover, these measures are all supported on the limit set
$\Lambda_{\Gamma}$. We call them $(\Gamma,F)$-Patterson Sullivan densities.
###### Remark 2.11.
Notice that Patterson-Sullivan densities only depend on the normalized
potential $F-\delta_{\Gamma,F}$. Since
$\delta_{\Gamma,F+\kappa}=\delta_{\Gamma,F}+\kappa$, replacing $F$ by
$F-\delta_{\Gamma,F}$ allows us to work without loss of generality with
potential satisfying $\delta_{\Gamma,F}=0$. We call such potential
_normalized_.
The next estimate tells us, in a sense, that we can think of $\mu_{0}$ as a
measure of a fractal solid angle, pondered by the potential. This is better
understood by recalling that since the area of a hyperbolic sphere of large
radius $r$ is a power of $\sim e^{r}$, then the solid angle of an object of
diameter $1$ lying in that sphere is a power of $\sim e^{-r}$. In the
following Shadow lemma, the object is a ball $B(y,R)$, at distance $d(x,y)$
from an observer at $x$.
###### Proposition 2.12 (Shadow Lemma, [PPS15] Lemma 3.10).
Let $R>0$ be large enough. There exists $C>0$ such that, for all
$x,y\in\text{Hull}(\Lambda_{\Gamma})$:
$C^{-1}e^{\int_{x}^{y}(\tilde{F}-\delta_{\Gamma,F})}\leq\mu_{x}\left(\mathcal{O}_{x}B(y,R)\right)\leq
Ce^{\int_{x}^{y}(\tilde{F}-\delta_{\Gamma,F})}.$
###### Definition 2.13.
The shadow lemma calls for the following hypothesis: we say that the potential
$F$ satisfy the regularity assumptions (R) if $F$ is Hölder regular and if
$\sup_{\Omega\Gamma}F<\delta_{\Gamma,F}$.
###### Remark 2.14.
By Lemma 3.3 in [PPS15], we see that we can construct potentials satisfying
(R) as follow: choose some potential $F_{0}$ satisfying (R) (for example, the
constant potential) and then choose any Holder map
$E:T^{1}M\rightarrow\mathbb{R}$ satisfying
$2\sup_{\Omega\Gamma}|E|<\delta_{\Gamma,F}-\sup_{\Omega{\Gamma}}F$. Then
$F:=F_{0}+E$ satisfies the assumption (R). A similar assumption is introduced
in [GS14].
The point of the assumption (R) is to ensure that the Patterson-Sullivan
densities exhibit some regularity. This is possible because we have a tight
control over the geometry of shadows.
###### Lemma 2.15.
Let $\Gamma\subset Iso^{+}(\mathbb{H}^{d})$ be convex-cocompact, and let
$F:T^{1}({\mathbb{H}^{d}/\Gamma})\rightarrow\mathbb{R}$ satisfy the regularity
assumptions (R). Let $\delta_{reg}\in(0,1)$ such that
$\delta_{reg}<\delta_{\Gamma,F}-\sup_{\Omega\Gamma}F$. Let $\mu$ denote some
$(\Gamma,F)$-Patterson-Sullivan density. Then
$\exists C>0,\ \forall\xi\in\partial_{\infty}\mathbb{H}^{d},\ \forall r>0,\
\mu(B(\xi,r))\leq Cr^{\delta_{reg}},$
where the ball is in the sense of some visual distance.
###### Proof.
First of all, since for all $p,q\in\mathbb{H}^{d}$,
$\xi\in\partial_{\infty}\mathbb{H}^{d}\mapsto e^{C_{\xi}(p,q)}$ is continuous
and since the ideal boundary is compact, we can easily reduce our statement to
the case where $\mu$ is a Patterson-Sullivan density based on the center of
the ball $o$. Moreover, since all the visual distances are equivalent, one can
suppose that we are working for the visual distance based at $o$ too. Finally,
since the support of $\mu_{o}$ is $\Lambda_{\Gamma}$, we can suppose without
loss of generality that $B(\xi,r)\cap\Lambda_{\Gamma}\neq\emptyset$. Since in
this case there exists some $\tilde{\xi}\in\Lambda_{\Gamma}$ such that
$B(\xi,r)\subset B(\tilde{\xi},2r)$, we may further suppose without loss of
generality that $\xi\in\Lambda_{\Gamma}$.
Now fix $\xi\in\Lambda_{\Gamma}$ and $r>0$. Let
$x\in\text{Hull}(\Lambda_{\Gamma})$ lay in the ray starting from $o$ and
ending at $\xi$. Let $\rho\in[0,1]$, let $y\in\mathbb{H}^{d}$ such that
$[o,y]$ is tangent to the sphere $S(x,\rho)$, and note $\eta$ the ending of
the ray starting from $o$ and going through $y$. The hyperbolic law of sine
(see Lemma A.4 and A.5 in [LNP19]) allows us to compute directly:
$d_{o}(\xi,\eta)=\frac{1}{2}\cdot\frac{\sinh(\rho)}{\sinh(d(o,x))}.$
It follows that there exists $C>0$ such that for all $\xi\in\Lambda_{\Gamma}$,
for all $r>0$, there exists $x\in\text{Hull}(\Lambda_{\Gamma})$ such that
$e^{-d(o,x)}\leq Cr$ and $B_{o}(\xi,r)\subset\mathcal{O}_{o}B(x,1)$. The
desired bound follows from the shadow lemma, since the geodesic segment
joining $o$ and $x$ lays in $\text{Hull}(\Lambda_{\Gamma})$. ∎
The regularity of the Patterson-Sullivan densities is going to allow us to
state a second version of the contraction lemma. First, let us introduce a bit
of notations. We fix, for all the duration of the paper, a large enough
constant $C_{\Gamma}>0$. For $\gamma\in\Gamma$, we define
$\kappa(\gamma):=d(o,\gamma o)$, $r_{\gamma}:=e^{-\kappa(\gamma)}$ and
$B_{\gamma}:=\mathcal{O}_{o}B(\gamma o,C_{\Gamma})$. By the hyperbolic law of
sine, the radius of $B_{\gamma}$ is $\sinh(C_{\Gamma})/\sinh(r_{\gamma})$
(Lemma A.5 in [LNP19].) If $C_{\Gamma}$ is chosen large enough and when
$\kappa(\gamma)$ is large, we get a radius of $\sim
e^{C_{\Gamma}}r_{\gamma}\geq r_{\gamma}$. We have the following covering
result.
###### Lemma 2.16 ([LNP19], Lemma A.8).
Define $r_{n}:=e^{-4C_{\Gamma}n}$, and let $S_{n}:=\\{\gamma\in\Gamma\
,e^{-2C_{\Gamma}}r_{n}\leq r_{\gamma}<r_{n}\\}$. For all $n\geq 1$, the family
$\\{B_{\gamma}\\}_{\gamma\in S_{n}}$ cover $\Lambda_{\Gamma}$. Moreover, there
exists $C>0$ such that:
$\forall n,\forall\xi\in\Lambda_{\Gamma},\ \\#\\{\gamma\in S_{n}\ ,\ \xi\in
B_{\gamma}\\}\leq C.$
Now, we are ready to state our second contraction lemma. Since the potential
is not supposed bounded, a lot of technical bounds will only be achieved by
working on the limit set or on its convex hull. One of the main goal of the
second contraction lemma is then to replace $x_{\gamma}^{m}$ by a point
$\eta_{\gamma}$ lying in the limit set.
###### Lemma 2.17 (Second contraction lemma).
Let $\Gamma\subset\text{Iso}^{+}(\mathbb{H}^{d})$ be convex-cocompact. Let
$F:T^{1}(\mathbb{H}^{d}/\Gamma)\rightarrow\mathbb{R}$ be a potential
satisfying (R). Denote by $\mu_{o}\in\mathcal{P}(\Lambda_{\Gamma})$ the
associated Patterson-Sullivan density at $o$. Then there exists a family of
points $(\eta_{\gamma})_{\gamma\in\Gamma}$ such that, for any
$\gamma\in\Gamma$ with large enough $\kappa(\gamma)$, we have
$\eta_{\gamma}\in\Lambda_{\Gamma}\cap B_{\gamma}$, and moreover:
1. 1.
there exists $c>0$ independant of $\gamma$ such that
$d_{o}(\xi,\eta_{\gamma})\geq r_{\gamma}/2\Rightarrow
d_{o}(\gamma^{-1}\xi,\gamma^{-1}\eta_{\gamma})\geq c$,
2. 2.
for all $\varepsilon_{0}\in(0,\delta_{reg})$, there exists $C$ independant of
$\gamma$ such that:
$\int_{\Lambda_{\Gamma}}d_{o}(\gamma(\xi),\eta_{\gamma})^{\varepsilon_{0}}d\mu_{o}(\xi)\leq
Cr_{\gamma}^{\varepsilon_{0}}.$
###### Proof.
Recalling that the visual distance and the euclidean distance are equivalent
on the unit sphere, if we forget about $\eta_{\gamma}$ and replace it by
$x_{\gamma}^{m}$ instead, then the first point is a direct corollary of the
first contraction lemma. We just have to check two points: first, since
$\Gamma$ is discrete without torsion, there exists $c_{0}>0$ such that for all
$\gamma\in\Gamma\setminus\\{Id\\}$, $d(o,\gamma o)>c_{0}$. The second point is
to check that the orders of magnitude of $r_{\gamma}$ and
$\varepsilon_{\gamma}$ (quantity introduced in the first contraction lemma)
are compatible. This can be checked using an explicit formula relating the
hyperbolic distance with the euclidean one in the ball model:
$r_{\gamma}=e^{-\kappa(\gamma)}=\frac{1-|\gamma(o)|}{1+|\gamma(o)|}\sim\varepsilon_{\gamma}/2$
(see [Ra06], exercise 4.5.1). We even have a large security gap for the first
statement to hold (recall that the critical scale is
$\sim\varepsilon_{\gamma}^{2}$).
We will use the strong contraction properties of $\Gamma$ to construct a point
$\eta_{\gamma}\in\Lambda_{\Gamma}$ very close to $x_{\gamma}^{m}$. Since
$\Gamma$ is convex-cocompact, we know in particular that
$\text{diam}(\Lambda_{\Gamma})>0$. Now let $\gamma\in\Gamma$ such that
$\kappa(\gamma)$ is large enough. The first contraction lemma says that there
exists $A_{\gamma}\subset\partial_{\infty}\mathbb{H}^{d}$ with
$\text{diam}(A_{\gamma})\leq Cr_{\gamma}$ such that
$\text{diam}(\gamma(A_{\gamma}^{c}))\leq r_{\gamma}/10$. It follows that we
can find a point $\widehat{\eta}_{\gamma}\in\Lambda_{\Gamma}$ such that
$d(A_{\gamma},\widehat{\eta}_{\gamma})>\text{diam}(\Lambda_{\Gamma})/3$.
Fixing $\eta_{\gamma}:=\gamma(\widehat{\eta}_{\gamma})$ gives us a point
satisfying $\eta_{\gamma}\in\Lambda_{\Gamma}$ and
$d_{o}(\eta_{\gamma},x_{\gamma}^{m})\lesssim r_{\gamma}^{2}$. Hence
$\eta_{\gamma}\in B_{\gamma}$, and moreover, any point $\xi$ satisfying
$d_{o}(\xi,\eta_{\gamma})\geq r_{\gamma}/2$ will satisfy $\gamma^{-1}(\xi)\in
A_{\gamma}$, and so
$d_{o}(\gamma^{-1}(\xi),\gamma^{-1}(\eta_{\gamma}))\geq\text{diam}(\Lambda_{\Gamma})/3$.
This proves the first point.
For the second point: since the set
$A_{\gamma}\subset\partial_{\infty}\mathbb{H}^{d}$ is of diameter $\leq
Cr_{\gamma}$ and satisfy that, for all $\xi\notin A_{\gamma}$,
$d_{o}(\gamma(\xi),x_{\gamma}^{m})\leq r_{\gamma}/10$, the upper regularity of
$\mu_{0}$ yields
$\int_{\Lambda_{\Gamma}}d_{o}(\gamma(\xi),x_{\gamma}^{m})^{\varepsilon_{0}}d\mu_{o}(\xi)\leq
C\mu_{o}(A_{\gamma})+\int_{A_{\gamma}^{c}}Cr_{\gamma}^{\varepsilon_{0}}d\mu_{o}\leq
Cr_{\gamma}^{\varepsilon_{0}}.$
The desired bound follows from $d_{o}(x_{\gamma}^{m},\eta_{\gamma})\lesssim
r_{\gamma}$, using the triangle inequality.∎
## 3 Patterson-Sullivan densities are stationary measures
### 3.1 Stationary measures
In this subsection we define stationary measures and state our main theorem.
###### Definition 3.1.
Let $\nu\in\mathcal{P}(\Gamma)$ be a probability measure on $\Gamma\subset
SO(n,1)$. Let $\mu\in\mathcal{P}(\partial_{\infty}\mathbb{H}^{d})$. We say
that $\mu$ is $\nu$-stationary if:
$\mu=\nu*\mu:=\int_{\Gamma}\gamma_{*}\mu\ d\nu(\gamma).$
Moreover, we say that the measure $\nu$ has exponential moment if there exists
$\varepsilon>0$ such that
$\int_{\Gamma}\|\gamma\|^{\varepsilon}d\nu(\gamma)<\infty$. Finally, we denote
by $\Gamma_{\nu}$ the subgroup of $\Gamma$ generated by the support of $\nu$.
###### Theorem 3.2.
Let $\Gamma\subset\text{Iso}^{+}(\mathbb{H}^{d})$ be a convex-cocompact group,
and let $F:T^{1}(\mathbb{H}^{d}/\Gamma)\rightarrow\mathbb{R}$ be a potential
on the unit tangent bundle satisfying (R). Let $x\in\mathbb{H}^{d}$ and let
$\mu_{x}\in\mathcal{P}(\Lambda_{\Gamma})$ denotes the $(\Gamma,F)$ Patterson-
Sullivan density from $x$. Then there exists $\nu\in\mathcal{P}(\Gamma)$ with
exponential moment (seen as a random walk in $SO(d,1)$) such that $\mu$ is
$\nu$-stationary and such that $\Gamma_{\nu}=\Gamma$.
###### Remark 3.3.
This result for $d=2$ was announced without proof by Jialun Li in [Li18] (see
remark 1.9). A proof in the case of constant potentials is done in the
appendix of [LNP19]. Our strategy is inspired by this appendix. For more
details on stationary measures, see the references therein.
First of all, a direct computation allows us to see that if
$(\mu_{x})_{x\in\mathbb{H}^{d}}$ are $(\Gamma,F)$ Patterson-Sullivan
densities, then, for any $\eta\in SO_{0}(d,1)$ ,
$(\eta_{*}\mu_{\eta^{-1}x})_{x\in\mathbb{H}^{d}}$ are
$(\eta\Gamma\eta^{-1},\eta_{*}F)$ Patterson-Sullivan densities. This remark
allows us to reduce our theorem to the case where the basepoint $x$ is the
center of the ball $o$. Our goal is to find $\nu\in\mathcal{P}(\Gamma)$ such
that $\nu*\mu_{o}=\mu_{o}$. Assuming that $F$ is normalized, this can be
rewritten as follows:
$d\mu_{o}(\xi)=\sum_{\gamma}\nu(\gamma)d(\gamma_{*}\mu_{o})(\xi)=\sum_{\gamma}\nu(\gamma)e^{C_{F,\xi}(o,\gamma
o)}d\mu_{o}(\xi).$
Hence, $\mu_{o}$ is $\nu$-stationary if
$\sum_{\gamma\in\Gamma}\nu(\gamma)f_{\gamma}=1\ \text{ on }\Lambda_{\Gamma},$
where
$f_{\gamma}(\xi):=e^{C_{F,\xi}(o,\gamma o)}.$
###### Remark 3.4.
Our main goal is to find a way to decompose the constant function $1$ as a sum
of $f_{\gamma}$. Here is the intuition behind our proof.
Define $r_{\gamma}^{-F}:=e^{-\int_{o}^{\gamma
o}\tilde{F}}=f_{\gamma}(x^{m}_{\gamma})\simeq f_{\gamma}(\eta_{\gamma})$. The
first thing to notice is that $f_{\gamma}$ looks like an approximation of
unity centered at $x^{m}_{\gamma}$. Renormalizing yields the intuitive
statement $r_{\gamma}^{F}f_{\gamma}\sim\mathbb{1}_{B_{\gamma}}$. The idea is
that this approximation gets better as $\kappa(\gamma)$ becomes large. Once
this observation is done, there is a natural n-th approximation operator that
can be defined. For some positive function $R$, one can write:
$R\simeq\sum_{\gamma\in
S_{n}}R(\eta_{\gamma})\mathbb{1}_{B_{\gamma}}\simeq\sum_{\gamma\in
S_{n}}R(\eta_{\gamma})r_{\gamma}^{F}f_{\gamma}=:P_{n}R.$
Proving that the operator $P_{n}$ does a good enough job at approximating some
functions $R$ is the content of the approximation lemma 3.8. In particular, it
is proved that, under some assumptions on $R>0$, we have $cR\leq P_{n}R\leq
CR$.
The conclusion of the proof is then easy. We fix a constant $\beta>0$ small
enough so that $cR\leq\beta P_{n}R\leq R$. Then, we define by induction
$R_{0}=1$ and $0<R_{n+1}:=R_{n}-\beta P_{n+1}R_{n}\leq(1-c)R_{n}$. By
induction, this gives $R_{n}\leq(1-c)^{n}$, and hence
$1=R_{0}-\lim_{n}R_{n}=\sum_{k}(R_{k}-R_{k+1})$ is a decomposition of $1$ as a
sum of $f_{\gamma}$.
### 3.2 The approximation operator
First, we collect some results on $f_{\gamma}$ that will allows us to think of
it as an approximation of unity around $x_{m}^{\gamma}$ with width
$r_{\gamma}$. The first point studies $f_{\gamma}$ near $x_{\gamma}^{m}$, the
second point study the decay of $f_{\gamma}$ away from it, and the last point
is a regularity estimate at the scale $r_{\gamma}$. To quantify this decay, we
recall the notion of _potential gap_ (or gap map).
###### Definition 3.5.
The following potential gap
$D_{F}:\mathbb{H}^{d}\times\partial_{\infty}\mathbb{H}^{d}\times\partial_{\infty}\mathbb{H}^{d}$
is well defined and continuous:
$D_{F,x}(\eta,\xi):=\exp\frac{1}{2}\lim_{t\rightarrow\infty}\left(\int_{x}^{\xi_{t}}\tilde{F}+\int_{\eta_{t}}^{x}\tilde{F}-\int_{\eta_{t}}^{\xi_{t}}\tilde{F}\right)$
where $(\eta_{t})$ and $(\xi_{t})$ denotes any unit speed geodesic ray
converging to $\eta$ and $\xi$.
Under our assumptions, the gap map (for some fixed $x$) behaves like a
distance on $\Lambda_{\Gamma}$: see [PPS15], section 3.4 for details. Finally,
we denote by $\iota(v)=-v$ the _flip map_ on the unit tangent bundle.
###### Lemma 3.6 (Properties of $f_{\gamma}$).
There exists $C_{\text{reg}}\geq 2$ (independant of $C_{\Gamma}$) and
$C_{0}\geq 1$ (depending on $C_{\Gamma})$ such that, for all
$\gamma\in\Gamma$:
1. 1.
For all $\xi\in B_{\gamma}$,
$r_{\gamma}^{F}f_{\gamma}(\xi)\in[e^{-C_{0}},e^{C_{0}}].$
2. 2.
For all $\xi\in\Lambda_{\Gamma}$ such that $d_{o}(\xi,\eta_{\gamma})\geq
r_{\gamma}/2$,
$C_{0}^{-1}D_{o}(\xi,\eta_{\gamma})^{-2}\leq
r_{\gamma}^{-F\circ\iota}f_{\gamma}(\xi)\leq
C_{0}D_{o}(\xi,\eta_{\gamma})^{-2}.$
3. 3.
For all $\xi,\eta\in\Lambda_{\Gamma}$ such that $d_{o}(\xi,\eta)\leq
r_{\gamma}/e^{2}$,
$\left|f_{\gamma}(\xi)/f_{\gamma}(\eta)-1\right|\leq C_{\text{reg}}\cdot
d_{o}(\xi,\eta)^{\alpha}r_{\gamma}^{-\alpha}.$
###### Proof.
Recall that $f_{\gamma}(x_{\gamma}^{m})=r_{\gamma}^{-F}$: the first point is
then a consequence of Proposition 2.8. The third point also directly follows
without difficulty. For the second item, recall that $\eta_{\gamma}\in
B_{\gamma}$, so that by the same argument we get
$r_{\gamma}^{-F\circ\iota}\simeq e^{C_{F\circ\iota,\eta_{\gamma}}(o,\gamma
o)}.$
Then, a direct computation yields:
$f_{\gamma}(\xi)r_{\gamma}^{-F\circ\iota}\simeq\exp\lim_{t\rightarrow\infty}\left(\left(\int_{\gamma
o}^{\xi_{t}}\tilde{F}-\int_{o}^{\xi_{t}}\tilde{F}\right)-\left(\int_{(\eta_{\gamma})_{t}}^{o}\tilde{F}-\int_{(\eta_{\gamma})_{t}}^{\gamma
o}\tilde{F}\right)\right)$
$=\lim_{t\rightarrow\infty}\exp{\left(\left(-\int_{o}^{\xi_{t}}\tilde{F}-\int_{(\eta_{\gamma})_{t}}^{o}\tilde{F}+\int_{(\eta_{\gamma})_{t}}^{\xi_{t}}\right)+\left(\int_{\gamma
o}^{\xi_{t}}\tilde{F}+\int_{(\eta_{\gamma})_{t}}^{\gamma
o}\tilde{F}-\int_{(\eta_{\gamma})_{t}}^{\xi_{t}}\tilde{F}\right)\right)}$
$=\frac{D_{F,\gamma
o}(\eta_{\gamma},\xi)^{2}}{D_{F,o}(\eta_{\gamma},\xi)^{2}}.$
Under our regularity hypothesis (R), and because $\Gamma$ is convex6cocompact
and $\xi,\eta\in\Lambda_{\Gamma}$, it is known that there exists $c_{0}>0$
such that $d_{\gamma o}(\xi,\eta_{\gamma})^{c_{0}}\leq D_{F,\gamma
o}(\eta_{\gamma},\xi)\leq 1$ (see [PPS15], page 56). The second contraction
lemma allows us to conclude, since $d_{\gamma
o}(\xi,\eta_{\gamma})=d_{o}(\gamma^{-1}\xi,\gamma^{-1}(\eta_{\gamma}))\geq c$
under the hypothesis $d_{o}(\xi,\eta_{\gamma})\geq r_{\gamma}/2$. ∎
To get further control over the decay rate of $f_{\gamma}$ away from
$\eta_{\gamma}$, the following role reversal result will be helpful.
###### Lemma 3.7 (symmetry).
Let $n\in\mathbb{N}$. Since $S_{n}$ is a covering of $\Lambda_{\Gamma}$, and
by choosing $C_{\Gamma}$ larger if necessary, we know that for every
$\eta\in\Lambda_{\Gamma}$ there exists $\tilde{\gamma}_{\eta}\in S_{n}$ such
that $\eta\in S_{\tilde{\gamma}_{\eta}}$ and
$d(\eta,\eta_{\tilde{\gamma}_{\eta}})\leq C_{reg}^{-2/\alpha}r_{\gamma}$.
Suppose that $d_{o}(\eta,\eta_{\gamma})\geq r_{\gamma}$. Then:
$C^{-1}f_{\tilde{\gamma}_{\eta}}(\eta_{\gamma})\leq f_{\gamma}(\eta)\leq
Cf_{\tilde{\gamma}_{\eta}}(\eta_{\gamma})$
for some constant $C$ independant of $n$, $\gamma$ and $\eta$.
###### Proof.
First of all, by the third point of the previous lemma, and since
$d(\eta,\eta_{\tilde{\gamma}_{\eta}})\leq
C_{\text{reg}}^{-2/\alpha}r_{\gamma}$, we can write
$f_{\gamma}(\eta)/f_{\gamma}(\eta_{\tilde{\gamma}_{\eta}})\in[1/2,3/2]$. Then,
by the previous lemma again, we see that
$\frac{f_{\gamma}(\eta)}{f_{\tilde{\gamma}_{\eta}}(\eta_{\gamma})}\simeq\frac{f_{\gamma}(\eta_{\tilde{\gamma}_{\eta}})}{f_{\tilde{\gamma}_{\eta}}(\eta_{\gamma})}\simeq\frac{D_{F,o}(\eta_{\tilde{\gamma}_{\eta}},\eta_{\gamma})^{2}}{D_{F,o}(\eta_{\gamma},\eta_{\tilde{\gamma}_{\eta}})^{2}}\simeq
1,$
where we used the quasi symmetry of the gap map ([PPS15], page 47). ∎
We are ready to introduce the $n$-th approximation operator. For some positive
function $R:\Lambda_{\Gamma}\rightarrow\mathbb{R}^{*}_{+}$, define, on
$\partial_{\infty}\mathbb{H}^{d}$, the following positive function:
$P_{n}R(\eta):=\sum_{\gamma\in
S_{n}}R(\eta_{\gamma})r_{\gamma}^{F}f_{\gamma}(\eta).$
The function $P_{n}R$ has the regularity of $f_{\gamma}$ for $\gamma\in
S_{n}$.
###### Lemma 3.8.
Choosing $C_{\Gamma}$ larger if necessary, the following hold. Let
$n\in\mathbb{N}$ and let $\xi,\eta\in\Lambda_{\Gamma}$ such that
$d_{o}(\xi,\eta)\leq r_{n+1}$. Then:
$\left|\frac{P_{n}R(\xi)}{P_{n}R(\eta)}-1\right|\leq\frac{1}{2}d_{o}(\xi,\eta)^{\alpha}r_{n+1}^{-\alpha}.$
###### Proof.
The regularity estimates on $f_{\gamma}$ and the positivity of $R$ yields:
$\left|P_{n}R(\xi)-P_{n}R(\eta)\right|\leq\sum_{\gamma\in
S_{n}}R(\eta_{\gamma})r_{\gamma}^{F}|f_{\gamma}(\xi)-f_{\gamma}(\eta)|$ $\leq
C_{\text{reg}}\sum_{\gamma\in
S_{n}}R(\eta_{\gamma})r_{\gamma}^{F}f_{\gamma}(\eta)d_{0}(\xi,\eta)^{\alpha}r_{\gamma}^{-\alpha}=P_{n}R(\eta)\cdot
C_{\text{reg}}e^{-2\alpha
C_{\Gamma}}d_{0}(\xi,\eta)^{\alpha}r_{n+1}^{-\alpha}.$
The bound follows. ∎
Now is the time where we combine all of our preliminary lemma to prove our
main technical lemma: the approximation operator does a good enough job at
approximating on $\Lambda_{\Gamma}$. Some natural hypothesis on $R$ are
required: the function to approximate has to be regular enough at scale
$r_{n}$, and has to have mild global variations (so that the decay of
$f_{\gamma}$ away from $x_{\gamma}^{m}$ is still usefull).
###### Lemma 3.9 (Approximation lemma).
Let $\varepsilon_{0}\in(0,\delta_{reg})$ and $C_{0}\geq 1$. Let
$n\in\mathbb{N}$, and let $R:\Lambda_{\Gamma}\rightarrow\mathbb{R}$ be a
positive function satisfying:
1. 1.
For $\xi,\eta\in\Lambda_{\Gamma}$, if $d_{o}(\xi,\eta)\leq r_{n+1}$, then
$\left|\frac{R(\xi)}{R(\eta)}-1\right|\leq\frac{1}{2}\left(\frac{d_{o}(\xi,\eta)}{r_{n+1}}\right)^{\alpha}.$
2. 2.
For $\xi,\eta\in\Lambda_{\Gamma}$, if $d_{o}(\xi,\eta)>r_{n+1}$, then
$R(\xi)/R(\eta)\leq
C_{0}d_{o}(\xi,\eta)^{\varepsilon_{0}}r_{n}^{-\varepsilon_{0}}.$
Then there exists $A\geq 1$ that only depends on $\varepsilon_{0}$ and $C_{0}$
such that, for all $\eta\in\Lambda_{\Gamma}$:
$A^{-1}R(\eta)\leq P_{n+1}R(\eta)\leq AR(\eta).$
###### Proof.
Let $\eta\in\Lambda_{\Gamma}$. We have:
$P_{n+1}R(\eta)=\sum_{\gamma\in
S_{n+1}}R(\eta_{\gamma})r_{\gamma}^{F}f_{\gamma}(\eta)=\underset{\eta\in
B_{\gamma}}{\sum_{\gamma\in
S_{n+1}}}R(\eta_{\gamma})r_{\gamma}^{F}f_{\gamma}(\eta)+\underset{\eta\notin
B_{\gamma}}{\sum_{\gamma\in
S_{n+1}}}R(\eta_{\gamma})r_{\gamma}^{F}f_{\gamma}(\eta).$
The first sum is easily controlled: if $\eta\in B_{\gamma}$, then
$R(\eta_{\gamma})\simeq R(\eta)$ and $r_{\gamma}^{F}f_{\gamma}(\eta)\simeq 1$.
Since $\eta$ is in a (positive and) bounded number of $B_{\gamma}$, we find
$C^{-1}R(\eta)\leq\underset{\eta\in B_{\gamma}}{\sum_{\gamma\in
S_{n+1}}}R(\eta_{\gamma})r_{\gamma}^{F}f_{\gamma}(\eta)\leq CR(\eta),$
which gives the lower bound since $R$, $r_{\gamma}^{F}$ and $f_{\gamma}$ are
positive. To conclude, we need to get an upper bound on the residual term.
Using $\text{diam}(B_{\gamma})\lesssim r_{n}$, the symmetry lemma on
$f_{\gamma}(\eta)$, the regularity and mild variations of $R$, and using the
shadow lemma $r_{\gamma}^{F}\simeq\mu_{o}(B_{\gamma})$, we get:
$\underset{\eta\notin B_{\gamma}}{\sum_{\gamma\in
S_{n}}}R(\eta_{\gamma})r_{\gamma}^{F}f_{\gamma}(\eta)\lesssim
R(\eta)r_{n}^{-\varepsilon_{0}}\sum_{\eta\notin
B_{\gamma}}r_{\gamma}^{F}d_{o}(\eta_{\gamma},\eta)^{\varepsilon_{0}}f_{\tilde{\gamma}_{\eta}}(\eta_{\gamma})$
$\lesssim
R(\eta)\left(1+r_{n}^{-\varepsilon_{0}}\int_{B(\eta,r_{n+1})^{c}}d_{o}(\xi,\eta)^{\varepsilon_{0}}\
f_{\tilde{\gamma}_{\eta}}(\xi)d\mu_{o}(\xi)\right).$
Finally, the second contraction lemma and the bound
$d_{o}(\eta,\eta_{\tilde{\gamma}_{\eta}})\lesssim r_{n}$ yields:
$\int_{B(\eta,r_{n+1})^{c}}d_{o}(\xi,\eta)^{\varepsilon_{0}}\
f_{\tilde{\gamma}_{\eta}}(\xi)d\mu_{o}(\xi)\leq\int_{\Lambda_{\Gamma}}d_{o}(\xi,\eta)^{\varepsilon_{0}}\
f_{\tilde{\gamma}_{\eta}}(\xi)d\mu_{o}(\xi)$
$=\int_{\Lambda_{\Gamma}}d_{o}(\xi,\eta)^{\varepsilon_{0}}d\mu_{\tilde{\gamma}_{\eta}o}(\xi)=\int_{\Lambda_{\Gamma}}d_{o}(\tilde{\gamma}_{\eta}(\xi),\eta)^{\varepsilon_{0}}d\mu_{o}(\xi)$
$\lesssim
r_{n}^{\varepsilon_{0}}+\int_{\Lambda_{\Gamma}}d_{o}(\tilde{\gamma}_{\eta}(\xi),\eta_{\tilde{\gamma}_{\eta}})^{\varepsilon_{0}}d\mu_{o}(\xi)\lesssim
r_{n}^{\varepsilon_{0}},$
which concludes the proof. ∎
### 3.3 The construction of $\nu$
In this last subsection, we construct the measure $\nu$ and conclude that
$(\Gamma,F)$ Patterson-Sullivan densities are stationary measures for a random
walk on $\Gamma$ with exponential moment. We can conclude by following the end
of [LNP19] very closely, but we will recall the last arguments for the
reader’s convenience.
Recall that the large constant $C_{\Gamma}\geq 1$ was fixed just before Lemma
2.16, and that $r_{n}:=e^{-4C_{\Gamma}n}$. Recall also that $\alpha>0$ is
fixed by Lemma 2.8. We fix $\beta\in(0,1)$ small enough so that $1-\beta\geq
e^{-4C_{\Gamma}\alpha}+\beta$, and we choose $\varepsilon_{0}$ so that
$r_{n}^{\varepsilon_{0}}=(1-\beta)^{n}$. By taking $\beta$ even smaller, we
can suppose that $\varepsilon_{0}<\delta_{reg}$. For this choice of
$\varepsilon_{0}$, and for
$C_{0}(\varepsilon_{0}):=2(1-\beta)^{-2}e^{4C_{\Gamma}\varepsilon_{0}}$, the
approximation lemma gives us a constant $A>1$ such that, under the hypothesis
of Lemma 3.9:
$\frac{\beta}{A^{2}}R\leq\frac{\beta}{A}P_{n+1}R\leq\beta R.$
We then use $P_{n}$ to successively take away some parts of $R$. Define, by
induction, $R_{0}:=1$ and
$R_{n+1}:=R_{n}-\frac{\beta}{A}P_{n+1}R_{n}\leq R_{n}.$
For the process to work as intended, we need to check that $R_{n}$ satisfies
the hypothesis of the approximation lemma.
###### Lemma 3.10.
Let $n\in\mathbb{N}$. The function $R_{n}$ is positive on $\Lambda_{\Gamma}$,
and for any $\xi,\eta\in\Lambda_{\Gamma}$:
1. 1.
If $d_{o}(\xi,\eta)\leq r_{n+1}$ then
$\left|\frac{R_{n}(\xi)}{R_{n}(\eta)}-1\right|\leq\frac{1}{2}d_{o}(\xi,\eta)^{\alpha}r_{n+1}^{-\alpha}$
2. 2.
If $d_{o}(\xi,\eta)>r_{n+1}$, then
$R_{n}(\xi)/R_{n}(\eta)\leq C_{0}(\varepsilon_{0})\cdot
d_{o}(\xi,\eta)^{\varepsilon_{0}}(1-\beta)^{-n}.$
###### Proof.
The proof goes by induction on $n$. The case $n=0$ is easy: the first point
holds trivially and the second holds since
$C_{0}(\varepsilon_{0})r_{1}^{\varepsilon_{0}}=C_{0}(\varepsilon_{0})e^{-4C_{\Gamma}\varepsilon_{0}}\geq
1$. Now, suppose that the result hold for some $n$. In this case, the
approximation lemma yields
$R_{n+1}=R_{n}-\frac{\beta}{A}P_{n}R\geq(1-\beta)R_{n},$
and in particular $R_{n+1}$ is positive. Let us prove the first point:
consider $\xi,\eta\in\Lambda_{\Gamma}$ such that $d_{o}(\xi,\eta)\leq
r_{n+2}$. Then Lemma 3.8 gives
$\left|\frac{\beta}{A}P_{n+1}R_{n}(\xi)-\frac{\beta}{A}P_{n+1}R_{n}(\eta)\right|\leq\frac{1}{2}\left(\frac{\beta}{A}P_{n+1}R_{n}(\eta)\right)\cdot
d_{o}(\xi,\eta)^{\alpha}r_{n+2}^{-\alpha}\leq\frac{1}{2}\beta R_{n}(\eta)\cdot
d_{o}(\xi,\eta)^{\alpha}r_{n+2}^{-\alpha}.$
Hence, using the induction hypothesis:
$\left|R_{n+1}(\xi)-R_{n+1}(\eta)\right|\leq\left|R_{n}(\xi)-R_{n}(\eta)\right|+\left|\frac{\beta}{A}P_{n+1}R_{n}(\xi)-\frac{\beta}{A}P_{n+1}R_{n}(\eta)\right|$
$\leq\frac{1}{2}\left(r_{n+1}^{-\alpha}+\beta
r_{n+2}^{-\alpha}\right)R_{n}(\eta)d_{o}(\xi,\eta)^{\alpha}$
$\leq\frac{1}{2}\frac{e^{-4C_{\Gamma}\alpha}+\beta}{1-\beta}R_{n+1}(\eta)d_{o}(\xi,\eta)^{\alpha}r_{n+2}^{-\alpha}.$
Recalling the definition of $\beta$ gives the desired bound. It remains to
prove the second point. First of all, notice that, for any $\xi$ and $\eta$,
we have:
$\frac{R_{n+1}(\xi)}{R_{n+1}(\eta)}\leq(1-\beta)^{-1}\frac{R_{n}(\xi)}{R_{n}(\eta)}.$
Now, suppose that $d_{o}(\xi,\eta)\in(r_{n+2},r_{n+1}]$. The induction
hypothesis gives $R_{n}(\xi)/R_{n}(\eta)\leq 1+|R_{n}(\xi)/R_{n}(\eta)-1|\leq
2$, and so:
$\frac{R_{n+1}(\xi)}{R_{n+1}(\eta)}\leq\frac{2}{1-\beta}=\frac{2}{1-\beta}r_{n+2}^{\varepsilon_{0}}(1-\beta)^{-(n+2)}\leq\frac{2}{(1-\beta)^{2}}\cdot
d_{o}(\xi,\eta)^{\varepsilon_{0}}(1-\beta)^{-{n+1}},$
which proves the bound. Finally, suppose that $d_{o}(\xi,\eta)>r_{n+1}$. In
this case, the induction hypothesis directly yields
$\frac{R_{n+1}(\xi)}{R_{n+1}(\eta)}\leq\frac{1}{1-\beta}\frac{R_{n}(\xi)}{R_{n}(\eta)}\leq
C_{0}(\varepsilon_{0})d_{o}(\xi,\eta)^{\varepsilon_{0}}(1-\beta)^{-(n+1)},$
and the proof is done. ∎
We are ready to prove Theorem 3.2, following Li in [LNP19].
###### Proof.
The previous lemma ensure that for all $n$, the function $R_{n}$ satisfies the
hypothesis of the approximation lemma. Hence, we can write for all $n$
$R_{n+1}=R_{n}-\frac{\beta}{A}P_{n+1}R_{n}\leq\left(1-\frac{\beta}{A^{2}}\right)R_{n},$
so that by induction:
$R_{n}\leq\left(1-\frac{\beta}{A^{2}}\right)^{n}\longrightarrow 0.$
It follows that
$1=R_{0}-\lim_{n}R_{n}=\sum_{n=1}^{\infty}(R_{n-1}-R_{n})=\frac{\beta}{A}\sum_{n=1}^{\infty}P_{n}(R_{n-1}),$
in other words:
$1=\sum_{n=1}^{\infty}\sum_{\gamma\in
S_{n}}\frac{\beta}{A}R_{n-1}(\eta_{\gamma})r_{\gamma}^{F}\cdot f_{\gamma}.$
Letting
$\nu(\gamma):=\frac{\beta}{A}R_{n-1}(\eta_{\gamma})r_{\gamma}^{F}\ \text{if}\
\gamma\in S_{n},\quad\nu(\gamma):=0\ \text{if}\ \gamma\neq\bigcup_{k}S_{k}$
gives us a probability measure on $\Gamma$ (since $\int f_{\gamma}d\mu_{o}=1$)
satisfying $\nu*\mu_{0}=\mu_{0}$, by the remarks made section 3.1. Checking
that the measure has exponential moment is easy since $\|\gamma\|\lesssim
r_{\gamma}^{-1}$ by Remark 2.4. Hence, by the shadow lemma
$r_{\gamma}^{F}\simeq\mu_{o}(B_{\gamma})$ and since $S_{n}$ covers each point
a bounded number of time, we get:
$\int_{\Gamma}\|\gamma\|^{\varepsilon}d\nu\lesssim\sum_{n}\sum_{\gamma\in
S_{n}}\nu(\gamma)r_{\gamma}^{-\varepsilon}$
$\lesssim\sum_{n}\left(\sum_{\gamma}\mu_{o}(B_{\gamma})\right)(1-\beta/A^{2})^{n}e^{\varepsilon
4C_{\Gamma}n}<\infty$
if $\varepsilon$ is small enough. Finally, we show that group $\Gamma_{\nu}$
spanned by the support of $\nu$ is $\Gamma$. To see this, say that
$C_{\Gamma}$ was chosen so large that $C_{\Gamma}\geq
6\text{diam}(\text{Hull}(\Lambda_{\Gamma})/\Gamma)$. In this case, there
exists $\gamma_{1}\in S_{1}$ such that $d(o,\gamma_{1}o)\in[|\ln
r_{1}|+C_{\Gamma}/2,|\ln r_{1}|+3C_{\Gamma}/2]$. Then, any $\gamma\in\Gamma$
such that $d(o,\gamma o)\leq C_{\Gamma}/2$ satisfies $\gamma_{1}\gamma\in
S_{1}$. In particular:
$\\{\gamma\in\Gamma\ ,\ d(o,\gamma o)\leq C_{\Gamma}/2\\}\subset\Gamma_{\nu},$
and it is then well known (see for example Lemma A.14 in [LNP19]) that this
set spans the whole group $\Gamma$ as soon as $C_{\Gamma}/2$ is larger than 3
times the diameter of $\text{Hull}({\Lambda_{\Gamma}})/\Gamma$. ∎
## 4 Consequences on equilibrium states
Now that we have proved Theorem 1.2, Corollary 1.3 follows directly from
[Li20] (since, when $d=2$, being Zariski dense is equivalent to being non-
elementary). To see how this statement induce some knowledge over equilibrium
states, let us recall more precisely the link between the latter and
Patterson-Sullivan densities. First, recall that the Hopf coordinates
$\text{Hopf}:\left((\partial_{\infty}\mathbb{H}^{2}\times\partial_{\infty}\mathbb{H}^{2})\setminus\mathcal{D}\right)\times\mathbb{R}\longrightarrow
T^{1}\mathbb{H}^{2}$
allows us to smoothly identify the unit tangent bundle of $\mathbb{H}^{2}$
with a torus minus the diagonal times $\mathbb{R}$ by the following process.
For any $v^{+}\neq v^{-}\in\partial_{\infty}\mathbb{H}^{2}$, and for any
$t\in\mathbb{R}$, $\text{Hopf}(v^{+},v^{-},t):=v$ is the unique vector $v\in
T^{1}M$ lying on the geodesic $]v^{-},v^{+}[$ such that
$\tilde{p}(\phi_{-t}(v))$ is the closest point to $o$ on this geodesic. We
will denote by $(\partial_{v^{+}},\partial_{v^{-}},\partial_{t})$ the induced
basis of $T(T^{1}M)$ in these coordinates. Finally, recall that $\iota$
denotes the flip map.
###### Theorem 4.1 ([PPS15], Theorem 6.1).
Let $\Gamma\subset Iso^{+}(\mathbb{H}^{d})$ be convex cocompact,
$M:=\mathbb{H}^{d}/\Gamma$, and $F:T^{1}M\rightarrow\mathbb{R}$ be a
normalized and Hölder-regular potential. Denote by
$m_{F}\in\mathcal{P}(T^{1}M)$ the associated equilibrium state, and let
$\tilde{m}_{F}$ be its $\Gamma$-invariant lift on $T^{1}\mathbb{H}^{d}$.
Denotes by $\mu_{x}^{F}$ the $(\Gamma,F)$ Patterson-Sullivan density with
basepoint $x$. Then, for any choice of $x\in\mathbb{H}^{d}$, the following
identity hold in the Hopf coordinates (up to a multiplicative constant
$c_{0}>0$):
$c_{0}\cdot
d\tilde{m}_{F}(v^{+},v^{-},t)=\frac{d\mu_{x}^{F}(v^{+})d\mu^{F\circ\iota}_{x}(v^{-})dt}{D_{F,x}(v^{+},v^{-})^{2}}.$
We are now ready to prove Fourier decay for $m_{F}$. To do a clean proof, we
write down three lemmas corresponding to Fourier decay in the three directions
$(\partial_{v^{+}},\partial_{v^{-}},\partial_{t})$. We will then combine all
of them to get the desired result.
###### Lemma 4.2.
Under the conditions of Theorem 1.2, there exists $\varepsilon>0$ such that
the following hold. Let $R\geq 1$ and let
$\chi:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}$ be a Hölder map supported on
some compact $K$. There exists $C\geq 1$ such that for any $C^{2}$ function
$\varphi:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}$ satisfying
$\|\varphi\|_{C^{2}}+(\inf_{K}|\partial_{v^{+}}\varphi|)^{-1}\leq R$, we have:
$\forall\xi\in\mathbb{R}^{*},\
\left|\int_{T^{1}\mathbb{H}^{2}}e^{i\xi\varphi(v)}\chi(v)d\tilde{m}_{F}(v)\right|\leq\frac{C}{|\xi|^{\varepsilon}}.$
###### Proof.
Denotes $\tilde{\varphi}$ and $\tilde{\chi}$ the functions $\varphi,\chi$ seen
in the Hopf coordinates. We get, for some large $a>0$ depending only on the
support of $\chi$:
$c_{0}\int_{T^{1}\mathbb{H}^{2}}e^{i\xi\varphi(v)}\chi(v)d\tilde{m}_{F}(v)=\int_{-a}^{a}\int_{\Lambda_{\Gamma}}\left(\int_{\Lambda_{\Gamma}}e^{i\xi\tilde{\varphi}(v^{+},v^{-},t)}\frac{\tilde{\chi}(v^{+},v^{-},t)}{D_{F,o}(v^{+},v^{-})^{2}}d\mu_{o}^{F}(v^{+})\right)d\mu_{o}^{F\circ\iota}(v^{-})dt.$
Now, since $\tilde{\chi}$ is supported in a compact subset of
$\left((\partial_{\infty}\mathbb{H}^{2}\times\partial_{\infty}\mathbb{H}^{2})\setminus\mathcal{D}\right)\times\mathbb{R}$,
and since $D_{F}$ is uniformly Hölder (and doesn’t vanish) on a compact subset
of $\Lambda_{\gamma}\times\Lambda_{\Gamma}\setminus\mathcal{D}$ (see [PPS15],
Lemma 3.6 and Proposition 3.5), and finally since
$\partial_{v^{+}}\tilde{\varphi}\neq 0$ on the compact support of
$\tilde{\chi}$, we see that Corollary 1.3 applies to the inner integral.
(Notice that we can always extend $D_{F}$ outside of
$\Lambda_{\Gamma}\times\Lambda_{\Gamma}\setminus\mathcal{D}$ so that it
becomes Holder on all
$(\partial_{\infty}\mathbb{H}^{2}\times\partial_{\infty}\mathbb{H}^{2})\setminus\mathcal{D}$,
see [Mc34].) This gives the desired bound. ∎
###### Lemma 4.3.
Under the conditions of Theorem 1.2, there exists $\varepsilon>0$ such that
the following hold. Let $R\geq 1$ and let
$\chi:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}$ be a Hölder map supported on
some compact $K$. There exists $C\geq 1$ such that for any $C^{2}$ function
$\varphi:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}$ satisfying
$\|\varphi\|_{C^{2}}+(\inf_{K}|\partial_{v^{-}}\varphi|)^{-1}\leq R$, we have:
$\forall\xi\in\mathbb{R}^{*},\
\left|\int_{T^{1}\mathbb{H}^{2}}e^{i\xi\varphi(v)}\chi(v)d\tilde{m}_{F}(v)\right|\leq\frac{C}{|\xi|^{\varepsilon}}.$
###### Proof.
We need to check that when $F$ satisfies the regularity assumptions (R), then
$F\circ\iota$ satisfies them too. This is easy, since
$\sup_{\Omega\Gamma}F\circ\iota=\sup_{\Omega\Gamma}F<\delta_{\Gamma,F}=\delta_{\Gamma,F\circ\iota}$
by Lemma 3.3 in [PPS15]. Moreover, $F\circ\iota$ is still Hölder regular.
Hence, one can apply our previous lemma with $F$ replaced by $F\circ\iota$,
and conclude. ∎
###### Lemma 4.4.
Under the conditions of Theorem 1.2, let $R\geq 1$ and let
$\chi:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}$ be a $\alpha$-Hölder map
supported on some compact $K$. There exists $C\geq 1$ such that, for any
$C^{2}$ function $\varphi:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}$ satisfying
$\|\varphi\|_{C^{2}}+(\inf_{K}|\partial_{t}\varphi|)^{-1}\leq R$, we have:
$\forall\xi\in\mathbb{R}^{*},\
\left|\int_{T^{1}\mathbb{H}^{2}}e^{i\xi\varphi(v)}\chi(v)d\tilde{m}_{F}(v)\right|\leq\frac{C}{|\xi|^{\alpha}}$
###### Proof.
The proof is classic. We have, for some compact
$\tilde{K}\subset\left(\partial_{\infty}\mathbb{H}^{2}\times\partial_{\infty}\mathbb{H}^{2}\right)\setminus\mathcal{D}$
and for some large enough $a>0$ depending only on the support of $\chi$:
$c_{0}\int_{T^{1}\mathbb{H}^{2}}e^{i\xi\varphi(v)}\chi(v)d\tilde{m}_{F}(v)=\iint_{\tilde{K}}\left(\int_{-a}^{a}e^{i\xi\tilde{\varphi}(v^{+},v^{-},t)}{\tilde{\chi}(v^{+},v^{-},t)}dt\right)D_{F,o}(v^{+},v^{-})^{-2}d(\mu_{o}^{F}\otimes\mu_{o}^{F\circ\iota})(v^{+},v^{-}).$
We then work on the inner integral. When $\tilde{\chi}$ is $C^{1}$, we can
conclude by an integration by parts. So a way to conclude is to approximate
$\tilde{\chi}$ by a $C^{1}$ map. Fix some smooth bump function
$\rho:\mathbb{R}\rightarrow\mathbb{R}^{+}$ such that $\rho$ is zero outside
$[-2,2]$, one inside $[-1,1]$, increasing on $[-2,-1]$ and decreasing on
$[1,2]$. For any $\varepsilon>0$, set
$\tilde{\chi}_{\varepsilon}(\cdot,\cdot,t):=\int_{\mathbb{R}}\tilde{\chi}(\cdot,\cdot,t-x)\rho(x/\varepsilon)dx/\varepsilon.$
This function is smooth on the $t$-variable. Moreover, if we denote by
$\alpha$ a Hölder exponent for $\chi$, then a direct computation yields:
$\|\tilde{\chi}_{\varepsilon}-\tilde{\chi}\|_{\infty}\lesssim\varepsilon^{\alpha},\quad\|\partial_{t}\tilde{\chi}_{\varepsilon}\|_{\infty}\lesssim\varepsilon^{-(1-\alpha)}.$
Hence:
$\left|\int_{-a}^{a}e^{i\xi\tilde{\varphi}}{\tilde{\chi}}dt\right|\leq
2a\|\tilde{\chi}-\tilde{\chi}_{\varepsilon}\|_{\infty}+\left|\int_{-a}^{a}e^{i\xi\tilde{\varphi}}{\tilde{\chi}_{\varepsilon}}dt\right|.$
To control the integral on the right, we do our aforementioned integration by
parts:
$\int_{-a}^{a}e^{i\xi\tilde{\varphi}}{\tilde{\chi}_{\varepsilon}}dt=\int_{-a}^{a}\frac{i\xi\partial_{t}\tilde{\varphi}}{i\xi\partial_{t}\tilde{\varphi}}e^{i\xi\tilde{\varphi}}{\tilde{\chi}_{\varepsilon}}dt$
$=\left[\frac{\tilde{\chi}_{\varepsilon}}{i\xi\partial_{t}\tilde{\varphi}}e^{i\xi\tilde{\varphi}}\right]_{t=-a}^{t=a}-\frac{i}{\xi}\int_{-a}^{a}\partial_{t}\left(\frac{\tilde{\chi}_{\varepsilon}}{\partial_{t}\tilde{\varphi}}\right)e^{i\xi\tilde{\varphi}}dt,$
so that
$\left|\int_{-a}^{a}e^{i\xi\tilde{\varphi}}{\tilde{\chi}_{\varepsilon}}dt\right|\lesssim|\xi|^{-1}{\varepsilon^{-(1-\alpha)}}.$
Finally, choosing $\varepsilon=1/|\xi|$ yields
$\left|\int_{T^{1}\mathbb{H}^{2}}e^{i\xi\varphi(v)}\chi(v)d\tilde{m}_{F}(v)\right|\lesssim\varepsilon^{\alpha}+\varepsilon^{-(1-\alpha)}|\xi|^{-1}\lesssim|\xi|^{-\alpha},$
which is the desired bound. ∎
###### Theorem 4.5.
Under the conditions of Theorem 1.2, there exists $\varepsilon>0$ such that
the following holds. Let $R\geq 1$ and let $\chi:T^{1}M\rightarrow\mathbb{R}$
be a Hölder map supported on some compact $K$. There exists $C\geq 1$ such
that, for any $C^{2}$ function $\varphi:T^{1}M\rightarrow\mathbb{R}$
satisfying $\|\varphi\|_{C^{2}}+(\inf_{K}\|d\varphi\|)^{-1}\leq R$, we have:
$\forall\xi\in\mathbb{R}^{*},\ \left|\int_{T^{1}M}e^{i\xi\varphi}\chi
dm_{F}\right|\leq\frac{C}{|\xi|^{\varepsilon}}$
###### Proof.
First of all, choose $\tilde{\chi}:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}$ a
lift of $\chi$ supported on a fundamental domain of $\Gamma$. Denote by
$\tilde{K}\subset T^{1}\mathbb{H}^{2}$ its (compact) support. Lift $\varphi$
to a $\Gamma$-invariant map
$\tilde{\varphi}:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}$. We then have:
$\int_{T^{1}M}e^{i\xi\varphi}\chi
dm_{F}=\int_{T^{1}\mathbb{H}^{2}}e^{i\xi\tilde{\varphi}}\tilde{\chi}d\tilde{m}_{F}.$
Now, consider the map
$\mathcal{B}_{\varphi}:T^{1}\mathbb{H}^{2}\rightarrow\mathbb{R}^{3}$ defined
by
$\mathcal{B}_{\varphi}(v):=\left((d\tilde{\varphi})_{v}(\partial_{v^{+}}),(d\tilde{\varphi})_{v}(\partial_{v^{-}}),(d\tilde{\varphi})_{v}(\partial_{t})\right)$.
Since $((\partial_{v^{+}})_{v},(\partial_{v^{-}})_{v},(\partial_{t})_{v})$ is
a basis of $T_{v}(T^{1}\mathbb{H}^{2})$ for any $v\in T^{1}\mathbb{H}^{2}$,
and since $d\tilde{\varphi}$ doesn’t vanish on $\tilde{K}$, we see that
$\mathcal{B}_{\varphi}(\tilde{K})\subset\mathbb{R}^{3}\setminus\\{0\\}$. By
uniform continuity of $\mathcal{B}_{\varphi}$ on the compact $\tilde{K}$, it
follows that there exists $c_{0}>0$ such that we can cover $\tilde{K}$ by a
finite union of compact balls $(B_{j})_{j\in J}$ satisfying:
$\forall j\in J,\ \exists e\in\\{v^{+},v^{-},t\\},\ \forall v\in B_{j},\
|\partial_{e}\tilde{\varphi}(v)|>c_{0}.$
To conclude, we consider a partition of unity $(\widehat{\chi}_{j})_{j}$
adapted to the cover $(B_{j})_{j}$, and we write:
$\int_{T^{1}\mathbb{H}^{2}}e^{i\xi\tilde{\varphi}}\tilde{\chi}d\tilde{m}_{F}=\sum_{j\in
J}\int_{B_{j}}e^{i\xi\tilde{\varphi}}\tilde{\chi}\widehat{\chi}_{j}d\tilde{m}_{F}.$
Each of the inner integrals is then controlled by either Lemma 4.2, Lemma 4.3
or Lemma 4.4. ∎
###### Remark 4.6.
We recover our main Theorem 1.4 as a particular case of Theorem 4.5. Indeed,
if $\varphi:K\subset T^{1}M\rightarrow\mathbb{R}^{3}$ is a $C^{2}$ local
chart, then for any $\zeta\in\mathbb{R}^{3}\setminus\\{0\\}$, one may write:
$\int_{T^{1}M}e^{i\zeta\cdot\varphi(v)}\chi(v)dm_{F}(v)=\int_{T^{1}M}e^{i|\zeta|(\zeta/|\zeta|)\cdot\varphi(v)}\chi(v)dm_{F}(v)\leq
C|\zeta|^{-\varepsilon},$
since the map $(u,v)\in\mathbb{S}^{2}\times K\mapsto u\cdot(d\varphi)_{v}\in
T_{v}^{*}(T^{1}M)$ doesn’t vanish (because the range of $(d\varphi)_{v}$ isn’t
contained in a plane). Notice that we used the uniformity of the constants
$C\geq 1$ given by the phases $u\cdot(d\varphi)$.
## Appendix A On the Fourier dimension
### A.1 The upper and lower Fourier dimension
We naturally want to make sense of the Fourier dimension of the non-wandering
set of the geodesic flow, so that we can write a sentence of the form:
$\dim_{F}\text{NW}(\phi)>0$. But since $NW(\phi)$ is a subset of an abstract
manifold, the usual definition doesn’t apply. In this appendix, we suggest
some definitions that one could choose to talk about the Fourier dimension of
a compact set lying in an abstract manifold.
First of all, recall that the Fourier dimension of a probability measure
$\mu\in\mathcal{P}(E)$, supported on some compact set
$E\subset\mathbb{R}^{d}$, can be defined as:
$\dim_{F}(\mu):=\sup\\{\alpha\geq 0\ |\ \exists C\geq
1,\forall\xi\in\mathbb{R}^{d}\setminus\\{0\\},\ |\widehat{\mu}(\xi)|\leq
C|\xi|^{-\alpha/2}\\},$
where the Fourier transform of $\mu$ is defined by
$\widehat{\mu}(\xi):=\int_{E}e^{-2i\pi\xi\cdot x}d\mu(x).$
The Fourier dimension of a compact set $E\subset\mathbb{R}^{d}$ is then
defined as
$\dim_{F}(E):=\sup\\{\min(d,\dim_{F}\mu),\
\mu\in\mathcal{P}(E)\\}\leq\dim_{H}E.$
To define the Fourier dimension of a measure lying in a abstract manifold, a
natural idea is to look at our measure into local charts. But this suppose
that we have a meaningful way to localize the usual definition of the Fourier
dimension. This is the content of the next well known lemma.
###### Lemma A.1.
Let $E\subset\mathbb{R}^{d}$ be a compact set. Let $\mu\in\mathcal{P}(E)$. Let
$\varepsilon>0$. Denote by $\text{Bump}(\varepsilon)$ the set of smooth
functions with support of diameter at most $\varepsilon$. Then:
$\dim_{F}\mu=\inf\\{\dim_{F}(\chi d\mu)\ |\
\chi\in\text{Bump}(\varepsilon)\\}.$
###### Proof.
Let $E\subset\mathbb{R}^{d}$ be a fixed compact set, and let
$\mu\in\mathcal{P}(E)$ be a fixed (borel) probability measure supported on
$E$. First of all, consider a finite covering of the compact set $E$ by balls
$(B_{i})_{i\in I}$ of radius $\varepsilon$. Consider an associated partition
of unity $(\chi_{i})_{i\in I}$. Then, for all $\alpha<\inf_{\chi}\dim_{F}(\chi
d\mu)$, there exists $C\geq 1$ such that:
$|\widehat{\mu}(\xi)|=\left|\sum_{i\in
I}\widehat{\chi_{i}d\mu}(\xi)\right|\leq C|\xi|^{-\alpha}.$
Hence $\dim_{F}\mu\geq\alpha$. Since this hold for any
$\alpha<\inf\\{\dim_{F}(\chi d\mu)\ |\ \chi\in\text{Bump}(\varepsilon)\\}$,
this yields $\dim_{F}\mu\geq\inf\\{\dim_{F}(\chi d\mu)\ |\
\chi\in\text{Bump}(\varepsilon)\\}.$ Now we prove the other inequality.
Fix some smooth function with compact support $\chi$. Its Fourier transform
$\widehat{\chi}$ is in the Schwartz class: in particular, for all $N\geq d+1$,
there exists $C_{N}$ such that $\widehat{\chi}(\eta)\leq C_{N}|\eta|^{-N}$ for
all $\eta\in\mathbb{R}^{d}\setminus\\{0\\}$. Let
$\alpha<\alpha^{\prime}<\dim_{F}\mu$. Then there exists $C\geq 1$ such that
$|\widehat{\mu}(\xi)|\leq C|\xi|^{-\alpha^{\prime}}$ for all
$\xi\in\mathbb{R}^{d}\setminus\\{0\\}$. Now, notice that:
$\widehat{\chi
d\mu}(\xi)=\widehat{\chi}*\widehat{\mu}(\xi)=\int_{\mathbb{R}^{d}}\widehat{\chi}(\eta)\widehat{\mu}(\xi-\eta)d\eta.$
We cut the integral in two parts, depending on some radius $r>0$ that we
choose to be $r:=|\xi|^{1-\varepsilon}$, where
$\varepsilon:=1-\alpha/\alpha^{\prime}$. We suppose that $|\xi|\geq 2$. In
this case, a direct computation show that whenever $\eta\in B(0,r)$, we have
$|\xi|^{1-\varepsilon}\leq C|\xi-\eta|$. We are finally ready to conclude our
computation:
$\left|\widehat{\chi
d\mu}(\xi)\right|\leq\left|\int_{B(0,r)}\widehat{\chi}(\eta)\widehat{\mu}(\xi-\eta)d\eta\right|+\left|\int_{B(0,r)^{c}}\widehat{\chi}(\eta)\widehat{\mu}(\xi-\eta)d\eta\right|$
$\lesssim_{N}\int_{\mathbb{R}^{d}}|\widehat{\chi}(\eta)|d\eta\cdot\frac{C}{|\xi|^{(1-\varepsilon)\alpha^{\prime}}}+\int_{B(0,r)^{c}}\frac{1}{|\eta|^{N}}d\eta$
$\lesssim_{N}\frac{1}{|\xi|^{\alpha}}+r^{N-d}\int_{B(0,1)^{c}}\frac{1}{|\zeta|^{N}}d\zeta\lesssim\frac{1}{|\xi|^{\alpha}}$
if $N$ is choosen large enough. It follows that $\dim_{F}(\chi
d\mu)\geq\alpha$, and this for any $\alpha<\dim_{F}\mu$, so $\dim_{F}(\chi
d\mu)\geq\dim_{F}(\mu)$. Taking the infimum in $\chi$ yields the desired
inequality. ∎
Now we understand how the Fourier dimension of a measure $\mu$ can be computed
by looking at the local behavior of $\mu$. But another, much harder problem
arise now: the Fourier dimension of a measure depends very much on the
embedding of this measure in the ambiant space. In concrete terms, the Fourier
dimension is not going to be independant on the choice of local charts. A way
to introduce an "intrinsic" quantity related to the Fourier dimension of a
measure would be to take the supremum or the infimum under all those charts.
We directly give our definition in the context of a manifold.
###### Definition A.2.
Let $M$ be a smooth manifold of dimension $d$. Let $E\subset M$ be a compact
set. Let $\mu\in\mathcal{P}(E)$. Let $k\in\mathbb{N}^{*}$. Let
$\text{Bump}(E)$ denote the set of all smooth functions
$\chi:M\rightarrow\mathbb{R}$ such that $\text{supp}(\chi)$ is contained in a
local chart. We denote by $\text{Chart}(\chi,C^{k})$ the set of all $C^{k}$
local charts $\varphi:U\rightarrow\mathbb{R}^{d}$, where
$U\supset\text{supp}(\chi)$ is an open set containing the support of $\chi$.
Now, define the lower Fourier dimension of $\mu$ by $C^{k}$ charts of $M$ by:
$\underline{\dim}_{F,C^{k}}(\mu):=\inf_{\chi\in\text{Bump}(E)}\inf\\{\dim_{F}(\varphi_{*}(\chi
d\mu)),\ \varphi\in\text{Chart}(\chi,C^{k})\\}.$
Similarly, define the upper Fourier dimension of $\mu$ by $C^{k}$ charts of
$M$ by:
$\overline{\dim}_{F,C^{k}}(\mu):=\inf_{\chi\in\text{Bump}(E)}\sup\\{\dim_{F}(\varphi_{*}(\chi
d\mu)),\ \varphi\in\text{Chart}(\chi,C^{k})\\}.$
###### Definition A.3.
Let $M$ be a smooth manifold of dimension $d$. Let $E\subset M$ be a compact
set. Let $\mu\in\mathcal{P}(E)$. We define the lower Fourier dimension of
$\mu$ by:
$\underline{\dim}_{F}(\mu)=\underline{\dim}_{F,C^{\infty}}(\mu).$
###### Remark A.4.
The lower Fourier dimension test if, for any localization $\chi d\mu$ of
$\mu$, and for any smooth local chart $\varphi$, one has some decay of the
Fourier transform of $\varphi_{*}(\chi d\mu)$. We then take the infimum of all
the best decay exponents. This quantity is $C^{\infty}$-intrinsic in the
following sense: if $\Phi:M\rightarrow M$ is a $C^{\infty}$-diffeomorphism,
then $\underline{\dim}_{F}(\Phi_{*}\mu)=\underline{\dim}_{F}(\mu)$.
Symetrically, the $C^{k}$-upper Fourier dimension test if, for any
localization $\chi d\mu$ of $\mu$, there exists a $C^{k}$-chart $\varphi$ such
that one has some decay for the Fourier transform of $\varphi_{*}(\chi d\mu)$.
This quantity is also $C^{\infty}$-intrinsic. Still, beware that the upper and
lower Fourier dimensions depends on the dimension of the ambiant manifold.
###### Remark A.5.
Let $E\subset M$ be a compact set lying in a manifold $M$ of dimension $d$.
Fix a bump function $\chi$ and a local chart
$\varphi\in\text{Chart}(\chi,C^{k})$. For $\mu\in\mathcal{P}(E)$ a measure
supported in $E\subset M$, we have the following bounds:
$0\leq\underline{\dim}_{F,C^{k}}\mu\leq\dim_{F}\varphi_{*}(\chi
d\mu)\leq\overline{\dim}_{F,C^{k}}\mu.$
Moreover, if $\dim_{H}E<d$, then:
$\overline{\dim}_{F,C^{k}}\mu\leq\dim_{H}E.$
###### Example A.6.
Let $M$ be a manifold of dimension $d$, and consider any smooth hypersurface
$N\subset M$. Let $k\geq 1$. Let $\mu$ be any smooth and compactly supported
measure on $N$. Then:
$\underline{\dim}_{F,C^{k}}(\mu)=0,\quad\overline{\dim}_{F,C^{k}}(\mu)=d-1.$
The first fact is easily proved by noticing that, locally, $N$ is
diffeomorphic to a linear subspace of $\mathbb{R}^{d}$, which has zero Fourier
dimension. The second fact is proved by noticing that, locally, $N$ is
diffeomorphic to a half sphere, and any smooth measure supported on the half
sphere exhibit power decay of its Fourier transform with exponent $(d-1)/2$.
###### Remark A.7.
It seems that, for some well behaved measures $\mu\in\mathcal{P}(E)$ supported
on compacts $E$ with $\dim_{H}E<d$, one might expect the quantity
$\overline{\dim}_{F,C^{k}}\mu$ is be comparable to $\dim_{H}E$. For some
measures lying in a 1-dimensionnal curve, this is the content of Theorem 2 in
[Ek16].
###### Remark A.8.
Using this langage, the results of [BD17], [LNP19], [SS20], [Le21] and [Li20]
all implies positivity of the lower Fourier dimension by $C^{2}$ charts of
some measures (respectively: Patterson-Sullivan measures, Patterson-Sullivan
measures, equilibrium states, equilibrium states, and stationary measures).
This is a bit stronger than a related notion found in [LNP19], namely the
$C^{2}$-stable positivity of the Fourier dimension. The results in our paper
implies the following: under the conditions of Theorem 1.2, the equilibrium
state $m_{F}\in\mathcal{P}(NW(\phi))$ satisfies
$\underline{\dim}_{F,C^{2}}(m_{F})>0,$
where the non-wandering set $NW(\phi)$ of the geodesic flow is seen in the
unit tangent bundle $T^{1}M$. In particular, its lower Fourier dimension is
positive.
### A.2 A variation with real valued phases
For completeness, we suggest two variations for intrinsic notions of Fourier
dimension for a measure in an abstract manifold. The first is exposed in this
subsection, and the next will be discussed in the next subsection. Inspired by
the computations made in section 4, we may want to look at more general
oscillatory integrals involving $\mu$. A possibility is the following.
###### Definition A.9.
Let $M$ be a smooth manifold of dimension $d$. Let $E\subset M$ be a compact
set. Let $\mu\in\mathcal{P}(E)$. Let $k\in\mathbb{N}^{*}$. Let
$\text{Bump}(E)$ denote the set of all smooth functions
$\chi:M\rightarrow\mathbb{R}$ such that $\text{supp}(\chi)$ is contained in a
local chart. We denote by $\text{Phase}(\chi,C^{k})$ the set of all real
valued and $C^{k}$ maps $\psi:U\rightarrow\mathbb{R}$ with nonvanishing
differential, where $U\supset\text{supp}(\chi)$ is an open set containing the
support of $\chi$. Now, define the lower Fourier dimension of $\mu$ by $C^{k}$
phases of $M$ by:
$\underline{\dim}_{F,C^{k}}^{\text{real}}(\mu):=\inf_{\chi\in\text{Bump}(E)}\inf\\{\dim_{F}(\psi_{*}(\chi
d\mu)),\ \psi\in\text{Phase}(\chi,C^{k})\\}.$
Similarly, define the upper Fourier dimension of $\mu$ by $C^{k}$ phases of
$M$ by:
$\overline{\dim}_{F,C^{k}}^{\text{real}}(\mu):=\inf_{\chi\in\text{Bump}(E)}\sup\\{\dim_{F}(\psi_{*}(\chi
d\mu)),\ \psi\in\text{Phase}(\chi,C^{k})\\}.$
As before, we also denote
$\underline{\dim}_{F}^{\text{real}}(\mu):=\underline{\dim}_{F,C^{\infty}}^{\text{real}}(\mu)$.
###### Remark A.10.
First of all, notice that $\psi_{*}(\chi d\mu)$ is a measure supported in
$\mathbb{R}$, so its Fourier transform is a function from $\mathbb{R}$ to
$\mathbb{C}$. More precisely:
$\forall t\in\mathbb{R},\ \widehat{\psi_{*}(\chi
d\mu)}(t):=\int_{E}e^{it\psi(x)}\chi(x)d\mu(x).$
Like before, the lower/upper Fourier dimensions with real phases are
$C^{\infty}$-intrinsic in the sense that for any $C^{\infty}$-diffeomorphism
$\Phi:M\rightarrow M$, we have
$\underline{\dim}_{F,C^{k}}^{\text{real}}(\Phi_{*}\mu)=\underline{\dim}_{F,C^{k}}^{\text{real}}(\mu)$
and
$\overline{\dim}_{F,C^{k}}^{\text{real}}(\Phi_{*}\mu)=\overline{\dim}_{F,C^{k}}^{\text{real}}(\mu)$.
###### Example A.11.
Let $M$ be a smooth manifold, and let $N$ be a smooth submanifold of $M$. Let
$\mu$ be a smooth and compactly supported probability measure in $N$. Then:
$\underline{\dim}_{F,C^{k}}^{\text{real}}(\mu)=0,\quad\overline{\dim}_{F,C^{k}}^{\text{real}}(\mu)=\infty.$
These equalities can be proved as follow. Consider some smooth bump function
$\chi$ with small enough support. Now, there exists a phase $\psi$, defined on
a neighborhood $U$ of $\text{supp}(\chi)$, with nonvanishing differential on
$U$ but which is constant on $N$. The associated oscillatory integral
$\widehat{\psi_{*}(\chi d\mu)}$ doesn’t decay, hence the computation on the
lower Fourier dimension with real phases. There also exists smooth a phase
$\psi$ such that $(d\psi)_{|TN}$ doesn’t vanish. By the non-stationnary phase
lemma, the associated oscillatory integral decay more than $t^{-N}$, for any
$N\geq 0$, hence the computation on the upper Fourier dimension with real
phases.
Notice how, in particular, $\min(\overline{\dim}_{F,C^{k}}^{real}(\mu),d)$ may
be strictly larger than the Hausdorff dimension of the support of $\mu$. This
may be a sign that this variation of the upper dimension isn’t well behaving
as a Fourier dimension.
###### Lemma A.12.
We can compare this Fourier dimension with the previous one. We have:
$\underline{\dim}_{F,C^{k}}(\mu)\leq\underline{\dim}_{F,C^{k}}^{real}(\mu),\quad\overline{\dim}_{F,C^{k}}(\mu)\leq\overline{\dim}_{F,C^{k}}^{real}(\mu).$
###### Proof.
Let $\alpha<\underline{\dim}_{F,C^{k}}(\mu)$. Then, for any bump function
$\chi$ and for any associated local chart $\varphi$, there exists some
constant $C$ such that, for all $\xi\in\mathbb{R}^{d}\setminus\\{0\\}$, we
have $|\widehat{\varphi_{*}(\chi d\mu)}(\xi)|\leq C|\xi|^{-\alpha/2}$. Now fix
$\psi:U\rightarrow\mathbb{R}$ with nonvanishing differential, where
$\text{supp}(\chi)\subset U$. By the submersion theorem, there exists a local
chart $\varphi:U\rightarrow\mathbb{R}^{d}$ such that
$\varphi(x)=\psi(x)e_{1}+\sum_{j=2}^{d}f_{j}(x)e_{j}$ (where $(e_{i})_{i}$ is
the canonical basis of $\mathbb{R}^{d}$, and where $f_{j}$ are some real
valued functions). Hence, one can write:
$|\psi_{*}(\chi d\mu)(t)|=|\varphi_{*}(\chi d\mu)(te_{1})|\leq
C|t|^{-\alpha/2}.$
Hence $\underline{\dim}_{F,C^{k}}^{real}(\mu)\geq\alpha$, and this for any
$\alpha<\underline{\dim}_{F,C^{k}}(\mu)$, hence the desired bound.
The second bound is proved as follow. Let
$\alpha<\overline{\dim}_{F,C^{k}}(\mu)$. Let $\chi$ be a small bump function.
There there exists a local chart $\varphi:U\rightarrow\mathbb{R}^{d}$, with
$U\supset\text{supp}(\chi)$, such that $\widehat{\varphi_{*}(\chi
d\mu)}\lesssim|\xi|^{-\alpha/2}$. Let $u\in\mathbb{S}^{d-1}$ and consider
$\psi(x):=u\cdot\varphi(x)$. It is easy to check that $\psi$ has nonvanishing
differential, and since, for any $t\in\mathbb{R}\setminus\\{0\\}$,
$|\widehat{\psi_{*}(\chi d\mu)}(t)|=|\widehat{\varphi_{*}(\chi
d\mu)}(ut)|\lesssim|t|^{-\alpha/2},$
we get $\overline{\dim}_{F,C^{k}}^{real}(\mu)\geq\alpha$. The bound follow. ∎
In concrete cases, we expect the lower Fourier dimension and the lower Fourier
dimension with real phases to be equal. Unfortunately, our choices of
definitions doesn’t clearly make that happen all the time. We have to add a
very natural assumption for the equality to hold.
###### Definition A.13.
Let $\mu\in\mathcal{P}(E)$, where $E\subset M$ is a compact subset of a smooth
manifold. We say that $\mu$ admits reasonnable constants for $C^{k}$-phases
if, for any $\alpha<\underline{\dim}^{\text{real}}_{F,C^{k}}(\mu)$, and for
any $\chi\in\text{Bump}(E)$, the following hold:
$\forall R\geq 1,\ \exists C_{R}\geq 1,\
\forall\psi\in\text{Phase}(\chi,C^{k}),$ $\left(\|\psi\|_{C^{k}}+\sup_{x\in
U}\|(d\psi)_{x}\|^{-1}\leq R\right)\Longrightarrow\left(\forall
t\in\mathbb{R}^{*},\ |\widehat{\psi_{*}(\chi d\mu)}(t)|\leq
C_{R}t^{-\alpha/2}\right).$
Under this natural assumption, we have equality of the lower Fourier
dimensions.
###### Lemma A.14.
Let $\mu\in\mathcal{P}(E)$, where $E\subset M$ is a compact subset of some
smooth manifold $M$. Suppose that $\mu$ admits reasonnable constants for
$C^{k}$-phases. Then:
$\underline{\dim}_{F,C^{k}}(\mu)=\underline{\dim}_{F,C^{k}}^{real}(\mu)$
###### Proof.
An inequality is already known, we just have to prove the second one. The
proof of the other inequality is the same argument as the one explained in
Remark 4.6. ∎
### A.3 A directionnal variation
A second natural and intrinsic idea would be to fix some (spatial) direction
on which to look for Fourier decay. We quickly discuss these notions and then
we will move on to discuss some notions of Fourier dimensions for sets.
###### Definition A.15.
Let $E\subset M$ be a compact set in some smooth manifold. Let $V\subset TM$
be a continuous vector bundle on an open neighborhood $\tilde{E}$ of $E$.
Denote by $\text{Bump}^{V}(E)$ the set of all smooth bump functions with
support included in $\tilde{E}$, and included in some local chart. For some
$\chi\in\text{Bump}^{V}(E)$, denote by $\text{Phase}^{V}(\chi,C^{k})$ the set
of all $C^{k}$ maps $\psi:U\rightarrow\mathbb{R}$ such that $(d\psi)_{|V}$
doesn’t vanish on $U$, where $\text{supp}(\chi)\subset U\subset\tilde{E}$ is
some open set.
For $\mu\in\mathcal{P}(E)$, we define its lower Fourier dimension in the
direction $V$ for $C^{k}$ phases by:
$\underline{\dim}_{F,C^{k}}^{V}(\mu):=\inf_{\chi\in\text{Bump}^{V}(E)}\inf\\{\dim_{F}(\psi_{*}(\chi
d\mu)),\ \psi\in\text{Phase}^{V}(\chi,C^{k})\\}.$
Similarly, define its upper Fourier dimension in the direction $V$ for $C^{k}$
phases by:
$\overline{\dim}_{F,C^{k}}^{V}(\mu):=\inf_{\chi\in\text{Bump}^{V}(E)}\sup\\{\dim_{F}(\psi_{*}(\chi
d\mu)),\ \psi\in\text{Phase}^{V}(\chi,C^{k})\\}.$
###### Remark A.16.
Again, these notions of Fourier dimensions are $C^{\infty}$-intrinsic, in the
following sense: if $\Phi:M\rightarrow M$ is a $C^{k}$-diffeomorphism of $M$,
then
$\underline{\dim}_{F,C^{k}}^{\Phi_{*}V}(\Phi_{*}\mu)=\underline{\dim}_{F,C^{k}}^{V}(\mu)$,
and
$\overline{\dim}_{F,C^{k}}^{\Phi_{*}V}(\Phi_{*}\mu)=\overline{\dim}_{F,C^{k}}^{V}(\mu)$.
###### Remark A.17.
With these notations, the results found in [Le23] implies that, for any
nonlinear and sufficiently bunched solenoid $S$, and for any equilibrium state
$\mu$, one has $\underline{\dim}_{F,C^{1+\alpha}}^{E_{u}}(\mu)>0$, where
$E_{u}$ is the unstable line bundle associated to the dynamics on the
solenoid.
###### Lemma A.18.
Let $V_{1},\dots,V_{n}\subset TM$ be some continuous vector bundles defined on
some open neighborhood $\tilde{E}$ of $E$. Suppose that
$(V_{1})_{p}+\dots(V_{n})_{p}=T_{p}M$ for all $p\in\tilde{E}$. Then:
$\min_{j}\underline{\dim}_{F,C^{k}}^{V_{j}}(\mu)=\underline{\dim}_{F,C^{k}}^{\text{real}}(\mu),\quad\max_{j}\overline{\dim}_{F,C^{k}}^{V_{j}}(\mu)\leq\overline{\dim}_{F,C^{k}}^{\text{real}}(\mu).$
###### Proof.
Let $\alpha<\underline{\dim}_{F,C^{k}}^{\text{real}}(\mu)$. Then, for any bump
$\chi$ and associated phase $\psi$, one has $\widehat{\psi_{*}(\chi
d\mu)}(t)\lesssim|t|^{-\alpha/2}$. In paticular, for any phase
$\phi_{j}\in\text{Phase}^{V_{j}}(\chi,C^{k})$, the previous decay holds, and
so $\min_{j}\underline{\dim}_{F,C^{k}}^{V_{j}}(\mu)\geq\alpha$. Hence
$\min_{j}\underline{\dim}_{F,C^{k}}^{V_{j}}(\mu)\geq\underline{\dim}_{F,C^{k}}^{\text{real}}(\mu)$.
Now let $\alpha<\min_{j}\underline{\dim}_{F,C^{k}}^{V_{j}}(\mu)$. Then, for
all $j$, for any bump $\chi$, and for any phase
$\psi_{j}\in\text{Phase}^{V_{j}}(\chi,C^{k})$, the previous decay applies.
Now, if we fix some $\chi$ and some associated phase
$\psi\in\text{Phase}(\chi,C^{k})$, we know that at each point $p$,
$(d\psi)_{p}$ is nonzero. In particular, there exists $j(p)$ such that
$(d\psi)_{|V^{j(p)}_{p}}\neq 0$. Following the proof of Theorem 4.5, we can
show by using a partition of unity that this implies $\widehat{\psi_{*}(\chi
d\mu)}(t)\lesssim|t|^{-\alpha/2}$. Hence
$\underline{\dim}_{F,C^{k}}^{\text{real}}(\mu)\geq\alpha$, and we have prove
equality.
For our last bound, let $\alpha<\max_{j}\overline{\dim}_{F,C^{k}}^{V_{j}}\mu$.
Then there exists $j$ such that, for all bump $\chi$, there exists an
associated phase $\psi_{j}\in\text{Phase}^{V_{j}}(\chi,C^{k})$ such that
$\widehat{\psi_{j}(\chi d\mu)}(t)\lesssim|t|^{-\alpha/2}$. Since
$\psi_{j}\in\text{Phase}(\chi,C^{k})$, we get
$\overline{\dim}_{F,C^{k}}^{\text{real}}\mu\geq\max_{j}\overline{\dim}_{F,C^{k}}^{V_{j}}\mu$.
∎
###### Remark A.19.
The reverse bound for the upper dimensions is not clear: if for all bump
functions $\chi$, there exists a phase $\psi$ with good fourier decay
properties for $\mu$, then nothing allows us to think that $\psi$ is going to
have nonvanishing diffenrential in some fixed $V_{j}$ on all $E$.
### A.4 What about sets ?
We finally define some intrinsic notions of Fourier dimensions for sets. First
of all, recall that the usual definition for some $E\subset\mathbb{R}^{d}$ is:
$\dim_{F}(E):=\sup\\{\dim_{F}(\mu)\leq d\ ,\
\mu\in\mathcal{P}(E)\\}\leq\dim_{H}(E).$
In particular, in view of the proof of Lemma A.1, we see that any measure
$\mu$ with some Fourier decay properties may be localized anywhere on its
support to still yield a measure with large Fourier dimension. Hence we find
the following localized formula, for any $\varepsilon>0$:
$\dim_{F}(E)=\underset{U\text{ open}}{\sup_{U\cap
E\neq\emptyset}}\dim_{F}(E\cap U).$
Now, we have two main ways to define the up(per) and low(er) Fourier dimension
of a compact set in a manifold: directly computing the Fourier dimension of
parts of $E$ in local charts, or taking the sup over the previously defined
notions of Fourier dimension for measures.
###### Definition A.20.
Let $E\subset M$ be a compact set included in some smooth manifold. We define
its lower/upper Fourier dimension with $C^{k}$-charts by:
$\underline{\dim}_{F,C^{k}}(E):=\sup\\{\underline{\dim}_{F,C^{k}}(\mu)\leq d,\
\mu\in\mathcal{P}(E)\\},\quad\overline{\dim}_{F,C^{k}}(E):=\sup\\{\overline{\dim}_{F,C^{k}}(\mu)\leq
d,\ \mu\in\mathcal{P}(E)\\},$
We also define the $C^{k}$-low Fourier dimension and $C^{k}$-up Fourier
dimensions of $E$ by:
$\uwave{\dim}{}_{F,C^{k}}(E):=\underset{U\text{ open chart}}{\sup_{U\cap
E\neq\emptyset}}\inf\\{\dim_{F}(\varphi(E\cap U))\ ,\
\varphi:U\rightarrow\mathbb{R}^{d}\ C^{k}\text{ local chart}\\},$
${{\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}}_{F,C^{k}}(E):=\underset{U\text{ open
chart}}{\sup_{U\cap E\neq\emptyset}}\sup\\{\dim_{F}(\varphi(E\cap U))\ ,\
\varphi:U\rightarrow\mathbb{R}^{d}\ C^{k}\text{ local chart}\\}.$
###### Remark A.21.
The low and up Fourier dimension are $C^{k}$-intrinsic in the natural sense.
For exemple, if $\Phi:M\rightarrow M$ is a $C^{k}$-diffeomorphism, then
$\uwave{\dim}{}_{F,C^{k}}(\Phi(E))=\uwave{\dim}{}_{F,C^{k}}(E)$. The lower and
upper Fourier dimension are $C^{\infty}$-intrinsic.
###### Lemma A.22.
Let $E\subset M$ be a compact set in some smooth manifold $M$. Then:
$0\leq\underline{\dim}_{F,C^{k}}(E)\leq\uwave{\dim}{}_{F,C^{k}}(E)\leq{{\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}}{}_{F,C^{k}}(E)=\overline{\dim}_{F,C^{k}}(E)\leq\dim_{H}(E)\leq
d.$
###### Proof.
Let us prove all the inequalities in order, from left to right.
$0\leq\underline{\dim}_{F,C^{k}}(E)$ is trivial. Let us prove the second one.
Let $\alpha<\underline{\dim}_{F,C^{k}}(E)$. By definition, there exists some
probability measure $\mu\in\mathcal{P}(E)$ such that
$\underline{\dim}_{F,C^{k}}(\mu)\geq\alpha$. Now, since the support of $\mu$
is nonempty, there exists $U$ some small open set and a bump function $\chi$
supported in $U$ such that $\chi d\mu$ is a (localized) nonzero measure. Let
$\varphi:U\rightarrow\mathbb{R}^{d}$ a local chart. Then, by hypothesis on
$\mu$, $\dim_{F}\varphi_{*}(\chi d\mu)\geq\alpha$. In particular, since (up to
normalization) $\varphi_{*}(\chi d\mu)\in\mathcal{P}(\varphi(E\cap U))$, we
have $\dim_{F}\varphi(E\cap U)\geq\alpha$. This for any local chart $\varphi$,
and so $\inf_{\varphi}\dim_{F}(\varphi(E\cap U))\geq\alpha$. This yields
$\underline{\dim}_{F,C^{k}}(E)\geq\alpha$. Since this is true for any
$\alpha<\underline{\dim}_{F,C^{k}}(E)$, we get the desired inequality.
The inequality
$\uwave{\dim}{}_{F,C^{k}}(E)\leq{{\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}}{}_{F,C^{k}}(E)$ is trivial. Let us
prove the equality between
${{\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}}{}_{F,C^{k}}(E)$ and
$\overline{\dim}_{F,C^{k}}(E)$. Let
$\alpha<{{\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}}{}_{F,C^{k}}(E)$. Then, there exists
some small open set $U$ (such that $U\cap E\neq\emptyset$ and a local chart
$\varphi:U\rightarrow\mathbb{R}^{d}$ such that $\dim_{F}(\varphi(U\cap
E))\geq\alpha$. By definition, it means that there exists some measure
$\nu\in\mathcal{P}(\varphi(E\cap U))$ such that $\dim_{F}\nu\geq\alpha$.
Letting $\mu:=\varphi^{-1}_{*}\nu\in\mathcal{P}(E\cap U)$ yields a measure
supported in $E$ that satisfies $\overline{\dim}_{F,C^{k}}(\mu)\geq\alpha$ (in
view of the proof of Lemma A.1). Hence,
$\overline{\dim}_{F,C^{k}}(E)\geq\alpha$. This, for any
$\alpha<{{\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}}{}_{F,C^{k}}(E)$, so that
$\overline{\dim}_{F,C^{k}}(E)\geq{{\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}}{}_{F,C^{k}}(E)$.
Let us prove the other inequality. Let $\alpha<\overline{\dim}_{F,C^{k}}(E)$.
By definition, there exists $\mu\in\mathcal{P}(E)$ such that
$\overline{\dim}_{F,C^{k}}(\mu)\geq\alpha$. Now let $U$ be some small open set
with $\mu_{|U}\neq 0$, and let $\varphi:U\rightarrow\mathbb{R}^{d}$ be a local
chart. Let $\chi$ be some bump function supported in $U$. Then, by hypothesis
on $\mu$, we have $\dim_{F}\varphi_{*}(\chi d\mu)\geq\alpha$. In particular,
$\dim_{F}(\varphi(E\cap U))\geq\alpha.$ Hence
${\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}_{F,C^{k}}(E)\geq\alpha$. This proves
the other inequality, and hence concludes the proof that
${{\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}}{}_{F,C^{k}}(E)=\overline{\dim}_{F,C^{k}}(E)$.
Finally, the fact that the Hausdorff dimension is invariant under
$C^{1}$-diffeomorphisms implies
${\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}_{F,C^{k}}(E)=\sup_{U}\sup_{\varphi}\dim_{F}(\varphi(U\cap
E))\leq\sup_{U}\sup_{\varphi}\dim_{H}(\varphi(E\cap U))=\sup_{U}\dim_{H}(E\cap
U)=\dim_{H}(E)\leq d.$
∎
###### Example A.23.
Let $N\subset M$ be a hypersurface in some smooth manifold $M$. Then:
$\underline{\dim}_{F,C^{k}}(N)=\uwave{\dim}{}_{F,C^{k}}(N)=0\quad,{\mathchoice{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\displaystyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\textstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptstyle\dim$\cr}}}{\vbox{\offinterlineskip\halign{\hfil#\hfil\cr$\scriptscriptstyle\rotatebox{180.0}{\uwave{}}$\cr\vskip
0.2pt\cr$\scriptscriptstyle\dim$\cr}}}}_{F,C^{k}}(N)=\overline{\dim}_{F,C^{k}}(N)=\dim_{H}(N)=d-1.$
###### Example A.24.
We can finally state the result that we wanted to state. Let $M$ be a convex-
cocompact hyperbolic surface. Let $NW(\phi)\subset T^{1}M$ be the non-
wandering set of the geodesic flow $\phi$, seen as lying in the unit tangent
bundle of $M$. Then:
$\underline{\dim}_{F,C^{2}}(NW(\phi))>0.$
###### Example A.25.
Let $L$ be a 1-dimensionnal manifold, and let $E\subset L$ be any compact
subset. Then:
$\overline{\dim}_{F,C^{1}}E=\dim_{H}E.$
This very striking result is proved in [Ek16]. Also, Ekstrom proves that, for
any $k\geq 1$, we have $\overline{\dim}_{F,C^{k}}E\geq(\dim_{H}E)/k$. This
motivates the following question: do we have, for any compact set $E$ in any
manifold $M$, the formula $\overline{\dim}_{F,C^{1}}(E)=\dim_{H}(E)$ ?
###### Remark A.26.
Other natural questions are the following. Can we find an example of set
$E\subset\mathbb{R}^{d}$ such that
$\underline{\dim}_{F,C^{k}}(E)<\uwave{\dim}{}_{F,C^{k}}(E)$ ? Or is it always
an equality ? Is the lower Fourier dimension $C^{k}$-intrinsic ?
For completeness, we conclude by introducing the real variation for the lower
Fourier dimension. We will not introduce this variation for the upper Fourier
dimension, as we said earlier that these seems to behave quite badly with
respect to the Hausdorff dimension. To keep it concise, we will not discuss
the directionnal variations.
###### Definition A.27.
Let $E\subset M$ be a compact subset of some smooth manifold $M$. Define the
lower Fourier dimension with $C^{k}$-phases by:
$\underline{\dim}_{F,C^{k}}^{\text{real}}(E):=\sup\\{\underline{\dim}_{F,C^{k}}^{\text{real}}\mu\leq
d\ ,\ \mu\in\mathcal{P}(E)\\}.$
###### Remark A.28.
By Lemma A.11, we see that
$\underline{\dim}_{F,C^{k}}(E)\leq\underline{\dim}_{F,C^{k}}^{\text{real}}(E)$.
Is this an equality, or are we able to produce an exemple were this inequality
is strict ? A related question is: if we denote by
$\mathcal{P}_{reas,C^{k}}(E)$ the set of probability measures that admits
reasonnables constants for $C^{k}$-phases (see Definition A.12), do we have
$\underline{\dim}_{F,C^{k}}(E)=\sup\\{\underline{\dim}_{F,C^{k}}\mu\leq d,\
\mu\in\mathcal{P}_{reas,C^{k}}(E)\\}\quad?$
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|
# Understanding the Regularity of Self-Attention
with Optimal Transport
Valérie Castin DMA, École Normale Supérieure PSL Pierre Ablin Apple
Gabriel Peyré DMA, École Normale Supérieure PSL CNRS
###### Abstract
Transformers and their multi-head attention mechanism have completely changed
the machine learning landscape in just a few years, by outperforming state-of-
art models in a wide range of domains. Still, little is known about their
robustness from a theoretical perspective. We tackle this problem by studying
the local Lipschitz constant of self-attention, that provides an attack-
agnostic way of measuring the robustness of a neural network. We adopt a
measure-theoretic framework, by viewing inputs as probability measures
equipped with the Wasserstein distance. This allows us to generalize attention
to inputs of infinite length, and to derive an upper bound and a lower bound
on the Lipschitz constant of self-attention on compact sets. The lower bound
significantly improves prior results, and grows more than exponentially with
the radius of the compact set, which rules out the possibility of obtaining
robustness guarantees without any additional constraint on the input space.
Our results also point out that measures with a high local Lipschitz constant
are typically made of a few diracs, with a very unbalanced distribution of
mass. Finally, we analyze the stability of self-attention under perturbations
that change the number of tokens, which appears to be a natural question in
the measure-theoretic framework. In particular, we show that for some inputs,
attacks that duplicate tokens before perturbing them are more efficient than
attacks that simply move tokens. We call this phenomenon mass splitting.
###### Contents
1. 1 Introduction
1. 1.1 Contributions
2. 1.2 Related Work
3. 1.3 Notations
2. 2 Generalizing Self-Attention to Probability Measures
3. 3 Estimating the Lipschitz Constant of Self-Attention
1. 3.1 Lipschitz Constant and Local Lipschitz Constant
1. 3.1.1 Local Lipschitz Constant on Probability Measures versus on Matrices
2. 3.2 Lower Bound on the Lipschitz Constant of Self-Attention
3. 3.3 Tightness of the Lower Bound
1. 3.3.1 Weighted Self-Attention
2. 3.3.2 Experiments
4. 3.4 The Lipschitz Constant of Masked Attention
4. 4 Mass Splitting
1. 4.1 Finite Mass Splitting: a Proxy for Mass Splitting
1. 4.1.1 Duplication Is the Worst Case for Finite Mass Splitting
2. 4.2 Numerical Evidence of the Mass Splitting Phenomenon
5. A Optimal Transport Toolbox
1. A.1 Pushforward, Wasserstein Distance
2. A.2 Geodesics
6. B Proofs of Section 3
1. B.1 Proof of Lemma 3.1
2. B.2 Proof of Lemma 3.3
3. B.3 Proof of Proposition 3.7
4. B.4 Weighted Self-Attention
5. B.5 Proof of Proposition 3.5
7. C Proofs of Section 4
1. C.1 Effect of Duplication on the Jacobian of Self-Attention
2. C.2 Concentrated Measures Are Not Splitted
3. C.3 Proof of Theorem 4.2
## 1 Introduction
Introduced by [58], Transformers and their multi-head attention mechanism [2]
have completely changed the machine learning landscape in just a few years, by
outperforming state-of-art models on a wide variety of tasks, from natural
language processing [7, 47, 64] to computer vision [16, 67, 66, 30]. Despite
this great empirical success, however, little is known about the robustness of
Transformer architectures – and in particular of self-attention, their main
building block – from a theoretical perspective. We tackle this problem by
focusing on the Lipschitz properties of self-attention, especially on its
local Lipschitz constant, which controls how fast the output can change with
respect to the input in the neighborhood of each point of the domain.
Studying the (local) Lipschitz continuity of neural networks is of particular
interest for various questions [50]. Indeed, estimating the local Lipschitz
constant of a neural network provides guarantees of adversarial robustness, in
an attack-agnostic way [56, 13, 57, 1, 63]. Identifying inputs with a high
local Lipschitz constant and understanding which local perturbation triggers
the biggest change in the output also allows to robustify the network, for
example using adversarial training [20, 37, 40, 28]. The Lipschitz constant is
also involved in generalization bounds [54, 41, 3, 36]. From a different
perspective, Lipschitz constrained neural networks can be used to estimate
Wasserstein distances [45], stabilize training of GANs [38], and build
invertible neural networks [4, 11]. Finally, bounding the Lipschitz constant
of a neural network is an important step in the study of the associated neural
ODE [10], in particular of its well-posedness [35, 19].
Lipschitz continuity of feed-forward and convolutional neural networks has
been extensively studied and remains a hard problem, the main difficulty being
to estimate precisely the Lipschitz constant of a composition of several
linear maps and pointwise linearities [59, 17, 29]. Still, taken
independently, each linear map or activation function has a known Lipschitz
constant. This is however not the case for Transformers: the self-attention
map has an involved non-linear structure, which makes the estimation of its
local Lipschitz constant more challenging. [26] show that the self-attention
map is not globally Lipschitz continuous by proving a lower bound on its
Lipschitz constant on the closed ball $\,\overline{\\!{B}}(0,R)$, lower bound
that grows quadratically with $R$. On the other side, [19] and [60, 61] derive
an upper bound on the Lipschitz constant of self-attention on
$\,\overline{\\!{B}}(0,R)$, by viewing self-attention as a map acting on
probability measures. This approach, that builds on the fact that self-
attention, denoted $f$ in what follows, is permutation equivariant – i.e. does
not distinguish vectors according to their order in the input, which means
that for all permutations $\sigma$ of the set $\\{1,\dots,n\\}$ and inputs
$X\in\mathbb{R}^{n\times d}$, it holds
$f(X_{\circ\sigma})=f(X)_{\circ\sigma},$
with $Y_{\circ\sigma}:=(y_{\sigma(1)},\dots,y_{\sigma(n)})^{\top}$ for any
matrix $Y=(y_{1},\dots,y_{n})\in\mathbb{R}^{n\times d}$, allows them to use
tools from optimal transport to bound the Lipschitz constant of self-attention
in the sense of Wasserstein distance. Their upper bound grows more than
exponentially with $R$, so that the quadratic lower bound and the exponential
upper bound put together provide a very loose estimation of the Lipschitz
constant of self-attention on compact sets. It is therefore important to
understand which one of the two bounds can be improved, if not both.
### 1.1 Contributions
We make the following contributions.
* $\bullet$
We bridge the gap between the matrix formalism and the measure-theoretic
formalism of self-attention by proving that they are equivalent when
restricting to empirical measures with a fixed number of samples (Lemma 3.3).
Such a connection between these two viewpoints on self-attention is absent in
[26], that use the matrix formalism, as well as in [60, 61] and [19], that use
measure-theoretic frameworks.
* $\bullet$
We significantly improve the lower bound obtained by [26] on the Lipschitz
constant of self-attention on the closed ball $B_{R}$ (centered in 0 and of
radius $R$), that grows like $R^{2}$. More precisely, we find a family of two-
dirac measures $(\mu_{R})_{R>0}$ such that $\mu_{R}$ is supported in $B_{R}$
and the $W_{2}$ local Lipschitz constant of self-attention at $\mu_{R}$ is
equal to
$\frac{C}{2}R^{2}e^{CR^{2}}$
with $C$ depending on the parameters of self-attention, up to a constant
factor close to 1 (Theorem 3.4). This result shows that the upper bound
obtained by [19], which is of the form
$C^{\prime}R^{2}e^{C^{\prime}R^{2}}$
up to a constant factor close to 1, with $C^{\prime}\leq 16\,C$, cannot be
significantly improved, in the sense that it is necessarily exponential with
respect to $R^{2}$. Self-attention is therefore a very brittle function in the
Wasserstein sense, which is somewhat disappointing from the perspective of a
robustness analysis.
* $\bullet$
We show numerically that our lower bound is tight up to a constant factor
close to 1, as well-chosen configurations with two unbalanced clusters – one
with a vanishing amount of mass, and the other concentrating the rest of the
mass, appear to maximize the local Lipschitz constant when fixing the support.
(Subsection 3.3)
* $\bullet$
We generalize residual masked self-attention to probability measures by
introducing an innovative framework, where the order of points in input
measures is encoded in a supplementary coordinate, and find an upper bound of
the form
$1+CR^{2}e^{CR^{2}}$
on the Lipschitz constant of residual masked self-attention restricted to
measures supported in $[0,1]\times\,\overline{\\!{B}}(0,R)$. (Subsection 3.4)
* $\bullet$
We evidentiate numerically, and theoretically in some special cases, a _mass
splitting_ phenomenon: in order to induce the greatest change in the output,
the best strategy to perturb an input with $n$ points is sometimes to
duplicate points and perturb the duplicated version of the output, which
amounts to a very small perturbation in the Wasserstein sense but can induce a
significant change in the output. (Section 4)
### 1.2 Related Work
##### Robustness and local Lipschitz constant estimation.
Neural networks are vulnerable to adversarial attacks [56], and most of the
methods proposed to measure and increase their robustness focus on specific
attacks [20, 43]. It turns out, however, that such methods can be defeated by
well-chosen unseen attacks [8]. Measures of robustness that are agnostic to
attack methods have therefore been proposed, often relying on the notion of
Lipschitz constant of networks [56, 31, 57]. As robustness lower bounds that
rely on the (global) Lipschitz constant tend to be too loose, tighter
constraints have been proposed involving the local Lipschitz constant [22,
63]. The problem of evaluating the local Lipschitz constant of deep networks
is now at the heart of several recent articles [57, 31], in particular for
Transformers [26, 60, 19]. From a more practical viewpoint, several Lipschitz-
constrained variants of the Transformer architecture have been proposed, to
increase robustness and reliability [25, 21, 65, 46].
##### Neural networks acting on measures.
[14] and [44] are the first to define neural networks acting on probability
measures, followed by several other articles [60, 68, 51, 19]. Modeling neural
networks as maps on probability measures has multiple applications, such as
studying Wasserstein regularity [60, 19], proving generalization bounds [68]
and doing a mean-field limit analysis of the dynamics of particles as they go
through the network [19]. The measure-theoretic approach is particularly
suited to the case of Encoder-only Transformers [15], as the self-attention
map is permutation equivariant, i.e., ignores the order of vectors in its
input. This property can be leveraged to model any infinitely deep Encoder as
a partial differential equation (PDE) on the space of measures, following the
principle of neural ODEs [10]. Analyzing this PDE then provides information
about the dynamics of tokens as they go through the Transformer, showing for
instance the emergence of clusters [19, 18]. In contrast, masked self-
attention, which is crucial in Decoder-only [33] and Encoder-Decoder [58]
architectures, is not permutation equivariant, so cannot be cast as naturally
into a measure-theoretic framework.
##### Regularity of self-attention and its variants.
[26] show that the self-attention map is not globally Lipschitz continuous by
proving a lower bound on its Lipschitz constant on the closed ball
$\,\overline{\\!{B}}(0,R)$, lower bound that grows quadratically with $R$. To
gain regularity, they define a new self-attention map, called L2 self-
attention, that is globally Lipschitz continuous on the set of inputs of
length $n$, for all $n\geq 1$. Their upper bound on the Lipschitz constant of
L2 self-attention however diverges with $n$. [19] and [60] prove a length-
independent upper bound on the Lipschitz constant of self-attention on
$\,\overline{\\!{B}}(0,R)$, by viewing self-attention as a map acting on
probability measures. Their upper bound grows more than exponentially with
$R$, so that the quadratic lower bound and the exponential upper bound put
together provide a very loose estimation of the Lipschitz constant of self-
attention on compact sets. Finally, [51] propose a modification of the
attention kernel that builds on the Sinkhorn-Knopp algorithm, and provide
empirical evidence of the better properties of this new choice of kernel with
respect to the classical one.
### 1.3 Notations
For any vector $x\in\mathbb{R}^{d}$, the Euclidean norm of $x$ is denoted
$\left\lvert x\right\rvert$. For $R>0$, the open ball centered in $x$ and of
radius $R$ is denoted $B(x,R)$, and we write $\,\overline{\\!{B}}(0,R)$ or
simply $B_{R}$ the corresponding closed ball. The simplex
$\\{a\in[0,1]^{n}\mid\sum_{i=1}^{n}a_{i}=1\\}$ is denoted $\Sigma_{n}$. The
set $\mathcal{P}(\mathbb{R}^{d})$ is the set of probability measures on
$\mathbb{R}^{d}$. The set of probability measures with compact support is
denoted $\mathcal{P}_{c}(\mathbb{R}^{d})$, and
$\mathcal{M}_{n}(\mathbb{R}^{d})$ is the set of empirical measures on
$\mathbb{R}^{d}$ with $n$ samples. $\delta_{x}$ denotes the Dirac measure at
point $x\in\mathbb{R}^{d}$. The vector $1_{n}\in\mathbb{R}^{n}$ is the vector
of dimension $n$ with all coordinates equal to 1. For any vector
$x\in\mathbb{R}^{d}$, we denote
$\operatorname{\mathrm{diag}}(x)\in\mathbb{R}^{d\times d}$ the associated
diagonal matrix. Coordinatewise multiplication for vectors and matrices is
denoted $\odot$, and $\otimes$ is the Kronecker product between matrices. For
a block matrix $M=(M_{ij})_{1\leq i,j\leq n}\in\mathbb{R}^{dn\times dn}$ with
blocks $M_{i,j}$ of shape $d\times d$, and a matrix $A\in\mathbb{R}^{d\times
d}$, we denote $M\otimes_{d}A$ the block matrix $(M_{ij}A)_{1\leq i,j\leq
n}\in\mathbb{R}^{dn\times dn}$. For a function
$g\colon\mathcal{E}\to\mathcal{F}$ and a subset
$\mathcal{X}\subset\mathcal{E}$, the restriction of $g$ to $\mathcal{X}$ is
denoted $g_{\lvert\mathcal{X}}$. The Jacobian of $g$ at point $x$ is denoted
$D_{x}g$.
In the whole paper, the Jacobian of self-attention $f\colon\mathbb{R}^{n\times
d}\to\mathbb{R}^{n\times d}$ is to be understood as the Jacobian of the
vectorized form of $f$, defined by
$\operatorname{\mathrm{vec}}(X)\in\mathbb{R}^{nd}\mapsto\operatorname{\mathrm{vec}}(f(X))\in\mathbb{R}^{nd}$
where vec denotes row-wise vectorization. For example,
$\begin{pmatrix}1&2&3\\\ 4&5&6\\\ 7&8&9\end{pmatrix}$
gives the order of coordinates in our choice of vectorization for a $3\times
3$ matrix. We will sometimes write $\operatorname{\mathrm{vec}}X$ as $\vec{X}$
to alleviate notations.
## 2 Generalizing Self-Attention to Probability Measures
Transformers and self-attention. Introduced by [58], Transformers are
attention-based models that can process several types of data, from text [7,
47, 64] to images [16] and audio recordings [12]. When fed to a Transformer,
each data point is converted to a sequence of tokens (tokenization), then to a
sequence of vectors (input embedding). Then, positional encoding [58] adds
information about the order of vectors in the vectors themselves. The
resulting sequence of vectors $(x_{1},\dots,x_{n})\in(\mathbb{R}^{d})^{n}$,
that can be seen as a $n\times d$ matrix, is said to belong to the feature
space. In encoder-only models [15, 52, 34, 16],
$X\coloneqq(x_{1},\dots,x_{n})^{\top}$ then goes through a succession of
layers, each one made of two blocks. First, a residual multi-head self-
attention block of the form
$X\mapsto X+\sum_{h=1}^{H}f^{(h)}(X)W_{h}^{\top}$
with $W_{h}\in\mathbb{R}^{d\times k}$ and $f^{(h)}$ (single-head) self-
attention maps, where the self-attention map is defined as
$f\colon X\in\mathbb{R}^{n\times
d}\mapsto\operatorname{\mathrm{softmax}}\left(\frac{1}{\sqrt{k}}XQ^{\top}KX^{\top}\right)XV^{\top}\in\mathbb{R}^{n\times
d}$ (1)
where $k$ is a positive integer, $Q,K,V$ are $k\times d$ matrices that depend
on $h$ and on the layer, and
$\operatorname{\mathrm{softmax}}(w_{1},\dots,w_{n})\coloneqq\left(\frac{\exp(w_{i})}{\sum_{j=1}^{d}\exp(w_{j})}\right)_{1\leq
i\leq n}$
for $w_{1},\dots,w_{n}\in\mathbb{R}^{d}$. Then, a residual multilayer
perceptron applied vectorwise, i.e. of the form
$(x_{1},\dots,x_{n})\mapsto(x_{1},\dots,x_{n})+(g(x_{1}),\dots,g(x_{n}))$ with
$g$ a multilayer perceptron. Decoder-only models [33, 42, 9, 24] have the same
structure, but replace self-attention with masked self-attention, defined as
$f^{masked}(X)_{i}\coloneqq\operatorname{\mathrm{softmax}}\left(\frac{1}{\sqrt{k}}x_{i}Q^{\top}KX_{1:i}^{\top}\right)X_{1:i}V^{\top}$
(2)
for all $1\leq i\leq n$, where $X_{1:i}=(x_{1},\dots,x_{i})^{\top}$. Finally,
encoder-decoder models [58, 62, 48, 49, 32] couple self-attention, masked
self-attention and cross-attention, that is not considered in this work. It is
important to notice that the self-attention map introduced in Equation (1) is
permutation equivariant, which means that for all permutations $\sigma$ of the
set $\\{1,\dots,n\\}$, it holds
$f(X_{\circ\sigma})=f(X)_{\circ\sigma},$
with $Y_{\circ\sigma}:=(y_{\sigma(1)},\dots,y_{\sigma(n)})^{\top}$ for any
matrix $Y=(y_{1},\dots,y_{n})\in\mathbb{R}^{n\times d}$. Intuitively, a
permutation equivariant map does not use any information on the order of its
input vectors. On the contrary, the masked self-attention map has a sequential
structure: it is not permutation equivariant. We show in Subsection 3.4 that
it can still be cast into a measure-theoretic framework, by adding a
coordinate to the input space to encode the order of points.
##### Pushforward and Wasserstein distances.
Before introducing the measure-theoretic framework, we need a few notions from
optimal transport. For a probability measure $\mu$ on $\mathbb{R}^{d}$ and a
measurable map $\varphi\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$, the pushforward
of $\mu$ by $\varphi$, denoted $\varphi_{\sharp}\mu$, is the probability
measure given by
$\left(\varphi_{\sharp}\mu\right)(B)\coloneqq\mu(\varphi^{-1}(B))$
for any Borel set $B\subset\mathbb{R}^{d}$, where
$\varphi^{-1}(B)\coloneqq\\{x\in\mathbb{R}^{d}:\varphi(x)\in B\\}$.
Intuitively, $\varphi_{\sharp}\mu$ is obtained by transporting each element of
mass $\mu(\mathrm{d}x)$ from $x$ to $\varphi(x)$. We also have the following
useful property of the pushforward. For all probability measures
$\mu\in\mathcal{P}(\mathbb{R}^{d})$ and maps
$\varphi\colon\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$g\colon\mathbb{R}^{d}\to\mathbb{R}$, it holds, when both members of the
equation are well-defined:
$\int g(x)\mathrm{d}\left(\varphi_{\sharp}\mu\right)(x)=\int
g(\varphi(x))\mathrm{d}\mu(x).$
Let $p\geq 1$. Denote
$\mathcal{P}_{p}(\mathbb{R}^{d})\coloneqq\\{\mu\in\mathcal{P}(\mathbb{R}^{d}):\int\left\lvert
x\right\rvert^{p}\mathrm{d}\mu(x)<+\infty\\}.$
For $\mu$ and $\nu$ in $\mathcal{P}_{p}(\mathbb{R}^{d})$, the $p$-Wasserstein
distance between $\mu$ and $\nu$ is defined as
$W_{p}(\mu,\nu)\coloneqq\left(\inf_{\pi\in\Pi(\mu,\nu)}\int\left\lvert
x-y\right\rvert^{p}\mathrm{d}\pi(x,y)\right)^{1/p},$
where $\Pi(\mu,\nu)$ is the set of couplings between $\mu$ and $\nu$, i.e. of
probability measures $\pi\in\mathcal{P}(\mathbb{R}^{d}\times\mathbb{R}^{d})$
such that $\int\pi(\cdot,y)\mathrm{d}y=\mu$ and
$\int\pi(x,\cdot)\mathrm{d}x=\nu$. More details on the aforementioned objects
can be found for example in [53].
##### Neural networks acting on measures.
The self-attention map defined in Equation (1) is permutation equivariant,
which means that permuting $x_{i}$ and $x_{j}$ in the input of $f$ amounts to
permuting $f(X)_{i}$ and $f(X)_{j}$ in the output. This remark allows to view
self-attention as acting on point clouds, or on empirical measures [14, 68,
51, 19] by mapping the matrix input $X=(x_{1},\dots,x_{n})^{\top}$ to the
associated empirical measure
$m(X)\coloneqq\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}.$ (3)
Following [51], we can then extend self-attention to more general measures in
the following way. We can rewrite Equation (1) as
$f(X)=(\Gamma_{X}(x_{1}),\dots,\Gamma_{X}(x_{n})),$ (4)
with
$\Gamma_{X}\colon
x\mapsto\frac{\sum_{i=1}^{n}\exp\left(\frac{1}{\sqrt{k}}x^{\top}Q^{\top}Kx_{i}\right)Vx_{i}}{\sum_{i=1}^{n}\exp\left(\frac{1}{\sqrt{k}}x^{\top}Q^{\top}Kx_{i}\right)}.$
Seeing the ratio of sums as a ratio of integrals against the empirical measure
$m(X)$, and Equation (4) as a pushforward, we can define measure-theoretic
self-attention as
$F:\mu\in\mathcal{P}_{c}(\mathbb{R}^{d})\mapsto\left(\Gamma_{\mu}\right)_{\sharp}\mu$
(5)
where $\mathcal{P}_{c}(\mathbb{R}^{d})$ is the set of compactly supported
probability measures on $\mathbb{R}^{d}$, with
$\Gamma_{\mu}\colon
x\in\mathbb{R}^{d}\mapsto\frac{\int\exp\left(\frac{1}{\sqrt{k}}x^{\top}Q^{\top}Ky\right)Vy\,\mathrm{d}\mu(y)}{\int\exp\left(\frac{1}{\sqrt{k}}x^{\top}Q^{\top}Ky\right)\mathrm{d}\mu(y)}$
for all $\mu$ such that the integrals are well-defined, which is in particular
the case when $\mu$ is compactly supported. One sees that measure-theoretic
self-attention is a generalization of self-attention. As opposed to measure-
theoretic self-attention, we will call $f$ matrix self-attention in the
following. In order to study the Lipschitz constant of measure-theoretic self-
attention, we equip $\mathcal{P}_{c}(\mathbb{R}^{d})$ with the $p$-Wasserstein
distance $W_{p}$ for any $p\geq 1$. 111Note that this choice of distance is
more suited to the problem than the Hausdorff distance on point clouds: in
dimension 1, for example, the Hausdorff distance between the point clouds
$(1,1,0)$ and $(1,0)$ is zero, but self-attention does not map these point
clouds to the same point cloud.
## 3 Estimating the Lipschitz Constant of Self-Attention
Let us start with some preliminaries on (local) Lipschitz constants.
### 3.1 Lipschitz Constant and Local Lipschitz Constant
We first introduce the notions of Lipschitz constant and Lipschitz ratio, in a
general framework that is suited for measure-theoretic self-attention as well
as for matrix self-attention.
###### Definition 3.1 (Lipschitz constant, Lipschitz ratio).
Let
$\varphi\colon(\mathcal{E},d_{\mathcal{E}})\to(\mathcal{F},d_{\mathcal{F}})$
be a map between two metric spaces. The Lipschitz constant of $\varphi$ is
defined as
$\operatorname{\mathrm{Lip}}(\varphi)\coloneqq\sup_{\underset{x\neq
y}{x,y\in\mathcal{E}}}\frac{d_{\mathcal{F}}(\varphi(x),\varphi(y))}{d_{\mathcal{E}}(x,y)}.$
When $\mathcal{E}=\mathcal{F}$ and $d_{\mathcal{E}}=d_{\mathcal{F}}$, we may
denote
$\operatorname{\mathrm{Lip}}^{d_{\mathcal{E}}}(\varphi)\coloneqq\operatorname{\mathrm{Lip}}(\varphi)$
to stress on the choice of distance. The Lipschitz ratio associated to a pair
of distinct points $x,y\in\mathcal{E}$ is
$\frac{d_{\mathcal{F}}(\varphi(x),\varphi(y))}{d_{\mathcal{E}}(x,y)}.$
Bounding the Lipschitz constant of a function
$\varphi\colon(\mathcal{E},d_{\mathcal{E}})\to(\mathcal{F},d_{\mathcal{F}})$
gives control of how fast the output can change with a small perturbation of
the input. Sometimes, however, the function $\varphi$ is not globally
Lipschitz continuous: its Lipschitz constant is infinite. In that case, it is
more informative to look at the local Lipschitz constant, defined in the
following way.
###### Definition 3.2 (Local Lipschitz constant).
Let $\varphi\colon\mathcal{E}\to\mathcal{F}$ be a map between two metric
spaces. We define the local Lipschitz constant of $\varphi$ at point
$x\in\mathcal{E}$ as
$\operatorname{\mathrm{Lip}}_{x}(\varphi)\coloneqq\lim_{\varepsilon\to
0^{+}}\operatorname{\mathrm{Lip}}\varphi_{\lvert B(x,\varepsilon)}.$
The limit exists, as $\operatorname{\mathrm{Lip}}\varphi_{\lvert
B(x,\varepsilon)}$ decreases with $\varepsilon$.
Definition 3.2 is interesting, as it captures more information than the global
Lipschitz constant. More precisely, we have the following connection between
the two notions.
###### Lemma 3.1.
Let $\varphi\colon\mathcal{E}\to\mathcal{F}$ be a map between two metric
spaces. We have
$\operatorname{\mathrm{Lip}}(\varphi)\geq\sup_{x\in\mathcal{E}}\operatorname{\mathrm{Lip}}_{x}(\varphi).$
Assume moreover that the space $\mathcal{E}$ admits geodesics (see Appendix A
for the definition of a geodesic), which is the case for $\mathbb{R}^{d}$ and
$\mathcal{P}_{2}(\mathbb{R}^{d})$ equipped with $W_{2}$. Then, this inequality
becomes an equality.
See Appendix B.1 for the proof of Lemma 3.1. It is also important to notice
that bounding the local Lipschitz constant of $\varphi$ is equivalent to
bounding its Lipschitz constant on all possible compact sets
$K\subset\mathcal{E}$. We will therefore focus on proving Lipschitz bounds on
closed balls of the form $B_{R}$. Finally, let us point out that the local
Lipschitz constant has a more explicit formula in the particular case where
$\varphi$ is a differentiable function defined between normed vector spaces,
which is the case for matrix self-attention.
###### Lemma 3.2 ([55]).
Let $\varphi\colon\mathbb{R}^{d_{1}}\to\mathbb{R}^{d_{2}}$ be a differentiable
map. For all $x\in\mathbb{R}^{d_{1}}$, the local Lipschitz constant of
$\varphi$ at $x$ is equal to $\left\lVert D_{x}\varphi\right\rVert_{2}$, where
$D_{x}\varphi$ is the differential of $\varphi$ at $x$, and
$\left\lVert\cdot\right\rVert_{2}$ is the operator norm induced by the
Euclidean norm. Moreover, if $D_{x}\varphi$ is non zero and $u$ is its first
right singular vector, we have
$\frac{\left\lvert\varphi(x+\varepsilon
u)-\varphi(x)\right\rvert}{\varepsilon\left\lvert
u\right\rvert}\underset{\varepsilon\to 0^{+}}{\rightarrow}\left\lVert
D_{x}\varphi\right\rVert_{2}.$ (6)
###### Remark 3.3.
We are not aware of any result that would express the local Lipschitz constant
of measure-theoretic self-attention $F$ in terms of a notion of differential
of $F$, as Lemma 3.2 does for matrix self-attention. This seems an interesting
perspective for future work.
Let us now compare Lipschitz constants of matrix self-attention and measure-
theoretic self-attention.
#### 3.1.1 Local Lipschitz Constant on Probability Measures versus on
Matrices
Let $p\geq 1$. To study the Lipschitz properties of self-attention, we use the
following norm:
$\left\lVert X\right\rVert_{F,p}\coloneqq\left(\sum_{i=1}^{n}\left\lvert
x_{i}\right\rvert^{p}\right)^{1/p}$
for $X=(x_{1},\dots,x_{n})^{\top}$, as [26] do, while we equip the set
$\mathcal{P}_{c}(\mathbb{R}^{d})$, on which $F$ operates, with the
$p$-Wasserstein distance $W_{p}$. We have the following link between the
Lipschitz constants of $f$ and $F$.
###### Lemma 3.3.
Let $p\geq 1$. For any matrix $X\in\mathbb{R}^{n\times d}$, we have
$\operatorname{\mathrm{Lip}}_{X}^{\left\lVert\cdot\right\rVert_{F,p}}(f)=\operatorname{\mathrm{Lip}}_{m(X)}^{W_{p}}(F_{\lvert\mathcal{M}_{n}(\mathbb{R}^{d})})\leq\operatorname{\mathrm{Lip}}_{m(X)}^{W_{p}}(F).$
The proof can be found in Appendix B.2. The main idea behind this statement is
that if two matrices $X$ and $Y$ are close enough (in the sense of the
$\left\lVert\cdot\right\rVert_{F,p}$ norm), then we have $\left\lVert
X-Y\right\rVert_{F,p}=W_{p}(m(X),m(Y))$. Note however that
$\operatorname{\mathrm{Lip}}_{X}^{\left\lVert\cdot\right\rVert_{F,p}}(f)\neq\operatorname{\mathrm{Lip}}_{m(X)}^{W_{p}}(F)$
in general, as we will see in Section 4.
### 3.2 Lower Bound on the Lipschitz Constant of Self-Attention
We have the following lower bound on the Lipschitz constant of measure-
theoretic self-attention.
###### Theorem 3.4.
Let $p\geq 1$, and $R>0$. The self-attention map
$F\colon\mu\mapsto(\Gamma_{\mu})_{\sharp}\mu$
is $W_{p}$-Lipschitz continuous on the space $\mathcal{P}(B_{R})$ of
probability measures supported in $B_{R}$. Moreover, assume that:
1. (i)
$k=d$ and $V=I_{d}$,
2. (ii)
$A\coloneqq\frac{1}{\sqrt{d}}K^{\top}Q$ is symmetric,
and denote $\gamma_{1}\geq\dots\geq\gamma_{d}$ the ordered eigenvalues of $A$.
Then, the following claims hold true.
1. 1.
If $\gamma_{1}\geq-8\gamma_{d}$, there exists a function
$\theta\colon[0,+\infty)\to[0,+\infty)$ such that
$\theta(R)\to_{R\to+\infty}1$ and:
$\operatorname{\mathrm{Lip}}^{W_{2}}(F_{\lvert\mathcal{P}(B_{R})})\geq\theta(R)\frac{C}{2}R^{2}e^{CR^{2}}$
with $C\coloneqq\frac{\gamma_{1}}{8}$.
2. 2.
If $\gamma_{1}<-8\gamma_{d}$, there exists a function
$\theta\colon[0,+\infty)\to[0,+\infty)$ such that
$\theta(R)\to_{R\to+\infty}1$ and:
$\operatorname{\mathrm{Lip}}^{W_{2}}(F_{\lvert\mathcal{P}(B_{R})})\geq\theta(R)\frac{C^{\prime}}{2}R^{2}e^{C^{\prime}R^{2}}$
with $C^{\prime}\coloneqq\left\lvert\gamma_{d}\right\rvert$.
Moreover, in both cases, the right-hand side is equivalent when $R\to+\infty$
to the local Lipschitz constant of measure-theoretic self-attention at a well-
chosen pair of two-dirac probability measures, i.e. of the form
$p\delta_{x}+(1-p)\delta_{y}$.
The fact that $F$ is Lipschitz continuous on $\mathcal{P}(B_{R})$ comes from
the upper bound proven by [19], which tells us that
$\operatorname{\mathrm{Lip}}^{W_{2}}(F_{\lvert\mathcal{P}(B_{R})})\leq
2R^{2}\left\lVert V\right\rVert_{2}\left\lVert
A\right\rVert_{2}\left(1+e^{2R^{2}\left\lVert A\right\rVert_{2}}\right).$
Note that the lower bounds given in Theorem 3.4 are much closer to this upper
bound than to the lower bound obtained by [26], which grows like $R^{2}$ up to
a constant factor.
Assumption $(i)$ amounts to studying self-attention before multiplication by
$V$, as [26] do. In other words, the upper and lower bounds that we have on
the Lipschitz constant of the function
$\tilde{F}\colon\mu\mapsto(\tilde{\Gamma}_{\mu})_{\sharp}\mu$ with
$\tilde{\Gamma}_{\mu}(x)\coloneqq\frac{\int\exp\left(\frac{1}{\sqrt{k}}x^{\top}A^{\top}y\right)y\,\mathrm{d}\mu(y)}{\int\exp\left(\frac{1}{\sqrt{k}}x^{\top}A^{\top}y\right)\mathrm{d}\mu(y)}$
are nearly sharp. The situation is more complicated with a general matrix $V$,
as the matrices $V$ and $A$ may not interact properly: to take an extreme
example, probability measures leading to a high local Lipschitz constant for
$\tilde{F}$ may for instance have a support included in the kernel of $V$.
Moreover, our numerical experiments suggest that assumption $(ii)$ is too
strong: requiring the diagonalizability of $A$ instead seems to be enough for
the result to hold.
From the perspective of a robustness analysis, having a more than exponential
explosion of the Lipschitz constant with the radius is bad news. Indeed, to
ensure for example that the Lipschitz constant stays smaller than 1 on a ball
of radius 20 – which is a realistic value for the case of a trained Vision
Transformer [16] on CIFAR-10 [27] for example – with $V=I_{d}$, one should
take $A$ with an operator norm of the order of $10^{-2}$, which would shrink
all keys and queries to very small values, thus losing precision in the model.
In future work, it would therefore be interesting to find a relevant
constraint on the input space, that provides better guarantees on the
Lipschitzianity of self-attention.
Let us now precise which two-dirac measures have a local Lipschitz constant
equal to the lower bound. The following result implies Theorem 3.4.
###### Proposition 3.5.
Let $R>0$. Denote $\mathcal{P}^{2\mathrm{d}}(\mathbb{R}^{d})$ the set of two-
dirac probability measures. Assume that
1. (i)
$k=d$ and $V=I_{d}$,
2. (ii)
$A\coloneqq\frac{1}{\sqrt{d}}K^{\top}Q$ is symmetric.
Denote $\gamma_{1}\geq\dots\geq\gamma_{d}$ the eigenvalues of $A$, and
$u_{1},\dots,u_{d}$ corresponding unit eigenvectors.
1. 1.
If $\gamma_{1}\geq-8\gamma_{d}$, define
$p_{R}\coloneqq e^{-2CR^{2}}\mbox{ and }\mu_{R}\coloneqq
p_{R}\delta_{Ru_{1}}+(1-p_{R})\delta_{\frac{R}{2}u_{1}}$
for $R>0$, with $C\coloneqq\frac{\gamma_{1}}{8}$. Then
$\operatorname{\mathrm{Lip}}_{\mu_{R}}^{W_{2}}(F)\geq\operatorname{\mathrm{Lip}}_{\mu_{R}}^{W_{2}}(F_{\lvert\mathcal{P}^{2\mathrm{d}}(\mathbb{R}^{d})})\sim_{R\to+\infty}\frac{C}{2}R^{2}e^{CR^{2}}.$
2. 2.
If $\gamma_{1}<-8\gamma_{d}$, define
$p_{R}\coloneqq e^{-2C^{\prime}R^{2}}\mbox{ and }\mu_{R}\coloneqq
p_{R}\delta_{Ru_{d}}+(1-p_{R})\delta_{-Ru_{d}}$
for $R>0$, with $C^{\prime}\coloneqq\left\lvert\gamma_{d}\right\rvert$. Then
$\operatorname{\mathrm{Lip}}_{\mu_{R}}^{W_{2}}(F)\geq\operatorname{\mathrm{Lip}}_{\mu_{R}}^{W_{2}}(F_{\lvert\mathcal{P}^{2\mathrm{d}}(\mathbb{R}^{d})})\sim_{R\to+\infty}\frac{C^{\prime}}{2}R^{2}e^{C^{\prime}R^{2}}.$
The proof can be found in Appendix B.5. Note that Theorem 3.4 is an immediate
consequence of Proposition 3.5, using Lemma 3.1. The condition
$\gamma_{1}\geq-8\gamma_{d}$ can be interpreted as follows. Negative
eigenvalues induce a faster growth of the local Lipschitz constant than
positive eigenvalues: if $\gamma_{i}$ is a negative eigenvalue of $A$
associated to a unit eigenvector $u_{i}$, choosing
$p_{R}\coloneqq e^{-2\left\lvert\gamma_{i}\right\rvert R^{2}}\mbox{ and
}\mu_{R}\coloneqq p_{R}\delta_{Ru_{i}}+(1-p_{R})\delta_{-Ru_{i}}$
gives
$\operatorname{\mathrm{Lip}}_{\mu_{R}}^{W_{2}}(F_{\lvert\mathcal{P}^{2\mathrm{d}}(\mathbb{R}^{d})})\sim_{R\to+\infty}\frac{\left\lvert\gamma_{i}\right\rvert}{2}R^{2}e^{\left\lvert\gamma_{i}\right\rvert
R^{2}},$ (7)
whereas if $\gamma_{i}$ is a positive eigenvalue of $A$ associated to a unit
eigenvector $u_{i}$, choosing
$p_{R}\coloneqq e^{-\frac{\gamma_{i}}{8}R^{2}}\mbox{ and }\mu_{R}\coloneqq
p_{R}\delta_{Ru_{i}}+(1-p_{R})\delta_{\frac{R}{2}u_{i}}$
gives
$\operatorname{\mathrm{Lip}}_{\mu_{R}}^{W_{2}}(F_{\lvert\mathcal{P}^{2\mathrm{d}}(\mathbb{R}^{d})})\sim_{R\to+\infty}\frac{\gamma_{i}}{16}R^{2}e^{\frac{\gamma_{i}}{8}R^{2}}.$
(8)
We can see that when $\gamma_{1}>-8\gamma_{d}$, either $A$ has only positive
eigenvalues, or Equation (8) with $\gamma_{1}$ provides a faster growth of the
local Lipschitz constant than Equation (7) with $\gamma_{d}$. On the contrary,
when $\gamma_{1}<-8\gamma_{d}$, either $A$ has only negative eigenvalues, or
Equation (7) provides a better rate. Finally, in the edge case where
$\gamma_{1}=-8\gamma_{d}$, both equations grow at the same rate. From a high-
level perspective, the optimal strategy to make the local Lipschitz constant
explode is therefore to concentrate most of the mass on one point that goes to
infinity and is aligned with a well-chosen eigenvector of $A$, and to keep the
exponentially vanishing complement of mass on another point that also goes to
infinity, either twice faster in the same direction (if
$\gamma_{1}\geq-8\gamma_{d}$), or at the same speed in the opposite direction
(if $\gamma_{1}<-8\gamma_{d}$). This strategy differs from the one of [26],
who have one particle stuck at 0, and the others maximizing their empirical
variance, which leads to a local Lipschitz constant that is quadratic with the
radius (see Lemma C.4).
### 3.3 Tightness of the Lower Bound
To analyse the tightness of the lower bound given in Theorem 3.4, let us look
for the probability measure supported in $\mathcal{P}(B_{R})$ that induces the
highest local Lipschitz constant, and tackle this problem numerically. We
focus on probability measures with finite support:
$\mathcal{P}^{\mathrm{fin}}\coloneqq\\{\sum_{i=1}^{n}a_{i}\delta_{x_{i}}\mid
n\in\mathbb{N}^{*},\ x_{1},\dots,x_{n}\in\mathbb{R}^{d}\mbox{ and
}(a_{1},\dots,a_{n})\in\Sigma_{n}\\},$
with
$\Sigma_{n}:=\\{(a_{1},\dots,a_{n})\in\mathbb{R}_{+}^{n}:\sum_{i=1}^{n}a_{i}=1\\}$
the simplex of $\mathbb{R}^{n}$. This is still very general, as continuous
measures can be approximated arbitrarily well by elements of the set
$\mathcal{P}^{\mathrm{fin}}$, and Transformers can only handle a finite number
of points in practice. As we cannot compute numerically the local Lipschitz
constant of $F_{\lvert\mathcal{P}^{\mathrm{fin}}}$ at a measure
$\sum_{i=1}^{n}a_{i}\delta_{x_{i}}$, let us use a lower bound instead, by
introducing a generalization of matrix self-attention, called weighted self-
attention.
#### 3.3.1 Weighted Self-Attention
Weighted self-attention implements a matrix representation of probability
measures of the form $\sum_{i=1}^{n}a_{i}\delta_{x_{i}}$, just as matrix self-
attention does for empirical measures. It is defined by the following formula.
###### Definition 3.4 (Weighted self-attention).
For any vector $a\in\Sigma_{n}$, denote
$\mathcal{P}_{a}(\mathbb{R}^{d})\coloneqq\\{\sum_{i=1}^{n}a_{i}\delta_{x_{i}}\mid
x_{1},\dots,x_{n}\in\mathbb{R}^{d}\\}.$
We define the matrix version of the restriction of self-attention to
$\mathcal{P}_{a}(\mathbb{R}^{d})$ in the following way:
$f_{a}\colon X\in\mathbb{R}^{n\times
d}\mapsto\operatorname{\mathrm{softmax}}_{a}(XQ^{\top}KX^{\top})XV^{\top},$
where
$\operatorname{\mathrm{softmax}}_{a}(w_{1},\dots,w_{n})\coloneqq\left(\frac{a_{i}e^{w_{i}}}{\sum_{j}a_{j}e^{w_{j}}}\right)_{1\leq
i\leq n}.$
The function $f_{a}$ is called weighted self-attention associated to the
coefficients $a$.
Definition 3.4 is designed so that for any matrix $X\in\mathbb{R}^{n\times
d}$, it holds
$F(m_{a}(X))=m_{a}(f_{a}(X)),$
with
$m_{a}(X)\coloneqq\sum_{i=1}^{n}a_{i}\delta_{X_{i}}.$
Weighted self-attention proves very useful for numerical experiments: it
provides a representation that is much more convenient than the traditional
matrix setting. For instance, measures of the form
$\mu_{R}=e^{-2R^{2}}\delta_{R}+(1-e^{-2R^{2}})\delta_{-R}$, which are of
interest in our work, require a very large number of samples $n$ to be well
approximated by an empirical measure, especially when $R$ is large – in fact,
we need $n$ to be of the order of $e^{2R^{2}}$ to put the right amount of mass
on the first dirac, whereas weighted self-attention allows us to represent
$\mu_{R}$ with just two vectors, coupled with a vector of coefficients
$a=(e^{-2R^{2}},1-e^{-2R^{2}})$. To study the Lipschitz continuity of $f_{a}$,
we equip the space $\mathbb{R}^{n\times d}$ with a new scalar product
$\left\langle\cdot,\cdot\right\rangle_{a}$, defined as follows:
$\left\langle X,Y\right\rangle_{a}\coloneqq\sum_{i=1}^{n}a_{i}\left\langle
x_{i},y_{i}\right\rangle,$
where $X=(x_{1},\dots,x_{n})^{\top}$ and $Y=(y_{1},\dots,y_{n})^{\top}$. This
allows us to connect the local Lipschitz constant of $f_{a}$ for the norm
$\left\lVert\cdot\right\rVert_{a}$ associated to the scalar product
$\left\langle\cdot,\cdot\right\rangle_{a}$ to the local Lipschitz constant of
measure-theoretic self-attention in the Wasserstein 2 sense.
###### Lemma 3.6.
Let $X\in\mathbb{R}^{n\times d}$ be an input matrix, and $a\in\Sigma_{n}$ a
vector of coefficients. Then, we have
$\operatorname{\mathrm{Lip}}_{m_{a}(X)}^{W_{2}}F_{\lvert\mathcal{P}_{a}(\mathbb{R}^{d})}=\left\lVert
D_{X}^{a}f_{a}\right\rVert_{2,a},$
where $D^{a}$ is the Jacobian in the space $(\mathbb{R}^{n\times
d},\left\langle\cdot,\cdot\right\rangle_{a})$ and
$\left\lVert\cdot\right\rVert_{2,a}$ is the corresponding operator norm.
This is a nice property, as we can compute numerically the local Lipschitz
constant of $f_{a}$, just as for matrix self-attention. Lemma 3.6 can be
proven with the same steps as for Lemma 3.2. Now that we are equipped with
weighted self-attention, we can perform a gradient ascent on its local
Lipschitz constant, in order to estimate the Lipschitz constant of $F$
restricted to $\mathcal{P}(B_{R})$.
#### 3.3.2 Experiments
To gain some intuition about which probability measures with finite support
have a high local Lipschitz constant, we perform a joint gradient ascent on
$X$ and $a$ to maximize the local Lipschitz constant $\left\lVert
D_{X}^{a}f_{a}\right\rVert_{2,a}$ of $f_{a}$. More precisely, we do a
projected gradient ascent on $X$, to ensure that all lines of $X$ stay in the
ball $B_{R}$, and a mirror ascent on $a$, because $a$ has to belong to the
simplex $\Sigma_{n}$ at each step. We also choose $R=1$ for the experiments.
So the iterations are
$\displaystyle X_{0}=\pi(Z)\mbox{ with }Z_{ij}\sim\mathcal{N}(0,1)\mbox{ for
all }i,j\ \ \mbox{ and }\ \ a_{0}=\frac{1}{n}1_{n},$ (9) $\displaystyle
X_{k+1}=\pi\left(X_{k}-\eta\nabla_{X_{k}}\left\lVert
D_{X_{k}}^{a_{k}}f_{a_{k}}\right\rVert_{2,{a_{k}}}\right),$ (10)
$\displaystyle\tilde{a}_{k+1}=a_{k}\odot\exp\left(\eta\nabla_{a_{k}}\left\lVert
D_{X_{k+1}}^{a_{k}}f_{a_{k}}\right\rVert_{2,{a_{k}}}\right),$ (11)
$\displaystyle a_{k+1}=\tilde{a}_{k+1}/\sum_{i=1}^{n}(\tilde{a}_{k+1})_{i},$
(12)
where $\pi$ is the projection on the closed unit ball
$\,\overline{\\!{B}}(0,1)$ and $\odot$ denotes coordinatewise multiplication.
We use autodifferentiation in JAX [6]. Numerics show that the optimization
problem is non-convex: the algorithm can get stuck in local minima.
Nevertheless, an interesting dynamics of clustering of the particles can be
observed along the gradient ascent, as shown in Figure 1. This suggests that
there are measures with a high local Lipschitz constant that are made of only
a few diracs – which confirms our theoretical results, where the highest
Lipschitz constant is obtained with two-dirac measures. With this in mind, one
can focus on two-dirac measures, and perform a grid search on the position of
diracs and their weights to find the two-dirac measure supported in $B_{R}$
with the highest local Lipschitz constant. Doing this, we recover a measure
that is very close to $\mu_{R}$, defined in Proposition 3.5.
Figure 1: Evolution of the input (left column) and output (right column)
measures along a gradient ascent on the local Lipschitz constant of weighted
self-attention (see Equation (9)), with $Q^{\top}K=-I_{2}$ and $V=I_{2}$.
Darker points concentrate more mass: we set a transparency parameter equal to
$\sqrt{a_{i}}$ for each point $x_{i}$. We can see two distinct clusters
emerge, diametrally opposed, with one of them concentrating most of the mass.
We also plot the first right singular vector of the Jacobian of self-
attention, that shows in which direction to move input points to obtain the
biggest change in the output.
Remarkably, the experiments are much more difficult to interpret if, instead
of doing a gradient ascent on the local Lipschitz constant of only one point
cloud, one performs a gradient ascent on the Lipschitz ratio of two point
clouds, by making both point clouds vary at each iteration while constraining
them to stay in the ball $\,\overline{\\!{B}}(0,1)$. Should the optimization
converge to the global maximum, this second method would give a good
approximation of the Lipschitz constant of self-attention restricted to
empirical measures supported in $\,\overline{\\!{B}}(0,1)$ and with a fixed
number of diracs. Unfortunately, we observe that iterations systematically get
trapped in local minima, and the particles do not cluster.
### 3.4 The Lipschitz Constant of Masked Attention
Let us conclude this section with an analysis of the smoothness of _masked_
self-attention. We focus on the residual form of masked self-attention,
defined for $X=(x_{1},\dots,x_{n})^{\top}\in\mathbb{R}^{n\times d}$ as
$f^{m}(X)_{i}\coloneqq
x_{i}+\operatorname{\mathrm{softmax}}\left(\frac{1}{\sqrt{k}}x_{i}Q^{\top}KX_{1:i}^{\top}\right)X_{1:i}V^{\top}$
(13)
for all $1\leq i\leq n$, where
$f^{m}(X)\eqqcolon(f^{m}(X)_{1},\dots,f^{m}(X)_{n})^{\top}$ and
$X_{1:i}\coloneqq(x_{1},\dots,x_{i})^{\top}$. Residual masked self-attention,
which is a central building block of the architecture of Decoders [58],
processes inputs in a sequential way, so it is not permutation equivariant. As
a consequence, the map $f^{m}$ does not directly induce a map on empirical
measures as for traditional self-attention. Still, we can introduce a
convenient measure-theoretic framework for studying masked self-attention, in
the following way. Instead of viewing an input
$(x_{1},\dots,x_{n})\in(\mathbb{R}^{d})^{n}$ as the empirical measure
$\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$, let us add a coordinate
$s_{i}\in[0,1]$ — the choice of the interval is arbitrary — to each input
vector, to encode the position of the $x_{i}$. We can then represent
$(x_{1},\dots,x_{n})$ as the empirical measure over
$[0,1]\times\mathbb{R}^{d}$
$\bar{\mu}=\frac{1}{n}\sum_{i=1}^{n}\delta_{(s_{i},x_{i})},$
with $0\leq s_{1}<\dots<s_{n}\leq 1$ any choice of increasing real numbers
between 0 and 1, for example $s_{i}=i/n$ for $i=1,\dots,n$. Masked self-
attention can be computed from the measure $\bar{\mu}$ only, where the
position of the vectors $x_{i}$ is recovered by looking at the order of the
first coordinates $s_{i}$. To extend this idea to more general probability
measures, we define the measure-theoretic generalization of residual masked
self-attention as taking as input probability measures on the product space
$[0,1]\times\mathbb{R}^{d}$.
###### Definition 3.5 (Measure-theoretic masked residual self-attention).
For any probability measure $\bar{\mu}\in{\cal
P}_{c}([0,1]\times\mathbb{R}^{d})$, denote $\mu$ the second marginal of
$\bar{\mu}$, i.e.
$\mu(A)\coloneqq\int_{0}^{1}\int_{A}\mathrm{d}\bar{\mu}(s,x)$
for all Borel sets $A\subset\mathbb{R}^{d}$. We define residual masked self-
attention on ${\cal P}_{c}([0,1]\times\mathbb{R}^{d})$ as
$F^{m}\colon\bar{\mu}\mapsto\left(\mathrm{Id}+\Gamma_{\bar{\mu}}\right)_{\sharp}\bar{\mu}$
with
$\Gamma_{\bar{\mu}}(s,x)\coloneqq\left(0,\frac{\int_{\mathbb{R}^{d}}G(x,y)Vy\mathbf{1}_{\tau\leq
s}\mathrm{d}\bar{\mu}(\tau,y)}{\int_{\mathbb{R}^{d}}G(x,y)\mathbf{1}_{\tau\leq
s}\mathrm{d}\bar{\mu}(\tau,y)}\right).$
This definition generalizes Equation (13) in the following sense: denoting
$\mathrm{ord}$ the transformation
$\mathrm{ord}\colon X=(x_{1},\dots,x_{n})^{\top}\in\mathbb{R}^{n\times
d}\mapsto\frac{1}{n}\sum_{i=1}^{n}\delta_{(i/n,x_{i})}\in{\cal
P}_{c}([0,1]\times\mathbb{R}^{d}),$
we have
$F^{m}(\mathrm{ord}(X))=\mathrm{ord}(f^{m}(X))$
for all $X\in\mathbb{R}^{n\times d}$.
###### Remark 3.6.
For empirical measures, $\mathrm{ord}$ gives a canonical way of representing
$\mu:=1/n\sum_{i=1}^{n}\delta_{x_{i}}$ by $\bar{\mu}\in{\cal
P}_{c}([0,1]\times\mathbb{R}^{d})$, via $X=(x_{1},\dots,x_{n})^{\top}$. In the
case where $\mu$ has infinitely many diracs, we simply assume that the
coupling $\bar{\mu}$ is already given as input.
Let us now bound from above the Lipschitz constant of measure-theoretic masked
self-attention. To this aim, we introduce the following distance on ${\cal
P}_{c}([0,1]\times\mathbb{R}^{d})$, for $p\geq 1$.
###### Definition 3.7.
Let $\bar{\mu}$ and $\bar{\nu}$ be two probability measures in ${\cal
P}_{c}([0,1]\times\mathbb{R}^{d})$, and $p\geq 1$. If $\bar{\mu}$ and
$\bar{\nu}$ have the same marginal with respect to $s$, i.e.
$\int_{s_{1}}^{s_{2}}\int_{\mathbb{R}^{d}}\mathrm{d}\bar{\mu}(s,x)=\int_{s_{1}}^{s_{2}}\int_{\mathbb{R}^{d}}\mathrm{d}\bar{\nu}(s,x)$
for all $0\leq s_{1}<s_{2}\leq 1$, denote $\theta$ this marginal distribution,
and write with the disintegration theorem [5]
$\mathrm{d}\bar{\mu}(\tau,x)\eqqcolon\mathrm{d}\theta(\tau)\mathrm{d}\mu^{\tau}(x)$
and
$\mathrm{d}\bar{\nu}(\tau,x)\eqqcolon\mathrm{d}\theta(\tau)\mathrm{d}\nu^{\tau}(x).$
The measures $\mu^{\tau}$ and $\nu^{\tau}$ can be seen intuitively as the
restriction of $\mu$ and $\nu$ to the mass located at position $\tau$,
rescaled to obtain probability measures. We then measure the distance between
$\bar{\mu}$ and $\bar{\nu}$ with
$d_{p}(\bar{\mu},\bar{\nu})\coloneqq\left(\int_{0}^{1}W_{p}(\mu^{\tau},\nu^{\tau})^{p}\mathrm{d}\theta(\tau)\right)^{1/p}.$
If $\bar{\mu}$ and $\bar{\nu}$ do not have the same first marginal, we set
$d_{p}(\bar{\mu},\bar{\nu})\coloneqq+\infty.$
The distance $d_{p}$ derives from the conditional optimal transport problem,
studied by [23]. We have the following bound on the $d_{p}$-Lipschitz constant
of masked self-attention.
###### Proposition 3.7.
Let $R>0$ and $p\geq 1$. The measure-theoretic residual masked self-attention
map $F^{m}$ is Lipschitz continuous on the space of measures supported in
$[0,1]\times B_{R}$, with a Lipschitz constant upper-bounded by
$1+\left\lVert V\right\rVert_{2}(1+3\left\lVert
A\right\rVert_{2}R^{2})e^{2\left\lVert A\right\rVert_{2}R^{2}}.$
Here, we consider that two measures with different first marginals induce an
infinite Lipschitz ratio.
The proof can be found in Appendix B.3. This result shows that similar upper
bounds hold for traditional self-attention and for masked self-attention – in
particular, the dependency in the radius is more than exponential.
We have seen that the local Lipschitz constant of self-attention grows
extremely fast when increasing the support, and that it seems to be the case
for masked self-attention as well. Another interesting perspective is to
understand which local perturbations induce the largest variations of the
output of $F$. In other words, in which direction should one perturb a
probability measure $\mu$ to induce a maximal variation of the output of $F$?
This question brings into play the notion of mass splitting, that is the
object of the following section.
## 4 Mass Splitting
Let $a\in\Sigma_{n}$ be a vector of coefficients, and consider a measure
$\mu=\sum_{i=1}^{n}a_{i}\delta_{x_{i}}\in\mathcal{P}_{a}(\mathbb{R}^{d}).$
Does it hold that
$\operatorname{\mathrm{Lip}}_{\mu}^{W_{2}}(F)=^{?}\operatorname{\mathrm{Lip}}_{\mu}^{W_{2}}\left(F_{\lvert\mathcal{P}_{a}(\mathbb{R}^{d})}\right)?$
(14)
If this equality holds true, it means intuitively222The following claim is not
rigorously true, but gives the right intuition. that among all probability
measures in $B^{W_{2}}(\mu,\varepsilon)$, the one (or one of those) that
induces the highest Lipschitz ratio with respect to $\mu$ is itself in
$\mathcal{P}_{a}(\mathbb{R}^{d})$, for $\varepsilon>0$ small enough. In other
words, by slightly moving the diracs of $\mu$ in the right direction, one can
perturb the output of $F$ in the worst way possible among all perturbations
that stay in the ball $B^{W_{2}}(\mu,\varepsilon)$. As we will see in what
follows, Equation (14) is not always true. We call this phenomenon _mass
splitting_.
###### Definition 4.1 (Mass splitting, magnitude).
Let $F\colon\mathcal{U}\to\mathcal{F}$, where $\mathcal{F}$ is a metric space,
and $\mathcal{U}$ is an open set of the space of probability measures
$\mathcal{P}(\mathbb{R}^{d})$. For a probability measure
$\mu=\sum_{i=1}^{n}a_{i}\delta_{x_{i}}\in\mathcal{U}$ with finite support, we
say that $F$ splits the mass at $\mu$, or equivalently that $\mu$ is splitted
by $F$, if
$\operatorname{\mathrm{Lip}}_{\mu}^{W_{2}}(F)>\operatorname{\mathrm{Lip}}_{\mu}^{W_{2}}\left(F_{\lvert\mathcal{P}_{a}(\mathbb{R}^{d})}\right),$
where
$\mathcal{P}_{a}(\mathbb{R}^{d})\coloneqq\\{\sum_{i=1}^{n}a_{i}\delta_{x_{i}}\mid
x_{1},\dots,x_{n}\in\mathbb{R}^{d}\\}$. The gap
$\operatorname{\mathrm{Lip}}_{\mu}^{W_{2}}(F)-\operatorname{\mathrm{Lip}}_{\mu}^{W_{2}}\left(F_{\lvert\mathcal{P}_{a}(\mathbb{R}^{d})}\right)>0$
is called magnitude of the splitting.
In what follows, we will take $F$ to be measure-theoretic self-attention, and
restrict our theoretical analysis to the case of empirical measures instead of
probability measures with finite support, for simplicity. Our aim is to
understand what kind of perturbations induce the highest change in the output
of $F$ at a splitted measure: do the diracs of $\mu$ split in a finite number
of points? Or on the contrary, is the worst perturbation continuous with
respect to Lebesgue?
### 4.1 Finite Mass Splitting: a Proxy for Mass Splitting
We lack tools to analyze mass splitting as it is defined. Let us therefore
focus on the following sufficient condition for mass splitting, which is
easier to handle and serves as a proxy for studying mass splitting.
###### Definition 4.2 (Finite mass splitting).
Let $F\colon\mathcal{U}\to\mathcal{F}$, where $\mathcal{F}$ is a metric space,
and $\mathcal{U}$ is an open set of the space of probability measures
$\mathcal{P}(\mathbb{R}^{d})$. For an empirical measure
$\mu=1/n\sum_{i=1}^{n}\delta_{x_{i}}\in\mathcal{U}$, we say that $F$ finitely
splits the mass at $\mu$, or equivalently that $\mu$ is finitely splitted by
$F$, if
$\sup_{N\geq
1}\operatorname{\mathrm{Lip}}^{W_{2}}\left(F_{\lvert\mathcal{M}_{nN}(\mathbb{R}^{d})}\right)>\operatorname{\mathrm{Lip}}^{W_{2}}\left(F_{\lvert\mathcal{M}_{n}(\mathbb{R}^{d})}\right).$
It is straightforward that if $\mu$ is finitely splitted, then $\mu$ is
splitted. Let us now focus on the specific case of self-attention. We have the
following equivalent formulation of finite mass splitting, which is a direct
consequence of Lemma 3.2.
###### Lemma 4.1.
Let $F$ be measure-theoretic self-attention. Let $X\in\mathbb{R}^{d}$. Then
$F$ finitely splits the empirical measure $m(X)$ if and only if we have
$\sup_{N\geq 1}\left\lVert D_{1_{N}\otimes X}f\right\rVert_{2}>\left\lVert
D_{X}f\right\rVert_{2}.$
#### 4.1.1 Duplication Is the Worst Case for Finite Mass Splitting
The following result shows that to reach the magnitude of finite mass
splitting at an empirical measure $m(X)$ with $X\in\mathbb{R}^{n\times d}$, it
suffices to split only one well-chosen dirac of $\mu$ in just two equal parts.
###### Theorem 4.2.
Let $X=(x_{1},\dots,x_{n})^{\top}$ be a $n\times d$ matrix, and denote
$Y^{N}\coloneqq 1_{N}\otimes X$ the $N$-replication of $X$ for any positive
integer $N$. Assume that $V=I_{d}$. Then it holds
$\left\lVert D_{Y^{N}}f\right\rVert_{2}=\left\lVert
D_{Y^{2}}f\right\rVert_{2}$
for all $N\geq 2$. Furthermore, if the mass splits at $X$, then the maximal
magnitude of splitting at $X$ can be reached by splitting equally the mass of
the $i$-th particle of $X$ along the first singular vector of the matrix
$(\operatorname{\mathrm{Var}}^{(i)}Z)A$, with
$i\coloneqq\operatorname{\mathrm{argmax}}_{i^{\prime}\in\\{1,\dots,n\\}}\left\lVert(\operatorname{\mathrm{Var}}^{(i^{\prime})}Z)A\right\rVert_{2}^{2},$
where $\operatorname{\mathrm{Var}}^{(i)}Z$ is the variance of the random
variable $Z$ following the law $\mathbb{P}(Z=x_{j})\coloneqq P_{ij}$.
The proof of Theorem 4.2 can be found in Appendix C, and relies on a formula
for the Jacobian of $Y^{2}$ in terms of $D_{X}f$. Theorem 4.2 gives us a
simple way of checking numerically if a given empirical measure $m(X)$ is
finitely splitted.
### 4.2 Numerical Evidence of the Mass Splitting Phenomenon
Focusing on two-dirac measures, we compare
$\left\lVert D_{X}^{a}f\right\rVert_{2}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \left\lVert D_{1_{2}\otimes
X}^{a}f\right\rVert_{2}$
with $X=(x,y)$ for $(x,y)$ on a grid, and for various choices of coefficients
$a=(p,1-p)$. It turns out that splitted measures lie in well-defined areas, as
we can see on Figure 2 in dimension 1, and that there are infinitely many of
them – at least for suitable choices of parameters.
Figure 2: Visualisation of two-dirac measures splitted by self-attention. A
point $(x,y)$ belongs to the graph if the measure
$p\delta_{x}+(1-p)\delta_{y}$ is splitted by $f$, namely if $\left\lVert
D_{(x,y,x,y)^{\top}}^{a}f\right\rVert_{2}>\left\lVert
D_{(x,y)^{\top}}^{a}f\right\rVert_{2}$ with $a=(p,1-p)$. The colorbar
indicates the magnitude of the splitting. The first row corresponds to
$A=0.01$, the second one to $A=-0.01$, and $V=1$ in both cases. We make the
repartition of mass vary from 0.25 to 0.95, and observe that it can change the
magnitude of the splitting, and the shape of the zone containing the splitted
measures.
## Conclusion and Perspectives
We have shown that self-attention is a highly irregular function in the
Lipschitz sense, and more precisely that its Lipschitz constant on the space
of measures supported in $B_{R}$ grows like
$CR^{2}e^{CR^{2}}$
up to a factor close to $\frac{1}{2}$, where $C$ depends on the parameters of
self-attention. It is important to notice that the particular measures that
are responsible for the explosion of this local Lipschitz bound, that are two-
dirac measures with an extremely unbalanced display of mass, require a massive
number of samples to be well approximated by empirical measures. Therefore,
one can ask the question of deriving input length-dependent bounds –
hopefully, such bounds could grow significantly slower than our general bound.
Moreover, we have pointed out a mass splitting phenomenon for self-attention,
which opens promising perspectives for future work. Can we understand better
which type of functions are subject to mass splitting? Is finite mass
splitting equivalent to regular mass splitting? The link between mass
splitting and adversarial attacks seems also interesting to explore. The idea
of perturbing a duplication of the input brings into play a new family of
attacks, but these take place in the feature space, i.e. just before the
encoder, and not in the token space. Is it possible to invert the positional
encoding and the tokenization embedding to derive attacks in the token space?
From a different perspective, it could also be efficient to do virtual
adversarial training [39] to robustify the network against this kind of
attacks.
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## Appendix A Optimal Transport Toolbox
This section gathers some useful definitions and lemmas from optimal
transport. In what follows, $\mathcal{X}$ is a Borel subset of
$\mathbb{R}^{d}$.
### A.1 Pushforward, Wasserstein Distance
Let us start with the notion of pushforward.
###### Definition A.1 (Pushforward).
Set $\mu$ a probability measure on $\mathcal{X}$ and
$\varphi\colon\mathcal{X}\to\mathcal{X}$ a measurable function. The
pushforward of $\mu$ by $\varphi$, denoted $\varphi_{\sharp}\mu$, is the
probability measure given by
$\left(\varphi_{\sharp}\mu\right)(B)=\mu(\varphi^{-1}(B))$
for any Borel set $B\subset\mathcal{X}$, where
$\varphi^{-1}(B)\coloneqq\\{x\in\mathbb{R}^{d}:\varphi(x)\in B\\}$.
The pushforward measure $\varphi_{\sharp}\mu$ can be seen as the result of a
transportation of the mass of $\mu$ by $\varphi$. Intuitively,
$\varphi_{\sharp}\mu$ is obtained by transporting each element of mass
$\mu(\mathrm{d}x)$ from $x$ to $\varphi(x)$.
Another crucial tool is the notion of Wasserstein distance.
###### Definition A.2 (Wasserstein space, Wasserstein distance).
Let $p\geq 1$. Denote
$\mathcal{P}_{p}(\mathcal{X})\coloneqq\\{\mu\in\mathcal{P}(\mathcal{X}):\int_{\mathcal{X}}\left\lvert
x\right\rvert^{p}\mathrm{d}\mu(x)<\infty\\}$
the $p$-Wasserstein space. Then, the $p$-Wasserstein distance between two
probability measures $\mu,\nu\in\mathcal{P}_{p}(\mathcal{X})$ is defined as
$W_{p}(\mu,\nu)\coloneqq\left(\inf_{\pi\in\Pi(\mu,\nu)}\int\left\lvert
x-y\right\rvert^{p}\mathrm{d}\pi(x,y)\right)^{1/p}$
with $\Pi(\mu,\nu)$ the set of all couplings between $\mu$ and $\nu$, i.e. of
all probability measures $\pi\in\mathcal{P}(\mathcal{X}\times\mathcal{X})$
such that $\int\pi(\cdot,y)\mathrm{d}y=\mu$ and
$\int\pi(x,\cdot)\mathrm{d}x=\nu$.
Wasserstein distances have the following nice property, that is a direct
consequence of Jensen inequality.
###### Lemma A.1.
For every $p\geq 1$, it holds
$W_{1}\leq W_{p}.$
The distance $W_{1}$ has also a simple dual formulation.
###### Lemma A.2 ($W_{1}$ duality formula).
The distance $W_{1}$ can be rewritten with the so-called duality formula: for
all $\mu,\nu\in\mathcal{P}_{1}(\mathcal{X})$, it holds
$W_{1}(\mu,\nu)=\sup_{\operatorname{\mathrm{Lip}}(\varphi)\leq
1}\int\varphi\,\mathrm{d}(\mu-\nu),$ (15)
where the supremum is taken over all functions
$\varphi\colon\mathcal{X}\to\mathbb{R}$ with a Lipschitz constant bounded by
one.
The following result is useful to bound the Wasserstein distance between two
probability measures that are pushed forward by the same map.
###### Lemma A.3.
Let $p\geq 1$. Consider a measurable function
$\varphi\colon\mathcal{X}\to\mathcal{X}$, and probability measures
$\mu,\nu\in\mathcal{P}_{p}(\mathcal{X})$ such that
$\varphi_{\sharp}\mu\in\mathcal{P}_{p}(\mathcal{X})$ and
$\varphi_{\sharp}\nu\in\mathcal{P}_{p}(\mathcal{X})$. Then, it holds
$W_{p}(\varphi_{\sharp}\mu,\varphi_{\sharp}\nu)\leq\operatorname{\mathrm{Lip}}(\varphi)W_{p}(\mu,\nu).$
###### Proof.
We have
$\displaystyle W_{p}(\varphi_{\sharp}\mu,\varphi_{\sharp}\nu)^{p}$
$\displaystyle=\inf_{\pi^{\prime}\in\Pi(\varphi_{\sharp}\mu,\varphi_{\sharp}\nu)}\int\left\lVert
x-y\right\rVert^{p}\mathrm{d}\pi^{\prime}(x,y)$
$\displaystyle\leq\inf_{\pi\in\Pi(\mu,\nu)}\int\left\lVert\varphi(x)-\varphi(y)\right\rVert^{p}\mathrm{d}\pi(x,y)$
$\displaystyle\leq\operatorname{\mathrm{Lip}}(\varphi)^{p}\inf_{\pi\in\Pi(\mu,\nu)}\int\left\lVert
x-y\right\rVert^{p}\mathrm{d}\pi(x,y)$
$\displaystyle=\operatorname{\mathrm{Lip}}(\varphi)^{p}W_{p}(\mu,\nu)^{p},$
where the first inequality derives from the fact that every
$\pi\in\Pi(\mu,\nu)$ induces a coupling
$\pi^{\prime}\in\Pi(\varphi_{\sharp}\mu,\varphi_{\sharp}\nu)$ by setting
$\pi^{\prime}(B_{1}\times
B_{2})\coloneqq\pi(\varphi^{-1}(B_{1})\times\varphi^{-1}(B_{2}))$
for all Borel sets $B_{1},B_{2}\subset\mathcal{X}$, and that with this choice
of $\pi^{\prime}$ it holds
$\int\left\lVert
x-y\right\rVert^{p}\mathrm{d}\pi^{\prime}(x,y)=\int\left\lVert\varphi(x)-\varphi(y)\right\rVert^{p}\mathrm{d}\pi(x,y).$
∎
Let us now bound the Wasserstein distance between two different pushforwards
of the same probability measure.
###### Lemma A.4.
Let $p\geq 1$. Consider two measurable functions
$\varphi,\psi\colon\mathcal{X}\to\mathcal{X}$, and a probability measure
$\mu\in\mathcal{P}_{p}(\mathcal{X})$ such that
$\varphi_{\sharp}\mu\in\mathcal{P}_{p}(\mathcal{X})$ and
$\psi_{\sharp}\mu\in\mathcal{P}_{p}(\mathcal{X})$. Then, it holds
$W_{p}(\varphi_{\sharp}\mu,\psi_{\sharp}\mu)\leq\left\lVert\varphi-\psi\right\rVert_{L^{p}(\mu)}.$
###### Proof.
Recall that
$W_{p}(\varphi_{\sharp}\mu,\psi_{\sharp}\mu)^{p}=\inf_{\pi^{\prime}\in\Pi(\varphi_{\sharp}\mu,\psi_{\sharp}\mu)}\int\left\lVert
x-y\right\rVert^{p}\mathrm{d}\pi^{\prime}(x,y).$
Now consider the following coupling between $\varphi_{\sharp}\mu$ and
$\psi_{\sharp}\mu$, defined by the relation
$\pi^{\prime}(B\times C)\coloneqq\mu(\varphi^{-1}(B)\cap\psi^{-1}(C))$
for every Borel sets $B,C\subset\mathcal{X}$. In other words, we set
$\mathrm{d}\pi^{\prime}(y,z)\coloneqq\int_{\varphi^{-1}(y)\cap\psi^{-1}(z)}\mathrm{d}\mu$,
and $\mathrm{d}\pi^{\prime}(y,z)=0$ if
$\varphi^{-1}(y)\cap\psi^{-1}(z)=\emptyset$. With this definition of
$\pi^{\prime}$, we have
$\displaystyle W_{p}(\varphi_{\sharp}\mu,\psi_{\sharp}\mu)^{p}$
$\displaystyle\leq\int\left\lVert
x-y\right\rVert^{p}\mathrm{d}\pi^{\prime}(x,y)$
$\displaystyle=\int\left\lVert\varphi(x)-\psi(x)\right\rVert^{p}\mathrm{d}\mu(x).$
∎
### A.2 Geodesics
The notion of geodesic is useful for the following section of the Appendix.
###### Definition A.3 (Geodesic).
Let $(\mathcal{E},d_{\mathcal{E}})$ be a metric space. A curve
$\gamma\colon[0,1]\to\mathcal{E}$ is called a geodesic if there exists a
constant $v\geq 0$ such that for all $t_{1},t_{2}\in[0,1]$ we have
$d_{\mathcal{E}}(\gamma(t_{1}),\gamma(t_{2}))=v\left\lvert
t_{2}-t_{1}\right\rvert.$
We say that the space $\mathcal{E}$ is a geodesic space if for any
$x,y\in\mathcal{E}$, there exists a geodesic between $x$ and $y$.
One important example of geodesic space is the 2-Wasserstein space
$\mathcal{P}_{2}(\mathbb{R}^{d})$.
###### Lemma A.5.
The space $\mathcal{P}_{2}(\mathbb{R}^{d})$ is a geodesic space.
For a proof of this result, see for instance [53].
## Appendix B Proofs of Section 3
### B.1 Proof of Lemma 3.1
###### Proof of Lemma 3.1.
The first inequality is straightforward. Let us assume that the space
$\mathcal{E}$ admits geodesics, and show that
$\operatorname{\mathrm{Lip}}(\varphi)=\sup_{x\in\mathcal{E}}\operatorname{\mathrm{Lip}}_{x}(\varphi).$
Recall that
$\operatorname{\mathrm{Lip}}_{x}(\varphi)\coloneqq\lim_{\varepsilon>0}\operatorname{\mathrm{Lip}}\left(\varphi_{\lvert
B(x,\varepsilon)}\right)$. Let us first show that for all $\varepsilon>0$, it
holds
$\operatorname{\mathrm{Lip}}(\varphi)=\sup_{x\in\mathcal{E}}\operatorname{\mathrm{Lip}}\left(\varphi_{\lvert
B(x,\varepsilon)}\right).$ (16)
Let $\varepsilon>0$ and $\eta>0$. Pick $x,y\in\mathcal{E}$ such that
$\frac{d_{\mathcal{E}}(\varphi(x),\varphi(y))}{d_{\mathcal{E}}(x,y)}\geq\operatorname{\mathrm{Lip}}(\varphi)-\eta$.
Let $\gamma\colon[0,1]\to\mathcal{E}$ a geodesic between $x$ and $y$, and
denote $z_{i}\coloneqq\gamma(i/n)$ for $0\leq i\leq n$, with $n$ chosen large
enough to have $d_{\mathcal{E}}(z_{i+1},z_{i})<\varepsilon$ for all $0\leq
i\leq n$. Then
$\displaystyle d_{\mathcal{E}}(\varphi(z_{n}),\varphi(z_{0}))$
$\displaystyle\leq\sum_{i=0}^{n-1}d_{\mathcal{E}}(\varphi(z_{i+1}),\varphi(z_{i}))$
$\displaystyle\leq\sup_{z\in\mathcal{E}}\operatorname{\mathrm{Lip}}\left(\varphi_{\lvert
B(z,\varepsilon)}\right)\sum_{i=0}^{n-1}d_{\mathcal{E}}(z_{i+1},z_{i})$
$\displaystyle=\sup_{z\in\mathcal{E}}\operatorname{\mathrm{Lip}}\left(\varphi_{\lvert
B(z,\varepsilon)}\right)d_{\mathcal{E}}(x,y)$
by definition of a geodesic. This implies that
$\sup_{z\in\mathcal{E}}\operatorname{\mathrm{Lip}}\left(\varphi_{\lvert
B(z,\varepsilon)}\right)\geq\operatorname{\mathrm{Lip}}(\varphi)-\eta$, and
therefore
$\sup_{z\in\mathcal{E}}\operatorname{\mathrm{Lip}}\left(\varphi_{\lvert
B(z,\varepsilon)}\right)\geq\operatorname{\mathrm{Lip}}(\varphi)$
by taking the limit $\eta\to 0^{+}$. The reverse inequality is
straightforward, which concludes the proof of Equation (16).
Now Equation (16) gives
$\operatorname{\mathrm{Lip}}(\varphi)=\sup_{x\in\mathcal{E}}\operatorname{\mathrm{Lip}}(\varphi_{\lvert
B(x,1)})\geq\sup_{x\in\mathcal{E}}\inf_{\varepsilon>0}\operatorname{\mathrm{Lip}}(\varphi_{\lvert
B(x,\varepsilon)})=\sup_{x\in\mathcal{E}}\operatorname{\mathrm{Lip}}_{x}(\varphi).$
It remains to prove the reverse inequality. Set $\eta>0$, and let $K_{\eta}$
be a compact set such that $\operatorname{\mathrm{Lip}}(\varphi_{\lvert
K_{\eta}})\geq\operatorname{\mathrm{Lip}}(\varphi)-\eta$. For all integers
$n\geq 1$, define $x_{n}\in K_{\eta}$ such that
$\operatorname{\mathrm{Lip}}(\varphi_{\lvert B(x_{n},1/n)})\geq\sup_{x\in
K_{\eta}}\operatorname{\mathrm{Lip}}(\varphi_{\lvert B(x_{n},1/n)\cap
K_{\eta}})-1/n.$
The sequence $(x_{n})$ lives in a compact set, so it admits a subsequential
limit: $x_{\varphi(n)}\to x^{*}$. Then, we have
$\sup_{x\in K_{\eta}}\lim_{\varepsilon\to
0+}\operatorname{\mathrm{Lip}}(\varphi_{\lvert
B(x,\varepsilon)})\geq\lim_{n\to+\infty}\operatorname{\mathrm{Lip}}(\varphi_{\lvert
B(x^{*},1/\varphi(n))})\geq\operatorname{\mathrm{Lip}}(\varphi_{\lvert
K_{\eta}}),$
which allows us to conlude by doing $\eta\to 0^{+}$. ∎
### B.2 Proof of Lemma 3.3
###### Proof of Lemma 3.3.
Set $X\in\mathbb{R}^{n\times d}$. One can choose $\varepsilon_{1}>0$ small
enough (for example $\varepsilon_{1}<\min_{x_{i}\neq x_{j}}\left\lVert
x_{i}-x_{j}\right\rVert/2$) to have
$\left\lVert X-Y\right\rVert_{F,p}\leq\varepsilon_{1}\Rightarrow\left\lVert
X-Y\right\rVert_{F,p}=W_{p}(m(X),m(Y)).$
Indeed, if
$\left\lVert X-Y\right\rVert_{F,p}\leq\varepsilon_{1}<\min_{x_{i}\neq
x_{j}}\left\lVert x_{i}-x_{j}\right\rVert/2,$
then for all $1\leq i\leq n$, we have $\left\lVert
x_{i}-y_{i}\right\rVert\leq\varepsilon_{1}$, and thus $x_{i}$ is the nearest
neighbour (or one of the nearest neighbours) of $y_{i}$ among the $x_{j}$.
Similarly, one can choose $\varepsilon_{2}>0$ small enough to have
$\left\lVert
f(X)-f(Y)\right\rVert_{F,p}\leq\varepsilon_{2}\Rightarrow\left\lVert
f(X)-f(Y)\right\rVert_{F,p}=W_{p}(m(f(X)),m(f(Y))).$
Then, we can set $\varepsilon\leq\varepsilon_{1}$ small enough to have
$\left\lVert X-Y\right\rVert_{F,p}\leq\varepsilon\Rightarrow\left\lVert
f(X)-f(Y)\right\rVert_{F,p}\leq\varepsilon_{2},$
and it holds, for all $\eta\leq\varepsilon$ and all $Y$ such that $\left\lVert
X-Y\right\rVert_{F,p}\leq\eta$, that
$\left\lVert X-Y\right\rVert_{F,p}=W_{p}(m(X),m(Y))$
and
$\left\lVert f(X)-f(Y)\right\rVert_{F,p}=W_{p}(m(f(X)),m(f(Y))).$
Now for all $\eta\leq\varepsilon$, we have
$\displaystyle\operatorname{\mathrm{Lip}}^{\left\lVert\cdot\right\rVert_{F,p}}f_{\lvert
B_{\left\lVert\cdot\right\rVert_{F,p}}(X,\eta)}$ $\displaystyle=\sup_{Y\in
B_{\left\lVert\cdot\right\rVert_{F,p}}(X,\eta)}\frac{\left\lVert
f(X)-f(Y)\right\rVert_{F,p}}{\left\lVert X-Y\right\rVert_{F,p}}$
$\displaystyle=\sup_{Y\in
B_{\left\lVert\cdot\right\rVert_{F,p}}(X,\eta)}\frac{W_{p}(m(f(X)),m(f(Y)))}{W_{p}(m(X),m(Y))}$
$\displaystyle=\sup_{Y\in
B_{\left\lVert\cdot\right\rVert_{F,p}}(X,\eta)}\frac{W_{p}(F(m(X)),F(m(Y)))}{W_{p}(m(X),m(Y))},$
by definition of $\varepsilon$ and $F$. We conclude the proof by noticing
that:
* $\bullet$
$Y\in B_{\left\lVert\cdot\right\rVert_{F,p}}(X,\eta)$ implies $m(Y)\in
B_{W_{p}}(m(X),\eta)$, which shows that
$\operatorname{\mathrm{Lip}}^{\left\lVert\cdot\right\rVert_{F,p}}f_{\lvert
B_{\left\lVert\cdot\right\rVert_{F,p}}(X,\eta)}\leq\operatorname{\mathrm{Lip}}^{W_{p}}F_{\lvert\mathcal{M}_{n}(\mathbb{R}^{d})}$,
* $\bullet$
$\mu\in B_{W_{p}}(m(X),\eta)$ with $\mu\in\mathcal{M}_{n}(\mathbb{R}^{d})$
implies the existence of $Y\in\mathbb{R}^{n\times d}$ such that $\mu=m(Y)$ and
$\left\lVert X-Y\right\rVert_{F,p}=W_{p}(m(X),m(Y))$, so that $Y\in
B_{\left\lVert\cdot\right\rVert_{F,p}}(X,\eta)$. Indeed, take $Y$ such that
$\mu_{n}=m(Y)$ and then permute its coordinates so that $x_{i}$ becomes a
nearest neighbour of $y_{i}$. This shows the reverse inequality, and concludes
the proof.
∎
### B.3 Proof of Proposition 3.7
###### Proof of Proposition 3.7.
Let $\bar{\mu}$ and $\bar{\nu}$ be two distinct measures in ${\cal
P}([0,1]\times B_{R})$. Assume that $\bar{\mu}$ and $\bar{\nu}$ have the same
first marginal, denoted $\theta$ (otherwise, we consider that they are
associated to an infinite Lipschitz ratio). We have
$\displaystyle d_{p}(F^{m}(\bar{\mu}),F^{m}(\bar{\nu}))$ $\displaystyle\leq
d_{p}((\mathrm{Id}+\Gamma_{\bar{\mu}})_{\sharp}\bar{\mu},(\mathrm{Id}+\Gamma_{\bar{\mu}})_{\sharp}\bar{\nu})+d_{p}((\mathrm{Id}+\Gamma_{\bar{\mu}})_{\sharp}\bar{\nu},(\mathrm{Id}+\Gamma_{\bar{\nu}})_{\sharp}\bar{\nu})$
$\displaystyle\leq\left(\int_{0}^{1}W_{p}\left((\mathrm{Id}+\Gamma_{\bar{\mu}}^{s})_{\sharp}\mu^{s},(\mathrm{Id}+\Gamma_{\bar{\mu}}^{s})_{\sharp}\nu^{s}\right)^{p}\right)^{1/p}$
$\displaystyle\phantom{space}+\left(\int_{0}^{1}W_{p}\left(\mathrm{Id}+\Gamma_{\bar{\mu}}^{s})_{\sharp}\nu^{s},(\mathrm{Id}+\Gamma_{\bar{\nu}}^{s})_{\sharp}\nu^{s}\right)\right)^{1/p},$
where we denote
$\Gamma_{\bar{\mu}}^{s}(x):=\frac{\int VyG(x,y)\mathbf{1}_{\tau\leq
s}\mathrm{d}\bar{\mu}(\tau,y)}{\int G(x,y)\mathbf{1}_{\tau\leq
s}\mathrm{d}\bar{\mu}(\tau,y)}.$
Using Lemma A.3, we get
$W_{p}\left((\mathrm{Id}+\Gamma_{\bar{\mu}}^{s})_{\sharp}\mu^{s},(\mathrm{Id}+\Gamma_{\bar{\mu}}^{s})_{\sharp}\nu^{s}\right)\leq\operatorname{\mathrm{Lip}}(\mathrm{Id}+\Gamma_{\bar{\mu}}^{s})W_{p}(\mu^{s},\nu^{s})$
where the Lipschitz constant is taken on $B_{R}$. A similar computation as for
traditional self-attention shows that for all $0\leq s\leq 1$ and $x\in B_{R}$
we have
$\left\lVert D_{x}\Gamma^{s}\right\rVert_{2}\leq\left\lVert
V\right\rVert_{2}\left\lVert A\right\rVert_{2}R^{2},$
so that
$\operatorname{\mathrm{Lip}}(\mathrm{Id}+\Gamma_{\bar{\mu}}^{s})\leq
1+\left\lVert V\right\rVert_{2}\left\lVert A\right\rVert_{2}R^{2}.$
To bound the second term in the previous inequality, we use Lemma A.4, to get
$W_{p}\left((\mathrm{Id}+\Gamma_{\bar{\mu}}^{s})_{\sharp}\nu^{s},(\mathrm{Id}+\Gamma_{\bar{\nu}}^{s})_{\sharp}\nu^{s}\right)\leq\left\lVert\Gamma_{\bar{\mu}}^{s}-\Gamma_{\bar{\nu}}^{s}\right\rVert_{L^{\infty}(B_{R})}.$
Again, a similar computation as for traditional self-attention shows that
$\left\lVert\Gamma_{\bar{\mu}}^{s}-\Gamma_{\bar{\nu}}^{s}\right\rVert_{L^{\infty}(B_{R})}\leq\left\lVert
V\right\rVert_{2}(1+2\left\lVert A\right\rVert_{2}R^{2})e^{2\left\lVert
A\right\rVert_{2}R^{2}}W_{p}(\mu^{s},\nu^{s}),$
which concludes the proof. ∎
### B.4 Weighted Self-Attention
Introducing $f_{a}$ amounts to looking at $f$ on the space $\mathbb{R}^{nd}$
equipped with a new scalar product $\left\langle\cdot,\cdot\right\rangle_{a}$,
defined as follows.
$\left\langle X,Y\right\rangle_{a}\coloneqq\sum_{i=1}^{n}a_{i}\left\langle
x_{i},y_{i}\right\rangle,$
with $X=(x_{1},\dots,x_{n})^{\top}$ and $Y=(y_{1},\dots,y_{n})^{\top}$. This
change of scalar product also changes the definition of the norm, the operator
norm and the Jacobian, which are central in the analysis of the Lipschitz
properties of $f_{a}$. We have the following expressions for these quantities
in the case of weighted self-attention in its vectorized form.
###### Lemma B.1.
Let $a\in\Sigma_{n}$. Equip the vector space $\mathbb{R}^{nd}$ with the
following scalar product:
$\left\langle\vec{X},\vec{Y}\right\rangle_{a}\coloneqq\sum_{i=1}^{n}a_{i}\left\langle
x_{i},y_{i}\right\rangle$
where $\vec{X}$ and $\vec{Y}$ are the vectorized forms of the matrices
$X=(x_{1},\dots,x_{n})^{\top}$ and $Y=(y_{1},\dots,y_{n})^{\top}$, with row-
wise vectorization. We have the following properties.
1. (i)
The norm associated with the scalar product
$\left\langle\cdot,\cdot\right\rangle_{a}$, denoted
$\left\lVert\cdot\right\rVert_{a}$, reads
$\left\lVert\vec{X}\right\rVert_{a}^{2}=\sum_{i=1}^{n}a_{i}\left\lVert
x_{i}\right\rVert^{2}=\left\lVert\operatorname{\mathrm{diag}}(\sqrt{a}\otimes
1_{d})\vec{X}\right\rVert^{2},$
where the square root is taken coordinatewise.
2. (ii)
The scalar product $\left\langle\cdot,\cdot\right\rangle_{a}$ induces the
following action $\cdot_{a}$ of linear operators on vectors of
$\mathbb{R}^{nd}$:
$M\cdot_{a}\vec{X}=M\operatorname{\mathrm{diag}}(a\otimes 1_{d})\vec{X},$
for all $M\in\mathbb{R}^{nd\times nd}$.
3. (iii)
The induced operator norm is
$\left\lVert
M\right\rVert_{2,a}=\sup_{\left\lVert\vec{X}\right\rVert_{a}=1}\left\lVert
M\cdot_{a}\vec{X}\right\rVert_{a}=\left\lVert\operatorname{\mathrm{diag}}(\sqrt{a}\otimes
1_{d})M\operatorname{\mathrm{diag}}(\sqrt{a}\otimes 1_{d})\right\rVert_{2}.$
4. (iv)
If all coordinates of $a$ are non zero, the Jacobian of $f_{a}$ on the space
$(\mathbb{R}^{nd},\left\langle\cdot,\cdot\right\rangle_{a})$, denoted
$D^{a}f_{a}$, is equal to
$D_{\vec{X}}^{a}f_{a}=(D_{\vec{X}}f_{a})\operatorname{\mathrm{diag}}(a\otimes
1_{d})^{-1}.$
The proof consists in short computations we do not report here.
### B.5 Proof of Proposition 3.5
We start with the following lemma, that derives from a simple computation.
###### Lemma B.2.
Let $\mu\coloneqq p\delta_{x}+(1-p)\delta_{y}$, with $p\in(0,1)$ and
$x,y\in\mathbb{R}^{d}$. Denote $a\coloneqq(p,1-p)$ and
$P=:\left(\begin{smallmatrix}\pi_{1}&1-\pi_{1}\\\
1-\pi_{2}&\pi_{2}\end{smallmatrix}\right)$, recalling that
$P\coloneqq\operatorname{\mathrm{softmax}}(XA^{\top}X^{\top})$ with
$X=\left(\begin{smallmatrix}x^{\top}\\\ y^{\top}\end{smallmatrix}\right)$.
Then
$D_{X}f_{a}=\begin{pmatrix}B_{1}&B_{2}\\\ B_{3}&B_{4}\end{pmatrix}$
with
$\displaystyle B_{1}=\pi_{1}(1-\pi_{1})(x-y)(2x-y)^{\top}A+\pi_{1}I_{d},$
$\displaystyle B_{2}=-\pi_{1}(1-\pi_{1})(x-y)x^{\top}A+(1-\pi_{1})I_{d},$
$\displaystyle B_{3}=-\pi_{2}(1-\pi_{2})(y-x)y^{\top}A+(1-\pi_{2})I_{d},$
$\displaystyle B_{4}=\pi_{2}(1-\pi_{2})(y-x)(2y-x)^{\top}A+\pi_{2}I_{d}.$
Now, as we have assumed symmetry of the matrix $A$, we can write
$A=\sum\gamma_{i}u_{i}u_{i}^{\top},$
with $\gamma_{1}\geq\dots\geq\gamma_{d}$ the eigenvalues of $A$, and
$(u_{1},\dots,u_{d})$ an orthonormal basis of associated eigenvectors. Let us
prove Proposition 3.5 only in the first case, i.e. when
$p=p_{R}=e^{-2CR^{2}}$, and $x=Ru_{1}$ and $y=\frac{R}{2}u_{1}$. The other
case can be proven with the same method. Observing that $\pi_{1}\to 1$ and
$\pi_{2}\to 1/2$, we compute an equivalent of each block of $D_{X}f_{a}$ when
$R\to+\infty$.
$\displaystyle B_{1}$ $\displaystyle\to I_{d}$ $\displaystyle B_{2}$
$\displaystyle\sim-\frac{\gamma R^{2}}{2}e^{-\frac{\gamma
R^{2}}{4}}u_{1}u_{1}^{\top}\to 0_{d}$ $\displaystyle B_{3}$
$\displaystyle\sim\frac{\gamma R^{2}}{16}u_{1}u_{1}^{\top}$ $\displaystyle
B_{4}$ $\displaystyle\to\frac{1}{2}I_{d}.$
Equations (iii) and (iv) of Lemma B.1 allow us to conclude the proof, noticing
that
$\left\lVert\begin{pmatrix}\mathcal{O}(1)&{\cal O}(1)\\\ \frac{\gamma
R^{2}}{16}e^{\frac{\gamma R^{2}}{8}}u_{1}u_{1}^{\top}&{\cal
O}(1)\end{pmatrix}\right\rVert_{p}\sim\frac{\gamma R^{2}}{16}e^{\frac{\gamma
R^{2}}{8}}.$
## Appendix C Proofs of Section 4
### C.1 Effect of Duplication on the Jacobian of Self-Attention
Let us start by stating a formula for the Jacobian of $f$ at
$X\coloneqq(x_{1},\dots,x_{n})^{\top}$.
###### Lemma C.1 ([26]).
The Jacobian of (the vectorized form of) matrix self-attention $f$ is of the
following form:
$D_{X}f=(I_{n}\otimes V)\left(P\otimes
I_{d}+D\otimes_{d}A+R\otimes_{d}A^{\top}\right),$ (17)
with
* $\bullet$
$P\coloneqq\operatorname{\mathrm{softmax}}(XA^{\top}X^{\top})$,
* $\bullet$
$D$ a block-diagonal matrix, with $n$ diagonal blocks of shape $d\times d$
given by $\operatorname{\mathrm{Var}}^{(i)}Z$ for $1\leq i\leq n$, where
$\operatorname{\mathrm{Var}}^{(i)}Z$ is the variance of the random variable
$Z$ following the law $\mathbb{P}(Z=x_{j})=P_{ij}$,
* $\bullet$
$R=(R_{ij})_{1\leq i,j\leq n}$ a block matrix with $R_{ij}\coloneqq
P_{ij}\left(x_{j}-\sum_{k=1}^{n}P_{ik}x_{k}\right)x_{i}^{\top}\in\mathbb{R}^{d\times
d}$.
We can then write the Jacobian of $f$ at
$Y\coloneqq\left(\begin{smallmatrix}1\\\ 1\end{smallmatrix}\right)\otimes X$
in function of $D_{X}f$.
###### Lemma C.2.
Let $X\in\mathbb{R}^{n\times d}$, and denote
$Y\coloneqq\left(\begin{smallmatrix}1\\\ 1\end{smallmatrix}\right)\otimes X$.
The point cloud $Y$ is simply the duplicated version of $X$, and is associated
to the same measure as $X$. Assume that $V=I_{d}$. Then, we have
$D_{Y}f_{2n}=\frac{1}{2}\begin{pmatrix}1&1\\\ 1&1\end{pmatrix}\otimes
D_{X}f_{n}+\frac{1}{2}\begin{pmatrix}1&-1\\\
-1&1\end{pmatrix}\otimes(D\otimes_{d}A),$ (18)
where the matrix $D$ is defined in Lemma C.1. Therefore, we have
$\left\lVert D_{Y}f_{2n}\right\rVert_{2}>\left\lVert
D_{X}f_{n}\right\rVert_{2}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{if and only
if}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \left\lVert D\otimes_{d}A\right\rVert_{2}>\left\lVert
D_{X}f_{n}\right\rVert_{2}.$
### C.2 Concentrated Measures Are Not Splitted
Lemma C.2 allows us to show that empirical measures that are concentrated
enough around their expectation cannot be splitted by matrix self-attention.
###### Proposition C.3.
Assume that $k=d$, and that $V=A=I_{d}$. Let a matrix $X\in\mathbb{R}^{n\times
d}$ whose rows belong to a ball of radius 1. Denote
$Y\coloneqq(\begin{smallmatrix}1\\\ 1\end{smallmatrix})\otimes X$. Then
$\operatorname{\mathrm{Lip}}_{X}f_{n}=\operatorname{\mathrm{Lip}}_{Y}f_{2n}.$
###### Proof.
We start with a useful lemma.
###### Lemma C.4.
Let $Z$ be a random variable supported in
$\,\overline{\\!{B}}(x_{0},r)\subset\mathbb{R}^{d}$, with any
$x_{0}\in\mathbb{R}^{d}$ and $r>0$. Then, denoting
$\operatorname{\mathrm{Var}}Z=\mathbb{E}[(Z-\mathbb{E}Z)(Z-\mathbb{E}Z)^{\top}]$
we have
$\left\lVert\operatorname{\mathrm{Var}}Z\right\rVert_{2}\leq r^{2},$
with equality when $\mu_{Z}=\frac{1}{2}(\delta_{x_{0}+x}+\delta_{x_{0}-x})$
with $x\in\mathbb{R}^{d}$ such that $\left\lVert x\right\rVert=r$.
###### Proof.
Let us assume without loss of generality that $x_{0}=0$. It is straightforward
to check that if $\mu_{Z}=\frac{1}{2}(\delta_{x}+\delta_{-x})$ then
$\left\lVert\operatorname{\mathrm{Var}}Z\right\rVert_{2}=r^{2}$. To show that
this is the maximal value the variance can take, we use the triangle
inequality:
$\displaystyle\left\lVert\mathbb{E}[(Z-\mathbb{E}Z)(Z-\mathbb{E}Z)^{\top}]\right\rVert_{2}$
$\displaystyle\leq\mathbb{E}\left\lVert(Z-\mathbb{E}Z)(Z-\mathbb{E}Z)^{\top}\right\rVert_{2}$
$\displaystyle=\mathbb{E}[\left\lVert Z-\mathbb{E}Z\right\rVert^{2}]$
$\displaystyle=\mathbb{E}[\left\lVert
Z\right\rVert^{2}]-\left\lVert\mathbb{E}Z\right\rVert^{2}.$
Now let us pick any $x\in B_{R}\setminus B(0,r)$. We have
$\mathbb{E}(Z-x)^{\top}(Z+x)\leq 0$, as the angle between the vectors $Z-x$
and $Z+x$ is at least $\pi/2$ for $Z$ in $B(0,r)$. By expanding this relation
we get
$E[\left\lVert Z\right\rVert^{2}]-\left\lVert x\right\rVert^{2}\leq 0,$
which yields the result.
∎
Let us now move on to the proof of the Proposition. We first show that for any
number of samples $n$ and any matrix $X\in\mathbb{R}^{n\times d}$, we have
$\left\lVert D_{X}f_{n}\right\rVert_{2}\geq 1$. Recall Lemma C.1:
$D_{X}f=P\otimes I_{d}+D+R.$
One can easily see that $R(1_{n}\otimes I_{d})=0$, where $1_{n}$ is the vector
of size $n$ with all coordinates equal to $1$, so that
$\displaystyle D_{X}f_{n}(1_{n}\otimes I_{d})$ $\displaystyle=P(1_{n}\otimes
I_{d})+D(1_{n}\otimes I_{d})$ $\displaystyle=1_{n}\otimes
I_{d}+\begin{pmatrix}(\operatorname{\mathrm{Var}}^{(1)}Z)\\\ \vdots\\\
(\operatorname{\mathrm{Var}}^{(n)}Z)\end{pmatrix}.$
Then for example $\left\lVert D_{X}f_{n}(1_{n}\otimes
e_{1})\right\rVert^{2}\geq\sum_{i=1}^{n}\left(1+D_{11}\right)^{2}\geq
n=\left\lVert 1_{n}\otimes e_{1}\right\rVert^{2}$, where $e_{1}$ is the first
vector of the canonical basis $(e_{1},\dots,e_{d})$ of $\mathbb{R}^{d}$,
because $(\operatorname{\mathrm{Var}}^{(i)}Z)_{11}\geq 0$ for any index
$i$.333This quantity is indeed the variance of a scalar random variable. This
proves that $\left\lVert D_{X}f_{n}\right\rVert_{2}\geq 1$.
Now thanks to Lemma C.2, to conclude that $X$ is not splitted by self-
attention, it is enough to show that for all $1\leq i\leq n$, we have
$\max_{1\leq i\leq
n}\left\lVert\operatorname{\mathrm{Var}}^{(i)}Z\right\rVert_{2}\leq\left\lVert
D_{X}f_{n}\right\rVert_{2},$ (19)
so that the maximal eigenvalue of $(D_{Y}f_{2n})^{\top}D_{Y}f_{2n}$ comes from
$(D_{X}f_{n})^{\top}D_{X}f_{n}$, and
$\operatorname{\mathrm{Lip}}_{Y}f_{2n}=\left\lVert
D_{Y}f_{2n}\right\rVert_{2}=\left\lVert
D_{X}f_{n}\right\rVert_{2}=\operatorname{\mathrm{Lip}}_{X}f_{n}$. And property
(19) follows from Lemma C.4, together with the fact that $\left\lVert
D_{X}f_{n}\right\rVert_{2}\geq 1$. ∎
### C.3 Proof of Theorem 4.2
###### Proof of Theorem 4.2.
Using decomposition (17), we define the matrices $D^{N}$ and $P^{N}$ and
$R^{N}$ such that
$D_{Y^{N}}f=D^{N}\otimes_{d}A+P^{N}\otimes I_{d}+R^{N}\otimes_{d}A^{\top}$
for any positive integer $N$. One can check that
$\begin{cases}P^{N}&=\frac{1}{N}1_{N\times N}\otimes P^{1},\\\
D^{N}&=\operatorname{\mathrm{diag}}(1_{N})\otimes D^{1},\\\
R^{N}&=\frac{1}{N}1_{N\times N}\otimes R^{1},\end{cases}$
where $1_{N}$ and $1_{N\times N}$ are respectively is the vector of dimension
$N$ and the matrix of shape $N\times N$ with all coordinates equal to 1.
Hence,
$\displaystyle D_{Y^{N}}f$
$\displaystyle=\operatorname{\mathrm{diag}}(1_{N})\otimes
D^{1}\otimes_{d}A+\frac{1}{N}1_{N\times N}\otimes(P^{1}\otimes
I_{d}+R^{1}\otimes A^{\top})$ $\displaystyle=G_{N}\otimes
D^{1}\otimes_{d}A+\frac{1}{N}1_{N\times N}\otimes D_{X}f,$
with
$G_{N}=\frac{N-1}{N}\begin{pmatrix}1&-\frac{1}{N-1}&\cdots&-\frac{1}{N-1}\\\
-\frac{1}{N-1}&\ddots&\ddots&\vdots\\\ \vdots&\ddots&\ddots&-\frac{1}{N-1}\\\
-\frac{1}{N-1}&\cdots&-\frac{1}{N-1}&1\end{pmatrix}.$
Noticing that
$G_{N}1_{N\times N}=1_{N\times N}G_{N}=0,$
and that
$G_{N}^{2}=G_{N}\mbox{ and }\frac{1}{N^{2}}1_{N\times
N}^{2}=\frac{1}{N}1_{N\times N},$
we get
$D_{Y^{N}}f^{\top}D_{Y^{N}}f=G_{N}\otimes(D^{1}\otimes_{d}A)^{\top}(D^{1}\otimes_{d}A)+\frac{1}{N}1_{N\times
N}\otimes D_{X}f^{\top}D_{X}f.$
Computing the characteristic polynomial of $G_{N}$, one sees that this matrix
has two eigenvalues: $1$ and $\frac{N-1}{N}$. The eigenspace associated to $1$
has dimension $N-1$, and is equal to the set
$\\{x\in\mathbb{R}^{N}\mid\sum_{i=1}^{N}x_{i}=0\\}$, which is included in the
kernel of the matrix $1_{N\times N}$. This leads to the following
observations:
* $\bullet$
The largest eigenvalue of the matrix $D_{Y^{N}}f^{\top}D_{Y^{N}}f$, which is
also equal to the squared operator norm $\left\lVert
D_{Y^{N}}\right\rVert_{2}^{2}$, is equal to
$\max\left(\left\lVert D_{X}f\right\rVert_{2}^{2},\max_{1\leq i\leq
n}\left\lVert(\operatorname{\mathrm{Var}}^{(i)}Z)A\right\rVert_{2}^{2}\right).$
* $\bullet$
When there is mass splitting at $X$, namely when $\max_{1\leq i\leq
n}\left\lVert(\operatorname{\mathrm{Var}}^{(i)}Z)A\right\rVert_{2}^{2}>\left\lVert
D_{X}f\right\rVert_{2}^{2}$, we have $\left\lVert
D_{Y^{N}}f\right\rVert_{2}=\left\lVert D_{Y^{2}}f\right\rVert_{2}>\left\lVert
D_{X}f\right\rVert_{2}$ for all $N\geq 2$.
* $\bullet$
If the mass splits at $X$, then the local Lipschitz constant at $X$ can be
reached by splitting equally the mass of the $i$-th particle of $X$ along the
first singular vector of the matrix $(\operatorname{\mathrm{Var}}^{(i)}Z)A$,
with
$i\coloneqq\operatorname{\mathrm{argmax}}_{i^{\prime}\in\\{1,\dots,n\\}}\left\lVert(\operatorname{\mathrm{Var}}^{(i^{\prime})}Z)A\right\rVert_{2}^{2}.$
∎
|
††thanks: The indicated authors are joint first authors††thanks: The indicated
authors are joint first authors
# Optical and atomic decoherence in entangled atomic ensembles generated by
quantum nondemolition measurements
Shuai Gao Joint International Research Laboratory of Information Display and
Visualization, School of Electronic Science and Engineering, Southeast
University, Nanjing 210096, China State Key Laboratory of Precision
Spectroscopy, School of Physical and Material Sciences, East China Normal
University, Shanghai 200062, China Shuang Li State Key Laboratory of
Precision Spectroscopy, School of Physical and Material Sciences, East China
Normal University, Shanghai 200062, China Manish Chaudhary State Key
Laboratory of Precision Spectroscopy, School of Physical and Material
Sciences, East China Normal University, Shanghai 200062, China New York
University Shanghai, 1555 Century Ave, Pudong, Shanghai 200122, China Matthew
Prest New York University Shanghai, 1555 Century Ave, Pudong, Shanghai
200122, China Ebubechukwu O. Ilo-Okeke New York University Shanghai, 1555
Century Ave, Pudong, Shanghai 200122, China Department of Physics, School of
Science, Federal University of Technology, P. M. B. 1526, Owerri, Imo state
460001, Nigeria Valentin Ivannikov New York University Shanghai, 1555
Century Ave, Pudong, Shanghai 200122, China Tim Byrnes<EMAIL_ADDRESS>New York University Shanghai, 1555 Century Ave, Pudong, Shanghai 200122, China
State Key Laboratory of Precision Spectroscopy, School of Physical and
Material Sciences, East China Normal University, Shanghai 200062, China NYU-
ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road North, Shanghai
200062, China Center for Quantum and Topological Systems (CQTS), NYUAD
Research Institute, New York University Abu Dhabi, UAE. Department of
Physics, New York University, New York, NY 10003, USA
###### Abstract
We study the effects of decoherence in the form of optical phase diffusion,
photon loss and gain, and atomic dephasing in entangled atomic ensembles
produced via quantum nondemolition (QND) measurements. For the optical
decoherence channels, we use the technique of integration within ordered
operators (IWOP) to obtain the Kraus operators that describe the decoherence.
We analyze the effect of different decoherence channels on a variety of
quantities such as the variances of the spin operators, entanglement and
correlation criteria, logarithmic negativity, and the Bell-CHSH inequality. We
generally find a smooth decay of correlations and entanglement in the presence
of decoherence. We find that various quantities retain showing non-classical
properties under all three types of decoherence, in the short interaction time
range. Our results show that such QND measurements are one of the most
promising methods for entanglement generation between two Bose-Einstein
condensates.
## I Introduction
Quantum entanglement [1, 2, 3, 4] is one of the central concepts in quantum
mechanics, and in quantum information science is considered to be an
indispensable resource to perform tasks that exceed the performance of
classical physics [5, 6, 7, 8, 9, 10, 11, 12, 13]. Quantum entanglement has
been generally associated with the microscopic world, but with the progress of
technology, quantum entanglement has been observed increasingly in the
macroscopic world [14, 15, 16, 17] Spin squeezing is a prime example of a
practical application of such macroscopic entanglement [18, 19, 20, 21], where
the quantum noise of an observable is suppressed below the standard quantum
limit [22, 14, 15, 23, 24, 25]. Optical systems were the first system where
such squeezed states were realized in experiment [26, 27, 28]. One well-
studied way that such squeezed states can be produced is through nonlinear
effects such as the Kerr effect [29, 30]. There are however other ways in
which squeezing can be generated, including quantum nondemolition (QND)
measurements [31], adiabatic transitions [32], state-dependent forces [33],
Rydberg excitations [34], splitting a single squeezed BEC [35], cavity QED
[17, 36, 32, 37, 38]. Squeezed states have also been obtained in some other
systems such as atomic ensembles [39, 40]. Using QND measurements squeezed
states have been generated in atomic ensembles [41, 42, 43, 44, 45]. There are
some other systems like mechanical resonators where squeezing has also been
experimentally realized [46, 47, 48, 49, 50, 51].
Squeezed states are a type of multipartite entangled state [52] where the
collective degrees of freedom, rather than the individual identity of the
underlying particles is the quantity of interest. In the context of Bell
correlations, however, the identity and the location of the particules is a
crucial property and underlies quantification of the non-local correlations
using the Clauser-Horne-Shimony-Holt (CHSH) inequality[53]. For multipartite
states, the two-mode squeezed state is one of the most important type of
squeezed states which exhibit Einstein-Podolsky-Rosen (EPR) correlations. Such
states can also be produced in other systems in addition to optical systems,
where the first demonstrations of atomic ensembles were pioneered by Polzik
and co-workers [54, 55, 56, 57, 58, 59, 60]. In these experiments, two
spatially separated Cesium gas clouds [61] were entangled using QND
measurements [62, 63, 64, 58, 65, 66]. Applications of the entanglement, such
as quantum teleportation for continuous variables [67] was successfully
realized [68, 54, 69]. For Bose-Einstein condensates (BECs), the creation of
many-particle entanglement in single BECs has been studied in various contexts
[70, 71, 72, 73]. Entanglement in different spatial regions of a single BEC
has been observed [14, 15, 16, 17]. However, the entanglement between two
spatially distinct BECs has not been realized in experiments to date. Numerous
entangling schemes between BECs have been proposed, ranging from methods using
adiabatic transitions [32], state-dependent forces [33], Rydberg excitations
[34], splitting a single squeezed BEC [35], and cavity QED [17, 36, 32, 37,
38]. Generating entanglement is elementary to various quantum information
tasks based on atomic ensembles, such as quantum teleportation [74, 75],
remote state preparation, clock synchronization [76, 77], and quantum
computing [78, 79, 38].
In this paper, we calculate the effects of decoherence in QND measurement-
based entanglement generation between atomic ensembles [80, 81, 82].
Previously, a protocol generating entanglement between BECs based on QND
measurement induced entanglement was theoretically analyzed [80]. The key
difference of this theory of QND measurements in comparison to past theories
[23, 56, 59, 83, 84] is that the QND dynamics is exactly solved such that
Holstein-Primakoff (HP) approximation is not used, and non-Gaussian effects
can be examined. Another difference is the geometry of the light that is used
(see Fig. 1) where a Mach-Zehnder configuration is used, instead of the
sequential configuration of [85, 56]. This is advantageous in the case where
there are many ensembles, as the entanglement can be generated in a scalable
all-to-all configuration. We investigate the effect of optical phase
diffusion, loss and gain of the light, and atomic dephasing on the entangled
state that is generated by the QND scheme. For the optical decoherence
channels, we use the integral within the ordered operator (IWOP) approach [86,
87] to analytically evaluate the decoherence effects. This allows for a direct
derivation of the Kraus operators for the decoherence channels, giving a
powerful way of evaluating the effects. We note that we have previously
performed a study of decoherence on QND-induced entangled states of BECs [81].
This work differs from our previous work in that we analyze phase diffusion
and incoherent gain which was not done before. For the atomic dephasing, we
also analyze several different quantities that were not calculated before,
such as the violation of CHSH inequality [88, 89].
The structure of this paper is as follows. Sec. II briefly reviews the QND
entangling protocol, defining the physical model and describing the basic
system we are working on. Sec. III examines the effect of optical phase
diffusion on the optical modes involved in the QND measurement. In Sec. IV,
the effects of photonic loss and gain are examined. In Sec. V, we analyze the
effects of atomic dephasing and show its effect on the Wineland squeezing
parameter and the Bell-CHSH inequality. Finally, the results are then
summarized in Sec. VI.
Figure 1: Experimental scheme for generating entanglement between two atomic
ensembles via QND measurements. A coherent light pulse $|\alpha\rangle$ enters
the first beam splitter of a Mach-Zehnder interferometer. Each atomic ensemble
or BEC has two relevant internal energy states that are populated by atoms,
with corresponding bosonic annihilation operators $g_{j},e_{j}$, where
$j\in\\{1,2\\}$ labels each atomic ensemble or BEC. The optical mode is split
into two modes labeled by $a_{j}$, $j\in\\{1,2\\}$ after the first beam
splitter. Then each optical mode interacts with the atoms via the QND
Hamiltonian. After the second beam splitter, the photons are detected with
counts $n_{c}$ and $n_{d}$.
## II QND Entangling scheme
In this section, we briefly summarize the theory for QND-induced entanglement
generation between two atomic ensembles or BECs [80].
### II.1 QND Measurement Model and Hamiltonian
We first give a brief description of the QND entangling scheme as shown in
Fig. 1. Two atomic ensembles or BECs are placed in well-separated traps such
as separate magnetic traps on an atom chip or two optical dipole traps [90,
91, 38]. The atoms have two internal energy states that are populated. For
instance, a suitable choice of the two internal states for ${}^{87}\text{Rb}$
could be two hyperfine ground states ($F=1,m_{F}=-1$ and $F=2,m_{F}=+1$) of an
atom. These have bosonic annihilation operators $g_{j},e_{j}$ respectively,
where $j\in\\{1,2\\}$ is used to distinguish the two atomic ensembles or BECs.
We introduce the collective spin using the Schwinger boson operators, which
are defined by the form as follows
$\displaystyle S^{x}_{j}$
$\displaystyle=e^{{\dagger}}_{j}g_{j}+g^{{\dagger}}_{j}e_{j},$ $\displaystyle
S^{y}_{j}$ $\displaystyle=-ie^{{\dagger}}_{j}g_{j}+ig^{{\dagger}}_{j}e_{j},$
$\displaystyle S^{z}_{j}$
$\displaystyle=e^{{\dagger}}_{j}e_{j}-g^{{\dagger}}_{j}g_{j}.$ (1)
The commutation relations for Schwinger boson operators are
$[S^{l},S^{m}]=2i\epsilon_{lmn}S^{n}$, where $\epsilon_{lmn}$ is the
completely anti-symmetric Levi-Civita tensor with $l,m,n\in\\{x,y,z\\}$. The
spin operators (II.1) most directly describe atoms within a BEC, where the
atoms are indistinguishable bosons. The algebra of spin operators for an
ensemble of distinguishable atoms is equivalent as long as the state of the
system remains in the subspace that is symmetric under particle interchange
[40]. Hence we will use the above formalism to equally describe an atomic
ensemble as well as a BEC.
We define the spin coherent state of the atomic ensemble or BECs to be
$\displaystyle|\theta,\phi\rangle\rangle_{j}$
$\displaystyle\equiv\frac{1}{\sqrt{N!}}\left(e^{\dagger}_{j}\cos\frac{\theta}{2}+e^{i\phi}g^{\dagger}_{j}\sin\frac{\theta}{2}\right)^{N}|\text{vac}\rangle$
$\displaystyle=\sum_{k=0}^{N}\sqrt{\binom{N}{k}}\cos^{k}(\frac{\theta}{2})\sin^{N-k}(\frac{\theta}{2})e^{i(N-k)\phi}|k\rangle_{j}.$
(2)
The direction of the polarization is given by the angles $0\leq\theta\leq\pi$,
$-\pi\leq\phi\leq\pi$ on the Bloch sphere. The initial state of atoms is taken
to be spins polarized in the $x$-direction
$\displaystyle|\frac{\pi}{2},0\rangle\rangle_{1}|\frac{\pi}{2},0\rangle\rangle_{2}.$
(3)
Meanwhile, the Fock states on the $j$th atomic ensemble are defined as
$\displaystyle{|k\rangle}_{j}=\frac{\left(e_{j}^{\dagger}\right)^{k}\left(g_{j}^{\dagger}\right)^{N-k}}{\sqrt{k!(N-k)!}}|\operatorname{vac}\rangle.$
(4)
The initial state of optical mode $b$ is a coherent light that could be
defined as
$|\alpha\rangle_{b}\equiv e^{-|\alpha|^{2}/2}e^{\alpha
b^{\dagger}}|\text{vac}\rangle.$ (5)
The coherent light enters the first beam splitter and is split into two modes
labeled by $a_{j}$, where $j\in\left\\{1,2\right\\}$, evolving to the state
$\displaystyle|\frac{\alpha}{\sqrt{2}}\rangle_{1}|\frac{\alpha}{\sqrt{2}}\rangle_{2}.$
(6)
Meanwhile, the initial state of the atomic ensembles is prepared in the
$S^{x}$ direction
$\displaystyle|\Psi_{0}\rangle$
$\displaystyle=\left.|\frac{\pi}{2},0\rangle\rangle_{1}\right.\left.|\frac{\pi}{2},0\rangle\rangle_{2}\right.$
$\displaystyle=|k_{x}=N\rangle_{1}|k_{x}=N\rangle_{2}.$ (7)
Each optical mode interacts with a BEC that is separated in two arms of a
Mach-Zehnder interferometer via the QND Hamiltonian
$\displaystyle H=\frac{\hbar\Omega}{2}\left(S_{1}^{z}-S_{2}^{z}\right)J^{z}.$
(8)
This entangles the light and the atomic spin degrees of freedom.
### II.2 Entangled wavefunction after QND measurement
Evolving the initial states (3) and (6) with the Hamiltonian (8), and evolving
the photons through the second beamsplitter gives the state [80]
$\displaystyle|\psi(\tau)\rangle=$
$\displaystyle\frac{1}{2^{N}}\sum_{k_{1},k_{2}=0}^{N}\sqrt{\binom{N}{k_{1}}\binom{N}{k_{2}}}|k_{1},k_{2}\rangle$
$\displaystyle\times\left|\alpha\cos\left(k_{1}-k_{2}\right)\tau\right\rangle_{c}\left|-i\alpha\sin\left(k_{1}-k_{2}\right)\tau\right\rangle_{d}.$
(9)
Then projecting the light on Fock states gives rise to a modified atomic
wavefunction [80]
$\displaystyle|{\psi}_{n_{c}n_{d}}(\tau)\rangle$
$\displaystyle=\frac{1}{\sqrt{\cal
N}}\sum_{k_{1},k_{2}=0}^{N}\sqrt{{{N}\choose{k_{1}}}{{N}\choose{k_{2}}}}$
$\displaystyle\times
C_{n_{c}n_{d}}^{\alpha}[(k_{1}-k_{2})\tau]|k_{1},k_{2}\rangle.$ (10)
Here the dimensionless time is defined by $\tau=\Omega t$. The $C$-function is
defined by
$\displaystyle
C_{n_{c}n_{d}}^{\alpha}(\chi)\equiv\frac{\alpha^{n_{c}+n_{d}}e^{-|\alpha|^{2}/2}}{\sqrt{n_{c}!n_{d}!}}\cos^{n_{c}}\chi\sin^{n_{d}}\chi,$
(11)
and the normalization factor ${\cal N}$ is
$\displaystyle{\cal
N}=\sum_{k_{1},k_{2}=0}^{N}{{N}\choose{k_{1}}}{{N}\choose{k_{2}}}|C_{n_{c}n_{d}}[(k_{1}-k_{2})\tau]|^{2}.$
(12)
The most likely photon counts that are measured are distributed around
$n_{c}+n_{d}\approx|\alpha|^{2}$, due to photon number conservation. The
$C$-function takes the form of a Gaussian and can be approximated in the short
interaction time regime as [92]
$\displaystyle
C_{n_{c}n_{d}}(\chi)\propto\exp\left(-\frac{[|\chi|-\frac{1}{2}\arccos(\frac{n_{c}-n_{d}}{n_{c}+n_{d}})]^{2}}{2\sigma_{n_{c}n_{d}}^{2}}\right),$
(13)
where
$\displaystyle\sigma_{n_{c}n_{d}}\approx\frac{1}{\sqrt{n_{c}+n_{d}}}.$ (14)
For outcomes $n_{c}\gg n_{d}$ and large photon numbers $|\alpha|^{2}\gg 1$,
the state is then well approximated by
$\displaystyle|{\psi}_{n_{c}\gg n_{d}}(\tau)\rangle\approx\left(\frac{4}{\pi
N}\right)^{1/4}\sum_{k=0}^{N}e^{-\frac{2}{N}\left(k-\frac{N}{2}\right)^{2}}|k\rangle|k\rangle.$
(15)
This is an entangled state due to the correlations in Fock states that are
present.
## III Optical phase diffusion
The first type of decoherence that we will analyze is phase diffusion for the
optical modes. During the QND scheme as shown in Fig. 1, the optical modes may
undergo phase diffusion. The state prior to photon detection thus becomes a
mixed state, which results in a source of decoherence for the final atomic
wavefunction. We will use the IWOP technique to obtain the exact density
matrix after applying the phase diffusion channel. We illustrate the effects
of the decoherence on various quantities such as logarithmic negativity,
variances of spin correlators, probability distributions, correlation-based
criteria, and Bell-CHSH inequalities.
### III.1 Phase diffusion master equation
An optical mode $c$ undergoing phase diffusion is described by the master
equation [93, 94]
$\displaystyle\frac{d}{dt}\rho=-\kappa\left(c^{\dagger}c\rho+\rho
cc^{\dagger}-c\rho c^{\dagger}-c^{\dagger}\rho c\right),$ (16)
where $\kappa$ is the phase diffusion rate. We use the methods of Ref. [95] to
exactly solve the master equations. This is done by transforming the density
operators into ordinary functional equations by using the representation of
thermal entanglement states. Exact solutions to the master equations are
obtained using the technique of integration within ordered operators (IWOP).
The density matrix including the effects of phase diffusion is given as [96]
$\displaystyle\rho(t)=\sum_{m,n=0}^{\infty}M_{m,n}^{\text{PD}}(t)\rho(0){M_{m,n}^{\text{PD}}}^{\dagger}(t)$
(17)
where the Kraus operators for phase diffusion are defined as
$\displaystyle M_{m,n}^{\text{PD}}(t)=\sqrt{\frac{1}{m!n!}\frac{(\kappa
t)^{m+n}}{(\kappa t+1)^{m+n+1}}}c^{\dagger m}\left(\frac{1}{1+\kappa
t}\right)^{c^{\dagger}c}c^{n}.$ (18)
Here, $\rho(0)$ is the initial density matrix without decoherence.
In our case, phase diffusion is applied at the end of the QND entanglement
preparation process. We note that due to the two modes $c,d$ there are two
sets of Kraus operators that need to be applied. The initial density matrix in
our case is therefore
$\displaystyle\rho_{0}$ $\displaystyle=|\psi(\tau)\rangle\langle\psi(\tau)|,$
(19)
where $|\psi(\tau)\rangle$ is the state as defined in (9). Applying the phase
diffusion and performing the photonic measurements gives the density matrix
$\displaystyle\rho_{\text{PD}}=$ $\displaystyle\langle n_{c}|\langle
n_{d}|\sum_{m,n,m^{\prime},n^{\prime}=0}^{\infty}M_{m,n}^{\text{PD},c}(t)M_{m^{\prime},n^{\prime}}^{\text{PD},d}(t)$
$\displaystyle\times\rho_{0}{M_{m^{\prime},n^{\prime}}^{\text{PD},d}}^{\dagger}(t){M_{m,n}^{\text{PD},c}}^{\dagger}(t)|n_{c}\rangle|n_{d}\rangle,$
(20)
where we have labeled a mode on the Kraus operator with a superscript. The
explicit evaluated expression is given in Appendix A. The above expression
assumes that the time evolution of the QND interaction $\tau$ is equal to the
time that phase diffusion occurs for in (16). The rate at which the QND
interaction occurs $\Omega$ and the dephasing rate $\kappa$ can be adjusted
according to the ratio $\tilde{\kappa}=\kappa/\Omega$. Using this density
matrix we calculate various quantities of interest as below.
### III.2 Effect of decoherence on various quantities
#### III.2.1 Entanglement
We first examine the effect of phase diffusion on the entanglement between the
atomic ensembles. We use logarithmic negativity to quantify the entanglement,
which is an entanglement monotone for mixed states [97, 98]
$\displaystyle E=\log_{2}\left\|\rho^{T_{2}}\right\|,$ (21)
where $\rho^{T_{2}}$ is partial transpose on the second BEC. We calculate the
normalized logarithmic negativity, in relation to the maximum value that is
possible between two $N+1$ dimensional systems
$\displaystyle E_{\max}=\log_{2}(N+1).$ (22)
In Fig. 2(a), we show the dependence of the entanglement for various values of
the dimensionless phase diffusion rate $\tilde{\kappa}$. We observe that for
small values of $\tilde{\kappa}\leq 0.1$, the entanglement is relatively
unaffected for all interaction times. The curve shows a characteristic
“devil’s crevasse” form as observed in previous works [99, 85]. With
increasing $\tilde{\kappa}$, the entanglement shows an exponential decay, and
the sharp dips are smoothed out. The entanglement rises sharply corresponding
to the generation of entanglement using the QND scheme within the dephasing
time. For longer times, the entanglement is increasingly affected by the
dephasing, due to the longer times that the decoherence has to degrade the
state. Beyond $\tilde{\kappa}\geq 10$, the devil’s crevasse structure is no
longer visible. We note that there are no decoherence effects at times
$\tau=n\pi$, due to such times having no entanglement even in the decoherence-
free case. This does not necessarily mean that the state is unaffected since
the entanglement is only one aspect of the state and may still affect the
state.
Figure 2: Entanglement as quantified by the normalized logarithmic negativity
(21) for the state (9) in the presence of different kinds of decoherence
channels. (a) Entanglement in the phase diffusion channel versus time with
decoherence $\tilde{\kappa}=\kappa/\Omega$ as marked. (b) Entanglement for the
photonic loss/gain channel versus time with decoherence rates as marked. We
set the photonic gain rate to be $g/\Omega=0.1$ and show various photon loss
rates $\tilde{\gamma}=\gamma/\Omega$. The parameters used in (a) are
$N=10,n_{c}=20,n_{d}=0,\alpha=\sqrt{20}$ while
$N=10,n_{c}=100,n_{d}=0,\alpha=10$ for other plots.
#### III.2.2 Probability distribution
We next consider the probability distribution of measuring state (20) in
various bases
$\displaystyle p_{l}(k_{1},k_{2})=\langle
k_{1},k_{2}|^{(l)}\rho|k_{1},k_{2}\rangle^{(l)}.$ (23)
where we use the notation
$\displaystyle|k_{1},k_{2}\rangle^{(l)}=|k_{1}\rangle^{(l)}\otimes|k_{2}\rangle^{(l)}$
(24)
for $l\in\\{x,y,z\\}$. The Fock states in the $x$ and $y$ bases are defined as
$\displaystyle|k\rangle^{(x)}$ $\displaystyle=e^{-iS^{y}\pi/4}|k\rangle^{(z)}$
$\displaystyle|k\rangle^{(y)}$
$\displaystyle=e^{-iS^{z}\pi/4}e^{-iS^{y}\pi/4}|k\rangle^{(z)}.$ (25)
where the Fock state $|k\rangle^{(z)}$ is the same as that defined in (4). The
Fock states in the $x$ and $y$-basis can be written in terms of the $z$-basis
Fock states using the relations given in Appendix B.
Figure 3: Probability distributions after measuring state (10) with
decoherence in various bases, as defined in (23). (a) The phase diffusion
channel with dissipative coefficient $\tilde{\kappa}\in\\{0,1,20\\}$ from top
to bottom, for the state (20). (b) The photonic loss/gain channel with loss
coefficient $\tilde{\gamma}\in\\{0.2,1,10\\}$ from top to bottom, and
$g/\Omega=0.1$. The parameters for all plots are
$N=10,n_{c}=100,n_{d}=0,\tau=0.1$.
Figure 3(a) shows the probabilities with phase diffusion channel for various
amounts of decoherence $\tilde{\kappa}$ at times $\tau=0.1$, corresponding to
times where two-spin squeezing is visible. For the decoherence-free case, we
see a pattern of correlations for $l=x,z$ and anti-correlations for $l=y$. As
the dephasing is increased, the pattern of correlations and anti-correlations
are modified. The spin correlations in the $z$-direction are decreased, and
the diagonal probability distribution is modified to circular distribution
which exhibits no correlations. However, the correlations do not become
completely removed even for very large amounts of phase diffusion, such as
$\tilde{\kappa}=20$. Interestingly, the $y$-correlations appear to be improved
under phase diffusion, where the width of the anti-correlations is reduced. We
confirm this effect in the variances in the following section. For the
$x$-correlations, from the probability distribution, there is little visible
change under the phase diffusion, except for the removal of interference
fringes. Overall, we observe that even under very strong phase diffusion, the
correlations remain relatively unaffected for the probability distributions
that are shown in Fig. 3(a). This is likely due to the relatively short
interaction time that is examined, where the type of entangled state is
relatively robust in the presence of decoherence [99].
#### III.2.3 Variances
Variances of spin correlators $S^{x}_{1}-S^{x}_{2}$, $S^{y}_{1}+S^{y}_{2}$,
and $S^{z}_{1}-S^{z}_{2}$ under the phase diffusion channel are shown as a
function of interaction time in Fig. 4. We choose these observables since they
are expected to show squeezing effects. Comparing the $\tilde{\kappa}=1$ with
the decoherence-free case of $\tilde{\kappa}=0$, we observe that there is very
little difference for all spin correlators at most evolution times. The
largest differences arise at times $\tau=\pi$, where the state involves
entangled Schrodinger cat states [81], which are particularly susceptible to
decoherence. The variance of $S_{1}^{x}-S_{2}^{x}$ remains large, whereas this
drops to zero in the decoherence-free case. Similarly, the variance of
$S_{1}^{y}+S_{2}^{y}$ in the decoherence-free case reaches a large value at
$\tau=\pi$ but is reduced when phase diffusion is introduced. Meanwhile, the
variance of $S_{1}^{z}-S_{2}^{z}$ is relatively affected at all times when
comparing the $\tilde{\kappa}=1$ and decoherence-free cases. However, as the
phase diffusion rate is further increased, generally the variance of
$S_{1}^{z}-S_{2}^{z}$ is increased, which can be understood as the loss of
correlations. Meanwhile, the $S_{1}^{x}-S_{2}^{x}$ and $S_{1}^{y}+S_{2}^{y}$
correlations are relatively unaffected with a further increase of phase
diffusion beyond $\tilde{\kappa}=1$.
Figure 4: The variance of the state (20) with phase diffusion decoherence for
the operators $S_{1}^{x}-S_{2}^{x},S_{1}^{y}+S_{2}^{y}$ and
$S_{1}^{z}-S_{2}^{z}$. Variances are plotted for (a)(c)(e) long time scales;
(d)(e)(f) zoomed into short time scales. The parameters are
$N=10,n_{c}=100,n_{d}=0,\alpha=\sqrt{n_{c}}=10$.
#### III.2.4 Correlation-based criteria
While logarithmic negativity is a well-established method for detecting
entanglement, in an experimental setting it requires full tomography of the
quantum state which has a large overhead and may not be practically feasible
in many cases. A preferable approach is to detect entanglement with low-order
spin expectation values, which are more readily measured and are relatively
insensitive to imperfect spin measurements. Here we describe and calculate
three correlation-based methods to detect entanglement and EPR steering.
The first correlation-based entanglement criterion is the Wineland squeezing
parameter
$\displaystyle\xi$ $\displaystyle=\frac{{2(\Delta
S_{\bot}^{2})}_{\text{min}}}{\left|\left\langle S\right\rangle\right|}$
$\displaystyle\geq 1\hskip 28.45274pt\text{(separable)}$ (26)
where $S_{\bot}$ is any component perpendicular to the average total spin and
$\Delta S_{\bot}^{2}$ is minimum fluctuation in the direction of the vertical
component. The coefficient $\xi$ corresponds to the degree of squeezing. A
smaller value means a better degree of squeezing. The inequality (26) is
derived assuming a separable state, hence $\xi<1$ signals the presence of
entanglement.
In a previous study [80], it was found that the Hofmann-Takeuchi criterion
[100] is one of the most effective spin correlation-based detection methods
for entanglement. In our case, it is defined as
$\displaystyle{\cal{C}}_{\text{ent }}$
$\displaystyle\equiv\frac{\operatorname{Var}\left(S_{1}^{x}-S_{2}^{x}\right)+\operatorname{Var}\left(S_{1}^{y}+S_{2}^{y}\right)+\operatorname{Var}\left(S_{1}^{z}-S_{2}^{z}\right)}{4N}$
$\displaystyle\geq 1.\hskip 28.45274pt\text{(separable)}$ (27)
Again, the inequality (27) is derived assuming separable states, hence
detecting ${\cal{C}}_{\text{ent }}<1$ signals the presence of entanglement.
Finally, EPR steering is a subclass of entangled states where one party can
nonlocally affect the other party’s state through measurements [101, 102,
103]. A correlated-based inequality that detects EPR steering is [104]
$\displaystyle{\cal{C}}_{\text{steer }}^{1\rightarrow 2}$
$\displaystyle\equiv\frac{\operatorname{Var}\left(S_{1}^{y}+S_{2}^{y}\right)\operatorname{Var}\left(S_{1}^{z}-S_{2}^{z}\right)}{\left\langle
S_{1}^{x}\right\rangle^{2}}$ $\displaystyle\geq 1\hskip 28.45274pt\text{(un-
steerable)}$ (28)
A violation of the inequality (28) implies the existence of EPR steering from
BEC 1 to BEC 2.
In Fig.5, we compare the above three different correlation-based criteria.
Fig.5(a) shows Wineland squeezing parameter in the region where entanglement
is shown. The time region of sustained squeezing for which $\xi<1$ is reduced
as the phase diffusion rate $\tilde{\kappa}$ is increased. This is consistent
with the logarithmic negativity calculations of Fig. 2(a), where the phase
diffusion has the effect of reducing entanglement. Now turning to the Hofmann-
Takeuchi criterion, we find it can detect entanglement for a wide range of
times except for some particular points where ${\cal{C}}_{\text{ent }}\approx
1$. As with the Wineland squeezing criterion, generally, the degree of
violation of the inequality (27) is reduced with an increasing phase diffusion
rate. It however remains a powerful method to detect entanglement, and in
comparison to the Wineland squeezing criterion, it detects entanglement in a
much wider range.
Finally, for the EPR steering criterion, the effect of phase diffusion is to
decrease the region where violation of the inequality (28) occurs. As would be
expected, for EPR steering it is generally more difficult to find a violating
region, which we attribute to the fact that EPR steerable states are a more
specialized class of entangled states. Beyond $\tilde{\kappa}>10$, the
criterion fails to detect any entangled states, while the Wineland squeezing
and the Hofmann-Takeuchi criterion continue to detect entangled states.
Figure 5: Detection of entanglement and EPR-steering using correlation-based
criteria. (a) The Wineland squeezing criterion (26) for the state (20) with
phase diffusion decoherence. (b) The Hofmann-Takeuchi criterion (27) for the
state (20) with phase diffusion decoherence. (c) The EPR steering criterion
(28) for the state (20) with phase diffusion decoherence. The parameters used
here are $N=10,n_{c}=100,n_{d}=0,\alpha=\sqrt{n_{c}+n_{d}}=10$. The
dissipative coefficient is as marked. We set $\Omega=1$ such that the
decoherence rate is in units of $\Omega$, and the time units are $1/\Omega$.
#### III.2.5 Bell-CHSH inequality
Figure 6: Bell-CHSH correlations for the state (10) with two types of optical
deocherence. (a) Time dependence of (29) for phase diffused state (20) for
$N=5$ and $\theta_{B}=0.37$. (b) Optimal values of (29) concerning $\tau$ and
$\theta_{B}$ for various $N$ with phase diffused state (20). (c) Time
dependence of (29) for the state (34) which has undergone photon loss and
gain. (d) Optimal values of (29) concerning $\tau$ and $\theta_{B}$ for
various $N$ under photon loss and gain (34). Shaded regions indicate the
regions where there is a violation of the CHSH inequality. The common
parameters are $n_{c}=100,n_{d}=0,\alpha=10$.
In Ref.[105], it was shown that the Bell-CHSH inequality can be violated using
the state (10). The form of the CHSH inequality that was used, reads as
$\displaystyle\mathcal{C}_{\text{CHSH}}\equiv$
$\displaystyle\Big{|}\left\langle
M_{1}^{(1)}M_{2}^{(1)}\right\rangle+\left\langle
M_{1}^{(1)}M_{2}^{(2)}\right\rangle$ $\displaystyle-\left\langle
M_{1}^{(2)}M_{2}^{(1)}\right\rangle+\left\langle
M_{1}^{(2)}M_{2}^{(2)}\right\rangle\Big{|}$ $\displaystyle\leq$ $\displaystyle
2\hskip 28.45274pt\text{(local HV)},$ (29)
which is valid for a local hidden variable (HV) theory and
$\displaystyle M_{1}^{(1)}$ $\displaystyle=\text{sgn}(S^{z}_{1})$
$\displaystyle M_{1}^{(2)}$
$\displaystyle=\text{sgn}(S^{z}_{1}\cos\theta_{B}+S^{y}_{1}\sin\theta_{B})$
$\displaystyle M_{2}^{(1)}$
$\displaystyle=\text{sgn}(S^{z}_{2}\cos\frac{\theta_{B}}{2}+S^{y}_{2}\sin\frac{\theta_{B}}{2})$
$\displaystyle M_{2}^{(2)}$
$\displaystyle=\text{sgn}(S^{z}_{2}\cos\frac{\theta_{B}}{2}-S^{y}_{2}\sin\frac{\theta_{B}}{2}).$
(30)
Here, $\text{sgn}(x)$ takes the sign of the eigenvalue of the operator. Using
the sign of spin operators is an experimentally viable observable since it is
largely insensitive to atom number fluctuations [106]. The superscripts
(1),(2) are the two measurement choices that can be made on the two atomic
ensembles. The angle $\theta_{B}$ must be optimized to find the largest
violation and was found to be well-approximated by the empirical relation
$\theta_{B}\approx(3.2/N+1.7/N^{2})/(1+2.1/N)$ [105].
In Ref. [105], the Bell-CHSH inequality was studied without considering
decoherence effects. Figure 6(a) shows the time dependence of the left-hand
side of the criterion (29) with phase channel decoherence for $N=5$. The
shaded area indicates the classical region of Bell-CHSH inequality.
Remarkably, we see very little change in the values of
$\mathcal{C}_{\text{CHSH}}$, and above a particular interaction time, CHSH
violations are seen despite the presence of strong phase diffusion of the
optical modes. In Fig. 6(b), the optimal values of (29) are as a function of
$1/N$. We observe the violations occur for all $N$ in the presence of strong
phase diffusion. We again find that decoherence has little effect on the
optimal Bell-CHSH inequality. While this is a positive from an experimental
context, we note that for large ensemble sizes the level of violation is
reduced, hence an increased precision of $\mathcal{C}_{\text{CHSH}}$ will be
required to observe any violation.
## IV Photonic Loss and Gain
We now consider a loss and gain channel for the photons during the QND
measurement induced entanglement generation as described in Sec. II.2.
Physically, the loss of photons may occur during the transmission of the light
through the Mach-Zehnder interferometer and is potentially a major source of
decoherence to the scheme. Photonic gain may also occur where ambient
incoherent photons may enter the modes, also acting as a source of
decoherence.
### IV.1 Photon loss and gain master equation
The master equation for photonic loss and gain is written as [107, 108, 109]
$\displaystyle\frac{d\rho}{dt}=\gamma\left(2c\rho
c^{\dagger}-c^{\dagger}c\rho-\rho c^{\dagger}c\right)+g\left(2c^{\dagger}\rho
c-cc^{\dagger}\rho-\rho cc^{\dagger}\right),$ (31)
where $\gamma$ is the photon loss rate and $g$ is the photon gain rate.
Solving this master equation using the IWOP technique can be written in terms
of Kraus operators as [110, 111]
$\displaystyle\rho(t)$
$\displaystyle=\sum_{p,q=0}^{\infty}M_{p,q}^{\text{LG}}(t)\rho(0){M_{p,q}^{\text{LG}}}^{\dagger}(t)$
(32)
where
$\displaystyle M_{p,q}^{\text{LG}}(t)$
$\displaystyle=\sqrt{\frac{\gamma^{p}g^{q}T_{1}^{p+q}T_{3}}{p!q!T_{2}^{2q}}}e^{c^{\dagger}c\ln
T_{2}}c^{\dagger q}c^{p}$ $\displaystyle T_{1}$
$\displaystyle=\frac{1-e^{-2(\gamma-g)t}}{\gamma-ge^{-2(\gamma-g)t}}$
$\displaystyle T_{2}$ $\displaystyle=\frac{(\gamma-g)e^{-(\gamma-g)t}}{\gamma-
ge^{-2(\gamma-g)t}}$ $\displaystyle T_{3}$
$\displaystyle=\frac{\gamma-g}{\gamma-ge^{-2(\gamma-g)t}}=1-gT_{1}.$ (33)
The above expression reduces to the standard form of the Kraus operator for
loss [81] when $g=0$, where the probability of the photon loss is
$1-e^{-2\gamma t}$.
Using the above Kraus operator we may derive the effect of photonic loss and
gain on the state by applying it to (19)
$\displaystyle\rho_{\text{LG}}=$ $\displaystyle\langle n_{c}|\langle
n_{d}|\sum_{p,q,p^{\prime},q^{\prime}=0}^{\infty}M_{p,q}^{\text{LG},c}(t)M_{p^{\prime},q^{\prime}}^{\text{LG},d}(t)$
$\displaystyle\times\rho_{0}{M_{p^{\prime},q^{\prime}}^{\text{LG},d}}^{\dagger}(t){M_{p,q}^{\text{LG},c}}^{\dagger}(t)|n_{c}\rangle|n_{d}\rangle.$
(34)
The evaluated expression is given in Appendix C. As with phase diffusion, the
above expression assumes that the time evolution of the QND interaction $\tau$
is equal to the time that phase diffusion occurs for in (16). We note that
this is different to the assumption made in Ref. [81], where the probability
of photon loss was set to be a constant, and hence was a time-independent
quantity. The rate at which the QND interaction occurs $\Omega$ and the loss
rate $\gamma$ can be adjusted according to the ratios
$\tilde{\gamma}=\gamma/\Omega$ and $\tilde{g}=g/\Omega$. Using this density
matrix we calculate various quantities of interest as below.
### IV.2 Effect of decoherence on various quantities
#### IV.2.1 Entanglement
In Fig. 2(b), we show the logarithmic negativity for a fixed photon gain rate
$\tilde{g}$ and various rates of loss $\tilde{\gamma}$. We choose such a
regime since it is the most physically relevant regime, where the dominant
decoherence mechanism is photon loss. The constant photon gain accounts for
the possibility that there is an incoherent source of photons entering the
interferometer, but we assume that this can be suitably controlled such that
is a relatively smaller effect. We again see the characteristic devil’s
crevasse structure to the entanglement as the time is varied. Generally, the
amount of entanglement decreases with longer times due to the assumption here
that the time of evolving the photonic loss and gain master equation is equal
to the QND interaction time. This results in a probability of the photon loss
that exponentially approaches 1, such that the entanglement exponentially
reduces with time.
The results shown in Fig. 2(b) are qualitatively different from what was shown
in Ref. [81], where there was little difference between the ideal entanglement
curves with and without photon loss. We attribute this to the effectively
larger photon loss that our current calculations include, compared to Ref.
[81]. In Ref. [81], the largest photon loss probability that was considered
was 0.95. In comparison, for example, the photon loss probability for $\xi=1$
at $\tau=\pi$ is 0.998. Once the photonic population is entirely removed
through photon loss, it is clear that the entangling scheme cannot work since
the state (9) no longer involves any photons and the $C$-function (11) no
longer produces an amplitude on the states. We also note that the dependence
of the phase diffusion plots in Fig. 2(a), are somewhat different from the
photon loss and gain dependence, which tends to maintain high levels of
entanglement followed by a sharp drop off. This is in contrast with phase
diffusion has the effect of decreasing the overall level of entanglement. We
again interpret this as the QND measurement being relatively robust in the
presence of photon loss until all the photons are removed, at which point the
protocol no longer is capable of generating entanglement.
#### IV.2.2 Probability distribution
Fig. 3(b) shows the probability distribution (23) for the state (34). The
ideal case without any decoherence is shown in the top row of Fig. 3(a). We
see that there is very little difference between the ideal case and the case
with photon loss $\tilde{\gamma}=0.2,1$ and gain $\tilde{g}=0.1$. At very
large loss rates $\tilde{\gamma}=10$, some difference starts to develop in the
$x$ correlations and $y$ anti-correlations but the $z$-correlations remain
visually the same. As with the phase diffusion case, the $y$ anti-correlations
tend to improve slightly for the larger photon loss cases. We do not have a
good interpretation of this effect, but this appears to be a feature of both
types of photonic decoherence that the $y$ anti-correlations are improved.
#### IV.2.3 Variances
Fig. 7(a)(c)(e) shows the variances of the state (34) for the operators
$S_{1}^{x}-S_{2}^{x},S_{1}^{y}+S_{2}^{y}$, and $S_{1}^{z}-S_{2}^{z}$
respectively for the loss and gain channel. We again consider the case with
fixed incoherent photon gain and vary the loss rate. We see that for the $x$\-
and $y$-basis correlators, there is relatively little change to the dependence
despite relatively large amounts of loss present. Some of the small structure
in the variations is removed for $\tilde{\kappa}=10$, but this is a small
effect with respect to the overall variance. For the $z$-basis correlators,
for early times, all values of the variance remain similar, and squeezing is
observed. For longer times, the variance becomes more affected by the loss,
and at $\tilde{\kappa}=10$ the variance returns to its initial value. This can
be attributed due to the fact that the states at longer evolution times tend
to be more sensitive to the decoherence such that it minimizes the
correlations.
Figure 7: Variances and correlation-based criteria for the QND entangled state
under photon loss and gain. (a)(b)(c) Expectation values
$S_{1}^{x}-S_{2}^{x},S_{1}^{y}+S_{2}^{y}$ and $S_{1}^{z}-S_{2}^{z}$; (d)
Wineland squeezing; (e) Hofmann-Takeuchi criterion; (f) EPR steering
criterion. The parameters are $N=10,n_{c}=100,n_{d}=0,\alpha=\sqrt{n_{c}}=10$.
We set $\Omega=1$ such that $\tilde{\kappa}$ and $g$ is in units of $\Omega$,
and the time units are $1/\Omega$.
#### IV.2.4 Entanglement criteria and Bell-CHSH inequality
In Fig.7(b)(d)(f), we show the three different correlation-based criteria of
the Wineland squeezing parameter (26), Hofmann-Takeuchi criterion (27), and
the EPR steering criterion (28). For the Wineland squeezing parameter, we find
a generally similar behavior to phase diffusion, where the region where
entanglement is detected tends to reduce with increasing photon loss.
Generally, the criterion only detects entanglement in the short time regime
where the initial squeezing of states occurs with respect to the spin
variances. Fig.7(d) shows the Hofmann-Takeuchi criterion, which tends to
follow the time dependence of the $S_{1}^{z}-S_{2}^{z}$ variance. This is due
to the fact that variances of $S_{1}^{x}-S_{2}^{x}$ and $S_{1}^{y}+S_{2}^{y}$
show a perfect symmetry, which means that the sum
$\text{Var}(S_{1}^{x}-S_{2}^{x})+\text{Var}(S_{1}^{y}+S_{2}^{y})$ is a
constant. We find that this is again a powerful way of detecting entanglement,
showing that entanglement is present in the system for nearly all times for
most values of $\tilde{\gamma}$. The EPR steering criterion in Fig.7(f) shows
again a similar trend where the increased photon loss causes a more limited
region where EPR steering is detected. Figure.6(c) shows the Bell-CHSH
criterion (29) as a function of time including the effects of photon loss and
gain. Again we see that there is great robustness in observing a violation,
even in the presence of strong photon loss. In Fig.6(d) we see the dependence
of the level of violation as a function of $1/N$. We again see that for all
$N$ values a Bell violation can be observed and there is very little
difference between the levels of violation including photon loss and gain and
the ideal case. The main difficulty in seeing such correlations will then be
in observing the Bell’s violation for larger atomic ensembles where the level
of violation approaches the classical bound of ${\cal C}_{\text{CHSH}}=2$.
## V Atomic Dephasing
We now examine a third channel of decoherence, atomic dephasing, which
directly acts on the atoms to remove coherence in the atomic state. This may
physically arise due to noise sources that the atoms may experience,
originating from technical noise in the atomic traps, or due to photonic
scattering due to the QND Hamiltonian (8). In a previous study [81], the
effect of $S^{z}$ dephasing on the state (10) was examined, where quantities
such as the expectation values, variances, entanglement criteria, and
distribution were investigated. However, several quantities such as the
Wineland criterion on Bell-CHSH inequality were not examined. We discuss some
of the quantities that were not considered in Ref. [81] here, to analyze the
impact of decoherence.
### V.1 Dephasing master equation
The master equation for atomic dephasing reads
$\displaystyle\frac{d\rho}{dt}=-\Gamma\sum_{n=1}^{2}\left[\left(S_{n}^{z}\right)^{2}\rho-2S_{n}^{z}\rho
S_{n}^{z}+\rho\left(S_{n}^{z}\right)^{2}\right],$ (35)
where $\Gamma$ is the dephasing rate and it has been assumed to be the same
for both atomic ensembles. Solving the master equation (35) exactly, the
density matrix elements evolve according to
$\displaystyle\rho_{\text{D}}=$
$\displaystyle\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}}|k_{1},k_{2}\rangle\langle
k_{1}^{\prime},k_{2}^{\prime}|\langle
k_{1},k_{2}|\psi_{n_{c}n_{d}}(\tau)\rangle$
$\displaystyle\times\langle\psi_{n_{c}n_{d}}(\tau)|k_{1}^{\prime},k_{2}^{\prime}\rangle
e^{-2\tilde{\Gamma}\tau[(k_{1}-k_{1}^{\prime})^{2}+(k_{2}-k_{2}^{\prime})^{2}]},$
(36)
where the atomic wavefunction after the photonic measurement is given by (10).
As before, we set the dephasing time to be equal to the QND interaction time.
The relative strengths of the dephasing and the interaction can be adjusted by
the dimensionless ratio $\tilde{\Gamma}=\Gamma/\Omega$.
### V.2 Effect of decoherence on various quantities
#### V.2.1 Wineland squeezing criteria
Figure 8(a) shows the Wineland squeezing parameter (26) as a function of time
$\tau$ for different decoherence factors $\tilde{\Gamma}$. For small dephasing
rates, the Wineland squeezing parameter detects entanglement (shaded region)
in the short time regime. In comparison to the Hofmann-Takeuchi and EPR
steering criterion as calculated in Ref. [81], the region is reduced,
indicating that this is a less sensitive detector of entanglement. As was
found in Ref. [81], entanglement is present for all times $0<\tau<\pi$ for
dephasing rates $\Gamma=0,0.01,0.1$ as detected by logarithmic negativity. The
various entanglement witnesses are only a sufficient condition for
entanglement, and a lack of violation of the criterion does not indicate that
entanglement is not present. For small dephasing rates, there is an optimal
interaction time $\tau_{\text{opt}}$ which minimizes the Wineland squeezing
parameter (Fig. 8(b)). This shows generally a linear relationship with $1/N$.
In Fig. 8(c), the relationship between optimal Wineland squeezing parameter
$\xi_{opt}=\xi(\tau_{\text{opt}})$ with atom numbers $1/N$ are shown for
various dephasing rates. We observe that $\xi_{opt}$ remains in the
entanglement detection region for the dephasing rates $\tilde{\Gamma}=0,0.01$.
For $\tilde{\Gamma}=0.1$, $\xi_{\text{opt}}=1$ for larger values of $N$ and
hence no entanglement is detected. For this regime, the dephasing is strong
enough to remove the minimum as seen in Fig. 8(a). Hence we see that it
becomes more difficult to detect entanglement using the Wineland squeezing
criterion for large ensembles and larger dephasing rates. Hence as an
entanglement detection tool, other methods such as the Hoffman-Takeuchi
criterion appear to be a better choice.
Figure 8: The Wineland squeezing parameter for the state (36) with
decoherence. (a) Wineland squeezing parameter versus QND interaction time
$\tau$ with decoherence rates $\tilde{\Gamma}$ as marked. The atom number is
$N=20$. (b) The optimal squeezing time $\tau_{\text{opt}}$ determined by
minimizing $\xi(\tau)$ plotted as a function of $1/N$. A fit of the optimal
times for $\Gamma=0$ is given by
$\tau_{\text{opt}}^{(\tilde{\Gamma}=0)}=0.104+\frac{0.0413}{N}$. (c) The
optimal Wineland squeezing $\xi(\tau_{\text{opt}})$ versus $1/N$ for different
decoherence rates $\tilde{\Gamma}$. For all plots, we choose parameters
$n_{c}=100,n_{d}=0,\alpha=10$.
#### V.2.2 Bell-CHSH inequality
In Fig. 9(a), we calculate the left-hand side of the Bell-CHSH inequality (29)
for the state (36) as a function of the QND interaction time with different
dephasing rates. For each dephasing rate, there is a slightly different
dependence with $\tau$, where larger dephasing rates tend to diminish the
violation for longer times. Fig. 9(b) shows the optimized value of (29) as a
function of $N$. We see that the Bell violations are observed for all $N$ even
in the presence of dephasing. The surprisingly robust nature of the Bell
violations in the presence of dephasing is a positive sign that such
correlations may be observable in experimental situations. We note however
that the violations tend to diminish for larger ensembles so that for larger
systems, such that alternative criteria such as the Hoffman-Takeuchi criterion
may be more sensitive detectors of entanglement.
Figure 9: Bell-CHSH correlations for the state (36) with atomic dephasing. (a)
Time dependence of (29) for different decoherence ratios as marked. Here we
choose parameters $\theta_{B}=0.251,N=11$. (b) Optimal values of (29) with
respect to $\tau$ and $\theta_{B}$ for various $N$. Common parameters
throughout are $n_{c}=100,n_{d}=0,\alpha=10$.
## VI Conclusion
We have studied the effect of three types of decoherence (optical phase
diffusion, photon loss and gain, and atomic dephasing) on entangled states
produced by QND measurements in atomic ensembles. Generally, we find a similar
conclusion to Ref. [81] where the states are relatively robust for all types
of decoherence in the short time regime. For example, in the entanglement
plots of Fig. 2, even for very large decoherence rates there is a significant
amount of entanglement in the short time regime but decays for longer times.
The correlations, as shown in Figs. 3,4,7 are all robust in the short time
regime but can degrade at longer interaction and decoherence times. As such,
correlation-based detection of entanglement and EPR steering tend to be
robustly observed in the short time regime as seen in Figs. 5,7,8.
Interestingly, Bell correlations are also very robust under various types of
decoherence, as seen in Figs. 6,9. The robustness of Bell’s violation can be
attributed to the fact that the optimal interaction time occurs for the short
time regime. The main issue here with experimental observation is that for
larger $N$ the level of violation diminishes, making its detection more
challenging.
Our results suggest that producing entangled states based on QND measurements
should be one of the most experimentally viable methods in the context of
BECs. As mentioned in the introduction, while such methods are well
established for entanglement generation in hot atomic ensembles [56, 31],
there is no corresponding experiment that has been realized for BECs. While
there have been numerous approaches that have been theoretically investigated
for entanglement generation in BECs already [33, 34, 35, 17, 36, 32, 37, 38],
so far none have been realized experimentally. The QND approach allows for a
robust and versatile method to generate such entanglement. The theory that has
been developed in [80] and summarized in this paper allows for the
investigation beyond the short time regime, to produce more exotic types of
entanglement. The generation of such highly entangled many-body states is of
fundamental interest to the macroscopic nature of such states, and also has
been shown to have applications in quantum information [74, 75, 76, 77, 78,
79].
## Acknowledgments
This work is supported by the National Natural Science Foundation of China
(62071301); NYU-ECNU Institute of Physics at NYU Shanghai; the Joint Physics
Research Institute Challenge Grant; the Science and Technology Commission of
Shanghai Municipality (19XD1423000,22ZR1444600); the NYU Shanghai Boost Fund;
the China Foreign Experts Program (G2021013002L); the NYU Shanghai Major-
Grants Seed Fund; Tamkeen under the NYU Abu Dhabi Research Institute grant
CG008.
## Data availability statement
The data that supports the findings of this study are available within the
article (and any supplementary material).
## Appendix A Density matrix for optical phase diffusion
In this section, we give details of the explicit expression for the density
matrix including optical phase diffusion. Starting from the initial state
(19), we evaluate
$\displaystyle\rho_{\text{PD}}=\langle n_{c}|\langle
n_{d}|\sum_{m,n,m^{\prime},n^{\prime}=0}^{\infty}M_{m,n}^{\text{PD},c}(t)M_{m^{\prime},n^{\prime}}^{\text{PD},d}(t)$
$\displaystyle\times\rho_{0}{M_{m^{\prime},n^{\prime}}^{\text{PD},d}}^{\dagger}(t){M_{m,n}^{\text{PD},c}}^{\dagger}(t)|n_{c}\rangle|n_{d}\rangle$
$\displaystyle=\frac{1}{4^{N}}\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}=0}^{N}\sqrt{\binom{N}{k_{1}}\binom{N}{k_{2}}\binom{N}{k_{1}^{\prime}}\binom{N}{k_{2}^{\prime}}}z^{2}(\tau)$
$\displaystyle\times
e^{-|\alpha|^{2}}e^{u(\tau){|\alpha|}^{2}\cos[(k_{1}-k_{2}-k_{1}^{\prime}+k_{2}^{\prime})\tau]}$
$\displaystyle\times
z^{2(n_{c}-m+n_{d}-m^{\prime})}(\tau)\sum_{m=0}^{n_{c}}\sum_{m^{\prime}=0}^{n_{d}}\frac{u^{m+m^{\prime}}(\tau)}{m!m^{\prime}!}$
$\displaystyle\times\left\\{\alpha^{2}\cos[(k_{1}-k_{2})\tau]\cos[(k_{1}^{\prime}-k_{2}^{\prime})\tau]\right\\}^{n_{c}-m}$
$\displaystyle\times\left\\{\alpha^{2}\sin[(k_{1}-k_{2})\tau]\sin[(k_{1}^{\prime}-k_{2}^{\prime})\tau]\right\\}^{n_{d}-m^{\prime}}$
$\displaystyle\times\frac{n_{c}!n_{d}!}{[(n_{c}-m)!]^{2}[(n_{d}-m^{\prime})!]^{2}}|k_{1}\rangle|k_{2}\rangle\langle
k_{1}^{\prime}|\langle k_{2}^{\prime}|$
$\displaystyle=\frac{1}{4^{N}}\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}=0}^{N}\sqrt{\binom{N}{k_{1}}\binom{N}{k_{2}}\binom{N}{k_{1}^{\prime}}\binom{N}{k_{2}^{\prime}}}$
$\displaystyle\times\frac{[-u(\tau)]^{n_{c}}U\left(-{n_{c}},1,-\frac{{n_{c}}\cos[({k_{1}}-{k_{2}})\tau]\cos[({k_{1}^{\prime}}-{k_{2}^{\prime}})\tau]z^{2}(\tau)}{u(\tau)}\right)}{n_{c}!}$
$\displaystyle\times\frac{[-u(\tau)]^{n_{d}}U\left(-{n_{d}},1,-\frac{{n_{d}}\sin[({k_{1}}-{k_{2}})\tau]\sin[({k_{1}^{\prime}}-{k_{2}^{\prime}})\tau]z^{2}(\tau)}{u(\tau)}\right)}{n_{d}!}$
$\displaystyle\times
z^{2}(\tau)D[(k_{1}-k_{2}-k_{1}^{\prime}+k_{2}^{\prime})\tau]|k_{1}\rangle|k_{2}\rangle\langle
k_{1}^{\prime}|\langle k_{2}^{\prime}|.$ (37)
Here
$\displaystyle z\left(\tau\right)$
$\displaystyle=\frac{1}{\tilde{\kappa}\tau+1}$ $\displaystyle u(\tau)$
$\displaystyle=\frac{\tilde{\kappa}\tau}{\tilde{\kappa}\tau+1}$ $\displaystyle
D(\chi)$ $\displaystyle=e^{-|\alpha|^{2}[1-u(\tau)\cos(\chi)]}$ (38)
and $U$ is the confluent hyper-geometric function of the second kind.
## Appendix B Fock state matrix elements
The matrix elements of the $S^{y}$ rotation is given by [40]
$\displaystyle\langle
k|e^{-iS^{y}\theta/2}|k^{\prime}\rangle=\sqrt{k^{\prime}!(N-k^{\prime})!k!(N-k)!}$
$\displaystyle\times\sum_{n}\frac{(-1)^{n}\cos^{k-k^{\prime}+N-2n}(\theta/2)\sin^{2n+k^{\prime}-k}(\theta/2)}{(k-n)!(N-k^{\prime}-n)!n!(k^{\prime}-k+n)!},$
(39)
where $|k\rangle$ are the eigenstates of $S^{z}$. The matrix elements of
$S^{x}$ are accordingly given by
$\displaystyle\langle
k|e^{-iS^{x}\theta/2}|k^{\prime}\rangle=i^{k^{\prime}-k}\langle
k|e^{-iS^{y}\theta/2}|k^{\prime}\rangle,$ (40)
using the fact $S^{x}=e^{-iS^{z}\pi/4}S^{y}e^{iS^{z}\pi/4}$.
## Appendix C Density matrix for photon loss and gain
In this section, we give the explicit expression of the density matrix
including photonic gain and loss. Starting from the initial state (19), we
evaluate
$\displaystyle\rho_{\text{LG}}=\langle n_{c}|\langle
n_{d}|\sum_{p,q,p^{\prime},q^{\prime}=0}^{\infty}M_{p,q}^{\text{LG}}(c)M_{p^{\prime},q^{\prime}}^{\text{LG}}(d)$
$\displaystyle\times\rho_{0}{M_{p^{\prime},q^{\prime}}^{\text{LG}}}^{\dagger}(d){M_{p,q}^{\text{LG}}}^{\dagger}(c)|n_{c}\rangle|n_{d}\rangle$
$\displaystyle=\frac{1}{4^{N}}\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}=0}^{N}\sqrt{\binom{N}{k_{1}}\binom{N}{k_{2}}\binom{N}{k_{1}^{\prime}}\binom{N}{k_{2}^{\prime}}}$
$\displaystyle{T_{3}}^{2}R[(k_{1}-k_{2}-k_{1}^{\prime}+k_{2}^{\prime})\tau]\sum_{p=0}^{n_{c}}\sum_{q=0}^{n_{d}}\frac{(gT_{1})^{p+q}}{p!q!}$
$\displaystyle\times\\{\alpha^{2}\cos[(k_{1}-k_{2})\tau]\cos[(k_{1}^{\prime}-k_{2}^{\prime})\tau]\\}^{n_{c}-p}$
$\displaystyle\times\\{\alpha^{2}\sin[(k_{1}-k_{2})\tau]\sin[(k_{1}^{\prime}-k_{2}^{\prime})\tau]\\}^{n_{d}-q}$
$\displaystyle\times{T_{2}}^{2(n_{c}-p+n_{d}-q)}\frac{n_{c}!}{{[(n_{c}-p)!]}^{2}}\frac{n_{d}!}{{[(n_{d}-q)!]}^{2}}|k_{1}\rangle|k_{2}\rangle\langle
k_{1}^{\prime}|\langle k_{2}^{\prime}|$
$\displaystyle=\frac{1}{4^{N}}\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}=0}^{N}\sqrt{\binom{N}{k_{1}}\binom{N}{k_{2}}\binom{N}{k_{1}^{\prime}}\binom{N}{k_{2}^{\prime}}}$
$\displaystyle\frac{(-g{T_{1}})^{{n_{c}}}U\left(-{n_{c}},1,-\frac{\alpha^{2}\cos[({k_{1}}-{k_{2}})\tau]\cos[({k_{1}^{\prime}}-{k_{2}^{\prime}})\tau]{T_{2}}^{2}}{g{T_{1}}}\right)}{{n_{c}}!}$
$\displaystyle\frac{(-g{T_{1}})^{{n_{d}}}U\left(-{n_{d}},1,-\frac{\alpha^{2}\sin[({k_{1}}-{k_{2}})\tau]\sin[({k_{1}^{\prime}}-{k_{2}^{\prime}})\tau]{T_{2}}^{2}}{g{T_{1}}}\right)}{{n_{d}}!}$
$\displaystyle\times{T_{3}}^{2}R[(k_{1}-k_{2}-k_{1}^{\prime}+k_{2}^{\prime})\tau]|k_{1}\rangle|k_{2}\rangle\langle
k_{1}^{\prime}|\langle k_{2}^{\prime}|,$ (41)
where $R(\chi)=e^{-|\alpha|^{2}[1-\gamma T_{1}\cos(\chi)]}$.
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|
# On Simplices with a Given Barycenter That Are Enclosed by the Standard
Simplex
Brent Austgen<EMAIL_ADDRESS>John J. Hasenbein Erhan Kutanoglu
Operations Research & Industrial Engineering Program, The University of Texas
at Austin, Austin, TX, USA
###### Abstract
We present an optimization model defined on the manifold of the set of
stochastic matrices. Geometrically, the model is akin to identifying a
maximum-volume $n$-dimensional simplex that has a given barycenter and is
enclosed by the $n$-dimensional standard simplex. Maximizing the volume of a
simplex is equivalent to maximizing the determinant of its corresponding
matrix. In our model, we employ trace maximization as a linear alternative to
determinant maximization. We identify the analytical form of a solution to
this model. We prove the solution is optimal and present necessary and
sufficient conditions for it to be the unique optimal solution. Additionally,
we show the identified optimal solution is an inverse $M$-matrix, and that its
eigenvalues are the same as its diagonal entries. We demonstrate how the model
and its solutions apply to the task of synthesizing conditional cumulative
distribution functions (CDFs) that, in tandem with a given discrete marginal
distribution, coherently preserve a given CDF.
###### keywords:
barycenter, eigenvalue, optimization, simplex
††journal: Operations Research Letters
We present an optimization model that is akin to identifying a maximum-volume
simplex that has a given barycenter and is enclosed by the standard simplex.
We provide the analytical form of a particular solution to the model, a matrix
$\overline{U}$, and prove that it is optimal. Additionally, we prove necessary
and sufficient conditions for $\overline{U}$ to be the unique optimal
solution.
We provide the analytical form of $\overline{V}=\overline{U}^{-1}$. We prove
that $\overline{V}$ is an $M$-matrix and that $\overline{U}$ is thus an
inverse $M$-matrix. We also prove that the eigenvalues of $\overline{U}$ and
$\overline{V}$ are the same as their diagonal entries.
We demonstrate how the optimization model and its solutions apply to the task
of synthesizing conditional cumulative distribution functions (CDFs) that, in
tandem with a given discrete marginal distribution, coherently preserve a
given CDF.
## Introduction
### 1.1 Overview
We present an optimization model defined on the manifold of the set of
stochastic matrices. Geometrically, the model is akin to identifying a
maximum-volume $n$-dimensional simplex. The identification of maximum- and
minimum-volume simplices under various conditions has many applications
including but not limited to hyperspectral unmixing [1, 2, 3], system control
[4, 5], computer-aided design [6], and signal processing [7]. In our model,
the conditions we consider are that the simplex has a given barycenter and
that the simplex is enclosed by the $n$-dimensional standard simplex.
Maximizing the volume of a simplex is equivalent to maximizing the determinant
of its corresponding matrix. In our model, we employ the trace (the sum of
eigenvalues) as a linear alternative to the determinant (the product of
eigenvalues).
We introduce the model and its dual in Section 2. In Section 3, we identify
the analytical form of primal and dual solutions and prove that the solutions
are optimal using the Karush-Kuhn-Tucker (KKT) conditions. We additionally
prove a necessary and sufficient condition for the identified optimal primal
solution, a matrix, to be uniquely optimal. In Section 4, we present the
analytical inverse of that and then classify the two matrices. We additionally
identify and discuss the eigenvalues of these matrices. In Section 5, we
present the application that motivates the optimization model and its
solutions. The application involves synthesizing conditional cumulative
distribution functions (CDFs) that, in tandem with a given discrete marginal
distribution, coherently preserve a given CDF. We present our conclusions in
Section 6.
### 1.2 Notation, Conventions, & Definitions
In this paper, all matrices and vectors are indexed using 1-based indexing.
All vectors are column vectors unless transposed, and all transposed vectors
are row vectors. We let $\bm{0}$ and $\bm{1}$ be vectors of all zeros or ones,
respectively, and we let $\bm{e}_{i}$ be the vector with a 1 in the
$i\textsuperscript{th}$ position and zeros elsewhere. We define
$[c]^{+}=\max\\{c,0\\}$ and $[c]^{-}=\min\\{c,0\\}$. We additionally define
$\mathbb{I}_{[\,\cdot\,]}$ as the indicator function that takes a value of one
if the subscripted condition is true and zero otherwise.
Our analysis revolves around stochastic matrices and vectors, terms we now
define.
###### Definition 1.
A square matrix $A\in\mathbb{R}^{n\times n}$ is a (right) stochastic matrix if
$A\bm{1}=\bm{1}$ and $A\geq 0$ elementwise.
###### Definition 2.
A vector $\bm{a}\in\mathbb{R}^{n}$ is a stochastic vector if
$\bm{a}^{\top}\bm{1}=1$ and $\bm{a}\geq\bm{0}$ elementwise.
In Section 5, we discuss the 1-Wasserstein distance, also known as the Earth-
Mover’s distance. We now define that distance as it appears in Rachev and
Rüschendorf [8].
###### Definition 3.
The _1-Wasserstein distance_ between probability measure $\mu_{1}$ with
associated CDF $F_{1}(x)$ and probability measure $\mu_{2}$ with associated
CDF $F_{2}(x)$ is
$W_{1}(\mu_{1},\mu_{2})=\int_{\mathbb{R}}|F_{1}(x)-F_{2}(x)|\mathop{}\\!\mathrm{d}x.$
## Optimization Model
Let $U\in\mathbb{R}^{n\times n}$ be a stochastic matrix and
$\bm{q}\in\mathbb{R}^{n}$ a stochastic vector. Note that $U\geq 0$ and
$\bm{q}\geq\bm{0}$ together imply that $U^{\top}\bm{q}\geq\bm{0}$, and that
$\bm{1}^{\top}U^{\top}\bm{q}=\bm{1}^{\top}\bm{q}=1$. That is, $U^{\top}$ maps
one stochastic vector to another.
Given two $n$-dimensional stochastic vectors ${\bm{q}>\bm{0}}$ and
${\bm{p}>\bm{0}}$, we consider as a candidate any stochastic matrix $U$ whose
transpose maps $\bm{q}$ to $\bm{p}$. That is, our candidate set is
$\mathcal{U}=\\{U\in\mathbb{R}^{n\times
n}:U^{\top}\bm{q}=\bm{p},U\bm{1}=\bm{1},U\geq 0\\}.$ (1)
Among these matrices, we seek a matrix whose trace is maximized. The problem
we describe may be formulated as a linear program (LP):
$\max\\{\operatorname{tr}(U):U\in\mathcal{U}\\}.$ (P)
Recall that the trace of a matrix is the sum of its diagonal entries, which is
also equal to the sum of its eigenvalues. As such, (P) is an eigenvalue
optimization problem.
The geometry of (P) may be interpreted as follows. Take the rows of $U$ to be
vertices of an $(n-1)$-simplex and the entries of $\bm{q}$ to be weights at
those vertices. The feasible space is the set of $(n-1)$-simplices that have a
$\bm{q}$-weighted barycenter of $\bm{p}$ and are enclosed by the standard
${(n-1)}$-simplex. The goal is to identify a simplex for which the sum of
$L^{1}$ distances between the $i\textsuperscript{th}$ vertex and $\bm{e}_{i}$
is minimized.
To obtain the dual of (P), consider its Lagrangian:
$\mathcal{L}(U,\bm{\alpha},\bm{\beta},\Gamma)=\mathrm{tr}(U)-\bm{\alpha}^{\top}(U^{\top}\bm{q}-\bm{p})-\bm{\beta}^{\top}(U\bm{1}-\bm{1})+\mathrm{tr}(\Gamma
U^{\top}).$
From this, we see that
$\nabla_{U}\mathcal{L}(U,\bm{\alpha},\bm{\beta},\Gamma)=I-\bm{q}\bm{\alpha}^{\top}-\bm{\beta}\bm{1}^{\top}+\Gamma.$
Hence, the dual function is
$g(\bm{\alpha},\bm{\beta},\Gamma)=\begin{cases}\bm{\alpha}^{\top}\bm{p}+\bm{\beta}^{\top}\bm{1}~{}&\text{if}~{}\bm{q}\bm{\alpha}^{\top}+\bm{\beta}\bm{1}^{\top}-\Gamma=I,\Gamma\geq
0,\\\ \infty~{}&\text{otherwise}.\end{cases}$ (2)
The dual LP, derived from (2), is α^⊤p\+ β^⊤1 qα^⊤+ β1 ^⊤- Γ= I, Γ≥0.
We summarize the KKT conditions [9, 10] for (P) and (2) in Table 1. Since (P)
and (2) are LPs, the KKT conditions are necessary and sufficient for
optimality.
Table 1: KKT Conditions for (P) and (2). Stationarity | $\bm{q}\bm{\alpha}^{\top}+\bm{\beta}\bm{1}^{\top}-\Gamma=I$
---|---
Primal Feasibility | $\bm{p}^{\top}=\bm{q}^{\top}U$
$U\bm{1}=\bm{1}$
$U\geq 0$
Dual Feasibility | $\Gamma\geq 0$
Complementary Slackness | $\operatorname{tr}(U\Gamma^{\top})=0$
## Optimal Primal & Dual Solutions
For stochastic vectors $\bm{p}>\bm{0}$ and $\bm{q}>\bm{0}$, let $\overline{U}$
be the corresponding matrix with entries
$\overline{u}_{ij}\triangleq\begin{cases}1-[1-p_{i}/q_{i}]^{+},&i=j,\\\\[2.5pt]
0,&i\neq j~{}\text{and}~{}\bm{p}=\bm{q},\\\\[2.5pt]
\displaystyle\frac{[1-p_{i}/q_{i}]^{+}[p_{j}-q_{j}]^{+}}{\sum_{k=1}^{n}[p_{k}-q_{k}]^{+}},&i\neq
j~{}\text{and}~{}\bm{p}\neq\bm{q}.\end{cases}$ (2)
Additionally, let $\overline{\bm{\alpha}}$ and $\overline{\bm{\beta}}$ be
$n$-dimensional vectors and $\overline{\Gamma}$ an $n\times n$ matrix with
entries
$\displaystyle\overline{\alpha}_{j}$
$\displaystyle\triangleq\mathbb{I}_{[p_{j}<q_{j}]}/q_{j},$
$\displaystyle\overline{\beta}_{i}$
$\displaystyle\triangleq\mathbb{I}_{[p_{i}\geq q_{i}]},$
$\displaystyle\overline{\gamma}_{ij}$
$\displaystyle\triangleq\mathbb{I}_{[p_{i}\geq
q_{i}]}+\mathbb{I}_{[p_{j}<q_{j}]}\cdot q_{i}/q_{j}-\mathbb{I}_{[i=j]}.$
We present these values on a case-by-case basis in Table 2.
Table 2: Values of $\overline{\bm{\alpha}}$, $\overline{\bm{\beta}}$, $\overline{\Gamma}$, and related quantities on a case-by-case basis. Conditions | $\overline{\alpha}_{j}$ | $\overline{\beta}_{i}$ | $\overline{\gamma}_{ij}$ | $\left[\bm{q}\overline{\bm{\alpha}}^{\top}\right]_{ij}$ | $\left[\overline{\bm{\beta}}\bm{1}^{\top}\right]_{ij}$
---|---|---|---|---|---
$i=j$ | $p_{i}<q_{i}$ | $p_{j}<q_{j}$ | $1/q_{j}$ | $0$ | $0$ | $1$ | $0$
$i=j$ | $p_{i}\geq q_{i}$ | $p_{j}\geq q_{j}$ | $0$ | $1$ | $0$ | $0$ | $1$
$i\neq j$ | $p_{i}<q_{i}$ | $p_{j}<q_{j}$ | $1/q_{j}$ | $0$ | $q_{j}/q_{i}$ | $q_{j}/q_{i}$ | $0$
$i\neq j$ | $p_{i}<q_{i}$ | $p_{j}\geq q_{j}$ | $0$ | $0$ | $0$ | $0$ | $0$
$i\neq j$ | $p_{i}\geq q_{i}$ | $p_{j}<q_{j}$ | $1/q_{j}$ | $1$ | $q_{j}/q_{i}+1$ | $q_{j}/q_{i}$ | $1$
$i\neq j$ | $p_{i}\geq q_{i}$ | $p_{j}\geq q_{j}$ | $0$ | $1$ | $1$ | $0$ | $1$
We now work toward proving that $\overline{U}$ and
$(\overline{\bm{\alpha}},\overline{\bm{\beta}},\overline{\Gamma})$ are an
optimal primal-dual pair.
###### Lemma 1.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, the corresponding matrix $\overline{U}$ is feasible in (P).
###### Proof.
For $\overline{U}$ to be feasible in (P), it must exist in $\mathcal{U}$. If
$\bm{p}=\bm{q}$, then $\overline{U}=I\in\mathcal{U}$, trivially. Otherwise,
consider as preliminaries that
$[1-p_{i}/q_{i}]^{+}=[q_{i}-p_{i}]^{+}/q_{i},$
and
$q_{j}\overline{u}_{jj}=q_{j}-q_{j}[1-p_{j}/q_{j}]^{+}=q_{j}-[q_{j}-p_{j}]^{+}=p_{j}-[p_{j}-q_{j}]^{+}.$
Now first,
$\displaystyle\sum_{j=1}^{n}\overline{u}_{ij}$
$\displaystyle=\overline{u}_{ii}+\sum_{j\neq i}\overline{u}_{ij}$
$\displaystyle=1-[1-p_{i}/q_{i}]^{+}+\sum_{j\neq
i}\frac{[1-p_{i}/q_{i}]^{+}[p_{j}-q_{j}]^{+}}{\sum_{k=1}^{n}[p_{k}-q_{k}]^{+}}$
$\displaystyle=1-\frac{[q_{i}-p_{i}]^{+}[p_{j}-q_{j}]^{+}}{q_{i}\sum_{k=1}^{n}[p_{k}-q_{k}]^{+}}$
$\displaystyle=1.$
And second,
$\displaystyle\sum_{i=1}^{n}q_{i}\overline{u}_{ij}$
$\displaystyle=q_{j}\overline{u}_{jj}+\sum_{i\neq j}q_{i}\overline{u}_{ij}$
$\displaystyle=p_{j}-[p_{j}-q_{j}]^{+}+\sum_{i\neq
j}\frac{q_{i}[1-p_{i}/q_{i}]^{+}[p_{j}-q_{j}]^{+}}{\sum_{k=1}^{n}[p_{k}-q_{k}]^{+}}$
$\displaystyle=p_{j}-\frac{[p_{j}-q_{j}]^{+}[q_{j}-p_{j}]^{+}}{\sum_{k=1}^{n}[p_{k}-q_{k}]^{+}}$
$\displaystyle=p_{j}.$
Thus, $\overline{U}\bm{1}=\bm{1}$ and $\overline{U}^{\top}\bm{q}=\bm{p}$.
Finally, $\overline{U}\geq 0$ is evident from the definition of
$\overline{u}_{ij}$. Hence, $\overline{U}\in\mathcal{U}$. ∎
###### Lemma 2.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, the corresponding vectors $\overline{\bm{\alpha}}$ and
$\overline{\bm{\beta}}$ and matrix $\overline{\Gamma}$ are feasible in (2).
###### Proof.
One may verify that $\overline{\bm{\alpha}},\overline{\bm{\beta}}$, and
$\overline{\Gamma}$ comprise a feasible solution to (2) from Table 2. ∎
###### Lemma 3.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, the corresponding matrices $\overline{U}$ and
$\overline{\Gamma}$ satisfy the complementary slackness condition
$\operatorname{tr}(U\Gamma^{\top})=0$.
###### Proof.
Observe that $\overline{\gamma}_{ij}=0$ only when $i=j$ or when $i\neq
j,p_{i}<q_{i}$, and $p_{j}\geq q_{j}$. In any other case,
$\overline{u}_{ij}=0$. As such $\overline{\gamma}_{ij}\overline{u}_{ij}=0$ for
all $i,j=1,\ldots,n$. ∎
###### Theorem 4.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, the corresponding $\overline{U}$ is optimal in (P) and the
corresponding $\overline{\bm{\alpha}}$, $\overline{\bm{\beta}}$, and
$\overline{\Gamma}$ are optimal in (2).
###### Proof.
The KKT conditions in Table 1 are necessary and sufficient for LP optimality.
For $\overline{U}$, $\overline{\bm{\alpha}}$, $\overline{\bm{\beta}}$, and
$\overline{\Gamma}$, Lemma 1 proves primal feasibility, Lemma 2 proves dual
feasibility and stationarity, and Lemma 3 proves complementary slackness. ∎
It follows, of course, that
$\operatorname{tr}(\overline{U})=\overline{\bm{\alpha}}^{\top}\bm{p}+\overline{\bm{\beta}}^{\top}\bm{1}$.
No matter, showing how this equality holds shines light on the relationship
between $[\,\cdot\,]^{+}$ in the construction of the primal optimal solution
and $\mathbb{I}_{[\,\cdot\,]}$ in the construction of the dual optimal
solution:
$\displaystyle\overline{\bm{\alpha}}^{\top}\bm{p}+\overline{\bm{\beta}}^{\top}\bm{1}$
$\displaystyle=\sum_{i=1}^{n}\left(p_{i}/q_{i}\cdot\mathbb{I}_{[p_{i}<q_{i}]}+\mathbb{I}_{[p_{i}\geq
q_{i}]}\right)$
$\displaystyle=\sum_{i=1}^{n}\left(p_{i}/q_{i}\cdot(1-\mathbb{I}_{[p_{i}\geq
q_{i}]})+\mathbb{I}_{[p_{i}\geq q_{i}]}\right)$
$\displaystyle=\sum_{i=1}^{n}\left((1-p_{i}/q_{i})\mathbb{I}_{[p_{i}\geq
q_{i}]}+p_{i}/q_{i}\right)$
$\displaystyle=\sum_{i=1}^{n}\left(1-[1-p_{i}/q_{i}]^{+}\right)$
$\displaystyle=\sum_{i=1}^{n}\overline{u}_{ii}$
$\displaystyle=\operatorname{tr}(\overline{U}).$
To conclude this section, we discuss conditions under which $\overline{U}$ is
uniquely optimal in (P).
###### Lemma 5.
Every optimal solution to (P) possesses
$u_{ii}=\min\\{p_{i}/q_{i},1\\}=1-[1-p_{i}/q_{i}]^{+},\quad\forall
i=1,\ldots,n.$
Moreover, if an optimal solution possesses $u_{ii}=1$ then its
$i\textsuperscript{th}$ row is $\bm{e}_{i}^{\top}$, and if it possesses
$u_{ii}=\frac{p_{i}}{q_{i}}$ then its $i\textsuperscript{th}$ column is
$\frac{p_{i}}{q_{i}}\bm{e}_{i}$.
###### Proof.
For each $i=1,\ldots,n$,
$\displaystyle U\bm{1}=\bm{1},U\geq 0$ $\displaystyle\implies u_{ii}\leq 1,$
$\displaystyle U^{\top}\bm{q}=\bm{p},U\geq 0$ $\displaystyle\implies
u_{ii}\leq p_{i}/q_{i}.$
As such, the constraints of (P) imply for each $i=1,\ldots,n$ that
${u_{ii}\leq\min\\{p_{i}/q_{i},1\\}=1-[1-p_{i}/q_{i}]^{+}}$ independently. The
objective of (P) is to maximize $\sum_{i=1}^{n}u_{ii}$, and $\overline{U}$
does so by satisfying all of these constraints at equality. Hence, every
optimal solution to (P) must satisfy these constraints at equality.
If $u_{ii}=1$, then $U\bm{1}=\bm{1},U\geq 0$ imply that $u_{ij}=0$ for all
$j\neq i$. Similarly, if $u_{ii}=p_{i}/q_{i}$, then
$U^{\top}\bm{q}=\bm{p},U\geq 0$ imply that $u_{ji}=0$ for all $j\neq i$. ∎
The following lemma, which provides a necessary and sufficient condition for
the uniqueness of an optimal LP solution, comes from Mangasarian [11].
###### Lemma 6.
Let $\overline{\bm{x}}$ be an optimal solution to the linear program
$\min\\{\bm{p}^{\top}\bm{x}:A\bm{x}=\bm{b},C\bm{x}\geq d\\}.$
Let $\bm{c}_{i}^{\top}$ be the $i\textsuperscript{th}$ row of $C$, and let
$C_{\mathcal{E}}$ be the matrix whose rows are all the $\bm{c}_{i}$ satisfying
$\bm{c}_{i}^{\top}\overline{\bm{x}}=d_{i}$. Then $\overline{\bm{x}}$ is
uniquely optimal if and only if
$\widetilde{\mathcal{X}}=\\{\tilde{\bm{x}}:\bm{p}^{\top}\tilde{\bm{x}}\leq
0,\tilde{\bm{x}}\neq\bm{0},A\tilde{\bm{x}}=\bm{0},C_{\mathcal{E}}\tilde{\bm{x}}\geq\bm{0}\\}=\varnothing.$
Lemma 6 essentially states that for any $\tilde{\bm{x}}\in\widetilde{X}$ and
sufficiently small $\epsilon>0$, $\overline{\bm{x}}$ and
$\overline{\bm{x}}+\epsilon\tilde{\bm{x}}$ are both feasible and optimal
solutions. With this in mind, we consider the conditions under which
$\displaystyle\widetilde{\mathcal{U}}\triangleq\\{\widetilde{U}\in\mathbb{R}^{n\times
n}:\operatorname{tr}(\widetilde{U})\geq 0,\widetilde{U}\neq
0,\widetilde{U}^{\top}\bm{q}=\bm{0},\widetilde{U}\bm{1}=\bm{0},$
$\displaystyle\tilde{u}{ij}\geq 0,\forall
i,j=1,\ldots,n~{}\text{s.t.}~{}\overline{u}_{ij}=0$
$\displaystyle\\}=\varnothing.$
###### Theorem 7.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, define
$\displaystyle\mathcal{G}$ $\displaystyle=\\{i:p_{i}>q_{i}\\},$
$\displaystyle\mathcal{E}$ $\displaystyle=\\{i:p_{i}=q_{i}\\},$
$\displaystyle\mathcal{L}$ $\displaystyle=\\{i:p_{i}<q_{i}\\}.$
The corresponding matrix $\overline{U}\in\mathbb{R}^{n\times n}$ is uniquely
optimal in (P) if and only if $|\mathcal{G}|\leq 1$ or $|\mathcal{L}|\leq 1$.
###### Proof.
We first consider the case of $|\mathcal{G}|\leq 1$. Note that
$|\mathcal{G}|=0$ if and only if $\bm{p}=\bm{q}$, and it follows from Lemma 5
that $\overline{U}=I$ is uniquely optimal. If $|\mathcal{G}|=1$, then
$|\mathcal{E}\cup\mathcal{L}|=n-1$. In this case, $n-1$ columns of an optimal
to solution to (P) are determined by Lemma 5, and the remaining column is
determined by $U\bm{1}=\bm{1}$. Hence, $\overline{U}$ is uniquely optimal. For
$|\mathcal{L}|\leq 1$, a similar argument applies to the rows of an optimal
solution.
Now if neither $|\mathcal{G}|\leq 1$ nor $|\mathcal{L}|\leq 1$, then
${|\mathcal{G}|\geq 2}$ and ${|\mathcal{L}|\geq 2}$. Let
$i,i^{\prime}\in\mathcal{L},i\neq i^{\prime}$, let
$j,j^{\prime}\in\mathcal{G},j\neq j^{\prime}$, and let $\widetilde{U}$ be a
matrix with ${\tilde{u}_{ij}=q_{i^{\prime}}}$,
${\tilde{u}_{ij^{\prime}}=-q_{i^{\prime}}}$,
${\tilde{u}_{i^{\prime}j}=-q_{i}}$,
${\tilde{u}_{i^{\prime}j^{\prime}}=q_{i}}$, and all other entries zero. Note
that $\mathcal{G}\cap\mathcal{L}=\varnothing$, so
$\operatorname{tr}(\widetilde{U})=0$. The matrix $\widetilde{U}$ trivially
satisfies $\widetilde{U}\neq 0$, $\widetilde{U}^{\top}\bm{q}=\bm{0}$, and
$\widetilde{U}\bm{1}=\bm{0}$. Finally, owing to the fact that $p_{i}<q_{i}$,
$p_{i^{\prime}}<q_{i^{\prime}}$, $p_{j}>q_{j}$, and
$p_{j^{\prime}}>q_{j^{\prime}}$, all of $\overline{u}_{ij}$,
$\overline{u}_{ij^{\prime}}$, $\overline{u}_{i^{\prime}j}$, and
$\overline{u}_{i^{\prime}j^{\prime}}$ are strictly positive. Accordingly, the
corresponding entries of $\widetilde{U}$ are not subject to the only other
constraint of $\widetilde{\mathcal{U}}$. Hence,
$\widetilde{U}\in\widetilde{\mathcal{U}}\neq\varnothing$. Following Lemma 6,
$\overline{U}$ is not uniquely optimal in (P). ∎
###### Corollary 8.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, the corresponding $\overline{U}$ is uniquely optimal in (P)
if $n=1,2,3$.
###### Example 1.
Let
$\bm{p}^{\top}=[\nicefrac{{1}}{{2}},\nicefrac{{1}}{{4}},\nicefrac{{1}}{{8}},\nicefrac{{1}}{{8}}]$
and
$\bm{q}^{\top}=[\nicefrac{{1}}{{20}},\nicefrac{{1}}{{5}},\nicefrac{{7}}{{20}},\nicefrac{{2}}{{5}}]$.
Then $\mathcal{G}=\\{1,2\\}$, $\mathcal{E}=\varnothing$,
$\mathcal{L}=\\{3,4\\}$, and
$\overline{U}=\left[\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
\nicefrac{{3}}{{8}}&\nicefrac{{1}}{{8}}&\nicefrac{{1}}{{2}}&0\\\
\nicefrac{{9}}{{16}}&\nicefrac{{3}}{{16}}&0&\nicefrac{{1}}{{4}}\end{array}\right].$
Using the construction in the proof for Theorem 7, we obtain
$\widetilde{U}=\left[\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
\nicefrac{{-1}}{{2}}&\nicefrac{{1}}{{2}}&0&0\\\
\nicefrac{{1}}{{4}}&\nicefrac{{-1}}{{4}}&0&0\end{array}\right]\in\widetilde{\mathcal{U}}.$
From this, we see that
$\overline{U}+\min\\{\nicefrac{{\overline{u}_{31}}}{{q_{4}}},\nicefrac{{\overline{u}_{42}}}{{q_{3}}}\\}\widetilde{U}=\left[\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&\nicefrac{{1}}{{2}}&\nicefrac{{1}}{{2}}&0\\\
\nicefrac{{3}}{{4}}&0&0&\nicefrac{{1}}{{4}}\end{array}\right]$
is likewise feasible and optimal in (P).
## Properties of $\overline{U}$ and its Inverse
In this section, we identify the matrix $\overline{V}$ that is the inverse of
$\overline{U}$, then we discuss the properties of these matrices. First, we
show that $\overline{V}$ is a $Z$-matrix and moreover an $M$-matrix, and that
$\overline{U}$ is accordingly an inverse $M$-matrix (i.e., a matrix that is
the inverse of an $M$-matrix). These classes of matrices are noteworthy for
their numerous and varied characterizations [12, 13]. Independently, we show
that the diagonal entries of $\overline{U}$ and $\overline{V}$ are the
eigenvalues of those matrices.
For stochastic vectors $\bm{p}>\bm{0}$ and $\bm{q}>\bm{0}$, let $\overline{V}$
be the corresponding matrix with entries
$\overline{v}_{ij}\triangleq\begin{cases}1-[1-q_{i}/p_{i}]^{-},&i=j,\\\\[5.0pt]
0,&i\neq j~{}\text{and}~{}\bm{p}=\bm{q},\\\\[2.5pt]
\displaystyle\frac{[1-q_{i}/p_{i}]^{-}[p_{j}-q_{j}]^{+}}{\sum_{k=1}^{n}[p_{k}-q_{k}]^{+}},&i\neq
j~{}\text{and}~{}\bm{p}\neq\bm{q}.\end{cases}$ (3)
We now prove that $\overline{V}$ is the inverse of $\overline{U}$.
###### Theorem 9.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, the corresponding matrices $\overline{U}$ and
$\overline{V}$ satisfy $\overline{U}\overline{V}=I$.
###### Proof.
If $\bm{p}=\bm{q}$, then $\overline{U}=\overline{V}=I$, and the result is
trivially true. Otherwise, observe that
$\displaystyle\overline{u}_{kk}=1-[1-p_{k}/q_{k}]^{+}=\min\\{p_{k}/q_{k},1\\},$
$\displaystyle\overline{v}_{kk}=1-[1-q_{k}/p_{k}]^{-}=\max\\{q_{k}/p_{k},1\\}.$
If $p_{k}\geq q_{k}$, then $\overline{u}_{kk}=1$ and $\overline{v}_{kk}=1$. If
$p_{k}<q_{k}$, then $\overline{u}_{kk}=p_{k}/q_{k}$ and
$\overline{v}_{kk}=q_{k}/p_{k}$. Hence,
$\overline{u}_{kk}\overline{v}_{kk}=1$. Additionally, observe that
$[p_{k}-q_{k}]^{+}[1-q_{k}/p_{k}]^{-}=0$, so
$\overline{u}_{ik}\overline{v}_{kj}=\frac{[1-p_{i}/q_{i}]^{+}[p_{k}-q_{k}]^{+}}{\sum_{l=1}^{n}[p_{l}-q_{l}]^{+}}\cdot\frac{[1-q_{k}/p_{k}]^{-}[p_{j}-q_{j}]^{+}}{\sum_{l=1}^{n}[p_{l}-q_{l}]^{+}}=0$
for all $i,j,k=1,\ldots,n,i\neq k,k\neq j$. As such,
$\left[\overline{U}\overline{V}\right]_{ij}=\sum_{k=1}^{n}\overline{u}_{ik}\overline{v}_{kj}=\begin{cases}1~{}\text{if}~{}i=j,\\\
0~{}\text{otherwise.}\end{cases}$
∎
Since $\bm{1}$ is a fixed point of $\overline{U}$, it is also a fixed point of
its inverse $\overline{V}$. Hence, $\overline{V}\bm{1}=\bm{1}$. Also,
$\overline{U}^{\top}\bm{p}=\bm{q}\implies\overline{V}^{\top}\overline{U}^{\top}\bm{p}=\overline{V}^{\top}\bm{q}\implies\overline{V}^{\top}\bm{q}=\bm{p}.$
Geometrically, $\overline{U}$ is the matrix representation of a simplex that
has a $\bm{p}$-weighted barycenter of $\bm{q}$ and is enclosed by the standard
simplex. Its inverse $\overline{V}$ is the matrix representation of a simplex
that has a $\bm{q}$-weighted barycenter of $\bm{p}$ and is an enclosure of the
standard simplex.
### 4.1 Classification of $\overline{U}$ and $\overline{V}$
Using the fact that $\overline{U}\overline{V}=I$, we now classify these
matrices.
###### Lemma 10.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, the corresponding $\overline{V}$ is a $Z$-matrix.
###### Proof.
A $Z$-matrix is a real-valued square matrix whose off-diagonal entries are
nonpositive [13]. From (3), we see that the offdiagonal entries of
$\overline{V}$ are either zero (if $\bm{p}=\bm{q}$) or nonpositive (if
$\bm{p}\neq\bm{q}$). ∎
###### Theorem 11.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, the corresponding $\overline{V}$ is an $M$-matrix.
###### Proof.
For a matrix $A$ to be an $M$-matrix, it is necessary and sufficient for $A$
to be a $Z$-matrix that is nonsingular with ${A^{-1}\geq 0}$ [13], which for
$\overline{V}$ follows from Theorem 1, Theorem 9, and Lemma 10. ∎
###### Corollary 12.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, the corresponding $\overline{U}$ is an inverse $M$-matrix.
### 4.2 Eigenvalues of $\overline{U}$ and $\overline{V}$
In Section 3 we showed that $\overline{U}$ is an optimal solution to the
matrix optimization problem (P), which we noted is an eigenvalue optimization
problem. Interestingly, the eigenvalues of $\overline{U}$ and $\overline{V}$
are their diagonal entries, which we now prove.
###### Lemma 13.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, for all $i=1,2,\ldots,n,$ either the
$i\textsuperscript{th}$ row or $i\textsuperscript{th}$ column of the
corresponding $\overline{U}$ consists entirely of zeros except for
$\overline{u}_{ii}>0$, and either the $i\textsuperscript{th}$ row or
$i\textsuperscript{th}$ column of the corresponding $\overline{V}$ consists
entirely of zeros except for $\overline{v}_{ii}>0$.
###### Proof.
If $\bm{p}=\bm{q}$, then $\overline{U}=\overline{V}=I$. In such a case, both
the $i\textsuperscript{th}$ row and $i\textsuperscript{th}$ column of
$\overline{U}$ are all zeros except for $\overline{u}_{ii}=1$. Similarly, both
the $i\textsuperscript{th}$ row and $i\textsuperscript{th}$ column of
$\overline{V}$ are all zeros except for $\overline{v}_{ii}=1$.
Otherwise, if $\bm{p}\neq\bm{q}$, recall for $i\neq j$ that
$\displaystyle\overline{u}_{ij}=\frac{[1-p_{i}/q_{i}]^{+}[p_{j}-q_{j}]^{+}}{\sum_{k=1}^{n}[p_{k}-q_{k}]^{+}},$
$\displaystyle\overline{v}_{ij}=\frac{[1-q_{i}/p_{i}]^{-}[p_{j}-q_{j}]^{+}}{\sum_{k=1}^{n}[p_{k}-q_{k}]^{+}}.$
Now observe that for all $i,j=1,\ldots,n,i\neq j$, either
$p_{i}\geq
q_{i}\implies[1-p_{i}/q_{i}]^{+}=[1-q_{i}/p_{i}]^{-}=0\implies\overline{u}_{ij}=\overline{v}_{ij}=0,$
or
$p_{i}\leq
q_{i}\implies[p_{i}-q_{i}]^{+}\implies\overline{u}_{ji}=\overline{v}_{ji}=0.$
∎
###### Theorem 14.
For any two $n$-dimensional stochastic vectors ${\bm{p}>\bm{0}}$ and
${\bm{q}>\bm{0}}$, the eigenvalues of the corresponding $\overline{U}$ are
$\overline{u}_{11},\overline{u}_{22},\ldots,\overline{u}_{nn}$, and the
eigenvalues of the corresponding $\overline{V}$ are similarly
$\overline{v}_{11},\overline{v}_{22},\ldots,\overline{v}_{nn}$.
###### Proof.
From Lemma 13, for $i=1,\ldots,n$ either the $i\textsuperscript{th}$ row or
$i\textsuperscript{th}$ column of $\overline{U}$ consists of all zeros except
for $\overline{u}_{ii}>0$. The same is true of $\overline{V}$. Using Laplace
expansion, one may thus expand strategically on rows or columns to get
characteristic equations
$\displaystyle\det(\overline{U}-\lambda
I)=\prod_{i=1}^{n}(\overline{u}_{ii}-\lambda),$
$\displaystyle\det(\overline{V}-\lambda
I)=\prod_{i=1}^{n}(\overline{v}_{ii}-\lambda).$
Ergo, the eigenvalues of $\overline{U}$ are
$\overline{u}_{11},\overline{u}_{22},\ldots,\overline{u}_{nn}$ and, similarly,
the eigenvalues of $\overline{V}$ are
$\overline{v}_{11},\overline{v}_{22},\ldots,\overline{v}_{nn}$. ∎
While $\overline{U}$, among $U\in\mathcal{U}$, maximizes
$\operatorname{tr}(U)$, it does not necessarily maximize $|\det(U)|$. That is,
among $U\in\mathcal{U}$, the matrix $\overline{U}$ does not necessarily
correspond to a simplex of maximum unsigned volume. The reason is that trace
maximization inherently promotes a simplex orientation that sites the
$i\textsuperscript{th}$ vertex near to $\bm{e}_{i}$. Consider the following
relaxation of (P) involving the joint optimization of $U\in\mathcal{U}$ and
$M\in\mathcal{P}$, the set of all $n\times n$ permutation matrices:
$\max\\{\operatorname{tr}(MU):U\in\mathcal{U},M\in\mathcal{P}\\}.$ (P-Perm)
The goal of this problem is to identify a $U\in\mathcal{U}$ whose
premultiplication with an appropriate $M\in\mathcal{P}$ yields a trace-
maximizing matrix. This relaxation of (P) promotes no particular simplex
orientation. Rather, it allows the simplex associated with $U$ to be oriented
freely via $M\in\mathcal{P}$.
For fixed $M\in\mathcal{P}$, a matrix $U\in\mathcal{U}$ that maximizes
$\operatorname{tr}(MU)$ is the $\overline{U}$ corresponding to stochastic
vectors $\bm{p}>0$ and $M^{\top}\bm{q}>0$. An optimal $M$ to (P-Perm) may thus
be identified via ∑_i ∈R ∑_j ∈R (1 - [1 - p_j / q_i]^+) m_ij M1 = 1 M^⊤1 = 1
M∈{0, 1}^n ×n This assignment problem features discrete variables. However,
the variable coefficients in the constraints comprise a totally unimodular
matrix and the constant terms in the constraints are all integral. As such,
every optimal extreme point solution to the linear programming relaxation is
an optimal $M$ to (P-Perm) [14].
That (P-Perm) is a relaxation of (P) means there may exist a
$\widehat{U}\in\mathcal{U}$ and $\widehat{M}\in\mathcal{P}$ for which
$\operatorname{tr}(\widehat{M}\widehat{U})\geq\operatorname{tr}(\overline{U})$.
Of course, if
$\operatorname{tr}(\widehat{M}\widehat{U})>\operatorname{tr}(\overline{U})$,
it is possible that $\det(\widehat{M}\widehat{U})>\det(\overline{U})$ and,
since the absolute value of the determinant of a matrix is invariant under
permutation, $|\det(\widehat{U})|>|\det(\overline{U})|$. We demonstrate this
now in an example.
###### Example 2.
Let
$\bm{p}^{\top}=[\nicefrac{{3}}{{10}},\nicefrac{{2}}{{5}},\nicefrac{{1}}{{10}},\nicefrac{{1}}{{5}}]$
and
$\bm{q}^{\top}=[\nicefrac{{1}}{{8}},\nicefrac{{3}}{{8}},\nicefrac{{3}}{{10}},\nicefrac{{1}}{{5}}]$.
Then
$\overline{U}=\left[\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
\nicefrac{{7}}{{12}}&\nicefrac{{1}}{{12}}&\nicefrac{{1}}{{3}}&0\\\
0&0&0&1\end{array}\right]$
is the optimal solution to (P), and
$\widehat{M}=\left[\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\
0&0&0&1\end{array}\right],~{}\widehat{U}=\left[\begin{array}[]{cccc}0&\nicefrac{{1}}{{5}}&\nicefrac{{4}}{{5}}&0\\\
0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\end{array}\right]$
are the optimal solution to (P-Perm). For this example, we have
$\operatorname{tr}(\widehat{M}\widehat{U})=\nicefrac{{19}}{{5}}>\nicefrac{{10}}{{3}}=\operatorname{tr}(\overline{U})$
and
$|\det(\widehat{U})|=\nicefrac{{4}}{{5}}>\nicefrac{{1}}{{3}}=|\det(\overline{U})|$.
The simplices associated with $\overline{U}$ and $\widehat{U}$ are illustrated
in Figure 1. Indeed, the simplex associated with $\widehat{U}$ has a visibly
larger volume than that associated with $\overline{U}$.
Figure 1: The simplices associated with $\overline{U}$ (left) and
$\widehat{U}$ (right) from Example 2.
## Application
Let $X$ be a random variable (RV) with support $\Omega$ and associated CDF
$F(x)$. We wish to synthesize conditional CDFs with respect to a discrete RV
$Y$ with support $\mathcal{N}=\\{1,2,\ldots,n\\}$ according to
$F(x\,|\,Y=i)=S_{i}(F(x)),\quad\forall i\in\mathcal{N}.$ (4)
Now define $p_{i}=\operatorname{Pr}(Y=i)$. Also, define
$\bm{p}=[p_{1},p_{2},\ldots,p_{n}]^{\top}$, and
$\bm{S}(z)=[S_{1}(z),S_{2}(z),\ldots,S_{n}(z)]^{\top}$. For such synthesized
conditional CDFs to be coherent, $\bm{p}$ and $\bm{S}(z)$ must satisfy
$\displaystyle\bm{p}^{\top}\geq\bm{0},$ (5)
$\displaystyle\bm{p}^{\top}\bm{1}=1,$ (6) $\displaystyle\bm{S}(0)=\bm{0},$ (7)
$\displaystyle\bm{S}(1)=\bm{1},$ (8)
$\displaystyle\bm{S}(z)\leq\bm{S}(\hat{z}),$ $\displaystyle\forall
z,\hat{z}\in[0,1],z\leq\hat{z},$ (9) $\displaystyle\bm{p}^{\top}\bm{S}(z)=z,$
$\displaystyle\forall z\in[0,1].$ (10)
Conditions (5) and (6) are simply the first two axioms of probability.
Conditions (7)-(9) ensure for each $i\in\mathcal{N}$ that $S_{i}(F(x))$ is the
CDF of an RV with support $\Omega_{i}\subseteq\Omega$. Condition (10),
equivalent to $\bm{p}^{\top}\bm{S}(F(x))=F(x),\forall x\in\Omega$, is the law
of total probability.
Suppose there exist independent RVs $X_{1},X_{2},\ldots,X_{m}$ all identically
distributed according to $F(x)$. Essentially, we wish to partition the RVs
into sets enumerated $i=1,\ldots,n$ with relative sizes $p_{1},\ldots,p_{n}$,
and then synthesize conditional CDFs to force a distinction between RVs from
different sets while retaining $F(x)$ as the marginal CDF.
As we showed in our original work on this application [15], and as we show in
the sequel, one may identify with little effort special cases of $\bm{p}$ and
$S(z)$ that satisfy (5)-(10). However, the question that motivates
$\mathcal{U}$ and $\overline{U}$ pertains to more general cases. Specifically,
for any given $\bm{p}$ that satisfies (5) and (6), do there exist unique
$S_{i}(z),\forall i\in\mathcal{N}$ that satisfy (7)-(10)? If so, which exhibit
maximum distinguishability (a notion we discuss further after the following
theorem)?
###### Theorem 15.
Suppose $\bm{p}>\bm{0}$ and $\bm{S}(z)$ satisfy (5)-(10), and let
$\bm{q}>\bm{0}$ be a given vector that satisfies (5) and (6). Then for any
$U\in\mathcal{U}$, $\bm{q}$ and $\bm{T}(z)=U\bm{S}(z)$ satisfy (7)-(10).
###### Proof.
Given $\bm{S}(0)=\bm{0}$, it follows that ${\bm{T}(0)=U\bm{S}(0)=\bm{0}}$.
Given $\bm{S}(1)=\bm{1}$ and $U\bm{1}=\bm{1}$, it follows that
${\bm{T}(1)=U\bm{S}(1)=U\bm{1}=\bm{1}}$. That $U\bm{1}=\bm{1}$ and $U\geq 0$
means that each $T_{i}(z)$ is a convex combination of functions that are
nondecreasing on $z\in[0,1]$. Each $T_{i}(z)$ is, as such, also nondecreasing
on $z\in[0,1]$. Finally, recall that ${\bm{q}^{\top}U=\bm{p}^{\top}}$. Given
$\bm{p}^{\top}\bm{S}(z)=z$, it follows that
${\bm{q}^{\top}\bm{T}(z)=\bm{q}^{\top}U\bm{S}(z)=\bm{p}^{\top}\bm{S}(z)=z}$. ∎
This theorem guarantees that for any given $\bm{q}$ that satisfies (5) and (6)
there exists a $\bm{T}(z)$ that satisfies (7)-(10) provided
${\mathcal{U}\neq\varnothing}$ and there exist $\bm{p}$ and $\bm{S}(z)$ that
satisfy (5)-(10). Of course,
$\overline{U}\in\mathcal{U}\implies\mathcal{U}\neq\varnothing$. Additionally,
note that $\overline{U}$ maximizes the contribution of $S_{i}(z)$ to
$T_{i}(z)$ in the mapping $\bm{T}(z)=U\bm{S}(z)$. So if
$S_{i}(z),i=1,\ldots,n$ are distinguishable in the sense that $S_{i}(z)\geq
S_{i+1},\forall z\in[0,1]$ and
$\int_{0}^{1}\left(S_{i}(z)-S_{i+1}(z)\right)\mathop{}\\!\mathrm{d}z>0$ for
all $1\leq i<n$, mapping $\bm{S}(z)$ to $\bm{T}(z)$ via $\overline{U}$ in a
sense maximizes the distinguishability of $T_{i}(z),i=1,\ldots,n$. Regarding
the existence of $\bm{p}$ and $\bm{S}(z)$ that satisfy (5)-(10), we now
present a pair of examples.
First, for $i=1,\ldots,n$ let $p_{i}=1/n$ such that $\bm{p}$ satisfies (5) and
(6), and let $S_{i}(z)$ be the CDF of the continuous uniform distribution with
support interval $[\frac{i-1}{n},\frac{i}{n})$:
$S_{i}(z)=\begin{cases}0,&z<\frac{i-1}{n},\\\ nz-(i-1),&\frac{i-1}{n}\leq
z<\frac{i}{n},\\\ 1,&\frac{i}{n}\leq z.\end{cases}$ (11)
Since each $S_{i}(z)$ is a CDF, $\bm{S}(z)$ satisfies (7)-(9). For any
$\hat{z}\in[0,1)$, ${\exists!~{}\hat{i}\in\\{1,2,\ldots,n\\}}$ such that
$\hat{z}\in[\frac{\hat{i}-1}{n},\frac{\hat{i}}{n})$. So ${S_{i}(\hat{z})=1}$
for ${i\in\\{1,\ldots,\hat{i}-1\\}}$,
${S_{\hat{i}}(\hat{z})=n\hat{z}-(\hat{i}-1)}$, and ${S_{i}(\hat{z})=0}$ for
${i\in\\{\hat{i}+1,\ldots,n\\}}$. Ergo,
$\bm{p}^{\top}\bm{S}(z)=\frac{1}{n}\left[(\hat{i}-1)\cdot
1+nz-(\hat{i}-1)+(n-\hat{i})\cdot 0\right]=z,$
so $\bm{p}$ and $\bm{S}(z)$ also satisfy (10).
Clearly, ${S_{1}(z)\geq S_{2}(z)\geq\cdots\geq S_{n}(z)}$. The 1-Wasserstein
distance measured between consecutive transformations is
$\displaystyle\int_{0}^{1}\\!\left(S_{i}(z)-S_{i+1}(z)\right)\mathop{}\\!\mathrm{d}z$
$\displaystyle=\int_{\frac{i-1}{n}}^{\frac{i}{n}}\\!\left(nz-(i-1)\right)\mathop{}\\!\mathrm{d}z+\int_{\frac{i}{n}}^{\frac{i+1}{n}}\\!\left(nz-i\right)\mathop{}\\!\mathrm{d}z$
$\displaystyle=n\int_{\frac{i-1}{n}}^{\frac{i}{n}}\\!\left(z-\tfrac{i-1}{n}\right)\mathop{}\\!\mathrm{d}z+n\int_{\frac{i}{n}}^{\frac{i+1}{n}}\\!\left(z-\tfrac{i}{n}\right)\mathop{}\\!\mathrm{d}z$
$\displaystyle=2n\int_{0}^{\frac{1}{n}}z\mathop{}\\!\mathrm{d}z$
$\displaystyle=\frac{1}{n}.$
That is, $S_{1}(z),S_{2}(z),\ldots,S_{n}(z)$ are equally spaced according to
the $1$-Wasserstein distance.
Second, we consider $S_{i}(z)$ based on the CDFs of beta distributions. Note
that the probability density function (PDF) for a beta distribution with
integer parameters $i$ and $n-i+1$ is
$\displaystyle
f(z;i,n-i+1)=\\!\binom{n}{i}iz^{i-1}(1-z)^{n-i}=\\!\sum_{j=0}^{n-i}\binom{n}{i}\binom{n-i}{j}i(-1)^{j}z^{i+j-1},$
For $i=1,\ldots,n$ again let $p_{i}=1/n$, and let
$S_{i}(z)=F(z;i,n-i+1)=\sum_{j=0}^{n-1}\binom{n}{i}\binom{n-i}{j}\left(\frac{i}{i+j}\right)(-1)^{j}z^{i+j}.$
(12)
Again, since each $S_{i}(z)$ is a CDF, $\bm{S}(z)$ satisfies (7)-(9). Also,
$\displaystyle\bm{p}^{\top}\bm{S}(z)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=0}^{n-i}\binom{n}{i}\binom{n-i}{j}\left(\frac{i}{i+j}\right)(-1)^{j}z^{i+j}$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\sum_{k=i}^{n}\binom{n}{i}\binom{n-i}{k-i}\left(\frac{i}{k}\right)(-1)^{k-i}z^{k}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\sum_{i=1}^{k}\binom{n}{i}\binom{n-i}{k-i}\left(\frac{i}{k}\right)(-1)^{k-i}z^{k}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\left(\sum_{i=1}^{k}\binom{k-1}{i-1}(-1)^{-i}\right)\binom{n}{k}(-1)^{k}z^{k}$
$\displaystyle=z$
Above, the final equality follows from the inner sum being an alternating sum
of binomial coefficients. For $k>1$, the inner sum is zero. For $k=1$, the
inner sum is $-1$. Hence, $\bm{p}$ and $\bm{S}(z)$ satisfy (10).
Since $F(z;\alpha,\beta)\geq F(z;\alpha^{\prime},\beta^{\prime}),\forall
z\in[0,1]$ if and only if $\alpha\leq\alpha^{\prime}$ and
$\beta\geq\beta^{\prime}$ [16, 17], we have $S_{1}(z)\geq
S_{2}(z)\geq\cdots\geq S_{n}(z)$. The beta distribution is related to the
Bernstein basis polynomials. Let
$b_{i}^{n}(z)=\binom{n}{i}z^{i}(1-z)^{n-i},i=0,\ldots,n$ be the $n+1$
degree-$n$ Bernstein basis polynomials [18]. Then,
${S_{i}(z)=F(z;i,n-i+1)=n\int_{0}^{z}b_{i-1}^{n-1}(z^{\prime})\mathop{}\\!\mathrm{d}z^{\prime}}$.
Following properties of indefinite and definite integrals of Bernstein basis
polynomials, the 1-Wasserstein distance measured between consecutive
transformations is
$\displaystyle\int_{0}^{1}\left(S_{i}(z)-S_{i+1}(z)\right)\mathop{}\\!\mathrm{d}z$
$\displaystyle=\int_{0}^{1}\int_{0}^{z}\left(nb_{i-1}^{n-1}(z^{\prime})-nb_{i}^{n-1}(z^{\prime})\right)\mathop{}\\!\mathrm{d}z^{\prime}\mathop{}\\!\mathrm{d}z$
$\displaystyle=\int_{0}^{1}\left(\sum_{j=i}^{n}b_{j}^{n}(z)-\sum_{j=i+1}^{n}b_{j}^{n}(z)\right)\mathop{}\\!\mathrm{d}z$
$\displaystyle=\int_{0}^{1}b_{i}^{n}(z)\mathop{}\\!\mathrm{d}z$
$\displaystyle=\frac{1}{n+1}.$
Again, $S_{1}(z),S_{2}(z),\ldots,S_{n}(z)$ are equally spaced according to the
$1$-Wasserstein distance.
We now conclude this section with an example that demonstrates how solutions
to (P) or (P-Perm) are applied to synthesize conditional probability
distributions as we described.
###### Example 3.
Let
$\bm{p}^{\top}\\!=\\![\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}},\nicefrac{{1}}{{4}}]^{\top}$,
and
$\bm{q}^{\top}\\!=\\![\nicefrac{{1}}{{3}},\nicefrac{{1}}{{9}},\nicefrac{{1}}{{2}},\nicefrac{{1}}{{18}}]^{\top}$.
For these vectors, the matrices
$\overline{U}=\left[\begin{array}[]{cccc}\nicefrac{{3}}{{4}}&\nicefrac{{5}}{{48}}&0&\nicefrac{{7}}{{48}}\\\
0&1&0&0\\\ 0&\nicefrac{{5}}{{24}}&\nicefrac{{1}}{{2}}&\nicefrac{{7}}{{24}}\\\
0&0&0&1\end{array}\right]~{}\text{and}~{}\widehat{U}=\left[\begin{array}[]{cccc}0&0&\nicefrac{{3}}{{4}}&\nicefrac{{1}}{{4}}\\\
0&0&0&1\\\ \nicefrac{{7}}{{18}}&\nicefrac{{1}}{{2}}&0&\nicefrac{{1}}{{9}}\\\
1&0&0&0\end{array}\right]$
are optimal solutions to (P) and (P-Perm), respectively. To accompany
$\bm{p}$, for $n=4$ we consider $\bm{S}^{\text{UD}}(z)$ based on continuous
uniform distributions as defined in (11) and also $\bm{S}^{\text{BD}}(z)$
based on beta distributions as defined in (12). Finally, let $F(x)=\Phi(x)$,
the CDF of a standard normal distribution.
We show $\bm{S}\textsuperscript{UD}(z)$,
${\overline{\bm{T}}\textsuperscript{UD}(z)=\overline{U}\bm{S}\textsuperscript{UD}(z)}$,
${\widehat{\bm{T}}\textsuperscript{UD}(z)=\widehat{U}\bm{S}\textsuperscript{UD}(z)}$,
$\bm{S}\textsuperscript{BD}(z)$,
${\overline{\bm{T}}\textsuperscript{BD}(z)=\overline{U}\bm{S}\textsuperscript{BD}(z)}$,
${\widehat{\bm{T}}\textsuperscript{BD}(z)=\widehat{U}\bm{S}\textsuperscript{BD}(z)}$,
as well as their application to synthesizing conditional probability
distributions based on $F(x)$ in Figure 2.
The piecewise-linearity of each $S_{i}^{\text{UD}}(z)$ leads to the
perturbations of $F(x)$, a smooth function, being non-smooth. In contrast,
that each $S_{i}^{\text{UD}}(z)$ is a polynomial leads to the perturbations of
$F(x)$ being smooth. Additionally, each $S_{i}^{\text{UD}}(z)$ is non-
increasing on portions of the unit interval whereas each
$S_{i}^{\text{BD}}(z)$ is strictly increasing. In the context of our
application, the consequence is that the support set associated with
$S_{i}^{\text{UD}}(F(x))$ is $\Omega_{i}\subset\Omega$ whereas that associated
with $S_{i}^{\text{BD}}(F(x))$ is $\Omega_{i}=\Omega$.
Applying $\widehat{U}$ leads to relatively larger 1-Wasserstein distances
between the consecutive conditional CDFs, as seen on the right side of Figure
2. However, the orientation of $\widehat{U}$ leads to those CDFs having a
different order, which may not ideal for certain applications of the method.
While $\overline{U}$ does a more modest job of separating the conditional
CDFs, its orientation maintains the ordering.
Figure 2: The transformations (left column) from Example 3 and their
application to synthesizing conditional CDFs based on $F(x)$ (right column).
## Conclusion
We presented an optimization model involving simplices that have a given
barycenter and that are enclosed by the standard simplex. We presented the
analytical form of an optimal solution to the model, and the conditions under
which it is the unique optimal solution. We showed the solution is an inverse
$M$-matrix whose eigenvalues are the same as its diagional entries. Finally,
we demonstrated how the model and its solutions apply to the task of
synthesizing conditional cumulative distribution functions that, in tandem
with a given discrete marginal distribution, coherently preserve a given CDF.
## Competing Interests
The authors have no competing interests to declare.
## Data Availability
No data was used for the research described in the article.
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|
# Protecting President Zelenskyy against Deep Fakes
Matyáš Boháček
Gymnasium of Johannes Kepler
Parléřova 2/118, 169 00 Praha 6, Czech Republic
<EMAIL_ADDRESS>
Hany Farid
Department of Electrical Engineering and Computer Sciences
School of Information
University of California, Berkeley
<EMAIL_ADDRESS>
###### Abstract
The 2022 Russian invasion of Ukraine is being fought on two fronts: a brutal
ground war and a duplicitous disinformation campaign designed to conceal and
justify Russia’s actions. This campaign includes at least one example of a
deep-fake video purportedly showing Ukrainian President Zelenskyy admitting
defeat and surrendering. In anticipation of future attacks of this form, we
describe a facial and gestural behavioral model that captures distinctive
characteristics of Zelenskyy’s speaking style. Trained on over eight hours of
authentic video from four different settings, we show that this behavioral
model can distinguish Zelenskyy from deep-fake imposters.This model can play
an important role – particularly during the fog of war – in distinguishing the
real from the fake.
_Keywords_ Deep fakes $\cdot$ Disinformation $\cdot$ Digital Forensics $\cdot$
Facial Mannerisms $\cdot$ Gestural Mannerisms
## 1 Introduction
In the early days of the Russian invasion of Ukraine, President Zelenskyy
warned the world that Russia’s digital disinformation machinery would create a
deep fake of him admitting defeat and surrendering. A few weeks later in mid-
March of 2022, a deep fake of Zelenskyy appeared with just this message
(Allyn, 2022). This video, Figure 1, was quickly debunked thanks to the rather
crude audio and video and to Zelenskyy’s pre-bunking. This type of deep fake,
however, is likely just the beginning of a new digital front that we might
expect in this and future conflicts.
|
---|---
Figure 1: One video frame of the real (left) and deep-fake version (right) of
Ukrainian President Zelenskyy.
A recent set of perceptual studies (Groh et al., 2022) examined the ability of
untrained observers to distinguish between real and deep-fake videos. In one
condition, participants viewed a single video and categorized it as real or
fake. Participants correctly identified $66\%$ of the deep-fake videos, as
compared to chance performance of $50\%$ (pooled responses from all
participants – so-called crowd wisdom – yields an improved accuracy of
$80\%$). In a second condition, participants were shown the prediction by the
top-performing DFDC computational model (Ferrer, 2020) and given the
opportunity to update their response. In this collaborative condition,
individual participant accuracy improved to $73\%$.
While we may have the ability to perceptually detect some deep-fake videos,
our ability is not terribly reliable and this task will become increasingly
more difficult as deep fakes continue to improve in quality and
sophistication. We must, therefore, turn to computational methods to assist in
the task of distinguishing the real from the fake.
The computational detection of deep-fake videos can be partitioned into three
basic categories: (1) learning-based, in which features that distinguish real
from fake content are explicitly learned by any of a range of different
machine-learning techniques (Zhou et al., 2017; Afchar et al., 2018; Li et
al., 2019); (2) artifact-based, in which a range of low-level (pixel based) to
high-level (semantic based) features are explicitly designed to distinguish
between real and fake content (Li et al., 2018; Agarwal and Farid, 2021;
Agarwal et al., 2020a); and (3) identity-based, in which biometric-style
features are used to identify if the person depicted in a video is who it
purports to be (Agarwal et al., 2019, 2020b, 2021; Cozzolino et al., 2021).
The advantage of learning-based approaches is they are able to learn detailed
and subtle video-synthesis artifacts. The disadvantage is these techniques
often struggle to generalize to new content not explicitly part of the
training data set, and can be vulnerable to adversarial attacks (Carlini and
Farid, 2020), and simple laundering attacks where the synthesized media is
trans-coded or resized (Barni et al., 2018). In the 2019-2020 Deepfake
Detection Challenge (Ferrer, 2020), for example, $2116$ teams competed for one
million dollars (USD) in prizes. Teams were provided $23,654$ real videos and
$104,500$ deep-fake videos created from the provided real videos. The top
performing learning-based detector achieved a detection accuracy of only
$65\%$ on a set of $4000$ holdout videos, half of which were real and half of
which were deep fakes (i.e., chance performance is $50\%$). These results
reveal that fully automatic detection of deep fakes in the wild remains a
challenging problem.
On the other hand, the advantage of artifact-based techniques is they can
exploit inconsistencies that are difficult to circumvent or launder. The
disadvantage is these techniques are typically narrowly applicable to a subset
of deep-fake videos, and often require human annotation as part of the
process.
The advantage of identity-based techniques is they are also resilient to
adversarial and laundering attacks and are typically applicable to many
different forms of deep fakes. The disadvantage of these approaches is an
identity-specific model must be constructed for each individual, typically
from hours of authentic video footage. This may be practical when it comes to,
for example, protecting a few world leaders from deep fakes – for which hours
of video can typically be found online – but is otherwise impractical. The
other disadvantage is that the learned mannerisms are somewhat context
dependent: when a world leader is giving a public address, for example, she
may be more formal than when she is giving an unscripted interview, and so the
specific mannerisms may not generalize across different contexts.
Because we are focused here on protecting one world leader – Ukrainian
President Zelenskyy – and because we can easily acquire hours of video of
Zelenskyy, we contend that an identity-based approach is the most sensible and
robust approach. We start with the identity-based technique of (Agarwal et
al., 2019), leveraging distinct patterns of facial and head movements, to
distinguish Zelenskyy from an imposter or deep fake. We then augment this
identity-based model with new gestural features capturing how a speaker uses
their hands when speaking.
After reviewing the facial mannerisms portion of the model and describing the
new gestural mannerisms portion, we evaluate the efficacy of our model in
distinguishing Zelenskyy from deep-fake Zelenskyy and a range of other
identities.
(a) | (b) | (c) | (d)
---|---|---|---
| | |
Figure 2: Four representative examples of President Zelenskyy in different
contexts: (a) public address; (b) press briefing; (c) bunker; and (d)
armchair.
## 2 Methods
### 2.1 Data Set
We downloaded $506$ minutes of video of Zelenskyy from YouTube and the
official website of the office of the Ukranian
president111https://www.president.gov.ua/en/videos/videos-archive in four
different contexts: (a) public address ($91$ min); (b) press briefing ($207$
min); (c) bunker ($47$ min); and (d) armchair ($161$ min). Shown in Figure 2
are representative examples from each of these settings.
Portions of each video with large camera motions (e.g., zoom, translation,
cross-fade) were automatically detected and removed from the data set. In
particular, the inter-frame difference was computed between each successive
pair of video frames. Assuming each video depicts a speaker in the center of
the frame, a camera motion was detected if the absolute difference on the left
and right margin (defined as $10\%$ of the frame width) was above a specified
threshold.
A total of $57$ minutes of interview-style videos of seven world leaders
(Jacinda Ardern, Joe Biden, Kamala Harris, Boris Johnson, Wladimir Klitschko,
Angela Merkel, and Vladimir Putin) were used as decoys (i.e., not Zelenskyy).
Our deep-fake detection is designed to distinguish Zelenskyy’s behavioral and
gestural mannerisms from imposters driving the creation of a deep fake, and so
these decoy videos – regardless of the identities – serve as proxies for deep
fakes. An additional $50$ minutes of video across $27$ distinct individuals
taken from the FaceForensics++ (Rössler et al., 2019) dataset were used as
additional decoys. In addition to these proxies, three commissioned lip-sync
deep fakes ($2$ min) created by the team at
Colossyan222https://www.colossyan.com, and one in-the-wild deep fake ($1$ min)
were added to this decoy dataset (Figure 1).
| | | |
---|---|---|---|---
---
Figure 3: Shown above are five equally-spaced and cropped frames from a
$10$-second video clip annotated with the estimated facial landmarks (red
markers) and head pose (blue box). Shown below are two of the $16$ action
units as a function of time (inner brow raiser [AU01] and lid tightener
[AU07]).
### 2.2 Facial Mannerisms
The identity-based forensic technique of (Agarwal et al., 2019) is based on
the observation that individuals have distinct speaking styles in terms of
facial expressions and head movements. Former President Obama, for example,
tends to tilt his head upwards when he smiles, and downwards when he frowns.
Starting with a single video as input, the OpenFace2 toolkit (Baltrusaitis et
al., 2018) extracts facial landmark positions, facial action units, head pose,
and eye gaze on a per-frame basis. Facial muscle movement and expression are
encoded using facial action units (AU) (Ekman and Friesen, 1976). The
OpenFace2 toolkit provides – on a per-frame basis – the strength of $17$
different AUs: inner brow raiser (AU01), outer brow raiser (AU02), brow
lowerer (AU04), upper lid raiser (AU05), cheek raiser (AU06), lid tightener
(AU07), nose wrinkler (AU09), upper lip raiser (AU10), lip corner puller
(AU12), dimpler (AU14), lip corner depressor (AU15), chin raiser (AU17), lip
stretcher (AU20), lip tightener (AU23), lip part (AU25), jaw drop (AU26), and
eye blink (AU45).
The forensic facial model incorporates $16$ AUs (AU45, eye blink, was found
not to be distinctive and therefore eliminated from consideration) and four
additional features: (1) head rotation about the x-axis; (2) head rotation
about the z-axis (as with AU45, head rotation about the y-axis was found not
to be distinctive); (3) the horizontal distance between the corners of the
mouth (mouthh); and (4) the vertical distance between the lower and upper lip
(mouthv), yielding a total of $20$ facial-mannerism features. Shown in Figure
3 are several frames of the facial and head tracking and two representative
examples of the measured action units across a $10$-second clip.
These features are combined with gestural mannerisms, described next, to form
a person-specific behavioral model.
| | | |
---|---|---|---|---
---
Figure 4: Shown above are five equally-spaced frames from a $10$-second video
clip annotated with the estimated gestural tracking. Shown below are two of
the $12$ gestural features corresponding to the vertical position of the left
and right wrist (the spatial position of the wrists are reported in normalized
units relative to a body-centric action plane).
### 2.3 Gestural Mannerisms
Across cultures and languages, hand gestures provide additional information
not always captured by a speaker’s words alone (Church et al., 2017). In
addition to distinct gestural patterns found across age (Özer D. et al.,
2017), sex (Özçalışkan and Goldin-Meadow, 2010), and culture (Pika et al.,
2006), recent work also finds that individuals exhibit distinct gestural
patterns (Özer D and T., 2020). We, therefore, hypothesize that hand gestures,
in addition to the facial expressions and head movements described above, will
improve our ability to identify an individual’s distinct speaking patterns.
Arm and hand position and movement are estimated in each input video frame
using Blazepose (Bazarevsky et al., 2020) from the MediaPipe library (Lugaresi
et al., 2019). Because we are interested only in the upper body, we consider
the image $x$-, $y$-coordinates corresponding to the shoulder, elbow, and
wrist of both arms, Figure 4, yielding a total of $12$ individual
measurements. These upper-body coordinates, initially specified relative to
the video-frame size, are normalized into a speaker-centric action plane
(Boháček and Hrúz, 2022). This action plane is a rectangular bounding box
centered on the speaker’s chest with a width $8\times$ and height $6\times$
the measured head height (De Silva, 2008; Bauer, 2014). In this normalized
bounding box, the upper left-hand corner is $(0,0)$ and the lower right-hand
corner is $(1,1)$. This normalization ensures that the tracked upper-body
coordinates can be compared across different speaker locations and sizes.
Shown in Figure 4 are several frames of the upper-body tracking and
representative examples of the extracted gestural features across a
$10$-second clip.
Whereas the tracked $x,y$ facial features are converted into a higher-level
representation in the form of action units, we find that a similar approach
with the hand gestures was less effective than simply considering the
normalized $x,y$ locations of the tracked shoulders, elbows, and wrists.
### 2.4 Behavioral Model
Correlations between all pairs of the $20$ facial features and $12$ gestural
features are used to capture individualized mannerisms (e.g., head tilt and
smiling/frowning). A total of $~{}_{32}C_{2}=(32\times 31)/2=496$ correlations
are extracted from overlapping $10$-second video clips extracted from an input
video in question.
Trained on authentic video of a person of interest, a novelty detection model
in the form of a one-class, non-linear support vector machine (SVM) (Schölkopf
et al., ; Pedregosa et al., 2011) is used to distinguish an individual from
imposters and deep fakes. An advantage of this classifier is that it only
requires examples of authentic videos.
The $506$ minutes of Zelenskyy video is partitioned into overlapping (by $5$
seconds) $10$-second video clips, yielding a total of $157,752$ clips. The
$110$ minutes of other identities in the World Leaders, FaceForensics++, and
Deep-Fake Zelenskyy videos are similarly partitioned, yielding a total of
$25,077$ clips.
These clips are randomly partitioned into a $80/20$ training/testing split.
The SVM is trained on the $496$ facial- and gestural-feature pairwise
correlations. The SVM hyper-parameters, consisting of the Gaussian kernel
width ($\gamma$) and outlier percentage ($\nu$), are optimized by performing a
grid search over these parameters across the training set. The trained
classifier is then evaluated against the hold-out testing set. This entire
process is repeated $100$ times with randomized training/testing splits, from
which we report average classification accuracy.
Three different classifiers are trained on facial features only, gestural
features only, and facial and gestural features combined. The SVM
classification threshold for the individual features is selected to yield a
$95\%$ training accuracy of correctly classifying real Zelenskyy clips. The
classification threshold for the combined features is selected to yield a
$99\%$ training accuracy.
## 3 Results
Shown in Table 1 is the classification accuracy (averaged over $100$ random
training/testing data splits) of our behavioral model evaluated against the
$10$-second video clips of seven different world-leaders, $28$ distinct
identities in the FaceForensics++ dataset (Rössler et al., 2019), and real and
deep-fake versions of Zelenskyy.
We find that the facial features and gestural features alone are insufficient
to consistently detect deep-fake version of Zelenskyy (see the last two
columns of Table 1). The combination of facial and gestural, however, yields
significant improvements in detection accuracy. Because deep-fake techniques
are – rightfully – focused on high-quality facial and audio synthesis, and
because of the expected difficulty in synthesizing realistic hands and hand
gestures, we posit that the combination of facial and gestural signals will
prove reliable for at least a few years.
As compared to the best-performing DFDC model (last row of Table 1)
(Seferbekov, 2020), our model achieves significantly higher classification
across all non-Zelenskyy data sets. This comparison, however, is not entirely
fair as our behavioral model is trained to detect deep-fake versions of just
one identity, whereas the DFDC model is a generic deep-fake detector. On the
other hand, our classifier operates on $10$-second video clips whereas the
DFDC model has the advantage of operating on the entire video. This comparison
does, nevertheless, show the power of identity-specific models.
| | | | Lip-Sync | In-The-Wild
---|---|---|---|---|---
| World | | Real | Deep-Fake | Deep-Fake
Model | Leaders | FF++ | Zelenskyy | Zelenskyy | Zelenskyy
facial | $91.7$ | $91.1$ | $94.7$ | $17.4$ | $83.9$
gestural | $77.4$ | $95.7$ | $95.0$ | $12.1$ | $33.3$
facial + gestural | $100.0$ | $100.0$ | $97.1$ | $94.9$ | $100.0$
DFDC | $73.1$ | $84.5$ | $93.5$ | $13.3$ | $1.7$
Table 1: Classification accuracy (reported as percentages) for our behavioral
model with facial, gestural, and these two features combined evaluated against
seven different world leaders, $28$ identities in the FaceForensics++ dataset
and against both real and deep-fake versions of Zelenskyy. By comparison, our
model significantly outperforms the best-performing DFDC model (Seferbekov,
2020).
### 3.1 Ablation
To determine how many of the $496$ pairwise facial and gestural correlations
are needed to achieve the classification accuracy reported in Table 1, we
trained a series of one-class SVMs on randomly selected subsets – ranging in
size between $10$ and $400$ – of all facial and gestural features. Shown in
Figure 5 is the median accuracy of classifying the identities in the world
leaders and deep-fake Zelenskyy videos. In this figure, each data point
corresponds to the median accuracy ($50\%$ quantile) from $25$ independent and
randomly selected features of each subset size; the error bars correspond to
the $25\%$ and $75\%$ quantile.
With the full set of $496$ facial and gestural features, detection accuracy is
$99.88\%$. Detection accuracy grows relatively linearly between feature
subsets of size $10$ and $300$ plateauing at $99.52\%$ with $400$ features.
Here we see that a significant fraction of the facial and gestural features
are, collectively, rich and informative.
Figure 5: Each data point corresponds to the median ($50\%$ quantile) accuracy
for classifiers trained on between $10$ and $496$ randomly selected facial and
gestural features; the error bars correspond to the $25\%$ and $75\%$
quantile.
To determine which specific facial and gestural features are most
discriminative, we next trained $500$ classifiers on random feature subsets of
size $10$. The discriminatory power of each feature is computed from the
average accuracy of each classifier to which a feature contributed. Across all
$500$ classifiers, the detection accuracy on the world leaders and deep-fake
Zelenskyy data sets ranges from $44.4\%$ to $4.3\%$. The top $20$ most
discriminative correlation features and respective classifier accuracy are:
Feature 1 | | Feature 2 | Classifier Accuracy ($\%$)
---|---|---|---
head-pose-Rx | $\Leftrightarrow$ | right-elbow-y | $44.4$
head-pose-Rx | $\Leftrightarrow$ | right-wrist-y | $36.9$
head-pose-Rx | $\Leftrightarrow$ | left-elbow-y | $33.7$
left-elbow-x | $\Leftrightarrow$ | left-shoulder-x | $32.8$
head-pose-Rx | $\Leftrightarrow$ | left-shoulder-y | $32.5$
head-pose-Rx | $\Leftrightarrow$ | lip-vertical | $29.3$
left-elbow-x | $\Leftrightarrow$ | right-elbow-y | $27.4$
right-elbow-y | $\Leftrightarrow$ | right-shoulder-y | $27.4$
right-elbow-x | $\Leftrightarrow$ | right-shoulder-x | $27.3$
head-pose-Rx | $\Leftrightarrow$ | left-wrist-y | $26.9$
AU14 | $\Leftrightarrow$ | AU17 | $26.2$
AU06 | $\Leftrightarrow$ | right-elbow-y | $25.6$
mouthv | $\Leftrightarrow$ | AU14 | $25.5$
AU12 | $\Leftrightarrow$ | AU15 | $25.4$
AU12 | $\Leftrightarrow$ | AU14 | $25.0$
right-elbow-y | $\Leftrightarrow$ | left-shoulder-y | $25.0$
pose-Rz | $\Leftrightarrow$ | right-shoulder-x | $24.4$
mouthv | $\Leftrightarrow$ | AU15 | $23.7$
pose-Rx | $\Leftrightarrow$ | right-shoulder-y | $23.0$
AU06 | $\Leftrightarrow$ | AU14 | $22.7$
where *-Rx and *-Rz correspond to 3-D head rotations, *-x and *-y correspond
to the horizontal and vertical image position, and AU* corresponds to specific
facial action units (see Section 2.2). Here we see that the correlation
between head rotation and hand gestural features are the most discriminative,
highlighting the importance of the addition of gestural features to the
original facial-based model. For President Zelenskyy, in particular, head
rotation (as in nodding affirmatively) is highly correlated to his hand
movements.
As compared to the median accuracy of $8.4\%$ across random features of subset
size $10$ (Figure 5), these top-ranked features achieve accuracies between
three and five times higher. A single classifier trained on the top $10$ and
$20$ features, however, only yields a prediction accuracy on the world leaders
and deep-fake Zelenskyy data sets of $59.2\%$ and $63.4\%$, providing further
evidence that a full set of facial and gestural features are necessary to
achieve a high classification accuracy.
## 4 Discussion
Although the term deep fakes first splashed on the screen in 2017, the
precursor to what we now call deep fakes dates back two decades. In the
seminal video-rewrite work Bregler et al. (1997), a video of a person speaking
is automatically modified to yield a video of them saying things not found in
the original footage. The resulting video quality and resolution were
generally lower than today’s deep-fake videos, but the results were
nevertheless impressive. Some $25$ years later, deep neural networks, GANs,
massive data sets, and unlimited compute cycles have led to increasingly more
realistic and sophisticated deep-fake videos.
While the democratization of access to techniques for manipulating and
synthesizing videos has led to interesting and entertaining applications, they
have also given rise to complex ethical and legal question Chesney and Citron
(2019). In the fog of war, in particular, deep fakes pose a significant threat
to our ability to understand and respond to rapidly evolving events.
While our approach to protecting a single individual – Ukrainian President
Zelenskyy – does not address the broader issue of deep fakes, it does bring
some level of digital protection to the arguably most important Ukrainian
voice at this time of war.
## Acknowledgement
We are grateful to Zoltan Kovacs, Muhammad Shahzaib Aslam, and the rest of the
Colossyan team for creating the Zelenskyy lip-sync deep fakes.
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# On the coupling of the Curved Virtual Element Method with the one-equation
Boundary Element Method for 2D exterior Helmholtz problems
L. Desiderio<EMAIL_ADDRESS>S. Falletta<EMAIL_ADDRESS>M. Ferrari<EMAIL_ADDRESS>L. Scuderi<EMAIL_ADDRESS>Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di
Parma,
Parco Area delle Scienze, 53/A, 43124, Parma, Italia Dipartimento di Scienze
Matematiche “G.L. Lagrange”, Politecnico di Torino,
Corso Duca degli Abruzzi, 24, 10129, Torino, Italia
###### Abstract
We consider the Helmholtz equation defined in unbounded domains, external to
2D bounded ones, endowed with a Dirichlet condition on the boundary and the
Sommerfeld radiation condition at infinity. To solve it, we reduce the
infinite region, in which the solution is defined, to a bounded computational
one, delimited by a curved smooth artificial boundary and we impose on this
latter a non reflecting condition of boundary integral type. Then, we apply
the curved virtual element method in the finite computational domain, combined
with the one-equation boundary element method on the artificial boundary. We
present the theoretical analysis of the proposed approach and we provide an
optimal convergence error estimate in the energy norm. The numerical tests
confirm the theoretical results and show the effectiveness of the new proposed
approach.
###### keywords:
Exterior Helmholtz problems, Curved Virtual Element Method, Boundary Element
Method, Non Reflecting Boundary Condition.
## 1 Introduction
Frequency-domain wave propagation problems defined in unbounded regions,
external to bounded obstacles, turn out to be a difficult physical and
numerical task due to the issue of determining the solution in an infinite
domain. One of the typical techniques to solve such problems is the Boundary
Integral Equation (BIE) method, which allows to reduce by one the dimension of
the problem, requiring only the discretization of the obstacle boundary. Once
the boundary distribution is retrieved by means of a Boundary Element Method
(BEM) [36], the solution of the original problem at each point of the exterior
domain is obtained by computing a boundary integral. However, this procedure
may result not efficient, especially when the solution has to be evaluated at
many points of the infinite domain.
During the last decades much effort has been concentrated on developing
alternative approaches. Among these we mention those based on the coupling of
domain methods, such as Finite Difference Method (FDM), Finite Element Method
(FEM) and the recent Virtual Element Method (VEM), with the BEM. These are
obtained by reducing the unbounded domain to a bounded computational one,
delimited by an artificial boundary, on which a suitable Boundary Integral-Non
Reflecting Boundary Condition (BI-NRBC) is imposed. This latter guarantees
that the artificial boundary is transparent and that no spurious reflections
arise from the resolution of the original problem by means of the interior
domain method applied in the finite computational domain.
The most popular approaches for such a coupling, associated to the use of the
FEM in the interior domain, involve the Green representation of the solution
and are often referred to as the Johnson & Nédélec Coupling (JNC) [29] or the
Costabel & Han Coupling (CHC) [19, 27]. Since the JNC is based on a single
BIE, involving both the single and the double layer integral operators
associated with the fundamental solution, it is known as the _one equation
BEM-FEM coupling_ and it gives rise to a non-symmetric final linear system.
On the contrary, the CHC is based on a couple of BIEs, one of which involves
the second order normal derivative of the fundamental solution (hence a
hypersingular integral operator), and it yields to a symmetric scheme. Despite
the fact that an integration by parts strategy can be applied to weaken the
hypersingularity, the approach turns out to be quite onerous from the
computational point of view, especially in the case of frequency-domain wave
problems for which the accuracy of the BEM is strictly connected to the
frequency parameter and to the density of discretization points per
wavelength. Even if the CHC has been applied in several contexts, among which
we mention the recent paper [25], where the theoretical analysis has been
derived for the solution of the Helmholtz problem by means of a VEM, from the
engineering point of view the JNC is the most natural and appealing way to
deal with unbounded domain problems (for very recent real-life applications
see, for example, [2] and [23]).
In this paper we propose a new approach based on the JNC between the Galerkin
BEM and the Curved Virtual Element Method (CVEM) in the interior of the
computational domain. This choice is based on the fact that the VEM allows to
broaden the classical family of the FEM for the discretization of partial
differential equations for what concerns both the decomposition of domain with
complex geometry and the definition of local high order discrete spaces. In
the standard VEM formulation the discrete spaces, built on meshes made of
polygonal or polyhedral elements, are similar to the usual finite element
spaces with the addition of suitable non-polynomial functions. The novelty of
the VEM consists in defining discrete spaces and degrees of freedom in such a
way that the elementary stiffness and mass matrices can be computed using only
the degrees of freedom, without the need of explicitly knowing the non-
polynomial functions (from which the “virtual” word descends), with a
consequent easiness of implementation even for high approximation orders.
Originally developed as a variational reformulation of the nodal Mimetic
Finite Difference (MFD) method [6, 15, 31], the VEM has been applied to a wide
variety of interior problems (among the most recent papers we refer the reader
to [3, 9, 10, 16]). On the contrary, only few papers deal with VEM applied to
exterior problems, among which the already mentioned [25] and [23]. In this
latter the JNC between the collocation BEM and the VEM has been numerically
investigated for the approximation of the solution of Dirichlet boundary value
problems defined by the 2D Helmholtz equation.
The very satisfactory results we have obtained in [23] have stimulated us to
further investigate on the application of the VEM to the solution of exterior
problems. For this reason, we propose here a novel approach in the CVEM-
Galerkin context that we have studied both from the theoretical and the
numerical point of view. In particular, the choice of the CVEM instead of the
standard (polygonal) VEM relies on the fact that the use of curvilinear
elements allows to avoid the sub-optimal rate of convergence for orders of
accuracy higher than 2, when curvilinear obstacles are considered. Further,
due to the arbitrariness of the choice of the artificial boundary, and dealing
with curved virtual elements, we choose the latter of curvilinear type. It is
worth mentioning that the choice of the CVEM space refers in particular to
that proposed in [8]. For the discretization of the BI-NRBC, we consider a
classical BEM associated to Lagrangian nodal basis functions. As already
remarked, the main challenge in the theoretical analysis is the lack of
ellipticity of the associated bilinear form. However, using the Fredholm
theory for integral operators, it is possible to prove the well-posedness of
the problem in case of computational domains with smooth artificial
boundaries. Moreover, the analysis of the Helmholtz problem and of the
proposed numerical method for its solution, is carried out by interpreting the
new main operators as perturbations of the Laplace ones. We present the
theoretical analysis of the method in a quite general framework and we provide
an optimal error estimate in the energy norm. Since the analysis is based on
the pioneering paper by Jonhson and Nédélec, the smoothness properties of the
artificial boundary represent a key requirement. We remark that, for the
classical Galerkin approach, the breakthrough in the theoretical analysis that
validates the stability of the JNC also in case of non-smooth boundaries, was
proved by Sayas in [37]. However, since we deal with a generalized Galerkin
method, the same analysis can not be straightforwardly applied and needs
further investigations.
The paper is organized as follows: in the next section we present the model
problem for the Helmholtz equation and its reformulation in a bounded region,
by the introduction of the artificial boundary and the associated one equation
BI-NRBC. In Section 3 we introduce the variational formulation of the problem
restricted to the finite computational domain, recalling the corresponding
main theoretical issues, among which existence and uniqueness of the solution.
In Section 4 we apply the Galerkin method providing an error estimate in the
energy norm, for a quite generic class of approximation spaces. Then, in
Section 5 we describe the choice of the CVEM-BEM approximation spaces and we
prove the validity of the error analysis in this specific context.
Additionally, we detail the algebraic formulation of the coupled global
scheme. Finally, in Section 6 we present some numerical results highlighting
the effectiveness of the proposed approach and the validation of the
theoretical results. Furthermore, in the last example we show that the optimal
convergence order of the scheme is guaranteed also when polygonal
computational domains are considered. Finally, some conclusions are drawn in
Section 7.
## 2 The model problem
In a fixed Cartesian coordinates system $\mathbf{x}=\left(x_{1},x_{2}\right)$,
we consider an open bounded domain $\Omega_{\tiny{0}}\subset\mathbf{R}^{2}$
with a Lipschitz boundary $\Gamma_{\tiny{0}}$ having positive Lebesgue
measure. We denote by
$\Omega_{e}:=\mathbf{R}^{2}\setminus\overline{\Omega}_{\tiny{0}}$ the exterior
unbounded domain (see Figure 1 (a)) and we consider the following frequency-
domain wave propagation problem:
$\displaystyle\Delta
u_{e}(\mathbf{x})+\kappa^{2}u_{e}(\mathbf{x})=-f(\mathbf{x})$
$\displaystyle\mathbf{x}\in\Omega_{e},$ (2.1a) $\displaystyle
u_{e}(\mathbf{x})=g(\mathbf{x})$ $\displaystyle\mathbf{x}\in\Gamma_{0},$
(2.1b)
$\displaystyle\lim\limits_{\|\mathbf{x}\|\rightarrow\infty}\|\mathbf{x}\|^{\frac{1}{2}}\left(\nabla
u_{e}(\mathbf{x})\cdot\frac{\mathbf{x}}{\|\mathbf{x}\|}-\imath\kappa
u_{e}(\mathbf{x})\right)=0.$ (2.1c)
In the above problem, Equation (2.1a) is known as the Helmholtz equation, with
source term $f\in L^{2}(\Omega_{e})$, Equation (2.1b) represents a boundary
condition of Dirichlet type with datum $g$, and Equation (2.1c) is the
Sommerfeld radiation condition, that ensures the appropriate behaviour of the
complex-valued unknown function $u_{e}$ at infinity. Furthermore, $\nabla$ and
$\Delta$ denote the nabla and Laplace operators, respectively, and $\imath$
stands for the imaginary unit.
We recall that the wave number $\kappa$ is often real and constant, and it is
complex if the propagation medium is energy absorbing, or a function of the
space if the medium is inhomogeneous. Here, we suppose that $\kappa$ is real,
positive and constant.
In the sequel we assume that $g\in H^{\nicefrac{{1}}{{2}}}(\Gamma_{0})$, to
guarantee existence and uniqueness of the solution $u_{e}$ of Problem (2.1) in
the Sobolev space $H^{1}_{\text{loc}}(\Omega_{e})$ (see [18]).
As many practical situations require, we aim at determining the solution
$u_{e}$ of Problem (2.1) in a bounded subregion of $\Omega_{e}$ surrounding
$\Omega_{0}$. To this end, we introduce an artificial boundary $\Gamma$ which
allows decomposing $\Omega_{e}$ into a finite computational domain $\Omega$,
bounded internally by $\Gamma_{0}$ and externally by $\Gamma$, and an infinite
residual one, denoted by $\Omega_{\infty}$, as depicted in Figure 1 (b). We
choose $\Gamma$ such that $\text{supp}(f)$ is a bounded subset of $\Omega$. We
assume that $\Gamma_{0}$ and $\Gamma$ are made up of a finite number of curves
of class $C^{m+1}$, with $m\geq 0$, so that $\Omega$ is a domain with piece-
wise smooth boundaries. Moreover, we assume that $\Gamma$ is a Lyapunov
regular contour, i.e. the gradient of any local parametrization is Hölder
continuous.
Figure 1: Model problem setting.
Denoting by $u$ and $u_{\infty}$ the restrictions of the solution $u_{e}$ to
$\Omega$ and $\Omega_{\infty}$ respectively, and by $\mathbf{n}$ and
$\mathbf{n}_{\infty}$ the unit normal vectors on $\Gamma$ pointing outside
$\Omega$ and $\Omega_{\infty}$, we impose the following compatibility and
equilibrium conditions on $\Gamma$ (recall that
$\mathbf{n}_{\infty}=-\mathbf{n}$):
$u(\mathbf{x})=u_{\infty}(\mathbf{x}),\qquad\frac{\partial
u}{\partial{\mathbf{n}}}(\mathbf{x})=-\frac{\partial
u_{\infty}}{\partial{\mathbf{n}_{\infty}}}(\mathbf{x}),\qquad\mathbf{x}\in\Gamma.$
(2.2)
In the above relations and in the sequel we omit, for simplicity, the use of
the trace operator to indicate the restriction of $H^{1}$ functions to the
boundary $\Gamma$ from the exterior or interior. In order to obtain a well
posed problem in $\Omega$, we need to impose a proper boundary condition on
$\Gamma$. It is known that the solution $u_{\infty}$ in $\Omega_{\infty}$ can
be represented by the following Kirchhoff formula:
$u_{\infty}(\mathbf{x})=\int_{\Gamma}G_{\kappa}(\mathbf{x},\mathbf{y})\frac{\partial
u_{\infty}}{\partial\mathbf{n}_{\infty}}(\mathbf{y})\,\differential\Gamma_{\mathbf{y}}-\int_{\Gamma}\frac{\partial
G_{\kappa}}{\partial\mathbf{n}_{\infty,\mathbf{y}}}(\mathbf{x},\mathbf{y})u_{\infty}(\mathbf{y})\,\differential\Gamma_{\mathbf{y}}\qquad\mathbf{x}\in\Omega_{\infty}\setminus\Gamma,$
(2.3)
in which $G_{\kappa}$ is the fundamental solution of the 2D Helmholtz problem
and $\mathbf{n}_{\infty,\mathbf{y}}$ denotes the normal unit vector with
initial point in $\mathbf{y}\in\Gamma$. The expression of $G_{\kappa}$ and of
its normal derivative in (2.3) are given by
$G_{\kappa}(\mathbf{x},\mathbf{y}):=\frac{\imath}{4}H_{0}^{(1)}(\kappa
r)\quad\text{and}\quad\frac{\partial
G_{\kappa}}{\partial\mathbf{n}_{\infty,\mathbf{y}}}(\mathbf{x},\mathbf{y})=\frac{\imath\kappa}{4}\frac{\mathbf{r}\cdot\mathbf{n}_{\infty,\mathbf{y}}}{r}H_{1}^{(1)}(\kappa
r),$
where $r=|\mathbf{r}|=|\mathbf{x}-\mathbf{y}|$ represents the distance between
the source point $\mathbf{x}$ and the field point $\mathbf{y}$, and
$H_{m}^{(1)}$ denotes the $m$-th order Hankel function of the first kind.
We introduce the single-layer integral operator $\text{V}_{\kappa}\colon
H^{-\nicefrac{{1}}{{2}}}(\Gamma)\to H^{\nicefrac{{1}}{{2}}}(\Gamma)$
$\text{V}_{\kappa}\psi(\mathbf{x}):=\int_{\Gamma}G_{\kappa}(\mathbf{x},\mathbf{y})\psi(\mathbf{y})\,\differential\Gamma_{\mathbf{y}},\qquad\mathbf{x}\in\Gamma$
and the double-layer integral operator $\text{K}_{\kappa}\colon
H^{\nicefrac{{1}}{{2}}}(\Gamma)\to H^{\nicefrac{{1}}{{2}}}(\Gamma)$
$\text{K}_{\kappa}\varphi(\mathbf{x}):=-\int_{\Gamma}\frac{\partial
G_{\kappa}}{\partial\mathbf{n}_{\infty,\mathbf{y}}}(\mathbf{x},\mathbf{y})\varphi(\mathbf{y})\,\differential\Gamma_{\mathbf{y}},\qquad\mathbf{x}\in\Gamma,$
which are continuous for all $\kappa>0$ (see [28]). Then, the trace of (2.3)
on $\Gamma$ reads (see [18])
$\frac{1}{2}u_{\infty}(\mathbf{x})-\text{V}_{\kappa}\frac{\partial
u_{\infty}}{\partial\mathbf{n}_{\infty}}({\mathbf{x}})-\text{K}_{\kappa}u_{\infty}(\mathbf{x})=0,\qquad\mathbf{x}\in\Gamma.$
(2.4)
Equation (2.4), which expresses the natural relation that $u_{\infty}$ and its
normal derivative have to satisfy at each point of the artificial boundary, is
imposed on $\Gamma$ as an exact (non local) BI-NRBC to solve Problem (2.1) in
the finite computational domain. Thus, taking into account the compatibility
and equilibrium conditions (2.2), and introducing the notation
$\displaystyle\lambda:=\frac{\partial u}{\partial\mathbf{n}}$, the new problem
defined in the domain of interest $\Omega$ takes the form:
$\displaystyle\Delta u(\mathbf{x})+\kappa^{2}u(\mathbf{x})=-f(\mathbf{x})$
$\displaystyle\mathbf{x}\in\Omega$ (2.5a) $\displaystyle
u(\mathbf{x})=g(\mathbf{x})$ $\displaystyle\mathbf{x}\in\Gamma_{0}$ (2.5b)
$\displaystyle\frac{1}{2}u(\mathbf{x})+\text{V}_{\kappa}\lambda({\mathbf{x}})-\text{K}_{\kappa}u(\mathbf{x})=0$
$\displaystyle\mathbf{x}\in\Gamma.$ (2.5c)
We point out that $\lambda$, which is defined on the boundary $\Gamma$ in
general by means of a trace operator (see [34]), is an additional unknown
function.
For the theoretical analysis we will present in the forthcoming section, we
further need to introduce the fundamental solution $G_{0}$ of the Laplace
equation and its normal derivative:
$G_{0}(\mathbf{x},\mathbf{y}):=-\frac{1}{2\pi}\log
r\quad\text{and}\quad\frac{\partial
G_{0}}{\partial\mathbf{n}_{\infty,\mathbf{y}}}(\mathbf{x},\mathbf{y})=\frac{1}{2\pi}\frac{\mathbf{r}\cdot\mathbf{n}_{\infty,\mathbf{y}}}{r^{2}}.$
Denoting by $\text{V}_{0}$ and $\text{K}_{0}$ the associated single and double
layer operators, the following regularity property of the operators
$\text{V}_{\kappa}-\text{V}_{0}$ and $\text{K}_{\kappa}-\text{K}_{0}$ holds.
###### Lemma 2.1.
The operators
$\text{V}_{\kappa}-\text{V}_{0}:H^{-\nicefrac{{1}}{{2}}}(\Gamma)\rightarrow
H^{\nicefrac{{5}}{{2}}}(\Gamma)$ and
$\text{K}_{\kappa}-\text{K}_{0}:H^{\nicefrac{{1}}{{2}}}(\Gamma)\rightarrow
H^{\nicefrac{{3}}{{2}}}(\Gamma)$ are continuous.
###### Proof.
We preliminary recall that the Hankel functions $H_{m}^{(1)}$, with $m=0,1$,
have the following asymptotic behaviour when $r\rightarrow 0$ (see formulae
(2.14) and (2.15) in [35]):
$\displaystyle H_{0}^{(1)}(r)=\frac{\imath 2}{\pi}\log{(r)}+1+\frac{\imath
2}{\pi}\left(\gamma-\log{(2)}\right)+O(r^{2}),$ (2.6a) $\displaystyle
H_{1}^{(1)}(r)=-\frac{\imath 2}{\pi r}+O(1)$ (2.6b)
where $\gamma\simeq 0.577216$ is the Euler constant. Then it easily follows
that, when $r\rightarrow 0$
$\displaystyle
G_{\kappa}(\mathbf{x},\mathbf{y})-G_{0}(\mathbf{x},\mathbf{y})=\frac{\imath}{4}-\frac{1}{2\pi}\left(\gamma-\log{\left(\frac{\kappa}{2}\right)}\right)+O(r^{2}),$
$\displaystyle\frac{\partial
G_{\kappa}}{\partial\mathbf{n}_{\infty,\mathbf{y}}}(\mathbf{x},\mathbf{y})-\frac{\partial
G_{0}}{\partial\mathbf{n}_{\infty,\mathbf{y}}}(\mathbf{x},\mathbf{y})=O(1).$
Following [28] (see Section 7.1), we can therefore deduce that
$G_{\kappa}-G_{0}$ and $\displaystyle\frac{\partial
G_{\kappa}}{\partial\mathbf{n}_{\infty,\mathbf{y}}}-\frac{\partial
G_{0}}{\partial\mathbf{n}_{\infty,\mathbf{y}}}$ are kernel functions with
pseudo-homogeneous expansions of degree 2 and 0, respectively. From these
properties, and proceeding as in [36] (see Remark 3.13), the thesis is proved.
∎
###### Remark 1.
Similarly, since $G_{0}$ is a kernel function with a pseudo-homogeneous
expansion of degree 0, we can deduce that the operator
$\text{V}_{0}:H^{s}(\Gamma)\rightarrow H^{s+1}(\Gamma)$ is continuous for all
$s\in\mathbf{R}$.
## 3 The weak formulation of the model problem
We start by noting that, as usual, we can reduce the non homogeneous boundary
condition on $\Gamma_{0}$ in (2.1) to a homogeneous one by splitting $u_{e}$
as the sum of a suitable fixed function in
$H^{1}_{g,\Gamma_{0}}(\Omega):=\\{u\in H^{1}(\Omega):u=g\,\text{ on
}\Gamma_{0}\\}$ satisfying the Sommerfeld radiation condition and of an
unknown function belonging to the space $H^{1}_{0,\Gamma_{0}}(\Omega)$.
Therefore, from now on, we consider Problem (2.5) with $g=0$.
In order to derive the weak form of Problem (2.5), we introduce the bilinear
forms $a:H^{1}(\Omega)\times H^{1}(\Omega)\rightarrow\mathbf{C}$ and
$m:L^{2}(\Omega)\times L^{2}(\Omega)\rightarrow\mathbf{C}$ given by
$a(u,v):=\int\limits_{\Omega}\nabla u(\mathbf{x})\cdot\nabla
v(\mathbf{x})\,\differential\mathbf{x}\qquad\text{and}\qquad
m(u,v):=\int\limits_{\Omega}u(\mathbf{x})v(\mathbf{x})\,\differential\mathbf{x},$
(3.1)
and the $L^{2}(\Gamma)$-inner product
$(\cdot,\cdot)_{\Gamma}:L^{2}(\Gamma)\times
L^{2}(\Gamma)\rightarrow\mathbf{C}$
$(\lambda,v)_{\Gamma}=\int\limits_{\Gamma}\lambda(\mathbf{x})v(\mathbf{x})\differential\Gamma_{\mathbf{x}},$
extended to the duality pairing $\langle\cdot,\cdot\rangle_{\Gamma}$ on
$H^{-\nicefrac{{1}}{{2}}}(\Gamma)\times H^{\nicefrac{{1}}{{2}}}(\Gamma)$.
The variational formulation of Problem (2.5) consists in finding $u\in
H_{0,\Gamma_{0}}^{1}(\Omega)$ and $\lambda\in
H^{-\nicefrac{{1}}{{2}}}(\Gamma)$ such that
$\displaystyle
a(u,v)-\kappa^{2}m(u,v)-\langle\lambda,v\rangle_{\Gamma}=m(f,v)$
$\displaystyle\forall\,v\in H^{1}_{0,\Gamma_{0}}(\Omega),$ (3.2a)
$\displaystyle\langle\mu,\left(\frac{1}{2}I-\text{K}_{\kappa}\right)u\rangle_{\Gamma}+\langle\mu,\text{V}_{\kappa}\lambda\rangle_{\Gamma}=0$
$\displaystyle\forall\,\mu\in H^{-\nicefrac{{1}}{{2}}}(\Gamma),$ (3.2b)
where $I$ stands for the identity operator. In order to reformulate the above
problem as an equation in operator form, following [29], we consider the
Hilbert space $V:=H^{1}_{0,\Gamma_{0}}(\Omega)\times
H^{-\nicefrac{{1}}{{2}}}(\Gamma)$, equipped with the norm
$\left\|\hat{u}\right\|_{V}^{2}:=\left\|u\right\|_{H^{1}(\Omega)}^{2}+\left\|\lambda\right\|_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)}^{2},\quad\text{for}\
\hat{u}=(u,\lambda),$
induced by the scalar product
$\left(\hat{u},\hat{v}\right)_{V}:=\left(u,v\right)_{H^{1}(\Omega)}+\left(\lambda,\mu\right)_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)},\quad\text{for}\
\hat{u}=(u,\lambda),\,\hat{v}=(v,\mu).$
We introduce the bilinear form $\mathcal{A}_{\kappa}:V\times
V\rightarrow\mathbf{C}$ defined, for $\hat{u}=(u,\lambda)$ and
$\hat{v}=(v,\mu)$, by
$\mathcal{A}_{\kappa}(\hat{u},\hat{v}):=a(u,v)-\kappa^{2}m(u,v)-\langle\lambda,v\rangle_{\Gamma}+\langle\mu,u\rangle_{\Gamma}+2\langle\mu,\text{V}_{\kappa}\lambda\rangle_{\Gamma}-2\langle\mu,\text{K}_{\kappa}u\rangle_{\Gamma},$
(3.3)
and the linear continuous operator $\mathcal{L}_{f}:V\rightarrow\mathbf{C}$
$\mathcal{L}_{f}(\hat{v}):=m(f,v),\quad\hat{v}=(v,\mu).$
Thus, Problem (3.2) can be rewritten as follows: find $\hat{u}\in V$ such that
$\mathcal{A}_{\kappa}(\hat{u},\hat{v})=\mathcal{L}_{f}(\hat{v})\qquad\forall\,\hat{v}\in
V.$ (3.4)
The well-posedness of the above problem has been proved in [32] (see Theorem
3.2), provided that $\kappa^{2}$ is not an eigenvalue of the Dirichlet-Laplace
problem in $\Omega$.
For what follows, it will be useful to rewrite $\mathcal{A}_{\kappa}$ by means
of the bilinear forms $\mathcal{B}_{\kappa},\mathcal{K}_{\kappa}:V\times
V\rightarrow\mathbf{C}$, defined as:
$\displaystyle\mathcal{A}_{\kappa}(\hat{u},\hat{v}):=\mathcal{B}_{\kappa}(\hat{u},\hat{v})+\mathcal{K}_{\kappa}(\hat{u},\hat{v})$
(3.5a)
$\displaystyle\mathcal{B}_{\kappa}(\hat{u},\hat{v}):=a(u,v)-\kappa^{2}m(u,v)-\langle\lambda,v\rangle_{\Gamma}+\langle\mu,u\rangle_{\Gamma}+2\langle\mu,\text{V}_{\kappa}\lambda\rangle_{\Gamma}$
(3.5b)
$\displaystyle\mathcal{K}_{\kappa}(\hat{u},\hat{v}):=-2\langle\mu,\text{K}_{\kappa}u\rangle_{\Gamma}$
(3.5c)
for $\hat{u}=(u,\lambda),\hat{v}=(v,\mu)\in V$. Due to the continuity property
of both the operators $\text{V}_{\kappa}$ and $\text{K}_{\kappa}$ when
$\kappa$ is real and non-negative (see [25]), by using the trace theorem and
the Cauchy-Schwarz inequality, it is easy to prove that the corresponding
linear mappings
$\mathcal{A}_{\kappa},\mathcal{B}_{\kappa},\mathcal{K}_{\kappa}:V\rightarrow
V^{{}^{\prime}}$, defined by
$\left(\mathcal{A}_{\kappa}\hat{u}\right)(\hat{v}):=\mathcal{A}_{\kappa}(\hat{u},\hat{v}),\qquad\left(\mathcal{B}_{\kappa}\hat{u}\right)(\hat{v}):=\mathcal{B}_{\kappa}(\hat{u},\hat{v}),\qquad\left(\mathcal{K}_{\kappa}\hat{u}\right)(\hat{v}):=\mathcal{K}_{\kappa}(\hat{u},\hat{v}),$
are continuous from $V$ to its dual $V^{{}^{\prime}}$. Finally, we introduce
the adjoint operators
$\mathcal{A}^{*}_{\kappa},\mathcal{B}^{*}_{\kappa}:V\rightarrow
V^{{}^{\prime}}$ defined by:
$\displaystyle\left(\mathcal{A}^{*}_{\kappa}\hat{v}\right)(\hat{u}):=\left(\mathcal{A}_{\kappa}\hat{u}\right)(\hat{v})=\mathcal{A}_{\kappa}(\hat{u},\hat{v})$
$\displaystyle\left(\mathcal{B}^{*}_{\kappa}\hat{v}\right)(\hat{u}):=\left(\mathcal{B}_{\kappa}\hat{u}\right)(\hat{v})=\mathcal{B}_{\kappa}(\hat{u},\hat{v}).$
In the following remarks we recall classical results about the afore
introduced maps.
###### Remark 2.
Theorem 3.2 in [32] and the closed graph theorem ensure that, if $\kappa^{2}$
is not an eigenvalue of the Dirichlet-Laplace problem in $\Omega$, the inverse
linear mappings
$\mathcal{A}_{\kappa}^{-1},\mathcal{A}_{\kappa}^{*-1}:V^{{}^{\prime}}\rightarrow
V$ are continuous.
###### Remark 3.
Denoting by $H_{0}^{-\nicefrac{{1}}{{2}}}(\Gamma):=\left\\{\lambda\in
H^{-\nicefrac{{1}}{{2}}}(\Gamma)\ :\
\langle\lambda,1\rangle_{\Gamma}=0\right\\}$, we set
$\widetilde{V}:=H^{1}_{0,\Gamma_{0}}(\Omega)\times
H^{-\nicefrac{{1}}{{2}}}_{0}(\Gamma)$. It has been proved in [29] (see Lemmas
1, 2 and 3) that the mappings
$\mathcal{A}_{0},\mathcal{A}^{*}_{0},\mathcal{B}_{0},\mathcal{B}^{*}_{0}:\widetilde{V}\rightarrow\widetilde{V}^{{}^{\prime}}$
are isomorphisms. Moreover, for $s\geq 0$, the mappings
$\mathcal{A}_{0}^{-1},\mathcal{A}_{0}^{*-1},\mathcal{B}_{0}^{-1},\mathcal{B}_{0}^{*-1}:H^{s-1}(\Omega)\times
H^{s-\nicefrac{{1}}{{2}}}(\Gamma)\times H^{s+\nicefrac{{1}}{{2}}}(\Gamma)\to
H^{s+1}(\Omega)\times H^{s-\nicefrac{{1}}{{2}}}(\Gamma)$ are continuous.
Finally, we recall that $\mathcal{B}_{0}$ is coercive in the
$\widetilde{V}$-norm.
## 4 The Galerkin method
In what follows, the notation $Q_{1}\apprle Q_{2}$ (resp. $Q_{1}\apprge
Q_{2}$) means that the quantity $Q_{1}$ is bounded from above (resp. from
below) by $c\,Q_{2}$, where $c$ is a positive constant that, unless explicitly
stated, does not depend on any relevant parameter involved in the definition
of $Q_{1}$ and $Q_{2}$.
In order to describe the Galerkin approach applied to (3.4), we introduce a
sequence of unstructured meshes $\\{\mathcal{T}_{h}\\}_{h>0}$, that represent
coverages of the domain $\Omega$ with a finite number of elements $E$, having
diameter $h_{E}$. The mesh width $h>0$, related to the spacing of the grid, is
defined as $h:=\underset{E\in\mathcal{T}_{h}}{\max}h_{E}$. Moreover, we denote
by $\mathcal{T}_{h}^{\Gamma}$ the decomposition of the artificial boundary
$\Gamma$, inherited from $\mathcal{T}_{h}$, which consists of curvilinear
parts joined with continuity. We suppose that for each $h$ and for each
element $E\in\mathcal{T}_{h}$ there exists a constant $\varrho>0$ such that
the following assumptions are fulfilled:
1. ($A.{1}$)
$E$ is star-shaped with respect to a ball of radius greater than $\varrho
h_{E}$;
2. ($A.{2}$)
the length of any (eventually curved) edge of $E$ is greater than $\varrho
h_{E}$.
We introduce the splitting of the bilinear forms $a$ and $m$ defined in (3.1)
into a sum of local bilinear forms
$a^{\text{\tiny{E}}},m^{\text{\tiny{E}}}:H^{1}(E)\times
H^{1}(E)\rightarrow\mathbf{C}$, associated to the elements $E$ of the
decomposition of $\Omega$:
$\displaystyle a(u,v)$
$\displaystyle=\sum\limits_{E\in\mathcal{T}_{h}}a^{\text{\tiny{E}}}(u,v):=\sum\limits_{E\in\mathcal{T}_{h}}\int\limits_{E}\nabla
u(\mathbf{x})\cdot\nabla v(\mathbf{x})\,\differential\mathbf{x},$
$\displaystyle m(u,v)$
$\displaystyle=\sum\limits_{E\in\mathcal{T}_{h}}m^{\text{\tiny{E}}}(u,v):=\sum\limits_{E\in\mathcal{T}_{h}}\int\limits_{E}u(\mathbf{x})v(\mathbf{x})\,\differential\mathbf{x}.$
Then, for any $k\in\mathbf{N}$, denoting by $P_{k}(E)$ the space of
polynomials of degree $k$ defined on $E$, we introduce the local polynomial
$H^{1}$-projection $\Pi_{k}^{\nabla,E}:H^{1}(E)\rightarrow P_{k}(E)$, defined
such that for every $v\in H^{1}(E)$:
$\begin{cases}\displaystyle\int_{E}\nabla\Pi_{k}^{\nabla,E}v\cdot\nabla
q\,\differential E=\displaystyle\int_{E}\nabla v\cdot\nabla q\,\differential
E\qquad\forall\,q\in P_{k}(E),\\\\[10.0pt] \displaystyle\int_{\partial
E}\Pi_{k}^{\nabla,E}v\,\differential s=\displaystyle\int_{\partial
E}v\,\differential s\end{cases}$ (4.1)
and the local polynomial $L^{2}$-projection operator
$\Pi_{k}^{0,E}:L^{2}(E)\rightarrow P_{k}(E)$, defined for all $v\in L^{2}(E)$
such that
$\int_{E}\Pi_{k}^{0,E}v\,q\,\differential E=\int_{E}v\,q\,\differential
E\qquad\forall\,q\in P_{k}(E).$ (4.2)
From the definition of $\Pi_{k}^{\nabla,E}$ and of $\Pi_{k}^{0,E}$, it
follows:
$\displaystyle
a^{\text{\tiny{E}}}\left(\Pi_{k}^{\nabla,E}v,q\right)=a^{\text{\tiny{E}}}\left(v,q\right)$
$\displaystyle\forall\,q\in P_{k}(E),$ $\displaystyle
m^{\text{\tiny{E}}}\left(\Pi_{k}^{0,E}v,q\right)=m^{\text{\tiny{E}}}\left(v,q\right)$
$\displaystyle\forall\,q\in P_{k}(E).$
Moreover, since $\Omega$ is the union of star-shaped domains
$E\in\mathcal{T}_{h}$, the local polynomial projectors $\Pi_{k}^{\nabla,E}$
and $\Pi_{k}^{0,E}$ can be extended to the global projectors
$\Pi_{k}^{\nabla}:H^{1}(\Omega)\rightarrow P_{k}(\mathcal{T}_{h})$ and
$\Pi_{k}^{0}:L^{2}(\Omega)\rightarrow P_{k}(\mathcal{T}_{h})$ as follows:
$\displaystyle\left(\Pi_{k}^{\nabla}v\right)_{|_{E}}:=\Pi_{k}^{\nabla,E}v_{|_{E}}$
$\displaystyle\forall\,v\in H^{1}(\Omega)$
$\displaystyle\left(\Pi_{k}^{0}v\right)_{|_{E}}:=\Pi_{k}^{0,E}v_{|_{E}}$
$\displaystyle\forall\,v\in L^{2}(\Omega),$
$P_{k}(\mathcal{T}_{h})$ being the space of piecewise polynomials with respect
to the decomposition $\mathcal{T}_{h}$ of $\Omega$.
In the following lemma we prove a polynomial approximation property of the
above defined projectors. To this aim, since we shall deal with functions
belonging to the space
$H^{1}(\mathcal{T}_{h}):=\underset{E\in\mathcal{T}_{h}}{\prod}H^{1}(E)$, we
need to introduce the following broken $H^{1}$ norm
$\|v\|_{H^{1}(\mathcal{T}_{h})}:=\left(\sum_{E\in\mathcal{T}_{h}}\|v\|^{2}_{H^{1}(E)}\right)^{1/2}.$
###### Lemma 4.1.
Assuming ($A.{1}$), for all $v\in H^{s+1}(\Omega)$ with $0\leq s\leq k$, it
holds:
$\left\|v-\Pi_{k}^{0}v\right\|_{L^{2}(\Omega)}\apprle
h^{s+1}\left\|v\right\|_{H^{s+1}(\Omega)}.$ (4.4)
Moreover, for all $v\in H^{s+1}(\Omega)$ with $1\leq s\leq k$, it holds:
$\left\|v-\Pi_{k}^{\nabla}v\right\|_{H^{1}(\mathcal{T}_{h})}\apprle
h^{s}\left\|v\right\|_{H^{s+1}(\Omega)}.$ (4.5)
###### Proof.
Let us denote by $R_{E}>0$ the radius of the ball in $E\in\mathcal{T}_{h}$
satisfying ($A.{1}$). For any $v\in L^{2}(E)$ and $q\in P_{k}(E)$ we can write
$\left\|v-\Pi_{k}^{0,E}v\right\|_{L^{2}(E)}^{2}=\left(v-\Pi_{k}^{0,E}v,v-q\right)_{L^{2}(E)}+\left(v-\Pi_{k}^{0,E}v,q-\Pi_{k}^{0,E}v\right)_{L^{2}(E)}=\left(v-\Pi_{k}^{0,E}v,v-q\right)_{L^{2}(E)},$
where we have used (4.2) together with $q-\Pi_{k}^{0,E}v\in P_{k}(E)$. Then,
by applying the Cauchy-Schwarz inequality, we easily get
$\left\|v-\Pi_{k}^{0,E}v\right\|_{L^{2}(E)}\leq\left\|v-q\right\|_{L^{2}(E)}\qquad\forall\,q\in
P_{k}(E).$
Now, assuming $v\in H^{s+1}(E)$ with $0\leq s\leq k$ and using the Bramble-
Hilbert Lemma (see Lemma 4.3.8 in [14]), we have
$\left\|v-\Pi_{k}^{0,E}v\right\|_{L^{2}(E)}\leq\underset{q\in
P_{k}(E)}{\text{inf}}\left\|v-q\right\|_{L^{2}(E)}\apprle
C\hskip-2.84544pt\left(\frac{h_{E}}{R_{E}}\right)h_{E}^{s+1}\left\|v\right\|_{H^{s+1}(E)},$
where the implicit constant depends only on $k$ and
$C:\mathbf{R}^{+}\rightarrow\mathbf{R}^{+}$ is an increasing function. Since,
by virtue of Assumption ($A.{1}$), the function
$C\hskip-2.27626pt\left(\nicefrac{{h_{E}}}{{R_{E}}}\right)$ is uniformly
bounded, we can easily get (4.4). Finally, inequality (4.5) can be proved
similarly. ∎
### 4.1 The discrete variational formulation
We present here a class of Galerkin type discretizations of Problem (3.4),
which includes, but is not limited to, VEMs. In Section 5 we will give an
example of CVEM that falls in the framework considered.
Let $Q_{h}^{k}\subset H_{0,\Gamma_{0}}^{1}(\Omega)$ and $X_{h}^{k}\subset
H^{-\nicefrac{{1}}{{2}}}(\Gamma)$ denote two finite dimensional spaces
associated to the meshes $\mathcal{T}_{h}$ and $\mathcal{T}_{h}^{\Gamma}$,
respectively. Introducing the discrete space $V_{h}^{k}:=Q_{h}^{k}\times
X_{h}^{k}$, the Galerkin method applied to Problem (3.2) reads: find
$\hat{u}_{h}\in V_{h}^{k}$ such that
$\mathcal{A}_{\kappa,h}(\hat{u}_{h},\hat{v}_{h}):=\mathcal{B}_{\kappa,h}(\hat{u}_{h},\hat{v}_{h})+\mathcal{K}_{\kappa}(\hat{u}_{h},\hat{v}_{h})=\mathcal{L}_{f,h}(\hat{v}_{h})\quad\forall\,\hat{v}_{h}\in
V_{h}^{k},$ (4.6)
where $\mathcal{A}_{\kappa,h},\mathcal{B}_{\kappa,h}:V_{h}^{k}\times
V_{h}^{k}\rightarrow\mathbf{C}$ and
$\mathcal{L}_{f,h}:V_{h}^{k}\rightarrow\mathbf{C}$ are suitable approximations
of $\mathcal{A}_{\kappa}$, $\mathcal{B}_{\kappa}$ and $\mathcal{L}_{f}$,
respectively.
In order to prove existence and uniqueness of the solution $\hat{u}_{h}\in
V_{h}^{k}$ and to derive a priori error estimates, we preliminary introduce
some assumptions on the discrete spaces, on the bilinear form
$\mathcal{B}_{\kappa,h}$ and on the linear operator $\mathcal{L}_{f,h}$.
We assume that the following properties for $Q_{h}^{k}$, $X_{h}^{k}$ and
$\widetilde{X}_{h}^{k}:=X_{h}^{k}\cap H_{0}^{-\nicefrac{{1}}{{2}}}(\Gamma)$
hold: for $1\leq s\leq k$
1. ($H1.{a}$)
$\underset{v_{h}\in
Q_{h}^{k}}{\text{inf}}\left\|v-v_{h}\right\|_{H^{1}(\Omega)}\apprle
h^{s}\left\|v\right\|_{H^{s+1}(\Omega)}\hskip 5.69046pt\quad\quad\forall\,v\in
H^{s+1}(\Omega)$;
2. ($H1.{b}$)
$\underset{\mu_{h}\in
X_{h}^{k}}{\text{inf}}\left\|\mu-\mu_{h}\right\|_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)}\apprle
h^{s}\left\|\mu\right\|_{H^{s-\nicefrac{{1}}{{2}}}(\Gamma)}\hskip
5.69046pt\quad\forall\,\mu\in H^{s-\nicefrac{{1}}{{2}}}(\Gamma)$;
3. ($H1.{c}$)
$\underset{\mu_{\scriptscriptstyle
0h}\in\tilde{X}_{h}^{k}}{\text{inf}}\left\|\mu_{\scriptscriptstyle
0}-\mu_{\scriptscriptstyle
0h}\right\|_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)}\apprle
h^{s}\left\|\mu_{\scriptscriptstyle
0}\right\|_{H^{s-\nicefrac{{1}}{{2}}}(\Gamma)}\hskip
5.69046pt\quad\forall\,\mu_{\scriptscriptstyle 0}\in
H^{s-\nicefrac{{1}}{{2}}}(\Gamma)\cap H_{0}^{-\nicefrac{{1}}{{2}}}(\Gamma)$.
According to the definition of the $\|\cdot\|_{V}$ norm, the above assumptions
ensure the following approximation property for the product spaces $V_{h}^{k}$
and $\tilde{V}_{h}^{k}:=Q_{h}^{k}\times\tilde{X}_{h}^{k}$:
* •
given $\hat{v}=(v,\mu)\in H^{s+1}(\Omega)\times
H^{s-\nicefrac{{1}}{{2}}}(\Gamma)$, there exists
$\hat{v}_{h}^{I}=(v_{h}^{I},\mu_{h}^{I})\in V_{h}^{k}$ such that
$\left\|\hat{v}-\hat{v}_{h}^{I}\right\|_{V}\apprle
h^{s}\left(\left\|v\right\|_{H^{s+1}(\Omega)}+\left\|\mu\right\|_{H^{s-\nicefrac{{1}}{{2}}}(\Gamma)}\right);$
(4.7)
* •
given $\hat{v}_{\scriptscriptstyle 0}=(v,\mu_{\scriptscriptstyle 0})\in
H^{s+1}(\Omega)\times(H^{s-\nicefrac{{1}}{{2}}}(\Gamma)\cap
H_{0}^{-\nicefrac{{1}}{{2}}}(\Gamma))$, there exists
$\hat{v}_{\scriptscriptstyle 0h}^{I}=(v_{h}^{I},\mu_{\scriptscriptstyle
0h}^{I})\in\tilde{V}_{h}^{k}$ such that
$\left\|\hat{v}_{\scriptscriptstyle 0}-\hat{v}_{\scriptscriptstyle
0h}^{I}\right\|_{V}\apprle
h^{s}\left(\left\|v\right\|_{H^{s+1}(\Omega)}+\left\|\mu_{\scriptscriptstyle
0}\right\|_{H^{s-\nicefrac{{1}}{{2}}}(\Gamma)}\right).$ (4.8)
Concerning the bilinear form $\mathcal{B}_{\kappa,h}$, we assume that for all
$\kappa\geq 0$:
1. ($H2.{a}$)
$k$-consistency:
$\mathcal{B}_{\kappa,h}(\hat{q},\hat{v}_{h})=\mathcal{B}_{\kappa}(\hat{q},\hat{v}_{h})\quad\forall\,\hat{q}\in
P_{k}(\mathcal{T}_{h})\times X_{h}^{k}\ \text{and}\ \forall\,\hat{v}_{h}\in
V_{h}^{k}$;
2. ($H2.{b}$)
continuity in $V$-norm:
$\left|\mathcal{B}_{\kappa,h}(\hat{v}_{h},\hat{w}_{h})\right|\apprle\left\|\hat{v}_{h}\right\|_{V}\left\|\hat{w}_{h}\right\|_{V}\quad\forall\hat{v}_{h},\hat{w}_{h}\in
V_{h}^{k}$.
###### Remark 4.
It is worth noting that, in Assumption ($H2.{a}$), the evaluation of the
bilinear form $\mathcal{B}_{\kappa}$ is well defined provided that the
computation of the bilinear form $a(\cdot,\cdot)$ is split into the sum of the
local contributions associated to the elements $E$ of $\mathcal{T}_{h}$. For
simplicity of notation, here and in what follows, we assume that such
splitting is considered whenever necessary. Moreover, we assume that the
approximated bilinear form $\mathcal{B}_{\kappa,h}$ is well defined on the
space $H^{1}(\mathcal{T}_{h})$.
Assumptions ($H2.{a}$) and ($H2.{b}$) allow to prove the following consistency
result.
###### Lemma 4.2.
Let $\hat{v}:=\left(v,\mu\right)\in H^{s+1}(\Omega)\times
H^{s-\nicefrac{{1}}{{2}}}(\Gamma)$, $1\leq s\leq k$, and
$\hat{v}_{h}^{I}:=\left(v_{h}^{I},\mu_{h}^{I}\right)\in V_{h}^{k}$ the
interpolant of $\hat{v}$ in $V_{h}^{k}$ such that relation (4.7) holds. Then
$\left|\mathcal{B}_{\kappa}(\hat{v}_{h}^{I},\hat{w}_{h})-\mathcal{B}_{\kappa,h}(\hat{v}_{h}^{I},\hat{w}_{h})\right|\apprle
h^{s}\left\|v\right\|_{H^{s+1}(\Omega)}\left\|\hat{w}_{h}\right\|_{V}\qquad\forall\,\hat{w}_{h}\in
V_{h}^{k}.$
###### Proof.
By abuse of notation, let
$\Pi^{\nabla}_{k}\hat{v}:=(\Pi^{\nabla}_{k}v,\mu_{h}^{I})$. We start from the
inequality
$\left|\mathcal{B}_{\kappa}(\hat{v}_{h}^{I},\hat{w}_{h})-\mathcal{B}_{\kappa,h}(\hat{v}_{h}^{I},\hat{w}_{h})\right|\leq\left|\mathcal{B}_{\kappa}(\hat{v}_{h}^{I},\hat{w}_{h})-\mathcal{B}_{\kappa,h}(\Pi_{k}^{\nabla}\hat{v},\hat{w}_{h})\right|+\left|\mathcal{B}_{\kappa,h}(\Pi_{k}^{\nabla}\hat{v},\hat{w}_{h})-\mathcal{B}_{\kappa,h}(\hat{v}_{h}^{I},\hat{w}_{h})\right|=:I+II.$
(4.9)
By using Assumption ($H2.{a}$) and the continuity of $\mathcal{B}_{\kappa}$ in
the $V$-norm, we get
$I=\left|\mathcal{B}_{\kappa}(\Pi_{k}^{\nabla}\hat{v}-\hat{v}_{h}^{I},\hat{w}_{h})\right|\apprle\left\|\Pi_{k}^{\nabla}\hat{v}-\hat{v}_{h}^{I}\right\|_{H^{1}(\mathcal{T}_{h})\times
H^{-\nicefrac{{1}}{{2}}}(\Gamma)}\left\|\hat{w}_{h}\right\|_{V}.$ (4.10)
Concerning the term $II$, from ($H2.{b}$) we obtain
$II=\left|\mathcal{B}_{\kappa,h}(\Pi_{k}^{\nabla}\hat{v}-\hat{v}_{h}^{I},\hat{w}_{h})\right|\apprle\left\|\Pi_{k}^{\nabla}\hat{v}-\hat{v}_{h}^{I}\right\|_{H^{1}(\mathcal{T}_{h})\times
H^{-\nicefrac{{1}}{{2}}}(\Gamma)}\left\|\hat{w}_{h}\right\|_{V}.$ (4.11)
Finally, by definition of $\Pi_{k}^{\nabla}\hat{v}$, using (4.5) and
Assumption ($H1.{a}$) we can write
$\left\|\Pi_{k}^{\nabla}\hat{v}-\hat{v}_{h}^{I}\right\|_{H^{1}(\mathcal{T}_{h})\times
H^{-\nicefrac{{1}}{{2}}}(\Gamma)}=\left\|\Pi_{k}^{\nabla}v-v_{h}^{I}\right\|_{H^{1}(\mathcal{T}_{h})}\leq\left\|\Pi_{k}^{\nabla}v-v\right\|_{H^{1}(\mathcal{T}_{h})}+\left\|v-v_{h}^{I}\right\|_{H^{1}(\Omega)}\apprle
h^{s}\left\|v\right\|_{H^{s+1}(\Omega)},$
from which, combining (4.10) and (4.11) with (4.9), the thesis follows. ∎
Similarly, the following lemma can be proved.
###### Lemma 4.3.
Let $\hat{v}_{\scriptscriptstyle 0}:=\left(v,\mu_{\scriptscriptstyle
0}\right)\in H^{s+1}(\Omega)\times(H^{s-\nicefrac{{1}}{{2}}}(\Gamma)\cap
H_{0}^{-\nicefrac{{1}}{{2}}}(\Gamma))$, $1\leq s\leq k$, and
$\hat{v}_{\scriptscriptstyle 0h}^{I}:=\left(v_{h}^{I},\mu_{\scriptscriptstyle
0h}^{I}\right)\in\tilde{V}_{h}^{k}$ the interpolant of
$\hat{v}_{\scriptscriptstyle 0}$ in $\tilde{V}_{h}^{k}$ such that relation
(4.8) holds. Then
$\left|\mathcal{B}_{\scriptscriptstyle 0}(\hat{v}_{\scriptscriptstyle
0h}^{I},\hat{w}_{\scriptscriptstyle 0h})-\mathcal{B}_{\scriptscriptstyle
0,h}(\hat{v}_{\scriptscriptstyle 0h}^{I},\hat{w}_{\scriptscriptstyle
0h})\right|\apprle
h^{s}\left\|v\right\|_{H^{s+1}(\Omega)}\left\|\hat{w}_{\scriptscriptstyle
0h}\right\|_{V}\qquad\forall\,\hat{w}_{\scriptscriptstyle
0h}\in\tilde{V}_{h}^{k}.$
In order to prove the main results of our theoretical analysis, we need to
introduce further assumptions on the approximate bilinear form
$\mathcal{B}_{\kappa,h}$. Denoting by
$\mathcal{D}_{\kappa,h}:=\mathcal{B}_{\kappa,h}-\mathcal{B}_{0,h}$, we
require:
1. ($H3.{a}$)
$\mathcal{D}_{\kappa,h}$ is continuous in the weaker $W$-norm, with
$W:=L^{2}(\Omega)\times H^{-\nicefrac{{1}}{{2}}}(\Gamma)$:
$\left|\mathcal{D}_{\kappa,h}(\hat{v}_{h},\hat{w}_{h})\right|\apprle\left\|\hat{v}_{h}\right\|_{W}\left\|\hat{w}_{h}\right\|_{W}\qquad\forall\,\hat{v}_{h},\hat{w}_{h}\in
V_{h}^{k};$
2. ($H3.{b}$)
$\mathcal{B}_{0,h}$ is $\tilde{V}_{h}^{k}$-elliptic:
$\mathcal{B}_{0,h}(\hat{w}_{\scriptscriptstyle 0h},\hat{w}_{\scriptscriptstyle
0h})\apprge\left\|\hat{w}_{\scriptscriptstyle
0h}\right\|_{V}^{2}\qquad\forall\,\hat{w}_{\scriptscriptstyle
0h}\in\tilde{V}_{h}^{k};$
3. ($H3.{c}$)
$k$-consistency in the second term of $\mathcal{B}_{0,h}$:
$\mathcal{B}_{0,h}(\hat{w}_{h},\hat{q})=\mathcal{B}_{0}(\hat{w}_{h},\hat{q})\quad\forall\,\hat{q}\in
P_{k}(\mathcal{T}_{h})\times X_{h}^{k}\ \text{and}\ \forall\,\hat{w}_{h}\in
V_{h}^{k}.$
###### Remark 5.
We remark that Assumption ($H3.{a}$) is the discrete counterpart of the
continuity property of the bilinear form
$\mathcal{D}_{\kappa}:=\mathcal{B}_{\kappa}-\mathcal{B}_{0}$. Indeed,
according to the continuity of $\text{V}_{\kappa}-\text{V}_{0}$ and using the
Cauchy-Schwarz inequality, we obtain: for $\hat{v}=(v,\mu)\in V$ and
$\hat{w}=(w,\nu)\in V$
$\left|\mathcal{D}_{\kappa}(\hat{v},\hat{w})\right|=\left|\kappa^{2}m(v,w)-2\langle\nu,(\text{V}_{\kappa}-\text{V}_{0})\mu\rangle\right|\apprle\left\|\hat{v}\right\|_{W}\left\|\hat{w}\right\|_{W}.$
(4.12)
Assumptions ($H3.{a}$)–($H3.{c}$) are used to prove the following Lemmas 4.4,
4.5 and 4.6, which are then crucial to obtain the Ladyzhenskaya-
Babu$\check{\text{s}}$ka-Brezzi condition for the discrete bilinear form
$\mathcal{A}_{\kappa,h}$.
###### Lemma 4.4.
Let $\hat{v}_{h}=(v_{h},\mu_{h})\in\left(H^{s+1}(\Omega)\times
H^{-\nicefrac{{1}}{{2}}}(\Gamma)\right)\cap V_{h}^{k}$ with $0\leq s\leq k$.
Then
$\left|\mathcal{D}_{\kappa}(\hat{v}_{h},\hat{w}_{h})-\mathcal{D}_{\kappa,h}(\hat{v}_{h},\hat{w}_{h})\right|\apprle
h^{s+1}\left\|v_{h}\right\|_{H^{s+1}(\Omega)}\left\|\hat{w}_{h}\right\|_{W},\quad\forall\,\hat{w}_{h}\in
V_{h}^{k}.$ (4.13)
###### Proof.
Let us denote, by abuse of notation,
$\Pi_{k}^{0}\hat{v}_{h}:=\left(\Pi_{k}^{0}v_{h},\mu_{h}\right)$. By adding and
subtracting the term
$\mathcal{D}_{\kappa,h}(\Pi_{k}^{0}\hat{v}_{h},\hat{w}_{h})$ and using
Assumption ($H2.{a}$), we get
$\displaystyle\left|\mathcal{D}_{\kappa}(\hat{v}_{h},\hat{w}_{h})-\mathcal{D}_{\kappa,h}(\hat{v}_{h},\hat{w}_{h})\right|$
$\displaystyle\leq\left|\mathcal{D}_{\kappa}(\hat{v}_{h}-\Pi_{k}^{0}\hat{v}_{h},\hat{w}_{h})\right|+\left|\mathcal{D}_{\kappa,h}(\hat{v}_{h}-\Pi_{k}^{0}\hat{v}_{h},\hat{w}_{h})\right|$
$\displaystyle\apprle\left\|\hat{v}_{h}-\Pi_{k}^{0}\hat{v}_{h}\right\|_{W}\left\|\hat{w}_{h}\right\|_{W},$
the last inequality directly following from (4.12) and ($H3.{a}$). Finally, by
definition of $\Pi_{k}^{0}\hat{v}_{h}$ and using (4.4), we can write
$\left\|\hat{v}_{h}-\Pi_{k}^{0}\hat{v}_{h}\right\|_{W}=\left\|v_{h}-\Pi_{k}^{0}v_{h}\right\|_{L^{2}(\Omega)}\apprle
h^{s+1}\left\|v_{h}\right\|_{H^{s+1}(\Omega)},$
from which the thesis follows. ∎
###### Lemma 4.5.
Let $\hat{v}_{\scriptscriptstyle 0}=(v,\mu_{\scriptscriptstyle
0})\in\tilde{V}$. There exists one and only one $\hat{v}_{\scriptscriptstyle
0h}=(v_{h},\mu_{\scriptscriptstyle 0h})\in\tilde{V}_{h}^{k}$ such that
$\mathcal{B}_{0,h}(\hat{w}_{\scriptscriptstyle 0h},\hat{v}_{\scriptscriptstyle
0h})=\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle 0})\quad\forall\,\hat{w}_{\scriptscriptstyle
0h}=(w_{h},\nu_{\scriptscriptstyle 0h})\in\tilde{V}_{h}^{k}.$ (4.14)
Moreover, it holds:
$\displaystyle\left\|{\hat{v}}_{\scriptscriptstyle
0h}\right\|_{V}\apprle\left\|{\hat{v}_{\scriptscriptstyle 0}}\right\|_{V},$
(4.15a) $\displaystyle\left\|\mu_{\scriptscriptstyle
0h}-\mu_{\scriptscriptstyle
0}\right\|_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)}\apprle
h\left\|\hat{v}_{\scriptscriptstyle 0}\right\|_{V},$ (4.15b)
$\displaystyle\left\|v_{h}-v\right\|_{L^{2}(\Omega)}\apprle
h\left\|\hat{v}_{\scriptscriptstyle 0}\right\|_{V}.$ (4.15c)
###### Proof.
Existence and uniqueness of $\hat{v}_{\scriptscriptstyle
0h}\in\tilde{V}_{h}^{k}$, solution of (4.14), follow from Assumptions
($H2.{b}$) and ($H3.{b}$). Moreover, (4.15a) holds according to ($H3.{b}$) and
the continuity of the bilinear form $\mathcal{B}_{0}$ (see Remark 3).
In order to prove (4.15b), by using a duality argument, it is sufficient to
show that:
$\left|\langle\mu_{\scriptscriptstyle 0h}-\mu_{\scriptscriptstyle
0},\eta\rangle_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)\times
H^{\nicefrac{{3}}{{2}}}(\Gamma)}\right|\apprle
h\left\|\hat{v}_{\scriptscriptstyle
0h}\right\|_{V}\left\|\eta\right\|_{H^{\nicefrac{{3}}{{2}}}(\Gamma)}\quad\forall\eta\in
H^{\nicefrac{{3}}{{2}}}(\Gamma).$ (4.16)
Starting from $\eta\in H^{\nicefrac{{3}}{{2}}}(\Gamma)$, we consider
$\widetilde{w}:=\left(0,0,\eta\right)\in L^{2}(\Omega)\times
H^{\nicefrac{{1}}{{2}}}(\Gamma)\times H^{\nicefrac{{3}}{{2}}}(\Gamma)$ and we
set $\hat{w}_{\scriptscriptstyle 0}:=\mathcal{B}_{0}^{-1}\widetilde{w}$. Then,
we have:
$\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle 0},\hat{z}_{\scriptscriptstyle
0})=\mathcal{B}_{0}(\mathcal{B}_{0}^{-1}\widetilde{w},\hat{z}_{\scriptscriptstyle
0})=(\mathcal{B}_{0}\mathcal{B}_{0}^{-1}\widetilde{w})(\hat{z}_{\scriptscriptstyle
0})=\langle\zeta_{\scriptscriptstyle
0},\eta\rangle_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)\times
H^{\nicefrac{{3}}{{2}}}(\Gamma)}\quad\forall\,\hat{z}_{\scriptscriptstyle
0}=(z,\zeta_{\scriptscriptstyle 0})\in\tilde{V}.$ (4.17)
From the continuity of $\mathcal{B}_{0}^{-1}:L^{2}(\Omega)\times
H^{\nicefrac{{1}}{{2}}}(\Gamma)\times
H^{\nicefrac{{3}}{{2}}}(\Gamma)\rightarrow H^{2}(\Omega)\times
H^{\nicefrac{{1}}{{2}}}(\Gamma)$ (see Remark 3), it follows that:
$\left\|\hat{w}_{\scriptscriptstyle 0}\right\|_{H^{2}(\Omega)\times
H^{\nicefrac{{1}}{{2}}}(\Gamma)}\apprle\left\|\eta\right\|_{H^{\nicefrac{{3}}{{2}}}(\Gamma)}.$
(4.18)
Therefore, by choosing $\hat{z}_{\scriptscriptstyle
0}=\hat{v}_{\scriptscriptstyle 0h}-\hat{v}_{\scriptscriptstyle 0}$ in (4.17),
we can write
$\langle\mu_{\scriptscriptstyle 0h}-\mu_{\scriptscriptstyle
0},\eta\rangle_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)\times
H^{\nicefrac{{3}}{{2}}}(\Gamma)}=\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0},\hat{v}_{\scriptscriptstyle 0h}-\hat{v}_{\scriptscriptstyle
0})=\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle 0}-\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle 0h}-\hat{v}_{\scriptscriptstyle
0})+\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle
0h})-\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle 0}).$
Since $\hat{v}_{\scriptscriptstyle 0h}\in\tilde{V}_{h}^{k}$ is the solution of
(4.14), we rewrite the previous identity as follows:
$\displaystyle\left|\langle\mu_{\scriptscriptstyle 0h}-\mu_{\scriptscriptstyle
0},\eta\rangle_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)\times
H^{\nicefrac{{3}}{{2}}}(\Gamma)}\right|=\left|\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0}-\hat{w}_{\scriptscriptstyle 0h},\hat{v}_{\scriptscriptstyle
0h}-\hat{v}_{\scriptscriptstyle
0})+\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle
0h})-\mathcal{B}_{0,h}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle 0h})\right|$
$\displaystyle\leq\left|\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0}-\hat{w}_{\scriptscriptstyle 0h},\hat{v}_{\scriptscriptstyle
0h}-\hat{v}_{\scriptscriptstyle
0})\right|+\left|\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle
0h})-\mathcal{B}_{0,h}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle 0h})\right|=:I+II.$ (4.19)
Choosing in (4.1) $\hat{w}_{\scriptscriptstyle 0h}=\hat{w}_{\scriptscriptstyle
0h}^{I}$, the interpolant of $\hat{w}_{\scriptscriptstyle 0}\in\tilde{V}$ in
$\tilde{V}_{h}^{k}$ such that (4.8) holds, due to Lemma 4.3 with $s=1$ and
(4.15a), we can estimate $II$ as follows:
$II\apprle h\left\|\hat{v}_{\scriptscriptstyle
0}\right\|_{V}\left\|\hat{w}_{\scriptscriptstyle
0}\right\|_{H^{2}(\Omega)\times H^{\nicefrac{{1}}{{2}}}(\Gamma)}.$ (4.20)
Combining the continuity of $\mathcal{B}_{0}$, (4.8) with $s=1$ and (4.15a),
we have:
$I\apprle\left\|\hat{v}_{\scriptscriptstyle 0h}-\hat{v}_{\scriptscriptstyle
0}\right\|_{V}\left\|\hat{w}_{\scriptscriptstyle
0}-\hat{w}_{\scriptscriptstyle 0h}^{I}\right\|_{V}\apprle
h\left\|\hat{v}_{\scriptscriptstyle 0h}-\hat{v}_{\scriptscriptstyle
0}\right\|_{V}\left\|\hat{w}_{\scriptscriptstyle
0}\right\|_{H^{2}(\Omega)\times H^{\nicefrac{{1}}{{2}}}(\Gamma)}\apprle
h\left\|\hat{v}_{\scriptscriptstyle
0}\right\|_{V}\left\|\hat{w}_{\scriptscriptstyle
0}\right\|_{H^{2}(\Omega)\times H^{\nicefrac{{1}}{{2}}}(\Gamma)}.$ (4.21)
Finally, from (4.18) and (4.1) together with (4.20) and (4.21), inequality
(4.16) directly follows.
Inequality (4.15c) can be proved similarly to (4.15b). Indeed, if we consider
$\widetilde{w}:=\left(v_{h}-v,0,0\right)\in L^{2}(\Omega)\times
H^{\nicefrac{{1}}{{2}}}(\Gamma)\times H^{\nicefrac{{3}}{{2}}}(\Gamma)$ and
$\hat{w}_{\scriptscriptstyle 0}=\mathcal{B}_{0}^{-1}\widetilde{w}$ in (4.17),
we get
$\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle 0},\hat{z}_{\scriptscriptstyle
0})=\left(v_{h}-v,z\right)_{L^{2}(\Omega)}\qquad\forall\,\hat{z}_{\scriptscriptstyle
0}=(z,\zeta_{\scriptscriptstyle 0})\in\tilde{V}.$ (4.22)
Then, choosing $\hat{z}_{\scriptscriptstyle 0}=\hat{v}_{\scriptscriptstyle
0h}-\hat{v}_{\scriptscriptstyle 0}$ in (4.22), we have
$\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle 0},\hat{v}_{\scriptscriptstyle
0h}-\hat{v}_{\scriptscriptstyle
0})=\left(v_{h}-v,v_{h}-v\right)_{L^{2}(\Omega)}=\left\|v_{h}-v\right\|_{L^{2}(\Omega)}^{2}.$
Finally, by taking into account the continuity of $\mathcal{B}_{0}^{-1}$, we
obtain
$\left\|\hat{w}_{\scriptscriptstyle 0}\right\|_{H^{2}(\Omega)\times
H^{\nicefrac{{1}}{{2}}}(\Gamma)}\apprle\left\|v_{h}-v\right\|_{L^{2}(\Omega)}$
and, proceeding as we did to estimate (4.1), we write
$\left\|v_{h}-v\right\|_{L^{2}(\Omega)}^{2}\apprle
h\left\|\hat{w}_{\scriptscriptstyle 0}\right\|_{H^{2}(\Omega)\times
H^{\nicefrac{{1}}{{2}}}(\Gamma)}\left\|\hat{v}_{\scriptscriptstyle
0}\right\|_{V}\apprle
h\left\|v_{h}-v\right\|_{L^{2}(\Omega)}\left\|\hat{v}_{\scriptscriptstyle
0}\right\|_{V},$
from which (4.15c) follows. ∎
###### Lemma 4.6.
Let $\hat{v}=(v,\mu)\in V$. There exists $\hat{v}_{h}=(v_{h},\mu_{h})\in
V_{h}^{k}$ such that
$\mathcal{B}_{0,h}(\hat{w}_{h},\hat{v}_{h})=\mathcal{B}_{0}(\hat{w}_{h},\hat{v})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v}_{h}-\hat{v})\quad\forall\hat{w}_{h}\in(w_{h},\eta_{h})\in
V_{h}^{k}$
where $\bar{\eta}_{h}=\nicefrac{{\langle\eta_{h},1\rangle}}{{\langle
1,1\rangle}}$. Moreover, it holds
$\displaystyle\lVert\hat{v}_{h}\rVert_{V}\apprle\left\|\hat{v}\right\|_{V},$
(4.23a)
$\displaystyle\lVert\mu_{h}-\mu\rVert_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)}\apprle
h\lVert\hat{v}\rVert_{V},$ (4.23b)
$\displaystyle\left\|v_{h}-v\right\|_{L^{2}(\Omega)}\apprle
h\lVert\hat{v}\rVert_{V}.$ (4.23c)
###### Proof.
Let consider $\hat{v}_{\scriptscriptstyle 0}=(v,\mu_{0}):=(v,\mu-\bar{\mu})$,
with $\bar{\mu}=\nicefrac{{\langle\mu,1\rangle}}{{\langle 1,1\rangle}}$, and
$\hat{w}_{\scriptscriptstyle
0h}:=(w_{h},\eta_{h}-\bar{\eta}_{h})\in\tilde{V}_{h}^{k}$. Then we have
$\mathcal{B}_{0}(\hat{w}_{h},\hat{v})=\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0h},\hat{v})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v})=\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle
0})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v})+\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0h},(0,\bar{\mu})).$
According to Lemma 4.5 applied to the first term at the right hand side of the
above equality, there exists a unique $\hat{v}_{\scriptscriptstyle
0h}=(v_{h},{\mu_{\scriptscriptstyle 0h}})\in\tilde{V}_{h}^{k}$ such that
$\mathcal{B}_{0}(\hat{w}_{h},\hat{v})=\mathcal{B}_{0,h}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle
0h})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v})+\mathcal{B}_{0}(\hat{w}_{\scriptscriptstyle
0h},(0,\bar{\mu})).$
Using ($H3.{c}$) with $\hat{q}=(0,\bar{\mu})$, we can write
$\mathcal{B}_{0}(\hat{w}_{h},\hat{v})=\mathcal{B}_{0,h}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{\scriptscriptstyle
0h})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v})+\mathcal{B}_{0,h}(\hat{w}_{\scriptscriptstyle
0h},(0,\bar{\mu}))=\mathcal{B}_{0,h}(\hat{w}_{\scriptscriptstyle
0h},\hat{v}_{h})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v}),$
where we have set
$\hat{v}_{h}=(v_{h},\mu_{h}):=(v_{h},{\mu_{\scriptscriptstyle
0h}}+\bar{\mu})\in V_{h}^{k}$. Moreover, by adding and subtracting in this
latter the term $\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v}_{h})$ and using
($H2.{a}$), we get
$\mathcal{B}_{0}(\hat{w}_{h},\hat{v})=\mathcal{B}_{0,h}(\hat{w}_{h},\hat{v}_{h})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v}-\hat{v}_{h}).$
By applying the Cauchy-Schwarz inequality to estimate the term
$\lVert\bar{\mu}\rVert_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)}\apprle\lVert\mu\rVert_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)},$
and (4.15a), we prove (4.23a) as follows:
$\lVert\hat{v}_{h}\rVert_{V}=\lVert\hat{v}_{\scriptscriptstyle
0h}+(0,\bar{\mu})\rVert_{V}\apprle\lVert\hat{v}_{\scriptscriptstyle
0}\rVert_{V}+\lVert\bar{\mu}\rVert_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)}\apprle\left\|\hat{v}\right\|_{V}+\lVert\mu\rVert_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)}\apprle\left\|\hat{v}\right\|_{V}.$
Finally, by using (4.15b)-(4.15c) we easily prove (4.23b) and (4.23c):
$\displaystyle\lVert\mu_{h}-\mu\rVert_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)}=\left\|\mu_{\scriptscriptstyle
0h}-\mu_{0}\right\|_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)}\apprle
h\left\|\hat{v}_{\scriptscriptstyle 0}\right\|_{V}\apprle
h\lVert\hat{v}\rVert_{V},$
$\displaystyle\left\|v_{h}-v\right\|_{L^{2}(\Omega)}\apprle
h\left\|\hat{v}_{\scriptscriptstyle 0}\right\|_{V}\apprle
h\lVert\hat{v}\rVert_{V}.$
∎
### 4.2 Error estimate in the energy norm
In the present section we show the validity of the inf-sup condition for the
discrete bilinear form $\mathcal{A}_{\kappa,h}$, with $\kappa>0$, and we prove
that the discrete scheme has the optimal approximation order, providing for
the optimal error estimate in the $V$-norm.
###### Theorem 4.7.
Assume that $\kappa^{2}$ is not an eigenvalue of the Laplacian in $\Omega$
endowed with a Dirichlet boundary condition on $\Gamma$. Then, for $h$ small
enough,
$\sup_{\begin{subarray}{c}\hat{v}_{h}\in V_{h}\\\ \hat{v}_{h}\neq
0\end{subarray}}\frac{\mathcal{A}_{\kappa,h}(\hat{w}_{h},\hat{v}_{h})}{\lVert\hat{v}_{h}\rVert_{V}}\apprge\lVert\hat{w}_{h}\rVert_{V}\quad\forall\,\hat{w}_{h}\in
V_{h}.$
###### Proof.
Given $\hat{w}_{h}\in V_{h}^{k}$, let
$\hat{v}:=\mathcal{A}_{\kappa}^{*-1}J\hat{w}_{h}\in V$ where $J:V\to
V^{\prime}$ denotes the canonical continuous map
$(J\hat{w})(\hat{z}):=(\hat{w},\hat{z})_{V}$. Therefore we can write:
$\displaystyle\mathcal{A}_{\kappa}(\hat{z},\hat{v})$
$\displaystyle=\mathcal{A}_{\kappa}(\hat{z},\mathcal{A}_{\kappa}^{*-1}J\hat{w}_{h})=(\mathcal{A}_{\kappa}\hat{z})(\mathcal{A}_{\kappa}^{*-1}J\hat{w}_{h})=(\mathcal{A}_{\kappa}^{*}\mathcal{A}_{\kappa}^{*-1}J\hat{w}_{h})(\hat{z})$
$\displaystyle=(J\hat{w}_{h})(\hat{z})=(\hat{w}_{h},\hat{z})_{V},\qquad\forall\,\hat{z}\in
V.$ (4.24)
Moreover, according to the continuity of $\mathcal{A}_{\kappa}^{*-1}$ (see
Remark 2) and of $J$, we obtain
$\lVert\hat{v}\rVert_{V}\apprle\lVert\hat{w}_{h}\rVert_{V}.$ (4.25)
Now, by virtue of Lemma 4.6, writing $\hat{v}=(v,\mu)\in V$, there exists
$\hat{v}_{h}=(v_{h},\mu_{h})\in V_{h}^{k}$ such that
$\mathcal{B}_{0,h}(\hat{w}_{h},\hat{v}_{h})=\mathcal{B}_{0}(\hat{w}_{h},\hat{v})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v}_{h}-\hat{v})\quad\forall\,\hat{w}_{h}=(w_{h},\eta_{h})\in
V_{h}^{k}$ (4.26)
where $\bar{\eta}_{h}=\nicefrac{{\langle\eta_{h},1\rangle}}{{\langle
1,1\rangle}}$ and such that (4.23a)-(4.23c) hold. Proceeding as in Theorem 5.2
of [25], and recalling the definitions of $\mathcal{A}_{0,h}$ and
$\mathcal{A}_{\kappa,h}$ and of $\mathcal{D}_{\kappa,h}$ and
$\mathcal{D}_{\kappa}$ (see assumption ($H3.{a}$) and Remark 5), we rewrite
$\mathcal{A}_{\kappa,h}$ as follows:
$\displaystyle\mathcal{A}_{\kappa,h}$
$\displaystyle=\mathcal{A}_{0,h}+(\mathcal{A}_{k}-\mathcal{A}_{0})+(\mathcal{A}_{0}-\mathcal{A}_{0,h})+(\mathcal{A}_{\kappa,h}-\mathcal{A}_{\kappa})$
$\displaystyle=\mathcal{A}_{0,h}+(\mathcal{A}_{k}-\mathcal{A}_{0})+(\mathcal{B}_{0}-\mathcal{B}_{0,h})+(\mathcal{B}_{\kappa,h}-\mathcal{B}_{\kappa})$
$\displaystyle=\mathcal{A}_{0,h}+(\mathcal{A}_{k}-\mathcal{A}_{0})+(\mathcal{D}_{\kappa,h}-\mathcal{D}_{\kappa}).$
(4.27)
Using Lemma 4.4 with $s=0$, we have
$\left*(\mathcal{D}_{\kappa,h}-\mathcal{D}_{\kappa})(\hat{w}_{h},\hat{v}_{h})\right\lvert\apprle
h\lVert\hat{w}_{h}\rVert_{V}\lVert\hat{v}_{h}\rVert_{W}\apprle
h\lVert\hat{w}_{h}\rVert_{V}\lVert\hat{v}_{h}\rVert_{V}.$ (4.28)
Recalling (3.5a)–(3.5c) and (4.6), and using (4.26), we get:
$\displaystyle\mathcal{A}_{0,h}(\hat{w}_{h},\hat{v}_{h})$
$\displaystyle=\mathcal{B}_{0,h}(\hat{w}_{h},\hat{v}_{h})+\mathcal{K}_{0}(\hat{w}_{h},\hat{v}_{h})=\mathcal{B}_{0}(\hat{w}_{h},\hat{v})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v}_{h}-\hat{v})+\mathcal{K}_{0}(\hat{w}_{h},\hat{v}_{h})$
$\displaystyle=\mathcal{B}_{0}(\hat{w}_{h},\hat{v})+\mathcal{K}_{0}(\hat{w}_{h},\hat{v})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v}_{h}-\hat{v})+\mathcal{K}_{0}(\hat{w}_{h},\hat{v}_{h})-\mathcal{K}_{0}(\hat{w}_{h},\hat{v})$
$\displaystyle=\mathcal{A}_{0}(\hat{w}_{h},\hat{v})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v}_{h}-\hat{v})-2\langle\mu_{h}-\mu,\text{K}_{0}w_{h}\rangle_{\Gamma}.$
(4.29)
By applying the Hölder inequality and (4.23b), we can estimate the last term
in (4.29) as follows
$\displaystyle\left*\langle\mu_{h}-\mu,\text{K}_{0}w_{h}\rangle_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)\times
H^{\nicefrac{{3}}{{2}}}(\Gamma)}\right\lvert$
$\displaystyle\apprle\lVert\mu_{h}-\mu\rVert_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)}\lVert\text{K}_{0}w_{h}\rVert_{H^{\nicefrac{{3}}{{2}}}(\Gamma)}$
$\displaystyle\apprle
h\lVert\hat{v}\rVert_{V}\lVert\text{K}_{0}w_{h}\rVert_{H^{\nicefrac{{3}}{{2}}}(\Gamma)}.$
(4.30)
Then, using the continuity of $\text{K}_{0}:H^{\nicefrac{{1}}{{2}}}(\Gamma)\to
H^{\nicefrac{{3}}{{2}}}(\Gamma)$ (see [29], formula (2.11)) and the trace
theorem, we obtain
$\norm{\text{K}_{0}w_{h}}_{H^{\nicefrac{{3}}{{2}}}(\Gamma)}\apprle\norm{w_{h}}_{H^{\nicefrac{{1}}{{2}}}(\Gamma)}\apprle\lVert
w_{h}\rVert_{H^{1}(\Omega)}\leq\lVert\hat{w}_{h}\rVert_{V},$
and, hence, combining this latter with (4.30), it follows that
$\left*\langle\mu_{h}-\mu,\text{K}_{0}w_{h}\rangle_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)\times
H^{\nicefrac{{3}}{{2}}}(\Gamma)}\right\lvert\apprle
h\lVert\hat{v}\rVert_{V}\lVert\hat{w}_{h}\rVert_{V}.$ (4.31)
Then, from (4.29) and (4.31), we obtain
$\displaystyle\mathcal{A}_{0,h}(\hat{w}_{h},\hat{v}_{h})$
$\displaystyle\apprge\mathcal{A}_{0}(\hat{w}_{h},\hat{v})+\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v}_{h}-\hat{v})-h\lVert\hat{v}\rVert_{V}\lVert\hat{w}_{h}\rVert_{V}.$
(4.32)
By explicitly writing
$\displaystyle\mathcal{B}_{0}((0,\bar{\eta}_{h}),\hat{v}_{h}-\hat{v})$
$\displaystyle=-\langle\bar{\eta}_{h},v_{h}-v\rangle_{\Gamma}+2\langle\mu_{h}-\mu,\text{V}_{0}\bar{\eta}_{h}\rangle_{\Gamma}$
$\displaystyle=-\bar{\eta}_{h}\langle
1,v_{h}-v\rangle_{\Gamma}+2\bar{\eta}_{h}\langle\mu_{h}-\mu,\text{V}_{0}1\rangle_{\Gamma}=:I+II,$
and using the Cauchy-Schwarz inequality to bound
$\left*\bar{\eta}_{h}\right\lvert\apprle\lVert\eta_{h}\rVert_{H^{-\nicefrac{{1}}{{2}}}}$,
we can estimate $II$ by using Hölder inequality, the continuity of
$\text{V}_{0}:H^{\nicefrac{{1}}{{2}}}(\Gamma)\to
H^{\nicefrac{{3}}{{2}}}(\Gamma)$ (see Remark 1) and (4.23b):
$\displaystyle\left*II\right\lvert$
$\displaystyle\apprle\lVert\eta_{h}\rVert_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)}\lVert\mu_{h}-\mu\rVert_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)}\lVert
V_{0}1\rVert_{H^{\nicefrac{{3}}{{2}}}(\Gamma)}\apprle
h\lVert\hat{v}\rVert_{V}\lVert\hat{w}_{h}\rVert_{V}.$
To estimate the term $I$, we use Hölder inequality and the trace theorem (see
e.g. Eq. (2.1) of [20]) and we obtain, for
$0<\varepsilon<\nicefrac{{1}}{{2}}$:
$\displaystyle\left*I\right\lvert$
$\displaystyle\apprle\lVert\eta_{h}\rVert_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)}\left*\langle
1,v_{h}-v\rangle_{\Gamma}\right\lvert\apprle\lVert\hat{w}_{h}\rVert_{V}\left*\langle
1,v_{h}-v\rangle_{\Gamma}\right\lvert\apprle\lVert\hat{w}_{h}\rVert_{V}\lVert
v_{h}-v\rVert_{H^{\varepsilon}(\Gamma)}$
$\displaystyle\apprle\lVert\hat{w}_{h}\rVert_{V}\lVert
v_{h}-v\rVert_{H^{\nicefrac{{1}}{{2}}+\varepsilon}(\Omega)}.$
Then, using the characterization of the fractional Sobolev space
$H^{\nicefrac{{1}}{{2}}+\varepsilon}(\Omega)$ as the real interpolation
between $L^{2}(\Omega)$ and $H^{1}(\Omega)$, by a standard result concerning
the norm of real interpolation spaces (see Prop. 2.3 of [30]), it holds that
$\lVert v_{h}-v\rVert_{H^{\nicefrac{{1}}{{2}}+\varepsilon}(\Omega)}\leq\lVert
v_{h}-v\rVert^{\nicefrac{{1}}{{2}}-\varepsilon}_{L^{2}(\Omega)}\lVert
v_{h}-v\rVert^{\nicefrac{{1}}{{2}}+\varepsilon}_{H^{1}(\Omega)}$. Hence, by
applying (4.23a) and (4.23c), we finally get:
$\displaystyle\left*I\right\lvert$ $\displaystyle\apprle
h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{v}\rVert_{V}^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert
v_{h}-v\rVert^{\nicefrac{{1}}{{2}}+\varepsilon}_{H^{1}(\Omega)}\lVert\hat{w}_{h}\rVert_{V}\apprle
h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{v}\rVert_{V}^{\nicefrac{{1}}{{2}}-\varepsilon}(\lVert
v\rVert_{H^{1}(\Omega)}+\lVert
v_{h}\rVert_{H^{1}(\Omega)})^{\nicefrac{{1}}{{2}}+\varepsilon}\lVert\hat{w}_{h}\rVert_{V}$
$\displaystyle\apprle
h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{v}\rVert_{V}^{\nicefrac{{1}}{{2}}-\varepsilon}(\lVert\hat{v}\rVert_{V}+\lVert\hat{v}_{h}\rVert_{V})^{\nicefrac{{1}}{{2}}+\varepsilon}\lVert\hat{w}_{h}\rVert_{V}\apprle
h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{v}\rVert_{V}^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{v}\rVert_{V}^{\nicefrac{{1}}{{2}}+\varepsilon}\lVert\hat{w}_{h}\rVert_{V}=h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{v}\rVert_{V}\lVert\hat{w}_{h}\rVert_{V}.$
Combining (4.32) with the bounds for $I$ and $II$, we can write
$\displaystyle\mathcal{A}_{0,h}(\hat{w}_{h},\hat{v}_{h})$
$\displaystyle\apprge\mathcal{A}_{0}(\hat{w}_{h},\hat{v})-h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{v}\rVert_{V}\lVert\hat{w}_{h}\rVert_{V}-h\lVert\hat{v}\rVert_{V}\lVert\hat{w}_{h}\rVert_{V}$
$\displaystyle\apprge\mathcal{A}_{0}(\hat{w}_{h},\hat{v})-h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{v}\rVert_{V}\lVert\hat{w}_{h}\rVert_{V}.$
(4.33)
Starting from (4.27), using (4.28) and (4.2), it follows
$\displaystyle\mathcal{A}_{\kappa,h}(\hat{w}_{h},\hat{v}_{h})$
$\displaystyle\apprge\mathcal{A}_{0,h}(\hat{w}_{h},\hat{v}_{h})+(\mathcal{A}_{k}-\mathcal{A}_{0})(\hat{w}_{h},\hat{v}_{h})-h\lVert\hat{w}_{h}\rVert_{V}\lVert\hat{v}_{h}\rVert_{V}$
$\displaystyle\apprge\mathcal{A}_{0}(\hat{w}_{h},\hat{v})-h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{w}_{h}\rVert_{V}\lVert\hat{v}\rVert_{V}+(\mathcal{A}_{k}-\mathcal{A}_{0})(\hat{w}_{h},\hat{v}_{h})$
$\displaystyle=\mathcal{A}_{\kappa}(\hat{w}_{h},\hat{v})-h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{v}\rVert_{V}\lVert\hat{w}_{h}\rVert_{V}+(\mathcal{A}_{\kappa}-\mathcal{A}_{0})(\hat{w}_{h},\hat{v}_{h})+(\mathcal{A}_{0}-\mathcal{A}_{\kappa})(\hat{w}_{h},\hat{v})$
$\displaystyle=\lVert\hat{w}_{h}\rVert_{V}^{2}-h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{v}\rVert_{V}\lVert\hat{w}_{h}\rVert_{V}+(\mathcal{A}_{\kappa}-\mathcal{A}_{0})(\hat{w}_{h},\hat{v}_{h}-\hat{v})$
(4.34)
having used (4.24) in the last equality.
Concerning the last term in (4.2), we explicitly write:
$(\mathcal{A}_{\kappa}-\mathcal{A}_{0})(\hat{w}_{h},\hat{v}_{h}-\hat{v})=-\kappa^{2}m(w_{h},v_{h}-v)+2\langle\mu_{h}-\mu,(\text{V}_{\kappa}-\text{V}_{0})\eta_{h}-(\text{K}_{\kappa}-\text{K}_{0})w_{h}\rangle$
and, by using the continuity of $m$, the Hölder inequality and the continuity
of $\text{V}_{\kappa}-\text{V}_{0}:H^{-\nicefrac{{1}}{{2}}}(\Gamma)\to
H^{\nicefrac{{3}}{{2}}}(\Gamma)$ and of
$\text{K}_{\kappa}-\text{K}_{0}:H^{\nicefrac{{1}}{{2}}}(\Gamma)\to
H^{\nicefrac{{3}}{{2}}}(\Gamma)$ (see Lemma 2.1),
$\displaystyle\left*(\mathcal{A}_{\kappa}-\mathcal{A}_{0})(\hat{w}_{h},\hat{v}_{h}-\hat{v})\right\lvert$
$\displaystyle\apprle\lVert\hat{w}_{h}\rVert_{V}\lVert
v_{h}-v\rVert_{L^{2}(\Omega)}+\lVert\hat{w}_{h}\rVert_{V}\lVert\mu_{h}-\mu\rVert_{H^{-\nicefrac{{3}}{{2}}}(\Gamma)}\apprle
h\lVert\hat{w}_{h}\rVert_{V}\lVert\hat{v}_{h}\rVert_{V},$ (4.35)
having used, in the last bound, (4.23b) and (4.23c). Finally, combining (4.2)
with (4.35) and (4.25) we get
$\mathcal{A}_{\kappa,h}(\hat{w}_{h},\hat{v}_{h})\apprge\lVert\hat{w}_{h}\rVert_{V}^{2}-h^{\nicefrac{{1}}{{2}}-\varepsilon}\lVert\hat{w}_{h}\rVert_{V}\lVert\hat{v}\rVert_{V}\apprge(1-h^{\nicefrac{{1}}{{2}}-\varepsilon})\lVert\hat{w}_{h}\rVert_{V}^{2}$
from which, for $h$ small enough, the claim follows. ∎
We conclude this section by proving the convergence error estimate for Problem
(4.6).
###### Theorem 4.8.
Assume that $\kappa^{2}$ is not an eigenvalue of the Laplacian in $\Omega$
endowed with a Dirichlet boundary condition on $\Gamma$. Furthermore, assume
that there exist $k\in\mathbf{N}$ such that for all $1\leq s\leq k$ and
$\kappa>0$, Assumptions ($H1.{a}$)-($H1.{c}$), ($H2.{a}$), ($H2.{b}$),
($H3.{a}$)-($H3.{c}$) hold, and $\sigma:L^{2}(\Omega)\to\mathbf{R}^{+}$ such
that
1. ($H4.{a}$)
$\left|\mathcal{L}_{f}(\hat{v}_{h})-\mathcal{L}_{f,h}(\hat{v}_{h})\right|\apprle
h^{s}\left\|\hat{v}_{h}\right\|_{V}\,\sigma(f)\qquad\forall\,\hat{v}_{h}\in
V_{h}^{k}$.
Then, for $h$ small enough, Problem (4.6) admits a unique solution
$\hat{u}_{h}\in V_{h}^{k}$ and if $\hat{u}$, solution of Problem (3.4),
satisfies $\hat{u}\in H^{s+1}(\Omega)\times
H^{s-\nicefrac{{1}}{{2}}}(\Gamma)$, it holds
$\lVert\hat{u}-\hat{u}_{h}\rVert_{V}\apprle h^{s}\left(\lVert
u\rVert_{H^{s+1}(\Omega)}+\sigma(f)\right).$
###### Proof.
Existence and uniqueness of $\hat{u}_{h}$ follows from the discrete inf-sup
condition of Theorem 4.7. Let $\hat{u}_{h}^{I}\in V_{h}^{k}$ be the
interpolant of $\hat{u}$. By virtue of Theorem 4.7 there exists
$\hat{v}_{h}^{*}=(v_{h}^{*},\mu_{h}^{*})\in V_{h}^{k}$ such that
$\lVert\hat{u}_{h}-\hat{u}_{h}^{I}\rVert_{V}\apprle\frac{\mathcal{A}_{\kappa,h}(\hat{u}_{h}-\hat{u}_{h}^{I},\hat{v}_{h}^{*})}{\lVert\hat{v}_{h}^{*}\rVert_{V}}.$
Since $\hat{u}$ and $\hat{u}_{h}$ are solution of (3.4) and (4.6)
respectively, we have
$\displaystyle\lVert\hat{u}_{h}-\hat{u}_{h}^{I}\rVert_{V}\lVert\hat{v}_{h}^{*}\rVert_{V}$
$\displaystyle\apprle\mathcal{A}_{\kappa,h}(\hat{u}_{h}-\hat{u}_{h}^{I},\hat{v}^{*}_{h})=\mathcal{A}_{\kappa,h}(\hat{u}_{h},\hat{v}^{*}_{h})-\mathcal{A}_{\kappa,h}(\hat{u}_{h}^{I},\hat{v}^{*}_{h})$
$\displaystyle=\mathcal{L}_{f,h}(\hat{v}^{*}_{h})-\mathcal{A}_{\kappa,h}(\hat{u}_{h}^{I},\hat{v}^{*}_{h})+[\mathcal{A}_{\kappa}(\hat{u},\hat{v}^{*}_{h})-\mathcal{L}_{f}(\hat{v}^{*}_{h})]$
$\displaystyle=[\mathcal{L}_{f,h}(\hat{v}^{*}_{h})-\mathcal{L}_{f}(\hat{v}^{*}_{h})]+\mathcal{A}_{\kappa}(\hat{u}-\hat{u}_{h}^{I},\hat{v}^{*}_{h})+[\mathcal{A}_{\kappa}(\hat{u}_{h}^{I},\hat{v}^{*}_{h})-\mathcal{A}_{\kappa,h}(\hat{u}_{h}^{I},\hat{v}^{*}_{h})]$
$\displaystyle=[\mathcal{L}_{f,h}(\hat{v}^{*}_{h})-\mathcal{L}_{f}(\hat{v}^{*}_{h})]+\mathcal{A}_{\kappa}(\hat{u}-\hat{u}_{h}^{I},\hat{v}^{*}_{h})+[(\mathcal{B}_{\kappa}-\mathcal{B}_{\kappa,h})(\hat{u}_{h}^{I},\hat{v}^{*}_{h})].$
Then, by using Assumption ($H4.{a}$), the continuity of $\mathcal{A}_{\kappa}$
and Lemma 4.2, we obtain
$\displaystyle\lVert\hat{u}_{h}-\hat{u}_{h}^{I}\rVert_{V}\lVert\hat{v}_{h}^{*}\rVert_{V}$
$\displaystyle\apprle
h^{s}\left\|\hat{v}^{*}_{h}\right\|_{V}\,\sigma(f)+\lVert\hat{u}-\hat{u}_{h}^{I}\rVert_{V}\lVert\hat{v}_{h}^{*}\rVert_{V}+h^{s}\lVert
u\rVert_{H^{s+1}(\Omega)}\lVert\hat{v}_{h}^{*}\rVert_{V},$
from which it easily follows
$\lVert\hat{u}-\hat{u}_{h}\rVert_{V}\leq\lVert\hat{u}-\hat{u}_{h}^{I}\rVert_{V}+\lVert\hat{u}_{h}-\hat{u}_{h}^{I}\rVert_{V}\apprle\lVert\hat{u}-\hat{u}_{h}^{I}\rVert_{V}+h^{s}\lVert
u\rVert_{H^{s+1}(\Omega)}+h^{s}\sigma(f).$ (4.36)
Finally, combining (4.7) and (4.36) we obtain the thesis. ∎
## 5 The discrete scheme
In this section we introduce the discrete CVEM-BEM scheme for the coupling
procedure (3.2). We start by briefly describing the main tools of the VEM; we
refer the interested reader to [1, 4, 8] for a deeper presentation. In what
follows, we denote by $\mathbf{V}_{1},\ldots,\mathbf{V}_{n_{\text{\tiny{E}}}}$
the $n_{\text{\tiny{E}}}$ vertices of an element $E\in\mathcal{T}_{h}$ and by
$e_{1},\ldots,e_{n_{\text{\tiny{E}}}}$ the edges of its boundary $\partial E$.
For simplicity of presentation, we assume that at most one edge is curved
while the remaining ones are straight. We identify the curved edge by
$e_{1}\subset\partial\Omega$, to which we associate a regular invertible
parametrization $\gamma_{E}:I_{E}\rightarrow e_{1}$, where
$I_{E}\subset\mathbf{R}$ is a closed interval. Furthermore, we denote by
$\mathbf{V}_{E}$, $h_{E}$ and $|E|$ the mass center, the diameter and the
Lebesgue measure of $E$, respectively. Additionally, we call
$N_{\text{\tiny{V}}}$ and $N_{\text{{e}}}$ the numbers of total vertices and
edges of $\mathcal{T}_{h}$, respectively.
In what follows we will show that all the assumptions, used to obtain the
theoretical results in Section 4.1, are satisfied.
### 5.1 The discrete spaces $Q_{h}^{k}$, $X_{h}^{k}$ and
$\widetilde{X}_{h}^{k}$: validity of Assumptions ($H1.{a}$)–($H1.{c}$)
In order to describe the discrete space $Q_{h}^{k}$, introduced in Section 4.1
in a generic setting, we preliminarily consider for each $E\in\mathcal{T}_{h}$
the following local finite dimensional _augmented_ virtual space
$\widetilde{Q}^{k}_{h}(E)$ and the local _enhanced_ virtual space
$Q^{k}_{h}(E)$ defined respectively,
$\widetilde{Q}^{k}_{h}(E):=\left\\{v_{h}\in H^{1}(E)\cap C^{0}(E)\ :\ \Delta
v_{h}\in P_{k}(E),\
v_{h}\,\raisebox{-5.0pt}{$|_{e_{1}}$}\in\widetilde{P}_{k}(e_{1}),\
v_{h}\,\raisebox{-5.0pt}{$|_{e_{i}}$}\in P_{k}(e_{i})\
\forall\,i=2,\ldots,n_{\text{\tiny{E}}}\right\\}$
and
$Q^{k}_{h}(E):=\left\\{v_{h}\in\widetilde{Q}^{k}_{h}(E)\ :\
m^{\text{\tiny{E}}}\left(\Pi_{k}^{\nabla,E}v_{h},q\right)=m^{\text{\tiny{E}}}\left(v_{h},q\right)\
\forall\,q\in P_{k}(E)/P_{k-2}(E)\right\\},$
where $\widetilde{P}_{k}(e_{1}):=\left\\{\widetilde{q}:=q\circ\gamma_{E}^{-1}\
:\ q\in P_{k}(I_{E})\right\\}$ and $P_{k}(E)/P_{k-2}(E)$ stands for the space
of all polynomials of degree $k$ on $E$ that are $L^{2}$-orthogonal to all
polynomials of degree $k-2$ on $E$.
For details on such spaces, we refer the reader to [8] (see Remark 2.6) and to
[1] (see Section 3).
It is possible to prove (see Proposition 2 in [1] and Proposition 2.2 in [8])
that the dimension of $Q^{k}_{h}(E)$ is
$n:=\dim(Q^{k}_{h}(E))=kn_{\text{\tiny{E}}}+\frac{k(k-1)}{2}$
and that a generic element $v_{h}$ of $Q^{k}_{h}(E)$ is uniquely determined by
the following $n$ conditions (see [8], Proposition 2.2):
* •
its values at the $n_{\text{\tiny{E}}}$ vertices of $E$;
* •
its values at the $k-1$ internal points of the $(k+1)-$point Gauss-Lobatto
quadrature rule on every straight edge $e_{2},\ldots,e_{n_{E}}\in\partial E$;
* •
its values at the $k-1$ internal points of $e_{1}$ that are the images,
through $\gamma_{E}$, of the $(k+1)-$point Gauss-Lobatto quadrature rule on
$I_{E}$;
* •
the internal $k(k-1)/2$ moments of $v_{h}$ against a polynomial basis
$\mathcal{M}_{k-2}(E)$ of $P_{k-2}(E)$ defined for $k\geq 2$, as:
$\frac{1}{|E|}\int\limits_{E}v_{h}(\mathbf{x})p(\mathbf{x})\,\differential\mathbf{x}\qquad\forall\,p\in\mathcal{M}_{k-2}(E)\
\text{with}\ \|p\|_{L^{\infty}(E)}\apprle 1.$ (5.1)
According to the definition of $\widetilde{Q}^{k}_{h}(E)$, it is easy to check
that $P_{0}(E)\subset Q^{k}_{h}(E)$ while, in general, $P_{k}(E)\not\subset
Q^{k}_{h}(E)$, for $k>0$. Now, choosing an arbitrary but fixed ordering of the
degrees of freedom such that these are indexed by $i=1,\cdots,n$, we introduce
as in [4] the operator $\text{dof}_{i}:Q^{k}_{h}(E)\longrightarrow\mathbf{C}$,
defined as
$\text{dof}_{i}(v_{h}):=\text{the value of the $i$-th local degree of freedom
of}\,v_{h}.$
The basis functions $\left\\{\Phi_{j}\right\\}_{j=1}^{n}$ chosen for the space
$Q^{k}_{h}(E)$ are the standard Lagrangian ones, such that
$\text{dof}_{i}(\Phi_{j})=\delta_{ij},\qquad i,j=1,\ldots,n,$
$\delta_{ij}$ being the Kronecker delta.
On the basis of the definition of the local enhanced virtual space
$Q^{k}_{h}(E)$, we are allowed to construct the global enhanced virtual space
$Q^{k}_{h}:=\left\\{v_{h}\in H^{1}_{0,\Gamma_{0}}(\Omega)\ :\
v_{h}\,\raisebox{-5.0pt}{$|_{E}$}\in Q^{k}_{h}(E)\quad\forall
E\in\mathcal{T}_{h}\right\\}.$
We remark that the global degrees of freedom for a generic element $v_{h}\in
Q^{k}_{h}$ are:
* •
its values at each of the $\widetilde{N}_{\text{\tiny{V}}}$ vertices of
$\mathcal{T}_{h}$ that do not belong to $\Gamma_{0}$;
* •
its values at the $k-1$ internal points of the $(k+1)-$point Gauss-Lobatto
quadrature rule on each of the $\bar{N}_{\text{\tiny{e}}}$ straight edges of
$\mathcal{T}_{h}$;
* •
its values at the $k-1$ internal points of the
$\widetilde{N}_{\text{\tiny{e}}}$ curved edge of $\mathcal{T}_{h}$, that do
not belong to $\Gamma_{0}$ and that are the images through the parametrization
$\gamma_{E}$ of the the $(k+1)-$point Gauss-Lobatto quadrature rule on
parametric interval $I_{E}$;
* •
its moments up to order $k-2$ in each of the $N_{\text{\tiny{E}}}$ elements of
$\mathcal{T}_{h}$, for $k\geq 2$:
$\frac{1}{|E|}\int\limits_{E}v_{h}(\mathbf{x})p(\mathbf{x})\,\differential\mathbf{x}\qquad\forall\,p\in\mathcal{M}_{k-2}(E)(E)\
\text{with}\ \|p\|_{L^{\infty}(E)}\apprle 1.$
Consequently, $Q^{k}_{h}$ has dimension
$N:=\dim(Q^{k}_{h})=\widetilde{N}_{\text{\tiny{V}}}+(k-1)(\widetilde{N}_{\text{\tiny{e}}}+\bar{N}_{\text{\tiny{e}}})+\frac{k(k-1)}{2}N_{\text{\tiny{E}}}.$
(5.2)
###### Remark 6.
We remark that the global enhanced virtual space $Q^{k}_{h}$ defined above is
slightly different from that introduced in the pioneering paper on CVEM [8],
the latter being defined for the solution of the Laplace problem. However, as
highlighted in Remark 2.6 in [8], the theoretical analysis therein contained
can be extended to our context by following the ideas of [1].
In the following lemma we prove that Assumption ($H1.{a}$) holds for the space
$Q_{h}^{k}$.
###### Lemma 5.1.
Let $v\in H^{s+1}(\Omega)$ with $\nicefrac{{1}}{{2}}<s\leq k$. Then
$\inf_{v_{h}\in Q_{h}^{k}}\|v-v_{h}\|_{H^{1}(\Omega)}\apprle
h^{s}\|v\|_{H^{s+1}(\Omega)}.$
###### Proof.
Let $E$ be an element of $\mathcal{T}_{h}$. By virtue of Theorem 3.7 in [8]
there exists $v_{h}^{I}$, interpolant of $v$ in $Q_{h}^{k}$, such that
$|v-v_{h}^{I}|_{H^{1}(E)}\apprle h_{E}^{s}\|v\|_{H^{s+1}(E)}.$
Moreover, by using the Poincaré-Friedrichs inequality (see (2.11) in [13]), we
can write
$\|v-v_{h}^{I}\|_{L^{2}(E)}\apprle
h_{E}|v-v_{h}^{I}|_{H^{1}(E)}+\left|\int_{\partial
E}\left[v(\mathbf{x})-v_{h}^{I}(\mathbf{x})\right]ds\right|\apprle
h_{E}^{s+1}\|v\|_{H^{s+1}(E)}+\int_{\partial
E}\left|v(\mathbf{x})-v_{h}^{I}(\mathbf{x})\right|ds.$
Then, by applying the Hölder inequality, Lemma 3.2 and (3.20) in [8], we can
estimate the second term at the right hand side of the above inequality as
follows
$\displaystyle\int_{\partial
E}\left|v(\mathbf{x})-v_{h}^{I}(\mathbf{x})\right|ds$
$\displaystyle=\sum_{e\subset\partial
E}\int_{e}\left|v(\mathbf{x})-v_{h}^{I}(\mathbf{x})\right|ds\leq\sum_{e\subset\partial
E}|e|^{\nicefrac{{1}}{{2}}}\|v-v_{h}^{I}\|_{L^{2}(e)}\leq
h_{E}^{\nicefrac{{1}}{{2}}}\sum_{e\subset\partial
E}\|v-v_{h}^{I}\|_{L^{2}(e)}$ $\displaystyle\apprle
h_{E}^{\nicefrac{{1}}{{2}}}\sum_{e\subset\partial
E}h_{E}^{s+\nicefrac{{1}}{{2}}}\|v\|_{H^{s+\nicefrac{{1}}{{2}}}(e)}\apprle
h_{E}^{s+1}\|v\|_{H^{s+1}(E)}.$
Combining the local bounds for the $L^{2}$-norm and for the $H^{1}$-seminorm
of $v-v_{h}^{I}$ on $E$, we obtain
$\|v-v_{h}^{I}\|_{H^{1}(\Omega)}\apprle h^{s}\|v\|_{H^{s+1}(\Omega)},$
from which the thesis easily follows. ∎
Finally, we introduce the boundary element space $X_{h}^{k}$ associated to the
artificial boundary $\Gamma$
$X_{h}^{k}:=\left\\{\lambda\in L^{2}(\Gamma)\ :\lambda_{\mkern 1.0mu\vrule
height=6.02777pt\mkern 2.0mue}\in\widetilde{P}_{k}(e),\ \forall
e\in\Gamma\right\\}\qquad\text{with}\qquad|e|<h,$
where $|e|$ denotes the length of the edge $e$. By virtue of Theorem 4.3.20 in
[36], we have that the space $X_{h}^{k}$ satisfies the interpolation property
($H1.{b}$). For what concerns the space $\widetilde{X}_{h}^{k}=X_{h}^{k}\cap
H_{0}^{-\nicefrac{{1}}{{2}}}(\Gamma)$ and the corresponding hypothesis
($H1.{c}$), we refer to (3.2b) in [29]. Moreover, a natural basis for the
space $X_{h}^{k}$ consists in the choice of the functions
$\Phi_{j_{{|_{\Gamma}}}}$, which are the restriction of $\Phi_{j}$ on
$\Gamma$.
### 5.2 The discrete bilinear forms $\mathcal{A}_{\kappa,h}$ and
$\mathcal{B}_{\kappa,h}$: validity of Assumptions ($H2.{a}$), ($H2.{b}$) and
($H3.{a}$)–($H3.{c}$)
In order to define computable discrete local bilinear forms
$a^{\text{\tiny{E}}}_{h}:Q^{k}_{h}(E)\times Q^{k}_{h}(E)\rightarrow\mathbf{C}$
and $m^{\text{\tiny{E}}}_{h}:Q^{k}_{h}(E)\times
Q^{k}_{h}(E)\rightarrow\mathbf{C}$, following [4] and by using the definition
of $\Pi_{k}^{\nabla,E}$ and $\Pi_{k}^{0,E}$, we first split
$a^{\text{\tiny{E}}}$ and $m^{\text{\tiny{E}}}$ in a part that can be computed
exactly (up to the machine precision) and in a part that will be suitably
approximated:
$\displaystyle a^{\text{\tiny{E}}}(u_{h},v_{h})$
$\displaystyle=a^{\text{\tiny{E}}}\left(\Pi_{k}^{\nabla,E}u_{h},\Pi_{k}^{\nabla,E}v_{h}\right)+a^{\text{\tiny{E}}}\left(\left(I-\Pi_{k}^{\nabla,E}\right)u_{h},\left(I-\Pi_{k}^{\nabla,E}\right)v_{h}\right)$
(5.3) $\displaystyle m^{\text{\tiny{E}}}(u_{h},v_{h})$
$\displaystyle=m^{\text{\tiny{E}}}\left(\Pi_{k}^{0,E}u_{h},\Pi_{k}^{0,E}v_{h}\right)+m^{\text{\tiny{E}}}\left(\left(I-\Pi_{k}^{0,E}\right)u_{h},\left(I-\Pi_{k}^{0,E}\right)v_{h}\right),$
(5.4)
$I$ being the identity operator. The implementation steps for the computation
of
$a^{\text{\tiny{E}}}\left(\Pi_{k}^{\nabla,E}u_{h},\Pi_{k}^{\nabla,E}v_{h}\right)$
and $m^{\text{\tiny{E}}}\left(\Pi_{k}^{0,E}u_{h},\Pi_{k}^{0,E}v_{h}\right)$
require the choice of a suitable basis of the space $P_{k}(E)$, that allows to
define in practice the projectors $\Pi_{k}^{\nabla,E}$ and $\Pi_{k}^{0,E}$. In
accordance with the standard literature on VEM (see [5], Section 3.1), we have
considered the basis of the scaled monomials, i.e.
$\mathcal{M}_{k}(E):=\left\\{p_{\bm{\alpha}}(\mathbf{x})=\left(\frac{\mathbf{x}-\mathbf{V}_{E}}{h_{E}}\right)^{\bm{\alpha}},\
\bm{\alpha}=(\alpha_{1},\alpha_{2})\ :\
|\bm{\alpha}|=\alpha_{1}+\alpha_{2}\leq k\right\\},$
where, we recall, ${\bf V}_{E}$ and $h_{E}$ denote the mass centre and the
diameter of $E$, respectively.
Following [5], the second term in (5.3) is approximated by the following
bilinear form which represents a stabilization term:
$\displaystyle
s^{\text{\tiny{E}}}\left(\left(I-\Pi_{k}^{\nabla,E}\right)u_{h},\left(I-\Pi_{k}^{\nabla,E}\right)v_{h}\right)$
$\displaystyle:=\sum\limits_{j=1}^{n}\text{dof}_{j}\left(\left(I-\Pi_{k}^{\nabla,E}\right)u_{h}\right)\text{dof}_{j}\left(\left(I-\Pi_{k}^{\nabla,E}\right)v_{h}\right).$
(5.5)
On the contrary, since the analysis of the method requires the ellipticity
property only for the bilinear form $\mathcal{B}_{0,h}$ (see Assumption
($H3.{b}$)), an analogous stabilizing term is not needed in (5.4). Therefore
we define the approximations $a^{\text{\tiny{E}}}_{h}$ and
$m^{\text{\tiny{E}}}_{h}$ of $a^{\text{\tiny{E}}}$ and $m^{\text{\tiny{E}}}$,
respectively, as follows:
$\displaystyle a^{\text{\tiny{E}}}_{h}(u_{h},v_{h})$
$\displaystyle:=a^{\text{\tiny{E}}}\left(\Pi_{k}^{\nabla,E}u_{h},\Pi_{k}^{\nabla,E}v_{h}\right)+s^{\text{\tiny{E}}}\left(\left(I-\Pi_{k}^{\nabla,E}\right)u_{h},\left(I-\Pi_{k}^{\nabla,E}\right)v_{h}\right)$
(5.6) $\displaystyle m^{\text{\tiny{E}}}_{h}(u_{h},v_{h})$
$\displaystyle:=m^{\text{\tiny{E}}}\left(\Pi_{k}^{0,E}u_{h},\Pi_{k}^{0,E}v_{h}\right).$
(5.7)
As shown in [7] and [8], the approximate bilinear forms satisfy the following
properties:
* •
$k$-consistency: for all $v_{h}\in Q^{k}_{h}(E)$ and for all $q\in P_{k}(E)$:
$a^{\text{\tiny{E}}}_{h}(v_{h},q)=a^{\text{\tiny{E}}}(v_{h},q)\qquad\text{and}\qquad
m^{\text{\tiny{E}}}_{h}(v_{h},q)=m^{\text{\tiny{E}}}(v_{h},q);$ (5.8)
* •
stability: for all $v_{h}\in Q^{k}_{h}(E)$:
$a^{\text{\tiny{E}}}(v_{h},v_{h})\apprle
a^{\text{\tiny{E}}}_{h}(v_{h},v_{h})\apprle
a^{\text{\tiny{E}}}(v_{h},v_{h})\qquad\text{and}\qquad
m^{\text{\tiny{E}}}_{h}(v_{h},v_{h})\apprle m^{\text{\tiny{E}}}(v_{h},v_{h}).$
(5.9)
The global approximate bilinear forms $a_{h},m_{h}:Q^{k}_{h}\times
Q^{k}_{h}\rightarrow\mathbf{C}$ are then defined by summing up the local
contributions as follows:
$a_{h}(u_{h},v_{h}):=\sum\limits_{E\in\mathcal{T}_{h}}a^{\text{\tiny{E}}}_{h}(u_{h},v_{h})\qquad\text{and}\qquad
m_{h}(u_{h},v_{h}):=\sum\limits_{E\in\mathcal{T}_{h}}m^{\text{\tiny{E}}}_{h}(u_{h},v_{h}).$
From the right hand side of (5.9), it immediately follows that
$m_{h}(v_{h},v_{h})\apprle\left\|v_{h}\right\|_{L^{2}(\Omega)}^{2}\qquad\forall\,v_{h}\in
Q^{k}_{h}$ (5.10)
while, combining (5.9) with the Poincaré-Friedrichs inequality (see (5.3.3) in
[14]), we have:
$\left\|v_{h}\right\|_{H^{1}(\Omega)}^{2}\apprle
a_{h}(v_{h},v_{h})\apprle\left\|v_{h}\right\|_{H^{1}(\Omega)}^{2}\qquad\forall
v_{h}\in Q^{k}_{h}.$ (5.11)
Moreover, the characterization of the virtual element space $Q_{h}^{k}$ and
the boundary element space $X_{h}^{k}$ allows us to formally define the
bilinear form $\mathcal{B}_{\kappa,h}:V^{k}_{h}\times
V^{k}_{h}\rightarrow\mathbf{C}$,
$\mathcal{B}_{\kappa,h}(\hat{u}_{h},\hat{v}_{h}):=a_{h}(u_{h},v_{h})-\kappa^{2}m_{h}(u_{h},v_{h})-\langle\lambda_{h},v_{h}\rangle_{\Gamma}+\langle\mu_{h},u_{h}\rangle_{\Gamma}+2\langle\mu_{h},\text{V}_{\kappa}\lambda_{h}\rangle_{\Gamma}$
for $\hat{u}_{h}=(u_{h},\lambda_{h}),\hat{v}_{h}=(v_{h},\mu_{h})\in
V^{k}_{h}$.
From the $k$-consistency of the discrete bilinear forms
$a^{\text{\tiny{E}}}_{h}$ and $m^{\text{\tiny{E}}}_{h}$ (see (5.8)), it
immediately follows that $\mathcal{B}_{\kappa,h}$ satisfies Assumption
($H2.{a}$). Furthermore, the continuity of the bilinear forms $a_{h}$ and
$m_{h}$, as well as the continuity of the boundary operator
$\text{V}_{\kappa}$, ensure the $V$-norm continuity of
$\mathcal{B}_{\kappa,h}$, i.e. Assumption ($H2.{b}$). Analogously, the
$W$-norm continuity of
$\mathcal{D}_{\kappa,h}=\mathcal{B}_{\kappa,h}-\mathcal{B}_{0,h}$, i.e.
Assumption ($H3.{a}$), is a consequence of the continuity of $m_{h}$ and of
Lemma 2.1.
Now, to prove the $\widetilde{V}_{h}^{k}$-ellipticity, i.e. assumption
($H3.{b}$), we focus on the term
$\mathcal{B}_{0,h}(\hat{v}_{\scriptscriptstyle 0h},\hat{v}_{\scriptscriptstyle
0h})=a_{h}(v_{h},v_{h})+2\langle\mu_{\scriptscriptstyle
0h},\text{V}_{0}\mu_{\scriptscriptstyle 0h}\rangle_{\Gamma}$
for $\hat{v}_{\scriptscriptstyle 0h}=(v_{h},\mu_{\scriptscriptstyle
0h})\in\widetilde{V}^{k}_{h}$. In order to bound the first and the second term
in the above sum, we use (5.11) and Theorem 6.22 in [40], respectively, and we
get
$\mathcal{B}_{0,h}(\hat{v}_{\scriptscriptstyle 0h},\hat{v}_{\scriptscriptstyle
0h})\apprge\left\|v_{h}\right\|_{H^{1}(\Omega)}^{2}+\left\|\mu_{\scriptscriptstyle
0h}\right\|_{H^{-\nicefrac{{1}}{{2}}}(\Gamma)}^{2}=\left\|\hat{v}_{h}\right\|_{V}^{2}.$
Thus, Assumption ($H3.{c}$) is a direct consequence of the $k$-consistency
(5.8).
### 5.3 The discrete linear operator $\mathcal{L}_{f,h}$: validity of
Assumption ($H4.{a}$)
In the present section, we define the discrete linear operator
$\mathcal{L}_{f,h}:V_{h}^{k}\rightarrow\mathbf{C}$ such that
$\mathcal{L}_{f,h}(\hat{v}_{h}):=\begin{cases}\sum\limits_{E\in\mathcal{T}_{h}}m^{E}(f,\Pi^{0,E}_{1}v_{h})&k=1,2\,,\\\
\sum\limits_{E\in\mathcal{T}_{h}}m^{E}(f,\Pi^{0,E}_{k-2}v_{h})&k\geq
3.\end{cases}$
Assuming $f\in H^{k-1}(\Omega)$, in [13] (see Lemma 3.4) it has been proved
that
$\left|\mathcal{L}_{f}(\hat{v}_{h})-\mathcal{L}_{f,h}(\hat{v}_{h})\right|\apprle
h^{k}\left|f\right|_{H^{k-1}(\Omega)}\left\|v_{h}\right\|_{H^{1}(\Omega)}.$
Hence, Assumption ($H4.{a}$) is fulfilled with
$\sigma(f)=\left|f\right|_{H^{k-1}(\Omega)}$.
### 5.4 Algebraic formulation of the discrete problem
For what follows, in order to detail the algebraic form of the coupled CVEM-
BEM method, it is convenient to re-order and split the complete index set
$\mathcal{S}:=\\{j=1,\cdots,N\\}$ of the basis functions
$\left\\{\Phi_{j}\right\\}_{j\in\mathcal{S}}$ of $Q^{k}_{h}$ as
$\mathcal{S}=\mathcal{S}^{I}\cup\mathcal{S}^{\Gamma},$ (5.12)
where $\mathcal{S}^{I}$ and $\mathcal{S}^{\Gamma}$ denote the sets of indices
related to the internal degrees of freedom and to the degrees of freedom lying
on $\Gamma$, respectively. With this choice, we have
$Q_{h}^{k}=\text{span}\left\\{\Phi_{j}\right\\}_{j\in\mathcal{S}^{I}\cup\mathcal{S}^{\Gamma}},\qquad
X^{k}_{h}=\text{span}\left\\{\Phi_{j_{{|_{\Gamma}}}}\right\\}_{j\in\mathcal{S}^{\Gamma}}.$
In order to derive the linear system associated to the discrete problem (4.6),
we expand the unknown function $\hat{u}_{h}=(u_{h},\lambda_{h})\in
Q_{h}^{k}\times X^{k}_{h}$ as
$\displaystyle
u_{h}(\mathbf{x})=:\sum\limits_{j\in\mathcal{S}}u_{h}^{j}\Phi_{j}(\mathbf{x})\qquad\text{with}\quad
u_{h}^{j}=\text{dof}_{j}(u_{h})$ (5.13)
$\displaystyle\lambda_{h}(\mathbf{x})=:\sum\limits_{j\in\mathcal{S}^{\Gamma}}\lambda_{h}^{j}\Phi_{j_{{|_{\Gamma}}}}(\mathbf{x})\qquad\text{with}\quad\lambda_{h}^{j}=\text{dof}_{j}(\lambda_{h}).$
Hence, using the basis functions of $Q^{k}_{h}$ to test the discrete
counterpart of equation (3.2a), we get for
$i\in\mathcal{S}^{I}\cup\mathcal{S}^{\Gamma}$
$\displaystyle\sum\limits_{j\in\mathcal{S}^{I}\cup\mathcal{S}^{\Gamma}}u_{h}^{j}$
$\displaystyle\sum\limits_{E\in\mathcal{T}_{h}}\left[a^{\text{\tiny{E}}}_{h}(\Phi_{j},\Phi_{i})-\kappa^{2}m^{\text{\tiny{E}}}_{h}(\Phi_{j},\Phi_{i})\right]-\sum\limits_{j\in\mathcal{S}^{\Gamma}}\lambda_{h}^{j}\langle\Phi_{j},\Phi_{i}\rangle_{\Gamma}=\mathcal{L}_{f,h}((\Phi_{i},0)).$
(5.14)
To write the matrix form of the above linear system, we introduce the
stiffness and mass matrices $\mathbb{A}$, $\mathbb{M}$ and the matrix
$\mathbb{Q}$ whose entries are respectively defined by
$\mathbb{A}_{ij}:=\sum\limits_{E\in\mathcal{T}_{h}}a^{\text{\tiny{E}}}_{h}(\Phi_{j},\Phi_{i}),\qquad\mathbb{M}_{ij}:=\sum\limits_{E\in\mathcal{T}_{h}}m^{\text{\tiny{E}}}_{h}(\Phi_{j},\Phi_{i}),\qquad\mathbb{Q}_{ij}:=\langle\Phi_{j},\Phi_{i}\rangle_{\Gamma}$
and the column vectors
$\mathbf{u}=\left[u_{h}^{j}\right]_{j\in\mathcal{S}^{I}\cup\mathcal{S}^{\Gamma}}$,
$\bm{\lambda}=\left[\lambda_{h}^{j}\right]_{j\in\mathcal{S}^{\Gamma}}$ and
$\mathbf{f}=\left[\mathcal{L}_{f,h}((\Phi_{i},0))\right]_{i\in\mathcal{S}^{I}\cup\mathcal{S}^{\Gamma}}$.
In accordance with the splitting of the set of the degrees of freedom (5.12),
we consider the block partitioned representation of the above matrices and
vectors (with obvious meaning of the notation), and we rewrite equation (5.14)
as follows:
$\displaystyle\left[\begin{array}[]{ll}\mathbb{A}^{II}-\kappa^{2}\mathbb{M}^{II}&\mathbb{A}^{I\Gamma}-\kappa^{2}\mathbb{M}^{I\Gamma}\\\
&\\\ \mathbb{A}^{\Gamma I}-\kappa^{2}\mathbb{M}^{\Gamma
I}&\mathbb{A}^{\Gamma\Gamma}-\kappa^{2}\mathbb{M}^{\Gamma\Gamma}\\\
\end{array}\right]\left[\begin{array}[]{l}\mathbf{u}^{I}\\\ \\\
\mathbf{u}^{\Gamma}\end{array}\right]-\left[\begin{array}[]{l}\mathbb{Q}^{I\Gamma}\bm{\lambda}\\\
\\\
\mathbb{Q}^{\Gamma\Gamma}\bm{\lambda}\end{array}\right]=\left[\begin{array}[]{l}\mathbf{f}^{I}\\\
\\\ \mathbf{f}^{\Gamma}\end{array}\right]$ (5.27)
noting that $\mathbb{Q}^{I\Gamma}$ is a null matrix since
$\langle\Phi_{j},\Phi_{i}\rangle_{\Gamma}=0$ for $i\in\mathcal{S}^{I}$,
$j\in\mathcal{S}^{\Gamma}$.
For what concerns the discretization of the BI-NRBC, by inserting (5.13) in
(3.2b) and testing with the functions $\Phi_{i}$, $i\in\mathcal{S}^{\Gamma}$,
we obtain
$\displaystyle\sum\limits_{j\in\mathcal{S}^{\Gamma}}$
$\displaystyle\left\\{u_{h}^{j}\left[\frac{1}{2}\int\limits_{\Gamma}\Phi_{j}(\mathbf{x})\Phi_{i}(\mathbf{x})\differential\Gamma_{\mathbf{x}}-\int\limits_{\Gamma}\left(\int\limits_{\Gamma}\frac{\partial
G}{\partial\mathbf{n}_{\mathbf{y}}}(\mathbf{x},\mathbf{y})\Phi_{j}(\mathbf{y})\,\differential\Gamma_{\mathbf{y}}\right)\Phi_{i}(\mathbf{x})\differential\Gamma_{\mathbf{x}}\right]\right.$
(5.28)
$\displaystyle\,\,\left.+\lambda_{h}^{j}\int\limits_{\Gamma}\left(\int\limits_{\Gamma}G(\mathbf{x},\mathbf{y})\Phi_{j}(\mathbf{y})\,\differential\Gamma_{\mathbf{y}}\right)\Phi_{i}(\mathbf{x})\differential\Gamma_{\mathbf{x}}\right\\}=0.$
To detail the computation of the integrals in (5.28), we start by splitting
the integral on the whole $\Gamma$ into the sum of the contributions
associated to each boundary edge $\Gamma_{{}_{\ell}}$,
$\ell=1,\cdots,N^{\Gamma}$
$\displaystyle\sum\limits_{j\in\mathcal{S}^{\Gamma}}$
$\displaystyle\left\\{u_{h}^{j}\left[\frac{1}{2}\sum_{\ell=1}^{N^{\Gamma}}\int\limits_{\Gamma_{{}_{\ell}}}\Phi_{j}(\mathbf{x})\Phi_{i}(\mathbf{x})\differential\Gamma_{\mathbf{x}}-\sum_{\ell=1}^{N^{\Gamma}}\int\limits_{\Gamma_{\ell}}\left(\sum_{r=1}^{N^{\Gamma}}\int\limits_{\Gamma_{r}}\frac{\partial
G}{\partial\mathbf{n}_{\mathbf{y}}}(\mathbf{x},\mathbf{y})\Phi_{j}(\mathbf{y})\,\differential\Gamma_{\mathbf{y}}\right)\Phi_{i}(\mathbf{x})\differential\Gamma_{\mathbf{x}}\right]\right.$
(5.29)
$\displaystyle\,\,\left.+\lambda_{h}^{j}\sum_{\ell=1}^{N^{\Gamma}}\int\limits_{\Gamma_{\ell}}\left(\sum_{r=1}^{N^{\Gamma}}\int\limits_{\Gamma_{r}}G(\mathbf{x},\mathbf{y})\Phi_{j}(\mathbf{y})\,\differential\Gamma_{\mathbf{y}}\right)\Phi_{i}(\mathbf{x})\differential\Gamma_{\mathbf{x}}\right\\}=0.$
Then, denoting by $E_{\ell}$, $\ell=1,\cdots,N^{\Gamma}$, the mesh element of
$\mathcal{T}_{h}$ that has one of its curved edges on $\Gamma$ and by
$\gamma_{E_{\ell}}:I_{E_{\ell}}\rightarrow\Gamma_{\ell}$ the associated
pameterization, we rewrite (5.29) by introducing $\gamma_{E_{\ell}}$ and hence
by reducing the integration over $\Gamma_{\ell}$ to that over the parametric
interval $I_{E_{\ell}}$:
$\displaystyle\sum\limits_{j\in\mathcal{S}^{\Gamma}}$
$\displaystyle\left\\{u_{h}^{j}\left[\frac{1}{2}\sum_{\ell=1}^{N^{\Gamma}}\int\limits_{I_{E_{\ell}}}\Phi_{j}\Big{(}\gamma_{E_{\ell}}(\vartheta)\Big{)}\Phi_{i}\Big{(}\gamma_{E_{\ell}}(\vartheta)\Big{)}\Big{|}\gamma^{\prime}_{E_{\ell}}(\vartheta)\Big{|}\differential\vartheta\right.\right.$
$\displaystyle\,\,\left.\left.-\sum_{\ell=1}^{N^{\Gamma}}\int\limits_{I_{E_{\ell}}}\left(\sum_{r=1}^{N^{\Gamma}}\int\limits_{I_{E_{r}}}\frac{\partial
G}{\partial\mathbf{n}_{\mathbf{y}}}\Big{(}\gamma_{E_{\ell}}(\vartheta),\gamma_{E_{r}}(\sigma)\Big{)}\Phi_{j}\Big{(}\gamma_{E_{r}}(\sigma)\Big{)}\,\Big{|}\gamma^{\prime}_{E_{r}}(\sigma)\Big{|}\differential\sigma\right)\Phi_{i}\Big{(}\gamma_{E_{\ell}}(\vartheta)\Big{)}\Big{|}\gamma^{\prime}_{E_{\ell}}(\vartheta)\Big{|}\differential\vartheta\right]\right.$
$\displaystyle\,\,\left.+\lambda_{h}^{j}\sum_{\ell=1}^{N^{\Gamma}}\int\limits_{I_{E_{\ell}}}\left(\sum_{r=1}^{N^{\Gamma}}\int\limits_{I_{E_{r}}}G\Big{(}\gamma_{E_{\ell}}(\vartheta),\gamma_{E_{r}}(\sigma)\Big{)}\Phi_{j}\Big{(}\gamma_{E_{r}}(\sigma)\Big{)}\,\Big{|}\gamma^{\prime}_{E_{r}}(\sigma)\Big{|}\differential\sigma\right)\Phi_{i}\Big{(}\gamma_{E_{\ell}}(\vartheta)\Big{)}\Big{|}\gamma^{\prime}_{E_{\ell}}(\vartheta)\Big{|}\differential\vartheta\right\\}=0.$
Finally, introducing the BEM matrices $\mathbb{V}$ and $\mathbb{K}$ whose
entries, for $i,j\in\mathcal{S}^{\Gamma}$, are respectively
$\displaystyle\mathbb{V}_{ij}$
$\displaystyle:=\sum_{\ell=1}^{N^{\Gamma}}\int\limits_{I_{E_{\ell}}}\left(\sum_{r=1}^{N^{\Gamma}}\int\limits_{I_{E_{r}}}G\Big{(}\gamma_{E_{\ell}}(\vartheta),\gamma_{E_{r}}(\sigma)\Big{)}\Phi_{j}\Big{(}\gamma_{E_{r}}(\sigma)\Big{)}\,\Big{|}\gamma^{\prime}_{E_{r}}(\sigma)\Big{|}\differential\sigma\right)\Phi_{i}\Big{(}\gamma_{E_{\ell}}(\vartheta)\Big{)}\Big{|}\gamma^{\prime}_{E_{\ell}}(\vartheta)\Big{|}\differential\vartheta$
$\displaystyle\mathbb{K}_{ij}$
$\displaystyle:=\sum_{\ell=1}^{N^{\Gamma}}\int\limits_{I_{E_{\ell}}}\left(\sum_{r=1}^{N^{\Gamma}}\int\limits_{I_{E_{r}}}\frac{\partial
G}{\partial\mathbf{n}_{\mathbf{y}}}\Big{(}\gamma_{E_{\ell}}(\vartheta),\gamma_{E_{r}}(\sigma)\Big{)}\Phi_{j}\Big{(}\gamma_{E_{r}}(\sigma)\Big{)}\,\Big{|}\gamma^{\prime}_{E_{r}}(\sigma)\Big{|}\differential\sigma\right)\Phi_{i}\Big{(}\gamma_{E_{\ell}}(\vartheta)\Big{)}\Big{|}\gamma^{\prime}_{E_{\ell}}(\vartheta)\Big{|}\differential\vartheta,$
the matrix form of the BI-NRBC reads
$\left(\frac{1}{2}\mathbb{Q}^{\Gamma\Gamma}-\mathbb{K}\right)\mathbf{u}^{\Gamma}+\mathbb{V}\bm{\lambda}=\mathbf{0}.$
(5.30)
By combining (5.27) with (5.30) we obtain the final linear system
$\displaystyle\left[\begin{array}[]{ccc}\mathbb{A}^{II}-\kappa^{2}\mathbb{M}^{II}&\mathbb{A}^{I\Gamma}-\kappa^{2}\mathbb{M}^{I\Gamma}&\mathbb{O}\\\
&\\\ \mathbb{A}^{\Gamma I}-\kappa^{2}\mathbb{M}^{\Gamma
I}&\mathbb{A}^{\Gamma\Gamma}-\kappa^{2}\mathbb{M}^{\Gamma\Gamma}&-\mathbb{Q}^{\Gamma\Gamma}\\\
&\\\
\mathbb{O}&\frac{1}{2}\mathbb{Q}^{\Gamma\Gamma}-\mathbb{K}&\mathbb{V}\end{array}\right]\left[\begin{array}[]{l}\mathbf{u}^{I}\\\
\\\ \mathbf{u}^{\Gamma}\\\ \\\
\bm{\lambda}\end{array}\right]=\left[\begin{array}[]{c}\mathbf{f}^{I}\\\ \\\
\mathbf{f}^{\Gamma}\\\ \\\ \mathbf{0}\end{array}\right]$ (5.46)
which represents the matrix form of the coupling of equations (5.14) and
(5.29).
## 6 Numerical results
In this section, we present some numerical test to validate the theoretical
results and to show the effectiveness of the proposed method. To this aim,
some preliminary features are addressed.
For the generation of the partitioning $\mathcal{T}_{h}$ of the computational
domain $\Omega$, we have used the GMSH software (see [26]). In particular, we
have built unstructured conforming meshes consisting of quadrilaterals, by
employing the Mesh.ElementOrder option within the GMSH code. If an element $E$
has a (straight) edge bordering with a curvilinear part of $\partial\Omega$,
we transform it into a curved boundary edge by means of a suitable
parametrization. In order to develop a convergence analysis, we start by
considering the coarse mesh associated to the level of refinement zero (lev.
0) and all the successive refinements are obtained by halving each side of its
elements.
To test our numerical approach, the order $k$ of the approximation spaces is
chosen equal to 1 (linear) and 2 (quadratic) for both spaces $Q_{h}^{k}$ and
$X_{h}^{k}$. Moreover, recalling that the approximate solution $u_{h}$ is not
known inside the polygons, as suggested in [8] we compute the $H^{1}$-seminorm
and $L^{2}$-norm errors, and the corresponding Estimated Order of Convergence
(EOC), by means of the following formulas:
* •
$H^{1}$-seminorm error
$\varepsilon^{\nabla,k}_{\text{lev}}:=\sqrt{\frac{\sum\limits_{E\in\mathcal{T}_{h}}\left|u-\Pi_{k}^{\nabla,E}u_{h}\right|^{2}_{H^{1}(E)}}{\sum\limits_{E\in\mathcal{T}_{h}}\left|u\right|^{2}_{H^{1}(E)}}}$
and
$\text{EOC}:=\log_{2}\left(\frac{\varepsilon^{\nabla,k}_{\text{lev}+1}}{\varepsilon^{\nabla,k}_{\text{lev}}}\right)$;
* •
$L^{2}$-norm error
$\varepsilon^{0,k}_{\text{lev}}:=\sqrt{\frac{\sum\limits_{E\in\mathcal{T}_{h}}\left\|u-\Pi_{k}^{0,E}u_{h}\right\|^{2}_{L^{2}(E)}}{\sum\limits_{E\in\mathcal{T}_{h}}\left\|u\right\|^{2}_{L^{2}(E)}}}$
and
$\text{EOC}:=\log_{2}\left(\frac{\varepsilon^{0,k}_{\text{lev}+1}}{\varepsilon^{0,k}_{\text{lev}}}\right)$.
In the above formulas the superscript $k=1,2$ refers to the linear or
quadratic order approximation of $u$, the subscript lev refers to the
refinement level and, we recall, $\Pi_{k}^{\nabla,E}$ and $\Pi_{k}^{0,E}$ are
the local $H^{1}$ and $L^{2}$-projector defined in (4.1) and (4.2),
respectively.
All the numerical computations have been performed on a cluster with two
Intel® XEON® E5-2683v4 CPUs (2.1 GHz clock frequency and 16 cores) by means of
parallel Matlab® codes.
### 6.1 On the computation of the integrals involved in the proposed approach
We start by describing the quadrature techniques adopted for the computation
of the integrals appearing in the local bilinear forms $a_{h}^{E}$ and
$m_{h}^{E}$ in (5.6) and (5.7). To this aim we point out that, if the element
$E$ is a polygon with straight edges, the choice of $\mathcal{M}_{k}(E)$
defined by (5.2) allows for an exact (up to the machine precision) and easy
computation of the first term in the right hand side of (5.6) and of (5.7)
(for the details we refer to formulas (27)–(30) in [23]). On the contrary, if
the element $E$ is a polygon with a curved edge, the corresponding integrals
can be computed by applying the $n$-point Gauss-Lobatto quadrature rule. This
latter, contrarily to the former, is affected by a quadrature error if the
involved parametrization is not of polynomial type. However, in our test, the
choice $n=8$ has guaranteed the optimal convergence order of the global
scheme.
For what concerns the evaluation of the $H^{1}$-seminorm and $L^{2}$-norm
errors, to compute the associated integrals over polygons we have used the
$n$-point quadrature formulas proposed in [38] and [39], which are exact for
polynomials of degree at most $2n$. For curved polygons, we have applied the
generalization of these formulas suggested in [38] (see Remark 1). In this
case too, we have chosen $n=8$.
Finally, for what concerns the computation of the integrals defining the BEM
matrix elements, since in the theoretical analysis we have assumed that the
boundary integral operators are not approximated, it is crucial to compute
them with a high accuracy. We recall that the numerical integration
difficulties spring from the asymptotic behaviour of the Hankel function
$H_{0}^{(1)}(r)$ near the origin (see (2.6a)), the latter being the kernel of
the single layer operator $\text{V}_{\kappa}$. To compute the corresponding
integrals with high accuracy by a small number of nodes, we have used the very
simple and efficient polynomial smoothing technique proposed in [33], referred
as the _q-smoothing_ technique. It is worth noting that such technique is
applied only when the distance $r$ approaches to zero. This case corresponds
to the matrix entries belonging to the main diagonal and to those co-
diagonals, for which the supports of the basis functions overlap or are
contiguous. After having introduced the _q-smoothing_ transformation, with
$q=3$, we have applied the $n$-point Gauss-Legendre quadrature rule with $n=9$
for the outer integrals, and $n=8$ for the inner ones (see [24] and Remark 3
in [23] for further details). For the computation of all the other integrals,
we have applied a $9\times 8$-point Gauss-Legendre product quadrature rule.
Incidentally, we point out that the integrals involving the Bessel function
$H^{(1)}_{1}(r)$, appearing in the kernel of the double layer operator
$\text{K}_{\kappa}$ (see (2.6b)), do not require a particular quadrature
strategy, since its singularity $1/r$ is factored out by the same behaviour of
the Jacobian near the origin. Hence, for the computation of the entries of the
matrix $\mathbb{K}$, we have directly applied a $9\times 8$-point Gauss-
Legendre product quadrature rule.
The above described quadrature strategy has guaranteed the computation of all
the mentioned integrals with a full precision accuracy (16-digit double
precision arithmetic) for both $k=1$ and $k=2$.
### 6.2 Definition of the test problem
We consider Problem (2.1) with source $f=0$ and Dirichlet condition
$g(\mathbf{x})=\frac{\imath}{4}H_{0}^{(1)}\left(\kappa|\mathbf{x}-\mathbf{x}_{0}|\right)\quad\text{with}\quad\mathbf{x}_{0}=(0,0),\
\mathbf{x}\in\Gamma_{0}$ (6.1)
prescribed on the boundary $\Gamma_{0}$, $H_{0}^{(1)}$ being the 0-th order
Hankel function of the first kind. In this case, the exact solution
$u(\mathbf{x})$ is known and it is the field produced by the point source
$\mathbf{x}_{0}$. Its expression is given by (6.1) for every
$\mathbf{x}\in\mathbf{R}^{2}$. In the sequel, we report the numerical results
corresponding to two choices of the wave number $\kappa=1$ and $\kappa=10$.
It is worth noting that the system (5.46) is associated to the discretization
of the model problem considered for the theoretical analysis in which, we
recall, the Dirichlet boundary condition is null. In the forthcoming examples,
dealing with null source $f$ and non vanishing Dirichlet datum $g$, the right
hand side term of (5.46) involves the function $g$ instead of $f$ (see, for
example, [23] for details on the corresponding algebraic linear system).
In Example 1 we test the CVEM-BEM approach for the choice of a computational
domain having both interior and artificial curved boundaries, for which the
theoretical analysis has been given. As we will show, the use of the CVEM
reveals to be crucial to retrieve the optimal convergence order of the global
scheme. Indeed, the standard VEM approach, defined on the polygon that
approximates the curved computational domain, allows retrieving only a sub-
optimal convergence order (see Figure 5).
However, even if we have not provided theoretical results for the CVEM-BEM
coupling in the case of a non sufficiently smooth artificial boundary
$\Gamma$, in Example 2 we show that the proposed method allows to obtain the
optimal convergence order also when a polygonal $\Gamma$ is considered.
### 6.3 Example 1. Computational domain with curved boundaries
Let us consider the unbounded region $\Omega_{\text{e}}$, external to the
unitary disk $\Omega_{0}:=\\{\mathbf{x}=(x_{1},x_{2})\in\mathbf{R}^{2}\ :\
x_{1}^{2}+x_{2}^{2}\leq 1\\}$. The artificial boundary $\Gamma$ is the
circumference of radius 2,
$\Gamma=\\{\mathbf{x}=(x_{1},x_{2})\in\mathbf{R}^{2}\ :\
x_{1}^{2}+x_{2}^{2}=4\\}$, so that the finite computational domain $\Omega$ is
the annulus bounded internally by $\Gamma_{0}=\partial\Omega_{0}$ and
externally by $\Gamma$.
In Table 1, we report the total number of the degrees of freedom associated to
the CVEM space, corresponding to each decomposition level of the computational
domain, and the approximation orders $k=1,2$. To give an idea of the
curvilinear mesh used, in Figure 2 we plot the meshes corresponding to level 0
(left plot) and level 2 (right plot). We remark that the maximum level of
refinement we have considered is lev. 7 for $k=1$, whose number of degrees of
freedom coincides with that of lev. 6 for $k=2$. Therefore, in the following
tables the symbol $\times$ means that the corresponding simulation has not
been performed.
Figure 2: Meshes of $\Omega$ for lev. 0 (left plot) and lev. 2 (right plot). | lev. 0 | lev. 1 | lev. 2 | lev. 3 | lev. 4 | lev. 5 | lev. 6 | lev. 7
---|---|---|---|---|---|---|---|---
$k=1$ | $104$ | $368$ | $1,376$ | $5,312$ | $20,864$ | $82,688$ | $329,216$ | $1,313,792$
$k=2$ | $368$ | $1,376$ | $5,312$ | $20,864$ | $82,688$ | $329,216$ | $1,313,792$ | $\times$
Table 1: Example 1. Total number of degrees of freedom for $k=1,2$ and for
different levels of refinement.
In Figures 3 and 4, we show the real and imaginary parts of the numerical
solution for the wave numbers $\kappa=1$ and $\kappa=10$ respectively,
obtained by the quadratic approximation associated to the minimum refinement
level for which the graphical behaviour is accurate and not wavy; this latter
is lev. 3 for Figure 3 and lev. 5 for Figure 4.
Figure 3: Example 1. Real (left plot) and imaginary (right plot) part of the
numerical solution for $\kappa=1$, lev. 3 and $k=2$. Figure 4: Example 1. Real
(left plot) and imaginary (right plot) part of the numerical solution for
$\kappa=10$, lev. 5 and $k=2$.
In Tables 2 and 3, we report the errors $\varepsilon^{\nabla,k}_{\text{lev}}$
and $\varepsilon^{0,k}_{\text{lev}}$ and the corresponding EOC. As we can see,
for both $\kappa=1$ and $\kappa=10$, the $H^{1}$-seminorm error confirms the
convergence order $k$ of the method. Although we did not provide the
$L^{2}$-norm error estimate, the reported numerical results show the expected
convergence order $k+1$.
| $L^{2}$-norm | $H^{1}$-seminorm
---|---|---
lev. | $h$ | $\varepsilon^{0,1}_{\text{lev}}$ | EOC | $\varepsilon^{0,2}_{\text{lev}}$ | EOC | $\varepsilon^{\nabla,1}_{\text{lev}}$ | EOC | $\varepsilon^{\nabla,2}_{\text{lev}}$ | EOC
$0$ | $8.02e-01$ | $1.64e-02$ | | $5.83e-04$ | | $5.22e-02$ | | $6.07e-03$ |
| | | $1.9$ | | $3.0$ | | $1.0$ | | $2.0$
$1$ | $4.28e-01$ | $4.52e-03$ | | $7.23e-05$ | | $2.59e-02$ | | $1.54e-03$ |
| | | $1.9$ | | $3.0$ | | $1.0$ | | $2.0$
$2$ | $2.22e-01$ | $1.18e-03$ | | $9.00e-06$ | | $1.29e-02$ | | $3.88e-04$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$3$ | $1.13e-01$ | $3.00e-04$ | | $1.12e-06$ | | $6.44e-03$ | | $9.72e-05$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$4$ | $5.68e-02$ | $7.56e-05$ | | $1.40e-07$ | | $3.22e-03$ | | $2.42e-05$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$5$ | $2.85e-02$ | $1.90e-05$ | | $1.75e-08$ | | $1.61e-03$ | | $6.07e-06$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$6$ | $1.43e-02$ | $4.75e-06$ | | $2.20e-09$ | | $8.04e-04$ | | $1.52e-06$ |
| | | $2.0$ | | $\times$ | | $1.0$ | | $\times$
$7$ | $7.14e-03$ | $1.19e-06$ | | $\times$ | | $4.02e-04$ | | $\times$ |
Table 2: Example 1. $L^{2}$-norm and $H^{1}$-seminorm relative errors and
corresponding EOC, for $\kappa=1$ and $k=1,2$.
| $L^{2}$-norm | $H^{1}$-seminorm
---|---|---
lev. | $h$ | $\varepsilon^{0,1}_{\text{lev}}$ | EOC | $\varepsilon^{0,2}_{\text{lev}}$ | EOC | $\varepsilon^{\nabla,1}_{\text{lev}}$ | EOC | $\varepsilon^{\nabla,2}_{\text{lev}}$ | EOC
$0$ | $8.02e-01$ | $6.03e-01$ | | $2.57e-01$ | | $5.77e-01$ | | $3.07e-01$ |
| | | $0.8$ | | $2.7$ | | $0.6$ | | $1.8$
$1$ | $4.28e-01$ | $3.52e-01$ | | $4.00e-02$ | | $3.92e-01$ | | $8.59e-02$ |
| | | $1.5$ | | $3.2$ | | $1.1$ | | $2.0$
$2$ | $2.22e-01$ | $1.33e-01$ | | $4.37e-03$ | | $1.84e-01$ | | $2.18e-02$ |
| | | $1.8$ | | $3.2$ | | $1.2$ | | $2.0$
$3$ | $1.13e-01$ | $3.76e-02$ | | $4.71e-04$ | | $7.88e-02$ | | $5.49e-03$ |
| | | $1.9$ | | $3.1$ | | $1.1$ | | $2.0$
$4$ | $5.68e-02$ | $9.74e-03$ | | $5.51e-05$ | | $3.65e-02$ | | $1.38e-03$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$5$ | $2.85e-02$ | $2.46e-03$ | | $6.75e-06$ | | $1.78e-02$ | | $3.44e-04$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$6$ | $1.43e-02$ | $6.16e-04$ | | $8.39e-07$ | | $8.86e-03$ | | $8.61e-05$ |
| | | $2.0$ | | $\times$ | | $1.0$ | | $\times$
$7$ | $7.14e-03$ | $1.54e-04$ | | $\times$ | | $4.42e-03$ | | $\times$ |
Table 3: Example 1. $L^{2}$-norm and $H^{1}$-seminorm relative errors and
corresponding EOC, for $\kappa=10$ and $k=1,2$.
For completeness, to highlight the importance of the use of CVEM for curved
domains, in Figure 5 we report the behaviour of the $H^{1}$-seminorm and
$L^{2}$-norm errors, obtained by applying the standard VEM defined on
polygonal approximations of the computational domain. As expected, the optimal
rate of convergence is confirmed for the $H^{1}$-seminorm, while the
approximation of the geometry affects the $L^{2}$-norm error only in the case
$k=2$.
Figure 5: Example 1. $H^{1}$-seminorm (left plot) and $L^{2}$-norm (right
plot) error for a sequence of “straight” meshes, for wave number $\kappa=1$
### 6.4 Example 2. Computational domain with piece-wise linear boundaries
Let now $\Omega_{0}:=[-1,1]^{2}$ and $\Gamma$ the contour of the square
$[-2,2]^{2}$ (see Figure 6). Since the domain of interest is a polygon, we can
apply the standard (non curvilinear) VEM on polygonal meshes without
introducing an approximation of the geometry. In Table 4, we report the total
number of degrees of freedom of the VEM space, associated to each
decomposition level of the computational domain, for $k=1,2$. In Figure 6, we
plot the meshes corresponding to lev. 0 (left plot) and lev. 3 (right plot).
We remark that the maximum level of refinement we have considered is lev. 7
for k = 1, whose number of degrees of freedom coincides with that of lev. 6
for k = 2.
Figure 6: Example 2. Meshes of $\Omega$ for lev. 0 (left plot) and lev. 3 (right plot). | lev. 0 | lev. 1 | lev. 2 | lev. 3 | lev. 4 | lev. 5 | lev. 6 | lev. 7
---|---|---|---|---|---|---|---|---
$k=1$ | $120$ | $432$ | $1,632$ | $6,336$ | $24,960$ | $99,072$ | $394,752$ | $1,575,940$
$k=2$ | $432$ | $1,632$ | $6,336$ | $24,960$ | $99,072$ | $394,752$ | $1,575,940$ | $-$
Table 4: Example 2. Total number of degrees of freedom for $k=1,2$ and for
different levels of refinement.
In Figures 7 and 8, we show the real and imaginary parts of the numerical
solution for the wave numbers $\kappa=1$ and $\kappa=10$ respectively,
obtained by the quadratic approximation associated to the minimum refinement
level for which the graphical behaviour is accurate and not wavy; this latter
is lev. 3 for Figure 7 and lev. 5 for Figure 8.
Figure 7: Example 1. Real (left plot) and imaginary (right plot) part of the
numerical solution for $\kappa=1$, lev. 3 and $k=2$. Figure 8: Example 1. Real
(left plot) and imaginary (right plot) part of the numerical solution for
$\kappa=10$, lev. 5 and $k=2$.
As we can see from Tables 5 and 6, the expected convergence order of the VEM-
BEM approach for both $H^{1}$-seminorm and $L^{2}$-norm errors are confirmed,
even if the assumption on the regularity of the artificial boundary $\Gamma$,
required by the theory, is not satisfied.
| $L^{2}$-norm | $H^{1}$-seminorm
---|---|---
lev. | $h$ | $\varepsilon^{0,1}_{\text{lev}}$ | EOC | $\varepsilon^{0,2}_{\text{lev}}$ | EOC | $\varepsilon^{\nabla,1}_{\text{lev}}$ | EOC | $\varepsilon^{\nabla,2}_{\text{lev}}$ | EOC
$0$ | $7.60e-01$ | $1.71e-02$ | | $8.34e-04$ | | $1.57e-01$ | | $1.66e-02$ |
| | | $2.0$ | | $3.0$ | | $1.1$ | | $2.0$
$1$ | $3.85e-01$ | $4.37e-03$ | | $1.01e-04$ | | $7.57e-02$ | | $4.07e-03$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$2$ | $1.94e-01$ | $1.10e-03$ | | $1.26e-05$ | | $3.78e-02$ | | $1.02e-03$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$3$ | $9.73e-02$ | $2.74e-04$ | | $1.57e-06$ | | $1.89e-02$ | | $2.56e-04$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$4$ | $4.87e-02$ | $6.86e-05$ | | $1.96e-07$ | | $9.46e-03$ | | $6.40e-05$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$5$ | $2.44e-02$ | $1.71e-05$ | | $2.46e-08$ | | $4.73e-03$ | | $1.60e-05$ |
| | | $2.0$ | | $2.9$ | | $1.0$ | | $2.0$
$6$ | $1.22e-02$ | $4.29e-06$ | | $3.35e-09$ | | $2.36e-03$ | | $4.03e-06$ |
| | | $2.0$ | | $\times$ | | $1.0$ | | $\times$
$7$ | $6.10e-03$ | $1.07e-06$ | | $\times$ | | $1.18e-03$ | | $\times$ |
Table 5: Example 2. $L^{2}$-norm and $H^{1}$-seminorm relative errors and
corresponding EOC, for $\kappa=1$ and $k=1,2$.
| $L^{2}$-norm | $H^{1}$-seminorm
---|---|---
lev. | $h$ | $\varepsilon^{0,1}_{\text{lev}}$ | EOC | $\varepsilon^{0,2}_{\text{lev}}$ | EOC | $\varepsilon^{\nabla,1}_{\text{lev}}$ | EOC | $\varepsilon^{\nabla,2}_{\text{lev}}$ | EOC
$0$ | $7.60e-01$ | $1.02e-00$ | | $4.21e-01$ | | $1.05e-00$ | | $5.54e-01$ |
| | | $1.0$ | | $3.7$ | | $0.7$ | | $1.7$
$1$ | $3.85e-01$ | $5.22e-01$ | | $3.25e-02$ | | $6.43e-01$ | | $1.25e-01$ |
| | | $1.7$ | | $3.1$ | | $1.2$ | | $2.0$
$2$ | $1.94e-01$ | $1.60e-01$ | | $3.78e-03$ | | $2.77e-01$ | | $3.24e-02$ |
| | | $1.9$ | | $3.1$ | | $1.2$ | | $2.0$
$3$ | $9.73e-02$ | $4.22e-02$ | | $4.55e-04$ | | $1.23e-01$ | | $8.16e-03$ |
| | | $2.0$ | | $3.0$ | | $1.1$ | | $2.0$
$4$ | $4.87e-02$ | $1.07e-02$ | | $5.62e-05$ | | $5.92e-02$ | | $2.04e-03$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$5$ | $2.44e-02$ | $2.67e-03$ | | $7.01e-06$ | | $2.93e-02$ | | $5.11e-04$ |
| | | $2.0$ | | $3.0$ | | $1.0$ | | $2.0$
$6$ | $1.22e-02$ | $6.68e-04$ | | $8.85e-07$ | | $1.46e-02$ | | $1.28e-04$ |
| | | $2.0$ | | $\times$ | | $1.0$ | | $\times$
$7$ | $6.10e-03$ | $1.67e-04$ | | $\times$ | | $7.29e-03$ | | $\times$ |
Table 6: Example 2. $L^{2}$-norm and $H^{1}$-seminorm relative errors and
corresponding EOC, for $\kappa=10$ and $k=1,2$.
## 7 Conclusions and perspectives
In this paper we have proposed a novel numerical approach for the solution of
2D Helmholtz problems defined in unbounded regions, external to bounded
obstacles. It consists in reducing the unbounded domain to a finite
computational one and in the coupling of the CVEM with the one equation BEM,
by means of the Galerkin approach. We have carried out the theoretical
analysis of the method in a quite abstract framework, providing an optimal
error estimate in the energy norm, under assumptions that can, in principle,
include a variety of discretization spaces wider than those we have
considered, for both the BIE and the interior PDE. While the VEM/CVEM has been
extensively and successfully applied to interior problems, its application to
exterior problems is still at an early stage and, to the best of the authors’
knowledge, the CVEM approach has been applied in this paper for the first time
to solve exterior frequency-domain wave propagation problems in the Galerkin
context.
We remark that the above mentioned coupling has been proposed in a conforming
approach context, so that the order of the CVEM and BEM approximation spaces
have been chosen with the same polynomial degree of accuracy, and the grid
used for the BEM discretization is the one inherited by the interior CVEM
decomposition.
It is worth noting that it is possible in principle to decouple the CVEM and
the BEM discretization, both in terms of degree of accuracy and of non-
matching grids. This would lead to a non-conforming coupling approach by
using, for example, a mortar type technique (see for instance [11]). Such an
approach would offer the further advantage of coupling different types of
approximation spaces and of using fast techniques for the discretization of
the BEM (see for example the very recent papers [17, 12, 21, 22]). This will
be the subject of a future investigation.
## Acknowledgments
This research benefits from the HPC (High Performance Computing) facility of
the University of Parma, Italy.
## Funding
This work was performed as part of the GNCS-IDAM 2020 research program
_“Metodologie innovative per problemi di propagazione di onde in domini
illimitati: aspetti teorici e computazionali”_. The second and the third
author were partially supported by MIUR grant _“Dipartimenti di Eccellenza
2018-2022”_ , CUP E11G18000350001.
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|
# Remark on using quantum states prepared by the adiabatic quantum
computation
Kazuto Oshima
National Institute of Technology, Gunma College,Maebashi 371-8530, Japan
E-mail<EMAIL_ADDRESS>
###### Abstract
We indicate that there are points to keep in mind in utilizing quantum states
prepared by the adiabatic quantum computation. Even if an instantaneous
expectation value of a physical quantity for the adiabatically prepared
quantum state is close to an expectation value for the true vacuum, this does
not assure us that the prepared vacuum is close to the true vacuum. In general
time average of the expectation value tend to systematically differ from the
true value. Using a simple model we discuss how to diminish this systematic
difference.
PACS numbers:03.65.-w, 03.67.-a
Recently, several quantum systems are analyzed by quantum computers or quantum
simulators with not so many qubits[1]. The quantum annealing[2] gives a
fundamental principle of the D-wave. The adiabatic quantum computation,
advocated by Farhi et.al.[3] more than two decades ago, can also be carried
out for quantum systems with not so many number of qubits. For some quantum
field theories the adiabatic quantum computation have been used for preparing
ground states[4, 5]. After preparing ground states, it has been observed that
an expectation value of certain physical quantity varies significantly under a
constant Hamiltonian[4]. This oscillation originates from the deviation of the
prepared vacuum from the true vacuum. It also has been observed that a time
average of the expectation value of the physical quantity systematically
differs from the exact value computed by another method.
The purpose of this paper is to indicate that, even if an instantaneous
expectation value of a physical quantity for the approximate vacuum prepared
by the adiabatic method is close to the expectation value for the true vacuum
by chance, it is inevitable in general that an expectation value of a physical
quantity for the approximate vacuum significantly oscillates in time around a
point that slightly differs from an expectation value for the true vacuum. We
also discuss how to diminish this systematic difference.
According to the adiabatic quantum computation[3], we start from a simple
Hamiltonian $H_{0}$ that has a non-generate trivial vacuum. We gradually
change the Hamiltonian in time to a target Hamiltonian $H_{T}$ that we should
analyze. The quantum adiabatic theorem[6, 7] assures us that the trivial
vacuum of the initial Hamiltonian $H_{0}$ approaches the vacuum of the target
Hamiltonian $H_{T}$ if the change of the Hamiltonian is very moderate and
there is a sufficient energy gap between the vacuum and excited states of the
time varying Hamiltonian. We simulate quantum adiabatic computation using the
quantum simulator by IBM for the simplest one-qubit case. We examine the
quantum state prepared by the quantum adiabatic computation.
First, we choose the initial Hamiltonian ${\hat{H}}_{0}=-JZ,J>0$, and the
target Hamiltonian ${\hat{H}}_{T}=-JX$. The initial ground state is
$|0\rangle$ and the desired final state is $|+\rangle$. The adiabatic
Hamiltonian ${\hat{H}}_{A}(s)$ that connects ${\hat{H}}_{0}$ and
${\hat{H}}_{T}$ is given by
${\hat{H}}_{A}(s)=(1-s){\hat{H}}_{0}+s{\hat{H}}_{T},\qquad 0\leq s\leq 1,$ (1)
where for example $s={t\over T},0\leq t\leq T$ for an adequate time period
$T$. The quantum adiabatic computation starts at the time $t=0$ and finishes
at the time $t=T$. For the ideal case, the ground state of the target
Hamiltonian ${\hat{H}}_{T}$ has been prepared at the time $t=T$. After the
time $t=T$, we observe a physical quantity under the constant Hamiltonian
${\hat{H}}_{T}$. If the true ground state $|+\rangle$ has been prepared, an
expectation value of the physical quantity is constant. In the actual
adiabatic quantum computation we have a quantum state that is slightly
different from the true vacuum $|+\rangle$. Let us represent the state we have
at the time $t=T$ as
$|\psi(t=T)\rangle=\alpha|+\rangle+\beta|-\rangle,\quad|\alpha|^{2}+|\beta|^{2}=1,$
(2)
where $|\beta|^{2}$ is supposed to be small. Under the total Hamiltonian
${\hat{H}}_{T}=-JX$, this quantum state time develops as
$|\psi(t)\rangle={\alpha}e^{iJ(t-T)}|+\rangle+{\beta}e^{-iJ(t-T)}|-\rangle,$
(3)
where we have set the Plank constant as $\hbar=1$ for simplicity. For this
state we measure the observable $Z$. The expectation value of $Z$ time
develops as
$\langle\psi(t)|Z|\psi(t)\rangle=2|\alpha\beta|\cos(2J(t-T)+\theta),$ (4)
where the angle $\theta$ is defined by
$\alpha\beta^{*}=|\alpha\beta^{*}|e^{i\theta}$. Thus at the time $t=T$ we get
the expectation value $2|\alpha\beta|\cos{\theta}$, which may be a good
approximation of the desired value $\langle+|Z|+\rangle=0$ by chance. The
deviation, however, reaches up to $2|\alpha\beta|$ in the time development. We
can obtain the precise value by time averaging
$\langle\psi(t)|Z|\psi(t)\rangle$ over a period. Fig.1(a) shows one of our
quantum simulation results. From the peak to peak value $4|\alpha\beta|$ we
compute the varance $2|\alpha\beta|^{2}$ as $0.000730$, and we get
$2|\beta|^{2}=0.000730$. By another simulation shots we observe $-X$ that
commutes with the Hamiltonian. After the time $t=T$ the expectation value of
$-X$ is almost constant(Fig.1(b)). Its average is $-0.999320$ and its variance
is $1.35\times 10^{-9}$ for one of our simulation result with $10^{6}$ shots.
The value $-0.999320$ almost agrees with the previous value
$-1+2|\beta|^{2}=-0.999270$.
Second, we examine another simple one-qubit model. We take the initial
Hamiltonian as ${\hat{H}}_{0}=-JZ,J>0$, and we take the target Hamiltonian as
${\hat{H}}_{T}=-JH$, where $H$ is the Hadamard gate. We again observe the
physical quantity $Z$. We represent the eigenstates of the target Hamiltonian
${\hat{H}}_{T}=-JH$ as $|h\pm\rangle$, where they satisfy
$H|h\pm\rangle=\pm|h\pm\rangle$. The explicit expressions of $|h\pm\rangle$
are
$|h+\rangle={1\over\sqrt{4-2\sqrt{2}}}(|0\rangle+(\sqrt{2}-1)|1\rangle),$ (5)
$|h-\rangle={1\over\sqrt{4+2\sqrt{2}}}(|0\rangle-(\sqrt{2}+1)|1\rangle),$ (6)
After the adiabatic state preparation process, the observable $Z$ time
develops as
$e^{i{\hat{H}}_{T}t}Ze^{-i{\hat{H}}_{T}t}=e^{-iJ{H}t}Ze^{iJ{H}t}={1\over\sqrt{2}}H-{1\over\sqrt{2}}Y\sin{2Jt}+{1\over
2}(Z-X)\cos{2Jt}.$ (7)
At the time $t=T$ if we have a state
$|\psi(t=T)\rangle=\alpha|h+\rangle+\beta|h-\rangle,|\alpha|^{2}+|\beta|^{2}=1$,
instead of the desired state $|h+\rangle$, we have at a time $t(\geq T)$
$\langle\psi(t)|Z|\psi(t)\rangle={1\over\sqrt{2}}(1-2|\beta|^{2})+\sqrt{2}|\alpha\beta|\cos(2J(t-T)+\theta),$
(8)
where we have again set $\alpha\beta^{*}=|\alpha\beta^{*}|e^{i\theta}$. Since
the physical quantity $Z$ does not anti-commute with the target Hamiltonian
${\hat{H}}_{T}=-JH$, the expectation value $\langle\psi(t)|Z|\psi(t)\rangle$
oscillate in time around the value ${1\over\sqrt{2}}(1-2|\beta|^{2})$ that
slightly less than the desired value ${1\over\sqrt{2}}=0.707107$.
Fig.2 shows one of our simulation results. We have used the second order
Suzuki-Trotter formula[8, 9]. In the result, a time average of
$\langle\psi(t)|Z|\psi(t)\rangle$ is 0.706690, which we have computed from the
average of the maximum value and the minimum value. We can find the value
$|\beta|^{2}$ from the variance of the values of
$\langle\psi(t)|Z|\psi(t)\rangle$ in the range $t\geq T$. In the result, the
variance $|\alpha|^{2}|\beta|^{2}$ is $0.0003222$ and we find
$|\beta|^{2}=0.0003223$. Thus the expectation value of $Z$ is slightly
improved to $0.707145$. Thus the systematic error for the expectation value of
$Z$ that is obtained from the time average has been diminished.
We have studied the quantum state preparation by the adiabatic quantum
computation. We have examined two simple 1-qubit models. For the first case,
the prepared quantum state is supposed to be a superposition of the true
vacuum and the excited state. The expectation value oscillates in time around
the expectation value of the true vacuum. This is rather special case that the
physical quantity anti-commute with the target Hamiltonian. The second model
will represent rather general case. We observe the physical quantity that does
not anti-commute with the target Hamiltonian. In this case the time average of
the expectation value differs from the expectation value for the true vacuum.
We can diminish this difference from the time behavior of the expectation
value. Although our models may be simple, our analysis would be useful to
grasp properties of adiabatically prepared quantum states for more complicated
systems, such as $(1+1)-$dimensional Schwinger model[4].
## References
* [1] E.A.Martinez, C.A. Muschik, P.Schindler, D.Nigg, A.Erhard, M.Heyl, P.Hauke, M.Dalmonte, T.Monz, P.Zoller and R.Blatt, 534, 516 (2016).
* [2] T.Kadowaki and H.Nishimori, Phys.Rev.E58, 5355(1998).
* [3] E.Farhi, J.Goldstone, S.Gutmann, J.Lapan, A.Lundgren and D.Preda, Science292, 472(2001).
* [4] B.Chakraborty, M.Honda, T.Izubuti, Y.Kikuchi and A.Tomiya, arXiv:2001.00485(hep-lat).
* [5] M.Honda, E.Itou, Y.Kikuchi, L.Nagano and T.Okuda, arXiv:2105.03276(hep-lat).
* [6] M. Born and V. A. Fock, Zeit. fur Physik A51, 165(1928),
* [7] T. Kato, J. Phys. Soc Japan. 5, 435(1950).
* [8] H.H.Trotter, Proc. Amer. Math. Soc. 10 ,545(1959).
* [9] M.Suzuki, Comm. Math. Phys. 51, 183 (1976).
Figure Captions
Fig.1(a)
Simulation result of the adiabatic state preparation for the Hamiltonian
${\hat{H}}_{T}=-JX$ by IBM qasm-simulator. We have started from the ground
state of ${\hat{H}}_{0}=-JZ$ and we have observed $Z$. We have set $J=1$, the
adiabatic time period $T=36$, and one time-step width $\delta{t}={1\over 8}$.
The number of shots is $10^{6}$. The orange line represent the theoretical
value. After the time $T$, a time average over a period precisly leads to 0.
Fig.1(b) An expectation value of $-X$. We have used another $10^{6}$ shots.
Fig.2
Simulation result of the adiabatic state preparation for the Hamiltonian
${\hat{H}}_{T}=-JH$ by IBM qasm-simulator. We have started from the ground
state of ${\hat{H}}_{0}=-JZ$ and we have observed $Z$. We have set
$J={\pi\over 4}$, the adiabatic time period $T=36$, and one time-step width
$\delta{t}={1\over 24}$. The number of shots is $10^{6}$. The orange line
represent the theoretical value. After the time $T$, a time average over a
period slightly less than the theoretical value ${1\over\sqrt{2}}$.
Fig.1(a)
Fig.1(b)
Fig.2
|
11institutetext: Insight SFI Centre for Data Analytics, Ireland
22institutetext: Dublin City University, Ireland
# Synthetic data for unsupervised polyp segmentation ††thanks: This project
has received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
765140. This publication has emanated from research supported by Science
Foundation Ireland (SFI) under Grant Number SFI/12/RC/2289_P2, co-funded by
the European Regional Development Fund.
Enric Moreu 1122 0000-0003-1336-6477 Kevin McGuinness 1122 0000-0003-1336-6477
Noel E. O’Connor 1122 0000-0002-4033-9135
###### Abstract
Deep learning has shown excellent performance in analysing medical images.
However, datasets are difficult to obtain due privacy issues, standardization
problems, and lack of annotations. We address these problems by producing
realistic synthetic images using a combination of 3D technologies and
generative adversarial networks. We use zero annotations from medical
professionals in our pipeline. Our fully unsupervised method achieves
promising results on five real polyp segmentation datasets. As a part of this
study we release Synth-Colon, an entirely synthetic dataset that includes 20
000 realistic colon images and additional details about depth and 3D geometry:
https://enric1994.github.io/synth-colon
###### Keywords:
Computer Vision Synthetic Data Polyp Segmentation Unsupervised Learning
## 1 Introduction
Colorectal cancer is one of the most commonly diagnosed cancer types. It can
be treated with an early intervention, which consists of detecting and
removing polyps in the colon. The accuracy of the procedure strongly depends
on the medical professionals experience and hand-eye coordination during the
procedure, which can last up to 60 minutes. Computer vision can provide real-
time support for doctors to ensure a reliable examination by double-checking
all the tissues during the colonoscopy.
Figure 1: Synth-Colon dataset samples include: synthetic image, annotation,
realistic image, depth map, and 3D mesh (from left to right).
The data obtained during a colonoscopy is accompanied by a set of issues that
prevent creating datasets for computer vision applications. First, there are
privacy issues because it is considered personal data that can not be used
without the consent of the patients. Second, there are a wide range of cameras
and lights used to perform colonoscopies. Every device has its own focal
length, aperture, and resolution. There are no large datasets with
standardized parameters. Finally, polyp segmentation datasets are expensive
because they depend on the annotations of qualified professionals with limited
available time.
We propose an unsupervised method to detect polyps that does not require
annotations by combining 3D rendering and a CycleGAN [23]. First, we produce
artificial colons and polyps based on a set of parameters. Annotations of the
location of the polyps are automatically generated by the 3D engine. Second,
the synthetic images are used alongside real images to train a CycleGAN. The
CycleGAN is used to make the synthetic images appear more realistic. Finally,
we train a HarDNeT-based model [3], a state-of-the-art polyp segmentation
architecture, with the realistic synthetic data and our self-generated
synthetic labels.
The contributions of this paper are as follows:
* •
To the best of our knowledge, we are the first to train a polyp segmentation
model with zero annotations from the real world.
* •
We propose a pipeline that preserves the self-generated annotations when
shifting the domain from synthetic to real.
* •
We release Synth-Colon (see Figure 1), the largest synthetic dataset for polyp
segmentation including additional data such as depth and 3D mesh.
The remainder of the paper is structured as follows: Section 2 reviews
relevant work, Section 3 explains our method, Section 4 presents the Synth-
Colon dataset, Section 5 describes our experiments, and Section 6 concludes
the paper.”
## 2 Related work
Here we briefly review some relevant works related to polyp segmentation and
synthetic data.
### 2.1 Polyp segmentation
Figure 2: Real samples from CVC-ColonDB with the corresponding annotation made
by a medical professionals indicating the location of cancerous polyps.
Early polyp segmentation was based in the texture and shape of the polyps. For
example, Hwang et al. [8] used ellipse fitting techniques based on shape.
However, some corectal polyps can be small (5mm) and are not detected by these
techniques. In addition, the texture is easily confused with other tissues in
the colon as can be seen in Figure 2.
With the rise of convolutional neural networks (CNNs) [10] the problem of the
texture and shape of the polyps was solved and the accuracy was substantially
increased. Several authors have applied deep convolutional networks to the
polyp segmentation problem. Brandao et al. [2] proposed to use a fully
convolutional neural network based on the VGG [16] architecture to identify
and segment polyps. Unfortunately, the small datasets available and the large
number of parameters make these large networks prone to overfitting. Zhou et
al. [22] used an encoder-decoder network with dense skip pathways between
layers that prevented the vanishing gradient problem of VGG networks. They
also significantly reduced the number of parameters, reducing the amount of
overfitting. More recently, Chao et al. [3] reduced the number of shortcut
connections in the network to speed-up inference time, a critical issue when
performing real-time colonoscopies in high-resolution. They focused on
reducing the memory traffic to access intermediate features, reducing the
latency. Finally, Huang et al. [7] improved the performance and inference time
by combining HarDNet [3] with a cascaded partial decoder [21] that discards
larger resolution features of shallower layers to reduce latency.
### 2.2 Synthetic data
The limitation of using large neural networks is that they often require large
amounts of annotated data. This problem is particularly acute in medical
imaging due to problems in privacy, standardization, and the lack of
professional annotators. Table 1 shows the limited size and resolution of the
datasets used to train and evaluate existing polyp segmentation models. The
lack of large datasets for polyp segmentation can be addressed by generating
synthetic data.
Thambawita et al. [18] used a generative adversarial network (GAN) to produce
new colonoscopy images and annotations. They added a fourth channel to SinGAN
[14] to generate annotations that are consistent with the colon image. They
then used style transfer to improve the realism of the textures. Their results
are excellent considering the small quantity of real images and professional
annotations that are used. Gao et al. [6] used a CycleGAN to translate
colonoscopy images to polyp masks. In their work, the generator learns how to
segment polyps by trying to fool a discriminator.
Table 1: Real polyp segmentation datasets size and resolution. Dataset | #Images | Resolution
---|---|---
CVC-T [19] | 912 | 574 x 500
CVC-ClinicDB [1] | 612 | 384 x 288
CVC-ColonDB [17] | 380 | 574 x 500
ETIS-LaribPolypDB [15] | 196 | 1225 x 966
Kvasir [9] | 1000 | Variable
Synthetic images combined with generative networks have also been widely used
in the depth prediction task [11, 12]. This task helps doctors to verify that
all the surfaces in the colon have been analyzed. Synthetic data is essential
for this task because of the difficulties to obtain depth information in a
real colonoscopy.
Unlike previous works, our method is entirely unsupervised and does not
require any human annotations. We automatically generate the annotations by
defining the structure of the colon and polyps and transferring the location
of the polyps to a 2D mask. The key difference between our approach and other
state-of-the-art is that we combine 3D rendering and generative networks.
First, the 3D engine defines the structure of the image and generates the
annotations. Second, the adversarial network makes the images realistic.
Similar unsupervised methods have also been successfully applied in other
domains like crowd counting. For example, Wang et al. [20] render crowd images
from a video game and then use a CycleGAN to increase the realism.
## 3 Method
Our approach is composed of three steps: first, we procedurally generate colon
images and annotations using a 3D engine; second, we feed a CycleGAN with
images from real colonoscopies and our synthetic images; finally, we use the
realistic images created by CycleGAN to train an image segmentation model.
### 3.1 3D colon generation
Figure 3: The structure of the colon is composed by 7 segments to simulate the
curvature of the intestinal tract.
The 3D colon and polyps are procedurally generated using Blender, a 3D engine
that can be automated via scripting.
Our 3D colons structure is a cone composed by 2454 faces. Vertices are
randomly displaced following a normal distribution in order to simulate the
tissues in the colon. Additionally, the colon structure is modified by
displacing 7 segments as in Figure 3. For the textures we used a base color
[0.80, 0.13, 0.18] (RGB). For each sample we shift the color to other tones of
brown, orange and pink. One single polyp is used on every image, which is
placed inside the colon. It can be either in the colon’s walls or in the
middle. Polyps are distorted spheres with 16384 faces. Samples with polyps
occupying less than 20,000 pixels are removed.
Lighting is composed by a white ambient light, two white dynamic lights that
project glare into the walls, and three negative lights that project black
light at the end of the colon. We found that having a dark area at the end
helps CycleGAN to understand the structure of the colon. The 3D scene must be
similar to real colon images because otherwise, the CycleGAN will not
translate properly the images to the real-world domain. Figure 4 illustrates
the images and ground truth generated by the 3D engine.
Figure 4: Synthetic colons with corresponding annotations rendered using a 3D
engine.
### 3.2 CycleGAN
A standard CycleGAN composed by two generators and two discriminators is
trained using real images from colonoscopies and synthetic images generated
using the 3D engine as depicted in Figure 6. We train a CycleGAN for 200
epochs and then we infer real images in the “Generator Synth to Real” model,
producing realistic colon images.
Figure 5 displays synthetic images before and after the CycleGAN domain
adaptation. Note that the position of the polyps is not altered. Hence, the
ground truth information generated by the 3D engine is preserved.
Figure 5: Synthetic images (first row) and realistic images generated by our
CycleGAN (second row). Figure 6: Our CycleGAN architecture. We train two
generator models that try to fool two discriminator models by changing the
domain of the images.
### 3.3 Polyp segmentation
After creating a synthetic dataset that has been adapted to the real colon
textures, we train an image segmentation model. We used the HarDNeT-MSEG [7]
model architecture because of its real-time performance and high accuracy. We
use the same hyperparameter configuration as in the original paper.
## 4 Synth-Colon
We publicly release Synth-Colon, a synthetic dataset for polyp segmentation.
It is the first dataset generated using zero annotations from medical
professionals. The dataset is composed of 20 000 images with a resolution of
500$\times$500\. Synth-Colon additionally includes realistic colon images
generated with our CycleGAN and the Kvasir training set images. Synth-Colon
can also be used for the colon depth estimation task [12] because we provide
depth and 3D information for each image. Figure 1 shows some examples from the
dataset. In summary, Synth-Colon includes:
* •
Synthetic images of the colon and one polyp.
* •
Masks indicating the location of the polyp.
* •
Realistic images of the colon and polyps. Generated using CycleGAN and the
Kvasir dataset.
* •
Depth images of the colon and polyp.
* •
3D meshes of the colon and polyp in OBJ format.
## 5 Experiments
### 5.1 Metrics
We use two common metrics for evaluation. The mean Dice score, given by:
$\mathrm{mDice}=\frac{2\times tp}{2\times tp+fp+fn},$ (1)
and the mean intersection over union (IoU):
$\mathrm{mIoU}=\frac{tp}{tp+fp+fn},$ (2)
where in both forumlae, $tp$ is the number of true positives, $fp$ the number
of false positives, and $fn$ the number of false negatives.
### 5.2 Evaluation on real polyp segmentation datasets
We evaluate our approach on five real polyp segmentation datasets. Table 2
shows the results obtained when training HarDNeT-MSEG [7] using our synthetic
data. Note that our method is not using any annotations. Results are
satisfactory considering the fact that labels have been generated
automatically. We found that training the CycleGAN with only the images from
the target dataset performs better than training the CycleGAN with all the
datasets combined, indicating a domain gap among the real-world datasets.
Table 2: Evaluation of our synthetic approach on real-world datasets. The
metrics used are mean Dice similarity index (mDice) and mean Intersection over
Union (mIoU).
| CVC-T | ColonDB | ClinicDB | ETIS | Kvasir
---|---|---|---|---|---
| mDice | mIoU | mDice | mIoU | mDice | mIoU | mDice | mIoU | mDice | mIoU
U-Net [13] | 0.710 | 0.627 | 0.512 | 0.444 | 0.823 | 0.755 | 0.398 | 0.335 | 0.818 | 0.746
SFA [5] | 0.467 | 0.329 | 0.469 | 0.347 | 0.700 | 0.607 | 0.297 | 0.217 | 0.723 | 0.611
PraNet [4] | 0.871 | 0.797 | 0.709 | 0.640 | 0.899 | 0.849 | 0.628 | 0.567 | 0.898 | 0.840
HarDNet-MSEG [7] | 0.887 | 0.821 | 0.731 | 0.660 | 0.932 | 0.882 | 0.677 | 0.613 | 0.912 | 0.857
Synth-Colon (ours) | 0.703 | 0.635 | 0.521 | 0.452 | 0.551 | 0.475 | 0.257 | 0.214 | 0.759 | 0.527
### 5.3 Study with limited real data
In this section we evaluate how our approach based on synthetic imagery and
domain adaptation compares with the fully supervised state-of-the-art HarDNeT-
MSEG network when there are fewer training examples available. We train the
CycleGAN used in the proposed approach, without ground truth segmentation
labels, on progressively larger sets of imagery, and compare this with the
supervised method trained on the same amount of labelled imagery. Table 3
shows the results of the experiment, which demonstrates that synthetic data is
extremely useful for domains where annotations are very scarce. While our
CycleGAN can produce realistic images with a small sample of only five real
images, supervised methods require many images and annotations to achieve good
performance. Table 3 shows that our unsupervised approach is useful when there
are less than 50 real images and annotations. Note that zero images here means
there is no domain adaptation via the CycleGAN.
Table 3: Evaluation of the proposed approach on the Kvasir dataset when few real images are available. The performance is measured using the mean Dice metric. | Synth-Colon (ours) | HarDNeT-MSEG [7]
---|---|---
0 images | 0.356 | -
5 images | 0.642 | 0.361
10 images | 0.681 | 0.512
25 images | 0.721 | 0.718
50 images | 0.735 | 0.781
900 (all) images | 0.759 | 0.912
## 6 Conclusions
We successfully trained a polyp segmentation model without annotations from
doctors. We used 3D rendering to generate the structure of the colon and
generative adversarial networks to make the images realistic, and demonstrated
that it can perform quite reasonably in several datasets, even outperforming
some fully supervised methods in some cases. We hope this study can help
aligning synthetic data and medical imaging in future. As future work, we will
explore how to include our synthetic annotations in the CycleGAN.
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|
§ METHOD
§.§ Wind Data
Real wind forecasts are gathered from the ECMWF's ERA5 global reanalysis dataset <cit.>. As mentioned by Bellemare et al., simplex noise <cit.> can be used to augment the wind data, thus emulating forecasting errors <cit.>. Specifically, wind forecasts between $1$st November $2022$ and $31$st January $2023$ are chosen as seasonal changes can have a big impact on the wind fields. Furthermore, wind fields within the tropics, at longitude $-113^{\circ}$ latitude $1^{\circ}$, are chosen as they are shown to contain a range of diverse winds <cit.>.
The wind-field contains wind vectors over a grid of points in the parameter space $\mathcal{X} \times \mathcal{Y} \times \mathcal{P} \times \mathcal{T}\times \mathcal{V}$, where the longitude $\mathcal{X}$ and and latitudinal $\mathcal{Y}$ positions are sampled with a resolution of $0.4^{\circ}$ at various pressure points $\mathcal{P}$ ranging from $2000$ Pa to $17500$ Pa. Furthermore, forecasts are collected every $6$ hours for the three day episode and each wind-vector has two components in the longitude and latitude direction $(v_{wx}, v_{wy})$.
§.§ High-level Soft Actor-Critic
The Soft Actor-Critic algorithm is selected since the maximization of both the value function and the policy entropy is expected to assist in exploring the wind field, preventing the controller from getting trapped in local minima.
The MDP state space ($\mathcal{S}$) consists of $50$ wind and $27$ ambient features. The wind features are collected at 25 equally-spaced points between the vertical pressure limits $[5000, 14000]$ Pa. For each point, the magnitude of the wind speed $|v|$ and bearing error $\theta$ with respect to the target is calculated, as in <cit.>. These form two individual vectors which then are concatenated $[|v_0|,...,|v_{24}|,\theta_0,...,\theta_{24}]\in\mathbb{R}^{50}$.
The ambient features consist of simulated onboard measurements. These measurements include the altitude $h_t$, ascent-rate $\dot{h}_t$, wind velocity $|v_h|$ and bearing error at the current altitude $\theta_h$, the balloon envelope drag area $A$ and volume $V$, total system mass $m_T$, distance $|x|$ and heading to the target $[\sin(\theta_x), \cos(\theta_x)]$. The past three altitudes $[h_{t-1}, h_{t-2}, h_{t-3}]$, ascent rates $[\dot{h}_{t-1}, \dot{h}_{t-2}, \dot{h}_{t-3}]$, and float actions $[a_{2, t-1}, a_{2, t-2}, a_{2, t-3}]$. Then finally the sand mass $m_s$, and helium mols $n$ are also included. Both the wind and ambient features are then concatenated together to form a feature set of 74 variables.
We utilise the distance-based reward function proposed by Bellemare et al. <cit.>, shown in Eq <ref>.
\begin{equation}
\label{eq:reward_function}
\begin{cases}
1.0, & \text{if }|x| < 50 \\
c2^{-(|x|-\rho)/\tau)}, & \text{otherwise},
\end{cases}
\end{equation}
where the cliff constant $c=0.4$ ensures there is a distinct difference in the value of the state inside the target region compared to outside.
The MDP action space ($\mathcal{A}$) consists of three actions $a\in[a_0, a_1, a_2]$. Representing the action space in this way decouples the ascent rate into desired altitude $a_0\in[14000, 21000]$ km and time-factor $a_1\in[1, 5]$, as a component of time-steps, to reach that altitude. A larger time-factor $a_1$ represents a slower ascent rate, as the agent decides lower urgency to reach the desired altitude. Decoupling the ascent rate augments the secondary function of resource conservation into the learning process. From this perspective, altitude $a_0$ relates to wind speeds propelling the balloon to the target direction. And the time-factor $a_1$ relates to the amount of resources used to reach that altitude.
Another reason for decoupling the action space in this way is to incorporate transparency into the agent's decision making. Compared to previous works on station-keeping, trajectories generated in the form of desired altitudes and time-factors are more transparent than low-level control inputs <cit.>. Further work is needed to discern policy explanations from a generated trajectory in relation to the wind-field <cit.>.
The third action $a_2\in[-1, 1]$ is the binary option to float vertically and takes precedence over the other two actions. To discretise the action, the controller chooses to float if the action is positive, $a_2>0$. In the event that action $a_2$ is selected, actions $a_0$ and $a_1$ become unnecessary and are subsequently disregarded.
An episode runs for a total of 3 days, where the balloon's initial conditions consists of the relative longitude, latitude and altitude from the target, date, time, and wind noise which are sampled uniformly from a random seed. Every six hours, wind forecasts are obtained from ECMWF's ERA5 dataset <cit.>. To determine the wind speed at a specific time and position, multidimensional interpolation is applied to gather data from the wind field. An episode ends when the agent reaches a flight time of 3 days or runs out of resources.
Both the actor and critic networks are parameterised as neural networks, with two hidden layers of size $256$. The actor network takes input $77$ which is the combination of the wind and ambient features and outputs a mean and covariance for each of the three actions, $\mu\in\mathbb{R}^{3}$, $\log\sigma\in\mathbb{R}^3$. The critic network instead takes the concatenated vector of both the state and action and calculates the state-action value $Q(s_t, a_t)$.
(52, 76)(0,0)Actor $\theta$
(54, 64)(0,0)$256$
(8, 36)(0,0)s
(32, 58)(0,0)$77$
(75.5, 62)(0,0)$3$
(75.5, 34)(0,0)$3$
(100, 20)(0,0)$\log \sigma$
(127, 50)(0,0)$\log \pi$
(95, 50)(0,0)$\mu$
(127, 21)(0,0)$\tilde{a}$
(190, 85)(0,0)Critic $\phi$
(156, 25)(0,0)$3$
(156, 72)(0,0)$77$
(185, 62)(0,0)$80$
(207, 67)(0,0)$256$
(229, 45)(0,0)$1$
(245, 38)(0,0)Q
(225, 9)(0,0)x$4$
Soft Actor-Critic <cit.> network architecture. The actor network is modelled as a Feed-forward parameterised Gaussian policy where the actions are represented as the hyperbolic tangent $\tanh$ applied to z values sampled from the mean $\mu$ and covariance given by the neural network. Whereas the critic is modeled as a soft Q-function.
§.§ Altitude Controller
The altitude controller takes the desired altitude $a_0$, time-factor $a_1$, and float condition $a_2$ from the SAC controller, and either drops or vents helium depending on the calculated desired ascent rate. Where the current altitude is denoted $h_t$ and the time between each action, stride time $\Delta t$, is set to 20 minutes.
\begin{equation}
\dot{h}_{\text{des}} = \frac{a_0 - h_t}{a_1 \Delta t}
\end{equation}
To prevent the agent from prioritizing receiving rewards by exploiting the environment <cit.>, a minimum limit for ballasting $m_{\text{s\_min}}$ and venting $n_{\text{min}}$ is set. If ascent rates are chosen that calculate masses less than these values then no action is taken. There is a loophole in this approach that could allow the agent to fly above or bellow the set altitude limits. To prevent this behaviour, the controller will float above or bellow the altitude limits unless an action is chosen that propels the balloon between the altitude limits is chosen. The full controller is shown in Eq <ref>.
\begin{equation}
\label{eq:fullcontroller}
\text{action} =
\begin{cases}\text{Float,} & \text{if } a_2 > 0 \\
\text{Do Nothing,} & \text{if } n_{\text{calc}} < n_{\text{min}} \text{ or } m_{\text{s\_calc}} < m_{\text{s\_min}}\\
\text{Ballast,} & \dot{h}_{\text{des}} > \dot{h}_t \\
\text{Vent,} & \dot{h}_{\text{des}} < \dot{h}_t \\
\end{cases}
\end{equation}
Where $n_{\text{calc}}$ and $m_{\text{s\_calc}}$ are the number of mols and sand mass calculated to vent or ballast, respectively, for the given action. The equations to calculate these are described below.
§.§ Resources used to float, vent and ballast
To calculate the desired mass to vent or ballast, we assume the balloon has reached steady state and equate Eq <ref> to 0. To vent, we solve Eq <ref> for the number of mols $n_{\text{calc}}$ of helium given the desired ascent rate $\dot{h}_{\text{des}}$ from the SAC controller. This results in a polynomial function which we can then solve for the real roots.
\begin{align}
\begin{split}
\rho g \left(\frac{RT}{P} - M\right)n_{\text{calc}}
- \frac{1}{2}\rho\left|\dot{h}\right|\dot{h}&C_d\pi\left(\frac{3RT}{4\pi P}\right)^{\frac{2}{3}}n_{\text{calc}}^\frac{2}{3} \\
& -\left(m_p + m_s\right)g=0
\end{split}
\end{align}
Similarly, to ballast we can solve for the mass of sand $m_{\text{s\_calc}}$ by rearranging Eq <ref>.
\begin{equation}
m_{\text{s\_calc}} = \rho V - \frac{1}{2g} C_d A \left|\dot{h}\right|\dot{h} - m_p - m_h
\end{equation}
When floating, the balloon has no vertical velocity, and hence drag is not acting on the balloon envelope. This simplifies Eq <ref> by setting the drag force $F_d$ equal to $0$. Furthermore, we expand the mass term $m$ to include all the components. This includes the mass of the payload (and envelope) $m_p$, sand $m_s$ and helium $nM_h$. Where $n$ and $M_h$ are the number of moles and molar mass of helium respectively.
\begin{equation}
0 = \rho V g - (m_p + m_s + nM_h)g
\end{equation}
Subsequently, we select the appropriate method to maintain equilibrium between the buoyancy force and weight of the balloon. If the buoyancy force exceeds the weight of the balloon, we opt to vent helium $n_{\text{calc}}$. Conversely, if the weight of the balloon is more dominant, we opt to ballast $m_{\text{s\_calc}}$:
\begin{equation}
\begin{cases}
n_{\text{calc}} = \frac{m_s + m_p}{\frac{\rho R T}{P}-M_h}, & \text{if } \rho gV > mg \\
m_{\text{s\_calc}} = \rho V - m_p - m_h, & \text{otherwise}\\
\end{cases}
\end{equation}
|
# FEDERATED SPATIAL REUSE OPTIMIZATION IN NEXT-GENERATION DECENTRALIZED IEEE
802.11 WLANS
Francesc Wilhelmi CTTC (Spain) Jernej Hribar CONNECT Centre, Trinity
College Dublin (Ireland) Selim F. Yilmaz Imperial College London (United
Kingdom) Emre Ozfatura Imperial College London (United Kingdom) Kerem
Ozfatura Imperial College London (United Kingdom) Ozlem Yildiz New York
University (USA) Deniz Gündüz Imperial College London (United Kingdom)
University of Modena and Reggio Emilia (Italy) Hao Chen Xiamen University
(China) Xiaoying Ye Xiamen University (China) Lizhao You Xiamen University
(China) Yulin Shao Imperial College London (United Kingdom) Paolo Dini
CTTC (Spain) Boris Bellalta Universitat Pompeu Fabra (Spain)
(NOTE: Corresponding author: Francesc Wilhelmi<EMAIL_ADDRESS>
)
###### Abstract
As wireless standards evolve, more complex functionalities are introduced to
address the increasing requirements in terms of throughput, latency, security,
and efficiency. To unleash the potential of such new features, artificial
intelligence (AI) and machine learning (ML) are currently being exploited for
deriving models and protocols from data, rather than by hand-programming. In
this paper, we explore the feasibility of applying ML in next-generation
wireless local area networks (WLANs). More specifically, we focus on the IEEE
802.11ax spatial reuse (SR) problem and predict its performance through
federated learning (FL) models. The set of FL solutions overviewed in this
work is part of the 2021 International Telecommunication Union (ITU) AI for 5G
Challenge.
Keywords – Federated learning, IEEE 802.11ax, ITU Challenge 2021, machine
learning, network simulator, spatial reuse
## 1\. Introduction
Wireless networks are evolving towards artificial intelligence (AI) / machine
learning (ML)-driven systems able to address the overwhelming requirements of
future mobile communications [1, 2], namely the fifth generation (5G), beyond
5G (B5G), and the sixth generation (6G). The application of ML for networking
can be found at different communication layers and parts of a network, e.g.,
network management to drive the self-organizing networks (SON) paradigm [3],
optimization of the medium access control (MAC) layer in decentralized channel
access [4], or AI-native physical communication protocols [5, 6]. The fact is
that AI/ML can leverage the vast amount of network and user data to generate
new knowledge that allows improving the network performance and, hence, making
progress in the development of novel network applications such as those based
on extended reality.
Nevertheless, the use of ML in communications also raises concerns of
different nature. First, ML-based solutions typically require a lot of energy
for training complex models (e.g., neural networks) and high bandwidth for
exchanging training data, which typically needs to be centralized to a single
point. Moreover, the massive usage of networking data for ML may threaten
security and users’ privacy. The privacy issue may be exacerbated in
decentralized networks such as IEEE 802.11 wireless local area networks
(WLANs), whereby the lack of a central network manager may make inter-WLAN
interactions unreliable.
To address some of the challenges posed by traditional ML training, Federated
Learning (FL) optimization was introduced in [7] as a distributed training
paradigm that allows keeping the data at its source. Since then, a significant
number of FL applications have flourished across different fields, such as
medicine [8], autonomous driving [9], UAV-based wireless networks [10]. FL has
become attractive to foster collaboration among different parties interested
in solving a common problem. Under the management of a central server
(typically, a neutral entity), FL participants contribute to building a common
ML model, by sharing model weights generated using its own local data, rather
than forwarding raw data for centralized training.
In this paper, we study the application of FL models to the IEEE 802.11
spatial reuse (SR) problem, which aims to enhance spectral efficiency by
adjusting the devices’ carrier sense area to increase the number of concurrent
transmissions in overlapping deployments. IEEE WLANs are an important part of
the B5G ecosystem as it represents a cost-effective but high-performance
solution for the access network. In particular, we overview the output of the
problem statement entitled “ITU-ML5G-PS-004: Federated Learning for Spatial
Reuse in a multi-BSS (Basic Service Set) scenario”, which was part of the 2021
International Telecommunication Union (ITU) AI for 5G Challenge [11].111The
ITU AI/ML challenge is a global competition that gathers professionals,
researchers, practitioners, and students from all around the globe to solve
relevant problems on ML for communications. The purpose of the challenge was
the exploration of federated solutions to predict the performance of IEEE
802.11ax (11ax) networks applying SR. Such a performance prediction solution
is called to be an essential part of ML-assisted networks, which overarching
goal is optimization. To address the performance prediction problem, a dataset
with simulated measurements on crowded 11ax deployments applying SR was
provided, which was used to develop the FL solutions presented in this paper.
The usage of simulated data for enriching training datasets is another
relevant topic for enabling ML in communications [12].
The main contributions of this paper are as follows:
* •
We overview the SR technology for both 11ax and future amendments and propose
the usage of FL to address it.
* •
We provide a dataset with 11ax SR measurements for next-generation WLANs. The
dataset is open and can be accessed at [13].
* •
We overview the set of FL solutions proposed by the participants of the 2021
ITU AI for 5G Challenge to predict the performance of novel IEEE 802.11ax SR
WLANs. Table 1 briefly summarizes the proposed models, as well as the main
motivation behind them.
Table 1: Summary of the ML models proposed by the participants of the
challenge.
Team | Proposed Model | Motivation | Ref.
---|---|---|---
FederationS | | DNN with two
---
parallel branches
| Exploit relationships among
---
training features
[14]
| FedIPC
---
| NN with a Multi-Output
---
Regression Objective
| Take advantage of knowledge
---
on wireless operation
[15]
| WirelessAI
---
| CNN with FCNN
---
| Exploit graph representations
---
in wireless networks
[16]
The rest of the paper is structured as follows. Section 2 introduces the SR
problem in 11ax and future WLANs. Section 3 provides some basics on FL with
special emphasis on its applications in networking. Section 4 overviews the
provided SR dataset for training ML models. The solutions proposed by the
challenge participants are described in detail in Section 5 and evaluated in
Section 6. Section 7 concludes the paper with some remarks and future
directions.
## 2\. Spatial Reuse in 802.11ax WLANs: overview and research gaps
IEEE 802.11 technology, commonly known as Wi-Fi, is one of the most popular
solutions for the access network due to its ease of deployment and low cost
(it operates on unlicensed bands). However, its fundamental operation is based
on carrier sense multiple access (CSMA), whose performance is well known to
degrade when dealing with a large number of concurrent users [17]. To address
the issues raised by network density and to meet the increasingly strict
requirements posed by next-generation applications (e.g., virtual reality),
802.11 amendments introduce novel functionalities and protocol enhancements.
For instance, standards 802.11n (2009) and 802.11ac (2013) provided high
throughput (HT) and very high throughput (VHT) devices by including, for
instance, the application channel bonding (CB), whereby basic channels could
be aggregated to increase the capacity of a single transmission.
As for the SR operation [18], it was recently introduced by the IEEE 802.11ax
(2021) standard [19] to increase the number of parallel transmissions in
overlapping basic service sets (OBSS). Among other features like orthogonal
frequency division multiple access (OFDMA), or downlink/uplink multi-user
multiple input-multiple-output (MU-MIMO), SR aims at enhancing the performance
and efficiency. To do so, it provides two different operational modes:
1. 1.
OBSS Packet Detect-based SR (OBSS/PD-based SR).
2. 2.
Parametrized Spatial Reuse (PSR).
The main difference between the two mechanisms lies in the way SR transmission
opportunities (TXOPs) are detected by devices implementing them. While
OBSS/PD-based SR operates in the downlink, PSR is designed for the uplink. In
what follows, we focus on OBSS/PD, which has gained more interest and is under
consideration for evolution in the future amendments, such as the IEEE
802.11be (11be) [20]. A comprehensive overview of these two mechanisms can be
found in [18].
In essence, OBSS/PD-based SR allows devices to transmit in parallel with
others that gained channel access beforehand. To do so, a new OBSS/PD
threshold is defined to be applied when an incoming detected transmission
marks the radio channel as busy through clear channel assessment (CCA)
operation. CCA allows overlapping devices to share a common channel and is
triggered when the preamble of a Wi-Fi transmission is identified. Provided
that the OBSS/PD threshold allows initiating a new SR transmission, a transmit
power limitation must be applied so that the generated interference does not
affect the original transmission.
(a)
(b)
Fig. 1: IEEE 802.11ax OBSS/PD-based SR operation: (a) signal reception areas,
(b) diagram of packet exchange.
The OBSS/PD SR operation is illustrated in Fig. 1 for two overlapping access
points (APs), AP${}_{\text{A}}$ and AP${}_{\text{B}}$. As shown in Fig. 1(a),
AP${}_{\text{A}}$ detects the signals from all the considered devices
(represented by the red area), including AP${}_{\text{B}}$ and station B
(STA${}_{\text{B}}$), which belong to a different BSS. In particular,
AP${}_{\text{B}}$ is inside the carrier sense area of AP${}_{\text{A}}$
(represented by the gray area), so they both must contend for the channel
whenever the other starts a transmission (e.g., to its associated STAs).
Nevertheless, thanks to the OBSS/PD-based SR operation, AP${}_{\text{A}}$ can
ignore AP${}_{\text{B}}$’s transmissions when applying the OBSS/PD threshold
(represented by the green area). At the packet level (shown in Fig. 1(b)),
AP${}_{\text{A}}$ starts decoding the preamble of a new transmission from
AP${}_{\text{B}}$, which has initially gained the access to the medium using
CSMA with collision avoidance (CSMA/CA). From the preamble reception,
AP${}_{\text{A}}$ determines that the channel is busy at the MAC layer due to
the CCA operation. But, using the SR mechanism, AP${}_{\text{A}}$ identifies
an SR TXOP because the incoming signal is below the OBSS/PD threshold. Hence,
AP${}_{\text{A}}$ can initiate a transmission before AP${}_{\text{B}}$ leaves
the channel, provided that a transmit power restriction is applied, denoted by
$\text{TX\\_PWR}_{\text{max}}$, for the sake of not affecting
AP${}_{\text{B}}$’s transmission [21]:
$\text{TX\\_PWR}_{\text{max}}=\text{TX\\_PWR}_{\text{ref}}-(\text{OBSS/PD}-\text{OBSS/PD}_{\text{min}}),$
((1))
where $\text{TX\\_PWR}_{\text{ref}}$ is the transmit power reference (set to
21 dBm or 25 dBm, depending on the device’s antenna capabilities), OBSS/PD is
the selected OBSS/PD threshold for detecting SR TXOPs, and
$\text{OBSS/PD}_{\text{min}}$ is the minimum OBSS/PD threshold (fixed to -82
dBm).
While SR promises to enhance spectral efficiency in dense OBSS deployments,
its actual performance is hindered by the proper selection of the OBSS/PD
threshold, which may not be trivial due to the complex inter-device
interactions in a WLAN. The fact is that the OBSS/PD-based SR operation is a
decentralized mechanism that only considers the interactions between principal
transmitters (i.e., devices gaining access to the channel for transmitting),
but does not account for either the interference at the recipients of such
transmissions or the impact of uplink control frames (e.g., acknowledgment
packets). Since the standard does not provide any method for selecting the
proper OBSS/PD threshold, there is an imperative need for finding effective
mechanisms to leverage the SR operation.
ML, in this context, is considered a promising tool to capture the complex
interactions among IEEE 802.11 devices applying SR. In general, ML has been
applied to a plethora of problems in IEEE 802.11 networks, including PHY
optimization (rate selection [22], resource allocation [23]), assisting
management operations (e.g., AP selection and handover [24], channel band
selection [25]), or supporting novel features like MU-MIMO or channel bonding
with enhanced monitoring, analytics, and decision-making [26, 27]. For further
details on ML application to Wi-Fi, we refer the interested reader to the
comprehensive survey in [28].
In the particular case of SR, most of the literature has so far focused on
reinforcement learning (RL) and online learning techniques, whereby agents
attempt to learn the best OBSS/PD configuration sequentially. In [29, 30], the
authors modeled the decentralized SR problem as multi-armed bandits (MAB), an
online learning framework whereby agents attempt to address the exploration-
exploitation trade-off. While [29] studied the problem by using selfish
rewards in a competitive environment, [30] considered shared rewards for the
sake of maximizing fairness. Other RL-based approaches can be found in [31]
and [32].
The online learning paradigm turns out to be a cost-effective solution to the
decentralized SR problem thanks to its ability for solving complex partial
information problems. In addition, WLANs typically experience a high
variability both in terms of devices’ mobility and activation/deactivation, so
past learned information may become easily outdated. However, as shown in
[30], online learning may have some pitfalls when applied to dense WLANs,
mainly raised by the high action-decision space, the non-stationarity of
agents’ rewards in competitive settings, or the complexity of finding a proper
shared reward that enables maximizing the overall network performance.
For those reasons, in this paper, we focus on the suitability of supervised
learning methods, mostly based on deep learning (DL), for the SR problem in
WLANs. To the best of our knowledge, this approach has not been studied before
in the context of SR. A centralized DL-based method was proposed in [33] to
jointly select the transmission power and the CCA, but not in the context of
11ax SR operation. DL was also applied in [34] to address the channel bonding
problem in dense WLANs. This and other DL solutions for the dynamic channel
bonding problem in IEEE 802.11ax WLANs were overviewed in [35].
We note that the overviewed works on DL consider centralized approaches, which
require data to be gathered at a single point for training a static model,
which is then used homogeneously across all the AI-enabled devices.
Nevertheless, in practice, some deployments (e.g., residential WLANs) may have
limitations in terms of computation, storage, or communication capabilities
(for instance, low-throughput connections, intermittent availability).
Moreover, separate WLAN deployments can be substantially different, thus
requiring specialized models (rather than general ones). To address these
limitations of centralized learning on heterogeneous deployments, we focus on
the FL paradigm, introduced in the next section.
## 3\. An Introduction to Federated Learning for Networking
The FL optimization paradigm was first introduced in [36] to address some
critical issues of traditional centralized ML mechanisms. In FL, training is
done at end devices (or clients), which do not share their training data with
others. Instead, ML model updates are provided and aggregated under the
management of a typically central server. By removing the procedures related
to data exchange, FL decreases the communication overhead and enhances user
privacy and security. Moreover, FL is an appealing solution for dealing with
heterogeneous sets of clients, thus allowing to create specialized models
according to clients’ characteristics. With that, FL has the potential to
revolutionize ML implementations, bringing them closer to practical
applications and use cases. Many examples of FL have emerged in recent years,
including, but not limited to, in medicine [37], finance [38], industry 4.0
[39], or telecommunications [40, 41].
In the telecommunications realm, novel ML solutions require handling a vast
amount of data, often highly distributed across the network. These kinds of
resource-demanding applications may threaten the stability of the network on
the one hand and may experience low performance due to the communication
bottleneck on the other hand. FL can potentially alleviate some of these
issues by reducing the overheads generated by the ML operation while providing
good performance. FL also contributes to enhancing privacy, which is a
critical issue in communications. FL applications in communications [42]
include autonomous driving [43], unmanned aerial vehicle (UAV)-based wireless
networks [44], edge computing [45], physical layer optimization [46], or
Internet-of-Things (IoT) intelligence [47].
The generic FL algorithm operates iteratively, generating a global model
update at each iteration with the help of a subset of clients. Each
algorithm’s iteration follows the following general steps (see Fig. 2):
1. 1.
A set of $\mathcal{K}=\\{1,2,...,K\\}$ clients download the current model
parameters, $w_{t}$, from the central server (also called the parameter
server).
2. 2.
Clients perform training in parallel using their local datasets
$\mathcal{D}^{(k)}$ (with size $N^{(k)}$) and update the model weights
accordingly, denoted by $w^{(k)}\in\mathbb{R}^{d}$.
3. 3.
The server pulls the model updates from the participating clients (a subset of
clients may be selected in each FL round for the sake of performance) and
orchestrates weight aggregation to generate an updated global model
$w_{t+1}\in\mathbb{R}^{d}$.
4. 4.
Above steps are repeated until convergence, i.e., until a time horizon is
completed or a certain accuracy goal is met.
Fig. 2: FL operation with WLAN contexts.
At this point, it is important to highlight the federated averaging (FedAvg)
method [48], which is based on stochastic gradient descent (SGD) optimization
and performs well for non-convex problems. In FedAvg (shown in Alg. 1),
clients perform several batch updates at each iteration using local data to
update the global model parameters. Unlike in classical federated stochastic
gradient descent (FedSGD), where gradients are exchanged, FedAvg considers
sharing model updates (e.g., the parameters of a neural network). By applying
multiple rounds of training, FL seeks to minimize a global finite-sum cost
function $l(w)$ by optimizing the global model parameters $w$:
$\min_{w\in\mathbb{R}^{d}}l(w)=\min_{w\in\mathbb{R}^{d}}\sum_{k=1}^{K}\frac{N^{(k)}}{N}l^{(k)}(w,\mathcal{D}^{(k)}),$
((2))
where $l^{(k)}(w,\mathcal{D}^{(k)})$ is the loss experienced by client $k$
when using the global model $w$ on its local data, and $N$ is the total size
of the distributed dataset, i.e., $N=\sum_{\forall k\in\mathcal{K}}N^{(k)}$.
To compute local updates, clients run $E$ epochs of SGD based on the target
local loss function $l^{(k)}$ and the batch size $B$ applied to local data
$\mathcal{D}^{(k)}$. Using a learning rate $\eta$, local updates are obtained
by:
$w_{t+1}^{(k)}\leftarrow w_{t}^{(k)}-\eta\nabla
l^{(k)}(w_{t},\mathcal{D}^{(k)})$ ((3))
Finally, being $\eta$ the learning rate, the server aggregates clients’
weights based on the importance $\alpha_{k}$ assigned to each client which may
be set according to local dataset lengths (as indicated in Eq. ((2))):
$w_{t+1}=\sum_{k=1}^{K}\alpha^{(k)}w_{t+1}^{(k)}$ ((4))
Algorithm 1 Federated Averaging (FedAvg)
1:for $t=1,2,\ldots,T$ do
2: for $k\in\mathbb{K}_{tr}$ do in parallel
3: Pull $\boldsymbol{w}_{t}$ from central server:
$\boldsymbol{w}^{(k)}_{t,0}=\boldsymbol{w}_{t}$
4: for $e=1,\ldots,E$ do
5: Update model:
$\boldsymbol{w}^{(k)}_{t,e}=\boldsymbol{w}_{t,e}^{(k)}-\eta_{t}\nabla
l^{(k)}_{t,e}$
6: end for
7: Push $\boldsymbol{w}_{t+1}^{(k)}\leftarrow\boldsymbol{w}_{t,E}^{(k)}$
8: end for
9: FedAvg:
$\boldsymbol{w}_{t+1}=\frac{1}{|\mathbb{K}_{tr}|}\sum_{k\in\mathbb{K}_{tr}}\boldsymbol{w}_{t+1}^{(k)}$
10:end for
Beyond FedAvg, other optimization mechanisms have been proposed to improve the
convergence and efficiency of FL [49, 50]. For further details on FL, we refer
the interested reader to the works in [51, 52], and to [53] for the
implementations of FL over wireless networks in particular.
## 4\. Open simulated dataset on IEEE 802.11ax SR
Supervised ML methods typically require a significant amount of high-quality
data to perform well. Training data is usually obtained either from network
activity [54] or from measurement campaigns [55]. However, obtaining real
traces from networks can be challenging due to proprietary limitations (data
owners are reluctant to share their assets), data privacy issues (most network
data is generated by final users), or difficulties in obtaining data from a
rich set of situations (anomalies are hard to reproduce and identify). In this
sense, the usage of synthetic datasets for model training is gaining attention
[12]. Such datasets can be obtained, for instance, from network simulators
(e.g., ns-3, OMNET++, OPNET). Simulators are a cost-effective solution for
generating comprehensive datasets. Some prominent examples of synthetic
datasets oriented to ML training can be found in [56, 57].
As for the provided dataset on 11ax SR [13], it has been generated with
Komondor [58], an open-source IEEE 802.11ax-oriented simulator that includes
features like channel bonding or SR. Komondor does not implement the targeted
functionalities, but its execution is also lightweight, thus allowing for
generating large datasets corresponding to massive WLAN deployments.
The dataset contains both training and test files, which include the results
obtained from several simulated random deployments applying 11ax SR (see the
example random deployment in Fig. 3). More specifically, a set of three
baseline scenarios was considered to represent different types of deployments.
Considering the features in each type of scenario (e.g., maximum number of
STAs per BSS, minimum distance between APs), 1,000 random deployments of each
type were generated for training. Each simulated deployment corresponds to a
context $k\in\mathbb{K}$, where the BSS of interest (namely,
BSS${}_{\text{A}}$) is used as a client for FL optimization. To enrich
contexts with data, each BSS includes information for each possible OBSS/PD
configuration $\tau$ (i.e., from -82 dBm to -62 dBm with 1 dBm precision).
Finally, for the test dataset, 1,000 more deployments were simulated using
more relaxed constraints. In this case, a single random OBSS/PD configuration
was selected in each deployment. Table 2 provides an overview of the entire
dataset.
Fig. 3: Example of a simulated WLAN deployment. Table 2: Summary of the
scenarios of the dataset.
| Sce id | Num. APs | Num. STAs | d_min(APs) | | Context
---
variations
Training | training1 | 2-6 | 1 | 10 m | None
training2 | 1-4 | 10 m | None
training3 | 1-4 | None | | Up to 20
---
locations
Test | test | 2-4 | None | None
Training scenario training1 considers BSSs with only one STA, which is useful
to minimize the impact of uplink transmissions, thus allowing to focus on
inter-AP interactions only. In contrast, scenarios training2 and training3
consider up to 4 STAs per AP, which contribute generating more traffic in the
uplink. As for the minimum distance between APs (d${}_{\text{min}}$), it is
set to 10 m in scenarios training1 and training2, whereas the rest have no
limitation. Furthermore, contexts in scenario training3 contain richer
datasets by simulating variations of the same deployments using different STA
locations.
The information included in simulated files is divided into features and
label. Concerning features for training, we find the following key elements:
1. 1.
Type of node: indicates whether the node is an AP or an STA.
2. 2.
BSS id: identifier of the BSS to which the node belongs.
3. 3.
Node location: {x,y,z} position of nodes in the map.
4. 4.
Primary channel: main frequency channel used for transmitting and for carrier
sensing.
5. 5.
Transmit power: default transmit power used for transmitting frames.
6. 6.
OBSS/PD threshold: sensitivity used within the OBSS/PD-based SR operation.
7. 7.
Received signal strength indicator (RSSI): average signal quality experienced
by STAs during reception phases.
8. 8.
Inter-BSS interference: average power sensed from devices belonging to other
BSSs.
9. 9.
Signal-to-interference-plus-noise ratio (SINR): average SINR experienced by
STAs when receiving data from their AP.
Fig. 4: Correlation between different input and output variables of the
dataset.
It is worth noting that most of the extracted information is typically
obtained on a continuous basis in a real system. Indeed, the RSSI, SINR and
throughput measurements can be reported periodically by STAs. Interference
powers can be measured during the listen-before-transmit (LBT) phase at the
AP, employing multi-antenna processing techniques to separate the different
interfering sources. In addition, time-of-arrival (TOA) ranging techniques can
determine the distance between STAs and APs.
As for the label, we provide the throughput $\gamma^{(k)}_{j,\tau}$ obtained
by each STA $j$ in context $k$ during the simulation, provided that the
OBSS/PD configuration $\tau$ is used. Predicting the throughput is the goal of
the implemented FL solutions described in the next section. Notice, as well,
that other Key Performance Indicators (KPIs) such as the average delay or the
number of SR TXOPs could have been considered.
To conclude this section, we show the correlation matrix between input and
output variables in Fig. 4, which is later used as a motivation for some of
the proposed ML solutions. Correlation values close to $0$ indicate a lack of
relations and structure between the data corresponding to these variables,
while correlation values close to $-1$ and $+1$ indicate a perfect negative
and positive correlation between variables, respectively. Finally, Fig. 5
shows the histogram of some of the most relevant features from the entire
dataset.
Fig. 5: Histogram of relevant features from the dataset.
## 5\. Federated Learning Solutions for Spatial Reuse
In this section, we describe the solutions proposed by the participants of the
2021 ITU AI for 5G Challenge: FederationS, FedIPC, and WirelessAI.
### 5.1 FederationS
This solution is designed in three stages. In the first stage, we analyze and
pre-process available datasets. In the second stage, using gained insights
from the data analysis, we define a deep neural network (DNN) model running in
each client. In the final stage, we describe the proposed FL algorithm.
In the data analysis stage, we consider scenarios training2 and training3,
containing $2000$ different IEEE 802.11ax deployments (see Table 2 for further
details). We extract several features available in the simulator’s output
files from these scenarios, namely the OBSS/PD configuration, the RSSI, the
interference at the reference AP from other APs, the SINR, and the throughput
of each STA. We also obtain additional information using available data in the
simulator’s input files. In particular, we extract the coordinates of APs and
STAs to compute the Euclidean distances among them. In addition, we obtain the
number of STA served by the reference AP and the number of interfering APs.
To obtain the final dataset to be used to train our model, we pre-process the
data through different steps. First, we clean the input and output data parsed
from the simulator files removing all non-numerical values from the dataset.
Then, we arrange the data of each STA to form 1-D vectors with $11$ numerical
entries used as the input of the model and containing all measurements and
system parameters. Conversely, we define the STA throughput as the target
variable and output of the model. To note that the features present entirely
different ranges between maximum and minimum values and are expressed with
different units of measurements, e.g. dBm for RSSI and interference power, dB
for SINR, and meters for distances. To balance each feature’s contribution to
the overall model predictions, we re-scale the features with the Min-Max
normalization method that transforms all features’ in the range $[0,1]$.
Finally, when input data are missing, like when the number of interfering AP
reported is less than the minimum recorded according to our system settings,
we assign those values with $0$s. The activation function that we will explain
later is chosen to keep neurons inactive when $0$s are present at the input.
To decide the ML method to be used, we make the following two main
observations from the correlation analysis done in Fig. 4:
1. 1.
Most features show a strong positive or negative correlation with the output
variable (throughput). As expected, only the OBSS/PD feature does not directly
affect the throughput. Otherwise, the problem would be trivial to model.
Indeed, the OBSS/PD value is correlated to the RSSI, which affects SINR and
throughput variables, showing that relationships between input and output
values of the system are not straightforward to characterize with domain-based
models. This justifies the adoption of DNN, which are extensively used for
their capabilities to model nonlinear relationships.
2. 2.
Two regions depicted with lighter colors at the top left and at the bottom
right of the correlation matrix identify two groups of features that show a
strong positive correlation between input variables. Thus, in the DNN
architecture design, these inputs of the model need to be fully connected. In
contrast, the two regions at the top right and bottom left are characterized
by elements with close to zero correlation values, meaning that the
relationship between features is weak. Therefore, these connections are
expected to bring a low contribution to the predictions and can be dropped in
the DNN architecture design.
Based on this, in the following design stage, we model DNN architecture as
represented in Fig. 6. First, we split the DNN model into two parallel
branches. The inputs of the first branch are the features constituting the
first block, i.e. RSSI, SINR, distance STA-AP, and OBSS-PD threshold. At the
same time, features like the number of STA, the power received from
interfering AP, and the number of interfering AP form the second block of
features and are used as input of the second branch. The input layers are
followed by two hidden layers defined for each branch separately. We use a
concatenation layer to merge the output of these two branches. The result of
the concatenation is then used as input of two additional hidden layers, which
are connected to the output layer of the model. We adopt the hyperbolic
tangent (_tanh_) activation function to provide positive and negative outputs
and keep neurons inactive when the inputs are $0$s. Finally, we add dropout
layers after each layer before the output layer to reduce overfitting.
[width=.5]tikz_figures/ann_small
Fig. 6: Visualization of FederationS’ DNN structure.
As for the FL solution, it is based on the implementation outlined in [59].
However, the FederationS algorithm combines the trained weights in a novel
way, which is designed specifically for the problem of performance prediction
in WLANs. Moreover, the aggregation process at the central server is tailored
to maximise the gain from contexts with more data samples. In particular, our
proposed FL solution follows the steps described in Algorithm 2.
Algorithm 2 FederationS solution.
1:Init set $\mathbb{K}$, i.e., init $K$ contexts with data samples
2:From $\mathbb{K}$, select $N_{eval}$ to create $\mathbb{K}_{val}$
3:Create a new set of contexts
$\mathbb{K}_{tr}=\mathbb{K}\cap\mathbb{K}_{val}$
4:Server initializes model parameters $\theta_{0}$ and $W_{0}$
5:The server transmits $\theta^{(k)}_{0},w^{(k)}_{0}$ to $k$-th contexts
6:for communication epoch $t=1,2,...,T$ do
7: Rand. select $N_{tr}$ contexts from $\mathbb{K}_{tr}$ to get
$\mathbb{K}_{ep}$
8: for $i$-th context in $\mathbb{K}_{ep}$ do
9: Split samples in $\beta$ ($\frac{n_{i}}{B}$ batches of size $B$)
10: Where $n^{(i)}$ is the number of data samples
11: for batch $b$ in $\beta$ do
12: $\theta^{(i)}_{t}\leftarrow\theta^{(i)}_{t-1}-\eta\nabla
l(\theta^{(i)}_{t-1};b)$
13: $w^{(i)}_{t}\leftarrow w^{(i)}_{t-1}-\eta\nabla l(w^{(i)}_{t-1};b)$
14: end for
15: Determine weight $\alpha^{(i)}=n^{(i)}/N_{STA}$
16: Transmit $\theta^{(i)}_{t}$, $w^{(i)}_{t}$, and $\alpha^{(i)}$ to central
server
17: end for
18: Calculate data samples weight $\alpha_{t}=\sum^{N_{tr}}_{i=1}\alpha^{(i)}$
19: Update model:
$\theta^{(k)}_{t}=\sum^{N_{tr}}_{i=1}\frac{\alpha^{(i)}*\theta^{(i)}_{t}}{\alpha_{t}}$,
$w^{(k)}_{t}=\sum^{N_{tr}}_{i=1}\frac{\alpha^{(i)}w^{(i)}_{t}}{\alpha_{t}}$
20: The server transmits $\theta_{k}^{t},w^{(k)}_{t}$ to $k$-th contexts
21:end for
22:Output: $\theta_{T}$ and $w_{T}$
In steps 1-5, we initialize the contexts and split them into train and
validation sets. After the initialization, the training takes place locally in
a subset of $N_{tr}$ contexts, which are selected randomly at every
communication epoch. The training results, i.e., the trained neural network
$\theta^{(i)}$ and its weights $w^{(i)}$ along with the number of data sample
$n^{(i)}$, are then transmitted from the $N_{tr}$ contexts to the central
server for aggregation. After the aggregation, the central server sends back
the global trained model to every context, which updates their local DNN
models. The training cycle repeats for $T$ communication epochs.
One of the most important aspects of the FL training consists of aggregating
the weights at the central server. Initially, we weighted the updates from
each context equally, but such an approach resulted in a skewed performance
toward contexts with more STA. Such behavior can be attributed to the fact
that contexts with four STA have twice as many samples as contexts with only
two STA. Instead, our solution proposed to weight each context update based on
the number of data samples used for the training in each context is normalized
by the number of STA in the contexts. We denote this normalization weights
with $\alpha$. This approach improves the accuracy of the predictions of the
throughput.
To evaluate the performance of the proposed solution (shown in Fig. 7), we
consider the mean average error (MAE). In particular, we use a neural network
trained for $T=250$ communication rounds. In Table 3, we report the list of
hyper-parameters used in the submitted solution and related pre-trained DNN
can be found in the GitHub repository [14]. We perform the validation using
five percent of available contexts, randomly sampled from $\mathbb{K}$. In
total, we had $K=1946$ available contexts. Five percent of available contexts
are used for evaluation, i.e., $N_{eval}=97$. At each communication round, we
select 500 contexts randomly to perform training on, i.e., $N_{tr}=500$.
Furthermore, the contexts we use during the training and validation set are
kept separated to prevent the data leakage and to recognize when the solution
starts to over-fitting to the training data, i.e., $\mathbb{K}_{val}$, where
$\mathbb{K}_{val}\cap\mathbb{K}_{tr}=\emptyset$.
[width=0.5]tikz_figures/mea_plot
Fig. 7: MAE obtained by FederationS’ over the number of communication epochs.
As shown in Fig. 7, the MAE decreases over the number of communication epochs.
The same trend occurs for the contexts that we use during the training process
(Training set) as well as for the contexts that we use only for validation
(Validation set). However, the validation throughput decrease is noisier as it
sometimes increases between two consecutive communication epochs. Such
behavior is due to the random sampling approach as not all randomly selected
sets $\mathbb{K}_{ep}$ wholesomely represent the system.
Table 3: FederationS hyper-parameters Neural Network training options | Solver | Adam [60]
---|---|---
Batch size ($B$) | $21$
Dropout | $10\%$
Learning rate | $10^{-4}$
L2 regularization | $10^{-5}$
### 5.2 FedIPC
We design an NN model that predicts the throughput of the STAs of a given BSS
for a chosen SR configuration. The goal of this solution is to find the
optimal OBSS/PD threshold maximizing the network throughput. To this end, we
aim to design a neural network model which predicts the throughput of each STA
in the given BSS for a chosen OBSS/PD threshold from a certain range, thus one
can tune the OBSS/PD threshold by using NN architecture. In the federated
learning setting, we model each context as a node, where their data consist of
simulations with different thresholds. We assume these nodes cannot
communicate with each other, but communicate with a parameter server in
rounds, which aggregates the weights of the nodes to update the global model.
Then, the parameter server distributes the updated global model to the
clients.
To train the NN model we employ the federated learning framework in the
following way; we call a Wi-Fi deployment with specific characteristics such
as node locations and number of interfering BSSs as a context. We consider $n$
contexts in total, where for each context, we have $s_{k}$ STAs per AP for
context $k$. We also define $a$ as the maximum number of access point and $b$
as the maximum number of STAs per AP. Note that all contexts may have
different number of STAs per AP. We also have interference sensed by APs, RSSI
of the STAs assigned to $\mathrm{AP}_{\mathrm{A}}$, and the average SINR of
each STA in $\mathrm{BSS}_{A}$. We can control threshold
$\tau_{k}\in\\{-82,-81,\ldots,-62\\}$. To simplify the problem, we can only
change the threshold of the $\mathrm{BSS}_{A}$, and all other BSSs’ thresholds
are fixed to -82 dBm. Thus, we only consider the STAs in $\mathrm{BSS}_{A}$.
Let $\gamma_{j,\tau_{k}}^{(k)}\in\mathbb{R}$ be the throughput of
$j^{\mathrm{th}}$ STA in the $\mathrm{BSS}_{A}$ of the $k^{\mathrm{th}}$
context, where threshold of the $\mathrm{BSS}_{A}$ is chosen as $\tau_{k}$.
For context i, our objective is to find $\tau_{k}$ that maximizes the
throughput for all STAs in the $\mathrm{BSS}_{A}$ of context $k$, i.e,
$\operatorname*{arg\,max}_{\boldsymbol{\tau}}\sum_{k=1}^{n}\sum_{j=1}^{s_{k}}\gamma_{j,\tau_{k}}^{(k)},$
((5))
where $\boldsymbol{\tau}=\left[\tau_{1},\tau_{2},\ldots,\tau_{n}\right]^{T}$,
$s_{k}$ is the number of STAs connected to each AP (or the number of STAs in
each BSS) in context $k$. Having the knowledge of throughput values
$\gamma_{j,\tau^{{}^{\prime}}}^{(k)}$, $\forall k,j,\tau_{{}^{\prime}}$, which
is not likely, one can easily calculate $\boldsymbol{\tau}$ using ((5)). Thus,
to determine the best threshold for each context $k$, we estimate
$\gamma_{j,\tau^{{}^{\prime}}}^{(k)}$ via $\hat{\gamma}_{j,\tau_{i}}^{(k)}$
for all STA $j$ and threshold $\tau^{{}^{\prime}}$ combinations.
Since we cannot directly calculate or know the throughput
$\gamma_{j,\tau}^{(k)}$, we estimate it via a model
$\hat{\gamma}^{(k)}_{j,\tau}=f^{(k)}_{j,\tau}$, where $i$ is the context
index, $j$ is the index of STA connected to $\mathrm{AP}_{\mathrm{A}}$ and
$\tau$ is the threshold. Moreover, estimating one STA’s throughput is highly
related to estimating another STA’s throughput in the same context. Thus, to
exploit this relation, we formulate the throughput regression problem as
multi-output regression, as the following:
$\boldsymbol{f}^{(k)}_{\tau}(\boldsymbol{x}_{\tau}^{(k)},\boldsymbol{W}_{i})=\left[f^{(k)}_{1,\tau}\,\,f^{(k)}_{2,\tau}\ldots
f^{(k)}_{b,\tau}\right]^{T},$ ((6))
where $k$ is the context index, $\boldsymbol{x}_{\tau}^{(k)}$ is the input
vector and $\boldsymbol{W}_{i}$ is the neural network weights of the model at
context $k$. The input vector
$\boldsymbol{x}_{\tau}^{(k)}\in\mathbb{R}^{4b+a}$ includes each STA’s features
in order (for STAs in $\mathrm{BSS}_{A}$). Each STA’s features are as the
following: interference sensed by APs, RSSI, the average SINR and the
threshold, respectively. When a context has less than $b$ STA per AP, we zero
pad for the remaining places until the vector reaches the maximum in the
dataset. This is possible since $a$ (the maximum number of APs) and $b$ (the
maximum number of STAs per AP) are fixed.
Since every context may have different number of STAs per AP, we mask the
nonexistent STAs as the following:
$f^{(k)}_{k,\tau}=\hat{\gamma}^{(k)}_{k,\tau}=\gamma^{(k)}_{k,\tau}=0,\,\,\,\,\forall
k\in\\{s_{k}+1,\,\ldots,b\\}.$
This way, we do not backpropagate any loss for nonexistent STAs, and the model
becomes suitable for variable number of STAs per AP for every context. Then,
we define the ground truth vector as:
$\boldsymbol{\gamma}^{(k)}_{\tau}=\left[\gamma_{1,\tau}^{(k)}\,\,\gamma_{2,\tau}^{(k)}\,\,\ldots\,\,\gamma_{b,\tau}^{(k)}\right]^{T}$
For the context $k$ (local node), our objective is to minimize mean-squared
error for regression task for any
$(\boldsymbol{x}_{\tau}^{(k)},\boldsymbol{\gamma}^{(k)}_{\tau})$ data point
among all contexts, i.e.,
$\operatorname*{arg\,min}_{\boldsymbol{W}_{k}}\sum_{\forall\tau,k}\left|\left|\boldsymbol{f}^{(k)}_{\tau}(\boldsymbol{x}_{\tau}^{(k)},\boldsymbol{W}_{i})-\boldsymbol{\gamma}^{(k)}_{\tau}\right|\right|_{2}^{2}.$
We use a feed-forward neural network with one hidden layer as our model
$\boldsymbol{f}^{(k)}_{\tau}(\boldsymbol{x}_{\tau}^{(k)},\boldsymbol{W}_{k})$
with weights
$\boldsymbol{W}_{k}=\left[\boldsymbol{W}_{k}^{(1)}\,\,\boldsymbol{W}_{k}^{(2)}\right]$,
where
$\boldsymbol{f}^{(k)}_{\tau}(\boldsymbol{x}_{\tau}^{(k)},\boldsymbol{W}_{i})=\boldsymbol{W}_{k}^{(2)}\mathrm{ReLU}(\boldsymbol{W}_{k}^{(1)}\boldsymbol{x}_{\tau}^{(k)})$,
$\boldsymbol{W}_{k}^{(2)}\in\mathbb{R}^{b\times h}$ and
$\boldsymbol{W}_{k}^{(1)}\in\mathbb{R}^{h\times(a+3b)}$. As seen, we use
rectified linear unit (ReLU) as our activation function. Note that this neural
network can easily be generalized to a neural network with multiple hidden
layers, but in our case, the neural network with only 1 hidden layer has
worked the best on the validation set.
To train the proposed NN architecture under the FL paradigm, FedAvg is applied
(see Section 3). We consider full participation during FL rounds, meaning that
all the users’ updates are used for averaging in each communication step.
Furthermore, we fix the batch size to $B=21$ (matching the size of local
datasets) and the number of local epochs to $E=1$. Regarding data splits, we
only use the scenario training3, as it is the one containing more complex and
complete data, and use the 80% of the contexts for training, the 10% of the
contexts for the first validation, and the remaining 10% for the second
validation. Notice that we use the first validation set for early stopping of
the global model, whereas the second one it to perform hyper-parameter tuning.
We tune our method by using Tree Parzen Estimator of the Optuna library [61]
and choose the model with the lowest MAE on the second validation set.
Finally, we evaluate our global model after every 20 rounds and stop training
if no improvement in validation MAE is achieved after $T=100$ rounds. We
evaluate the prediction results of our method via the MAE metric. Recall that
we estimate the throughputs by multi-output regression task and each context
may have different number of throughputs to be predicted. Thus, we flatten the
predictions for existing STAs before calculating the MAE. We normalize the
data by minimax normalization. We use the standard SGD implementation of
Pytorch [62] to implement federated averaging.
Table 4: Evaluation results for the best-performing neural networks with
different numbers of layers in FederationS’ hyperparameter optimization.
# hidden layers | Neurons per layer | MAE (Mbps)
---|---|---
1 | 256 | 5.10
2 | 256, 16 | 5.57
3 | 256, 32, 16 | 5.88
Table 4 shows the evaluation results for neural network architectures with
different number of layers. We only report the best configuration for each
different number of hidden layers. We choose the hyperparameters with the
least MAE on the first validation set and report the results on the second
validation set for different numbers of layers. The network with only 1 hidden
layer containing 256 neurons performs the best. As the number of layers
increases, we observe a decrease in performance. This is probably because our
network starts to overfit the data when the network becomes more complex. This
reasoning also supports that, although our network is much simpler, it is more
accurate than other participants’ networks, as shown in Table 7.
### 5.3 WirelessAI
We follow the FL framework to address the complex SR problem in multiple 11ax
WLAN cells. Individual agents first train their local NN models (with the same
network structure) using their local datasets, and then exchange and average
model weights through a centralized parameter server.
Typical NN models use simple data structures such as vectors to encode inputs
and outputs. However, in wireless networks characterized by graphs
$G=(V,~{}E)$, where $V$ is the set of nodes, and $E$ is the set of wireless
links, the number of nodes and the number of links can vary depending on the
networking scenarios. It is difficult to fix the vector dimension to fit all
networking scenarios. Even if we can fix the dimension and pad zeros to the
unused dimension fields, it is meaningless to use these fields.
To overcome the graph representation problem, we treat the whole network as an
image. More specially, we first fix a maximum range and treat the whole
network as a 1$\times$100$\times$100 gray-scale image with a default value of
0. Then we map nodes to values by their roles (i.e., AP role with value 1, the
target AP with OBSS/PDD value, other APs with value 1, and STAs with value 2),
and place the values to their corresponding locations. In this way, we can
represent any networks with arbitrary APs and STAs. Note that the topology
information is encoded into the image.
Then, we adopt two NNs to predict the performance: one part is a convolutional
neural network (CNN), and the other part is a fully connected neural network
(FCNN). We first use CNN to capture the interactions between STAs and APs to
predict the RSSI, SINR of the BSS of interest, and interference to the AP of
interest. The input of the CNN is the above processed gray-scale image, and
the used OBSS/PD value, and the output of the CNN is the RSSI, SINR, and the
caused interference. Then, we use the output of CNN as the input of FCNN to
predict the downlink throughput of the AP of interest.
The key rationale of using such architecture is to reduce computation
complexity. RSSI, SINR, and the caused interference are the key factors that
impact the final performance. Compared with using a whole FCNN to predict the
performance, if we can first use CNN to model the relationship among
{topology, OBSS/PD value} and {RSSI, SINR, and the caused interference}, and
then use a small dimension of FCNN to predict the performance, the computation
complexity is reduced.
To empower our FL algorithm, we treat each context as a local client and let
each local client use its own data to train the above two NNs. In particular,
we follow the standard FL training procedure, and run the training in rounds:
the above two NN models are trained by each local client, and the weights of
the local models are averaged to generate the global shared model, which is
used in the next round. For a dataset, there are overall 1000 local clients,
and we randomly choose 10 local clients in each round to generate the average
model. The global shared model has been updated using $T=20$ rounds in total.
The proposed NN model and FL algorithm are implemented in Pytorch.222The code
used to implement all the proposed methods by WirelessAI is available in
Github [16]. Table 5 and Table 6 summarize the architecture of the proposed NN
models. In particular, there are 13 layers in our CNN model including 5
convolution layers, 4 max-pooling layers, 1 adaptive average pooling layer,
and 3 fully-connected layers. For each convolution layer, the layers are
convolved with kernel size 3. In order to keep the size of the image after
each convolution operation and obtain more information on the image edge
position, we fill the images (i.e., padding) before each convolution
operation. After every convolution layer, a max-pooling operation is applied
to the feature maps. The kernel size of the max-pooling layer is 2. The
purpose of max-pooling is to reduce the size of the feature maps. The output
size of the adaptive average pooling layer is 1. The fully-connected layers
consist of respectively 512 and 64 and 17 output neurons. There are 1 input
layer, 2 hidden layers, and 1 output layer in our FCNN model. These layers
consist of respectively 512 and 128 and 64 and 6 output neurons. The ReLU is
used as an activation function for convolution layers and fully-connected
layers.
Table 5: A Summary Table of the Proposed CNN Model.
Layers | Type | Output Size | Kernel Size | Stride
---|---|---|---|---
1 | Convolution | 128$\times$100$\times$100 | 3 | 1
2 | Max-pooling | 128$\times$50$\times$50 | 2 | 2
3 | Convolution | 256$\times$50$\times$50 | 3 | 1
4 | Max-pooling | 256$\times$25$\times$25 | 2 | 2
5 | Convolution | 512$\times$25$\times$25 | 3 | 1
6 | Max-pooling | 512$\times$12$\times$12 | 2 | 2
7 | Convolution | 1024$\times$12$\times$12 | 3 | 1
8 | Max-pooling | 1024$\times$6$\times$6 | 2 | 2
9 | Convolution | 2048$\times$6$\times$6 | 3 | 1
10 | Adaptive average pooling | 1$\times$2048 | - | -
11 | Fully-Connected | 512 | - | -
12 | Fully-Connected | 64 | - | -
13 | Fully-Connected | 17 | - | -
Table 6: Summary Table of the WirelessAI FCNN Model. Layers | Type | Output Size
---|---|---
1 | Fully-Connected | 512
2 | Fully-Connected | 128
3 | Fully-Connected | 64
4 | Fully Connected | 6
## 6\. Performance Evaluation
In this section, we show the results obtained by the challenge participants’
models presented in Section 5. During the competition, the test dataset was
released without revealing the actual throughput of the simulated deployments.
Table 7 summarizes the performance accuracy obtained by each participating
team on the test dataset.
Table 7: Mean average error obtained by the solution proposed by each team. Team | MAE (Mbps)
---|---
FederationS | 6.5534
FedIPC | 5.8572
WirelessAI | 8.913
Next, we analyze the results obtained by each participant in more detail.
First, Fig. 8 showcases the empirical cumulative distribution function (CDF)
of the test error obtained by each solution. The results are compared to the
ones obtained by a vanilla centralized mechanism, which consists of a feed-
forward NN with $1024$, $512$, and $256$ neurons in each of its three layers,
with ReLU activation.333For further details on the centralized mechanism,
refer to the provided open-access repository [63]. In addition, to remark the
need for ML for the prediction problem in Wi-Fi, we provide the results of a
baseline analytical model, based on Continuous Time Markov Networks (CTMNs).
Such a baseline model implementation was presented in [64], as an extension of
the Spatial Flexible Continuous Time Markov Network (SFCTMN) framework [65].
To the best of our knowledge, the targeted analytical implementation of IEEE
802.11ax SR is one of the first of its kind and suits the SR operation because
it characterized both PHY and MAC phenomena in BSSs.
Fig. 8: Empirical CDF of the error obtained by each participants’ solution
over the test dataset. Results of baseline SFTCMN modeling and vanilla
centralized NN are included for comparison purposes.
As shown, all the proposed FL solutions improve the performance of both the
SFCTMN analytical model and the naive centralized method. The SFCTMN model,
while allowing to provide insights on the 11ax SR operation, is shown to fail
at faithfully representing the realistic phenomena observed in WLANs through
simulations. This result points to the need for ML models to capture the
complex phenomena in WLANs. Concerning the ML models presented in this paper,
FedIPC is the one providing the highest accuracy, with the 82.4% of the
predictions below a 10 Mbps error, compared to the 80.5% and the 60.3%
achieved by FederationS and WirelessAI, respectively.
Finally, to provide more insights on the generalization capabilities of each
model, Fig. 9 shows the error obtained by each solution, for each possible
number of APs and STAs in the test deployments. As shown, contrary to
intuition, all the models perform better as complexity increases, i.e., as the
number of APs and STAs is bigger. This result is mainly motivated by the fact
that DL allows capturing the most complex interactions in dense deployments,
which reinforces the role of AI-enabled solutions for network optimization.
(a)
(b)
Fig. 9: Generalization capabilities of the proposed models on test data with
respect to: (a) the number of STAs, and (b) the number of APs.
## 7\. Discussion
### 7.1 Contributions
In this paper, we have presented the main results gathered from problem
statement “ITU-ML5G-PS-004: Federated Learning for Spatial Reuse in a multi-
BSS (Basic Service Set) scenario” in the 2021 ITU AI for 5G Challenge. First,
we have overviewed the SR problem in IEEE 802.11ax and formulated a novel
optimization use case via FL. To evaluate the potential of this solution, we
have provided a dataset containing synthetic data on 11ax SR measurements in
random deployments. The dataset is open for the sake of reproducibility and to
engage other researchers to work on this topic. The provided dataset has been
used by the participants of the challenge to develop the models introduced in
this paper.
### 7.2 Lessons learned
We extract the following insights from the models and results overviewed in
this paper:
1. 1.
Predicting WLAN performance accurately is key to optimize these kinds of
networks. However, mechanisms like OFDMA or SR add further complexity to
accurately modeling WLANs. In this regard, DL-based models have shown a great
potential for capturing the complex interactions of IEEE 802.11ax WLANs
applying SR in dense deployments. This provides a paradigm shift with respect
to mostly adopted online learning mechanisms. Nevertheless, for the sake of
addressing spatial interactions in dynamic WLAN settings, both types of
mechanisms are envisioned to be combined.
2. 2.
FL suits the decentralized nature of WLANs and, despite contexts count on
limited data, its performance has been shown to outperform vanilla centralized
ML. Thus, FL provides opportunities for (i) training ML models
collaboratively, (ii) enhancing user privacy by not sharing data directly but
model weights, (iii) reducing the communication overhead of traditional
centralized ML mechanisms, and (iv) providing portability by reducing the
computation capabilities for the training of ML models.
3. 3.
Finally, using synthetic datasets for training ML models contributes to
enriching ground knowledge on certain network technologies and deployments.
Concerning this, we remark the importance of cost-effectively generating data
with network simulators, which can complement real networking data to, for
instance, reproduce anomalies.
### 7.3 Future research directions
The SR mechanism is expected to evolve towards a more sophisticated operation
in future IEEE 802.11 amendments. At the moment of writing this paper, task
group IEEE 802.11be (TGbe) is defining the coordinated SR (c-SR) mechanism as
part of the multi-AP operation [66]. Through c-SR, APs collaborate to further
improve the performance gains achieved by applying SR. More specifically, APs
exchange relevant information (e.g., measurements) to select the best SR
configuration, based on the recipient STAs to which transmissions are expected
to be held.
Fig. 10 illustrates the basics on c-SR. Considering the deployment shown in
Fig. 10(a), where two contending APs (namely APA and APB) are within the same
sensitivity area when using the default CCA/CS. Nevertheless, when applying
c-SR, both APs can transmit in parallel, thus enhancing spectral efficiency
and potentially reducing latency. To do so, the AP gaining channel access
after completing the backoff (BO) takes the role of sharing AP (in Fig. 10(b),
APA is the sharing AP). Likewise, APB acts as a shared AP. The sharing AP
sends a c-SR Trigger Frame (TF) to indicate the capabilities of the upcoming
transmission to selected STA (STAA), including the maximum acceptable
interference level (obtained through measurements). Based on the information
provided in the TF, the shared AP decides which STA to transmit and selects
the best configuration (e.g., transmit power, MCS) to that end. Notice, as
well, that a transmit power limitation is imposed by the sharing AP so that
its transmission is not affected by shared APs. Finally, once both
simultaneous transmissions take place, a block ACK (BA) is sent by STAs to
confirm successful downlink transmissions.
(a)
(b)
Fig. 10: c-SR operation: (a) deployment, and (b) exchange of packets.
Given the added complexity of evolved SR, the role of ML, and more
specifically FL, gains ground. Notice, as well, that in order to set the best
configuration that maximizes the overall network performance, both APs and
STAs perform measurements related to spectral utilization. Such a rich source
of data can be exploited by FL to drive intelligent-based network
optimization.
## Acknowledgement
The authors would like to thank enormously everyone that made possible the ITU
AI for 5G Challenge, with special mention to Vishnu Ram OV, Reinhard Scholl,
and Thomas Basikolo. Likewise, we would like to thank the valuable feedback
provided by Dr. Andrea Bonfante.
The present work has received funding from the European Union’s Horizon 2020
Marie Skłodowska Curie Innovative Training Network Greenedge (GA. No. 953775)
and has been partially supported by WINDMAL PGC2018-099959-B-I00
(MCIU/AEI/FEDER,UE). This work was also in part funded by the SFI-NSFC
Partnership Programme Grant Number 17/NSFC/5224.
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|
# The noise intensity of a Markov chain
Lukas Ramlow Benjamin Lindner Bernstein Center for Computational
Neuroscience Berlin, Philippstr. 13, Haus 2, 10115 Berlin, Germany Physics
Department of Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
###### Abstract
Stochastic transitions between discrete microscopic states play an important
role in many physical and biological systems. Often, these transitions lead to
fluctuations on a macroscopic scale. A classic example from neuroscience is
the stochastic opening and closing of ion channels and the resulting
fluctuations in membrane current. When the microscopic transitions are fast,
the macroscopic fluctuations are nearly uncorrelated and can be fully
characterized by their mean and noise intensity. We show how, for an arbitrary
Markov chain, the noise intensity can be determined from an algebraic
equation, based on the transition rate matrix. We demonstrate the validity of
the theory using an analytically tractable two-state Markovian dichotomous
noise, an eight-state model for a Calcium channel subunit (De Young-Keizer
model), and Markov models of the voltage-gated Sodium and Potassium channels
as they appear in a stochastic version of the Hodgkin-Huxley model.
## I Introduction
For models of many fluctuation phenomena in physics, biology, chemistry, and
other fields, it is important to properly characterize the noise that drives a
given dynamical system. Examples include the firing of a neuron, driven by
channel noise WhiRub00 ; DorWhi05 ; SchFis10 ; FisSch12 ; MoeIan16 ; PuTho21
and by shot-noise-like input from other neurons HohBur01 ; RicGer06 ; WolLin10
; RicSwa10 ; BriDes15 ; DroLin17 , the fluctuations in the intensity of an
excitable laser HakSau67 ; DubKra99 ; LamGuz03 ; SchTia12 , or chemical
reactions in mesoscopically small volumes Gar85 ; Van92 ; Gil00 . In the past
decades, strongly simplified noise models, such as white Gaussian noise,
Poissonian shot noise, dichotomous noise, or an exponentially correlated
Ornstein-Uhlenbeck noise have often been used to describe the input noise in
these systems. As the models for the driving noise process become more
complex, one would like to use characteristics of the noise process that can
be used to fairly compare different noise models (and their effect on a
dynamical system). This fair comparison is already possible for simple noise
processes, such as different exponentially correlated processes like the
Gaussian Ornstein-Uhlenbeck process UhlOrn30 ; Ris84 ; HanJun95 , the
dichotomous telegraph noise HorLef83 ; Gar85 ; Ben06 ; DroLin14 , or the
exponentially distributed noise FarLin21 (which is, however, only
approximately exponentially correlated).
Simple characteristics of a stochastic (noise) process $x(t)$ are its
stationary mean and variance
$\displaystyle\mu=\left\langle
x(t)\right\rangle,\;\;\;\sigma^{2}=\left\langle\Delta
x^{2}\right\rangle=\left\langle x^{2}(t)\right\rangle-\left\langle
x(t)\right\rangle^{2},$ (1)
its correlation time
$\displaystyle\tau=\int_{0}^{\infty}dt^{\prime}\frac{\left\langle
x(t)x(t+t^{\prime})\right\rangle-\left\langle
x(t)\right\rangle^{2}}{\left\langle\Delta x^{2}\right\rangle},$ (2)
and its noise intensity
$\displaystyle D=\int_{0}^{\infty}dt^{\prime}\left\langle
x(t)x(t+t^{\prime})\right\rangle-\left\langle x(t)\right\rangle^{2}.$ (3)
The meaning of mean and variance are quite obvious. The correlation time (here
defined by an integral over the normalized autocorrelation function) provide
an order-of-magnitude estimate of the periods over which the process changes
significantly. Last but not least, the intensity (here defined by an integral
over the _un_ normalized autocorrelation function) captures how much of an
effect the process would have when driving a dynamical system. More
specifically, if $x(t)$ were the velocity of a Brownian particle, $D$ would
correspond to the diffusion coefficient, which is a reasonable measure of the
effect of the velocity noise on the position dynamics.
It is clear from the above definitions that correlation time, variance, and
intensity are connected by
$\displaystyle D=\sigma^{2}\tau,$ (4)
i.e. if we know two of the characteristics, we can easily compute the third
one. For processes described by a nonlinear Langevin equation, all four
characteristics (including also the mean value) can be expressed by
quadratures Ris89 ; see also refs. Lin07 ; Lin08 , which make the above-
mentioned connection between noise intensity and diffusion coefficient more
explicit.
For discrete-valued processes, $x(t)\in\\{x_{1},x_{2},\dots\\}$, governed by a
master equation, the mean and the variance can easily be calculated in all
cases where the stationary probability can be obtained. The calculation of the
noise intensity is more involved but has recently been worked out in our
previous paper and applied for a specific model RamFal23 ; RamFal23b . It
turned out that, in complete analogy to the procedure for the Langevin case
studied in Ris89 ; Lin07 , we can obtain closed-form expressions for these
characteristics. Multiple integrals in the continuous Langevin case correspond
to multiple sums in the Markov chain case. Here, we show that the theory is
not limited to a specific model, but can be applied to any random process
where the state transitions are governed by a master equation.
Our paper is structured as follows. We begin in sec. (II) with the general
framework for calculating the noise intensity. Then we discuss three examples.
In sec. (III) we illustrate the general result for the (simple and well-known)
case of Markovian dichotomous noise. In sec. (IV) we study the more
complicated case of an eight-state Markov model, as used in the De Young-
Keizer model to describe a subunit of calcium release channel. In sec. (V) we
study a stochastic version of the sodium and potassium currents as they appear
in the Hodgkin-Huxley model with channel noise. Finally, in sec. (VI), we
discuss further applications of our results, such as a general white-noise
approximation in cases where the microscopic transitions are much faster than
the dynamics of the driven system. However, we also point out some limitations
for systems where the noise is _not_ purely external and/or very fast but also
depends on the state of the driven system.
## II Noise intensity of a Markov chain
We consider a random process $x(t)$ with discrete states, where the
probability $p_{i}(t)$ of finding a state $i$ at time $t$ is determined by the
(homogeneous) master equation:
$\displaystyle\dot{\bm{p}}(t)=W\bm{p}(t)$ (5)
with the probability vector
$\bm{p}(t)=\begin{pmatrix}p_{1}(t)&p_{2}(t)&\dots\end{pmatrix}$ and the
transition rate matrix $W=(w_{ij})$. The entries $w_{ij}>0$ for $i\neq j$ are
the transition rates from a state $j$ to state a $i$ and $w_{jj}=-\sum_{i\neq
j}w_{ij}$. To fully characterize the process $x(t)$, each state $i$ is
assigned a specific value $x_{i}$, which are not necessarily different from
each other.
For such a random process, the calculation of the mean and the variance
follows standard procedures Gar85 ; Van92 and is based on the stationary
probability vector $\lim_{t\to\infty}\bm{p}(t)=\bm{p}$ (we indicate the
stationary state by omitting the time argument). This vector can be obtained
from the stationary master equation
$\displaystyle 0=W\bm{p}$ (6)
together with the normalization condition $\sum_{i}p_{i}=1$; the additional
condition is needed because of the rank deficiency of the matrix $W$.
Practically, the normalization can be incorporated by replacing an arbitrary
row of the matrix $W$ with ones and the corresponding entry in the zero vector
on the l.h.s. by a one. This leads to a linear system of equations solvable by
standard methods. Given the stationary probabilities $p_{i}$, the mean and the
variance can be calculated by
$\displaystyle\mu$ $\displaystyle=\sum x_{i}p_{i},$ $\displaystyle\sigma^{2}$
$\displaystyle=\sum_{i}(x_{i}-\mu)^{2}p_{i}.$ (7)
The calculation of the noise intensity is more advanced. We recall the
definition of the noise intensity by the integral over the autocorrelation
function eq. (3). In principle, the correlation function can be determined by
solving the time-dependent master eq. (5). However, it turns out that the
calculation of the time-dependent probability vector $\bm{p}(t)$ is not
necessary in order to calculate the noise intensity. Instead, taking advantage
of the fact that the integrated correlation function is of interest, an
algebraic equation can be found that determines the noise intensity and is not
much more complicated to solve than the equations that determine the mean or
variance.
To show this, we relate the noise intensity to the probabilities of the Markov
chain:
$\displaystyle D$ $\displaystyle=\int_{0}^{\infty}dt^{\prime}\,\langle
x(t+t^{\prime})x(t)\rangle-\langle x(t)\rangle^{2}$ (8)
$\displaystyle=\int_{0}^{\infty}dt^{\prime}\,\sum_{i,j}[x_{i}x_{j}p_{ij}(t^{\prime})p_{j}-x_{i}x_{j}p_{i}p_{j}]$
$\displaystyle=\sum_{i,j}x_{i}f_{ij}x_{j}p_{j}.$
where $p_{ij}(t^{\prime})=p(i,t+t^{\prime}|j,t)$ is the transition
probability, i.e. the probability of finding the state $i$ at $t+t^{\prime}$
given the state $j$ at time $t$. Since we are considering a homogeneous
process, this conditional probability does not depend on the absolute time
$t$, but only on the difference $t^{\prime}$ and is determined by the master
eq. (5) with the initial condition $p_{k}(t)=\delta_{kj}$. The auxiliary
function introduced in the last line of eq. (8),
$\displaystyle f_{ij}=\int_{0}^{\infty}dt^{\prime}\,p_{ij}(t^{\prime})-p_{i},$
(9)
is given by the integral over the difference between the transition and
stationary probabilities. Eq. (8) is of course just a reformulation of the
problem. However, it turns out that the auxiliary functions $f_{ij}$ for a
given $j$ can be calculated from a system of algebraic equations together with
an additional condition, a calculation that is very similar to that of the
stationary probabilities $p_{i}$. To see this, we formulate the master
equation where the state at some reference time $t$ has been specified
($p_{kj}(0)=\delta_{kj}$):
$\displaystyle\dot{p}_{kj}(t^{\prime})$
$\displaystyle=\sum_{i}w_{ki}p_{ij}(t^{\prime}),$ (10)
$\displaystyle\dot{p}_{kj}(t^{\prime})$
$\displaystyle=\sum_{i}w_{ki}[p_{ij}(t^{\prime})-p_{i}],$ $\displaystyle
p_{k}-\delta_{kj}$ $\displaystyle=\sum_{i}w_{ki}f_{ij}.$
To get from the first to the second line we have subtracted the stationary
master equation $0=\textstyle\sum_{i}w_{ki}p_{i}$. To get from the second to
the third line we have integrated over $t^{\prime}$, used eq. (10), and
exploited that
$\int_{0}^{\infty}dt^{\prime}\,\dot{p}_{kj}(t^{\prime})=p_{k}-\delta_{kj}$.
The last line in eq. (10) looks like an equation that uniquely determines
$f_{ij}$. However, because of the rank deficiency of $W$ we need an additional
condition that is obtained by observing that
$\displaystyle\sum_{i}f_{ij}=\int_{0}^{\infty}dt^{\prime}\,\sum_{i}p_{ij}(t^{\prime})-p_{i}=0.$
(11)
This condition is independent of $j$ and reflects that for any $t^{\prime}$
both the transition probability and the stationary probability are normalized
over the states $i$.
Finally, while eqs. (8) - (11) allow for the calculation of noise intensity,
they can be expressed more conveniently. For this purpose, we write eq. (8) in
matrix notation
$\displaystyle D=\bm{x}^{T}F\bm{y},$ (12)
where $f_{ij}$ is the entry in the $i$-th row and $j$-th column of the matrix
$F$ and the two vectors are given by
$\bm{x}=\begin{pmatrix}x_{1}&x_{2}&...\end{pmatrix}^{T}$ and
$\bm{y}=\begin{pmatrix}x_{1}p_{1}&x_{2}p_{2}&...\end{pmatrix}^{T}$. Similarly,
the set of linear eqs. (10) can be combined into a single matrix equation
$\displaystyle P-\mathbb{1}=WF$ (13)
where $P$ is a matrix in which each entry in the $i$-th row is given by the
stationary probability $p_{i}$ and $\mathbb{1}$ is the identity matrix. The
additional condition eq. (11) can be written as
$\displaystyle\begin{pmatrix}1&1&\dots&1\end{pmatrix}F=0,$ (14)
implying that each column of the matrix $F$ adds up to zero. Again,
practically, these conditions can be incorporated by replacing an arbitrary
row in $W$ by ones and the corresponding row of the matrix $P-\mathbb{1}$ by
zeroes.
In the following, we put the theory to the test for different models. The
first model is the well-known Markovian two-state model, the second is a more
involved eight-state model for a subunit of a Calcium channel (De Young-Keizer
model), the third example comprises the Sodium and Potassium channels in a
stochastic version of the Hodgkin-Huxley model of an excitable nerve membrane.
## III A simple example:
Markovian dichotomous noise
As an introduction to the method, we consider a Markovian dichotomous noise
for which the noise intensity is known and can be calculated in several ways
HorLef83 . A dichotomous Markov noise $x(t)$ is a Markov process with two
levels $x_{1}$ and $x_{2}$, corresponding to two different states with
transition rates $\alpha$ and $\beta$ between them. A schematic representation
of the model is shown in Fig. 1.
Figure 1: State diagram of a dichotomous noise. The system consists of two
states $1$ and $2$ with corresponding levels $x_{1}$ and $x_{2}$. The
transitions between the two states occur at rates $\alpha$ and $\beta$.
To calculate the noise intensity according to eq. (12), we first determine the
stationary probabilities using the stationary master eq. (6)
$\displaystyle 0=\begin{pmatrix}-\alpha&\beta\\\
\alpha&-\beta\end{pmatrix}\begin{pmatrix}p_{1}\\\ p_{2}\end{pmatrix},$ (15)
together with the normalization condition $p_{1}+p_{2}=1$ and obtain
$p_{1}=\beta/(\alpha+\beta)$ and $p_{2}=\alpha/(\alpha+\beta)$. We can now
calculate $F$ using eq. (13)
$\displaystyle\begin{pmatrix}p_{1}&p_{1}\\\ p_{2}&p_{2}\\\ \end{pmatrix}$
$\displaystyle-\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}=$ (16)
$\displaystyle\begin{pmatrix}-\alpha&\beta\\\
\alpha&-\beta\end{pmatrix}\begin{pmatrix}f_{11}&f_{12}\\\ f_{21}&f_{22}\\\
\end{pmatrix}$
with the additional conditions that each column of $F$ sums to zero, i.e.
$f_{11}+f_{21}=0$ and $f_{12}+f_{22}=0$. This yields:
$\displaystyle F=\frac{1}{(\alpha+\beta)^{2}}\begin{pmatrix}\alpha&-\beta\\\
-\alpha&\beta\end{pmatrix}$ (17)
and allows to determine the noise intensity using eq. (12):
$\displaystyle D$
$\displaystyle=\frac{1}{(\alpha+\beta)^{3}}\begin{pmatrix}x_{1}&x_{2}\end{pmatrix}\begin{pmatrix}\alpha&-\beta\\\
-\alpha&\beta\end{pmatrix}\begin{pmatrix}x_{1}\beta\\\ x_{2}\alpha\\\
\end{pmatrix}$ (18)
$\displaystyle=\frac{\alpha\beta}{(\alpha+\beta)^{3}}(x_{1}-x_{2})^{2}$
an expression that is in agreement with the result presented in HorLef83 .
Figure 2: The noise intensity of a Markovian Dichotomous noise. Panels A and B
show the noise intensity as a function of the ratio between the transition
rates $\alpha/\beta$ or the correlation time $\tau=1/(\alpha+\beta)$. The
intensity has a maximum at $\alpha^{*}=\beta/2$ or $\tau^{*}=2/(3\beta)$.
Panels C and D show the noise intensity as a function of the ratio between the
values taken in the two states $x_{1}/x_{2}$ or the variance
$\sigma^{2}=\alpha\beta(x_{1}-x_{2})^{2}/(\alpha+\beta)^{2}$. The intensity
has a minimum at $x_{1}^{*}=x_{2}$ and scales linearly with the variance.
Parameters: $\beta=1$, $x_{2}=1$.
The noise intensity has a maximum as a function of the rate $\alpha$, keeping
the other rate $\beta$ fixed, at $\alpha^{*}=\beta/2$ (Fig. 2A). As a function
of the correlation time $\tau=1/(\alpha+\beta)$ has also a maximum at
$\tau^{*}=2/(3\beta)$ (Fig. 2B). If the intensity is plotted as a function of
the ratio $x_{1}/x_{2}$ (Fig. 2C), it has a minimum and vanishes for
$x_{1}^{*}=x_{2}$ because the variance vanishes in this case. When plotted as
a function of the variance $\sigma^{2}$ (Fig. 2D), the noise intensity
increases linearly according to eq. (4).
The calculation presented here serves only as a sanity check. The real
advantage of the method lies in the possibility of calculating the noise
intensity for more complicated transition rate matrices, as we will show in
the following.
## IV A biophysical example:
Stochastic Ca2+ channel model
In this section, we consider a biophysical example and calculate the noise
intensity for a eight-state Markov model as illustrated in Fig. 3 and used in
the De Young-Keizer model to describe a single subunit of an inositol
trisphosphate (IP3) receptor YouKei92 . For such a model, no closed-form
expression for the noise intensity is known.
The entire De Young-Keizer model describes the dynamics of the intracellular
calcium (Ca2+) concentration, which in many cells serves as a signaling
molecule to transmit information about extracellular stimuli (calcium
signaling) BerBoo98 ; Cla07 . The characteristic short periodic increases in
the intracellular Ca2+ concentration that carry the information can be caused
either by an influx of Ca2+ from the extracellular medium or by a release of
Ca2+ from an intracellular store, the endoplasmic reticulum (ER). In both
cases, stochastic transitions between discrete states of the ion channels give
rise to macroscopic fluctuations in the intracellular Ca2+ concentration. The
De Young-Keizer model covers the case where the Ca2+ signal is evoked by the
release of Ca2+ from the ER through the IP3 receptor channel. This receptor
channel in turn is assumed to consist of three independent and identical
subunits with three binding sites each: one for the second-messenger molecule
IP3, produced in the cell in response to an extracellular stimulus (IP3
pathway), and one activating Ca2+ binding site and one inhibiting Ca2+ binding
site. The Ca2+ current through a single IP3 receptor channel is given by
$\displaystyle
I_{\text{Ca}}=c_{1}x^{(1)}(t)x^{(2)}(t)x^{(3)}(t)([\text{Ca}\textsuperscript{2+}]_{\text{er}}-[\text{Ca}\textsuperscript{2+}]_{\text{i}}),$
(19)
where $[\text{Ca}\textsuperscript{2+}]_{\text{er}}$ and
$[\text{Ca}\textsuperscript{2+}]_{\text{i}}$ are the ER and intracellular
(cytosolic) Ca2+ concentrations, $c_{1}$ is the volume ratio between the ER
and the cytosol, and $x^{(n)}(t)$ are dichotomous (two-valued) stochastic
processes capturing the state of the three IP3 receptor subunits.
Figure 3: State diagram of a Ca2+ channel subunit YouKei92 . Panel A shows the eight-state model of a single IP3-receptor subunit. The states are denoted $ijk$, where each index represents one of the three binding sites for IP3 ($i$), activating Ca2+ ($j$), and inhibitory Ca2+ ($k$). An index is $1$ ($0$) if the binding site is occupied (unoccupied). The conducting state $110$ is highlighted in green. Panel B shows the transition rates on the front and back of the die. Panel C shows the transitions between the front and back faces. Binding rates are denoted $\alpha$ and depend linearly on the corresponding concentration ($\alpha_{i}=\hat{\alpha}_{i}[\text{IP}\textsubscript{3}]$ for $i=1,3$ and $\alpha_{i}=\hat{\alpha}_{i}[\text{Ca}\textsuperscript{2+}]_{\text{i}}$ for $i=2,4,5$), while unbinding rates are denoted $\beta$ and are constants. Parameters are according to Table 1. Table 1: Simulation parameters for a IP3-receptor subunit in the De Young-Keizer model YouKei92 Parameter | Value | Description
---|---|---
binding constants
$\hat{\alpha}_{1}$ / $\mu\text{M}^{-1}\text{s}^{-1}$ | 400 | IP3
$\hat{\alpha}_{2}$ / $\mu\text{M}^{-1}\text{s}^{-1}$ | 0.2 | Ca2+ inhibition
$\hat{\alpha}_{3}$ / $\mu\text{M}^{-1}\text{s}^{-1}$ | 400 | IP3
$\hat{\alpha}_{4}$ / $\mu\text{M}^{-1}\text{s}^{-1}$ | 0.2 | Ca2+ inhibition
$\hat{\alpha}_{5}$ / $\mu\text{M}^{-1}\text{s}^{-1}$ | 20 | Ca2+ activation
dissociation constants $\gamma_{i}=\beta_{i}/\hat{\alpha}_{i}$
$\gamma_{1}$ / $\mu\text{M}$ | 0.13 | IP3
$\gamma_{2}$ / $\mu\text{M}$ | 1.049 | Ca2+ inhibition
$\gamma_{3}$ / nM | 943.4 | IP3
$\gamma_{4}$ / nM | 144.5 | Ca2+ inhibition
$\gamma_{5}$ / nM | 82.34 | Ca2+ activation
The kinetics of a single subunit is described by the scheme shown in Fig. 3.
The eight possible states shown in Fig. 3A result from the fact that each of
the three binding sites can be in two possible states, occupied or unoccupied.
The subunit states are labeled $ijk$, where $i$, $j$, and $k$ indicate whether
the IP3, activating Ca2+, and inhibitory Ca2+ binding sites are occupied
($i,j,k=1$) or unoccupied ($i,j,k=0$). The entire channel is open when in all
subunits the IP3 and activating Ca2+ binding sites are occupied and the
inhibitory Ca2+ binding site is unoccupied, i.e. when all three subunits are
in the $110$ state, highlighted in green in Fig. 3A. Put differently, every
state is assigned a value according to
$x_{ijk}=\delta_{i1}\delta_{j1}\delta_{k0}$, i.e. the value of the conducting
state $110$ is $1$ and the value of every other state is $0$. The transition
rates between the states on the front and back faces of the cube are shown in
Fig. 3B, while the transition rates between the two faces are shown in Fig.
3C. Binding rates are denoted $\alpha$ and depend linearly on the IP3 or Ca2+
concentration according to the law of mass action
($\alpha_{i}=\hat{\alpha}_{i}[\text{IP}\textsubscript{3}]$ for $i=1,3$ and
$\alpha_{i}=\hat{\alpha}_{i}[\text{Ca}\textsuperscript{2+}]_{\text{i}}$ for
$i=2,4,5$), whereas unbinding rates are denoted $\beta$ and are constants (we
keep the original notation of De Young and Keizer in terms of $\hat{\alpha}$
and $\gamma=\beta/\hat{\alpha}$, see Table 1).
While De Young and Keizer considered the IP3 receptor and its subunits in the
thermodynamic limit, we calculate the variance $\sigma^{2}$, correlation time
$\tau$ and noise intensity $D$ for a single subunit $x^{(n)}(t)$. Although the
noise intensity for the Markov chain can be calculated analytically, the
expressions for the stationary probability vector $\bm{p}$ and the auxiliary
matrix $F$ are lengthy. Therefore, we determine these two statistics
numerically by inverting the transition rate matrix $W$. Since $W$ does not
have full rank, this requires some manipulation, that we mentioned already in
sec. (II) and are more detailed below.
Figure 4: Statistical measures of a stochastic Ca2+ channel subunit. Panels A,
B, and C show the variance $\sigma^{2}$, correlation time $\tau$, and noise
intensity $D$ as a function of the intracellular calcium concenntration
$[\text{Ca}\textsuperscript{2+}]_{\text{i}}$ for a stochastic IP3 receptor
subunit governed by the scheme illustrated in Fig. 3. Vertical lines indicate
the standard error calculated from ten simulations. The variance and noise
intensity are calculated according to eq. (7) and eq. (12), respectively. The
correlation time is determined as the ratio $\tau=D/\sigma^{2}$. Parameters
are according to Table 1.
To compute the stationary probability vector $\bm{p}$, we implement the
normalization condition by replacing all entries in an arbitrary row of $W$
with ones, and the corresponding entry in the zero vector on the l.h.s. of the
stationary master equation (eq. (6)) with a one. This removes a redundant row
in $W$, which can be obtained by linear combination of the other rows, and
replaces it with the normalization condition $\sum_{i}p_{i}=1$. The same trick
is used to compute the auxiliary matrix $F$, again replacing all entries in a
row of $W$ with ones and the corresponding row of the matrix
$P_{0}-\mathbb{1}$ on the l.h.s. of eq. (13) with zeros. This satisfies the
conditions $\sum_{i}f_{ij}=0$.
The results for three statistics are shown in Fig. 4. In all cases, the
numerically calculated values show excellent agreement with the theoretical
predictions, demonstrating that the method is applicable even when the
transition rate matrix is more complicated. Furthermore, the results show that
the variance alone is an insufficient measure to quantify the effect of a
random process on a driven variable. For example, while the variance is nearly
constant for low values of $[\text{Ca}\textsuperscript{2+}]_{\text{i}}$, the
noise intensity shows a pronounced maximum for an intermediate value.
## V Voltage-gated channels:
Models of stochastic K+ and Na+ channels
As a third and final example, we calculate the noise intensity for two Markov
chains, as used in stochastic variants of the Hodgkin-Huxley model to describe
the gating of the potassium (K+) and sodium (Na+) channels FoxLu94 ; Fox97 ;
Koc99 . Similar to the example discussed in the previous section, microscopic
transitions between different discrete states of the ion channels lead to
stochastic ion currents and eventually to macroscopic fluctuations, here in
the voltage of an excitable membrane. We emphasize that we now consider the
characteristics of the current through an entire ion channel (K+ or Na+), in
contrast to the subunit activity addressed in the previous section.
The classical Hodgkin-Huxley model describes the dynamics of the membrane
potential $V$ and the generation of an action potential in a neuron by means
of a passive leak current, a voltage-dependent K+ current, and a voltage-
dependent Na+ current Izh07 ; GerKis14 . In a stochastic formulation the
latter two currents can be expressed by:
$\displaystyle I_{\text{K}}$
$\displaystyle=g_{\text{K}}n^{(1)}(t)n^{(2)}(t)n^{(3)}(t)n^{(4)}(t)(V-E_{\text{K}})$
(20) $\displaystyle I_{\text{Na}}$
$\displaystyle=g_{\text{Na}}m^{(1)}(t)m^{(2)}(t)m^{(3)}(t)h^{(1)}(t)(V-E_{\text{Na}})$
where $g_{\text{K}}$ and $g_{\text{Na}}$ are the maximal conductances and
$E_{\text{K}}$ and $E_{\text{Na}}$ are the reversal potentials. In our case,
the variables $n^{(i)}$, $m^{(j)}$, and $h^{(k)}$ are Markovian dichotomous
processes that capture the state of the subunits in a single K+ or Na+ channel
111The state is described as activated or deactivated for $n$ and $m$, and
inactivated or deinactivated for $h$ DayAbb01 . Accordingly to eq. (20), the
K+ channel consists of four activation gates of type $n$, whereas the Na+
channel consists of three activation gates of type $m$ and one inactivation
gate of type $h$. Only when all subunits are open, the channel is open.
In the original Hodgkin-Huxley model, the gating variables are deterministic
quantities bounded between zero and one and governed by the differential
equation:
$\displaystyle\tau_{x}(V)\dot{x}=\alpha_{x}(V)(1-x)-\beta_{x}(V)x,$ (21)
with $x=n,m,h$. In this case, $n$, $m$, and $h$ represent the fraction of open
subunits in a large ensemble. However, eq. (21) can also be interpreted as
$x(t)$ describing the probability of a single subunit taking a certain state
in a two-state system with voltage-dependent transition rates $\alpha_{x}(V)$
and $\beta_{x}(V)$ (similar to Fig. 1). This insight allows to formulate
stochastic variants of the Hodgkin-Huxley model consistent with the
deterministic model in the thermodynamic limit, where the state of each
subunit is represented by a Markovian dichotomous noise with a mean governed
by eq. (21) FoxLu94 ; Fox97 . In this formulation, the gating variables
correspond to the fluctuating fraction of open subunits in a finite ensemble.
Figure 5: State diagram of a K+ channel and a Na+ channel Koc99 . Panel A shows the five-state model of a stochastic K+ channel. The five states represent the number of activated $n$-type subunits (0 to 4) of the K+ channel. The transition rates are given above/below the arrows. All four subunits must be activated ($n_{4}$) for the K+ channel to open. Panel B shows the eight-state model of a stochastic Na+ channel. The eight states represent the number of activated $m$-type (0 to 3) and deinactivated $h$-type (0 to 1) subunits of the Na+ channel. All three subunits of type $m$ must be activated and the subunit of type $h$ must be deinactivated ($m_{3}h_{1}$) for the Na+ channel to open. Table 2: Simulation parameters for K+ and Na+ channels in the Hodgkin-Huxley model DayAbb01 Parameter | Value
---|---
$\alpha_{n}$ / $\text{s}^{-1}$ | $0.01(V+55)/[1-\exp(-0.1(V+55))]$
$\alpha_{m}$ / $\text{s}^{-1}$ | $0.1(V+40)/[1-\exp(-0.1(V+40))]$
$\alpha_{h}$ / $\text{s}^{-1}$ | $0.07\exp(-0.05(V+65))$
$\beta_{n}$ / $\text{s}^{-1}$ | $0.125\exp(-0.0125(V+65))$
$\beta_{m}$ / $\text{s}^{-1}$ | $4\exp(-0.0556(V+65))$
$\beta_{h}$ / $\text{s}^{-1}$ | $1/[1+\exp(-0.1(V+35))]$
$g_{\text{K}}$ / mS | 36
$g_{\text{Na}}$ / mS | 120
$E_{\text{K}}$ / mV | -77
$E_{\text{Na}}$ / mV | 50
In the following we calculate the noise intensity of the ion current through a
single K+ channel or a single Na+ channel. We have already emphasized that the
kinetics of a single subunit ($n^{(i)}(t)$, $m^{(j)}(t)$, and $h^{(k)}(t)$)
can be described by a Markovian dichotomous noise with a transition rate
matrix similar to the one used in sec. (III). One could be tempted to think
that the noise intensity of the product of a number of independent random
processes can be easily found from the intensities of the single processes.
However, we are not aware of a simple relation between the former and the
latter.
Figure 6: Statistical measures of a stochastic K+ and Na+ channel. Panels A
and B show the open probability, variance $\sigma^{2}$, correlation time
$\tau$, and noise intensity $D$ as a function of the membrane potential $V$
for the stochastic K+ and Na+ channels governed by the scheme illustrated in
Fig. 5 A and B, respectively. Vertical lines indicate the standard error
calculated from ten simulations. The variance and noise intensity are
calculated according to eq. (7) and eq. (12), respectively. The correlation
time is determined as the ratio $\tau=D/\sigma^{2}$. Parameters are according
to Table 2.
To calculate the noise intensity for the product, we need to formulate the
transition rate matrix for the random processes
$x(t)=n^{(1)}(t)n^{(2)}(t)n^{(3)}(t)n^{(4)}(t)$ for the K+ channel or
$x(t)=m^{(1)}(t)m^{(2)}(t)m^{(3)}(t)h^{(1)}(t)$ for the Na+ channel. Here, we
follow the formulation of Koc99 and use transition rate matrices
corresponding to the state diagrams in Fig. 5. The five possible states for
the K+ channel (Fig. 5A) result from the fact that the subunits are identical
and independent. Therefore, it is sufficient to describe the number of
subunits in the activated state. In this formulation, the transition rate from
the state $n_{3}$ (three activated $n$-type subunits) to $n_{4}$ is
$\alpha_{n}$, the rate at which the last deactivated gate is activated, and
the transition from $n_{3}$ to $n_{2}$ is $3\beta_{n}$, the rate at which one
out of three activated gates deactivates (see Koc99 ; DayAbb01 ). The entire
K+ channel is considered open when all gates are activated, i.e. the value of
the state $n_{4}$ is $1$ and the value of every other state is $0$. Similarly,
a reduced state diagram can be formulated for the Na+ channel (Fig. 5B). In
this case, the number of activated $m$-type subunits and number of inactivated
$h$-type subunit must be distinguished, resulting in eight different states.
The K+ channel is considered open, when all three $m$-type subunits are
activated and the $h$-type subunits deinactivated, i.e. the value of the state
$m_{3}h_{1}$ is $1$ and the value of every other state is $0$.
In Fig. 6 we compare simulation results and theoretical predictions of the
open probability, variance, correlation time, and noise intensity of the
stochastic K+ and Na+ currents according to eq. (20) where the gating
variables are described by the Markov schemes illustrated in Fig. 5A and 5B,
respectively. In both cases, the numerically calculations agree with the
theoretical predictions, demonstrating that the method is applicable to the
stochastic Hodgkin-Huxley model. However, we note that our method relies on
the assumption of a clamped voltage.
Regarding the interpretation of the obtained curves, we first note that the
two upper panels agree with the deterministic open probability of the
classical Hodgkin-Huxley model: A monotonically increasing function for the K+
channel (Fig. 6A1) and a non-monotonic function for the Na+ channel (Fig. 6B1)
due to the interplay between activation and inactivation. The latter maximum
implies maxima in the variance (Fig. 6B2) and the noise intensity (Fig. 6B4).
We note that there are also maxima in the characteristics of the K+ channel at
different voltage values (Fig. 6A2-A4). The maximum of the variance is
plausible because the open probability reaches zero and one in the limit of
extreme voltage values. This maximum then also entails a maximum of the noise
intensity.
## VI Summary and discussion
In this paper, we have developed a general framework to characterize a noise
process that is described by a finite Markov chain, i.e. by a Master equation
with a finite number of states. More specifically, we demonstrated that the
calculation of the noise intensity and correlation time of the process is only
slightly more complicated than the computation of the steady state and its
mean and variance. We illustrated our general result by application to three
cases: (i) the dichotomous noise (for which all characteristics are, of
course, well known); (ii) a stochastic calcium channel subunit as it is used
in the De Young-Keizer model; (iii) sodium and potassium currents as used in
the Hodgkin-Huxley equation of action potential generation. In all these
cases, our comparison to stochastic simulations of the underlying discrete
dynamics agreed well with the analytical predictions of our formulas over a
wide range of tested parameters.
The computation of noise intensity and correlation time has particular
importance in the context of the so-called diffusion approximation. In several
situations of interest the discrete fluctuations described by the Master
equation can be well approximated by a white Gaussian noise. This stochastic
process is fully characterized by its mean value and its noise intensity, for
which we derived a simple expression above. Once this approximation has been
made, the apparatus of nonlinear diffusion processes, in particular, the
Fokker-Planck equation for the evolution of the probability density, can be
used. In order to learn whether this approximation is really justified for a
specific system, it is crucial to know the correlation time of the noise and
to test whether it is much shorter than all other time scales in the system -
only if this is the case, we are permitted to neglect the temporal
correlations of noise entirely. This may also apply in the more complicated
situation in which both the mean and intensity depend on the dynamical
variable(s) of the driven system, i.e. when there is a feedback between the
dynamical variable(s) and the noise statistics.
Let us revisit the dynamics of the calcium subunit, for which the correlation
time was shown in Fig. 4B as a function of the (clamped) calcium
concentration. The maximum correlation time is below 3 seconds in this case.
If we now take into account that calcium is _not_ clamped but in fact obeys a
dynamics on the time scale of tens of seconds to multiple minutes BouMar08 ;
ThuSmi11 , we may justify to approximate the stochastic activity of the single
subunit by a Gaussian white noise with a calcium-dependent mean value and a
calcium-dependent noise intensity. This is true when the Ca2+ concentration is
below some spiking threshold, and it does not include the dynamics that is
responsible for the spike shape. For another calcium channel (cluster) model,
this has been carried out in a integrate-and-fire type model of intracellular
calcium excitability and thoroughly tested by us and a collaborator RamFal23 ;
RamFal23b .
For the gating variables of the Hodgkin-Huxley model a similar argument may be
possible. A naive version of the approximation is not justified to describe
the generation of the action potential. It is exactly the nonlinear interplay
between the voltage and gating dynamics that gives the action potential its
characteristic shape, i.e. the upstroke of the spike caused by the positive
feedback of sodium-channel opening upon an initial (and sufficiently strong)
depolarization and the downstroke due to the slower inactivation of sodium
channels and the opening of potassium channels. The membrane time constant in
the original Hodgkin-Huxley model (roughly, the time scale of the voltage
dynamics) is of the order of 3ms HodHux52 and thus comparable to the
correlation time of the potassium channel fluctuations (according to Fig. 6b
around 2.5ms for voltage values around the resting potential). Hence, in this
case it is recommendable to abstain from a white-noise approximation. Indeed,
different approximation schemes that are based on having a large number of
channels, have been divised, see e.g. the classical studies by Fox et al.
FoxLu94 ; Fox97 , who approximated the gating dynamics by chemical Langevin
equations, and more recent contributions which use stochastic shielding to
obtain numerically efficient descriptions of the inherent stochasticity
PuTho21 . We note that our results are still useful, because in experiment the
voltage _can and is routinely_ clamped to a prescribed value and currents
through specific channels can be isolated (methods for this are for instance
discussed in the textbook by Izhikevich Izh07 ), and in this situation our
formulas give exact results for the characteristics of the respective current
fluctuations. The same method can be applied to more complicated kinetic
schemes of channel states, see e.g. the review on the many models of sodium
channels Pat91 .
Of course, the two above cases are more involved in the sense that not always
the dynamics of the Markov chain itself is affected by the variable it drives.
When the output of the Markov chain acts as an external noise on a system, no
white-noise approximation has to be made and our results then simply provide
the most important noise characteristics of this stochastic process, making it
comparable to simpler noise models (white or low-pass filtered Gaussian noise,
white Poissonian noise, or colored dichotomous noise).
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|
# A Neuro-Inspired Autoencoding Defense Against Adversarial Perturbations
Can Bakiskan, Metehan Cekic, Ahmet Dundar Sezer, Upamanyu Madhow
###### Abstract
Deep Neural Networks (DNNs) are vulnerable to adversarial attacks: carefully
constructed perturbations to an image can seriously impair classification
accuracy, while being imperceptible to humans. While there has been a
significant amount of research on defending against such attacks, most
defenses based on systematic design principles have been defeated by
appropriately modified attacks. For a fixed set of data, the most effective
current defense is to train the network using adversarially perturbed
examples. In this paper, we investigate a radically different, neuro-inspired
defense mechanism, starting from the observation that human vision is
virtually unaffected by adversarial examples designed for machines. We aim to
reject $\ell^{\infty}$ bounded adversarial perturbations before they reach a
classifier DNN, using an encoder with characteristics commonly observed in
biological vision: sparse overcomplete representations, randomness due to
synaptic noise, and drastic nonlinearities. Encoder training is unsupervised,
using standard dictionary learning. A CNN-based decoder restores the size of
the encoder output to that of the original image, enabling the use of a
standard CNN for classification. Our nominal design is to train the decoder
and classifier together in standard supervised fashion, but we also consider
unsupervised decoder training based on a regression objective (as in a
conventional autoencoder) with separate supervised training of the classifier.
Unlike adversarial training, all training is based on clean images.
Our experiments on the CIFAR-10 show performance competitive with state-of-
the-art defenses based on adversarial training, and point to the promise of
neuro-inspired techniques for the design of robust neural networks. In
addition, we provide results for a subset of the Imagenet dataset to verify
that our approach scales to larger images.
## 1 Introduction
The susceptibility of neural networks to small, carefully crafted input
perturbations raises great concern regarding their robustness and security,
despite their immense success in a wide variety of fields: computer vision (He
et al. 2015; Chen et al. 2018), game playing agents (Silver et al. 2017), and
natural language processing (Vaswani et al. 2017).
Since this vulnerability of DNNs was pointed out (Biggio et al. 2013; Szegedy
et al. 2014; Goodfellow, Shlens, and Szegedy 2015), there have been numerous
studies on how to generate these perturbations (adversarial attacks)
(Goodfellow, Shlens, and Szegedy 2015; Kurakin, Goodfellow, and Bengio 2017;
Carlini and Wagner 2017; Madry et al. 2018) and how to defend against them
(Madry et al. 2018; Wong and Kolter 2017; Guo et al. 2017; Yang et al. 2019;
Buckman et al. 2018). Existing defenses that attempt to employ systematic or
provable techniques either do not scale to large networks (Wong and Kolter
2017), or have been defeated by appropriately modified attacks (Guo et al.
2017; Yang et al. 2019; Buckman et al. 2018). State of the art defenses (Madry
et al. 2018; Zhang et al. 2019; Carmon et al. 2019) employ adversarial
training (i.e., training the model with adversarially perturbed examples), but
there is little insight into how DNNs designed in this end-to-end, “top down”
fashion provide robust performance, and how they might perform against a yet-
to-be-devised attack. Classification performance with attacked images is still
well below that with clean images, hence there remain fundamental security
concerns as we seek to deploy DNNs in safety-critical applications such as
vehicular automation, in addition to standard concerns regarding inference for
tail events not seen during training.
Approach: In this paper, we turn to neuro-inspiration for design insights for
defending against adversarial attacks, inspired by the observation that humans
barely register adversarial perturbations devised for machines. While neuro-
inspiration could ultimately provide a general framework for designing DNNs
which are robust to a variety of perturbations, in this paper, we take a first
step by focusing on the well-known $\ell^{\infty}$ bounded attack, which
captures the concept of “barely noticeable” perturbation. Our architecture,
illustrated in Figure 1, does not require adversarial training: it consists of
(a) a neuro-inspired encoder which is learnt in a purely unsupervised manner,
(b) a decoder which produces an output of the same size as the original image,
(c) a standard CNN for classification. The decoder and classifier are trained
in standard supervised fashion using clean images passed through our encoder.
The key features we incorporate into our encoder design are sparsity and
overcompleteness, long conjectured to be characteristic of the visual system
(Olshausen and Field 1997), lateral inhibition (Blakemore, Carpenter, and
Georgeson 1970), synaptic noise (Prescott and De Koninck 2003; Pattadkal et
al. 2018), and drastic nonlinearity (Prenger et al. 2004).
Figure 1: Proposed autoencoding defense. Decoder restores input size but does
not attempt to reconstruct the input in our nominal design (supervised
decoder+classifier training).
We use standard unsupervised dictionary learning (Mairal et al. 2009) to learn
a sparse, highly overcomplete (5-10X relative to ambient dimension) patch-
level representations. However, we use the learnt dictionary in a non-standard
manner in the encoder, not attempting patch-level reconstructions.
Specifically, we take the top $T$ coefficients from each patch (lateral
inhibition), randomly drop a fraction $p$ of them (synaptic noise and lateral
inhibition), and threshold and quantize them, retaining only their sign
(drastic nonlinearity). This encoder design is the key step in attenuating
adversarial perturbations, as we show via analysis of the empirical statistics
of the encoder outputs. We use overlapping patches (providing an additional
degree of overcompleteness). The patch-level outputs, which have ternary
quantized entries, are fed to a multi-layer CNN decoder whose output is the
same size as the original RGB image input. This is then fed to a standard
classifier DNN.
Rationale: The rationale behind our encoder design is summarized as follows:
* •
An overcomplete dictionary for sparse coding results in large activations for
a small fraction of the atoms, in contrast with filters learnt in the first
layer of a traditional convolutional neural network where activations are
clustered around zero; see Figure 2 and Appendix. We can therefore drop most
of the activations, reducing the effective subspace available to the attacker.
* •
An attacker can still perturb the subset of top $T$ coefficients in each
patch. Randomly dropping a large fraction $p$ of these coefficients allows the
decoder and classifier to learn to handle randomness in the sparse code, as
well as an attacker knocking a coefficient out of the top $T$.
* •
The thresholds for ternary quantization of the selected coefficients are
selected to provably guarantee that the attacker cannot flip the sign of any
nonzero entry in the sparse code. The hard thresholding ensures that the
perturbation cannot add to a coefficient which would have been selected for a
clean image. Rather, the attacker must invest the effort in pushing a smaller
coefficient into the top $T$, and gamble on it being randomly selected.
Results and Summary of Contributions: We report on experiments on the CIFAR-10
and a subset of the ImageNet dataset (“Imagenette”) that yield interesting
insights into both defense and attack strategies.
* •
We demonstrate the promise of a “bottom-up” neuro-inspired approach for design
of robust neural networks that does not require adversarial training, in
contrast to the top-down approaches that currently dominate adversarial
machine learning. For state of the art PGD attacks, after compensating for
gradient obfuscation, our adversarial accuracies are significantly better than
adversarial training as in (Madry et al. 2018).
Our results for ImageNette indicate that our patch-level sparse coding
approach generalizes across image sizes.
* •
Based on experiments with a variety of attacks adapted to our defense, we come
up with a novel transfer attack, based on an unsupervised version of our
decoder, which reduces our adversarial accuracy to slightly below that of
(Madry et al. 2018). This highlights the need for radically new attack
strategies for novel defenses such as ours, which combine unsupervised and
supervised learning.
* •
We have created our own attack library for PyTorch (Paszke et al. 2019), which
includes different versions of Expectation over Transformation (EOT) for
defenses utilizing stochasticity at test time, leveraging the substantial
effort we have invested in attacking our defense using techniques that combat
gradient obfuscation from nonlinearity and randomness. The implementation of
this defense and the adversarial attack library can be found at this link.
Figure 2: Histogram of correlations for a typical patch with atoms of an
overcomplete dictionary vs. that of activations through layer 1 filters of a
standard classifier CNN.
## 2 Autoencoding Defense
We now discuss the details of the approach outlined in Section 1, which is
illustrated in Figure 1. We discuss (standard) dictionary learning of
overcomplete representations for sparse coding at the patch level in Section
2.1. We then discuss, in Section 2.2, the highly non-standard way in which we
use this dictionary in the encoder. This represents the core innovation in the
defense: selection of top $T$ coefficients for each patch, dropout, and
quantization, where the quantization threshold is related to the adversarial
$\ell^{\infty}$ budget that we are designing for. The CNN-based decoder, which
we describe in Section 2.3, is a relatively standard architecture which
restores the dimension to that of the original image, allowing us to then use
a standard CNN architecture for the classifier. However, unlike a standard
autoencoder, our nominal defense is to train the decoder and classifier
together in supervised fashion. We do, however, also consider an unsupervised-
trained version of the decoder, trained using a regression loss prior to
supervised training of the classifier. This provides a benchmark, but also, as
we shall see, is instrumental in devising attacks adapted to our nominal
defense. Finally, the use of test-time dropout allows ensembling, as discussed
in Section 2.4.
### 2.1 Overcomplete Patch-Level Dictionary for Sparse Coding
We consider images of size $N\times N$ with 3 RGB channels, processed using
$n\times n$ patches with stride $S$, so that we process $M=m\times m$ patches,
where $m=\lfloor{(N-n)/S}\rfloor+1$. Learning at the patch level allows for
the extraction of sparse local features, effectively allowing reduction of the
dimension of the space over which the adversary can operate for each patch.
We use a standard algorithm (Mairal et al. 2009) (implemented in Python
library scikit-learn), which is a variant of K-SVD (Elad and Aharon 2006).
Given a set of clean training images
$\mathcal{X}=\\{\mathbf{X}^{(k)}\\}_{k=1}^{K}$, an overcomplete dictionary
$\mathbf{D}$ with $L$ atoms can be obtained by solving the following
optimization problem (Mairal et al. 2009)
$\min_{\mathbf{D}\in\mathcal{C},\\{\boldsymbol{\alpha^{(k)}}\\}_{k=1}^{K}}\sum_{k=1}^{K}\sum_{i,j}\Bigl{(}\frac{1}{2}\left\|\mathbf{R}_{ij}\mathbf{X}^{(k)}-\mathbf{D}\boldsymbol{\alpha}_{ij}^{(k)}\right\|_{2}^{2}\\\
+\lambda\left\|\boldsymbol{\alpha}_{ij}^{(k)}\right\|_{1}\Bigr{)}$ (1)
where
$\mathcal{C}\triangleq\bigl{\\{}\mathbf{D}=[\mathbf{d}_{1},\ldots,\mathbf{d}_{L}]\in\mathbb{R}^{\bar{n}\times
L}\mid\left\|\mathbf{d}_{l}\right\|_{2}=1\,,\forall
l\in\\{1,\ldots,L\\}\bigr{\\}}$, $\lambda$ is a regularization parameter,
$\boldsymbol{\alpha}^{(k)}$ is an $m\times m\times L$ tensor containing the
coefficients of the sparse decomposition, and
$\mathbf{R}_{ij}\in\mathbb{R}^{\bar{n}\times\bar{N}}$ with $\bar{n}\triangleq
3n^{2}$ and $\bar{N}\triangleq 3N^{2}$ extracts the $(ij)$-th patch from image
$\mathbf{X}^{(k)}$. The optimization problem in (1) is not convex, but its
convexity with respect to each of the two variables $\mathbf{D}$ and
$\\{\boldsymbol{\alpha}^{(k)}\\}_{k=1}^{K}$ allows for efficient alternating
minimization (Mairal et al. 2009; Elad and Aharon 2006).
### 2.2 Sparse Randomized Encoder
Based on the overcomplete dictionary obtained from (1), we encode the image
patch by patch. For given image $\mathbf{X}$, patch
$\mathbf{x}_{ij}\in\mathbb{R}^{\bar{n}}$ is extracted based on the $(ij)$-th
block of $\mathbf{X}$; that is, $\mathbf{x}_{ij}=\mathbf{R}_{ij}\mathbf{X}$,
and then projected onto dictionary $\mathbf{D}$ in order to obtain projection
vector $\mathbf{\bar{x}}_{ij}$, where
$\mathbf{\bar{x}}_{ij}=\mathbf{D}^{T}\mathbf{x}_{ij}$. Since the dictionary is
highly overcomplete, a substantial fraction of coefficients typically take
large values, and a sparse reconstruction of the patch can be constructed from
a small subset of these. However, our purpose is robust image-level inference
rather than patch-level reconstruction, hence we use the dictionary to obtain
a discrete sparse code for each patch using random “population coding,” as
follows.
1) Top $T$ selection: We keep only the $T$ elements of the projection vector
with largest absolute values and zero out the remaining elements. The
surviving coefficients are denoted by $\mathbf{\check{x}}_{ij}$.
Rationale: Keeping $T$ relatively large (but still a small fraction of the
number of atoms $L$) provides robustness to attacks which seek to change the
subset of nonzero coefficients.
2) Dropout: Each of the top $T$ coefficients is dropped with probability $p$,
leaving surviving outputs
$\mathbf{\tilde{x}}_{ij}(l)=\biggl{\\{}\begin{array}[]{ll}0,&\text{ with
probability }p\\\ \mathbf{\check{x}}_{ij}(l),&\text{ with probability
}1-p\end{array},$ (2)
for all $l\in\\{1,\ldots,L\\}$.
Rationale: Using a large dropout probability $p$ masks the effect of an
attacker “demoting” a coefficient from the top $T$ (the decoder and classifier
are already trained against such events). Similarly, if an attacker “promotes”
a coefficient to the top $T$, the chances of it making it into the encoder
output remain small.
We note that the dropout used in our encoder is different from the standard
use of dropout to prevent overfitting (Srivastava et al. 2014). In the latter,
neurons are dropped randomly at training, but all neurons are used during
testing. In our encoder, dropout is used for both training and testing, and is
applied after the lateral inhibition corresponding to all coefficients other
than the top $T$ being set to zero.
Figure 3: Progression of coefficients after each operation: (i) Each voxel
shows projection onto a dictionary atom, (ii) Projections after taking top
$T$, (iii) Remaining projections after dropout, (iv) Projections after
activation and quantization. Notice the saturation in color.
3) Activation/Quantization: Finally, we obtain sparse codes with discrete
values by applying binary quantization with a dead zone designed to reject
perturbations.
$\mathbf{\hat{x}}_{ij}(l)=\biggl{\\{}\begin{array}[]{ll}\operatorname{sign}{(\mathbf{\tilde{x}}_{ij}(l))}\left\|\mathbf{d}_{l}\right\|_{1},&\text{
if
}\frac{|\mathbf{\tilde{x}}_{ij}(l)|}{\epsilon\left\|\mathbf{d}_{l}\right\|_{1}}\geq\beta\\\
0,&\text{otherwise}\end{array},$ (3)
for all $l\in\\{1,\ldots,L\\}$, where $\beta>1$ is a hyperparameter.
Rationale: By Hölder’s inequality, an attacker with $\ell^{\infty}$ budget
$\epsilon$ can perturb the $k$th basis coefficient by at most
$\epsilon\left\|\mathbf{d}_{k}\right\|_{1}$. By choosing $\beta>1$, we
guarantee that an attacker can never change the sign of a nonzero element of
the sparse code. Thus, the attacker can only demote a nonzero element to zero,
or promote a zero element to a nonzero value. As discussed, a large dropout
probability alleviates the impact of both demotions and promotions.
Another consequence of choosing $\beta>1$ is that weak patches whose top $T$
coefficients are not large enough compared to the maximum perturbation
$\epsilon\left\|\mathbf{d}_{k}\right\|_{1}$ get killed, thereby denying the
adversary the opportunity to easily perturb the patch-level sparse code.
Finally, the scaling of the surviving $\pm 1$ outputs by
$\left\|\mathbf{d}_{l}\right\|_{1}$ acknowledges that, while the basis
elements have unit $\ell^{2}$ norm, their $\ell^{1}$ norms are allowed to
vary, hence we allow basis functions whose projections survive a larger
$\ell^{1}$ norm based threshold to contribute more towards the decoder input.
This is entirely optional, since the decoder can easily learn the appropriate
weights.
Following patch-level processing with stride $S$, the encoder outputs an image
level sparse code which is an $m\times m\times L$ tensor. A typical example of
how the coefficients advance through our encoder after dictionary projection
is presented in Figure 3.
### 2.3 CNN-based Decoder
We employ a CNN-based decoder architecture which can use redundancy across
overlapping patches to obtain image-level information. The decoder employs
three transposed convolutional layers, each followed by ReLU activation
function, clipped at the end to produce output with dimension $N\times N\times
3$ equal to that of the original RGB image. This allows us to deploy a
standard classifier network after the decoder, and allows for a direct
comparison between supervised and unsupervised decoder training.
### 2.4 Ensemble Processing
In order to utilize the full potential of the randomization employed in the
encoder, we allow for ensemble processing in which an input image is processed
multiple (i.e., say $E$) random realizations of our encoder at test time.
Viewing the encoder randomization as a parameter to be averaged over, we
average the softmax outputs across the $E$ realizations (see Appendix).
## 3 Adversarial Attacks and Defenses
Attacks: These can be broadly grouped into two categories (Papernot et al.
2017; Papernot, McDaniel, and Goodfellow 2016; Brendel, Rauber, and Bethge
2017): whitebox attacks, in which the attacker has access to both the
structure and the parameters of the neural network; and blackbox attacks,
which have access only to the network outputs. Given a classifier
$f:\mathbf{x}\in\mathbb{R}^{N}\rightarrow\mathbf{y}\in\mathbb{R}^{C}$, the
goal of an adversary is to find a perturbation $\mathbf{e}$ that maximizes the
given loss function $\mathcal{L}$ for classification under some constraints.
Typically, adversarial attacks are constrained in $\ell^{p}$ norm, with
$p=\infty$ receiving the greatest attention because it can be tuned to be
imperceptible to humans (Goodfellow, Shlens, and Szegedy 2015; Kurakin,
Goodfellow, and Bengio 2017; Carlini and Wagner 2017). Among the many attack
methods, Projected Gradient Descent (PGD) appears to be the most effective
first order $\ell^{\infty}$ bounded attack, and is therefore generally used to
evaluate defense methods. PGD computes the perturbation iteratively as
follows:
$\mathbf{e}_{i+1}=\text{clip}_{\epsilon}\big{[}\mathbf{e}_{i}+\delta\cdot\text{sign}(\nabla_{\mathbf{e}}\mathcal{L}(\mathbf{f}(\mathbf{x}+\mathbf{e}_{i}),\mathbf{y}))\big{]}$
(4)
where $\mathbf{e}_{i}$ corresponds to the value of the perturbation at
iteration $i$ with $\mathbf{e}_{0}=\mathbf{0}$ or $\mathbf{e}_{0}$ with each
element drawn from uniform distribution $\mathcal{U}(-\epsilon,\epsilon)$,
$\epsilon$ is the overall $\ell^{\infty}$ attack budget, and $\delta$ is the
step size for each iteration. Expectation Over Transformation (EOT) is
suggested in (Athalye et al. 2017) to make attacks robust against
transformations, and (Tramer et al. 2020) suggests using this method to
evaluate defenses utilizing evaluation-time stochasticity. With EOT, PGD
becomes:
$\mathbf{e}_{i+1}=\text{clip}_{\epsilon}\big{[}\mathbf{e}_{i}+\delta\cdot\text{sign}(\sum_{r=0}^{N_{E}-1}\nabla_{\mathbf{e}}\mathcal{L}_{r}(\mathbf{f}(\mathbf{x}+\mathbf{e}_{i}),\mathbf{y}))\big{]}$
(5)
where $\mathbf{e}_{0}=\mathbf{0}$ and $N_{E}$ corresponds to the number of
multiple runs of the model. Taking the average of gradients for models
utilizing randomness in evaluation time helps stabilize the gradient
directions.
Reference (Tramer et al. 2020) motivates defense papers to extensively
evaluate their defended neural networks with properly optimized threat models
for the defense. Accordingly, we expend extensive effort in devising attacks
optimized for our approach, incorporating EOT (Athalye et al. 2017) to obtain
useful gradients. (Athalye, Carlini, and Wagner 2018).
Defenses: While there are plenty of attempts to defend against adversarial
attacks (Madry et al. 2018; Wong and Kolter 2017; Yang et al. 2019; Buckman et
al. 2018) (this is only a small subset of recent papers), the only state of
the art defenses still standing are those based on adversarial training using
adversarial perturbations computed using variants of the original FGSM method
(Goodfellow, Shlens, and Szegedy 2015) of gradient ascent on a cost function:
the PGD attack (iterative FGSM with random restarts) (Madry et al. 2018) is
the most prominent benchmark that we compare against, but there are recent
enhancements, such as the faster single-step R+FGSM scheme in (Wong et al.
2020), and the use of a modified cost function aiming to trade off clean and
adversarial accuracy (called TRADES) in (Zhang et al. 2019). In addition,
substantially increasing the amount of training data using unlabeled data (and
using noisy labels for these using an existing classifier) has been shown to
improve the performance of adversarial training (Carmon et al. 2019). However,
we do not have insight as to what properties of adversarially trained networks
provide robustness, and whether these properties guarantee robustness against
other attacks (conforming to the same attack budget) that have not yet been
devised. Further, there is still a significant gap between clean and
adversarial accuracies for an adversarially trained network for image datasets
such as CIFAR-10, showing that the perturbation is not completely rejected by
the network. It is claimed in (Schmidt et al. 2018) that this phenomenon is
due to lack of data in datasets such as SVHN and CIFAR, but a more likely
explanation in our view is that adversarially trained networks are still
“excessively linear.”
Provably robust defenses have also been studied extensively (Wong and Kolter
2017; Croce, Andriushchenko, and Hein 2018; Raghunathan, Steinhardt, and Liang
2018). These methods provide lower bound for adversarial accuracies; however,
guarantees are provided mostly for small datasets, models, and low attack
budgets. References (Lecuyer et al. 2019; Cohen, Rosenfeld, and Kolter 2019;
Salman et al. 2019) report certified robustness for $\ell^{2}$ bounded attacks
which is able to scale to larger datasets such as ImageNet. Unfortunately,
these certified defenses do not perform as well as adversarial training
against current attack methods. Other recent defense approaches include (Guo
et al. 2017; Buckman et al. 2018; Dhillon et al. 2018; Xie et al. 2018;
Bakiskan et al. 2020; Gopalakrishnan et al. 2018). However, a large number of
defense methods designed based on systematic principles have been defeated by
properly modified attacks (Athalye et al. 2017), or have not been shown to
scale to larger datasets.
## 4 Adaptation of Attacks for Our Defense
In recent years, a majority of the proposed defense methods have been defeated
by subsequent attacks (Tramer et al. 2020; Athalye, Carlini, and Wagner 2018),
which has led to calls for each defense proposal to be evaluated not only for
existing attacks, but also for attacks adapted for that particular defense
(Carlini et al. 2019). We agree with such guidelines, and have tried a variety
of whitebox and transfer attacks adapted to our defense, all of which use EOT
on top of PGD to deal with the randomness in our encoder. We have experimented
extensively, and report only on the most effective attacks that we have found.
Whitebox - Near Full Gradient Approximation (W-NFGA): In this mode, we get
close to a full whitebox attack; every operation except
activation/quantization is differentiated (see Figure 7 in Appendix). For
taking top $T$ coefficients and dropout operations, the gradients are
propagated to earlier layers only through nonzero coefficients. This is
similar to how maxpooling operation propagates gradients. For
activation/quantization, we experiment with two different backward pass
approximations: in the first one we take the identity function as the
approximation, in the second we consider a smooth approximation to the
activation/quantization function and take the derivative of this function as
the backward pass approximation (see Appendix for details). Both of the
backward pass approximations result in similar adversarial accuracies.
Whitebox - Autoencoder Identity Gradient Approximation (W-AIGA): Here, for
each gradient computation, the entire autoencoder is treated as having
identity gradient (see Figure 8 in Appendix). This approximation works well
only when the operation defined by the autoencoder is indeed close to identity
in the forward pass, which holds for the unsupervised-trained decoder, but not
for the supervised-trained decoder which is our nominal scheme. However, as we
shall see, it is important in devising the transfer attack described next.
Pseudo-Whitebox - Transfer (PW-T): In this mode, we keep the encoder
dictionary fixed, but utilize an unsupervised-trained decoder, with supervised
training of classifier weights as usual. Adversarial perturbations are
generated using W-NFGA or W-AIGA versions of whitebox attack, and it turns out
that W-AIGA is actually more effective for when using an unsupervised-trained
decoder. Our experiments show that this yields a surprisingly strong transfer
attack against our defense.
Blackbox - Transfer (B-T): In this mode, the adversarial attack is generated
based on adversarially trained classifier without taking our autoencoder model
into account and then applied to our proposed autoencoder model. (See Appendix
for details)
In the evaluations of our model variants including randomness, we use
Expectation over Transformation (EOT) (Athalye et al. 2017) to mitigate the
effects of randomization as recommended in (Athalye, Carlini, and Wagner
2018). Specifically, we consider PGD with EOT to evaluate the versions of our
defense with randomized encoders. For deterministic encoders (considered in
detailed ablation studies in supplementary materials), we consider only PGD,
since EOT does not improve attack performance in these settings. Note that the
attack modes W-NFGA, W-AIGA and PW-T are designed specifically for our
defense, and do not apply to the benchmark defenses that we compare against.
We experiment with different variants of EOT and use the strongest one. We
check our attack implementations by cross-testing our attacks with the Foolbox
adversarial attack toolbox, and find that the same attacks perform comparably.
## 5 Experiments, Results and Discussion
### 5.1 Model Parameters and Settings
| | PGD with EOT | |
---|---|---|---|---
| Clean | W-NFGA | W-NFGA | PW-T | B-T
| | Identity | Smooth | |
Our defense | 80.06 | 63.72 | 61.28 | 39.53 | 57.76
Table 1: Accuracies for our defense method under different attacks (CIFAR-10)
Our main focus is on evaluating our defense on the CIFAR-10 dataset
(Krizhevsky, Hinton et al. 2009), for which there are well-established
benchmarks in adversarial ML. This has 50000 train and 10000 test RGB images
of size $32\times 32$ ($N=32$). In order to verify that our approach scales to
larger images, we also consider the Imagenette dataset: 9469 train and 3925
validation RGB images, cropped to size $160\times 160$ ($N=160$). Both
datasets contain images from 10 classes. For CIFAR-10, we use $4\times 4$
($n=4$) patches (i.e., $n=4$ and ambient dimension $3n^{2}=48$) and an
overcomplete dictionary with $L=500$ atoms. The stride $S=2$, so the encoder
output is a $15\times 15\times 500$ tensor ($m=15$, $L=500$). The
regularization parameter in (1) is set to $\lambda=1$ and the number of
iterations is chosen as $1000$ to ensure convergence. The hyperparameters for
Imagenette are: $8\times 8$ ($n=8$) patches and an overcomplete dictionary
with $L=1000$ atoms, stride $S=4$ which gives encoder outputs of size
$38\times 38\times 1000$ ($m=38$, $L=1000$). The regularization parameter
$\lambda$ is set to $0.5$, and the number of iterations to $10000$. The
guiding principle behind the choice of hyperparameters $n$ and $S$ is the
empirical observation of feature sizes in relation to the size of the images.
The number of dictionary atoms, $L$, is chosen to be $10$ times the ambient
dimension for CIFAR-10, and $5$ times the ambient dimension for Imagenette,
where the limiting factor was the amount of computer memory used in dictionary
learning. In order to promote sparsity, the regularization parameter $\lambda$
is chosen in the upper range of values that result in convergence of the
dictionary learning process.
We set $T=50$, $p=0.95$. These values were found to yield the highest worst-
case adversarial accuracy, based on ablation with various values of $T$ and
$p$. A basis coefficient makes a nonzero contribution to the sparse code for
the patch only if all three conditions are met: it is in the top $T$, it is
not dropped, and it exceeds the threshold in (3) (we set the hyperparameter
$\beta=3$). As mentioned, we train the CNN-based decoder in supervised fashion
in tandem with the classifier. For comparison and attack design, we also
consider unsupervised (US) training of the decoder. We use cross-entropy loss
for supervised training. For unsupervised decoder training, we use $\ell^{2}$
distance-squared as regression loss. For unsupervised training, we train the
decoder for $50$ epochs. Also, in order to train the decoder, we use a cyclic
learning rate scheduler (Smith 2017) with a minimum and maximum learning rate
of $\eta_{min}=0$ and $\eta_{max}=0.05$, respectively. In this scheduler, the
learning rate first increases linearly in the first half of the training
process and then decreases in the second half. In order to provide a
consistent evaluation, we employ the ResNet-32 classifier used in (Madry et
al. 2018) for CIFAR-10, and use EfficientNet-B0 (Tan and Le 2019) for
Imagenette. For supervised training (of classifier plus decoder for our
nominal design, and of classifier alone for the unsupervised-trained decoder),
we use the same cyclic learning rate scheduler with the same parameters. The
number of epochs is $70$ for CIFAR-10 and $100$ for Imagenette. The batch size
in training is set to $64$ for both unsupervised and supervised training.
For parallel ensemble processing, after trying values in $1\leq E\leq 10$, we
set $E=10$, which yields the best performance, increasing clean accuracies by
up to $5\%$ (see Appendix).
For attacks, we consider PGD and PGD with EOT if it is applicable. Different
from the existing EOT implementation, we use
$\delta\cdot\textrm{sign}\left(\mathbf{E}_{r}\left[\nabla_{x}/||\nabla_{x}||_{2}\right]\right)$
in each step to compute the expectation, since we find in our experiments that
it results in a stronger adversary.
Default attack parameters: Unless otherwise stated, we use the following
parameters for $\ell^{\infty}$ bounded PGD with EOT for CIFAR-10 trained
models: an attack budget of $\epsilon=8/255$ (as is typical in the benchmarks
we consider), a step size of $\delta=1/225$, a number of $N_{S}=20$ steps, a
number of $N_{R}=1$ restarts, and a number of $N_{E}=40$ realizations for EOT.
The same default attack parameters are used for attacking models trained on
Imagenette, but given the lack of benchmarks, we test several attack budgets
$\epsilon\in\\{2/255,4/255,8/255\\}$.
Computation time: On a computer with a 40-core CPU, learning the overcomplete
dictionary takes 0.2 hours for CIFAR-10 and 0.8 hours for Imagenette. On a
single 1080 Ti GPU, training the decoder and classifier, and computing the
attack with default settings take 1.2, 1.5, and 3.5 hours, respectively for
CIFAR-10. The same computations take 3, 4, and 7 hours, respectively, for
Imagenette.
Benchmarks: Our benchmarks are the PGD adversarially trained (AT) (Madry et
al. 2018), R+FGSM adversarially trained (Wong et al. 2020), and TRADES (Zhang
et al. 2019) defenses for the same classifier architecture. We reimplement
these, to enable stress-testing these defenses with attacks of varying
computational complexity. We train these models for 100 epochs with the same
cyclic learning rate that we use for our models, and verify, for ResNet-32
classifier for CIFAR-10 and EfficientNet-B0 for Imagenette, that we can
reproduce results obtained using the original code . For PGD AT, training
hyperparameters are $\epsilon=8/255$, $\delta=1/255$, $N_{S}=10$, $N_{R}=1$.
For RFGSM AT, they are $\epsilon=8/255$, $\alpha=10/255$. For TRADES, they are
$\epsilon=8/255$, $\delta=1/255$, $N_{S}=10$, $N_{R}=1$, and
$\lambda_{\text{TRADES}}=1/6$.
Note that the classifier CNN used in our paper is "simple" ResNet-32 rather
than the wide ResNet-32, both of which are utilized in (Madry et al. 2018) and
other studies in the literature. The choice of the smaller ResNet-32 network
makes evaluation of attacks computationally more feasible.
### 5.2 Results, Ablation, and Discussion
| Clean | W-NFGA | PW-T | W-AIGA
---|---|---|---|---
Our defense | | | |
Complete | 80.06 | 61.28 | 39.53 | 79.48
without A&Q | 81.68 | 38.48 | 37.95 | 74.05
without Dropout | 76.93 | 76.61 | 34.68 | 76.92
without Top T | 65.72 | 23.35 | 29.95 | 59.80
Our defense (US) | | | |
Complete | 80.03 | 65.83 | – | 30.01
Table 2: Accuracies for ablation study (CIFAR-10)
Robustness against Defense-Adapted Attacks: We first investigate the
performance of our defense under the different attack modes specified in
Section 4. Table 1 provides clean and adversarial accuracies for the different
attack types. We note that the worst-case attack for it is not a white box
attack. Rather, it is a pseudo-whitebox transfer (PW-T) attack using a network
employing the same encoder but an unsupervised decoder. While this result is
surprising at first, it is intuitively pleasing. An attack succeeds only to
the extent to which it can change the identities of the top $T$ coefficients
in the encoder. Since the latter is designed to preserve information about the
original image, providing an unsupervised decoder might provide better
guidance to the attacker by giving it a reproduction of the original image to
work with. This conjecture is supported by Figure 4, which shows the
distribution (see Appendix for how this is computed) of the expected fraction
of corrupted patches for W-NFGA and PW-T. We see that the PW-T attack results
in a higher fraction of corrupted patches.
Ablation: We now examine the efficacy of each component of our architecture on
robustness via an ablation study in which we selectively remove one encoder
component at a time and retrain the decoder and classifier, adapting attacks
for each version of our defense. We present the results of the ablation study
in Table 2. Both W-NFGA and W-AIGA results in Table 2 are obtained using the
default attack parameters. For W-NFGA, we employ smooth backward pass
approximation to the activation and quantization function with sharpness of
$\sigma=0.25$. For PW-T, we obtain W-AIGA and W-NFGA attacks with default
settings for all four different ablated unsupervised trained models and use
those to test each ablated supervised model. The reported results for PW-T are
for the worst-case scenarios in which the attack achieving the lowest accuracy
is considered. In all cases except the one without top $T$, the lowest
accuracy for PW-T attack is obtained when W-AIGA attack is applied to the
unsupervised model without A&Q with $T=50$, $p=0.95$ and transferred. For the
case without top $T$, the attack that attains minimum accuracy is W-NFGA
attack obtained based on the unsupervised model without top $T$ with $p=0.95$.
The overall results in Table 2 show that each component in our design
contributes to improving robustness. The last row shows accuracies for our
encoder with an unsupervised-trained decoder and separate supervised training
of the classifier. We note that our approach of joint supervised training of
decoder and classifier has superior performance.
We have also conducted a detailed ablation study showing that our proposed
design outperforms many other variants of our defense (see Appendix).
Figure 4: Histogram of $\mathbf{E}[$fraction of patches with corrupted coefficients$]$ over images for our defense under W-NFGA and PW-T attack. | Clean | Adversarial (Worst case) | Attack Details
---|---|---|---
| Mode | Method | Parameters
NT | 93.10 | 0.00 | – | PGD | $N_{S}=20$, $N_{R}=1$
PGD AT (Madry et al. 2018) | 79.41 | 42.05 | – | PGD (C&W Loss) | $N_{S}=100$, $N_{R}=50$
RFGSM AT (Wong et al. 2020) | 80.86 | 42.42 | – | PGD (C&W Loss) | $N_{S}=100$, $N_{R}=50$
TRADES (Zhang et al. 2019) | 75.17 | 45.79 | – | PGD | $N_{S}=100$, $N_{R}=50$
Our defense | 80.06 | 39.53 | PW-T | PGD with EOT | $N_{S}=20$, $N_{R}=1$, $N_{E}=40$
Table 3: Comparison of our defense with other defense techniques (CIFAR-10)
Comparison with benchmarks: We then compare our defense against naturally
trained networks and our three adversarially trained benchmark defenses on the
CIFAR-10 dataset. Table 3 lists worst-case accuracies for each defense, where
we vary the computational burden of attack on the benchmarks up to a point
that is comparable to the default settings for our own EOT/PGD attack. NT
denotes natural training (no defense). The worst-case adversarial accuracy for
our defense is 39.53%, which is comparable to the worst-case accuracies for
the benchmarks, which range between 42-46%. Comparing with Table 1, we see
that the worst-case whitebox PGD plus EOT attack for our defense is 60.28%,
about 18% better than the PGD attack on (Madry et al. 2018).
The results for the evaluation on the Imagenette dataset are given in Table 4.
For NT, PGD AT, and TRADES, we use PGD attack with default parameters.
For our defense, the worst-case attack is again PW-T with transfer from the
W-AIGA attack with the unsupervised-trained decoder. These experiments confirm
that our defense scales to larger images. Both clean and adversarial
accuracies are comparable to or exceed that of adversarial training, which is
potentially also hampered by the smaller size of the training dataset.
| Clean | Adversarial ($\epsilon=x/255$)
---|---|---
| $x=2$ | $x=4$ | $x=8$
NT | 89.35 | 11.44 | 0.28 | 0.00
PGD AT | 80.97 | 75.31 | 68.81 | 53.32
TRADES | 80.08 | 75.67 | 70.75 | 59.46
Our defense | 79.36 | 76.03 | 72.81 | 65.45
Table 4: Accuracies for Imagenette dataset
## 6 Conclusions
While our results demonstrate the potential of neuro-inspiration and bottom-up
design of robust DNNs, there is significant scope for further improvement.
For example, attenuating adversarial perturbations by randomization and
drastic quantization at a single step in the encoder does lead to information
loss, as seen from the reduction in clean accuracy. We can also visualize this
information loss due to the encoder by reconstruction of the input using the
unsupervised-trained decoder, which is seen to yield less than crisp images
(see Appendix). Spreading the burden of attenuating perturbations across more
network layers may help in better preserving information.
The key to attenuating adversarial perturbations is our encoder, hence there
are many possible inference architectures that can be layered on top of it.
Our separation of decoder and classifier enables reuse of standard classifier
architectures, but there might be better options. Our design also enables the
transfer attack from unsupervised-trained decoder to supervised-trained
decoder, which turns out to be more effective than whitebox PGD with EOT on
the original network. It may be more difficult to design such transfer attacks
with arbitrary inference architectures, which highlights the need for further
research on adaptive attacks, especially for novel defenses that combine
unsupervised and supervised learning, and employ concepts such as drastic
nonlinearity and stochasticity.
Finally, while top-down adversarial training remains the state of the art
defense, it inherits the inherent lack of interpretability and guarantees in
DNNs resulting from the curse of dimensionality for optimization in high
dimensions.
A compelling feature of a bottom-up approach to defense is that, by focusing
on attenuating perturbations over smaller segments of the input, it has the
potential for evading the curse of dimensionality.
## 7 Acknowledgements
This work was supported in part by the Army Research Office under grant
W911NF-19-1-0053, and by the National Science Foundation under grants
CIF-1909320 and CNS-1518812.
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## 8 Appendix
### 8.1 Correlations Between Dictionary Atoms and Patches
In order to plot Figure 2, the normalized correlations between a given patch
$\mathbf{x}_{ij}$ and dictionary atoms $\mathbf{d}_{l}$ are calculated by
$\rho_{l}=\frac{\langle\mathbf{x}_{ij}\,,\mathbf{d}_{l}\rangle}{\|\mathbf{d}_{l}\|^{2}}=\langle\mathbf{x}_{ij}\,,\mathbf{d}_{l}\rangle\,,\forall
l\in\\{1,\ldots,L\\}$ (6)
where $\langle\cdot\,,\cdot\rangle$ represents the inner product and
$\|\mathbf{d}_{l}\|^{2}=1$ for all $l\in\\{1,\ldots,L\\}$ by construction. The
normalized activations of layer 1 filters of the standard CNN (i.e.,
$\mathbf{f}_{p}$ for $p\in\\{1,\ldots,160\\}$) are calculated by
$\gamma_{p}=\frac{\langle\mathbf{x}_{ij}\,,\mathbf{f}_{p}\rangle}{\|\mathbf{f}_{p}\|^{2}}\,,\forall
p\in\\{1,\ldots,160\\}\,.$ (7)
The histograms of $\\{\rho_{l}\\}_{l=1}^{L}$ and
$\\{\gamma_{p}\\}_{p=1}^{160}$ are then plotted to obtain Figure 2.
In Figure 5, the histograms of correlations and activations are presented for
10 additional randomly chosen patches. As exemplified in Figure 5, most of the
correlations and activations histograms exhibit the same qualitative behavior
as the “typical” patch considered in Figure 2.
Figure 5: Histograms of dictionary-atom correlations and layer 1 filter
activations for 10 random patches in the style of Figure 2.
### 8.2 Ensemble Processing
In ensemble processing, multiple (i.e., say $E$) random realizations of our
encoder are considered for an input image $\mathbf{X}$ at test time. The
softmax outputs are averaged across $E$ random realizations of the autoencoder
as follows:
$h_{i}(\mathbf{X})=\frac{1}{E}\sum_{e=0}^{E-1}\frac{e^{g_{i}(\mathbf{f}_{e}(\mathbf{X}))}}{\sum_{j=0}^{C-1}e^{g_{j}(\mathbf{f}_{e}(\mathbf{X}))}}$
(8)
where $\mathbf{f}_{e}(\cdot)$ denotes the $e$th autoencoder realization,
$g_{i}(\cdot)$ is the function corresponding to the $i$th class output of the
classifier function $\mathbf{g}(\cdot)$, $h_{i}(\cdot)$ is the function
corresponding to the $i$th class output for the overall ensemble model, and
$C$ is the number of classes.
Figure 6 shows the clean and adversarial accuracies for different number of
encoder realizations used in ensemble processing. For our defense model, we
observe that both clean and adversarial accuracies increase as the number of
realizations employed in ensemble processing increases.
Figure 6: Clean and adversarial accuracies for different number of
realizations in ensemble processing.
### 8.3 Approximations to Activation/Quantization
In the computation of backward pass of W-NFGA attacks, we consider two
different approximations for the activation/quantization function in (3):
identity and smooth approximations. In the former, (3) is approximated as the
identity function. The latter is obtained by considering a differentiable
forward approximating function for (3) and then taking its derivative. Let
$f(x)$ denote the approximate of (3). The smooth backward approximation is
given by
$\frac{df(x)}{dx}=\frac{\|\mathbf{d}_{k}\|_{1}}{2\sigma}\biggl{(}\operatorname{sech}^{2}\Bigl{(}\frac{x-\beta\epsilon\|\mathbf{d}_{k}\|_{1}}{\sigma}\Bigr{)}\\\
+\operatorname{sech}^{2}\Bigl{(}\frac{x+\beta\epsilon\|\mathbf{d}_{k}\|_{1}}{\sigma}\Bigr{)}\biggr{)}$
(9)
where
$f(x)=\frac{\|\mathbf{d}_{k}\|_{1}}{2}\biggl{(}\tanh\Bigl{(}\frac{x-\beta\epsilon\|\mathbf{d}_{k}\|_{1}}{\sigma}\Bigr{)}\\\
+\tanh\Bigl{(}\frac{x+\beta\epsilon\|\mathbf{d}_{k}\|_{1}}{\sigma}\Bigr{)}\biggr{)}$
(10)
with $\sigma$ determining the sharpness of the approximation.
### 8.4 Attack Mode Details
In Figure 7, the backward and forward passes of W-NFGA attack mode are shown.
In the forward pass, all components of the defense are employed. In the
backward pass, the gradients of all components except Activation/Quantization
are calculated and used. In order to calculate the gradients of the
activation/quantization function, we consider identity and smooth
approximations as explained in Section 8.3.
Figure 7: W-NFGA attack forward/backward pass
Figure 8 shows the forward and backward passes used in W-AIGA attack mode. In
the forward pass, all components of the defense are considered. In the
backward pass, only the gradients of the classifier are used and carried to
the input layer in each step of the attack by bypassing the autoencoder.
Figure 8: W-AIGA attack forward/backward pass
For blackbox transfer attacks (B-T), we use the attacks generated with default
attack parameters for PGD adversarially trained model of (Madry et al. 2018).
### 8.5 Computation of the Histograms in Figure 4
For a given image with index $i$, let $q_{i}(j)$ represent the probability
that a total of $j$ coefficients in top $T$ are “toppled” by the attack before
dropout. $q_{i}(j)$ is found empirically by taking the histogram of number of
“toppled” coefficients over patches for the image $i$.
We can then calculate the probability of a randomly selected patch in image
$i$ not being corrupted by the attack as
$\zeta_{i}=\sum_{j=0}^{T}q_{i}(j)\,p^{j}$
where $p$ is the probability of dropping a coefficient (Dropout rate).
We note that $\zeta_{i}$ is simply the probability that coefficients that were
not in the top $T$ in the clean image $i$ for this patch are dropped by the
dropout mechanism. Of course, conditioned on this “no corruption” event, the
sparse code for the patch has a different distribution from that of the clean
image because of the coefficients which have been knocked out of the top $T$.
But it still belongs to the original ensemble of possible sparse codes for
that patch for a clean image. We ignore these subtleties, and compute the
expected fraction of patches whose sparse code has at least one nonzero
coefficient which would not have appeared in the clean image as
$z_{i}=1-\zeta_{i}$. The histogram of $z_{i}$ across all images is what is
plotted in Figure 4.
Figure 9: Clean and adversarial images before and after our autoencoder based
on unsupervised-trained decoder.
### 8.6 Additional Ablation Studies
In order to better assess the contributions played by the components of our
defense, we have tested variations of our defense with different
hyperparameters and architectures. While the encoder in our nominal defense
with parameter settings $T=50,p=0.95$ yields the best attacked accuracy, we
find that our results are remarkably resilient to the specific values of $T$
and $p$, as long as the expected number of selected coefficients, $T(1-p)$, is
roughly the same. We report here on three models that use the same
architecture as our defense but with different hyperparameters, namely, models
with ($T=1,p=0.00$), ($T=5,p=0.50$), and ($T=10,p=0.75$). These
hyperparameters are chosen to have the expected value of surviving
coefficients ($T(1-p)$) within the same order of magnitude as our original
defense. As reported in Table 5, these hyperparameters all yield similar
results. Intriguingly, the adversarial accuracy for a deterministic scheme
with no dropout ($T=1,p=0.00$) is closest to the robustness of our nominal
defense ($T=50,p=0.95$). However, for a version of our defense with
unsupervised decoder training, randomness is far more important: the attacked
accuracy (under the worst-case attack) is $30.01\%$ for our nominal encoder
($T=50,p=0.95$) versus $19.64\%$ for the deterministic encoder ($T=1,p=0.00$).
This highlights the need for further research into how best to optimize the
sparse codes produced by our encoder architecture.
| Clean | W-NFGA | PW-T
---|---|---|---
Our defense | | |
Nominal : $T=50$, $p=0.95$ | 80.06 | 61.28 | 39.53
$T=1$, $p=0.00$ | 81.48 | 65.82 | 39.42
$T=5$, $p=0.5$ | 85.26 | 67.70 | 38.63
$T=10$, $p=0.75$ | 84.78 | 64.14 | 36.40
Other architectures | | |
Sparse Autoencoder | 91.33 | 0.06 | 71.88
Classifier w/ dropout | 88.22 | 0.08 | 66.05
Gaussian blur prepr. | 91.59 | 0.00 | 72.80
Table 5: Accuracies for additional ablation studies in CIFAR-10
We also report on our experiments with three architectures which incorporate
different aspects of our defense. In the first, we use a k-sparse autoencoder
(Makhzani and Frey 2013) trained in supervised fashion. It uses $k=50$ out of
500 channels in the bottleneck layer. In the second architecture, we expand
the number of channels in the first layer of the standard classifier we use in
our earlier experiments and apply dropout (at inference time) with $p=0.95$ to
this layer. This model does not differ from the standard classifier in any
other way. In the third architecture, based on the observation in Figure 9
that our frontend results in a blurred image, we apply Gaussian blurring to
the images before they go into the classifier. The standard deviation of the
Gaussian blur filter is determined by minimizing the $\ell^{2}$ loss between
Gaussian blurred images and images processed by unsupervised version of our
defense and set to $\sigma_{G}=0.625$. As seen from the results presented in
Table 5, attacked accuracies drop to zero for all of these architectures,
indicating that the robustness of our proposed architecture results from the
unique combination of ideas incorporated in our defense.
### 8.7 Analysis of Computational Budget
In Table 3, we aim to keep the computational budget comparable across the
attacks to our and other defense methods. We can estimate the computational
budget $C$ as $C=K\times N_{S}\times N_{R}\times N_{E}$ where $K$ is the
computational cost of a single backward pass, which depends on the overall
model size. For our defense, the cost of a backward pass is $K=K_{ours}$,
which is 5.4 times the backward pass cost with $K=K_{AT}$ for PGD AT, RFGSM
AT, and TRADES; that is, $K_{ours}=5.4\times\,K_{AT}$. This is mainly due to
the large number of filters used in our decoder structure. In Table 3, PGD AT,
RFGSM AT, and TRADES take $C=5\times 10^{3}\times K_{AT}$ computation steps
whereas the default attack to our defense takes $C=8\times 10^{2}\times
K_{ours}$, which is roughly the same computational budget.
Since attacking our defense is so computationally intensive, we had used the
following settings for the attacks on our defense in Table 3:
$N_{S}=20,N_{R}=1,N_{E}=40$. These parameters were optimized so as to keep the
computational budget to within an order of magnitude of those typically needed
to effectively attack adversarially trained networks. In Table 6, we report on
adversarial accuracies for our defense as we increase the computational budget
further, by increasing each of the parameters in turn, keeping the others
fixed. The results in Table 6 confirm that increasing the computational budget
relative to the “default” attack reported on in Table 3 does not decrease the
adversarial accuracies for our defense significantly.
| PW-T
---|---
| Default | $N_{S}=100$ | $N_{E}=100$ | $N_{R}=10$
Our defense | 39.53 | 38.93 | 39.25 | 39.38
Table 6: Accuracies for different computational budget attacks to our defense
in CIFAR-10
### 8.8 Images Before and After Autoencoding
While our nominal defense employs a supervised-trained decoder, we can
visualize the information loss due to our randomized sparse coding strategy
using an unsupervised decoder trained based on reconstruction loss. We present
in Figure 9 clean and adversarial images, before and after the autoencoder,
for this unsupervised-trained version of our defense model. The encoder is as
for our nominal defense, with $T=50$, $p=0.95$.
We note that there is an appreciable degradation in sharpness due to the
autoencoder, implying that there is significant scope for improvement in our
design, despite the promising gains in robustness that have been demonstrated.
|
# Gravitational corrections to the Einstein-Scalar-QCD model
Huan Souza<EMAIL_ADDRESS>Faculdade de Física, Universidade Federal
do Pará, 66075-110, Belém, Pará, Brazil. L. Ibiapina Bevilaqua
<EMAIL_ADDRESS>Escola de Ciências e Tecnologia, Universidade
Federal do Rio Grande do Norte
Caixa Postal 1524, 59072-970, Natal, Rio Grande do Norte, Brazil. A. C. Lehum
<EMAIL_ADDRESS>Faculdade de Física, Universidade Federal do Pará, 66075-110,
Belém, Pará, Brazil.
###### Abstract
This study employs the effective field theory approach to quantum gravity to
investigate a non-Abelian gauge theory involving scalar particles coupled to
gravity. The study demonstrates explicitly that the Slavnov-Taylor identities
are maintained at one-loop order, which indicates that the universality of the
color charge is preserved. Additionally, the graviton corrections to the two-
loop gluon self-energy and its renormalization are computed.
## I Introduction
Although we are still in need of a consistent and generally accepted
description of quantum gravity at high energies, if we restrict ourselves to
low energies compared to the Planck scale, we can nevertheless draw some
trustful conclusions about the gravitational phenomena at quantum level using
the viewpoint and methods of effective field theories Donoghue:1994dn ;
Burgess:2003jk ; Shapiro . Thus, the well known nonrenormalizability of
Einstein’s theory coupled to other fields 'tHooft:1974bx ; PhysRevLett.32.245
; Deser:1974cy is not an impediment to study the influence of gravity in the
renormalization of other fields and parameters in a meaningful way. The
central idea is that we add to the action the high-order terms needed to
renormalize the parameters of the lower-order terms and the new parameters
introduced will be irrelevant to the low-energy behavior of the theory.
As it is well known, the renormalized quantities of a theory depend on an
arbitrary scale and the renormalization group is the theoretical tool to study
this dependence and allows us to describe how the coupling constants change
with this scale, establishing the so-called running of the coupling constants
Srednicki:2007qs . If this dependence is such that the coupling constant gets
weaker as we go to higher energies the theory is said to be asymptotically
free Gross:1973id ; Politzer:1973fx ; Gross:1974jv . The possibility that
gravitational corrections could render all gauge coupling constants
asymptotically free was suggested by Robinson and Wilczek, who used the
effective field theory approach of quantum gravity to reach this conclusion
Robinson:2005fj . However, this result was soon contested by Pietrykowski
Pietrykowski:2006xy , who showed that the result was gauge dependent.
Subsequently, many works investigate the use of the renormalization group in
quantum gravity as an effective field theory (See for instance Refs.
Felipe:2012vq ; Felipe:2013vq ; Ebert:2007gf ; Nielsen:2012fm ; Toms:2008dq ;
Toms:2010vy ; Ellis:2010rw ; Anber:2010uj ; Bevilaqua:2015hma ;
Bevilaqua:2021uzk ; Bevilaqua:2021uev ). In a previous work Bevilaqua:2015hma
, we used dimensional regularization to compute gravitational effects on the
beta function of the scalar quantum electrodynamics at one-loop order and
found that all gravitational contributions cancel out. The situation is
different at two-loop order, in which we do find nonzero gravitational
corrections to the beta function for both scalar and fermionic QED, as shown
in a latter work Bevilaqua:2021uzk . However, those corrections give a
positive contribution to the beta function and thus the electrical charge is
not asymptotically free neither has a nontrivial fixed point.
The use of renormalization group in the context of non-renormalizable field
theories raise some subtle questions. The universality of the coupling
constants in effective field theories was discussed by Anber et al. in
Anber:2010uj , where it was suggested that an operator mixing could make the
coupling constants dependent on the process under consideration and therefore
non-universal. That would imply that, unlike renormalizable field theories,
the concept of running coupling may not be useful in the effective field
theory approach to quantum gravity. This is indeed the case for the quartic
self-interaction of scalars in scalar-QED, as discussed in Bevilaqua:2015hma
but, as shown in Bevilaqua:2015hma for scalar-QED and in Bevilaqua:2021uev
for fermionic-QED it seems not to be the case for the gauge coupling because
of the Ward identity. The central role of the gauge symmetry in the
universality of the gauge coupling for QED led us to explore this issue in the
non-Abelian case. Using dimensional regularization, we showed that the
Slavnov-Taylor identities are satisfied in a non-Abelian gauge theory coupled
to fermions and gravity Souza:2022ovu . In the same work, we have also
calculated the gravitatinal correction for the beta function at one-loop thus
verifying directly the absence of contributions from the gravitational sector.
In previous studies, the coupling of non-Abelian gauge theories to gravity has
been investigated Souza:2022ovu ; Buchbinder:1983nug ; Tang:2008ah ;
Tang:2011gz . In this research, we extend our previous analysis by
investigating the asymptotic behavior of a non-Abelian gauge theory coupled to
complex scalars and gravity. This exploration is motivated by the significant
role scalar theories play in the advancement of high-energy theory. Over the
years, scalar models have been proposed to tackle issues such as
renormalization group theory for non-renormalizable theories Barvinsky:1993zg
, the study of dilatons Shapiro:1995yc , and potential candidates for dark
matter Cohen:2011ec ; Arkani-Hamed:2008hhe .. In fact, Ref. Calmet:2021iid
argue that quantum gravity might have crucial implications in a theory of dark
matter. Additionally, a recent study and:2022ttn investigated the interaction
between SU(2) Yang-Mills waves and gravitational waves. The results revealed
that while the problem can be perturbatively studied in the symmetric phase,
non-perturbative approaches are necessary in the broken phase. Hence, the
examination of a non-Abelian gauge theory coupled to complex scalars and
gravity is of particular interest due to the fundamental role scalar theories
have played in addressing diverse problems in high-energy theory.
The paper is structured as follows. Section II introduces the Lagrangian and
propagators of the model. In Section III, the one-loop renormalization of the
model is presented, highlighting the preservation of gauge invariance of the
gravitational interaction and respect for the Slavnov-Taylor identities.
Section IV utilizes the Tarasov algorithm to compute the two-loop counterterm
for the gluon wave-function. Finally, concluding remarks are provided in
Section V. The minimal subtraction (MS) scheme is used throughout this work to
handle the UV divergences, with $(+---)$ being the spacetime signature and
natural units of $c=\hbar=1$ are adopted.
## II The Einstein-Scalar-QCD model
To get an effective field theory description for our model, we add higher
order terms to the Lagrangian of a non-Abelian gauge theory with complex
scalars coupled to gravity:
$\displaystyle\mathcal{L}=$
$\displaystyle\sqrt{-g}\sum_{f}\Big{\\{}\frac{2}{\kappa^{2}}R-\frac{1}{4}g^{\mu\alpha}g^{\nu\beta}G_{\mu\nu}^{a}G_{\alpha\beta}^{a}+g^{\mu\nu}(D_{\mu}\phi^{i})^{\dagger}D_{\nu}\phi^{i}-m_{i}(\phi^{i})^{\dagger}\phi^{i}+\lambda((\phi^{i})^{\dagger}\phi^{i})^{2}+\mathcal{L}_{HO}\Big{\\}},$
(1)
where the index $i=1,2,\cdots,N_{s}$ runs over the scalars flavors,
$G^{a}_{\mu\nu}=\nabla_{\mu}A_{\nu}^{a}-\nabla_{\nu}A_{\mu}^{a}+gf^{abc}A^{b}_{\mu}A^{c}_{\nu}$
is the non-Abelian field-strength with $f^{abc}$ being the structure constants
of the $SU(N)$ group, and $D_{\mu}=\partial_{\mu}-igt^{a}A^{a}_{\mu}$ is the
covariant derivative. The higher order terms $\mathcal{L}_{HO}$ are written as
$\mathcal{L}_{HO}=\frac{\tilde{\lambda}_{1}}{M_{P}^{2}}\left[\mathrm{Re}((\phi^{i})^{\dagger}\partial_{\mu}\phi^{i})\right]^{2}+\frac{\tilde{\lambda}_{2}}{M_{P}^{2}}\left[\mathrm{Im}((\phi^{i})^{\dagger}\partial_{\mu}\phi^{i})\right]^{2}-\frac{\tilde{e}_{3}}{4}G_{a}^{\mu\nu}\frac{\Box}{M_{P}^{2}}G^{a}_{\mu\nu}.$
(2)
To obtain the usual quadratic term for the gravitational field, we need to
expand $g_{\mu\nu}$ around the flat metric as
$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu},$ (3)
such that
$g^{\mu\nu}=\eta^{\mu\nu}-\kappa
h^{\mu\nu}+\cdots\qquad\text{and}\qquad\sqrt{-g}=1+\frac{\kappa}{2}h+\cdots,$
(4)
where $h=\eta^{\mu\nu}h_{\mu\nu}$. The affine connection is written as
$\Gamma^{\lambda}_{\leavevmode\nobreak\
\mu\nu}=\frac{1}{2}\kappa(\eta^{\lambda\sigma}-\kappa
h^{\lambda\sigma})(\partial_{\mu}h_{\sigma\nu}+\partial_{\nu}h_{\sigma\mu}-\partial_{\sigma}h_{\mu\nu}).$
(5)
Organizing the Lagrangian as,
$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{h}+\mathcal{L}_{f}+\mathcal{L}_{A};$ (6a)
$\displaystyle\mathcal{L}_{h}$ $\displaystyle=$
$\displaystyle\frac{2}{\kappa^{2}}\sqrt{-g}R;$ (6b)
$\displaystyle\mathcal{L}_{s}$ $\displaystyle=$
$\displaystyle\sqrt{-g}[g^{\mu\nu}(D_{\mu}\phi^{i})^{\dagger}D_{\nu}\phi^{i}-m_{i}(\phi^{i})^{\dagger}\phi^{i}+\lambda((\phi^{i})^{\dagger}\phi^{i})^{2}];$
(6c) $\displaystyle\mathcal{L}_{A}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{-g}}{4}g^{\mu\alpha}g^{\nu\beta}G_{\mu\nu}^{a}G^{a}_{\alpha\beta}.$
(6d)
Using Eqs. (3)-(5), we write the pure gravity sector (6b) in terms of
$h_{\mu\nu}$. Moreover, it is convinient to organize $\mathcal{L}_{h}$ in
powers of $h$ as follows:
$\displaystyle\mathcal{L}_{h}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{h}^{0}+\kappa\mathcal{L}_{h}^{1}+\cdots$ (7a)
$\displaystyle\mathcal{L}_{h}^{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\partial_{\mu}h\partial^{\mu}h+\frac{1}{2}\partial_{\mu}h^{\sigma\nu}\partial^{\mu}h_{\sigma\nu};$
(7b) $\displaystyle\mathcal{L}_{h}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}h^{\alpha}_{\leavevmode\nobreak\
\beta}\partial^{\mu}h^{\beta}_{\leavevmode\nobreak\
\alpha}\partial_{\mu}h-\frac{1}{2}h^{\alpha}_{\leavevmode\nobreak\
\beta}\partial_{\alpha}h^{\mu}_{\leavevmode\nobreak\
\nu}\partial^{\beta}h^{\nu}_{\leavevmode\nobreak\
\mu}-h^{\alpha}_{\leavevmode\nobreak\
\beta}\partial_{\mu}h^{\nu}_{\leavevmode\nobreak\
\alpha}\partial^{\mu}h^{\beta}_{\leavevmode\nobreak\ \nu}$ (7c)
$\displaystyle+\frac{1}{4}h\partial^{\beta}h^{\mu}_{\leavevmode\nobreak\
\nu}\partial_{\beta}h^{\nu}_{\leavevmode\nobreak\
\mu}+h^{\beta}_{\leavevmode\nobreak\
\mu}\partial_{\nu}h^{\alpha}_{\leavevmode\nobreak\
\beta}\partial^{\mu}h^{\nu}_{\leavevmode\nobreak\
\alpha}-\frac{1}{8}h\partial^{\nu}h\partial_{\nu}h,$
where the indices are raised and lowered with the flat metric (here and
henceforth, we are following the results in Ref. Choi:1994ax ).
For the matter sector (6c), the expansion around the flat metric give us
$\displaystyle\mathcal{L}_{s}$ $\displaystyle=$
$\displaystyle(D^{\mu}\phi^{i})^{\dagger}D_{\mu}\phi^{i}-m^{2}_{i}((\phi^{i})^{\dagger}\phi^{i})-\frac{\lambda}{4}((\phi^{i})^{\dagger}\phi^{i})^{2}-\kappa
h^{\mu\nu}(D_{\mu}\phi^{i})^{\dagger}D_{\nu}\phi^{i}$ (8a)
$\displaystyle+\frac{\kappa}{2}h\left[(D^{\mu}\phi^{i})^{\dagger}D_{\mu}\phi^{i}-m^{2}_{i}(\phi^{i})^{\dagger}\phi^{i}-\frac{\lambda}{4}((\phi^{i})^{\dagger}\phi^{i})^{2}\right],$
which we organize as follows
$\displaystyle\mathcal{L}_{s}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{s}^{0}+\kappa\mathcal{L}_{s}^{1}+\cdots$ (9a)
$\displaystyle\mathcal{L}_{s}^{0}$ $\displaystyle=$
$\displaystyle(D^{\mu}\phi^{i})^{\dagger}D_{\mu}\phi^{i}-m_{i}^{2}((\phi^{i})^{\dagger}\phi^{i})-\frac{\lambda}{4}((\phi^{i})^{\dagger}\phi^{i})^{2}$
(9b) $\displaystyle\mathcal{L}_{s}^{1}$ $\displaystyle=$
$\displaystyle-h^{\mu\nu}(D_{\mu}\phi^{i})^{\dagger}D_{\nu}\phi^{i}+\frac{1}{2}h\left[(D^{\mu}\phi^{i})^{\dagger}D_{\mu}\phi^{i}-m_{i}^{2}(\phi^{i})^{\dagger}\phi^{i}-\frac{\lambda}{4}((\phi^{i})^{\dagger}\phi^{i})^{2}\right];$
(9c)
and finally, for the gauge sector,
$\displaystyle\mathcal{L}_{A}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{A}^{0}+\kappa\mathcal{L}_{A}^{1}+\cdots$ (10a)
$\displaystyle\mathcal{L}_{A}^{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}G_{\mu\nu}^{a}G^{\mu\nu}_{a}$ (10b)
$\displaystyle\mathcal{L}_{A}^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}h^{\tau}_{\leavevmode\nobreak\
\nu}G^{\mu\nu}_{a}G_{\mu\tau}^{a}+\frac{1}{2}h\mathcal{L}_{A}^{0}.$ (10c)
As usual for gauge theories, in order to quantize this model, we have to deal
with the excess of degrees of freedom in $A_{\mu}^{a}$ and $h_{\mu\nu}$ due to
their symmetries. In our calculations, we have followed the Faddeev-Popov
procedure that introduces gauge-fixing terms in the action that will modify
the propagators of both $A_{\mu}^{a}$ and $h_{\mu\nu}$. Moreover, we must also
introduce ghost fields for both vector and tensor fields. However, the ghost
field associated with the graviton will not appear in this text because, since
we are working with the one-graviton exchange approximation, the new term
containing the ghosts added to the action will not contribute to the
renormalization of the gauge coupling constant. Therefore, whenever we refer
to ghost field in what follows, we mean the one associated with $A_{\mu}^{a}$.
The propagators for scalars, ghosts, gluons and gravitons are given,
respectively, by
$\displaystyle\Delta_{s}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{p^{2}-m_{a}^{2}};$ (11a) $\displaystyle\Delta_{ab}(p)$
$\displaystyle=$ $\displaystyle\frac{i}{p^{2}}\delta_{ab};$ (11b)
$\displaystyle\Delta^{\mu\nu}_{ab}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{p^{2}}\left(\eta^{\mu\nu}-(1-\xi_{A})\frac{p^{\mu}p^{\nu}}{p^{2}}\right)\delta_{ab};$
(11c) $\displaystyle\Delta^{\alpha\beta\mu\nu}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{p^{2}}\left(P^{\alpha\beta\mu\nu}-(1-\xi_{h})\frac{Q^{\alpha\beta\mu\nu}}{p^{2}}\right).$
(11d)
The gauge-fixing parameters $\xi_{A}$ and $\xi_{h}$ will be carried out
through the whole calculation, since we do not want to choose any specific
gauge. The projectors $P^{\alpha\beta\mu\nu}$ and $Q^{\alpha\beta\mu\nu}$ in
the graviton propagator are given by
$\displaystyle P^{\alpha\beta\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\eta^{\alpha\mu}\eta^{\beta\nu}+\eta^{\alpha\nu}\eta^{\beta\mu}-\eta^{\alpha\beta}\eta^{\mu\nu}\right);$
$\displaystyle Q^{\alpha\beta\mu\nu}$ $\displaystyle=$
$\displaystyle(\eta^{\alpha\mu}p^{\beta}p^{\nu}+\eta^{\alpha\nu}p^{\beta}p^{\mu}+\eta^{\beta\mu}p^{\alpha}p^{\nu}+\eta^{\beta\nu}p^{\alpha}p^{\mu}).$
(12)
## III The one-loop renormalization
The Slavnov-Taylor identities are a set of relations that must be satisfied by
the n-point functions to ensure the gauge independence of the observables of
the theory. In this section we want to explicitly show that the Slavnov-Taylor
identities are respected at one-loop order for our model. To simplify our
computations, we will consider here that all the masses are the same, so we
drop the index $i$. As we will see, this will not affect our final result.
We start by computing the n-point functions. Namely, the self-energy of
scalar, vector and ghost fields ($\Sigma_{s},\Pi^{\mu\nu}_{ab}$ and
$\Sigma_{ab}$, respectively), also the scalar-gluon, ghost-gluon and gluon-
gluon three-point functions ($\Gamma^{\mu}_{a}$, $\Gamma^{\mu}_{abc}$ and
$\Pi^{\mu\nu\alpha}_{abc}$, respectively), the gluon four-point function
($\Gamma^{\mu\nu\rho\sigma}_{abcd}$), and finally the scalar-gluon four-point
function ($\Pi^{\mu\nu}_{abcd}$). All the computations were done using the
Mathematica packages: FeynRules to generate the models feynrules , FeynArts to
draw the diagrams Hahn:2000kx , and FeynCalc to simplify and compute the
amplitudes Shtabovenko:2020gxv .
At one-loop, the self-energy of the scalar field, Fig. 1, results in
$\displaystyle-i\Sigma_{s}(p)$
$\displaystyle=ip^{2}\left(\frac{C_{A}\left(\xi_{A}-3\right)g^{2}-(\xi_{h}-2)\kappa^{2}m^{2}}{16\pi^{2}\epsilon}+Z_{2s}^{(1)}\right)$
(13) $\displaystyle+im^{2}\left(\frac{-C_{A}\xi_{A}g^{2}+4\lambda
N_{s}-(\xi_{h}-2)\kappa^{2}m^{2}}{16\pi^{2}\epsilon}-Z_{m_{s}}^{(1)}\right)+\mathrm{finite},$
where $C_{A}=N$ for the $SU(N)$ group. By imposing finiteness to
$\Sigma_{s}(p)$, we find the following one-loop counterterms:
$\displaystyle Z_{2s}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}m^{2}\left(\xi_{h}-2\right)-C_{A}\left(\xi_{A}-3\right)g^{2}}{16\pi^{2}\epsilon},$
(14a) $\displaystyle Z_{m}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{-C_{A}\xi_{A}g^{2}+4\lambda
N_{s}-(\xi_{h}-2)\kappa^{2}m^{2}}{16\pi^{2}\epsilon}.$ (14b)
Figure 1: Feynman diagrams for the scalar self-energy. Continuous, wiggly,
dotted, and dashed lines represent the scalar, gluon, ghost, and graviton
propagators, respectively.
For the gluon self-energy, it is convenient to write the one-loop correction
(corresponding to the diagrams in Fig. 2) as
$\displaystyle\Pi^{\mu\nu}_{ab}(p)=\left(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu}\right)\Pi(p)\delta_{ab},$
(15)
where the function $\Pi(p)$ is found to be
$\displaystyle\Pi(p)=-iZ_{3}^{(1)}-ip^{2}\tilde{Z}_{3}^{(1)}+\frac{i\kappa^{2}p^{2}\left(2-3\xi_{h}\right)}{96\pi^{2}\epsilon}-\frac{iC_{A}g^{2}\left(2N_{s}+3\xi_{A}-13\right)}{96\pi^{2}\epsilon}+\mathrm{finite},$
(16)
and, imposing the finiteness on $\Pi(p)$, we find
$\displaystyle Z_{3}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{C_{A}g^{2}\left(2N_{s}+3\xi_{A}-13\right)}{96\pi^{2}\epsilon},$
(17a) $\displaystyle\tilde{Z}_{3}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{\kappa^{2}(3\xi_{h}-2)}{96\pi^{2}\epsilon}\leavevmode\nobreak\
.$ (17b)
We can see from Eq. (16) that $Z_{3}$ is the relevant counterterm to the beta
function of the color charge, since it is the renormalizing factor for the
quadratic term $G^{\mu\nu}_{a}G_{\mu\nu}^{a}$, while $\tilde{Z}_{3}$
renormalizes a higher derivative term like $G^{\mu\nu}_{a}\Box
G_{\mu\nu}^{a}$. Notice also that the UV divergent part of Eq. (16) is not
dependent on the masses of the scalars.
Figure 2: Feynman diagrams for the gluon self-energy. Figure 3: Feynman
diagrams for the ghost self-energy.
Contributions to the ghost self-energy up to one-loop order are depicted in
Fig. 3. The resulting expression is
$\displaystyle-i\Sigma_{ab}$ $\displaystyle=$
$\displaystyle\left(\frac{ip^{2}C_{A}\left(\xi_{A}-3\right)g^{2}}{64\pi^{2}\epsilon}+ip^{2}Z_{2c}^{(1)}\right)\delta_{ab}+\mathrm{finite},$
(18)
and, imposing finiteness, we find
$\displaystyle Z_{2_{c}}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{C_{A}g^{2}\left(\xi_{A}-3\right)}{64\pi^{2}\epsilon}.$
(19)
Notice that in Fig. 3 the gravitational interactions are not shown. Although
in the action there is a coupling of $h^{\mu\nu}$ to the kinetic term of the
ghosts associated with the gluons, the gravitational contributions to the
ghost self-energy will be renormalized by a higher-order term and is therefore
irrelevant for our purposes here. One way to see why this is happens is to
observe that both the ghosts and the graviton are massless, so the only
contribution proportional to $\kappa^{2}$ must be of the order $p^{4}$.
Figure 4: Feynman diagrams for the vertex interaction between gluons and
ghosts up to one-loop order.
For the 3-point functions, let’s first consider the ghost-ghost-gluon vertex
(Fig. 4), where again all the gravitational corrections are renormalized by
higher-order terms and are therefore omitted here. Also, in the following
expressions, we will use $p_{1}$ and $p_{2}$ to represent incoming external
momenta, and $p_{3}$ and $p_{4}$ for outgoing momenta. The expression obtained
for these diagrams is
$\Gamma_{abc}^{\mu}=-gp_{3}^{\mu}f_{abc}\left(\frac{C_{A}g^{2}\xi_{A}}{32\pi^{2}\epsilon}+Z_{1c}^{(1)}\right)+\text{finite},$
(20)
and the subtraction of the UV pole will give us
$Z_{1_{c}}^{(1)}=-\frac{C_{A}g^{2}\xi_{A}}{32\pi^{2}\epsilon}.$ (21)
Figure 5: Feynman diagrams to the vertex interaction between quarks top and
gluons up to one-loop order.
For the other 3-point function, the scalar-scalar-gluon vertex, the
gravitational interaction will be present in some diagrams, as we can see in
Fig. 5, where the relevant contributions to this function up to one-loop order
are shown. The resulting expression is
$\displaystyle-i\Gamma^{\mu}_{abc}$ $\displaystyle=$ $\displaystyle
gf_{abc}(p^{\mu}_{2}-p^{\mu}_{3})\left(\frac{C_{A}\left(9-5\xi_{A}\right)g^{2}+4\kappa^{2}m^{2}\left(\xi_{h}-2\right)}{64\pi^{2}\epsilon}-Z_{1}^{(1)}\right)$
(22) $\displaystyle+O(p^{3})+\text{finite},$
from which, through MS, we find
$Z_{1}^{(1)}=\frac{C_{A}\left(9-5\xi_{A}\right)g^{2}+4\kappa^{2}m^{2}\left(\xi_{h}-2\right)}{64\pi^{2}\epsilon}.$
(23)
The 3-point function describing the vertex with three gluons in shown in Fig.
6. We have used the projection
Figure 6: Feynman diagrams to the gluons vertex interaction at one-loop order.
$\Pi^{\mu\nu\alpha}_{abc}=\eta^{\mu\nu}\Pi^{\alpha}_{abc}\qquad\Rightarrow\qquad\Pi^{\alpha}_{abc}=\frac{1}{4}\eta_{\mu\nu}\Pi^{\mu\nu\alpha}_{abc}$
(24)
and used the fact that $p_{3}=p_{1}+p_{2}$, to get
$\displaystyle-i\Pi^{\alpha}_{abc}$ $\displaystyle=$
$\displaystyle\frac{g^{3}f_{abc}C_{A}\left(-9\xi_{A}-4N_{s}+17\right)(p_{1}-p_{2})^{\alpha}}{256\pi^{2}\epsilon}-\frac{3}{4}Z_{3g}^{(1)}g\left(p_{1}-p_{2}\right){}^{\alpha}f_{abc}$
(25) $\displaystyle+O(p^{2})+\mathrm{finite},$
Through MS, we impose finiteness and find
$Z_{3g}^{(1)}=-\frac{g^{2}C_{A}\left(9\xi_{A}-17-4N_{s}\right)}{192\pi^{2}\epsilon}.$
(26)
Now, we consider the scattering of four gluons (Fig. 7 showed at the end of
the paper for convenience). Since the interaction of four gluons has no
derivatives, the $Z_{4g}$ counterterm will renormalize terms proportional to
$p^{0}$ and therefore we can set external momentum equals to zero if we
restrict ourselves to the computation of this counterterm. Also, for
simplicity, we have used the scalar projection
$\Gamma_{abcd}=\frac{1}{16}\eta_{\mu\nu}\eta_{\rho\sigma}\Gamma^{\mu\nu\rho\sigma}_{abcd},$
(27)
to obtain the expression for the gluon 4-point function
$\displaystyle-i\Gamma_{abcd}$ $\displaystyle=$
$\displaystyle-\left(\frac{iC_{A}g^{4}\left(N_{s}+3\xi_{A}-2\right)}{32\pi^{2}\epsilon}+\frac{3}{2}iZ_{4g}^{(1)}g^{2}\right)\Bigr{(}\text{tr}(t_{a}t_{b}t_{c}t_{d})-2\text{tr}(t_{a}t_{c}t_{b}t_{d})-2\text{tr}(t_{b}t_{c}t_{a}t_{d})$
(28)
$\displaystyle+\text{tr}(t_{b}t_{a}t_{c}t_{d})+\text{tr}(t_{c}t_{a}t_{b}t_{d})+\text{tr}(t_{c}t_{b}t_{a}t_{d})\Bigr{)},$
Then, again imposing finiteness through MS, we have
$Z^{(1)}_{1_{4g}}=-\frac{C_{A}g^{2}\left(N_{s}+3\xi_{A}-2\right)}{48\pi^{2}\epsilon}.$
(29)
The other 4-point function involves two scalars and two gluons (Fig. 8, again
showed at the end of the paper for convenience). For this vertex, we use the
following projection
$\Pi^{\mu\nu}_{abcd}=\eta^{\mu\nu}\Pi_{abcd}\qquad\Rightarrow\qquad\Pi_{abcd}=\frac{1}{4}\eta_{\mu\nu}\Pi^{\mu\nu}_{abcd}$
(30)
and then we have
$\displaystyle\Pi_{abcd}$
$\displaystyle=\left(\frac{ig^{2}-3C_{A}\left(\xi_{A}-1\right)g^{2}-2(\xi_{h}-2)\kappa^{2}m^{2}}{16\pi^{2}\epsilon}-2iZ_{2g}^{(1)}g^{2}\right)\Bigr{(}2\text{tr}(t_{a}t_{b}t_{c}t_{d})-\text{tr}(t_{a}t_{c}t_{b}t_{d})$
(31)
$\displaystyle-\text{tr}(t_{b}t_{a}t_{c}t_{d})-\text{tr}(t_{b}t_{c}t_{a}t_{d})-\text{tr}(t_{c}t_{a}t_{b}t_{d})+2\text{tr}(t_{c}t_{b}t_{a}t_{d})\Bigr{)}.$
and the counterterm is found to be
$Z_{2g}^{(1)}=-\frac{3C_{A}\left(\xi_{A}-1\right)g^{2}-2(\xi_{h}-2)\kappa^{2}m^{2}}{32\pi^{2}\epsilon}.$
(32)
From Eqs. (14a), (17a), (19), (21), (23), (26), (29) we conclude that
$Z_{1}^{(1)}-Z_{2s}^{(1)}=Z_{3g}^{(1)}-Z_{3}^{(1)}=\frac{1}{2}\left(Z_{4g}^{(1)}-Z_{3}^{(1)}\right)=\frac{1}{2}\left(Z_{2g}^{(1)}-Z_{2s}^{(1)}\right)=Z_{1c}^{(1)}-Z_{2c}^{(1)}=-\frac{C_{A}g^{2}(3+\xi_{A})}{64\pi^{2}\epsilon}$
(33)
so the Slavnov-Taylor identities Slavnov:1972fg ; Taylor:1971ff are indeed
respected and thus gravitational interaction does not spoil the gauge
symmetry. This result allows us to define a global color charge.
Moreover, we can show that the beta function is independent of $\kappa$ and
$m$, as the expression the one-loop beta function of the color charge can be
found through the relations between the renormalized coupling constants and
the counterterms given by
$\displaystyle g$ $\displaystyle=$
$\displaystyle\mu^{-2\epsilon}\frac{Z_{2s}Z_{3}^{1/2}}{Z_{1}}g_{0};$ (34a)
$\displaystyle g$ $\displaystyle=$
$\displaystyle\mu^{-2\epsilon}\frac{Z_{3}^{3/2}}{Z_{3g}}g_{0};$ (34b)
$\displaystyle g$ $\displaystyle=$
$\displaystyle\mu^{-2\epsilon}\frac{Z_{3}}{Z_{4g}^{1/2}}g_{0};$ (34c)
$\displaystyle g$ $\displaystyle=$
$\displaystyle\mu^{-2\epsilon}\frac{Z_{2c}Z_{3}^{1/2}}{Z_{1c}}g_{0};$ (34d)
$\displaystyle g$ $\displaystyle=$
$\displaystyle\mu^{-2\epsilon}\frac{Z_{2}^{1/2}Z_{3}^{1/2}}{Z_{2g}^{1/2}}g_{0}.$
(34e)
Therefore, the beta function for the color charge is
$\displaystyle\beta(g)$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\rightarrow
0}\mu\frac{dg}{d\mu}=\lim_{\epsilon\rightarrow
0}\mu\frac{d}{d\mu}\left[g_{0}\left(1-Z_{1}^{(1)}+Z_{2s}^{(1)}+\frac{Z_{3}^{(1)}}{2}\right)\mu^{-2\epsilon}\right]$
(35) $\displaystyle=$
$\displaystyle-\frac{g^{3}}{(4\pi)^{2}}\left(\frac{11}{3}C_{A}-\frac{2}{6}N_{s}\right).$
The observed outcome is gauge-independent, a characteristic that was
previously established via a functional approach in Ref.Folkerts:2011jz . This
property has also been verified in the context of the Effective Field Theory
of gravity when coupled with fermionic QCD in Souza:2022ovu .
As we can see, it does not depend on the mass, so our choice to make all
masses the same does not affect our result for the beta function at one-loop
order. On the other hand, as discussed in Bevilaqua:2021uev , at two-loop we
would expect a $\sum_{i}\kappa^{2}m_{i}^{2}$ term.
It is needed to stress here the importance of a regularization scheme that
preserves the symmetries of the model. In fact, the authors in
Ref.Folkerts:2011jz showed that in the weak-gravity limit there is no
gravitational contribution at one-loop order if the regularization scheme
preserves the symmetries of the model, such as dimensional regularization. On
the other hand, if the regularization scheme does not preserve all the
symmetries, there will be a negative contribution to the beta function (as
seen in Robinson:2005fj ).
## IV Two-loop Gluon self-energy
This section presents the computation of the two-loop gluon self-energy and
its renormalization. TARCER Mertig:1998vk , in combination with previously
cited Mathematica packages, is utilized for this computation. TARCER
implements the Tarasov algorithm for the reduction of two-loop scalar
propagator type integrals with external momentum and arbitrary masses
Tarasov:1997kx . The Feynman and harmonic gauges ($\xi_{A}=\xi_{h}=1$) are
used for simplicity, and the analysis is limited to the case in which there is
only one scalar particle ($N_{s}=1$).
The Feynman diagrams we need to compute are showed in Fig. 9. Due to gauge
invariance, our result can be expressed as
$\Pi^{(2)}_{\mu\nu}=\left(p^{2}g_{\mu\nu}-p_{\mu}p_{\nu}\right)\Pi^{(2)},$
(36)
where the function $\Pi^{(2)}$ is a scalar function that can be expressed in
terms of a set of basic integrals. To present the results in a simplified
manner, we will adopt a notation similar to the one used in the original
TARCER paper Mertig:1998vk for the basic integrals that will be utilized,
$\displaystyle\textbf{A}_{\nu}(m)=\frac{1}{\pi^{D/2}}\int\frac{d^{D}k}{[k^{2}-m^{2}]^{\nu}}$
(37a)
$\displaystyle\textbf{B}_{\nu_{1},\nu_{2}}(m_{1},m_{2})=\frac{1}{\pi^{D/2}}\int\frac{d^{D}k}{[k^{2}-m_{1}^{2}]^{\nu_{1}}[(k-p)^{2}-m_{2}^{2}]^{\nu_{2}}}$
(37b)
$\displaystyle\textbf{J}_{\nu_{1},\nu_{2},\nu_{3}}(m_{1},m_{2},m_{3})=\frac{1}{\pi^{D}}\int\frac{d^{D}k_{1}d^{D}k_{2}}{[k_{1}^{2}-m_{1}^{2}]^{\nu_{1}}[k_{5}^{2}-m_{2}^{2}]^{\nu_{2}}[k_{4}^{2}-m_{3}^{2}]^{\nu_{3}}}$
(37c)
$\displaystyle\textbf{F}_{\nu_{1},...,\nu_{5}}(m_{1},...,m_{5})=\frac{1}{\pi^{D}}\int\frac{d^{D}k_{1}d^{D}k_{2}}{[k_{1}^{2}-m_{1}^{2}]^{\nu_{1}}[k_{2}^{2}-m_{2}^{2}]^{\nu_{2}}[k_{3}^{2}-m_{3}^{2}]^{\nu_{3}}[k_{4}^{2}-m_{4}^{2}]^{\nu_{4}}[k_{5}^{2}-m_{5}^{2}]^{\nu_{5}}},$
(37d)
in which $p$ is the external momentum and we introduced $k_{3}=k_{1}-p$,
$k_{4}=k_{2}-p$, and $k_{5}=k_{1}-k_{2}$.
Therefore, we can write
$\displaystyle\Pi^{(2)}$ $\displaystyle=$ $\displaystyle
c_{1}\leavevmode\nobreak\ \textbf{A}_{1}(m)\leavevmode\nobreak\
\textbf{B}_{1,1}(0,0)+c_{2}\leavevmode\nobreak\
\textbf{A}_{1}(m)\leavevmode\nobreak\
\textbf{B}_{1,1}(m,m)+c_{3}\leavevmode\nobreak\
\textbf{B}_{1,1}(0,0)\leavevmode\nobreak\
\textbf{B}_{1,1}(m,m)+c_{4}\left(\textbf{A}_{1}(m)\right)^{2}$ (38)
$\displaystyle
c_{5}\left(\textbf{B}_{1,1}(0,0)\right)^{2}+c_{6}\left(\textbf{B}_{1,1}(m,m)\right)^{2}+c_{7}\leavevmode\nobreak\
\textbf{J}_{1,1,1}(0,0,0)+c_{8}\leavevmode\nobreak\
\textbf{J}_{1,1,1}(m,m,0)+c_{9}\leavevmode\nobreak\ \textbf{J}_{2,1,1}(m,m,0)$
$\displaystyle c_{10}\leavevmode\nobreak\
\textbf{F}_{1,1,1,1,1}(0,m,0,m,m)+c_{11}\leavevmode\nobreak\
\textbf{F}_{1,1,1,1,1}(m,0,m,0,m).$
All of the aforementioned integrals are established and can be found in
Refs.Martin:2005qm ; Martin:2003qz , and the coefficients $c_{i}$ are
presented in appendix A. As we are only concerned with the renormalization of
the gluon wave-function, we expand Eq.(38) around $p=0$ and retain only terms
proportional to $p^{0}$. Higher powers in the external momentum will be
renormalized by higher-order terms. Thus, we obtain:
$\displaystyle\Pi^{(2)}$ $\displaystyle=$ $\displaystyle-\frac{i\lambda
C_{A}\leavevmode\nobreak\
g^{2}}{384\pi^{4}\epsilon}-\frac{i\kappa^{2}m^{2}C_{A}\leavevmode\nobreak\
g^{2}}{256\pi^{4}\epsilon}+\frac{iC_{A}^{2}\leavevmode\nobreak\
g^{4}\log\left(m^{2}\right)}{384\pi^{4}\epsilon}-\frac{iC_{A}^{2}\leavevmode\nobreak\
g^{4}\log\left(-p^{2}\right)}{64\pi^{4}\epsilon}-\frac{i\lambda
C_{A}\leavevmode\nobreak\ g^{2}}{384\pi^{4}\epsilon}+\frac{5i\gamma
C_{A}^{2}\leavevmode\nobreak\ g^{4}}{384\pi^{4}\epsilon}$ (39)
$\displaystyle+\frac{17iC_{A}^{2}\leavevmode\nobreak\
g^{4}}{576\pi^{4}\epsilon}+\frac{5i\log(4\pi)C_{A}^{2}\leavevmode\nobreak\
g^{4}}{384\pi^{4}\epsilon}+\frac{5iC_{A}^{2}\leavevmode\nobreak\
g^{4}}{768\pi^{4}\epsilon^{2}}+O(p)+\mathrm{finite}.$
Now, we should compute the 1-loop diagrams with counterterms insertion in Fig.
10. By doing so, we obtain
$\displaystyle\Pi^{(2)}_{\mu\nu
CT}=(p^{2}g_{\mu\nu}-p_{\mu}p_{\nu})\Pi^{(2)}_{CT},$ (40)
where
$\displaystyle\Pi^{(2)}_{CT}$ $\displaystyle=$
$\displaystyle-\frac{iC_{A}^{2}\leavevmode\nobreak\
g^{4}\log\left(m^{2}\right)}{384\pi^{4}\epsilon}+\frac{iC_{A}^{2}\leavevmode\nobreak\
g^{4}\log\left(-p^{2}\right)}{64\pi^{4}\epsilon}-\frac{5iC_{A}^{2}\leavevmode\nobreak\
g^{4}}{384\pi^{4}\epsilon^{2}}+\frac{i\lambda C_{A}\leavevmode\nobreak\
g^{2}}{192\pi^{4}\epsilon}-\frac{5i\gamma C_{A}^{2}\leavevmode\nobreak\
g^{4}}{384\pi^{4}\epsilon}$ (41)
$\displaystyle-\frac{59iC_{A}^{2}\leavevmode\nobreak\
g^{4}}{2304\pi^{4}\epsilon}-\frac{5i\log(4\pi)C_{A}^{2}\leavevmode\nobreak\
g^{4}}{384\pi^{4}\epsilon}+O(p)+\mathrm{finite}.$
Therefore, we obtain that the two-loop gluon wave-function counterterm is
given by
$Z_{3}^{(2)}=\frac{C_{A}^{2}g^{4}}{256\pi^{4}\epsilon}-\frac{5C_{A}^{2}g^{4}}{768\pi^{4}\epsilon^{2}}-\frac{\kappa^{2}m^{2}C_{A}g^{2}}{256\pi^{4}\epsilon}.$
(42)
## V Concluding remarks
In summary, we have evaluated the n-point functions for the Einstein-Scalar-
QCD model and demonstrated that there are no gravitational corrections to the
beta function of the color charge at one-loop order. Additionally, we have
explicitly verified that the Slavnov-Taylor identities are preserved at this
order of perturbation theory, indicating that the universality of the color
charge is maintained. Lastly, we have computed the counterterm for the gluon
wave-function at two-loop order.
It is important to contextualize our results and compare them with previous
research. To this end, we will follow the discussion in Donoghue:2019clr and
highlight some distinctions between our findings and theirs. One such
difference lies in the adoption of a distinct regularization scheme. In
reference Tang:2008ah , it is argued that there are three primary concerns
that should be considered when working with quantum gravity: gauge invariance,
gauge conditions introduced in the quantization process, and the ability of
the method to regulate any type of divergence. It was further argued that
although dimensional regularization (DR) satisfies the first two requirements,
it cannot handle more than logarithmic divergences. Therefore, Tang and Wu
employed the Loop Regularization method (LP) in their studies Tang:2008ah ;
Tang:2011gz to regulate the divergences. This method is capable of dealing
with the quadratic divergences that appear in the Feynman diagrams. The
authors used LP to compute the beta functions of the Einstein-Yang-Mills
theory and compared the results with those obtained using DR. They found that
while using DR leads to no gravitational contribution at one-loop, the use of
LP leads to a contribution that is proportional to $\mu^{2}$.
It is a fundamental requirement that physical results should not depend on the
choice of the regularization scheme. Anber pointed out in Anber:2010uj that
the quadratic divergences are not relevant when using the S-matrix, which is a
physical quantity. Moreover, Toms demonstrated in Toms:2011zza that it is
possible to define the electrical charge in quantum gravity using the
background field method in a physically meaningful way that is not influenced
by the quadratic divergences. Therefore, such contributions should be regarded
as unphysical and should not be included in the evaluation of the running
coupling.
An intriguing avenue for further investigation pertains to the existence of a
non-Abelian scalar particle serving as a potential dark matter candidate, as
well as the implications of quantum gravity for dark matter. In the study
conducted in Ref.Calmet:2021iid , the potential ramifications of quantum
gravity on dark matter models were explored. It was demonstrated that quantum
gravity would give rise to a fifth force-like interaction, setting a lower
limit on the masses of bosonic dark matter candidates. The authors also argued
that, due to the influence of quantum gravity, these potential candidates
would decay. However, given the ongoing observation of dark matter in the
present universe, the authors were able to calculate an upper bound on the
mass of a scalar singlet dark matter particle. In our future work, we intend
to investigate the mass range for a non-Abelian scalar dark matter candidate,
as presented in our study. In such a scenario, the fifth force-like
interaction would also be non-Abelian in nature. This particular scenario was
discussed in Arkani-Hamed:2008hhe .
In our future endeavors, we plan to investigate the dynamics of the
renormalized coupling constant in non-Abelian gauge theories, considering the
presence of fermions and scalars coupled to gravity at the two-loop level.
This investigation will involve an expansion of our research to incorporate
modified theories of gravity, such as quadratic gravity Odintsov:1991nd ;
Salvio:2014soa ; Donoghue:2018izj ; Donoghue:2021cza . Drawing on the
qualitative analysis presented in Souza:2022ovu , we expect that modified
theories of gravity, characterized by unconventional properties such as
repulsive gravity under specific regimes, could potentially impact the
behavior of the beta function. These modified gravity theories introduce
additional gravitational interactions and might influence the running of the
coupling constant in non-Abelian gauge theories, leading to intriguing and
novel phenomena.
###### Acknowledgements.
The work of HS is partially supported by Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior (CAPES).
## Appendix A Two-loop coefficients
In this section we present the two-loop coefficients for the two-loop gluon
self-energy from Eq. (38).
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle-\frac{i\left(D^{4}-10D^{3}+35D^{2}-50D+24\right)C_{A}g_{s}^{2}}{960(D-4)(D-3)(D-1)^{2}m^{4}}(-4C_{A}g_{s}^{2}(20\left(2D^{2}-3D-11\right)m^{2}$
(43a)
$\displaystyle+\left(2D^{2}-11D+12\right)p^{2})-5\left(D^{2}-8D+12\right)\kappa^{2}m^{2}\left((D-8)p^{2}-48m^{2}\right));$
$\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle-\frac{iC_{A}g_{s}^{2}}{16(D-4)(D-3)(D-1)^{2}m^{2}p^{2}}(-64(D-1)^{2}\left(D^{2}-7D+12\right)\lambda
m^{2}$
$\displaystyle+8(D-1)C_{A}g_{s}^{2}\left(4\left(D^{3}-8D^{2}+19D-16\right)m^{2}+(D-2)Dp^{2}\right)+2D^{6}\kappa^{2}m^{4}-18D^{5}\kappa^{2}m^{4}$
$\displaystyle-D^{5}\kappa^{2}m^{2}p^{2}+22D^{4}\kappa^{2}m^{4}-64D^{4}\lambda
m^{2}+23D^{4}\kappa^{2}m^{2}p^{2}+262D^{3}\kappa^{2}m^{4}+576D^{3}\lambda
m^{2}$
$\displaystyle-196D^{3}\kappa^{2}m^{2}p^{2}-1124D^{2}\kappa^{2}m^{4}-1728D^{2}\lambda
m^{2}+696D^{2}\kappa^{2}m^{2}p^{2}+8D^{2}\kappa^{2}p^{4}+1712D\kappa^{2}m^{4}$
$\displaystyle+1984D\lambda
m^{2}-1048D\kappa^{2}m^{2}p^{2}-24D\kappa^{2}p^{4}-928\kappa^{2}m^{4}-768\lambda
m^{2}+544\kappa^{2}m^{2}p^{2}+16\kappa^{2}p^{4});$ $\displaystyle c_{3}$
$\displaystyle=$
$\displaystyle-\frac{i\left(D^{3}-8D^{2}+19D-12\right)C_{A}g_{s}^{2}\left(2C_{A}g_{s}^{2}+\kappa^{2}\left(2(D-2)m^{2}-(D-4)p^{2}\right)\right)}{2(D-4)(D-3)(D-1)^{2}};$
(43b) $\displaystyle c_{4}$ $\displaystyle=$
$\displaystyle\frac{i\left(3D^{4}-40D^{3}+180D^{2}-320D+192\right)C_{A}g_{s}^{2}}{960(D-6)(D-5)(D-4)^{2}(D-3)(D-2)(D-1)^{2}(3D-4)m^{4}p^{4}}(-1920(D-1)^{2}(D^{4}-14D^{3}$
(43c) $\displaystyle+71D^{2}-154D+120)\lambda
m^{2}p^{2}+4\left(D^{2}-3D+2\right)C_{A}g_{s}^{2}(\left(2D^{3}-19D^{2}+54D-45\right)(D-4)^{2}p^{4}$
$\displaystyle+32\left(4D^{5}-48D^{4}+113D^{3}+616D^{2}-3099D+3470\right)m^{4}+4(8D^{5}-40D^{4}-281D^{3}+2224D^{2}$
$\displaystyle-4899D+3924)m^{2}p^{2})+5(D-5)m^{2}p^{2}(\left(D^{2}-3D+2\right)((D^{5}-23D^{4}+200D^{3}-820D^{2}+1584D$
$\displaystyle-1056)\kappa^{2}p^{2}-384\left(D^{3}-8D^{2}+19D-12\right)\lambda)+4(5D^{7}-113D^{6}+1052D^{5}-5122D^{4}+13896D^{3}$
$\displaystyle-20896D^{2}+16032D-4800)\kappa^{2}m^{2}));$ $\displaystyle
c_{5}$ $\displaystyle=$
$\displaystyle\frac{iC_{A}g_{s}^{2}}{128(D-4)(D-1)^{2}}(64\left(D^{3}-5D^{2}+2D+2\right)C_{A}g_{s}^{2}+(-24D^{5}+497D^{4}-3680D^{3}+12984D^{2}$
(43d) $\displaystyle-21560D+11840);\kappa^{2}p^{2})$ $\displaystyle c_{6}$
$\displaystyle=$
$\displaystyle\frac{iC_{A}g_{s}^{2}}{64(D-4)(D-1)^{2}p^{2}}(\kappa^{2}(16\left(D^{3}-10D^{2}+36D-36\right)m^{4}-8\left(D^{3}-10D^{2}+48D-48\right)m^{2}p^{2}$
(43e)
$\displaystyle+\left(D^{3}-10D^{2}+64D-64\right)p^{4})-128(D-1)C_{A}g_{s}^{2}\left(2m^{2}-p^{2}\right));$
$\displaystyle c_{7}$ $\displaystyle=$
$\displaystyle-\frac{iC_{A}g_{s}^{2}}{48(D-6)(D-4)^{2}(D-1)(3D-4)p^{2}}(24(9D^{6}-189D^{5}+1364D^{4}-4756D^{3}+9280D^{2}$
(43f)
$\displaystyle-10336D+4992)C_{A}g_{s}^{2}+(6D^{8}-35D^{7}-2454D^{6}+39327D^{5}-240012D^{4}+695044D^{3}$
$\displaystyle-915664D^{2}+366464D+98304)\kappa^{2}p^{2});$ $\displaystyle
c_{8}$ $\displaystyle=$
$\displaystyle-\frac{iC_{A}g_{s}^{2}}{480(D-4)(D-2)(D-1)m^{2}p^{4}}(4(D-2)C_{A}g_{s}^{2}(32\left(12D^{4}-92D^{3}-41D^{2}+1577D-2776\right)m^{4}$
(43g)
$\displaystyle+4\left(24D^{4}-172D^{3}+273D^{2}+193D-516\right)m^{2}p^{2}+\left(6D^{4}-67D^{3}+271D^{2}-468D+288\right)p^{4})$
$\displaystyle+5\kappa^{2}m^{2}p^{2}(4\left(6D^{6}-213D^{5}+2417D^{4}-12716D^{3}+34112D^{2}-45272D+23616\right)m^{2}$
$\displaystyle+\left(3D^{6}-63D^{5}+518D^{4}-2092D^{3}+4296D^{2}-3968D+1024\right)p^{2}));$
$\displaystyle c_{9}$ $\displaystyle=$
$\displaystyle\frac{iC_{A}g_{s}^{2}}{480(D-4)(D-3)(D-2)(D-1)m^{2}p^{4}}(4(D-2)C_{A}g_{s}^{2}(240\left(7D^{2}-57D+100\right)m^{4}p^{2}$
(43h)
$\displaystyle-(D-4)^{2}\left(2D^{2}-9D+9\right)p^{6}+128\left(4D^{4}-32D^{3}-7D^{2}+548D-1041\right)m^{6}$
$\displaystyle-4\left(6D^{4}-39D^{3}-22D^{2}+517D-876\right)m^{2}p^{4})+5\kappa^{2}m^{2}p^{2}(16(2D^{6}-69D^{5}+789D^{4}-4236D^{3}$
$\displaystyle+11684D^{2}-16012D+8664)m^{4}-4(D^{6}-44D^{5}+543D^{4}-3040D^{3}+8736D^{2}-12616D$
$\displaystyle+7296)m^{2}p^{2}-\left(D^{6}-25D^{5}+246D^{4}-1220D^{3}+3224D^{2}-4416D+2496\right)p^{4}));$
$\displaystyle c_{10}$ $\displaystyle=$
$\displaystyle\frac{i\kappa^{2}m^{2}C_{A}g_{s}^{2}\left(\left(D^{2}-6D+4\right)p^{2}-4(D-2)m^{2}\right)}{2(D-1)};$
(43i) $\displaystyle c_{11}$ $\displaystyle=$
$\displaystyle-\frac{iC_{A}g_{s}^{2}\left(C_{A}g_{s}^{2}\left(8m^{2}-p^{2}\right)+(D-2)\kappa^{2}m^{2}\left((D-4)p^{2}-8m^{2}\right)\right)}{2(D-1)}.$
(43j)
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Figure 7: Feynman diagrams to the scattering between gluons up to one-loop
order and one graviton exchange. Figure 8: Feynman diagrams to the scattering
between gluons and quarks up to one-loop order and one graviton exchange.
Figure 9: Feynman diagrams to the gluon self-energy involving only one
graviton exchange at two-loop order. Figure 10: Gluon self-energy 1-loop
diagrams with counterterms insertions.
|
CTPU-PTC-24-02
EPHOU-24-001
Finite modular axion and radiative moduli stabilization
Tetsutaro Higaki${}^{\ a}$<EMAIL_ADDRESS>and Junichiro
Kawamura${}^{\ b}$<EMAIL_ADDRESS>and Tatsuo Kobayashi${}^{\ c}$
<EMAIL_ADDRESS>
a Department of Physics, Keio University, Yokohama, 223-8522, Japan
b Center for Theoretical Physics of the Universe, Institute for Basic Science
(IBS), Daejeon 34051, Korea
c Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
We propose a simple setup which can stabilize a modulus field of the finite
modular symmetry by the Coleman-Weinberg potential. Our scenario leads to a
large hierarchy suppressing instanton-like corrections $e^{2\pi i\tau}$ and to
a light axion identified as $\mathrm{Re}\tau$, where $\tau$ is the modulus
field. This stabilization mechanism provides the axion solution to the strong
CP problem. The potential has a minimum at a large $\mathrm{Im}\tau$ which
suppresses explicit $U(1)_{\mathrm{PQ}}$ violation terms proportional to
$e^{-2\pi{\mathrm{Im}\tau}}$, and hence the quality of the axion is ensured by
the residual symmetry associated with the $T$-transformation, $\tau\to\tau+1$,
around the fixed point $\tau\sim i\infty$.
## 1 Introduction
The finite modular symmetry has been intensively studied to explain the flavor
structure of the Standard Model (SM) quarks and leptons [1, 2, 3, 4, 5, 6, 7,
8, 9]. (See for reviews [10, 11].) Under the symmetry, a Yukawa coupling
constant behaves as the so-called modular form which is a holomorphic function
of the modulus $\tau$. Interestingly, the finite modular symmetries
$\Gamma_{N}$, $N\in\mathbb{N}$, are isomorphic to the non-Abelian discrete
symmetries [12], which can explain the flavor structure of the SM fermions
[13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Such modular dependence of the
Yukawa coupling to a modulus was found in heterotic orbifold models [23, 24,
25, 26, 27, 28] and magnetized D-brane models [29, 30, 31, 32, 33, 34, 35] in
superstring theory. Moreover, the finite modular symmetry can explain the
hierarchies of the quark and lepton masses and mixing if the modulus is
stabilized near one of the fixed points where the residual symmetry remains
unbroken [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47].
In the models with finite modular flavor symmetries, the value of modulus
$\tau$ plays an essential role to fit to the data and is often treated as a
free parameter. Whereas in a UV theory, its value should be fixed dynamically
through moduli stabilization. There are studies on the moduli stabilization
with the three-form fluxes [48, 49], the negative power of modular form [50,
51] 111 The positive power of the trivial singlet is also studied in Ref.
[51]. See also Ref. [52]. , and the general superpotential invariant under the
modular symmetry $SL(2,\mathbb{Z})$ [53, 54, 55, 56, 57], utilizing the Klein
$j$ function [58]. The latter two mechanisms will need non-perturbative
dynamics to generate those potentials. Recently, effect of the radiative
correction to the tree-level stabilization is discussed in Ref. [59].
In this work, we point out that the modulus can be stabilized only by the
Coleman-Weinberg (CW) potential generated by couplings with matter fields.
This mechanism provides a perturbative way to stabilize a modulus through the
supersymmetry breaking. For illustration, we shall consider a simple model
with vector-like quarks which transforms as non-trivial singlets under the
finite modular flavor symmetry. The resultant potential has a global minimum
at $\mathrm{Im}\,\tau\gg 1$, where the residual $\mathbb{Z}^{T}_{N}$ symmetry
associated with the $T$ transformation, $\tau\to\tau+1$, is unbroken. That can
lead to a large hierarchy suppressing instanton-like correction, i.e.,
$e^{-2\pi{\mathrm{Im}\,\tau}}\ll 1$. An interesting feature of this mechanism
is that the $\mathrm{Re}\,\tau$ direction can be identified as the QCD axion
due to the accidental Peccei-Quinn (PQ) symmetry originated from the
$\mathbb{Z}^{T}_{N}$ residual symmetry. We shall discuss the condition that
the axion solution to the strong CP problem is not spoiled due to the CW
potential, i.e. the axion quality is good enough.
This paper is organized as follows. The finite modular symmetry is briefly
reviewed in Sec. 2. In Sec. 3, we discuss the mechanism of the modular
stabilization by utilizing the CW potential, and then we argue existence of
the QCD axion in the setup in Sec. 4. Section 5 concludes.
## 2 Finite modular symmetry
We consider the series of groups
$\displaystyle\Gamma(N):=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z}),\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\equiv\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\quad\mathrm{mod}~{}N\right\\},$ (2.1)
where $N\in\mathbb{N}$ is called level and
$\Gamma:=SL(2,\mathbb{Z})=\Gamma(1)$ is a group of $2\times 2$ matrices with
determinant unity. The modular transformation $\gamma\in\Gamma(N)$ for a
complex parameter $\tau$ is defined as
$\displaystyle\tau\to\gamma\tau=\frac{a\tau+b}{c\tau+d}.$ (2.2)
The modular group is generated by three generators
$\displaystyle S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\quad
T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\quad R=\begin{pmatrix}-1&0\\\
0&-1\end{pmatrix}.$ (2.3)
The finite modular symmetry is defined as the quotient group
$\Gamma_{N}:=\overline{\Gamma}/\Gamma(N)$ where
$\overline{\Gamma}:=\Gamma(1)/\mathbb{Z}^{R}_{2}$. Here $\mathbb{Z}^{R}_{2}$
is the $\mathbb{Z}_{2}$ symmetry generated by $R$. Under $\Gamma_{N}$ with
$N<6$, the generators satisfy
$\displaystyle S^{2}=(ST)^{3}=T^{N}=1.$ (2.4)
There are Abelian discrete symmetries $\mathbb{Z}^{S}_{2}$,
$\mathbb{Z}^{ST}_{3}$ and $\mathbb{Z}^{T}_{N}$ in associated with the $S$,
$ST$ and $T$ generators which are unbroken at $\tau=i$, $w:=e^{2\pi i/3}$ and
$i\infty$, respectively. The finite modular symmetries are isomorphic to the
non-Abelian discrete symmetries, e.g. $\Gamma_{3}\simeq A_{4}$. A modular form
$Y^{({k})}_{r}(\tau)$ with a modular weight $k$ and representation $r$
transforms under $\Gamma_{N}$ as
$\displaystyle Y^{({k})}_{r}(\tau)\to(c\tau+d)^{k}\rho(r)Y^{({k})}_{r}(\tau),$
(2.5)
where $\rho(r)$ is the representation matrix of $\Gamma_{N}$. We assume that
the chiral superfield $Q$ with weight $-k_{Q}$ and representation $r_{Q}$
transforms as
$\displaystyle Q\to(c\tau+d)^{-k_{Q}}\rho(r_{Q})Q.$ (2.6)
More detailed discussions can be found in e.g. Refs. [10, 11].
## 3 Radiative moduli stabilization
We consider a simple supersymmetric model with the following Kähler potneital
and superpotnetial:
$\displaystyle K=$ $\displaystyle\
-h\log(-i\tau+i\tau^{\dagger})+\sum_{i}\left(\frac{Q_{i}^{\dagger}Q_{i}}{(-i\tau+i\tau^{\dagger})^{k_{Q_{i}}}}+\frac{\overline{Q}_{i}^{\dagger}\overline{Q}_{i}}{(-i\tau+i\tau^{\dagger})^{k_{\overline{Q}_{i}}}}\right),$
(3.1) $\displaystyle W=$ $\displaystyle\
\sum_{i}M_{Q_{i}}Y^{({k_{i}})}_{r_{i}}(\tau)\overline{Q}_{i}Q_{i},$
where $h\in\mathbb{N}$. The reduced Planck mass $M_{p}=2.4\times
10^{18}~{}\mathrm{GeV}$ is set to unity. The chiral superfield $Q_{i}$
($\overline{Q}_{i}$) has a modular weight $k_{Q_{i}}$ ($k_{\overline{Q}_{i}}$)
and is assumed to be a singlet under $\Gamma_{N}$ for simplicity. Here,
$Q_{i}$ and $\overline{Q}_{i}$ are vector-like pairs under the SM gauge group.
The weight of the modular form $k_{i}=k_{Q_{i}}+k_{\overline{Q}_{i}}-h$ and
the representation $r_{i}$ are chosen so that the combination $e^{K}|W|^{2}$
in supergravity is invariant under $\Gamma_{N}$. We introduce the vector-like
mass parameter $M_{Q_{i}}$ which could be replaced by vacuum expectation
values of fields. Throughout this work, we assume that the SM fields are
trivial singlets under the modular symmetry $\Gamma_{N}$ for simplicity and
they have no effects in the following analysis.
We shall show that the modulus $\tau$ can be stabilized without introducing a
superpotential for the modulus $\tau$, but with the 1-loop CW potential
$\displaystyle V_{\mathrm{CW}}=\frac{1}{32\pi^{2}}\sum_{i}$
$\displaystyle\left[\left(m_{i}^{2}+m_{Q_{i}}^{2}(\tau)\right)^{2}\left(\log\left(\frac{m_{i}^{2}+m_{Q_{i}}^{2}(\tau)}{\mu^{2}}\right)-\frac{3}{2}\right)\right.$
(3.2) $\displaystyle\hskip
85.35826pt\left.-(m_{Q_{i}}^{2}(\tau))^{2}\left(\log\left(\frac{m_{Q_{i}}^{2}(\tau)}{\mu^{2}}\right)-\frac{3}{2}\right)\right],$
with
$\displaystyle
m_{Q_{i}}^{2}(\tau):=M_{Q_{i}}^{2}\left(-i\tau+i\tau^{\dagger}\right)^{k_{i}}\left|{Y^{({k_{i}})}_{r_{i}}(\tau)}\right|^{2}.$
(3.3)
Here, we consider the $\overline{\mathrm{MS}}$ scheme with $\mu$ being the
renormalization scale. The first (second) term in the parenthesis corresponds
to the scalar (fermion) contribution. The supersymmetric mass $m_{Q_{i}}$ is
multiplied by the factor $(-i\tau+i\tau^{\dagger})^{k_{i}/2}$ due to the
canonical normalization. We introduce the soft supersymmetry breaking mass
squared $m_{i}^{2}$ for the scalar component of $Q_{i}$ which is assumed to be
independent of $\tau$ for simplicity. This is the case if supersymmetry is
broken by a mechanism irrelevant to the modulus $\tau$. We shall first see the
simplest case analytically at large $\mathrm{Im}\,\tau$, and then see the
results numerically later. Throughout this paper, we assume that the mass
parameters are common for $Q_{i}$’s, i.e. $M_{Q_{i}}=:M_{Q}$, $m_{i}=:m_{0}$
and $k_{i}=:k$ for simplicity. Note that the tree-level scalar potential in
supergravity is irrelevant when $Q_{i}=\overline{Q}_{i}=0$.
### 3.1 Simplified analysis
We argue the model of radiative stabilization of the modulus $\tau$ where
there is only one pair of $(\overline{Q},Q)$, so we omit the index $i$. The
singlet modular form of $\Gamma_{N}$ can be expanded as
$\displaystyle
Y^{({k})}_{1_{t}}(\tau)=q^{t/N}\sum_{n=0}^{\infty}c_{n}q^{n},\quad q:=e^{2\pi
i\tau},$ (3.4)
where $r=1_{t}$ is a singlet whose charge is $0\leq t<N$ under the
$\mathbb{Z}^{T}_{N}$ symmetry 222 In the usual notation [1], $1_{0}=1$,
$1_{1}=1^{\prime}$ and $1_{2}=1^{{\prime\prime}}$ for $A_{4}$. . Assuming
$\left|{q}\right|=e^{-2\pi\mathrm{Im}\,\tau}\ll 1$ and $m_{0}^{2}\ll
M_{Q}^{2}$, the CW potential is given by
$\displaystyle
V_{\mathrm{CW}}=\frac{m_{0}^{2}\tilde{M}_{Q}^{2}}{16\pi^{2}}x^{k}e^{-px}\left(\log\frac{\tilde{M}_{Q}^{2}}{e\mu^{2}}+k\log
x-px\right)+\mathcal{O}\left({\left|{q}\right|,m_{0}^{4}}\right),$ (3.5)
where $x:=2\mathrm{Im}\,\tau$, $p:=2\pi t/N$ and $\tilde{M}_{Q}:=c_{0}M_{Q}$.
At the leading order, there is no potential for $\mathrm{Re}\,\tau$ which will
be important for the axion interpretation discussed in the next section. The
first derivative of the potential is given by
$\displaystyle\frac{dV_{\mathrm{CW}}}{dx}\sim\frac{m_{0}^{2}\tilde{M}_{Q}^{2}}{16\pi^{2}}x^{k-1}e^{-px}\left(k-px\right)\left(\log\frac{\tilde{M}_{Q}^{2}}{\mu^{2}}+k\log
x-px\right).$ (3.6)
The point $x=k/p$ is a maximum for sufficiently large $k/p$, and hence this
potential can have a minimum at
$\displaystyle
x_{0}=-\frac{k}{p}\mathcal{W}\left(-\frac{p}{k}\left(\frac{\mu}{\tilde{M}_{Q}}\right)^{2/k}\right),$
(3.7)
where the second parenthesis of Eq. (3.6) is vanishing. Here, $\mathcal{W}(z)$
is the Lambert function satisfying $\mathcal{W}(z)e^{\mathcal{W}(z)}=z$ which
has two real values for $-e^{-1}<z<0$, and the larger one can be a solution
where our approximation $\left|{q}\right|\ll 1$ is valid. Taking
$\mu=\tilde{M}_{Q}$ and $p=2\pi/3~{}$, the location of the minimum is
$x_{0}=3.9,~{}7.9,~{}12$ and $16$ for $k=6,8,10$ and $12$, respectively.
$x_{0}$ increases slightly by decreasing $\mu$. Thus we find that there is the
minimum at $\mathrm{Im}\,\tau\gg 1$ in the CW potential. Note that this
minimum exists only for $t>0$ where the modular form $Y^{({k})}_{1_{t}}$ is a
non-trivial singlet. For the CW potential from the trivial-singlet modular
form, the potential increases as $\mathrm{Im}\,\tau$ increases in contrast to
that from the non-trivial one, and thus there is no minimum at
$\mathrm{Im}\,\tau\gg 1$ 333 The potentials discussed in Refs. [53, 54, 55,
56, 57, 58] have a similar behavior, that is, $V\to\infty$ for
$\mathrm{Im}\,\tau\to\infty$. See also the potential in Ref. [52]. Therefore
the dominant CW potential should be generated from the non-trivial singlet one
for the existence of the minimum at $\mathrm{Im}\,\tau\gg 1$, so that the
potential tends to approach to zero for a sufficiently large ${\rm Im}\tau$.
### 3.2 Numerical analysis
Figure 1: The shape of the CW potential along the $\mathrm{Im}\,\tau$
direction with $\mathrm{Re}\,\tau=0$ (red solid) and $\mathrm{Re}\,\tau=-0.5$
(blue dashed).
We shall study the potential Eq. (3.2) numerically without relying on the
$q$-expansion in Eq. (3.4). For illustration, we consider $\Gamma_{3}\simeq
A_{4}$ as the finite modular group, and two pairs of vector-like quarks whose
superpotential is given by
$\displaystyle
W=M_{Q}\sum_{i=1,2}Y^{({12})}_{1_{i}}(\tau)\overline{Q}_{i}Q_{i}.$ (3.8)
We can easily choose the representations and modular weights such that only
those two terms are allowed but mixing terms like $\bar{Q}_{1}Q_{2}$ are
forbidden. An explicit assignment of representations and modular weights will
be shown in Eq. (4.13) when we shall discuss the vector-like quark decays. The
modular forms of weight 12 are given by
$\displaystyle
Y^{({12})}_{1_{1}}(\tau)=(Y_{1}^{2}+2Y_{2}Y_{3})^{2}(Y_{3}^{2}+2Y_{1}Y_{2}),\quad
Y^{({12})}_{1_{2}}(\tau)=(Y_{1}^{2}+2Y_{2}Y_{3})(Y_{3}^{2}+2Y_{1}Y_{2})^{2},$
(3.9)
where the functions $Y_{1,2,3}$ are defined as [1]
$\displaystyle Y_{1}(\tau)=$ $\displaystyle\
\frac{i}{2\pi}\left[\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}-27\frac{\eta^{\prime}(3\tau)}{\eta(3\tau)}\right],$
(3.10) $\displaystyle Y_{2}(\tau)=$ $\displaystyle\
\frac{-i}{\pi}\left[\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+w^{2}\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+w\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}\right],$
(3.11) $\displaystyle Y_{3}(\tau)=$ $\displaystyle\
\frac{-i}{\pi}\left[\frac{\eta^{\prime}(\tau/3)}{\eta(\tau/3)}+w\frac{\eta^{\prime}((\tau+1)/3)}{\eta((\tau+1)/3)}+w^{2}\frac{\eta^{\prime}((\tau+2)/3)}{\eta((\tau+2)/3)}\right],$
(3.12)
with $\eta(\tau)$ being the Dedekind eta function. The $q$-expansion of the
modular forms are given by 444 There is an ambiguity of the normalization of
the modular forms which can not be determined from the symmetry. We simply
employ the normalization used in Ref. [1]. See also Ref. [60].
$\displaystyle
Y^{({12})}_{1_{1}}=-12q^{1/3}\left(1+472q+\mathcal{O}\left({q^{2}}\right)\right),\quad
Y^{({12})}_{1_{2}}=144q^{2/3}\left(1+224q+\mathcal{O}\left({q^{2}}\right)\right).$
(3.13)
The modular weights are chosen to $k=12$ so that the strong CP problem is
solved with keeping the modulus mass heavy, as discussed in the next section
555 Since $1_{1}$ and $1_{2}$ can be in the $2^{\prime}$ representation under
$A_{4}^{\prime}$, the two fields could be embedded into a doublet. . Here, we
introduce the two vector-like pairs, so that the mixed anomaly of $A_{4}$ with
the QCD is canceled [61, 62]. The singlets are trivial under the
$S$-transformation, and transform as $Y^{({k})}_{1_{t}}\to
w^{t}Y^{({k})}_{1_{t}}$ under the $T$-transformation. Hence, in our setup,
$\det(\rho(T))=w^{1+2}=1$ and the anomaly is canceled. The anomaly associated
with the modular weight can be canceled by the Green-Schwarz mechanism with an
another gauge coupling modulus or by adding $\log Y^{({k})}_{1_{0}}(\tau)$ to
the gauge kinetic function with a certain coefficient, where
$Y^{({k})}_{1_{0}}(\tau)$ is a trivial singlet modular form and could be
explained by threshold corrections from heavy modes [63].
Figure 1 shows the shape of the potential along the $\mathrm{Im}\,\tau$
direction. Here, we take $m^{2}/M_{Q}^{2}=10^{-8}$ and $\mu/M_{Q}=10^{-2}$ so
that the minimum resides at $\mathrm{Im}\,\tau\simeq 13$. The red solid (blue
dashed) line corresponds to $\mathrm{Re}\,\tau=0$ ($-0.5$). Note that the
fundamental domain is $\mathrm{Im}\,\tau\geq 1$ ($\sqrt{3}/2$) at
$\mathrm{Re}\,\tau=0~{}(-0.5)$. In the left panel, the origin of the
horizontal axis is $\sqrt{3}/2$ and the global picture of the potential is
shown. The right panel is the shape of the potential around the global minimum
at large $\mathrm{Im}\,\tau$. For $\mathrm{Im}\,\tau\lesssim 1$, where the
$q$-expansion is not efficient, there is a local minimum at $\tau\sim w$, but
$V_{\mathrm{CW}}(\tau=w)=0$, since $Y^{({12})}_{1_{1,2}}(w)=0$, is shallower
than the global minimum at $\mathrm{Im}\,\tau\sim 12$ where the potential
value is negative. The potential clearly depends on $\mathrm{Re}\,\tau$ at
small $\mathrm{Im}\,\tau$, while it is independent of $\mathrm{Re}\,\tau$
within numerical precision. Thus, the CW potential in the $\mathrm{Re}\,\tau$
direction is approximately flat near the global minimum at
$\mathrm{Im}\,\tau\gg 1$. As a result, our scenario of modulus stabilization
can lead a light axion as well as a large hierarchy by
$|q|=e^{-2\pi\mathrm{Im}\,\tau}\sim 10^{-36}$. We will confirm this flatness
analytically in the next section to discuss the quality of the axion solution
to the strong CP problem.
## 4 Axion solution and its quality
We have seen that the CW potential Eq. (3.2) induced by the superpotential Eq.
(3.8) is very flat in the $\mathrm{Re}\,\tau$ direction. The stabilization
mechanism can realize the axion solution to the strong CP problem by assuming
that the matter fields are vector-like quarks as in the KSVZ axion model [64,
65]. The effective $\theta$-angle $\overline{\theta}$ in the QCD is given by
$\displaystyle\overline{\theta}(\tau)=\theta_{0}+\mathrm{Arg}\left(Y^{({12})}_{1_{1}}(\tau)Y^{({12})}_{1_{2}}(\tau)\right)=\theta_{0}+\phi+\mathcal{O}\left({\left|{q}\right|}\right),$
(4.1)
where $\theta_{0}$ is a constant and $\phi:=2\pi\mathrm{Re}\,\tau$. The $\phi$
dependence appears as a result of the chiral anomaly of the QCD with the
approximate $U(1)_{\rm PQ}$, which originates from the residual
$\mathbb{Z}^{T}_{3}$ symmetry as shown below. For $\mathrm{Im}\,\tau\gg 1$,
the Yukawa coupling is given by
$\displaystyle Y^{({k})}_{1_{t}}\overline{Q}_{t}Q_{t}\propto
e^{i(t/3)\phi}\;\overline{Q}_{t}Q_{t},$ (4.2)
so we can find an accidental $U(1)_{\mathrm{PQ}}$ symmetry as
$\displaystyle\phi\to\phi+\alpha,\quad\overline{Q}_{t}Q_{t}\to
e^{-it\alpha/3}\overline{Q}_{t}Q_{t},$ (4.3)
where $\alpha$ is a real transformation parameter. Hence the PQ charge of
$\overline{Q}_{t}Q_{t}$ is $-t$, and the $U(1)_{\rm PQ}$ is anomalous. This
symmetry is ensured from the residual $\mathbb{Z}^{T}_{3}$ symmetry
corresponding to $\alpha=2\pi$, and the explicit breaking effects of the
$U(1)_{\mathrm{PQ}}$ are from $q=e^{2\pi i\tau}$ which is invariant under
$\mathbb{Z}^{T}_{3}$ but violates $U(1)_{\mathrm{PQ}}$ 666 The corrections to
the axion from $\log Y^{({k})}_{1_{0}}$ in the gauge kinetic function which
may exist to cancel the weight anomaly will also appear at
$\mathcal{O}\left({q}\right)$. . Therefore, $\mathrm{Re}\,\tau$ can be
identified as the QCD axion. Since the kinetic term of the modulus is given by
777 We write explicitly the reduced Planck mass $M_{p}$ in this section.
$\displaystyle\frac{hM_{p}^{2}}{(2\mathrm{Im}\,\tau)^{2}}\partial_{\mu}\tau^{\dagger}\partial^{\mu}\tau,$
(4.4)
the axion decay constant is $f_{a}=\sqrt{h}M_{p}/(2\pi
x_{0})\sim\mathcal{O}\left({10^{16}}\right)$ GeV which requires a fine-tuning
of the initial condition or early matter domination [66], to avoid the
overabundance of the axion.
The axion potential is given by
$\displaystyle
V_{a}=-\Lambda_{\mathrm{QCD}}^{4}\cos\left(\overline{\theta}(\phi)\right)+\Delta
V.$ (4.5)
Here, $\Delta V:=V_{\mathrm{CW}}|_{x=x_{0}}$ is the axion-dependent part of
the CW potential which can spoil the axion solution. The shift of the
$\theta$-angle due to $\Delta V$ is estimated as
$\displaystyle\Delta\theta=\frac{1}{\Lambda_{\mathrm{QCD}}^{4}}\left(\frac{\partial\Delta
V}{\partial\phi}\right)\simeq\frac{m_{0}^{2}M_{Q}^{2}}{8\pi^{2}\Lambda_{\mathrm{QCD}}^{4}}x_{0}^{k}\epsilon^{4+N}\Omega\sin{\theta_{0}},$
(4.6)
with
$\displaystyle\Omega:=\left(\beta_{1}\frac{kN-4\pi x_{0}}{kN-2\pi
x_{0}}-\beta_{2}\right)\left(\log\frac{m^{2}_{0}+m_{Q_{2}}^{2}}{e\mu^{2}}+\frac{m_{Q_{2}}^{2}}{m_{0}^{2}}\log\left(1+\frac{m^{2}_{0}}{m_{Q_{2}}^{2}}\right)\right),$
(4.7)
where $\epsilon:=e^{-\pi x_{0}/N}$. In our model, $k=12$ and $N=3$. Here,
$m^{2}_{Q_{2}}$ is the mass squared of $Q_{2}$ defined in Eq. (3.3) with
taking $x=x_{0}$ and dropping the negligible corrections at
$\mathcal{O}\left({q}\right)$. $\beta_{t}$ is the ratio $c_{1}/c_{0}$ in the
expansion of the modular forms $Y^{({k})}_{1_{t}}$ in Eq. (3.4), which is
given by $\beta_{1}=472$ and $\beta_{2}=224$ for $Y^{({12})}_{1_{1}}$ and
$Y^{({12})}_{1_{2}}$, respectively. Note that the
$\mathcal{O}\left({q}\right)$ correction proportional to $m_{Q_{1}}^{2}$ is
canceled at the minimum $x=x_{0}$ and thus the leading term is
$\epsilon^{4+N}$ which is proportional to
$m_{Q_{2}}^{2}\sim\mathcal{O}\left({\epsilon^{4}x_{0}^{k}M_{Q}^{2}}\right)$.
This shift of the angle should be less than
$\mathcal{O}\left({10^{-10}}\right)$ to be consistent with the measurement of
the neutron electric dipole moment [67]. A possible modulus $\tau$-dependence
of $\Lambda_{\mathrm{QCD}}$ from the gauge kinetic function will also be
suppressed by $\mathcal{O}\left({q}\right)$, and thus is sub-dominant for the
stabilization. Similarly the $\phi$ dependence of
$\partial_{\phi}\overline{\theta}$ is also negligible. In the $A_{4}$ model
discussed in the previous section, we obtain
$\displaystyle\left|{\Delta\theta}\right|\simeq 2\times
10^{-10}\times\sin\theta_{0}\left(\frac{\left|{\Omega}\right|}{10^{4}}\right)\left(\frac{m_{0}}{10^{7}~{}\mathrm{GeV}}\right)^{2}\left(\frac{M_{Q}}{M_{p}}\right)^{2}\left(\frac{x_{0}}{28}\right)^{8}\left(\frac{\epsilon}{10^{-12}}\right)^{7}.$
(4.8)
Thus the axion quality is ensured if $\epsilon\lesssim 10^{-12}$, where
$\mathrm{Im}\,\tau\sim 13$ as shown in Fig. 1. The required tuning becomes
mild if $\sin\theta_{0}\ll 1$ which might happen when the (generalized) CP is
conserved at some level as discussed in e.g. Ref. [56] 888See Refs. [68, 69]
for recent studies on the strong CP problem relevant to the modular symmetry..
In such a case, the strong CP problem is partially solved by the modular axion
$\phi\sim\mathrm{Re}\,\tau$.
Figure 2: Masses of the modulus $X\sim\mathrm{Im}\,\tau$ (red) and vector-
like quarks $Q_{1}$ (blue dashed) and $Q_{2}$ (green dot-dashed). In this
plot, $m_{0}=10^{7}~{}\mathrm{GeV}$, $M_{Q}=10^{4}\times\mu=M_{p}$, and $h=3$.
Before closing, we discuss the masses of the modulus
$X\propto\mathrm{Im}\,\tau$ and the vector-like quarks $Q_{1,2}$. The modulus
mass may need to be heavier than
$\mathcal{O}\left({10~{}\mathrm{TeV}}\right)$, so that the modulus decay
occurs before the big-bang nucleosynthesis and the moduli problem is avoided
[70, 71, 72, 73]. After canonically normalizing the kinetic term Eq. (4.4),
the modulus mass is given by
$\displaystyle
m_{X}^{2}=\frac{m_{0}^{2}\tilde{M}_{Q}^{2}}{8\pi^{2}hM_{p}^{2}}x_{0}^{k}\epsilon^{2}(k-px_{0})^{2}.$
(4.9)
Hence the modulus mass is related to $\Delta\theta$ as
$\displaystyle m_{X}\simeq$ $\displaystyle\
c_{0}\left|{k-px_{0}}\right|\sqrt{\frac{\Delta\theta}{h\Omega\sin{\theta_{0}}}}\frac{\Lambda_{\mathrm{QCD}}^{2}}{\epsilon^{1+N/2}M_{p}}$
(4.10) $\displaystyle\sim$ $\displaystyle\
42~{}\mathrm{TeV}\times\frac{c_{0}\left|{k-2\pi
x_{0}/N}\right|}{\sqrt{h\sin{\theta_{0}}}}\left(\frac{\Delta\theta}{10^{-10}}\right)^{1/2}\left(\frac{10^{4}}{\Omega}\right)^{1/2}\left(\frac{\Lambda_{\mathrm{QCD}}}{100~{}\mathrm{MeV}}\right)^{2}\left(\frac{10^{-12}}{\epsilon}\right)^{5/2},$
in the $A_{4}$ model. Thus the modulus mass decreases as the axion quality is
higher by $\epsilon\to 0$. For $\Delta\theta/\sin\theta_{0}\sim 10^{-10}$,
$\epsilon\lesssim 10^{-12}$ is required to avoid the moduli problem 999 The
lighter modulus would not be a problem if there is the thermal inflation [74,
75, 76], large Hubble induced mass during the inflation exists [77, 78],
and/or an inflation scale is lower than the modulus mass. .
The mass of the lighter vector-like quark $Q_{2}$, whose $T$ charge is $2$, is
given by
$\displaystyle m_{Q_{2}}\sim\tilde{M}_{Q}x_{0}^{k/2}\epsilon^{2}\sim
1~{}\mathrm{TeV}\times\left(\frac{\tilde{M}_{Q}}{M_{p}}\right)\left(\frac{x_{0}}{28}\right)^{6}\left(\frac{\epsilon}{10^{-12}}\right)^{2},$
(4.11)
thus $M_{Q}\sim M_{p}$ is necessary for
$m_{Q_{2}}\gtrsim\mathcal{O}\left({\mathrm{TeV}}\right)$ to be consistent with
the LHC constraints [79, 80, 81, 82, 83]. Conversely, if we require
$m_{Q_{2}}\gtrsim\mathrm{TeV}$ and $\tilde{M}_{Q}\lesssim M_{p}$, the
hierarchy parameter is bounded from below as $\epsilon\gtrsim 10^{-8}\times
x_{0}^{-k/4}$, and thus larger weight is preferred to allow
$\epsilon\sim\mathcal{O}\left({10^{-12}}\right)$ without having the too light
vector-like quark. Figure 2 shows the masses of the modulus
$X\propto\mathrm{Im}\,\tau$ and the vector-like quarks. As $\mathrm{Im}\,\tau$
increases, the axion quality becomes better, whereas the modulus and vector-
like quark masses decrease, and hence there is the $lower$ bound
$\Delta\theta\gtrsim 10^{-13}$ for sufficiently heavy vector-like quark
$Q_{2}$.
The vector-like quarks should be able to decay into SM particles. For
illustration, if the vector-like quarks have the same quantum number as the
down-type quarks, denoted by $d$, we can write down the superpotential
$\displaystyle
W_{\mathrm{mix}}=\sum_{i=1,2}{\lambda}_{i}Y^{({14})}_{1_{1}}H_{d}qQ_{i},$
(4.12)
where $H_{d}$ and $q$ are respectively the down-type Higgs doublet and the
doublet SM quark. Here, the representations and weights of quarks are set to
$\displaystyle r_{q}=r_{d}=r_{\overline{Q}_{1}}=1_{0},\quad
r_{Q_{1,2}}=r_{\overline{Q}_{2}}=1_{2},\quad k_{q}=h-k_{d}=8,\quad
k_{Q_{1,2}}-h=k_{\overline{Q}_{1,2}}=6,$ (4.13)
and those of $H_{d}$ is the trivial-singlet with weight $0$. We omit the
flavor indices of the SM quarks. With this assignment, the interactions in
Eqs. (3.8) and (4.12), as well as the SM Yukawa coupling $yH_{d}qd$ with $y$
being a constant, are realized, while the $\overline{Q}_{i}d$ is vanishing
because of the negative weight. The size of mixing $s_{Q_{i}}$ is estimated as
$\displaystyle s_{Q_{i}}\sim$ $\displaystyle\
\frac{{\lambda}_{i}x_{0}^{7}Y^{({14})}_{1_{1}}{\langle{H_{d}}\rangle}}{M_{Q}x_{0}^{6}Y^{({12})}_{1_{i}}}\sim\frac{{\lambda}_{i}x_{0}{\langle{H_{d}}\rangle}}{M_{Q}\epsilon^{i-1}}$
(4.14) $\displaystyle\sim$ $\displaystyle\
10^{-3}\times{\lambda}_{2}\left(\frac{x}{28}\right)\left(\frac{{\langle{H_{d}}\rangle}}{100~{}\mathrm{GeV}}\right)\left(\frac{M_{p}}{M_{Q}}\right)\left(\frac{10^{-12}}{\epsilon}\right),$
where $i=2$ in the second line, and we used
$Y^{({k})}_{1_{t}}\sim\epsilon^{t}$. Thus the vector-like quarks decay
promptly at the collider scale. It is noted that the CW potential induced by
the mixing term is negligible because ${\langle{H_{d}}\rangle}\ll M_{Q}$.
## 5 Summary and discussions
In this work, we point out that the modulus of the finite modular symmetry can
be stabilized by the Coleman-Weinberg (CW) potential. For illustration, we
studied the model with $\Gamma_{3}\simeq A_{4}$ symmetry and two vector-like
quark pairs which transform as non-trivial singlets, namely $1_{1}$ and
$1_{2}$. This is the minimal possibility to cancel the mixed QCD anomaly of
the finite modular symmetry, but to induce that of the $U(1)_{\mathrm{PQ}}$
symmetry. The CW potential has the global minimum at $\mathrm{Im}\,\tau\gg 1$
if the corresponding modular form is non-trivial singlet under $A_{4}$, where
the residual symmetry $\mathbb{Z}^{T}_{3}$ remains unbroken. Since the
potential along the $\mathrm{Re}\,\tau$ direction has extremely flat due to
this residual symmetry, we can regard $\mathrm{Re}\,\tau$ as the QCD axion to
solve the strong CP problem. The accidental $U(1)_{\mathrm{PQ}}$ arises due to
the $\mathbb{Z}^{T}_{3}$ symmetry and hence the PQ breaking effects are
controlled by $\Gamma_{3}$ which is a different situation argued in Ref. [84].
We examined the condition to ensure the quality of this finite modular axion,
and correlate with the masses of the modulus $X\sim\mathrm{Im}\,\tau$ and
vector-like quarks. Interestingly, the modulus $X$ and the lighter quark
$Q_{2}$ are expected to be $\mathcal{O}\left({\mathrm{TeV}}\right)$ scale for
$\Delta\theta<10^{-10}$ and thus could be probed by cosmology and collider
experiments. This mechanism can be generalized to other modular forms as long
as the CW potential is dominated by that with non-zero $T$-charge and the
potential converges to zero for $\mathrm{Im}\,\tau\gg 1$.
The modular $A_{4}$ symmetry in our scenario can not be used to explain the
hierarchical structure of the SM fermions, as studied in Refs. [36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47], since the hierarchy parameter $\epsilon$
is $\mathcal{O}\left({10^{-12}}\right)$ and is too small. Such a tiny
$\epsilon$ is required due to the correlation of the axion quality
$\Delta\theta$ with the modulus mass in Eq. (4.10). This relation can be
relaxed if the $\tau$-independent angle $\theta_{0}$ is small or the level $N$
is large. In the latter case, there exist a small hierarchy $|e^{2\pi
i\tau/N}|$ in the SM sector and a large one $|e^{2\pi i\tau}|$ in the vector-
like quark sector.101010 In general, for example, if one could break the
periodicity $\tau\sim\tau+1$ to $\tau\sim\tau+1/n$ ($n>1$) in the vector-like
$(Q,\bar{Q})$ sector with keeping the periodicity $\tau\sim\tau+1$ in the SM
fermion sector, then modular forms may appear $Y^{k}_{r}(\tau)$ in the SM
fermion sector and $Y^{k^{\prime}}_{r^{\prime}}(n\tau)$ in the $(Q,\bar{Q})$
sector, and one could realize both a small hierarchy $|e^{2\pi i\tau}|$ in the
SM sector and a large one $|e^{2\pi in\tau}|$ in the vector-like quark pairs.
. Thus, we could construct a model which explains the SM fermion hierarchies
by the modulus stabilized by the mechanism proposed in this paper. An explicit
model building is subject of our future work.
We have proposed a new scenario to lead to a large hierarchy
$e^{-2\pi\mathrm{Im}\,\tau}\sim\mathcal{O}\left({10^{-36}}\right)$ and a light
axion by stabilizing the modulus at $\mathrm{Im}\,\tau\gg 1$. Although we have
applied it to solve the strong CP problem by assuming $Q$ and $\bar{Q}$ as
vector-like matter fields with the QCD colors, our scenario would be useful to
explain other large hierarchies, which would be required for the proton
stability, tiny neutrino masses, CP/flavor violations, quintessence and so on,
together with light axions by assuming $Q$ and $\bar{Q}$ as visible or hidden
matter fields. We would study it elsewhere.
In this work, we assume that the soft scalar masses of the vector-like quarks
are independent of the modulus $\tau$ and supersymmetry is dominantly broken
by other sector, so that the supersymmetry breaking sector does not contribute
to the stabilization of the modulus $\tau$. In addition, there would be
contributions from gravitational instantons [85] which would change the
potential structure. Those effects and other dynamical effects including
stabilization mechanisms of other moduli, and uplifting of the vacuum energy
if necessary, may become important in complete models such as superstring, but
beyond the scope of this paper.
## Acknowledgement
We thank Y.Abe and N.Aso for helpful discussions in the early stage of this
work. The work of J.K. is supported in part by the Institute for Basic Science
(IBS-R018-D1). This work is supported in part by he Grant-in-Aid for
Scientific Research from the Ministry of Education, Science, Sports and
Culture (MEXT), Japan No. JP22K03601 (T.H.) and JP23K03375 (T.K.).
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|
capbtabboxtable[][]
# Training Vision-Language Models with Less Bimodal Supervision
Elad Segal<EMAIL_ADDRESS>
Ben Bogin<EMAIL_ADDRESS>
Jonathan Berant<EMAIL_ADDRESS>
Blavatnik School of Computer Science
Tel Aviv University
###### Abstract
Standard practice in pretraining multimodal models, such as vision-language
models, is to rely on pairs of aligned inputs from both modalities, for
example, aligned image-text pairs. However, such pairs can be difficult to
obtain in low-resource settings and for some modality pairs (e.g., structured
tables and images). In this work, we investigate the extent to which we can
reduce the reliance on such parallel data, which we term _bimodal supervision_
, and use models that are pretrained on each modality independently. We
experiment with a high-performing vision-language model, and analyze the
effect of bimodal supervision on three vision-language tasks. We find that on
simpler tasks, such as VQAv2 and GQA, one can eliminate bimodal supervision
completely, suffering only a minor loss in performance. Conversely, for NLVR2,
which requires more complex reasoning, training without bimodal supervision
leads to random performance. Nevertheless, using only 5% of the bimodal data
(142K images along with their captions), or leveraging weak supervision in the
form of a list of machine-generated labels for each image, leads to only a
moderate degradation compared to using 3M image-text pairs:
74%$\rightarrow$$\sim$70%.
## 1 Introduction
Figure 1: The effect of unimodal and bimodal pretraining on downstream
performance after finetuning. In VQAv2 and GQA, pretraining on unimodal data
alone (without image-text pairs) is competitive with models pretrained on
image-text pairs. On NLVR2, bimodal supervision is necessary, but one can
reach reasonable performance using only 5% of the image-text pairs or training
on machine-generated object labels. Random initialization leads to poor
performance on all tasks.
Pretraining models on large amounts of raw data using self-supervision has
revolutionized machine learning, and is now standard practice across a wide
range of modalities Liu et al. (2019); Raffel et al. (2020); Dosovitskiy et
al. (2021); Liu et al. (2021); Herzig et al. (2020); Schneider et al. (2019);
Baevski et al. (2022). While typically pretrained models are trained on data
from a single modality (_unimodal data_), the success of pretraining has
spread to the _bimodal_ setup, where models are trained on pairs of inputs,
each from a different modality (e.g. text and audio, Li et al., 2021a). Most
notably, vision-language models, such as LXMERT Tan and Bansal (2019), ViLT
Kim et al. (2021), METER Dou et al. (2022), CLIP Radford et al. (2021), and
others Li et al. (2019); Lu et al. (2019); Li et al. (2021b), have been
pretrained on manually or automatically collected parallel data that consists
of aligned image-text pairs.
While effective, pretraining on bimodal data comes at a cost. First, gathering
high-quality pairs can be challenging, especially in low-resource languages
and domains, or for modality pairs where parallel data is scarce. Second,
expanding this approach to more than two modalities (as in, e.g.,
MultimodalQA, Talmor et al., 2021) is challenging. Last, pretraining is
computationally expensive (Strubell et al., 2019; Bommasani et al., 2021), and
thus relying on pretraining for all modality pairs is inefficient.
Given these shortcomings, a natural question is how far can we get with models
pretrained on unimodal data only (_unimodal models_), such as BERT Devlin et
al. (2019) and ViT (Dosovitskiy et al., 2021), to reduce or obviate the need
for _bimodal_ pretraining. Can we align unimodal representations without
resorting to pretraining over millions of input pairs? While past work Dou et
al. (2022); Li et al. (2021b); Zhai et al. (2022) used unimodal models as an
initialization point before bimodal pretraining, it did not investigate its
effect on the amount of necessary bimodal data.
In this work, we investigate to what extent we can reduce the burden of
bimodal pretraining and finetune models on vision-language applications
starting with models that were unimodally pretrained. We choose a high-
performing architecture Dou et al. (2022) – a transformer image encoder and a
transformer text encoder, which pass their representations through additional
transformer layers that capture the interaction between the image and the
text, before performing a final classification task.
We test performance on visual question answering and visual reasoning tasks in
the following setups: (a) randomly initialized image and text encoders, (b)
unimodally-pretrained image and text encoders, and (c) unimodally-pretrained
image and text encoders that are then pretrained with bimodal supervision. We
test different sources for bimodal pretraining, which require different
amounts of human effort: (a) automatically harvested image-caption pairs
(Conceptual Captions, Sharma et al., 2018), (b) images paired with machine-
generated object labels (CCIL, Ng et al., 2021), (c) manually annotated image-
object pairs (ImageNet-1K, Russakovsky et al., 2015), and (d) image-question-
answer triples from visual question answering tasks. We note that due to
computational constraints the size of our pretraining corpus is smaller
compared to those used by industry-based researchers Li et al. (2022); Radford
et al. (2021); Jia et al. (2021).
We find (Figure 1) that on some tasks, models that do not use any bimodal
supervision are only slightly worse than models that are pretrained on large
amounts of image-text pairs – 70.7$\rightarrow$69.5 on VQAv2, and
56.1$\rightarrow$53.6 on GQA. However, for a more complex reasoning task, such
as NLVR2, bimodal supervision is _crucial_. Nevertheless, we show that one can
dramatically reduce the number of bimodal image-text pairs and still obtain
reasonable performance – either by using only 5% of the pairs
(74.3$\rightarrow$70.2) or through machine-generated object labels
(74.3$\rightarrow$68.0). Our code is available at
https://github.com/eladsegal/less-bimodal-sup.
## 2 Overview
Figure 2: _Left_ : Architecture overview: an image encoder and a text encoder
followed by a few transformer fusion layers, capturing interaction between
modalities through cross-attention. _Center_ : We pretrain the VL encoder from
bimodal supervision by taking contextualized representations of the image
($h^{\text{img}}$) and text ($h^{\text{txt}}$) and applying the image-text
matching (ITM) and masked language modeling (MLM) loss functions. _Right_ : We
finetune the VL encoder on downstream classification tasks by concatenating
the image and text representations and passing them through an MLP classifier.
We provide an overview of the experimental settings explored in this work. As
our architecture-of-choice, we leverage one that has been shown to perform
well across multiple tasks Dou et al. (2022), namely, a Vision-Language (VL)
encoder, where a unimodal image encoder creates image representations, a
unimodal text encoder creates text representations, and these two
representations are passed through a few transformer (Vaswani et al., 2017)
layers that capture cross-modal interactions (Figure 2, Left).
We experiment with three initializations of the image and text encoders.
First, we use random initialization as a baseline. Second, we initialize from
pretrained unimodal models (RoBERTa and ViT, Liu et al., 2019; Dosovitskiy et
al., 2021), which can potentially reduce the amount of bimodal pretraining.
Last, we pretrain the entire VL encoder with bimodal supervision (Figure 2,
Center), and compare different data sources for pretraining, each requiring
different amounts of human effort.
In each experiment we finetune and evaluate the VL encoder on downstream VL
applications (Figure 2, Right), focusing on classification tasks (visual
question answering and visual reasoning).
## 3 Data
We now describe the datasets used during bimodal pretraining and finetuning.
For downstream applications, we put an emphasis on tasks that require
reasoning over image(s) and text. Table 1 provides an example from each
dataset, and Appendix A provides key statistics and details on the composition
of the training sets.
ImageNet | Conceptual Captions | Conceptual Captions Image Labels
---|---|---
| |
Class label: printer | Caption: snail on a branch isolated on white background | Computer-generated labels: room, interior design, furniture, blue, living room, green, property, turquoise, home, floor, yellow, table, building, wall, house
VQAv2 | GQA | NLVR2
---|---|---
| |
Question: How many chairs can you count? Answer: 2 | Question: What vegetable is to the left of the bag? Answer: cauliflower | Sentence: The sink in one of the images is set into a brown wood hanging counter. Label: false
Table 1: Examples from all datasets used in this work.
### 3.1 Pretraining Datasets
#### ImageNet-1K Russakovsky et al. (2015)
is a human-annotated dataset that consists of over 1.2M images, divided into
1,000 classes that are mapped to meaningful concepts according to the WordNet
hierarchy Fellbaum (1998). Each concept is described by one or more language
phrases, and accompanied by $\sim$1000 images to illustrate it. We consider
ImageNet-1K as a source of lightweight bimodal supervision, relatively cheap
to obtain, as images are paired with text describing a single concept rather
than a full-sentence.
#### Conceptual Captions (CC) Sharma et al. (2018)
is a programmatically-generated dataset of image-text pairs that consists of
3.3M examples. Prior work has demonstrated that CC is an effective resource
for vision-language pretraining Kim et al. (2021); Li et al. (2021b); Lu et
al. (2019); Hendricks et al. (2021). We use CC as a primary source of bimodal
supervision, since: (a) it does not involve manual annotations, (b) it is
small enough to be used by resource-constrained researchers, and (c) its
images are from a different origin than the downstream tasks. Therefore, it
provides a suitable test bed for estimating models’ ability to generalize to
new images.
#### Conceptual Captions Image Labels (CCIL) Ng et al. (2021)
is a subset of 2M images from CC that contains machine-generated labels using
the Google Cloud image labelling API. While labels are cheap since they are
automatically-generated, the API was presumably trained on large amounts of
bimodal data. Nevertheless, we examine pretraining on images paired with sets
of labels to investigate whether this provides a sufficiently rich source of
bimodal supervision despite lacking natural language sentences. Past work
indeed showed that VL pretraining benefits from masking object labels Bitton
et al. (2021).
### 3.2 Downstream Tasks
#### VQAv2 Goyal et al. (2017)
VQAv2 is a human-authored visual question answering (VQA) dataset that
consists of 1.1M natural language questions with 10 short answers per question
over 204K images from COCO Lin et al. (2014). It is standard to treat VQAv2 as
a classification task, by only keeping questions with the most common answers
(3,129 classes) Anderson et al. (2018); Tan and Bansal (2019); Zhai et al.
(2022).
#### GQA Hudson and Manning (2019) (balanced)
is a VQA dataset whose public version contains 1.1M questions over 83K images
from Visual Genome Krishna et al. (2017). Unlike VQAv2, questions are created
programmatically from scene graphs created by human annotators. Using scene
graphs allows GQA to generate questions that test various reasoning skills
such as comparisons, logical inference, spatial reasoning, etc.
#### NLVR2 Suhr et al. (2019)
is a benchmark for testing models’ ability to reason over text and images. The
dataset contains 107K examples, where each example contains an English
sentence and two web images (see Table 1). The goal is to determine whether
the sentence is true or false in the context of the pair of images, a binary
classification task.
## 4 Method
Our goal is to develop a classifier
$f:\mathcal{X}\times\mathcal{I}\rightarrow\mathcal{C}$ that given an utterance
$\bm{x}$ and an image $\bm{i}$ predicts a class $c\in\mathcal{C}$.
### 4.1 Architecture
We use a VL architecture, adapted from Dou et al. (2022). The tokens of the
utterance $\bm{x}=(x_{0},x_{1},\dots,x_{n})$ are fed into a transformer _text
encoder_ , where $x_{0}$ is the special symbol $\texttt{CLS}_{\text{txt}}$.
Similarly, the image is broken into patches
$\bm{i}=(i_{0},i_{1},\dots,i_{m})$, where $i_{0}$ is a special symbol
$\text{CLS}_{\text{img}}$, which are fed into a transformer _image encoder_.
The image and text encoders compute contextualized representations of the
image and text $(\hat{h}_{0}^{\text{txt}},\dots,\hat{h}_{n}^{\text{txt}})$ and
$(\hat{h}_{0}^{\text{img}},\dots,\hat{h}_{m}^{\text{img}})$, which are then
linearly projected with projection matrices
$W_{\text{proj}}^{\text{txt}}\in\mathbb{R}^{d_{\text{txt}}\times
d},W_{\text{proj}}^{\text{img}}\in\mathbb{R}^{d_{\text{img}}\times d}$, where
$d_{\text{txt}},d_{\text{img}}$ are the hidden state dimensions of the text
and image encoders respectively. The projected representations of each
modality are then passed through transformer _fusion_ layers, which include
both a self-attention sublayer, and a cross-attention sublayer. Namely, each
modality performs cross-attention on the other modality to fuse information
from its representations, capturing interaction between the modalities.
Overall, the VL encoder outputs the image-and-text contextualized
representations
$\bm{h}^{\text{img}}=(h_{0}^{\text{img}},\dots,h_{n}^{\text{img}})$ and
$\bm{h}^{\text{txt}}=(h_{0}^{\text{txt}},\dots,h_{m}^{\text{txt}})$. An
overview of our architecture is given in Figure 2, Left.
All model parameters are jointly trained by defining loss functions over
classification heads, which we describe next. Since some model parameters are
initialized from a pretrained model, while other are randomly initialized, we
use a higher learning rate for randomly initialized weights compared to
pretrained weights, similar to Dou et al. (2022).
### 4.2 Pretraining Objectives
For pretraining, we use two objectives: masked language modeling (MLM) Devlin
et al. (2019); Tan and Bansal (2019), and image-text matching (ITM) Tan and
Bansal (2019), which are the most common objectives for VL pretraining and
lead to state-of-the-art performance Dou et al. (2022). During training, we
sum the ITM loss and the MLM loss for each training instance.
In MLM, given a masked token $x_{i}$ the goal is to maximize the probability
of the gold token given the representation $h_{i}^{\text{txt}}$, using cross-
entropy loss. In ITM, given a image-text pair $(\bm{x},\bm{i})$, we
concatenate the special CLS tokens $h_{0}^{\text{img}}$ and
$h_{0}^{\text{txt}}$, and use a sigmoid layer to predict if the image matches
the text or not. We train with binary cross-entropy loss.
When pretraining on Conceptual Captions, we use the same masking scheme
employed by Dou et al. (2022), that is, randomly masking 15% of the tokens.
For ImageNet, we are given an image and a text label and mask all of its
tokens. For CCIL, we are given an image and a list of text labels,
concatenated with commas as separators, ordered by their machine-generated
confidence scores. We then mask all tokens of a randomly-sampled label.
In ITM, in 50% of the examples, given a positive pair $(\bm{x},\bm{i})$, we
substitute the true image with a random one and label it as a negative
example.
### 4.3 Finetuning
Since the downstream applications in §3.2 can all be framed as classification
tasks, we add a classification head to finetune the VL encoder. The
classification head is a two-layer MLP, as in Kim et al. (2021). Specifically,
we take as input the concatenation of all the image and text CLS
representations, i.e., $[h^{\text{img}}_{0};h^{\text{txt}}_{0}]$, and use the
MLP to map them to $|\mathcal{C}|$ logits based on the number of task classes.
The objective during training is to maximize the probability of the correct
class(es), and we use standard cross-entropy loss. At inference time, we
return the top-scoring class for all downstream tasks.
In NLVR2, where each example has two images, we consider each example as two
image-text pairs, duplicating the text, and pass them separately through the
VL encoder (dubbed ‘the pair setup’ in Chen et al. (2020)). We then pass four
CLS representations (two for the images, two for the text) to the MLP to
obtain the prediction.
## 5 Experiments
#### Experimental Setup
We use ViT-Base (Dosovitskiy et al., 2021) as the image encoder, pretrained
and finetuned on ImageNet-21K at a resolution of 224x224 with a patch size of
16x16. We use RoBERTa-Base (Liu et al., 2019) as the text encoder. For the
cross-modal transformer, we use only two layers to save computational
resources, as previous work (Lu et al., 2019; Hendricks et al., 2021), as well
as our own preliminary findings, have shown that the effect of depth is small
after finetuning.
We run pretraining (§4.2) for a maximum of 7,400 steps, and finetune each
downstream task for 10 epochs. We specify batch sizes and learning rates for
each case in Appendix B.1.
The evaluation score for VQAv2 is VQA score, and accuracy for GQA and NLVR2.
Each result for VQAv2 and GQA is a 3-run average on the test-dev split, and
for NLVR2 a 10-run average on the public test split.
#### Limitations
Our work is performed within a limited compute budget. Therefore, we choose
our largest pretraining dataset to be CC although there are datasets orders of
magnitude larger. Compared to other work, we use images in a lower resolution,
which has been shown to decrease performance Dou et al. (2022). Also, Dou et
al. (2022) showed that better image encoders can significantly improve
performance even before bimodal pretraining, but we did not experiment with
different text and image encoders nor with larger models. Additionally, even
though further pretraining in some setups results in small performance
improvements, we decided the computational cost was unjustified. Bugliarello
et al. (2021) showed pretraining variance exists when training on CC, but we
were only able to pretrain once in each setup, due to the high computational
costs. All of the above means that our work is self-contained, but cannot be
directly compared in numbers to other works.
### 5.1 Main Results
| VQAv2 | GQA | NLVR2
---|---|---|---
Random init. | 52.3$\pm$0.1 | 43.1$\pm$0.1 | random
Vision Random init. | 54.2$\pm$0.0 | 44.3$\pm$0.2 | random
Language Random init. | 66.3$\pm$0.1 | 51.2$\pm$0.1 | random
Unimodally-pretrained | 69.5$\pm$0.1 | 53.6$\pm$0.1 | random
Bimodally-pretrained with CCIL | 69.6$\pm$0.3 | 52.9$\pm$0.5 | 68.0$\pm$0.7
Bimodally-pretrained with CC | 70.7$\pm$0.0 | 56.1$\pm$0.3 | 74.3$\pm$0.3
Table 2: Main results for all downstream tasks.
Table 2 shows the results of finetuning on all downstream tasks for different
initializations of the image and text encoders.
In addition to finetuning a model that is initialized with ViT and RoBERTa
(‘Unimodally-pretrained’), and in order to verify the importance of unimodal
pretraining, we finetune our model when the image encoder, text encoder, or
both encoders are randomly initialized. We find that pretraining the vision
model is essential for good performance, and observe a smaller drop in
performance when the text encoder is randomly initialized, similar to Zhai et
al. (2022).
Comparing the unimodally-pretrained model to one that was further pretrained
on CC (‘Bimodally-pretrained with CC’), we see that for VQAv2 the gap is only
1 point, and for GQA it is just 2.5 points. However, on the more challenging
NLVR2, which requires complex reasoning operations, bimodal pretraining is
essential, and the model achieves random performance without it. Nevertheless,
training with a weaker form of supervision, namely, a list of machine-
generated object labels from CCIL is sufficient for non-random and reasonably
high performance on NLVR2 (but has no effect on VQAv2 and GQA).
### 5.2 Effect of CC Size on Pretraining
Table 2 showed that bimodal pretraining is essential for obtaining non-random
results on NLVR2. A natural question is whether this can be obtained with
fewer pretraining examples. To this end, we pretrain on different fractions of
CC and present the results after finetuning in Table 7 (in the Appendix) and
Figure 4.
Surprisingly, even when using only $\sim$1% of CC (30K examples), performance
on NLVR2 is far from random – 67.3. When using 5% of the data, performance is
only moderately lower than when using CC in its entirety – 70.2 vs. 74.3. When
using 25% of the data for pretraining, performance on all three datasets is
less than two points lower than when using 100%, showing that indeed the
amount of bimdal supervision can be considerably reduced with only a small hit
in performance.
[] [] Max # of labels Unique Labels NLVR2 1 5.3K 52.6$\pm$1.4 (55.3) 3 8.0K
67.8$\pm$0.5 (68.6) 15 14.3K 68.0$\pm$0.7 (68.9)
Figure 3: Effect of the fraction of examples from CC on downstream task
performance. Solid/dashed line – average/maximum score over seeds. Figure 4:
Performance on NLVR2 when restricting the number of labels per image during
pretraining on CCIL (max. value is in the parentheses).
The aforementioned results were obtained by finetuning on all of the
downstream data per task. However, an interesting variant to consider is a
low-resource setting where we have only _some_ of the downstream data – what
is the importance of bimodal pretraining then? Table 8 (in the Appendix) and
Figure 5 show for VQAv2 and GQA that when less data is used for finetuning,
the benefit of pretraining with 5% or more of CC is greater than the benefit
observed when 100% of the downstream data is used for finetuning. For NLVR2,
we see that pretraining is still very helpful even with 100% of the downstream
data. The reason for the difference might be that VQAv2 and GQA are much
larger than NLVR2.
Figure 5: Effect of the fraction of examples from CC on downstream task
performance when finetuning on less downstream data. Solid/dashed line –
average/max. over seeds.
### 5.3 Pretraining with ImageNet Labels
We have seen in §5.1 that image-caption pairs are useful for pretraining VL
models. Here, we investigate if a weaker source of language supervision,
namely image labels only, suffices for aligning text and vision
representations. Specifically, we pair each ImageNet image with its label,
treating it as a caption, and pretrain with MLM and ITM as described in §4.2.
We observe _no difference_ in results compared to unimodally-pretrained models
(Table 9 in the Appendix) – performance remains random for NLVR2, and similar
for VQAv2 and GQA. This suggests that ImageNet labels do not provide adequate
signal for VL pretraining.
### 5.4 Pretraining with CCIL
One hypothesis for the lack of improvement when pretraining with ImageNet is
that a single label per image is too limiting, since images typically contain
many objects. To test this, we pretrain with CCIL, where each image is paired
with machine-generated labels, providing a richer image representation. We
pretrain with MLM and ITM as described in §4.2.
While pretraining on CCIL does not improve performance on VQAv2 and GQA, it
leads to dramatic improvement on NLVR2, reaching an average accuracy of
68.0$\pm$0.7 and a maximum accuracy of 68.9. This shows that providing a set
of object labels lets the model better align image and text representations.
Table 4 further validates this by showing results when restricting the maximal
number of labels per image. We observe that having multiple labels per image
is crucial, as performance is roughly random when using a single label. Using
3 labels is already sufficient for bootstrapping the model, and performance is
barely lower compared to using all 15 labels.
### 5.5 Transfer Learning
Finally, we test whether a model finetuned on a source downstream task (VQAv2
and GQA) can improve performance on a target task, i.e., in a transfer
learning setup, where we vary the amount of annotated data in the source task.
Table 10 (in the Appendix) and Figure 6 (left) show results when VQAv2 is the
source task and GQA and NLVR2 are the target tasks. VQAv2 appears to be an
effective source of bimodal supervision for both tasks – when using all of
VQAv2, performance on GQA is even slightly higher compared to pretraining on
CC data, and 3 points lower on NLVR2 (74.3$\rightarrow$71.1). Nevertheless,
the amount of data in the source task is important, and performance on NLVR2
is much lower when using 5%-25% of the data.
Table 11 (in the Appendix) and Figure 6 (right) show results when GQA is the
source task and VQAv2 and NLVR2 are the target tasks. We observe that VQAv2 is
a better source task compared to GQA – GQA does not improve performance on
VQAv2, and its effect on NLVR2 is much more moderate. A possible explanation
is that VQAv2 has natural language questions, while questions in GQA are
automatically generated. Another potential factor is the fact that VQAv2
typically require less reasoning steps compared to GQA.
Overall, in both cases we find transfer learning on downstream tasks is
useful, and can even perform closely to bimodally-pretrained models.
Figure 6: Effect of the fraction of examples from VQAv2 (left) and GQA (right)
on downstream task performance. Solid/dashed line – average/maximum score over
seeds.
## 6 Analysis
To better understand what data properties are important for pretraining, we
train on small subsets of CC (1% of the data) and VQAv2 (5% of the data), with
particular characteristics:
* •
Min/max length: We create subsets that minimize/maximize the average input
length.
* •
Min/max vocabulary size \- We create subsets that minimize/maximize the size
of the vocabulary. To do so we use a greedy procedure, where (a) we initialize
an empty set of examples, and at each step (b) randomly sample a candidate set
of 10K examples, and (c) choose the example that minimizes/maximizes the
current vocabulary size.
| 1% CC | 5% VQAv2
---|---|---
Subset | Length | Vocab. | NLVR2 | Length | Vocab. | NLVR2
Min length | 5.0 | 8.0K | 67.1$\pm$0.3 (67.4) | 4.45 | 3.6K | 57.4$\pm$3.1 (60.8)
Max length | 30.3 | 19.1K | 64.1$\pm$1.4 (65.7) | 12.7 | 7.6K | 53.9$\pm$2.0 (57.0)
Min vocab. | 6.5 | 0.3K | 64.8$\pm$1.2 (65.9) | 5.8 | 0.2K | 57.4$\pm$3.8 (62.2)
Max vocab. | 14.0 | 44.4K | 67.3$\pm$0.3 (67.7) | 7.7 | 16.4K | 55.1$\pm$3.1 (58.2)
Random | 10.3 | 13.3K | 67.3$\pm$0.5 (68.4) | 7.3 | 5.8K | 56.6$\pm$3.5 (59.3)
Table 3: Analyzing the effect of pretraining on CC/VQAv2 subsets with
particular properties. After training on each subset, we finetune on NLVR2.
Results are in Table 3. No subset is noticeably better than a random subset.
For CC, results are similar. For VQAv2, while performance when minimizing
length and vocabulary is better on average, the differences seem negligible,
given the high standard deviation.
#### Effect of length on pretraining
Table 3 shows that pretraining on long inputs substantially hurts performance
– results are reduced by at least 3 points for both CC and VQAv2. This is
surprising as one might hypothesize that longer inputs should be better since
they contain more information. A possible explanation is that simple examples
are necessary to bootstrap the pretraining procedure and align the text and
image representations.
#### Effect of vocabulary size on pretraining
Pretraining on a subset with higher lexical diversity should expose the model
to more concepts, both in images and texts, and therefore improve its
performance. While for CC this is indeed the case, for VQAv2 results for the
max vocabulary size setup with 16.4K words are lower than the min vocabulary
size setup with only 0.2K words. A possible explanation is the amount of
yes/no questions in the min/max vocabulary size subsets which is 80.7% and
44.5%, respectively – Since NLVR2 is a yes/no task, training on more yes/no
questions might be closer to its distribution.
## 7 Related Work
Dou et al. (2022) investigated unimodally-pretrained models, finetuning
different image and text encoders on multiple VL tasks, recognizing it as
efficient and performant. However, they did not consider the effects of the
types and amount of bimodal supervision. Past work investigated bimodal
supervision on VL models, but for models that use _frozen_ features from an
object detection model Singh et al. (2020); Hendricks et al. (2021), which (a)
cannot be adapted to unseen concepts Zhang et al. (2021), (b) require heavily
annotated object-level data for the training of the object detection model
Krishna et al. (2017); Anderson et al. (2018), and (c) result in an
architectural inductive bias towards objects (which is very beneficial for VQA
tasks). Singh et al. (2020) compared performance between multiple pretraining
datasets, varying their sizes. Unlike us, for all tasks, the effect of
different usage of bimodal supervision was small, compared to our NLVR2
experiments. Hendricks et al. (2021) assessed the contribution of pretraining
datasets from a set of standard VL pretraining datasets, but focused on zero-
shot image retrieval tasks.
Li et al. (2021c) and Zhou et al. (2022) also share the motivation of reducing
bimodal pretraining for VL models. With some similarity to our CCIL
experiments in §5.4, they avoid pretraining on collected parallel image-text
data altogether by utilizing predictions of regions and tags from an object
detection model to create VL-specialized training objectives.
Opposite to our setup, a current trend is to pretrain models on vast amounts
of bimodal data Radford et al. (2021); Zhai et al. (2022); Alayrac et al.
(2022), and perform zero/few-shot evaluation. While remarkable results were
achieved, performance is lower than finetuned models pretrained on less
bimodal data, which is relatively cheap to obtain.
## 8 Conclusion
A current obstacle on the road to multimodal models is reliance on bimodal
supervision. In this work, we go in an opposite direction from current trends,
and instead of using increasing amounts of bimodal data, we examine whether
one can use less of it. We find that indeed this is the case, where for simple
tasks just finetuning unimodally-pretrained models leads to performance that
is similar to bimodally-pretrained models, at a much lower cost. For complex
tasks, while bimodal pretraining is still necessary, its amount
(100%$\rightarrow$5%) and source quality (CC$\rightarrow$CCIL) can be
significantly reduced with only a moderate degradation in performance. We also
find that models finetuned on one downstream task are useful in a transfer
learning setup, achieving results close to bimodally-pretrained models.
## Acknowledgements
This research was partially supported by The Yandex Initiative for Machine
Learning, and the European Research Council (ERC) under the European Union
Horizons 2020 research and innovation programme (grant ERC DELPHI 802800).
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## Appendix for “Training Vision-Language Models with Less Bimodal
Supervision”
## A Training Data
Since for some of datasets the official training splits aren’t used as-is, we
provide the exact details of the training data composition for each dataset
and also key statistics for all of the datasets in Table 4.
Dataset | Training instances | Unique texts | Training images
---|---|---|---
ImageNet | 656K | 738 | 656K
Conceptual Captions (CC) | 2.84M | 2M | 2.84M
Conceptual Captions Image Labels (CCIL) | 1.84M | 1.79M | 1.84M
VQAv2 | 620K | 210K | 118K
GQA | 943K | 538K | 72K
NLVR2 | 86K | 23K | 103K
Table 4: Key statistics for the training datasets.
#### ImageNet-1K Russakovsky et al. [2015]
Since ImageNet classes are often too fine-grained, we manually collapse fine-
grained classes into an ancestor WordNet
class,111https://observablehq.com/@mbostock/imagenet-hierarchy. e.g., dog
breeds are collapsed to “dog”. Then, we create a balanced training set
according to the updated classes of the images.
Following is a list of the classes we collapse sub-classes into:
dog, fox, wild dog, wolf, coyote, domestic cat, bear, monkey, snake, lizard,
turtle, frog, salamander, lobster, crab, beetle, butterfly, spider, rabbit,
bird, fungus
#### Conceptual Captions (CC) Sharma et al. [2018]
Out of the 3.3M examples in the official training set, we were able to
download 2.84M examples from the provided image URLs.
#### Conceptual Captions Image Labels (CCIL) Ng et al. [2021]
Out of the 2M examples in the official training set, we were able to download
1.84M examples from the provided image URLs.
#### VQAv2 Goyal et al. [2017]
We create our training set similar to previous works on VQAv2 Tan and Bansal
[2019], Dou et al. [2022], and use the same validation set as Tan and Bansal
[2019], which was constructed from the official validation set based on 5,000
randomly chosen images.
To create the training set, we first create an answer set that contains only
majority answers that occurred at least 9 times on the official training and
validation sets together. Then, out of the official training and validation
sets, we filter out all of the examples that doesn’t have any answer in the
created answer set. Finally, out of the remaining examples, we discard every
example that appears in our validation set.
#### GQA Hudson and Manning [2019]
We use the official training set.
#### NLVR2 Suhr et al. [2019]
We use the official training set.
## B Experimental Setup
### B.1 Additional Implementation Details
#### Image Preprocessing
Both in pretraining and finetuning, we apply center crop on the image and
resize it to 224x224. When training, we additionally use RandAugment Cubuk et
al. [2020] as in Kim et al. [2021] with the exclusion of color-changing
strategies (Invert, Posterize, Solarize, SolarizeAdd) and the coutout
strategy.
#### Model Architecture
We use the model from Dou et al. [2022], but we simplify it with the removal
of two of its components since we didn’t observe a performance difference: the
single-layer feedforward network before the feeding of the [CLS]
representations to a task-specific head (e.g. ITM, MLM, classifier), and the
image token type embeddings.
#### Pretraining
We run pretraining for 7,400 steps, except when training on 1%, 5% and 10% of
CC, as more training results in an increase of the validation loss. We train
for 1850 steps on 1% and %5 of CC, and 3700 steps for 10% of CC.
The batch size is 3,840 and learning rates of $1e^{-4}$ and $5e^{-4}$ are used
for the pretrained and randomly initialized weights respectively. The learning
rate is warmed up from zero during the first 10% steps, and then linearly
decays back to zero throughout the remaining steps.
We use 8 NVIDIA V100 GPUs, and training takes about 16 hours for 100% of CC.
#### Finetuning
For finetuning, we use a batch size of 96 for VQAv2 and GQA, and 48 for NLVR2.
We specify the learning rates for finetuning before and after bimodal
pretraining in Tables 5 and 6 respectively. The learning rate is warmed up
from zero during the first 10% steps, and then linearly decays back to zero
throughout the remaining steps.
We use a single NVIDIA RTX 3090 GPU, and training takes 10, 15 and 4 hours for
VQAv2, GQA and NLVR2 respectively.
Weights | VQAv2 | GQA | NLVR2
---|---|---|---
Image encoder, Text encoder | $2e-5$ | $1e-5$ | $1e-5$
Cross-modal transformer | $2e-4$ | $1e-4$ | $1e-4$
Classifier head | $2e-4$ | $1e-4$ | $1e-4$
Table 5: Learning rates per weights for finetuning before bimodal pretraining for each downstream task. Weights | VQAv2 | GQA | NLVR2
---|---|---|---
Image encoder, Text encoder | $2e-5$ | $1e-5$ | $1e-5$
Cross-modal transformer | $1e-4$ | $1e-4$ | $5e-5$
Classifier head | $1e-3$ | $1e-4$ | $5e-4$
Table 6: Learning rates per weights for finetuning after bimodal pretraining
for each downstream task.
## C Results
CC Data | VQAv2 | GQA | NLVR2
---|---|---|---
0% | 69.5$\pm$0.1 | 53.6$\pm$0.1 | random
1% | 69.2$\pm$0.1 | 53.8$\pm$0.6 | 67.3$\pm$0.5
5% | 69.8$\pm$0.0 | 55.3$\pm$0.3 | 70.2$\pm$0.3
10% | 70.1$\pm$0.1 | 55.5$\pm$0.2 | 71.2$\pm$0.4
25% | 70.5$\pm$0.1 | 55.6$\pm$0.2 | 72.9$\pm$0.4
50% | 70.6$\pm$0.1 | 56.1$\pm$0.4 | 73.8$\pm$0.4
100% | 70.7$\pm$0.0 | 56.1$\pm$0.3 | 74.3$\pm$0.3
Table 7: Effect of the fraction of examples from CC on downstream task performance. Visualized with Fig. 4. | VQAv2 | GQA | NLVR2
---|---|---|---
CC Data | 10% | 25% | 100% | 10% | 25% | 100% | 10% | 25% | 100%
0% | 54.4$\pm$0.0 | 62.1$\pm$0.6 | 69.5$\pm$0.1 | 45.6$\pm$0.5 | 48.0$\pm$0.1 | 53.6$\pm$0.1 | random | random | random
1% | 55.8$\pm$0.2 | 60.8$\pm$0.1 | 69.2$\pm$0.1 | 45.2$\pm$0.3 | 48.1$\pm$0.5 | 53.8$\pm$0.6 | 52.5$\pm$0.9 | 54.6$\pm$0.6 | 67.3$\pm$0.5
5% | 58.1$\pm$0.2 | 63.7$\pm$0.1 | 69.8$\pm$0.0 | 46.3$\pm$0.4 | 49.1$\pm$0.1 | 55.3$\pm$0.3 | 55.6$\pm$1.5 | 61.0$\pm$0.8 | 70.2$\pm$0.3
100% | 62.4$\pm$0.2 | 66.0$\pm$0.1 | 70.7$\pm$0.0 | 48.4$\pm$0.6 | 51.8$\pm$0.3 | 56.1$\pm$0.3 | 63.4$\pm$0.5 | 67.6$\pm$0.6 | 74.3$\pm$0.3
Table 8: Effect of the fraction of examples from CC on downstream task performance when finetuning on less downstream data. Visualized with Fig. 5. | VQAv2 | GQA | NLVR2
---|---|---|---
Unimodally-pretrained | 69.5$\pm$0.1 | 53.6$\pm$0.1 | random
Bimodally-pretrained with ImageNet | 69.3$\pm$0.0 | 53.5$\pm$0.2 | random
Table 9: Performance on all downstream tasks, with and without ImageNet pretraining. No difference is observed. VQAv2 Data | GQA | NLVR2
---|---|---
0% | 53.6$\pm$0.1 | random
5% | 54.6$\pm$0.3 | 56.6$\pm$3.5
10% | 55.1$\pm$0.2 | 61.4$\pm$1.8
25% | 55.1$\pm$0.4 | 68.3$\pm$0.4
50% | 55.7$\pm$0.5 | 70.0$\pm$0.5
100% | 56.3$\pm$0.2 | 71.1$\pm$0.5
Bimodally-pretrained with CC | 56.1$\pm$0.3 | 74.3$\pm$0.3
Table 10: Effect of the fraction of examples from VQAv2 on downstream task performance. Visualized with Fig. 6 (left). GQA Data | VQAv2 | NLVR2
---|---|---
0% | 69.5$\pm$0.1 | random
5% | 69.1$\pm$0.1 | 52.3$\pm$1.5
10% | 69.3$\pm$0.1 | 53.3$\pm$2.3
25% | 69.2$\pm$0.1 | 55.5$\pm$3.2
50% | 69.3$\pm$0.1 | 59.5$\pm$2.4
100% | 69.4$\pm$0.1 | 63.1$\pm$1.0
Bimodally-pretrained with CC | 70.7$\pm$0.0 | 74.3$\pm$0.3
Table 11: Effect of the fraction of examples from GQA on downstream task
performance. Visualized with Fig. 6 (right).
|
special case of Theorem 5.2 where $\alpha_{1}=\alpha_{2}=\alpha_{3}=\gamma$.
###### Lemma 5.3.
Theorem 5.2 holds for $\alpha_{1}=\alpha_{2}=\alpha_{3}=\gamma$.
###### Proof.
By the relation between $\mathrm{LF}_{\mathbbm{C}}^{(\gamma,z_{i})_{i}}$ and
$\mathrm{QS}_{3}$ from Theorem 2.9 and the relation between
$\mathrm{QD}_{1,0}(\ell)$ and $\mathcal{M}_{1}^{\mathrm{disk}}(\gamma;\ell)$
from Theorem 2.13, we see that Lemma 5.3 follows from Theorem 3.7. ∎
We first explain how to go from $\alpha_{1}=\alpha_{2}=\alpha_{3}=\gamma$ to
the following case.
###### Proposition 5.4.
Theorem 5.2 holds for $\alpha_{1}=\alpha\in(Q-\frac{\gamma}{4},Q)$ and
$\alpha_{2}=\alpha_{3}=\gamma$.
We will prove Proposition 5.4 from Lemma 5.3 by a reweighting procedure. By
the definition of $\mathcal{M}^{\mathrm{disk}}_{1}(\alpha;\ell)$, if we sample
a field $X$ from $\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)$, then the law
of $(\mathbbm{H},X,i)/{\sim_{\gamma}}$ is
$\mathcal{M}^{\mathrm{disk}}_{1}(\alpha;\ell)$. We now recall a fact from
[ARS21] about the reweighting of
$\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)$ by “$e^{(\alpha-\gamma)X}$”.
###### Lemma 5.5 ([ARS21, Lemma 4.6]).
For any $\ell>0,\varepsilon\in(0,1)$ and for any nonnegative measurable
function $f$ of $X$ that depends only on $X|_{\mathbbm{H}\setminus
B_{\varepsilon}(i)}$, we have
$\int f(X|_{\mathbbm{H}\setminus
B_{\varepsilon}(i)})\times\varepsilon^{\frac{1}{2}(\alpha^{2}-\gamma^{2})}e^{(\alpha-\gamma)X_{\varepsilon}(i)}\,d{\mathrm{LF}_{\mathbbm{H}}^{(\gamma,i)}(\ell)}=\int
f(X|_{\mathbbm{H}\setminus
B_{\varepsilon}(i)})\,d\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell),$
where $X$ is a sample in $H^{-1}(\mathbbm{H})$ and $X_{\varepsilon}(i)$ means
the average of $X$ on the boundary of the ball
$B_{\varepsilon}(i)=\\{z:|z-i|<\varepsilon\\}$.
The key to the proof of Proposition 5.4 is the following reweighting result on
$\mathrm{LF}_{\mathbbm{C}}^{(\gamma,z_{i})_{i}}$.
###### Lemma 5.6.
Let $\eta_{1}$ be a simple loop separating $z_{1}$ from $z_{2}$ and $z_{3}$.
Let $D_{\eta_{1}}$ be the connected component of
$\mathbbm{C}\setminus\eta_{1}$ containing $z_{1}=0$. Let $p$ be a point on
$\eta_{1}$ and let $\psi:\mathbbm{H}\to D_{\eta_{1}}$ be the conformal map
with $\psi(i)=z_{1}$ and $\psi(0)=p$. For $\varepsilon\in(0,\frac{1}{4})$, let
$\mathbbm{C}_{\eta_{1},p,\varepsilon}=\mathbbm{C}\setminus\psi(B_{\varepsilon}(i))$.
For $\phi$ sampled from
$\mathrm{LF}_{\mathbbm{C}}^{(\gamma,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}$,
let $X=\phi\circ\psi+Q\log|\psi^{\prime}|$ so that
$(\mathbbm{H},X,i,0)/{\sim_{\gamma}}=(D_{\eta_{1}},\phi,z_{1},p)/{\sim_{\gamma}}$.
Then for a fixed $\alpha\in(Q-\frac{\gamma}{4},Q)$ and for any nonnegative
measurable function $f$ of $\phi$ that depends only on
$\phi|_{\mathbbm{C}_{\eta_{1},p,\varepsilon}}$ we have
$\displaystyle\int
f(\phi)\times\varepsilon^{\frac{1}{2}(\alpha^{2}-\gamma^{2})}e^{(\alpha-\gamma)X_{\varepsilon}(i)}\,d\mathrm{LF}_{\mathbbm{C}}^{(\gamma,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}=\int
f(\phi)\mathopen{}\mathclose{{}\left(\frac{\mathrm{CR}(\eta_{1},z_{1})}{2}}\right)^{-\frac{\alpha^{2}}{2}+Q\alpha-2}d\mathrm{LF}_{\mathbbm{C}}^{(\alpha,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}.$
###### Proof.
Let $\theta_{\varepsilon}$ be the uniform probability measure on $\partial
B_{\varepsilon}(i)$ and
$\widehat{\theta}_{\varepsilon}:=\psi_{*}\theta_{\varepsilon}$. Recall
notations in Section 2.2.1, where $\mathbbm{P}_{\mathbbm{C}}$ is the
probability measure for the GFF on $\mathbbm{C}$. Let $\mathbbm{E}$ be the
expectation for $\mathbbm{P}_{\mathbbm{C}}$, then
$G_{\mathbbm{C}}(z,w)=\mathbbm{E}[h(x)h(y)]=-\log|z-w|+\log|z|_{+}+\log|w|_{+}$.
For a sample $h$ from $\mathbbm{P}_{\mathbbm{C}}$, we set
$\widetilde{h}:=h-2Q\log|\cdot|_{+}+\sum_{j=1}^{3}\gamma
G_{\mathbbm{C}}(\cdot,z_{j})$. Lemma 5.6 will follow from three identities.
$e^{(\alpha-\gamma)X_{\varepsilon}(i)}=\mathopen{}\mathclose{{}\left(\frac{\mathrm{CR}(\eta_{1},0)}{2}}\right)^{(\alpha-\gamma)Q}e^{(\alpha-\gamma)(\phi,\widehat{\theta}_{\varepsilon})}.$
(5.4) $\int f(\phi)\times
e^{(\alpha-\gamma)(\phi,\widehat{\theta}_{\varepsilon})}\,d\mathrm{LF}_{\mathbbm{C}}^{(\gamma,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}=\mathopen{}\mathclose{{}\left(\frac{\varepsilon\mathrm{CR}(\eta_{1},0)}{2}}\right)^{-(\alpha-\gamma)\gamma}\int\mathbbm{E}[e^{(\alpha-\gamma)(h,\widehat{\theta}_{\varepsilon})}f(\widetilde{h}+c)]e^{(\alpha+2\gamma-2Q)c}\,dc.$
(5.5)
$\int\mathbbm{E}[e^{(\alpha-\gamma)(h,\widehat{\theta}_{\varepsilon})}f(\widetilde{h}+c)]e^{(\alpha+2\gamma-2Q)c}\,dc=\mathopen{}\mathclose{{}\left(\frac{\varepsilon\mathrm{CR}(\eta_{1},0)}{2}}\right)^{-\frac{1}{2}(\alpha-\gamma)^{2}}\int
f(\phi)\mathrm{LF}_{\mathbbm{C}}^{(\alpha,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}.$
(5.6)
To prove (5.4), note that
$X_{\varepsilon}(i)=(X,\theta_{\varepsilon})=(\phi\circ\psi+Q\log|\psi^{\prime}|,\theta_{\varepsilon})$.
Since $\psi^{\prime}$ is holomorphic and $\log|\psi^{\prime}|$ is harmonic, we
have $(\log|\psi^{\prime}|,\theta_{\varepsilon})=\log|\psi^{\prime}(i)|$ hence
$(X,\theta_{\varepsilon})=(\phi,\widehat{\theta}_{\varepsilon})+Q\log|\psi^{\prime}(i)|$.
Since $|\psi^{\prime}(i)|=\frac{1}{2}\mathrm{CR}(\eta_{1},0)$, we get (5.4).
To prove (5.5), let $\eta^{\varepsilon}:=\psi(\partial B_{\varepsilon}(i))$.
Since $\widehat{\theta}_{\varepsilon}$ is the harmonic measure on
$\eta^{\varepsilon}$ viewed from $0$, we have
$(\log|\cdot|,\widehat{\theta}_{\varepsilon})=\log\mathrm{CR}(\eta^{\varepsilon},0)=\log\frac{\varepsilon\mathrm{CR}(\eta_{1},0)}{2}$.
Since $\varepsilon<\frac{1}{4}$, the curve $\eta^{\varepsilon}$ is contained
in the unit disk hence $(\log|\cdot|_{+},\widehat{\theta}_{\varepsilon})=0$.
Let
$G_{\mathbbm{C}}^{\varepsilon}(0,z):=(G_{\mathbbm{C}}(\cdot,z),\widehat{\theta}_{\varepsilon})$.
Then $G_{\mathbbm{C}}^{\varepsilon}(0,z)=G_{\mathbbm{C}}(0,z)$ for
$z\in\mathbbm{C}_{\eta_{1},p,\varepsilon}$ and
$G_{\mathbbm{C}}^{\varepsilon}(0,0)=-\log\frac{\varepsilon\mathrm{CR}(\eta_{1},0)}{2}$.
Since $(z_{1},z_{2},z_{3})=(0,1,e^{i\pi/3})$, we have
$(-2Q\log|\cdot|_{+}+\sum_{j}\gamma
G_{\mathbbm{C}}(\cdot,z_{j}),\widehat{\theta}_{\varepsilon})=-\gamma\log\frac{\varepsilon\mathrm{CR}(\eta_{1},0)}{2}$.
Therefore
$(\widetilde{h}+c,\widehat{\theta}_{\varepsilon})=(h,\widehat{\theta}_{\varepsilon})-\gamma\log\frac{\varepsilon\mathrm{CR}(\eta_{1},0)}{2}+c$.
Recall from Definition 2.4 that
$\mathrm{LF}_{\mathbbm{C}}^{(\gamma,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}$ is
the law of $\widetilde{h}+c$ under
$e^{(3\gamma-2Q)c}dc\mathbbm{P}_{\mathbbm{C}}(dh)$. This gives (5.5).
To prove (5.6), note that
$\mathbbm{E}[h(z)(h,\widehat{\theta}_{\varepsilon})]=(G_{\mathbbm{C}}(\cdot,z),\widehat{\theta}_{\varepsilon})=G_{\mathbbm{C}}^{\varepsilon}(0,z)$,
which equals $G_{\mathbbm{C}}(z,0)$ for
$z\in\mathbbm{C}_{\eta_{1},p,\varepsilon}$. By Girsanov’s theorem and the fact
that $f(\phi)$ depends only on $\phi|_{\mathbbm{C}_{\eta,p,\varepsilon}}$, we
have
$\displaystyle\int\mathbbm{E}[e^{(\alpha-\gamma)(h,\widehat{\theta}_{\varepsilon})}f(\widetilde{h}+c)]e^{(\alpha+2\gamma-2Q)c}\,dc$
$\displaystyle=\mathbbm{E}[e^{(\alpha-\gamma)(h,\widehat{\theta}_{\varepsilon})}]\int\mathbbm{E}[f(\widetilde{h}+(\alpha-\gamma)G^{\varepsilon}_{\mathbbm{C}}(\cdot,0)+c)]e^{(\alpha+2\gamma-2Q)c}\,dc$
$\displaystyle=\mathbbm{E}[e^{(\alpha-\gamma)(h,\widehat{\theta}_{\varepsilon})}]\int\mathbbm{E}[f(\widetilde{h}+(\alpha-\gamma)G_{\mathbbm{C}}(\cdot,0)+c)]e^{(\alpha+2\gamma-2Q)c}\,dc.$
Since
$\operatorname{Var}((h,\widehat{\theta}_{\varepsilon}))=(G_{\mathbbm{C}}^{\varepsilon}(\cdot,0),\widehat{\theta}_{\varepsilon})=-\log\frac{\varepsilon\mathrm{CR}(\eta_{1},0)}{2}$,
we have
$\mathbbm{E}[e^{(\alpha-\gamma)(h,\widehat{\theta}_{\varepsilon})}]=\mathopen{}\mathclose{{}\left(\frac{\varepsilon\mathrm{CR}(\eta_{1},0)}{2}}\right)^{-\frac{1}{2}(\alpha-\gamma)^{2}}$.
Since the law of $\widetilde{h}+c$ under
$e^{(\alpha+2\gamma-2Q)c}dc\mathbbm{P}_{\mathbbm{C}}(dh)$ is
$\mathrm{LF}_{\mathbbm{C}}^{(\alpha,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}$, we
get (5.6).
Combining (5.4), (5.5) and (5.6), and collecting the prefactors via
$\mathopen{}\mathclose{{}\left(\frac{\mathrm{CR}(\eta_{1},0)}{2}}\right)^{(\alpha-\gamma)Q}\mathopen{}\mathclose{{}\left(\frac{\varepsilon\mathrm{CR}(\eta_{1},0)}{2}}\right)^{-(\alpha-\gamma)\gamma}\mathopen{}\mathclose{{}\left(\frac{\varepsilon\mathrm{CR}(\eta_{1},0)}{2}}\right)^{-\frac{1}{2}(\alpha-\gamma)^{2}}=\varepsilon^{-\frac{1}{2}(\alpha^{2}-\gamma^{2})}\mathopen{}\mathclose{{}\left(\frac{\mathrm{CR}(\eta_{1},0)}{2}}\right)^{-\frac{\alpha^{2}}{2}+Q\alpha-2},$
we conclude the proof of Lemma 5.6. ∎
We also need the following fact when dealing with uniform weldings that
involve $\mathcal{M}^{\mathrm{disk}}_{1}(\alpha;\ell)$.
###### Lemma 5.7.
For $\alpha>\frac{\gamma}{2}$ and $\ell>0$, let $(D,h,z)$ be an embedding of a
sample from $\mathcal{M}^{\mathrm{disk}}_{1}(\alpha;\ell)$. Given $(D,h,z)$,
let $p$ be a point sampled from the harmonic measure on $\partial D$ viewed
from $z$, then the law of $(D,h,z,p)/{\sim_{\gamma}}$ equals that of
$(\mathbbm{H},X,i,0)/{\sim_{\gamma}}$ where $X$ is sampled from
$\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)$.
###### Proof.
We assume that $(D,h,z)=(\mathbbm{H},h,i)$ where $h$ is a sample from
$\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)$. Let
$\psi_{p}:\mathbbm{H}\to\mathbbm{H}$ be the conformal map with $\psi_{p}(i)=i$
and $\psi_{p}(p)=0$ and set
$X=h\circ\psi_{p}^{-1}+Q\log|(\psi_{p}^{-1})^{\prime}|$. Then by the
coordinate change for Liouville fields on $\mathbbm{H}$ (see e.g. [ARS21,
Lemma 2.4]), the law of $X$ is also
$\mathrm{LF}_{\mathbbm{H}}^{(\gamma,i)}(\ell)$. Since
$(D,h,z,p)/{\sim_{\gamma}}=(\mathbbm{H},X,i,0)/{\sim_{\gamma}}$ we are done. ∎
We are now ready to prove Proposition 5.4. For notational simplicity for
$\ell>0$ we let $\mathfrak{M}_{\ell}$ be the measure on decorated quantum
surfaces corresponding to
$\frac{\gamma^{2}}{4\pi^{4}(Q-\gamma)^{4}}\iint_{0}^{\infty}\ell_{2}\ell_{3}\mathrm{Weld}(\mathcal{M}_{1}^{\mathrm{disk}}(\gamma;\ell_{2}),\mathcal{M}_{1}^{\mathrm{disk}}(\gamma;\ell_{3}),\mathrm{QP}(\ell,\ell_{2},\ell_{3}))\,d\ell_{2}\,d\ell_{3},$
so that the relevant integral for Proposition 5.4 is
$\int_{0}^{\infty}\ell\mathrm{Weld}(\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell),\mathfrak{M}_{\ell})\,d\ell$.
We sample a decorated quantum surface from
$\int_{0}^{\infty}\ell\mathrm{Weld}(\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell),\mathfrak{M}_{\ell})\,d\ell$
(5.7)
and let $(\mathbbm{C},\phi,\eta_{1},\eta_{2},\eta_{3},z_{1},z_{2},z_{3})$ be
its embedding; since we specify the locations of the three marked points, the
tuple $(\phi,\eta_{1},\eta_{2},\eta_{3})$ is uniquely specified by the
decorated quantum surface. Let $D_{\eta_{1}}$ and $D^{c}_{\eta_{1}}$ be the
connected components of $\mathbbm{C}\backslash\eta_{1}$ such that
$D_{\eta_{1}}$ contains $z_{1}$. Let $p\in\eta_{1}$ be a point sampled from
the harmonic measure of $\partial D_{\eta_{1}}$ viewed from $z_{1}$ and set
$\mathcal{D}_{1}=(D_{\eta_{1}},\phi,z_{1},p)/{\sim_{\gamma}}$. Let
$\psi:\mathbbm{H}\to D_{\eta_{1}}$ be the conformal map with $\psi(i)=z_{1}$
and $\psi(0)=p$. Let $X=\phi\circ\psi+Q\log|\psi^{\prime}|$ so that
$\mathcal{D}_{1}=(\mathbbm{H},X,i,0)/{\sim_{\gamma}}$. Let
$\mathcal{D}^{c}_{1}$ be the decorated quantum surface
$(D^{c}_{\eta_{1}},\phi,\eta_{2},\eta_{3},z_{2},z_{3},p)/{\sim_{\gamma}}$. See
Figure 4 for an illustration. The next lemma describes the law of
$(X,\mathcal{D}^{c}_{1})$.
###### Lemma 5.8.
Given a decorated quantum surface $\mathcal{S}$ sampled from
$\mathfrak{M}_{\ell}$, we write $\mathcal{S}^{\bullet}$ as the decorated
quantum surface obtained by further sampling a point on the boundary of
$\mathcal{S}$ according to the probability measure proportional to its quantum
boundary length measure. Let $\mathfrak{M}_{\ell}^{\bullet}$ be the law of
$\mathcal{S}^{\bullet}$. Then the law of $(X,\mathcal{D}^{c}_{1})$ defined
right above is
$\int_{0}^{\infty}\ell\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)\times\mathfrak{M}_{\ell}^{\bullet}d\ell$.
###### Proof.
By the definition of uniform welding, conditioning on $\mathcal{D}_{1}$, the
conditional law of $\mathcal{D}^{c}_{1}$ is the probability measure
proportional to $\mathfrak{M}_{\ell}^{\bullet}$ with $\ell$ being the boundary
length of $\mathcal{D}_{1}$. Since the marked boundary point of
$\mathcal{D}_{1}$ is sampled from the harmonic measure, Lemma 5.7 now yields
Lemma 5.8. ∎
Figure 4: Illustration for Lemma 5.8 and the proof of Proposition 5.4. The
region $D_{\eta_{1}}$ surrounded by $\eta_{1}$ is colored grey. Both
$(D_{\eta_{1}},\phi,z_{1},p)$ and $(\mathbbm{H},X,0,i)$ are embeddings of
$\mathcal{D}_{1}$, which are related by $\psi$. The domain
$\mathbbm{C}_{\eta_{1},p,\varepsilon}$ equals
$\mathbbm{C}\setminus\psi(B_{\varepsilon}(i))$, the region outside the red
curve on the left.
###### Proof of Proposition 5.4.
In the setting of Lemma 5.8 with $\alpha=\gamma$, by Lemma 5.3 the law of
$(\phi,\eta_{1},\eta_{2},\eta_{3},p)$ is
$\mathrm{Harm_{z_{1},\eta_{1}}}(dp)\,\mathrm{LF}_{\mathbbm{C}}^{(\gamma,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}(d\phi)\,\mathsf{m}_{3}(d\eta_{1},d\eta_{2},d\eta_{3})$,
where $\mathrm{Harm}_{z_{1},\eta_{1}}$ means the harmonic measure on
$\eta_{1}$ viewed from $z_{1}$. Therefore Lemma 5.8 with $\alpha=\gamma$ can
be written as
$\mathrm{LF}_{\mathbbm{C}}^{(\gamma,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}(d\phi)\,\mathrm{Harm_{z_{1},\eta_{1}}}(dp)\,\mathsf{m}_{3}(d\eta_{1},d\eta_{2},d\eta_{3})=\int_{0}^{\infty}\ell\mathrm{LF}_{\mathbbm{H}}^{(\gamma,i)}(\ell)\times\mathfrak{M}_{\ell}^{\bullet}d\ell,$
(5.8)
in the sense that $(\phi,\eta_{1},\eta_{2},\eta_{3},p)$ and
$(X,\mathcal{D}^{c}_{1})$ determine each other, and the two sides of (5.8)
give the laws of $(\phi,\eta_{1},\eta_{2},\eta_{3},p)$ and
$(X,\mathcal{D}^{c}_{1})$, respectively.
Recall the notations in Lemma 5.6. For $\varepsilon\in(0,\frac{1}{4})$, for
any nonnegative measurable function $f$ of
$\phi|_{\mathbbm{C}_{\eta_{1},p,\varepsilon}}$, and any nonnegative measure
function $g$ of $(\eta_{1},\eta_{2},\eta_{3})$, we get from (5.8) that
$\displaystyle\int
f(\phi|_{\mathbbm{C}_{\eta_{1},p,\varepsilon}})g(\eta_{1},\eta_{2},\eta_{3})\varepsilon^{\frac{1}{2}(\alpha^{2}-\gamma^{2})}e^{(\alpha-\gamma)X_{\varepsilon}(i)}\mathrm{LF}_{\mathbbm{C}}^{(\gamma,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}(d\phi)\,\mathrm{Harm_{z_{1},\eta_{1}}}(dp)\,\mathsf{m}_{3}(d\eta_{1},d\eta_{2},d\eta_{3})$
$\displaystyle=\int_{0}^{\infty}\mathopen{}\mathclose{{}\left(\int
f(\phi|_{\mathbbm{C}_{\eta_{1},p,\varepsilon}})g(\eta_{1},\eta_{2},\eta_{3})\varepsilon^{\frac{1}{2}(\alpha^{2}-\gamma^{2})}e^{(\alpha-\gamma)X_{\varepsilon}(i)}\ell\mathrm{LF}_{\mathbbm{H}}^{(\gamma,i)}(\ell)\times\mathfrak{M}_{\ell}^{\bullet}}\right)d\ell.$
(5.9)
Recall that
$\mathsf{m}_{3}^{\alpha,\gamma,\gamma}=\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathrm{CR}(\eta_{1},z_{1})}\right)^{-\frac{\alpha^{2}}{2}+Q\alpha-2}\mathsf{m}_{3}$.
By Lemma 5.6, the left side of (5.9) equals
$\displaystyle\int
fg\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathrm{CR}(\eta_{1},z_{1})}\right)^{-\frac{\alpha^{2}}{2}+Q\alpha-2}\mathrm{LF}_{\mathbbm{C}}^{(\alpha,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}(d\phi)\,\mathrm{Harm_{z_{1},\eta_{1}}}(dp)\,\mathsf{m}_{3}(d\eta_{1},d\eta_{2},d\eta_{3})$
$\displaystyle=$ $\displaystyle\int
fg\mathrm{LF}_{\mathbbm{C}}^{(\alpha,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}(d\phi)\mathrm{Harm_{z_{1},\eta_{1}}}(dp)\,\mathsf{m}_{3}^{\alpha,\gamma,\gamma}\,(d\eta_{1},d\eta_{2},d\eta_{3}).$
(5.10)
Here we write $f=f(\phi|_{\mathbbm{C}_{\eta_{1},p,\varepsilon}})$ and
$g=g(\eta_{1},\eta_{2},\eta_{3})$ to ease the notation.
By Lemma 5.5, the right side of (5.9) equals
$\int_{0}^{\infty}\mathopen{}\mathclose{{}\left(\int
fg\ell\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)\times\mathfrak{M}_{\ell}^{\bullet}}\right)d\ell.$
(5.11)
Since $\varepsilon,f,g$ are arbitrary, comparing (5.10) and (5.11) we get in
the same sense as in (5.8) that
$\mathrm{LF}_{\mathbbm{C}}^{(\alpha,z_{1}),(\gamma,z_{2}),(\gamma,z_{3})}(d\phi)\mathrm{Harm_{z_{1},\eta_{1}}}(dp)\,\mathsf{m}_{3}^{\alpha,\gamma,\gamma}(d\eta_{1},d\eta_{2},d\eta_{3})=\int_{0}^{\infty}\ell\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)\times\mathfrak{M}_{\ell}^{\bullet}d\ell.$
Forgetting about the point $p$ we get Proposition 5.4. ∎
###### Proof of Theorem 5.2.
The case $(\alpha_{1},\alpha_{2},\alpha_{3})=(\alpha,\gamma,\gamma)$ was
proved in Proposition 5.4 by reweighting from the $(\gamma,\gamma,\gamma)$
case. The exact same argument in Lemma 5.6 gives the following extension:
$\displaystyle\int
f(\phi|_{\mathbbm{C}_{\eta_{1},p,\varepsilon}})\times\varepsilon^{\frac{1}{2}(\alpha^{2}-\alpha_{1}^{2})}e^{(\alpha-\alpha_{1})X_{\varepsilon}(i)}\,d\mathrm{LF}_{\mathbbm{C}}^{(\alpha_{1},z_{1}),(\alpha_{2},z_{2}),(\alpha_{3},z_{3})}$
$\displaystyle=$ $\displaystyle\int
f(\phi|_{\mathbbm{C}_{\eta_{1},p,\varepsilon}})\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathrm{CR}(\eta_{1},z_{1})}\right)^{-\frac{\alpha_{1}^{2}}{2}+Q\alpha_{1}-2}\mathrm{LF}_{\mathbbm{C}}^{(\alpha,z_{1}),(\alpha_{2},z_{2}),(\alpha_{3},z_{3})}.$
(5.12)
By symmetry (5.12) holds if the roles of $z_{1},z_{2},z_{3}$ are permutated.
This allows us to obtain the general $(\alpha_{1},\alpha_{2},\alpha_{3})$ case
from the $(\gamma,\gamma,\gamma)$ case by reweighing around the three points
sequentially. ∎
### 5.2 Matching the quantum area: proof of Theorem 1.2
Theorem 1.2 is an immediate consequence of Lemma 5.1 and the following
proposition.
###### Proposition 5.9.
Fix $(z_{1},z_{2},z_{3})=(0,1,e^{i\pi/3})$. There is a constant
$C=C(\gamma)\in(0,\infty)$ such that for
$\alpha_{1},\alpha_{2},\alpha_{3}\in(Q-\frac{\gamma}{4},Q)$, and
$\mathsf{m}_{3}^{\alpha_{1},\alpha_{2},\alpha_{3}}$ as defined in (5.1) we
have
$\mathopen{}\mathclose{{}\left(\mathrm{LF}_{\mathbbm{C}}^{(\alpha_{i},z_{i})_{3}}\times\mathsf{m}_{3}^{\alpha_{1},\alpha_{2},\alpha_{3}}}\right)[e^{-\mu_{\phi}(\mathbbm{C})}]=C\prod_{i=1}^{3}\frac{2^{\alpha_{i}^{2}/2-Q\alpha_{i}}\Gamma(\frac{\gamma\alpha_{i}}{2}-\frac{\gamma^{2}}{4})}{\Gamma(\frac{2}{\gamma}(Q-\alpha_{i}))\cos(\frac{2\pi}{\gamma}(Q-\alpha_{i}))}\mathopen{}\mathclose{{}\left(\frac{\pi\Gamma(\frac{\gamma^{2}}{4})}{\Gamma(1-\frac{\gamma^{2}}{4})}}\right)^{-\frac{\alpha_{i}}{\gamma}}.$
(5.13)
###### Proof.
Recall $\mathrm{QP}_{3}(\ell_{1},\ell_{2},\ell_{3})[e^{-A}]$ from Theorem 4.2.
Recall $\mathcal{M}_{1}^{\mathrm{disk}}(\alpha_{i};\ell_{i})[e^{-A_{i}}]$ from
Theorem 2.16, where $A_{i}$ is the quantum area of a sample from
$\mathcal{M}_{1}^{\mathrm{disk}}(\alpha_{i};\ell_{i})$. By Theorem 5.2, for
some $\gamma$-dependent constants $C_{1},C_{2},C_{3}$, we have that
$\mathopen{}\mathclose{{}\left(\mathrm{LF}_{\mathbbm{C}}^{(\alpha_{i},z_{i})_{3}}\times\mathsf{m}_{3}^{\alpha_{1},\alpha_{2},\alpha_{3}}}\right)[e^{-\mu_{\phi}(\mathbbm{C})}]$
equals
$\displaystyle C_{1}$
$\displaystyle\iiint_{0}^{\infty}\ell_{1}\ell_{2}\ell_{3}\mathrm{QP}_{3}(\ell_{1},\ell_{2},\ell_{3})[e^{-A}]\prod_{i=1}^{3}\mathcal{M}_{1}^{\mathrm{disk}}(\alpha_{i};\ell_{i})[e^{-A_{i}}]\,d\ell_{1}\,d\ell_{2}\,d\ell_{3}$
$\displaystyle=C_{2}\prod_{i=1}^{3}\frac{\overline{U}(\alpha_{i})(4\sin(\frac{\pi\gamma^{2}}{4}))^{\alpha_{i}/\gamma}}{2^{\alpha_{i}^{2}/2}\Gamma(\frac{2}{\gamma}(Q-\alpha_{i}))}\int_{0}^{\infty}\frac{1}{\sqrt{\ell_{i}}}e^{-\ell_{i}\sqrt{\frac{1}{\sin(\pi\gamma^{2}/4)}}}K_{\frac{2}{\gamma}(Q-\alpha_{i})}\mathopen{}\mathclose{{}\left(\ell_{i}\sqrt{\frac{1}{\sin(\pi\gamma^{2}/4)}}}\right)\,d\ell_{i}$
$\displaystyle=C_{3}\prod_{i=1}^{3}\frac{\overline{U}(\alpha_{i})(4\sin(\frac{\pi\gamma^{2}}{4}))^{\alpha_{i}/\gamma}}{2^{\alpha_{i}^{2}/2}\Gamma(\frac{2}{\gamma}(Q-\alpha_{i}))\cos(\frac{2\pi}{\gamma}(Q-\alpha_{i}))},$
where $\overline{U}$ is as in (2.4) and (2.7) is used to evaluate the integral
on the second line. Expanding $\overline{U}(\alpha_{i})$ from (2.4) and
substituting $\sin(\frac{\pi\gamma^{2}}{4})$ by
$\frac{\pi}{\Gamma(\frac{\gamma^{2}}{4})\Gamma(1-\frac{\gamma^{2}}{4})}$, we
get (5.13). ∎
###### Proof of Theorem 1.2.
We divide the proof into four cases depending on the parameter range.
Case I: $\kappa<4$ and
$\lambda_{i}\in(\frac{3\kappa}{32}-1+\frac{2}{\kappa},\frac{\kappa}{8}-1+\frac{2}{\kappa})$
for all $i=1,2,3$. In this case we can find
$\alpha_{i}\in(Q-\frac{\gamma}{4},Q)$ satisfying
$\lambda_{i}=-\frac{\alpha_{i}^{2}}{2}+Q\alpha_{i}-2$ for each $i$. Comparing
Lemma 5.1 and Proposition 5.9 yields that for some $\gamma$-dependent constant
$C(\gamma)$ we have
${C^{\operatorname{CLE}}_{\kappa}(\lambda_{1},\lambda_{2},\lambda_{3})}=\frac{C(\gamma)}{C_{\gamma}^{\mathrm{DOZZ}}(\alpha_{1},\alpha_{2},\alpha_{3})}\prod_{i=1}^{3}\frac{\Gamma(\frac{\gamma\alpha_{i}}{2}-\frac{\gamma^{2}}{4})}{\Gamma(\frac{2}{\gamma}(Q-\alpha_{i}))\cos(\frac{2\pi}{\gamma}(Q-\alpha_{i}))}\mathopen{}\mathclose{{}\left(\frac{\pi\Gamma(\frac{\gamma^{2}}{4})}{\Gamma(1-\frac{\gamma^{2}}{4})}}\right)^{-\frac{\alpha_{i}}{\gamma}}.$
Now we can evaluate $C(\gamma)$ by setting
$\alpha_{1}=\alpha_{2}=\alpha_{3}=\gamma$, in which case $\lambda_{i}=0$ and
$C_{\kappa}^{\operatorname{CLE}}(0,0,0)=1$. This gives (1.3) for
${C^{\operatorname{CLE}}_{\kappa}(\lambda_{1},\lambda_{2},\lambda_{3})}$ and
(1.4) for the factor $N_{\gamma}(\alpha)$.
For the next two cases set $(z_{1},z_{2},z_{3})=(0,1,e^{i\pi/3})$ so that
$C_{\kappa}^{\operatorname{CLE}}(\lambda_{1},\lambda_{2},\lambda_{3})=\mathbbm{E}[\prod_{i=1}^{3}(\mathrm{CR}(\eta_{i},z_{i}))^{\lambda_{i}}]$.
Since the distance between $z_{i}$ and $\eta_{i}$ is less than $1$, Koebe 1/4
theorem yields $\frac{1}{4}\mathrm{CR}(\eta_{i},z_{i})\leq 1$. Thus
$\mathbbm{E}[\prod_{i=1}^{3}(\frac{1}{4}\mathrm{CR}(\eta_{i},z_{i}))^{\lambda_{i}}]\leq\mathbbm{E}[\prod_{i=1}^{3}(\frac{1}{4}\mathrm{CR}(\eta_{i},z_{i}))^{\widetilde{\lambda}_{i}}]$
for any real $\lambda_{i},\widetilde{\lambda}_{i}$ such that
$\lambda_{i}\geq\widetilde{\lambda}_{i}$. Namely
$4^{-\lambda_{1}-\lambda_{2}-\lambda_{3}}C_{\kappa}^{\operatorname{CLE}}(\lambda_{1},\lambda_{2},\lambda_{3})\leq
4^{-\widetilde{\lambda}_{1}-\widetilde{\lambda}_{2}-\widetilde{\lambda}_{3}}C_{\kappa}^{\operatorname{CLE}}(\widetilde{\lambda}_{1},\widetilde{\lambda}_{2},\widetilde{\lambda}_{3}).$
(5.14)
Case II: $\kappa<4$ and all
$\lambda_{i}>\frac{3\kappa}{32}-1+\frac{2}{\kappa}$. By (5.14) on
$\\{(\lambda_{1},\lambda_{2},\lambda_{3})\in\mathbbm{C}^{3}:\operatorname{Re}\lambda_{i}>\frac{3\kappa}{32}-1+\frac{2}{\kappa}\\}$,
the function
${C^{\operatorname{CLE}}_{\kappa}(\lambda_{1},\lambda_{2},\lambda_{3})}$ is
finite, hence analytic. On the other hand, the right hand side of (1.3) is an
explicit meromorphic function in $(\alpha_{1},\alpha_{2},\alpha_{3})$. Now
Case II follows from Case I.
Case III: $\kappa<4$ and
$\lambda_{i}\leq\frac{3\kappa}{32}-1+\frac{2}{\kappa}$ for some $i$. We will
show that
${C^{\operatorname{CLE}}_{\kappa}(\lambda_{1},\lambda_{2},\lambda_{3})}=\infty$
in this case. By the monotonicity (5.14) and symmetry, it suffices to prove
$C_{\kappa}^{\operatorname{CLE}}(\lambda_{1},\lambda_{2},\lambda_{3})=\infty$
for $\lambda_{1}\leq\frac{3\kappa}{32}-1+\frac{2}{\kappa}$ and
$\lambda_{2},\lambda_{3}>\frac{3\kappa}{32}-1+\frac{2}{\kappa}$. For
$\lambda>\frac{3\kappa}{32}-1+\frac{2}{\kappa}$ we have
$4^{-\lambda_{1}-\lambda_{2}-\lambda_{3}}C_{\kappa}^{\operatorname{CLE}}(\lambda_{1},\lambda_{2},\lambda_{3})>4^{-\lambda-\lambda_{2}-\lambda_{3}}C_{\kappa}^{\operatorname{CLE}}(\lambda,\lambda_{2},\lambda_{3}),$
Suppose $(\alpha,\alpha_{2},\alpha_{3})\in(Q-\frac{\gamma}{4},Q)^{3}$
correspond to $(\lambda,\lambda_{2},\lambda_{3})$. Then
$\lambda\downarrow\frac{3\kappa}{32}-1+\frac{2}{\kappa}$ means
$\alpha\downarrow Q-\frac{\gamma}{4}$. Recall the explicit formula for
$C_{\kappa}^{\operatorname{CLE}}(\lambda,\lambda_{2},\lambda_{3})$ in (1.3)
proven in Case II. Since $\lim_{\alpha\downarrow
Q-\frac{\gamma}{4}}N_{\gamma}(\alpha)=\infty$ and
$C_{\gamma}^{\mathrm{DOZZ}}(\alpha,\alpha_{2},\alpha_{3})\in(0,\infty)$ as
$(\alpha,\alpha_{2},\alpha_{3})$ satisfies the Seiberg bounds (2.2), we must
have
$C_{\kappa}^{\operatorname{CLE}}(\lambda,\lambda_{2},\lambda_{3})\rightarrow\infty$
as $\lambda\downarrow\frac{3\kappa}{32}-1+\frac{2}{\kappa}$. Therefore
$C_{\kappa}^{\operatorname{CLE}}(\lambda_{1},\lambda_{2},\lambda_{3})=\infty$
as desired.
Case IV: $\kappa=4$. This follows from Lemma A.5 via the continuity as
$\kappa\uparrow 4$. ∎
## 6 Electrical thickness of the SLE loop via conformal welding
In this section we prove Theorem 1.3. We will work on the horizontal cylinder
$\mathcal{C}=\mathbbm{R}\times[0,2\pi]/{\sim}$ with $x\sim x+2\pi i$ for
$x\in\mathbbm{R}$. Recall the shape measure $\mathcal{L}_{\kappa}$ of
$\operatorname{SLE}^{\mathrm{loop}}_{\kappa}$. Let
$\mathcal{L}_{\kappa}(\mathcal{C})$ be the pullback of $\mathcal{L}_{\kappa}$
under the map $z\mapsto e^{-z}$, which is a probability measure on loops
$\eta$ in $\mathcal{C}$ separating $\pm\infty$ and satisfying
$\max_{z\in\eta}\operatorname{Re}z=0$. For a loop $\eta$ sampled from
$\mathcal{L}_{\kappa}(\mathcal{C})$, we write $\vartheta(\eta)$ for the
electrical thickness of $\exp(\eta)$, namely
$-\log\mathrm{CR}(\exp(\eta),0)-\log\mathrm{CR}(\exp(-\eta),0)$. Then Theorem
1.3 is equivalent to
$\mathbbm{E}[e^{\lambda\vartheta(\eta)}]=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{ll}\frac{\sin(\pi(1-\kappa/4))}{\pi(1-\kappa/4)}\frac{\pi\sqrt{(1-\kappa/4)^{2}+\lambda\kappa/2}}{\sin(\pi\sqrt{(1-\kappa/4)^{2}+\lambda\kappa/2})}&\mbox{if
}\lambda<1-\frac{\kappa}{8}.\\\ \infty&\mbox{if }\lambda\geq
1-\frac{\kappa}{8}\end{array}}\right.$ (6.1)
Consider $\alpha<Q$ and set $\lambda=\frac{\alpha^{2}}{2}-Q\alpha+2$. Let
$\mathcal{L}_{\kappa}^{\alpha}$ be defined by the following reweighting of
$\mathcal{L}_{\kappa}(\mathcal{C})$:
$\frac{d\mathcal{L}_{\kappa}^{\alpha}}{d\mathcal{L}_{\kappa}(\mathcal{C})}(\eta)=(\frac{1}{4}\mathrm{CR}(\exp(\eta),0)\mathrm{CR}(\exp(-\eta),0))^{-\frac{\alpha^{2}}{2}+Q\alpha-2}=2^{2\lambda}e^{\lambda\vartheta(\eta)}.$
(6.2)
Then proving Theorem 1.3 amounts to computing the total mass
$|\mathcal{L}_{\kappa}^{\alpha}|$. Similarly as in the proof of Theorem 1.2 in
Section 5, we need a conformal welding result, Proposition 6.2, which involves
$\mathcal{L}_{\kappa}^{\alpha}$. It is a generalization of Proposition 2.20
where marked points have $\alpha$-singularities. To state it we first recall
the $\alpha$-generalization of the two-pointed quantum sphere; see e.g.
[AHS20, Section 2].
###### Definition 6.1.
For $\gamma\in(0,2)$, let $(B_{s})_{s\geq 0}$ be a standard Brownian motion
conditioned on $B_{s}-(Q-\alpha)s<0$ for all $s>0$, and
$(\widetilde{B}_{s})_{s\geq 0}$ an independent copy of $(B_{s})_{s\geq 0}$.
Let
$Y_{t}=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{ll}B_{t}-(Q-\alpha)t&\mbox{if
}t\geq 0\\\ \widetilde{B}_{-t}+(Q-\alpha)t&\mbox{if }t<0\end{array}}\right.$
Let $h^{1}(z)=Y_{\operatorname{Re}z}$ for each $z\in\mathcal{C}$. Let
$h^{2}_{\mathcal{C}}$ be independent of $h^{1}$ and have the law of the
lateral component of the GFF on $\mathcal{C}$. Let
$\hat{h}=h^{1}+h^{2}_{\mathcal{C}}$. Let $\mathbf{c}\in\mathbbm{R}$ be sampled
from $\frac{\gamma}{2}e^{2(\alpha-Q)c}dc$ independent of $\hat{h}$. Let
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)$ be the infinite measure describing
the law of the decorated quantum surface
$(\mathcal{C},\hat{h}+\mathbf{c},-\infty,+\infty)/{\sim_{\gamma}}$.
Consider $\kappa\in(0,4)$ and $\gamma=\sqrt{\kappa}$. Sample a pair
$(\eta,\mathbf{t})$ from $\mathcal{L}_{\kappa}^{\alpha}\times dt$ where $dt$
is the Lebesgue measure on $\mathbbm{R}$. Let
$\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}$ be the law of the
translated loop $\eta+\mathbf{t}$. Let $\mathbb{F}$ be the law of
$\hat{h}+\mathbf{c}$ as in Definition 6.1, so that the law of
$(\mathcal{C},\hat{h}+\mathbf{c},-\infty,+\infty)/{\sim_{\gamma}}$ is
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)$. Now sample $(h,\eta)$ from
$\mathbb{F}\times\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}$ and write
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep,\alpha}}$
as the law of $(\mathcal{C},h,\eta,-\infty,+\infty)/{\sim_{\gamma}}$. We are
now ready to state the conformal welding result needed for the proof of
Theorem 1.3.
###### Proposition 6.2.
For $\alpha\in(\frac{\gamma}{2},Q)$ and some $\gamma$-dependent constant
$C\in(0,\infty)$ we have
$C(Q-\alpha)^{2}\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}=\int_{0}^{\infty}\ell\cdot\mathrm{Weld}(\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell),\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell))\,d\ell,$
(6.3)
where
$\mathrm{Weld}(\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell),\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell))$
is defined as $\mathrm{Weld}(\mathrm{QD}_{1,0}(\ell),\mathrm{QD}_{1,0}(\ell))$
in Proposition 2.20.
We postpone the proof of Proposition 6.2 to Section 6.1 and proceed to the
proof of Theorem 1.3. Similarly as in the proof of Theorem 1.2, we would like
to evaluate the average of $e^{-A}$ on both sides of (6.3) and compare the
expressions to obtain $|\mathcal{L}^{\alpha}_{\kappa}|$. However, since
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}[e^{-A}]=\infty$,
we have to consider an alternative observable. Note that
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}$
is a measure on quantum surfaces decorated by two (ordered) marked points and
a loop separating them. The loop separates the quantum surface into two
connected components. For $0<\varepsilon<\delta$, let $E_{\delta,\varepsilon}$
be the event that the connected component containing the first marked point
has quantum area at least 1 and the loop has quantum length in
$(\varepsilon,\delta)$. We use the size of $E_{\delta,\varepsilon}$ as our
obervable, whose asymptotic is easy to obtain using Proposition 6.2.
###### Lemma 6.3.
Let $\alpha\in(\frac{\gamma}{2},Q)$. Recall $\overline{U}(\alpha)$ from
Proposition 2.10. With $C$ from Proposition 6.2 and an error term
$o_{\delta,\varepsilon}(1)$ satisfying $\lim_{\delta\to
0}\limsup_{\varepsilon\to 0}|o_{\delta,\varepsilon}(1)|=0$, we have
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}[E_{\delta,\varepsilon}]=\frac{1}{C(Q-\alpha)^{2}}\times\frac{(1+o_{\delta,\varepsilon}(1))\log\varepsilon^{-1}}{\frac{2}{\gamma}(Q-\alpha)\Gamma(\frac{2}{\gamma}(Q-\alpha))}\mathopen{}\mathclose{{}\left(\frac{2}{\gamma}2^{-\frac{\alpha^{2}}{2}}\overline{U}(\alpha)}\right)^{2}\mathopen{}\mathclose{{}\left(4\sin\frac{\pi\gamma^{2}}{4}}\right)^{-\frac{2}{\gamma}(Q-\alpha)}.$
###### Proof.
By Proposition 6.2,
$C(Q-\alpha)^{2}\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}[E_{\delta,\varepsilon}]$
equals the mass of $E_{\delta,\varepsilon}$ under the measure
$\int_{0}^{\infty}\ell\cdot\mathrm{Weld}(\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell),\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell))\,d\ell$.
This mass can be expressed as
$\int_{\varepsilon}^{\delta}\ell\cdot\mathopen{}\mathclose{{}\left|\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell)}\right|\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell)[A>1]\,d\ell=\int_{\varepsilon}^{\delta}\ell\mathopen{}\mathclose{{}\left|\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell)}\right|^{2}\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell)^{\\#}[A>1]\;d\ell,$
(6.4)
where $A$ is the quantum area of a sample from
$\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell)$. From the scaling relation
between quantum area and length, and the explicit density of $A$ under
$\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;1)^{\\#}$ given by the FZZ formula
Theorem 2.16, we see that
$\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell)^{\\#}[A>1]=\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;1)^{\\#}[A>\ell^{-2}]=\frac{1+o_{\ell}(1)}{\frac{2}{\gamma}(Q-\alpha)\Gamma(\frac{2}{\gamma}(Q-\alpha))}\mathopen{}\mathclose{{}\left(\frac{\ell^{2}}{4\sin\frac{\pi\gamma^{2}}{4}}}\right)^{\frac{2}{\gamma}(Q-\alpha)}\textrm{
as }\ell\to 0.$
By Proposition 2.10,
$|\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell)|=\frac{2}{\gamma}2^{-\frac{\alpha^{2}}{2}}\overline{U}(\alpha)\ell^{\frac{2}{\gamma}(\alpha-Q)-1}$.
Therefore the right side of (6.4) equals
$\frac{1}{\frac{2}{\gamma}(Q-\alpha)\Gamma(\frac{2}{\gamma}(Q-\alpha))}\mathopen{}\mathclose{{}\left(\frac{2}{\gamma}2^{-\frac{\alpha^{2}}{2}}\overline{U}(\alpha)}\right)^{2}\mathopen{}\mathclose{{}\left(4\sin\frac{\pi\gamma^{2}}{4}}\right)^{-\frac{2}{\gamma}(Q-\alpha)}\int_{\varepsilon}^{\delta}\ell^{-1}(1+o_{\ell}(1))\,d\ell.$
Writing $\int_{\varepsilon}^{\delta}\ell^{-1}(1+o_{\ell}(1))\,d\ell$ as
$(1+o_{\delta,\varepsilon}(1))\log\varepsilon^{-1}$, we have $\lim_{\delta\to
0}\limsup_{\varepsilon\to 0}|o_{\delta,\varepsilon}(1)|=0$ as desired. ∎
Given Lemma 6.3, the following proposition is the key to our proof of Theorem
1.3. It gives the size of $E_{\delta,\varepsilon}$ in terms of
$|\mathcal{L}_{\kappa}^{\alpha}|$ and the reflection coefficient for LCFT.
###### Proposition 6.4.
Let $\alpha\in(\frac{\gamma}{2},Q)$. With an error term
$o_{\delta,\varepsilon}(1)$ satisfying $\lim_{\delta\to
0}\limsup_{\varepsilon\to 0}|o_{\delta,\varepsilon}(1)|=0$, we have
$(\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha})[E_{\delta,\varepsilon}]=(1+o_{\delta,\varepsilon}(1))\frac{\overline{R}(\alpha)}{2(Q-\alpha)^{2}}|\mathcal{L}_{\kappa}^{\alpha}|\log\varepsilon^{-1},$
where $\overline{R}(\alpha)$ is the unit-volume reflection coefficient for
LCFT on the sphere [KRV17, RV19]:
$\overline{R}(\alpha)=-\mathopen{}\mathclose{{}\left(\frac{\pi\Gamma(\frac{\gamma^{2}}{4})}{\Gamma(1-\frac{\gamma^{2}}{4})}}\right)^{\frac{2}{\gamma}(Q-\alpha)}\frac{1}{\frac{2}{\gamma}(Q-\alpha)}\frac{\Gamma(-\frac{\gamma}{2}(Q-\alpha))}{\Gamma(\frac{\gamma}{2}(Q-\alpha))\Gamma(\frac{2}{\gamma}(Q-\alpha))}.$
(6.5)
We postpone the proof of Proposition 6.4 to Section 6.2 and proceed to prove
Theorem 1.3.
###### Proposition 6.5.
There exists a $\gamma$-dependent constant $C\in(0,\infty)$ such that
$2^{-\alpha^{2}+2Q\alpha}|\mathcal{L}_{\kappa}^{\alpha}|=C\frac{\frac{\gamma}{2}(Q-\alpha)}{\sin(\frac{\gamma\pi}{2}(Q-\alpha))}\quad\textrm{for
all }\alpha\in(\frac{\gamma}{2},Q).$ (6.6)
###### Proof.
By Lemma 6.3 and Proposition 6.4, we get
$\overline{R}(\alpha)|\mathcal{L}_{\kappa}^{\alpha}|=\frac{C^{\prime}}{(Q-\alpha)\Gamma(\frac{2}{\gamma}(Q-\alpha))}\mathopen{}\mathclose{{}\left(2^{-\frac{\alpha^{2}}{2}}\overline{U}(\alpha)}\right)^{2}\mathopen{}\mathclose{{}\left(4\sin\frac{\pi\gamma^{2}}{4}}\right)^{-\frac{2}{\gamma}(Q-\alpha)}$
for some $\gamma$-dependent constant $C^{\prime}$. Plugging in
$\overline{R}(\alpha)$ and $\overline{U}(\alpha)$ from (6.5) and (2.4), and
simplifying via the identities $\Gamma(z+1)=z\Gamma(z)$ and
$\Gamma(1-z)\Gamma(z)=\frac{\pi}{\sin(\pi z)}$, we arrive at (6.6). ∎
###### Proof of Theorem 1.3.
We break the proof of (6.1) hence Theorem 1.3, into three cases.
Case I: $\kappa<4$ and
$\lambda\in(1-\frac{\kappa}{8}-\frac{2}{\kappa},1-\frac{\kappa}{8})$. Set
$\alpha=Q-\sqrt{Q^{2}-4+2\lambda}$. Then $\alpha\in(\frac{\gamma}{2},Q)$ in
this case. Since $\lambda=\frac{\alpha^{2}}{2}-Q\alpha+2$, by (6.2) and
Proposition 6.5 we have
$\mathbbm{E}[e^{\lambda\vartheta(\eta)}]=2^{-2\lambda}|\mathcal{L}_{\kappa}^{\alpha}|=C\frac{\frac{\gamma}{2}(Q-\alpha)}{\sin(\frac{\gamma\pi}{2}(Q-\alpha))}=C\frac{\sqrt{(1-\frac{\kappa}{4})^{2}+\frac{\lambda\kappa}{2}}}{\sin(\pi\sqrt{(1-\frac{\kappa}{4})^{2}+\frac{\lambda\kappa}{2}})}$
for some $\gamma$-dependent constant $C$. Since $\kappa\in(0,4)$, we have
$0\in(1-\frac{\kappa}{8}-\frac{2}{\kappa},1-\frac{\kappa}{8})$. Thus we can
obtain the value of $C$ by considering
$1=\mathbbm{E}[e^{0}]=\frac{C(1-\kappa/4)}{\sin(\pi(1-\kappa/4))}$. This
yields (6.1) in this case.
Case II: $\kappa<4$ and $\lambda\in\mathbbm{R}$. Since $\vartheta(\eta)\geq
0$, the function $\lambda\mapsto\mathbbm{E}[e^{\lambda\vartheta(\eta)}]$ is
increasing. Thus for $\lambda<0$ we have
$\mathbbm{E}[e^{\lambda\vartheta(\eta)}]\leq\mathbbm{E}[e^{0\cdot\vartheta(\eta)}]=1$.
Since $1-\frac{\kappa}{8}-\frac{2}{\kappa}<0$ we can use analytic continuation
to extend (6.1) from
$\lambda\in(1-\frac{\kappa}{8}-\frac{2}{\kappa},1-\frac{\kappa}{8})$ to
$\lambda\in(-\infty,1-\frac{\kappa}{8})$. On the other hand, for any
$\lambda\geq 1-\frac{\kappa}{8}$ we have
$\mathbbm{E}[e^{\lambda\vartheta(\eta)}]\geq\lim_{\lambda^{\prime}\uparrow
1-\frac{\kappa}{8}}\mathbbm{E}[e^{\lambda^{\prime}\vartheta(\eta)}]$, which
equals $\infty$ using the explicit formula of
$\mathbbm{E}[e^{\lambda^{\prime}\vartheta(\eta)}]$.
Case III: $\kappa=4$. By Lemma A.4,
$\vartheta(\eta_{\kappa})\to\vartheta(\eta_{4})$ in law as $\kappa\uparrow 4$,
where $\eta_{\kappa}$ is a sample from $\mathcal{L}_{\kappa}$. Fix
$\lambda<\frac{1}{2}$ and $\lambda^{\prime}\in(\lambda,\frac{1}{2})$. Since
$\mathbbm{E}[e^{\lambda\vartheta(\eta_{\kappa})}]\leq\mathbbm{E}[e^{\lambda^{\prime}\vartheta(\eta_{\kappa})}]$
and $\mathbbm{E}[e^{\lambda^{\prime}\vartheta(\eta_{\kappa})}]$ is uniformly
bounded in $\kappa\in[3,4)$, the family
$\\{e^{\lambda\vartheta(\eta_{\kappa})}:\kappa\in[3,4)\\}$ is uniformly
integrable. Therefore $\lim_{\kappa\uparrow
4}\mathbbm{E}[e^{\lambda\vartheta(\eta_{\kappa})}]=\mathbbm{E}[e^{\lambda\vartheta(\eta_{4})}]$
as desired. For $\lambda\geq\frac{1}{2}$, the same argument as in Case II
gives $\mathbbm{E}[e^{\lambda\vartheta(\eta)}]=\infty$. ∎
It remains to prove Propositions 6.2 and 6.4. The common starting point of the
proofs is the relation between $\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)$ and
the Liouville field on $\mathcal{C}$ that we now recall from [AHS21].
###### Definition 6.6.
Let $P_{\mathcal{C}}$ be the law of the GFF on the cylinder $\mathcal{C}$
defined above Definition 2.7. Let $\alpha\in\mathbbm{R}$. Sample
$(h,\mathbf{c})$ from $P_{\mathcal{C}}\times[e^{(2\alpha-2Q)c}\,dc]$ and let
$\phi(z)=h(z)-(Q-\alpha)\mathopen{}\mathclose{{}\left|\operatorname{Re}z}\right|+\mathbf{c}$.
We write $\mathrm{LF}_{\mathcal{C}}^{(\alpha,\pm\infty)}$ as the law of
$\phi$.
We have the following description of
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep,\alpha}}$
in terms of $\mathrm{LF}_{\mathcal{C}}^{(\alpha,\pm\infty)}$ and
$\mathcal{L}_{\kappa}^{\alpha}$.
###### Proposition 6.7.
For $(\phi,\eta)$ sampled from
$\mathrm{LF}_{\mathcal{C}}^{(\alpha,\pm\infty)}\times\mathcal{L}_{\kappa}^{\alpha}$,
the law of $(\mathcal{C},\phi,\eta,-\infty,+\infty)/{\sim_{\gamma}}$ is
$C(Q-\alpha)^{2}\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}\quad\text{for
some $\gamma$-dependent constant }C\in(0,\infty).$ (6.7)
###### Proof.
This is an immediate consequence of the definition of
$\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}$ and [AHS21, Theorem 2.11],
which says the following: let $h$ be the field $\hat{h}+\mathbf{c}$ in
Definition 6.1, let $T\in\mathbbm{R}$ be sampled from Lebesgue measure
independently of $h$, and set $\phi:=h(\cdot+T)$. Then $\phi$ has law
$\frac{\gamma}{4(Q-\alpha)^{2}}\mathrm{LF}_{\mathcal{C}}^{(\alpha,\pm\infty)}$.
∎
We now proceed to prove Propositions 6.2 and 6.4 in the next two subsections.
### 6.1 Conformal welding with two generic insertions: proof of Proposition
6.2
Our proof of Proposition 6.2 closely follows that of Theorem 5.2. We start
from the case $\alpha=\gamma$.
###### Lemma 6.8.
If $(\phi,\eta)$ is sampled from
$\mathrm{LF}_{\mathcal{C}}^{(\gamma,\pm\infty)}\times\mathcal{L}_{\kappa}(\mathcal{C})$,
then the law of $(\mathcal{C},\phi,\eta,-\infty,+\infty)/{\sim_{\gamma}}$ is
$C\int_{0}^{\infty}\ell\cdot\mathrm{Weld}(\mathcal{M}_{1}^{\mathrm{disk}}(\gamma;\ell),\mathcal{M}_{1}^{\mathrm{disk}}(\gamma;\ell))\,d\ell$
some $\gamma$-dependent constant $C\in(0,\infty)$.
###### Proof.
Recall $\operatorname{SLE}_{\kappa}^{\mathrm{sep}}$ from Proposition 2.20,
which is $\operatorname{SLE}_{\kappa}^{\mathrm{loop}}$ restricted to loops
separating $0$ and $\infty$. By the definition of $\mathcal{L}_{\kappa}$, for
some $\gamma$-dependent constant $C^{\prime}\in(0,\infty)$, the pull back of
$\operatorname{SLE}_{\kappa}^{\mathrm{sep}}$ via the map $z\mapsto e^{-z}$ is
$C^{\prime}\operatorname{SLE}_{\kappa}^{\mathrm{sep},\gamma}$. Therefore,
$\mathrm{QS}_{2}\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep}}=C^{\prime}\mathcal{M}_{2}^{\mathrm{sph}}(\gamma)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep,\gamma}}$.
Now Propositions 6.7 and 2.20 yield Lemma 6.8. ∎
Suppose $\eta$ is a simple curve in $\mathcal{C}$ separating $+\infty$ and
$-\infty$ with two marked points $p^{-},p^{+}\in\eta$. Let $D^{\pm}_{\eta}$ be
the connected component of $\mathcal{C}\backslash\eta$ containing $\pm\infty$,
and let $\psi^{\pm}_{\eta}:\mathbbm{H}\to D^{\pm}_{\eta}$ be the conformal
maps sending $(i,0)$ to $(\pm\infty,p^{\pm})$. We need the following variant
of Lemma 5.6.
###### Lemma 6.9.
Fix $\alpha\in(\frac{\gamma}{2},Q)$ and $\varepsilon\in(0,\frac{1}{4})$. Let
$\eta$ be a simple curve in $\mathcal{C}$ separating $\pm\infty$ with two
marked points $p^{-},p^{+}\in\eta$. Let
$\mathcal{C}_{\eta,p^{\pm},\varepsilon}=\mathcal{C}\setminus(\psi^{-}_{\eta}(B_{\varepsilon}(i))\cup\psi^{+}_{\eta}(B_{\varepsilon}(i)))$.
For $\phi$ sampled from $\mathrm{LF}_{\mathcal{C}}^{(\gamma,\pm\infty)}$, let
$X^{+}=\phi\circ\psi^{+}_{\eta}+Q\log|(\psi^{+}_{\eta})^{\prime}|$ and
$X^{-}=\phi\circ\psi^{-}_{\eta}+Q\log|(\psi^{-}_{\eta})^{\prime}|$. Then for
any nonnegative measurable function $f$ of $\phi$ that depends only on
$\phi|_{\mathcal{C}_{\eta,p^{\pm},\varepsilon}}$, we have
$\displaystyle\int
f(\phi)\times\varepsilon^{\alpha^{2}-\gamma^{2}}e^{(\alpha-\gamma)(X^{-}_{\varepsilon}(i)+X^{+}_{\varepsilon}(i))}\,d\mathrm{LF}_{\mathcal{C}}^{(\gamma,\pm\infty)}$
$\displaystyle=\int
f(\phi)\mathopen{}\mathclose{{}\left(\frac{1}{4}\mathrm{CR}(\exp(\eta),0)\mathrm{CR}(\exp(-\eta),0)}\right)^{-\frac{\alpha^{2}}{2}+Q\alpha-2}\,d\mathrm{LF}_{\mathcal{C}}^{(\alpha,\pm\infty)}.$
###### Proof.
Let $g:\mathbbm{C}\to\mathbbm{C}$ be given by $g(z)=\frac{z}{z-1}$ and let
$G:\mathcal{C}\to\mathbbm{C}$ be given by $G=g\circ\exp$. By [AHS21, Lemma
2.13], if $\phi$ is sampled from
$\mathrm{LF}_{\mathcal{C}}^{(\gamma,\pm\infty)}$ then $\hat{\phi}:=\phi\circ
G^{-1}+Q\log|(G^{-1})^{\prime}|$ has law
$\mathrm{LF}_{\mathbbm{C}}^{(\gamma,0),(\gamma,-1)}$, and the same is true
when $\gamma$ is replaced by $\alpha$. Let
$(\hat{\eta},\hat{p}^{-},\hat{p}^{+})=(G(\eta),G(p^{-}),G(p^{+}))$. Since
$g^{\prime}(0)=-1$ and $\frac{d}{dz}(g(\frac{1}{z}))|_{z=0}=1$, we see that
$\mathrm{CR}(\exp(\eta),0)=\mathrm{CR}(\hat{\eta},0)$ and
$\mathrm{CR}(\exp(-\eta),0)=\mathrm{CR}(\hat{\eta},-1)$. Let
$\mathbbm{C}_{\hat{\eta},\hat{p}^{\pm},\varepsilon}:=\mathbbm{C}\backslash(G(\psi^{-}(B_{\varepsilon}(i)))\cup
G(\psi^{+}(B_{\varepsilon}(i))))$. Then Lemma 6.9 is equivalent to the
following: for any nonnegative measurable function $\hat{f}$ of $\hat{\phi}$
that depends only on
$\hat{\phi}|_{\mathbbm{C}_{\hat{\eta},\hat{p}^{\pm},\varepsilon}}$, we have
$\displaystyle\int\hat{f}(\hat{\phi})\times\varepsilon^{\alpha^{2}-\gamma^{2}}e^{(\alpha-\gamma)(X^{-}_{\varepsilon}(i)+X^{+}_{\varepsilon}(i))}\,d\mathrm{LF}_{\mathbbm{C}}^{(\gamma,0),(\gamma,-1)}$
$\displaystyle=\int\hat{f}(\hat{\phi})\mathopen{}\mathclose{{}\left(\frac{1}{4}\mathrm{CR}(\hat{\eta},0)\mathrm{CR}(\hat{\eta},-1)}\right)^{-\frac{\alpha^{2}}{2}+Q\alpha-2}d\mathrm{LF}_{\mathbbm{C}}^{(\alpha,0),(\alpha,-1)}.$
(6.8)
Indeed, by the exact same argument, Lemma 5.6 holds with
$\mathrm{LF}_{\mathbbm{C}}^{(\alpha,u_{1}),(\gamma,u_{2}),(\gamma,u_{3})}$
replaced by $\mathrm{LF}_{\mathbbm{C}}^{(\gamma,0),(\gamma,-1)}$, namely
$\displaystyle\int\hat{f}(\hat{\phi})\times\varepsilon^{\frac{1}{2}(\alpha^{2}-\gamma^{2})}e^{(\alpha-\gamma)X^{-}_{\varepsilon}(i)}\,d\mathrm{LF}_{\mathbbm{C}}^{(\gamma,0),(\gamma,-1)}=\int\hat{f}(\hat{\phi})\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathrm{CR}(\hat{\eta},0)}\right)^{-\frac{\alpha^{2}}{2}+Q\alpha-2}d\mathrm{LF}_{\mathbbm{C}}^{(\alpha,0),(\gamma,-1)}.$
Applying the argument of Lemma 5.6 again to change the insertion at $-1$, we
get (6.8). ∎
Similarly as in the proof of Proposition 5.4, for a curve $\eta$ in
$\mathcal{C}$ separating $\pm\infty$, we let $\mathrm{Harm}_{-\infty,\eta}$
(resp. $\mathrm{Harm}_{+\infty,\eta}$) be the harmonic measure on $\eta$
viewed from $-\infty$ (resp., $+\infty$).
###### Lemma 6.10.
There is a $\gamma$-dependent constant $C\in(0,\infty)$ such that the
following holds. Suppose $\alpha\in(\frac{\gamma}{2},Q)$. Sample
$(\phi,\eta,p^{-},p^{+})$ from the measure
$C\cdot\mathrm{LF}_{\mathcal{C}}^{(\alpha,\pm\infty)}(d\phi)\,\mathcal{L}_{\kappa}^{\alpha}(d\eta)\,\mathrm{Harm}_{-\infty,\eta}(dp^{-})\,\mathrm{Harm}_{+\infty,\eta}(dp^{+}).$
Let $X^{+}=\phi\circ\psi^{+}_{\eta}+Q\log|(\psi^{+}_{\eta})^{\prime}|$ and
$X^{-}=\phi\circ\psi^{-}_{\eta}+Q\log|(\psi^{-}_{\eta})^{\prime}|$. Let $\tau$
be the quantum length of the clockwise arc from $p^{-}$ to $p^{+}$ in
$D^{+}_{\eta}$. Then the law of $(X^{-},X^{+},\tau)$ is
$\int_{0}^{\infty}\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)\times\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)\times[1_{\tau\in(0,\ell)}d\tau]\,d\ell.$
###### Proof.
We first prove the case $\alpha=\gamma$, namely there is a $\gamma$-dependent
constant $C$ such that
$C\cdot\mathrm{LF}_{\mathcal{C}}^{(\gamma,\pm\infty)}(d\phi)\,\mathcal{L}^{\gamma}_{\kappa}(d\eta)\,\mathrm{Harm}_{-\infty,\eta}(dp^{-})\,\mathrm{Harm}_{+\infty,\eta}(dp^{+})=\int_{0}^{\infty}\mathrm{LF}_{\mathbbm{H}}^{(\gamma,i)}(\ell)\times\mathrm{LF}_{\mathbbm{H}}^{(\gamma,i)}(\ell)\times[1_{\tau\in(0,\ell)}d\tau]\,d\ell,$
(6.9)
where we identify the left hand side as the measure on
$H^{-1}(\mathbbm{H})\times H^{-1}(\mathbbm{H})\times[0,\infty)$ describing the
law of $(X^{-},X^{+},\tau)$. Indeed, (6.9) is an immediate consequence of
Lemma 6.8 and 5.7.
Now, let $\varepsilon\in(0,\frac{1}{4})$ and
$\mathbbm{H}_{\varepsilon}:=\mathbbm{H}\backslash B_{\varepsilon}(i)$. Let $g$
be a nonnegative measurable function of
$(X^{-}|_{\mathbbm{H}_{\varepsilon}},X^{+}|_{\mathbbm{H}_{\varepsilon}},\tau)$.
Similarly as in the proof of Proposition 5.4, reweighting (6.9) gives
$\displaystyle C\int
g(X^{-}|_{\mathbbm{H}_{\varepsilon}},X^{+}|_{\mathbbm{H}_{\varepsilon}},\tau)\varepsilon^{\alpha^{2}-\gamma^{2}}e^{(\alpha-\gamma)(X^{-}_{\varepsilon}(i)+X^{+}_{\varepsilon}(i))}\mathrm{LF}_{\mathcal{C}}^{(\gamma,\pm\infty)}(d\phi)\mathcal{L}^{\gamma}_{\kappa}(d\eta)\mathrm{Harm}_{-\infty,\eta}(dp^{+})\mathrm{Harm}_{-\infty,\eta}(dp^{+})$
$\displaystyle=\int_{0}^{\infty}\mathopen{}\mathclose{{}\left(\int
g(X^{-}|_{\mathbbm{H}_{\varepsilon}},X^{+}|_{\mathbbm{H}_{\varepsilon}},\tau)\varepsilon^{\alpha^{2}-\gamma^{2}}e^{(\alpha-\gamma)(X^{-}_{\varepsilon}(i)+X^{+}_{\varepsilon}(i))}\mathrm{LF}_{\mathbbm{H}}^{(\gamma,i)}(\ell)\times\mathrm{LF}_{\mathbbm{H}}^{(\gamma,i)}(\ell)\times[1_{\tau\in(0,\ell)}d\tau]}\right)\,d\ell.$
(6.10)
By Lemma 6.9 and the reweighting definition of $\mathcal{L}_{\kappa}^{\alpha}$
in (6.1), the left hand side of (6.10) equals
$C\int
g(X^{-}|_{\mathbbm{H}_{\varepsilon}},X^{+}|_{\mathbbm{H}_{\varepsilon}},\tau)\mathrm{LF}_{\mathcal{C}}^{(\alpha,\pm\infty)}(d\phi)\mathcal{L}_{\kappa}^{\alpha}(d\eta)\mathrm{Harm}_{-\infty,\eta}(dp^{-})\mathrm{Harm}_{+\infty,\eta}(dp^{+}).$
By Lemma 5.5, the right hand side equals
$\int_{0}^{\infty}\mathopen{}\mathclose{{}\left(\int
g(X^{-}|_{\mathbbm{H}_{\varepsilon}},X^{+}|_{\mathbbm{H}_{\varepsilon}},\tau)\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)\times\mathrm{LF}_{\mathbbm{H}}^{(\alpha,i)}(\ell)\times[1_{\tau\in(0,\ell)}d\tau]}\right)\,d\ell.$
Since the above two expressions agree for every $\varepsilon$ and $g$, we
obtain the result. ∎
###### Proof of Proposition 6.2.
By Proposition 6.7, the law of
$(\mathcal{C},\phi,\eta,\pm\infty)/{\sim_{\gamma}}$ from Lemma 6.10 equals
$C(Q-\alpha)^{2}\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}$
. On the other hand, Lemma 6.10 implies that the joint law of
$((\mathbbm{H},X^{-},i)/{\sim_{\gamma}},(\mathbbm{H},X^{+},i)/{\sim_{\gamma}},\tau)$
is
$\int_{0}^{\infty}\ell\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell)\times\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell)\times[1_{\tau\in(0,\ell)}\ell^{-1}d\tau]\,d\ell$.
The uniform distribution $[1_{\tau\in(0,\ell)}\ell^{-1}d\tau]$ means that the
two surfaces are uniformly welded. Therefore the law of
$(\mathcal{C},\phi,\eta,\pm\infty)/{\sim_{\gamma}}$ is
$\int_{0}^{\infty}\ell\mathrm{Weld}(\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell),\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\ell))\,d\ell$
as desired. ∎
### 6.2 The appearance of $\overline{R}(\alpha)$ and
$|\mathcal{L}_{\kappa}^{\alpha}|$: proof of Proposition 6.4
We will prove Proposition 6.4 via a particular embedding of
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)$. Let $h$ be the field
$\hat{h}+\mathbf{c}$ in Definition 6.1 so that the law of
$(\mathcal{C},h,-\infty,+\infty)/{\sim_{\gamma}}$ is
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)$. Now we restrict to the event
$\\{\mu_{h}(\mathcal{C})>1\\}$ and set $\phi:=h(\cdot-a)$, where
$a\in\mathbbm{R}$ is such that $\mu_{h}((-\infty,a)\times[0,2\pi])=1$. Namely,
we shift $h$ horizontally such that
$\mu_{\phi}(\\{z\in\mathcal{C}:\operatorname{Re}z\leq 0\\})=1$. Let $M$ be the
law of $\phi$ under this restriction. Then we can represent
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}[E_{\delta,\varepsilon}]$
in Proposition 6.4 as follows.
###### Lemma 6.11.
Given a simple closed curve $\eta$ on $\mathcal{C}$ separating $\pm\infty$,
let $D_{\eta}^{+}$ (resp. $D_{\eta}^{-}$) be the connected component of
$\mathcal{C}\backslash\eta$ containing $+\infty$ (resp. $-\infty$). For
$t\in\mathbbm{R}$, let $\eta+t$ be the curve on $\mathcal{C}$ obtained by
shifting $\eta$ by $t$. Now sample $(\phi,\mathbf{t},\eta^{0})$ from $M\times
dt\times\mathcal{L}_{\kappa}^{\alpha}$ and set $\eta=\eta^{0}+\mathbf{t}$. Let
$E^{\prime}_{\delta,\varepsilon}:=\\{(\phi,\eta):\varepsilon<\ell_{\phi}(\eta)<\delta\\}\text{
and }\mu_{\phi}(D_{\eta}^{-})>1\\}$ (6.11)
where $\ell_{\phi}(\eta)$ is the quantum length of $\eta$. Then
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}[E_{\delta,\varepsilon}]=(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}].$
###### Proof.
Since the measure $\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}$ is
invariant under translations along the cylinder, the law of
$(\mathcal{C},\phi,\eta,+\infty,-\infty)/{\sim_{\gamma}}$ is the restriction
of
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}$
to the event that the total quantum area is larger than $1$. Now Lemma 6.11
follows from the definition of $E_{\delta,\varepsilon}$. ∎
The next lemma explains how the reflection coefficient $\overline{R}(\alpha)$
shows up in Proposition 6.4.
###### Lemma 6.12.
The law of the quantum area of a sample from
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)$ is
$1_{a>0}\frac{1}{2}\overline{R}(\alpha)a^{\frac{2}{\gamma}(\alpha-Q)-1}\,da.$
As a corollary,
$|M|=\int_{1}^{\infty}\frac{1}{2}\overline{R}(\alpha)a^{\frac{2}{\gamma}(\alpha-Q)-1}\,da=\frac{\gamma\overline{R}(\alpha)}{4(Q-\alpha)}$.
###### Proof.
For $0<a<a^{\prime}$, with $\widehat{h}$ as in Definition 6.1, we have
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)[\mu_{\widehat{h}+c}(\mathcal{C})\in(a,a^{\prime})]=\mathbbm{E}\mathopen{}\mathclose{{}\left[\int_{-\infty}^{\infty}\mathbf{1}_{e^{\gamma
c}\mu_{\widehat{h}}(\mathcal{C})\in(a,a^{\prime})}\frac{\gamma}{2}e^{2(\alpha-Q)c}\,dc}\right]=\mathbbm{E}\mathopen{}\mathclose{{}\left[\int_{a}^{a^{\prime}}\frac{\gamma}{2}\mathopen{}\mathclose{{}\left(\frac{y}{\mu_{\widehat{h}}(\mathcal{C})}}\right)^{\frac{2}{\gamma}(\alpha-Q)}\frac{1}{\gamma
y}\,dy}\right]$
where we have used the change of variables $y=e^{\gamma
c}\mu_{\widehat{h}}(\mathcal{C})$. By [KRV17, Theorem 3.5] and [RV19,
(1.10)–(1.12)], for $\alpha\in(\frac{\gamma}{2},Q)$ we have
$\mathbbm{E}[\mu_{\widehat{h}}(\mathcal{C})^{\frac{2}{\gamma}(Q-\alpha)}]=\overline{R}(\alpha)$.
Interchanging the expectation and integral gives the result. ∎
By Lemmas 6.11 and 6.12, Proposition 6.4 is reduced to the following
proposition.
###### Proposition 6.13.
Let $\alpha\in(\frac{\gamma}{2},Q)$. With an error term
$o_{\delta,\varepsilon}(1)$ satisfying $\lim_{\delta\to
0}\limsup_{\varepsilon\to 0}|o_{\delta,\varepsilon}(1)|=0$, we have
$(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}]=(1+o_{\delta,\varepsilon}(1))|M||\mathcal{L}_{\kappa}^{\alpha}|\frac{2\log\varepsilon^{-1}}{\gamma(Q-\alpha)}.$
(6.12)
###### Proof of Proposition 6.4 given Proposition 6.13.
By Lemma 6.11 and 6.12, we have
$(\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha})[E_{\delta,\varepsilon}]=(1+o_{\delta,\varepsilon}(1))|M||\mathcal{L}_{\kappa}^{\alpha}|\frac{2\log\varepsilon^{-1}}{\gamma(Q-\alpha)}=(1+o_{\delta,\varepsilon}(1))\frac{\overline{R}(\alpha)}{2(Q-\alpha)^{2}}|\mathcal{L}_{\kappa}^{\alpha}|\log\varepsilon^{-1}.\qed$
Figure 5: Illustration of the proof of Proposition 6.13. On the left
$(\phi,\mathbf{t},\eta^{0})$ is sampled from $M\times
dt\times\mathcal{L}^{\alpha}_{\kappa}$ and $\eta=\eta^{0}+\mathbf{t}$. The
curve $\eta^{0}$ satisfies $\sup_{z\in\eta_{0}}\operatorname{Re}z=0$ and the
field $\phi$ satisfies $\mu_{\phi}(\\{z\in\mathcal{C}:\operatorname{Re}z\leq
0\\})=1$. The domain $D^{-}_{\eta}$ is colored grey. The event
$E^{\prime}_{\delta,\varepsilon}$ is
$\\{\ell_{\phi}(\eta)\in(\varepsilon,\delta)\textrm{ and
}\mu_{\phi}(D_{\eta}^{-})>1\\}$. If $E^{\prime}_{\delta,\varepsilon}$ occurs,
we have $\ell_{\phi}(\eta)\approx e^{\frac{\gamma}{2}X_{\bf t}}$. Therefore
$\mathbf{t}$ is in an interval close to the red one on the right figure, which
is approxiately
$(\frac{2}{\gamma(Q-\alpha)}\log\delta^{-1},\frac{2}{\gamma(Q-\alpha)}\log\varepsilon^{-1})$.
The proofs of Lemmas 6.17 and 6.19 are done by quantifying this statement in
two directions.
The high level idea for proving Proposition 6.13 is the following. Suppose
$(\phi,\mathbf{t},\eta^{0})$ is sampled from $M\times
dt\times\mathcal{L}_{\kappa}^{\alpha}$. For $s\geq 0$, let $X_{s}$ be the
average of $\phi$ on $[s,s+2\pi i]/{\sim}$. For most realizations of
$(\phi,\eta^{0})$, the occurrence of $E^{\prime}_{\delta,\varepsilon}$ is
equivalent to the event that $\mathbf{t}$ lies in some interval of length
$(1+o_{\delta,\varepsilon}(1))\frac{2\log\varepsilon^{-1}}{\gamma(Q-\alpha)}$
determined by $X$. (See Figure 5.) Hence the mass of
$E^{\prime}_{\delta,\varepsilon}$ is this length times
$|M||\mathcal{L}_{\kappa}^{\alpha}|$. In the rest of this section, we first
prove a few properties for $X$ in Section 6.2.1 and then prove Proposition
6.13 in Section 6.2.2.
#### 6.2.1 The field average process
We need the following description of the law of $\phi|_{\mathcal{C}_{+}}$ and
$(X_{s})_{s\geq 0}$, where
$\mathcal{C}_{+}:=\\{z\in\mathcal{C}\>:\>\operatorname{Re}z>0\\}$ is the right
half cylinder.
###### Lemma 6.14.
Let $\phi$ be a sample from $M$. Conditioned on $\phi|_{\mathcal{C}_{-}}$, the
conditional law of $\phi|_{\mathcal{C}_{+}}$ is the law of
$\phi_{0}+\hat{\mathfrak{h}}-(Q-\alpha)\operatorname{Re}(\cdot)$, where
$\phi_{0}$ is a zero boundary GFF on $\mathcal{C}_{+}$, and
$\hat{\mathfrak{h}}$ is a harmonic function determined by
$\phi|_{\mathcal{C}_{-}}$ whose average on the circle $[s,s+2\pi i]/{\sim}$
does not depend on $s$. Moreover, let $X_{s}$ be the average of $\phi$ on
$[s,s+2\pi i]/{\sim}$. Conditioned on $X_{0}$, the conditional law of
$(X_{s}-X_{0})_{s\geq 0}$ is the law of $(B_{s}-(Q-\alpha)s)_{s\geq 0}$ where
$B_{s}$ is a Brownian motion.
###### Proof.
The first statement is the sphere analog of [AG21, Lemmas 2.10 and 2.11] based
on the domain Markov property of Gaussian free field. The proof is identical
so we omit it. The second statement on $(X_{s})_{s\geq 0}$ follows from the
first statement. ∎
Proposition 6.13 essentially follows from the fact that the field average
process $(X_{s})_{s\geq 0}$ looks like a line of slope $-(Q-\alpha)$. We now
introduce two random times to quantify this. For $y>0$, let
$\sigma_{y}=\inf\\{s>0\>:\>X_{s}<\frac{2}{\gamma}\log
y\\}\quad\textrm{and}\quad\tau_{y}=\sup\\{s>0\>:\>X_{s}>\frac{2}{\gamma}\log
y\\}.$ (6.13)
###### Lemma 6.15.
$M$-a.e. the field $\phi$ satisfies $\lim_{y\to 0}\frac{\sigma_{y}}{\log
y^{-1}}=\lim_{y\to 0}\frac{\tau_{y}}{\log y^{-1}}=\frac{2}{\gamma(Q-\alpha)}$.
###### Proof.
Given Lemma 6.14, this is a straightforward fact about drifted Brownian
motion. ∎
The proof of the upper bound in Proposition 6.13 will rely on the following
lemma.
###### Lemma 6.16.
Sample $(\phi,\mathbf{t},\eta^{0})$ from $M\times
dt\times\mathcal{L}_{\kappa}^{\alpha}$ and set $\eta=\eta^{0}+\mathbf{t}$. Let
$\mathfrak{l}=\ell_{\phi}(\eta)$, which is the boundary length of the quantum
surface $(D_{\eta}^{-},\phi)/{\sim_{\gamma}}$. Fix $\zeta\in(0,1)$. Then there
exists a constant $C>0$ and a function $\mathrm{err}(\ell)$ such that
$\lim_{\ell\downarrow 0}\mathrm{err}(\ell)=0$, and conditioned on
$(D_{\eta}^{-},\phi)/{\sim_{\gamma}}$, the conditional probability of
$\\{\mathbf{t}\in(\sigma_{{\mathfrak{l}}^{1-\zeta}}-C,\tau_{{\mathfrak{l}}^{1+\zeta}})\\}$
is at least $1-\mathrm{err}(\mathfrak{l})$.
###### Proof.
We first introduce $C$ and $\mathrm{err},$ and then show that they satisfy
Lemma 6.16. Recall that $\eta^{0}$ is a loop in $\mathcal{C}$ separating
$\pm\infty$ and with $\sup_{z\in\eta^{0}}\operatorname{Re}z=0$. Let
$D_{\eta^{0}}^{+}$ be the connected component of
$\mathcal{C}\backslash\eta^{0}$ containing $+\infty$. Let
$f:\mathcal{C}_{+}\to D_{\eta^{0}}^{+}$ be the unique conformal map such that
$f(+\infty)=+\infty$ and $\lim_{z\to+\infty}(f(z)-z)\in\mathbbm{R}$. By
standard conformal distortion estimates (e.g. [MS19, Lemma 2.4]), for
$C=e^{10}$ we have
$|f(z)-z|<\frac{C}{3}\quad\textrm{and}\quad|f^{\prime\prime}(z)|<\frac{1}{2}<|f^{\prime}(z)|<2\quad\textrm{for
}\operatorname{Re}z>\frac{C}{3}.$ (6.14)
Let $g$ be a fixed smooth function on $\mathcal{C}$ supported on
$\\{\operatorname{Re}z\in(C-1,C)\\}$ such that $\int g(z)\,dz=1$ and $g$ is
invariant under rotations against the axis of $\mathcal{C}$. Let $S$ be the
collection of smooth functions $\xi$ that are supported on
$\\{\operatorname{Re}z\in[\frac{2}{3}C-1,\frac{4}{3}C]\\}\subset\mathcal{C}_{+}$
and satisfy $\|\xi\|_{\infty}\leq 4\|g\|_{\infty}$ and
$\|\nabla\xi\|_{\infty}\leq 8(\|g\|_{\infty}+\|\nabla g\|_{\infty})$. Let
$\mathrm{err}(\ell):=\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;1)^{\\#}\Big{[}\sup_{\xi\in
S}|(h,\xi)|+Q\log 2\geq-\frac{2\zeta}{\gamma}\log\ell\Big{]}$ (6.15)
where $(\mathcal{C}_{+},h,+\infty)$ is an embedding of a sample from
$\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;1)^{\\#}$, the probablity measure
proportional to $\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;1)$. Since the space
$S$ is invariant under rotations against the axis of $\mathcal{C}$, the
probability in (6.15) does not depend on the choice of the embedding.
Moreover, since $h$ is a.s. in the Sobolev space of index $-1$, we see that
$\sup_{\xi\in S}|(h,\xi)|<\infty$ a.s. hence $\lim_{\ell\to
0}\mathrm{err}(\ell)=0$.
We now show that $C$ and $\mathrm{err}$ satisfy Lemma 6.16. Set
$\phi^{0}(\cdot)=\phi(\cdot+\mathbf{t})$ and $\hat{\phi}=\phi^{0}\circ
f+Q\log|f^{\prime}|-\frac{2}{\gamma}\log\mathfrak{l}$. Then
$(D_{\eta}^{+},\phi,+\infty)/{\sim_{\gamma}}=(D_{\eta_{0}}^{+},\phi^{0},+\infty)/{\sim_{\gamma}}$
hence
$(\mathcal{C}_{+},\hat{\phi},+\infty)/{\sim_{\gamma}}=(D_{\eta}^{+},\phi-\frac{2}{\gamma}\log\mathfrak{l},+\infty)/{\sim_{\gamma}}$.
Moreover,
$(\phi^{0},g)-\frac{2}{\gamma}\log\mathfrak{l}=(\hat{\phi}\circ
f^{-1}+Q\log|(f^{-1})^{\prime}|,g)=(\hat{\phi},|f^{\prime}|^{2}g\circ
f)+(Q\log|(f^{-1})^{\prime}|,g).$ (6.16)
By (6.14) and the definition of $S$, we have $|f^{\prime}|^{2}g\circ f\in S$.
Then by (6.14) and (6.16), we have
$\big{|}(\phi^{0},g)-\frac{2}{\gamma}\log\mathfrak{l}\big{|}\leq|(\hat{\phi},|f^{\prime}|^{2}g\circ
f)|+|(Q\log|(f^{-1})^{\prime}|,g)|\leq\sup_{\xi\in S}|(\hat{\phi},\xi)|+Q\log
2.$
Recall from the proof of Lemma 6.11 that the law of
$(\mathcal{C},\phi,\eta,+\infty,-\infty)/{\sim_{\gamma}}$ is the restriction
of
$\mathcal{M}_{2}^{\mathrm{sph}}(\alpha)\otimes\operatorname{SLE}_{\kappa}^{\mathrm{sep},\alpha}$
to the event that the total quantum area is larger than $1$. By Proposition
6.2, conditioning on $(D_{\eta}^{-},\phi)/{\sim_{\gamma}}$, the conditional
law of $(D_{\eta}^{+},\phi,+\infty)/{\sim_{\gamma}}$ is
$\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;\mathfrak{l})^{\\#}$, hence the
conditional law of $(\mathcal{C}_{+},\hat{\phi},+\infty)/{\sim_{\gamma}}$ is
$\mathcal{M}_{1}^{\mathrm{disk}}(\alpha;1)^{\\#}$. Therefore, by (6.15) and
(6.16), conditioning on $(D_{\eta}^{-},\phi)/{\sim_{\gamma}}$, the conditional
probability of the event
$\mathopen{}\mathclose{{}\left|(\phi^{0},g)-\frac{2}{\gamma}\log\mathfrak{l}}\right|<-\frac{2\zeta}{\gamma}\log\mathfrak{l}$
is at least $1-\mathrm{err}(\mathfrak{l})$. Since $g$ is supported on
$\\{\operatorname{Re}z\in(C-1,C)\\}$, if
$\mathopen{}\mathclose{{}\left|(\phi^{0},g)-\frac{2}{\gamma}\log\mathfrak{l}}\right|<-\frac{2\zeta}{\gamma}\log\mathfrak{l}$,
then there exists $s\in[C-1,C]$ such that the average of $\phi^{0}$ on
$[s,s+2\pi i]$ lies in
$((1+\zeta)\frac{2}{\gamma}\log\mathfrak{l},(1-\zeta)\frac{2}{\gamma}\log\mathfrak{l})$.
This gives
$X_{\mathbf{t}+s}\in((1+\zeta)\frac{2}{\gamma}\log\mathfrak{l},(1-\zeta)\frac{2}{\gamma}\log\mathfrak{l})$
hence
$\mathbf{t}+s\in(\sigma_{{\mathfrak{l}}^{1-\zeta}},\tau_{{\mathfrak{l}}^{1+\zeta}})$
for some $s\in[C-1,C]$. Therefore
$\mathbf{t}\in(\sigma_{{\mathfrak{l}}^{1-\zeta}}-C,\tau_{{\mathfrak{l}}^{1+\zeta}})$,
which gives Lemma 6.16 with our choice of $C$ and $\mathrm{err}$. ∎
#### 6.2.2 Proof of Proposition 6.13
We refer to Figure 5 and the paragraph above Section 6.2.1 for an illustration
of our proof ideas. We will write $o_{\delta,\varepsilon}(1)$ as an error term
satisfying $\lim_{\delta\to 0}\limsup_{\varepsilon\to
0}|o_{\delta,\varepsilon}(1)|=0$ which can change from place to place. We
first prove the lower bound for $(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}]$.
###### Lemma 6.17.
We have $(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}]\geq(1+o_{\delta,\varepsilon}(1))|M||\mathcal{L}_{\kappa}^{\alpha}|\frac{2\log\varepsilon^{-1}}{\gamma(Q-\alpha)}.$
We will prove Lemma 6.17 by a coupling of the probability measure
$M^{\\#}=M/|M|$ and cylindrical GFF measure $P_{\mathcal{C}}$. Recall that for
a sample $h$ from $P_{\mathcal{C}}$, it can be written as $h=h_{1}+h_{2}$,
where $h_{1}$ is constant on vertical circles and $h_{2}$ is the lateral
component that has mean zero on all such circles.
###### Lemma 6.18.
There exists a coupling of $h$ sampled from $P_{\mathcal{C}}$ and $\phi$
sampled from $M^{\\#}$ such that $h_{2}$ is independent of $(X_{s})_{s\geq
0}$, and moreover, $\sup_{\operatorname{Re}z>1}|{\mathfrak{h}}(z)|<\infty$
where $\mathfrak{h}(z)=\phi(z)-h_{2}(z)-X_{\operatorname{Re}z}$.
###### Proof.
By [MS17, Proposition 2.8], on $\mathcal{C}_{+}$ the field $h$ can be written
as $h_{0}+\tilde{\mathfrak{h}}$ where $h_{0}$ is a zero boundary GFF on
$\mathcal{C}_{+}$ and $\tilde{\mathfrak{h}}$ is an independent harmonic
function on $\mathcal{C}_{+}$ such that
$\sup_{\operatorname{Re}z>1}|\tilde{\mathfrak{h}}(z)|<\infty$. Similarly, by
Lemma 6.14, $\phi(z)+(Q-\alpha)\operatorname{Re}z$ on $\mathcal{C}_{+}$ can be
written as $\phi_{0}+\hat{\mathfrak{h}}$ with the same properties. Coupling
$h$ and $\phi$ such that $h_{0}=\phi_{0}$ and $\tilde{\mathfrak{h}}$ is
independent of $\phi$, we are done. ∎
###### Proof of Lemma 6.17.
Let $\mathbbm{P}$ be the probability measure corresponding to the law of
$(\phi,h)$ as coupled in Lemma 6.18. Let $(\phi,h,\mathbf{t},\eta^{0})$ be a
sample from $\mathbbm{P}\times dt\times\mathcal{L}_{\kappa}^{\alpha}$. Then
the law of $(\phi,\mathbf{t},\eta^{0})$ is $M^{\\#}\times
dt\times\mathcal{L}_{\kappa}^{\alpha}$, where $M^{\\#}=M/|M|$. Let $h_{2}$ and
$\mathfrak{h}$ be defined as in Lemma 6.18 so that
$\phi=h_{2}+X_{\operatorname{Re}\cdot}+\mathfrak{h}$. Fix $\zeta\in(0,0.1)$
which will be sent to zero later. Let $I_{\delta,\varepsilon}$ be the interval
$((1+3\zeta)\frac{2}{\gamma(Q-\alpha)}\log\delta^{-1}+\delta^{-1},(1-3\zeta)\frac{2}{\gamma(Q-\alpha)}\log\varepsilon^{-1})$.
Let
$G_{\delta,\varepsilon}:=\\{I_{\delta,\varepsilon}\subset(\tau_{\delta^{1+2\zeta}}+\delta^{-1},\sigma_{\varepsilon^{1-2\zeta}})\quad\textrm{and}\quad\sup_{\operatorname{Re}z>1}|{\mathfrak{h}}(z)|<\zeta\cdot\frac{2}{\gamma}\log\delta^{-1}\\},$
where the random times $\sigma_{y},\tau_{y}$ are as in (6.13). By Lemma 6.15
we have
$\mathbbm{P}[I_{\delta,\varepsilon}\subset(\tau_{\delta^{1+2\zeta}}+\delta^{-1},\sigma_{\varepsilon^{1-2\zeta}})]=1-o_{\delta,\varepsilon}(1)$.
Since $\sup_{\operatorname{Re}z>1}|{\mathfrak{h}}(z)|<\infty$ almost surely,
we have $\mathbbm{P}[G_{\delta,\varepsilon}]=1-o_{\delta,\varepsilon}(1)$.
Fix $n>0$. Let $A_{n}=\\{\inf_{z\in\eta^{0}}\operatorname{Re}z>-n\\}$. Then
$\mathcal{L}^{\alpha}_{\kappa}[A_{n}]<\infty$. Let $\eta=\eta^{0}+\mathbf{t}$
and $\ell_{h_{2}}(\eta)$ be the quantum length of $\eta$ with respect to
$h_{2}$. Define the event
$E^{\prime}_{\delta,\varepsilon}(n):=\\{A_{n}\textrm{
occurs},\quad\mathbf{t}\in
I_{\delta,\varepsilon},\quad\textrm{and}\quad\ell_{h_{2}}(\eta)\in(\varepsilon^{\zeta},\delta^{-\zeta})\\}.$
Suppose $n<\delta^{-1}<\frac{2\zeta}{\gamma(Q-\alpha)}\log\varepsilon^{-1}$
and the event $E^{\prime}_{\delta,\varepsilon}(n)\cap G_{\delta,\varepsilon}$
occurs. Since $\mathbf{t}\in I_{\delta,\varepsilon}$ and
$G_{\delta,\varepsilon}\cap A_{n}$ occurs, we have
$\operatorname{Re}z\in(\tau_{\delta^{1+2\zeta}},\sigma_{\varepsilon^{1-2\zeta}})$
for each $z\in\eta$, hence
$\mathcal{C}_{-}:=\\{z\in\mathcal{C}:\operatorname{Re}z\leq 0\\}\subset
D^{-}_{\eta}$. By the definition of $\phi$ we have
$\mu_{\phi}(D_{\eta}^{-})\geq\mu_{\phi}(\mathcal{C}_{-})\geq 1$. Moreover,
from the bound on $\sup_{\operatorname{Re}z>1}|{\mathfrak{h}}(z)|$ we have
$\frac{\gamma}{2}(X_{\operatorname{Re}z}+\mathfrak{h}(z))\in((1-\zeta)\log\varepsilon,(1+\zeta)\log\delta)$
for each $z\in\eta$. Now
$\ell_{h_{2}}(\eta)\in(\varepsilon^{\zeta},\delta^{-\zeta})$ yields
$\ell_{\phi}(\eta)\in(\varepsilon,\delta)$. This gives
$E^{\prime}_{\delta,\varepsilon}(n)\cap G_{\delta,\varepsilon}\subset
E^{\prime}_{\delta,\varepsilon}=\\{\mu_{\phi}(D_{\eta}^{-})\geq
1,\ell_{\phi}(\eta)\in(\varepsilon,\delta)\\}$. Therefore
$(\mathbbm{P}\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}]\geq(\mathbbm{P}\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}(n)]-(\mathbbm{P}\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}(n)\setminus
G_{\delta,\varepsilon}].$ (6.17)
We claim that
$(\mathbbm{P}\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}(n)]=(1-o_{\delta,\varepsilon}(1))|I_{\delta,\varepsilon}|\mathcal{L}^{\alpha}_{\kappa}[A_{n}]\textrm{
and }(\mathbbm{P}\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}(n)\setminus
G_{\delta,\varepsilon}]=o_{\delta,\varepsilon}(1)|I_{\delta,\varepsilon}|.$
(6.18)
Since the law of $h_{2}$ is translation invariant, namely
$h_{2}\stackrel{{\scriptstyle d}}{{=}}h_{2}(\cdot-t)$ for each
$t\in\mathbbm{R}$, we have
$\displaystyle(\mathbbm{P}\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}(n)]=|I_{\delta,\varepsilon}|(\mathbbm{P}\times\mathcal{L}^{\alpha}_{\kappa})[\ell_{h_{2}}(\eta^{0})\in(\varepsilon^{\zeta},\delta^{-\zeta}),A_{n}]$
$\displaystyle=|I_{\delta,\varepsilon}|(\mathbbm{P}\times\mathcal{L}^{\alpha}_{\kappa})[A_{n}]-|I_{\delta,\varepsilon}|(\mathbbm{P}\times\mathcal{L}^{\alpha}_{\kappa})[\ell_{h_{2}}(\eta^{0})\notin(\varepsilon^{\zeta},\delta^{-\zeta}),A_{n}]=(1-o_{\delta,\varepsilon}(1))|I_{\delta,\varepsilon}|\mathcal{L}^{\alpha}_{\kappa}[A_{n}].$
In the last line we used
$(\mathbbm{P}\times\mathcal{L}^{\alpha}_{\kappa})[A_{n}]=\mathcal{L}^{\alpha}_{\kappa}[A_{n}]<\infty$
and
$(\mathbbm{P}\times\mathcal{L}^{\alpha}_{\kappa})[\ell_{h_{2}}(\eta^{0})\notin(\varepsilon^{\zeta},\delta^{-\zeta})|A_{n}]=o_{\delta,\varepsilon}(1)$,
the latter of which holds because $\ell_{h_{2}}(\eta^{0})<\infty$ a.e. in
$\mathbbm{P}\times\mathcal{L}^{\alpha}_{\kappa}$. This gives the first
identity in (6.18).
The second identity in (6.18) is proved by as follows:
$\displaystyle(\mathbbm{P}\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}(n)\setminus
G_{\delta,\varepsilon}]\leq(\mathbbm{P}\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[A_{n}\setminus
G_{\delta,\varepsilon}\textrm{ and }\mathbf{t}\in I_{\delta,\varepsilon}]$
$\displaystyle=\mathbbm{P}[G_{\delta,\varepsilon}\textrm{ does not
occur}]\times|I_{\delta,\varepsilon}|\times\mathcal{L}^{\alpha}_{\kappa}[A_{n}]=o_{\delta,\varepsilon}(1)|I_{\delta,\varepsilon}|\mathcal{L}^{\alpha}_{\kappa}[A_{n}].$
By (6.17) and (6.18), for a fixed $n$, we have $(\mathbbm{P}\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}]\geq(1-o_{\delta,\varepsilon}(1))|I_{\delta,\varepsilon}||\mathcal{L}^{\alpha}_{\kappa}[A_{n}]$
hence
$\frac{1}{|M|}(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}]\geq(1-o_{\delta,\varepsilon}(1))|I_{\delta,\varepsilon}|\mathcal{L}^{\alpha}_{\kappa}[A_{n}]\geq(1-o_{\delta,\varepsilon}(1))\mathcal{L}^{\alpha}_{\kappa}[A_{n}](1-4\zeta)\frac{2\log\varepsilon^{-1}}{\gamma(Q-\alpha)}.$
Sending $\varepsilon\to 0$, $\delta\to 0$, and $\zeta\to 0$ in order, we see
that
$\lim_{\delta\to 0}\liminf_{\varepsilon\to 0}\frac{(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}]\times\gamma(Q-\alpha)}{2\log\varepsilon^{-1}}\geq|M|\mathcal{L}^{\alpha}_{\kappa}[A_{n}].$
Since
$\mathcal{L}^{\alpha}_{\kappa}[A_{n}]\rightarrow|\mathcal{L}^{\alpha}_{\kappa}|$,
sending $n\to\infty$ we conclude the proof. ∎
It remains to prove the upper bound for $(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}]$.
###### Lemma 6.19.
We have $(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}]\leq(1+o_{\delta,\varepsilon}(1))|M||\mathcal{L}_{\kappa}^{\alpha}|\frac{2\log\varepsilon^{-1}}{\gamma(Q-\alpha)}.$
###### Proof.
Recall $\sigma$ and $\tau$ defined in (6.13). For $C>0$, consider the event
$F_{x,y,C}=\\{(\phi,\mathbf{t},\eta^{0})\>:\>\mathbf{t}\in(\sigma_{x}-C,\tau_{y})\\}.$
(6.19)
We will prove Lemma 6.19 by first bounding $(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[F_{x,y,C}]$ and then comparing
$E^{\prime}_{\delta,\varepsilon}$ to
$F_{\delta^{1-\zeta},\varepsilon^{1+\zeta},C}$. More precisely, we will prove
two estimates. First, for each $x>0$, we have
$(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[F_{x,y,C}]\leq(1+o_{y}(1))|M||\mathcal{L}_{\kappa}^{\alpha}|(\frac{2\log
y^{-1}}{\gamma(Q-\alpha)}+C)\quad\textrm{as }y\to 0,$ (6.20)
where $o_{y}(1)\to 0$ uniformly in $C$. Moreover, for a fixed $\zeta\in(0,1)$
there exists some $C>0$ such that
$(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[F_{\delta^{1-\zeta},\varepsilon^{1+\zeta},C}\mid
E^{\prime}_{\delta,\varepsilon}]=1-o_{\delta,\varepsilon}(1).$ (6.21)
Given (6.20) and (6.21), we see that for any fixed $\zeta>0$ there exists some
$C>0$ such that
$(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[E^{\prime}_{\delta,\varepsilon}]\leq(1+o_{\delta,\varepsilon}(1))(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[F_{\delta^{1-\zeta},\varepsilon^{1+\zeta},C}]\leq(1+o_{\delta,\varepsilon}(1))|M||\mathcal{L}_{\kappa}^{\alpha}|\frac{(1+\zeta)2\log\varepsilon^{-1}}{\gamma(Q-\alpha)},$
Sending $\zeta\to 0$ we will get Lemma 6.19.
We first prove (6.20). Note that $(M\times
dt\times\mathcal{L}_{\kappa}^{\alpha})[F_{x,y,C}]=M^{\\#}[\tau_{y}-\sigma_{x}+C]|M||\mathcal{L}^{\alpha}_{\kappa}|$
from the definition of $F_{x,y,C}$. Hence (6.20) is equivalent to
$M^{\\#}[\tau_{y}-\sigma_{x}]\leq(1+o_{y}(1))\frac{2\log
y^{-1}}{\gamma(Q-\alpha)}$. By Lemma 6.14, the process $X_{s}$ evolves as
Brownian motion with drift $-(Q-\alpha)$. Let $T=\inf\\{s\geq
0:B^{x}_{s}-(Q-\alpha)s=\frac{2}{\gamma}\log y\\}$ where $(B^{x}_{s})_{s\geq
0}$ is a Brownian motion starting from $\frac{2}{\gamma}\log x$. Then
$M^{\\#}[\tau_{y}-\sigma_{x}]\leq\mathbbm{E}[T]$. Since
$\mathbbm{E}[T]=(1+o_{y}(1))\frac{2\log y^{-1}}{\gamma(Q-\alpha)}$, we get
(6.20).
Since the event $E^{\prime}_{\delta,\varepsilon}$ is determined by
$(D_{\eta}^{-},\phi)/{\sim_{\gamma}}$, and
$\ell_{\phi}(\eta)\in(\varepsilon,\delta)$ on
$E^{\prime}_{\delta,\varepsilon}$, Lemma 6.16 yields (6.21) with the constant
$C$ from Lemma 6.16. This concludes the proof. ∎
## Appendix A Continuity of $\operatorname{CLE}_{\kappa}$ as $\kappa\uparrow
4$
In this appendix we supply the continuity in $\kappa$ needed to extend
Theorems 1.2 and 1.3 from $\kappa<4$ to $\kappa=4$. We start with a
monotonicity statement for $\operatorname{CLE}_{\kappa}$ proved in [SW12].
###### Lemma A.1 ([SW12]).
There exists a coupling of $\operatorname{CLE}_{\kappa}$ on the unit disk
$\mathbbm{D}$ for $\kappa\in(\frac{8}{3},4]$ such that a.s. each outermost
loop of $\operatorname{CLE}_{\kappa_{1}}$ is surrounded by an outermost loop
of $\operatorname{CLE}_{\kappa_{2}}$ if $\kappa_{1}<\kappa_{2}\leq 4$.
###### Proof.
By [SW12, Theorems 1.5, 1.6], the law of the boundaries of outermost loop
clusters in a Brownian loop soup with intensity
$c_{\kappa}=(3\kappa-8)(6-\kappa)/2\kappa$ is given by the outermost loops of
$\operatorname{CLE}_{\kappa}$. Now the monotonicity of $c_{\kappa}$ in
$\kappa\in(\frac{8}{3},4]$ yields the desired monotonicity in Lemma A.1. ∎
We recall the formula from [SSW09] for the conformal radii of CLE.
###### Theorem A.2 ([SSW09]).
For $\kappa\in(8/3,8)$, let $\eta_{\kappa}$ be the outermost loop surrounding
$0$ of a $\operatorname{CLE}_{\kappa}$ on $\mathbbm{D}$. Let
$\mathrm{CR}(\eta_{\kappa},0)$ be the conformal radius of the region
surrounded by $\eta_{\kappa}$ viewed from 0. Then
$\mathbbm{E}[\mathrm{CR}(\eta_{\kappa},0)^{\lambda}]=\frac{-\cos(\frac{4\pi}{\kappa})}{\cos\mathopen{}\mathclose{{}\left(\pi\sqrt{(1-\frac{4}{\kappa})^{2}-\frac{8\lambda}{\kappa}}}\right)}\quad\textrm{for
}\lambda>\frac{3\kappa}{32}+\frac{2}{\kappa}-1.$
Recall that the Hausdorff distance between two closed sets $E_{1},E_{2}$ on a
metric space $(X,d)$ is given by $\max\\{\sup_{x\in
E_{1}}\\{d(x,E_{2})\\},\sup_{x\in E_{2}}\\{d(x,E_{1})\\}\\}$. Lemma A.1 and
Theorem A.2 yield the following continuity result.
###### Lemma A.3.
Suppose we are in the coupling in Lemma A.1. For each $z\in\mathbbm{D}$, let
$\eta_{\kappa}(z)$ be the outermost loop around $z$ of the
$\operatorname{CLE}_{\kappa}$. For any fixed $z$, viewed as closed sets,
$\eta_{\kappa}(z)$ converges almost surely to $\eta_{4}(z)$ in the Hausdorff
metric as $\kappa\uparrow 4$.
###### Proof.
By the conformal invariance of $\operatorname{CLE}_{\kappa}$ we assume $z=0$
because the same argument will work for a general $z$. In this case we simply
write $\eta_{\kappa}(0)$ as $\eta_{\kappa}$. Since $\eta_{\kappa_{1}}$ is
surrounded by $\eta_{\kappa_{2}}$ if $\kappa_{1}<\kappa_{2}\leq 4$. we see
that $\mathrm{CR}(\eta_{\kappa},0)$ is increasing in $\kappa$. By the explicit
formula in Theorem A.2, we have $\lim_{\kappa\uparrow
4}\mathbbm{E}[\mathrm{CR}(\eta_{\kappa},0)]=\mathbbm{E}[\mathrm{CR}(\eta_{\kappa},0)]$.
Thus we must have $\lim_{\kappa\uparrow
4}\mathrm{CR}(\eta_{\kappa},0)=\mathrm{CR}(\eta_{4},0)$ a.s.
Let $D_{\kappa}$ be the region surrounded by $\eta_{\kappa}$ and
$D_{4^{-}}=\cup_{\kappa<4}D_{\kappa}$. The conformal radius of $D_{4^{-}}$
must be between $\lim_{\kappa\uparrow 4}\mathrm{CR}(\eta_{\kappa},0)$ and
$\mathrm{CR}(\eta_{4},0)$, hence equals $\mathrm{CR}(\eta_{4},0)$ a.s. This
means that $D_{\kappa}\uparrow D_{4}$ a.s. hence $\eta_{\kappa}\to\eta_{4}$
a.s. in the Hausdorff metric in the coupling in Lemma A.1. ∎
We now prove the continuity of the shape measure $\mathcal{L}_{\kappa}$ of the
SLE loop in Theorem 1.3.
###### Lemma A.4.
We have $\lim_{\kappa\uparrow 4}\mathcal{L}_{\kappa}=\mathcal{L}_{4}$ weakly
with respect to the Hausdorff metric.
###### Proof.
We only explain that under the natural parameterization, chordal
$\operatorname{SLE}_{\kappa}$ converges to $\operatorname{SLE}_{4}$ as
$\kappa\uparrow 4$. Once this is done we get that the two-sided whole plane
$\operatorname{SLE}_{\kappa}$ curve
$\operatorname{SLE}_{\kappa}^{p\rightleftharpoons q}$ converges to
$\operatorname{SLE}_{4}^{p\rightleftharpoons q}$ under natural parametrization
as well, since the two-sided curve is characterized by its re-sampling
property: conditioned on one of the curve segments between $p$ and $q$, the
conditional law of the other is a chordal $\operatorname{SLE}_{\kappa}$ in the
complementary domain. From this and the definition of
$\operatorname{SLE}_{\kappa}^{\mathrm{loop}}$ we reviewed in Section 2.4, we
get the convergence of $\mathcal{L}_{\kappa}$.
We now show that the law of chordal $\operatorname{SLE}_{\kappa}$ on
$\mathbbm{H}$ from $0$ to $\infty$ under natural parametrization is continuous
as $\kappa\uparrow 4$. We first recall that this family of measures is tight
in the local uniform topology of parametrized curves thanks to their Hölder
regularity established by Zhan [Zha19]. On the other hand the natural
parametrization of $\operatorname{SLE}_{\kappa}$ is characterized by a
conformal invariance and domain Markov property considered by Lawler and
Sheffield [LS11]. Therefore all subsequential limits of the chordal
$\operatorname{SLE}_{\kappa}$ measure agree with $\operatorname{SLE}_{4}$. ∎
###### Lemma A.5.
Suppose Theorem 1.2 holds for $\kappa\in(8/3,4)$. Then it holds for $\kappa=4$
as well.
###### Proof.
The right hand side of (1.3) is continuous as $\kappa\uparrow 4$. Therefore
$\\{\mathrm{CR}(z_{i},\eta_{i})\\}_{1\leq i\leq 3}$ for $\kappa<4$ converges
in law as $\kappa\uparrow 4$ and the moment generating function of the limit
is given by
${C^{\operatorname{CLE}}_{\kappa}(\lambda_{1},\lambda_{2},\lambda_{3})}$ with
$\kappa=4$. It remains to show that the limit is given by
$\\{\mathrm{CR}(z_{i},\eta_{i})\\}_{1\leq i\leq 3}$ with $\kappa=4$.
Fix a small $\varepsilon>0$. Let $S$ be the set of simple loops in
$\widehat{\mathbbm{C}}$ separating $z_{1}$ from $\\{z_{2},z_{3}\\}$. Let
$S_{\varepsilon}=\\{\eta\in S:\mathrm{CR}(z_{1},\eta)>\varepsilon\\}$. For
$\eta\in S$, let $D_{\eta}$ be the connected component of
$\widehat{\mathbbm{C}}\backslash\eta$ containing $z_{2},z_{3}$. For
$\kappa\in(8/3,4]$, the law of $(\eta_{1},\Gamma)$ restricted to the event
$\eta_{1}\in S_{\varepsilon}$ is the same as that of $(\eta,\Gamma)$ under
$1_{\eta=\eta_{1}}\mathrm{Count}_{S_{\varepsilon}\cap\Gamma}(d\eta)\operatorname{CLE}_{\kappa}(d\Gamma)$,
where $\mathrm{Count}_{S_{\varepsilon}\cap\Gamma}(d\eta)$ is the counting
measure on $S_{\varepsilon}\cap\Gamma$. Given a sample of $(\eta,\Gamma)$ of
$\mathrm{Count}_{S_{\varepsilon}\cap\Gamma}(d\eta)\operatorname{CLE}_{\kappa}(d\Gamma)$,
let $\Gamma_{+}$ be the subset of $\Gamma$ in $D_{\eta}$ and $F_{\Gamma_{+}}$
be the event that no loop in $\Gamma_{+}$ surrounds both $z_{2}$ and $z_{3}$.
Then the event $\eta=\eta_{1}$ is the same as $F_{\Gamma_{+}}$. By Proposition
3.8, for $\kappa\in(8/3,4]$, the law of $\eta$ under
$1_{\eta=\eta_{1}}\mathrm{Count}_{S_{\varepsilon}\cap\Gamma}(d\eta)\operatorname{CLE}_{\kappa}(d\Gamma)$
is $\operatorname{CLE}_{\kappa}^{D_{\eta}}[F_{\Gamma_{+}}]1_{\eta\in
S_{\varepsilon}}\cdot\operatorname{SLE}_{\kappa}^{\mathrm{loop}}(d\eta)$,
where $\operatorname{CLE}_{\kappa}^{D_{\eta}}$ is the law of a
$\operatorname{CLE}_{\kappa}$ in $D_{\eta}$.
By Lemmas A.4 and A.3, as $\kappa\uparrow 4$, the measure
$\operatorname{CLE}_{\kappa}^{D_{\eta}}[F_{\Gamma_{+}}]1_{\eta\in
S_{\varepsilon}}\cdot\operatorname{SLE}_{\kappa}^{\mathrm{loop}}(d\eta)$ on
loops converges weakly with respect to the Hausdorff metric to
$\operatorname{CLE}_{4}^{D_{\eta}}[F_{\Gamma_{+}}]1_{\eta\in
S_{\varepsilon}}\cdot\operatorname{SLE}_{4}^{\mathrm{loop}}(d\eta)$. Therefore
the law of $\eta_{1}$ conditioned on the event $\eta_{1}\in S_{\varepsilon}$
as $\kappa\uparrow 4$ converges weakly to the same law with $\kappa=4$. Since
$\lim_{\varepsilon\to 0}\mathbbm{P}[\Gamma\cap
S_{\varepsilon}\neq\emptyset]=\lim_{\varepsilon\to
0}\mathbbm{P}[\mathrm{CR}(z_{1},\eta_{1})>\varepsilon]=1$ uniformly for
$\kappa\in(\kappa_{0},4]$ for a fixed $\kappa_{0}\in(8/3,4)$, we can remove
the restriction $\eta_{1}\in S_{\varepsilon}$ and conclude that as
$\kappa\uparrow 4$ the law of $\eta_{1}$ converges weakly to the same law when
$\kappa=4$.
The conditional law of $(\eta_{2},\eta_{3})$ given $\eta_{1}$ is the law of
the two outermost loops surrounding $z_{2},z_{3}$ under the CLE measure
$\operatorname{CLE}_{\kappa}^{D_{\eta_{1}}}$ conditioned on these two loops
being distinct. By Lemma A.3, the joint law of $(\eta_{1},\eta_{2},\eta_{3})$
has the desired continuity as $\kappa\uparrow 4$. ∎
## Appendix B Identities for the modified Bessel function of the second kind
###### Proof of Equation (2.7).
We will prove the $c=1$ case; the general case then follows from a change of
variables. Recall the integral definition of $K_{\nu}(x)$ in (2.6). Using
$1+\cosh t=2(\cosh\frac{t}{2})^{2}$ and the fact that $\cosh$ is even, we can
express $\int_{0}^{\infty}\frac{1}{\sqrt{x}}e^{-x}K_{\nu}(x)\,dx$ as
$\iint_{0}^{\infty}\frac{1}{\sqrt{x}}e^{-x(1+\cosh t)}\cosh(\nu
t)\,dx\,dt=\sqrt{\pi}\int_{0}^{\infty}\frac{\cosh(\nu t)}{\sqrt{1+\cosh
t}}\,dt=\frac{1}{2}\sqrt{\pi/2}\int_{-\infty}^{\infty}\frac{\cosh(\nu
t)}{\cosh(t/2)}\,dt.$
Since $\int\frac{e^{\nu t}}{\cosh(t/2)}\,dt=\int\frac{e^{-\nu
t}}{\cosh(t/2)}\,dt$, we have $\int\frac{\cosh(\nu
t)}{\cosh(t/2)}\,dt=\int\frac{e^{\nu t}}{\cosh(t/2)}\,dt$. For a random
variable $X$ following the _hyperbolic secant distribution_
$\frac{dx}{\pi\cosh x}$, it is known that
$\mathbbm{E}[e^{tX}]=\frac{1}{\cos(\pi t/2)}$ for $|t|<1$; see e.g. [Fis14,
Section 1.3]. Therefore $\int_{-\infty}^{\infty}\frac{e^{\nu
t}}{\cosh(t/2)}\,dt=\frac{2\pi}{\cos(\pi\nu)}.$ This concludes the proof. ∎
###### Proof of Equation (2.8).
For $r>0$, using the change of coordinates $s=\sqrt{b}$, we get
$\displaystyle\int_{0}^{\infty}\frac{1}{\sqrt{b}(1+b)(1+b+br)}\,db=\int_{0}^{\infty}\frac{1}{s(1+s^{2})(1+s^{2}(1+r))}2s\,ds=\int_{-\infty}^{\infty}\frac{1}{(1+s^{2})(1+s^{2}(1+r))}\,ds$
$\displaystyle=\frac{1}{r}\int_{-\infty}^{\infty}\frac{1+r}{1+s^{2}(1+r)}-\frac{1}{1+s^{2}}\,ds=\pi\frac{\sqrt{1+r}-1}{r}=\frac{\pi}{\sqrt{1+r}+1}.$
Therefore, using (2.6) and the identity $1+\cosh t=2(\cosh\frac{t}{2})^{2}$,
the integral we must evaluate is
$\displaystyle\iiint_{0}^{\infty}\frac{1}{\sqrt{b}(b+1)}e^{-x(b+1+b\cosh
t)}\cosh(\nu
t)\,dt\,db\,dx=\iint_{0}^{\infty}\frac{1}{\sqrt{b}(1+b)(1+b+b\cosh
t)}\cosh(\nu t)\,db\,dt$ $\displaystyle=\pi\int_{0}^{\infty}\frac{\cosh(\nu
t)}{\sqrt{1+\cosh
t}+1}\,dt=\frac{\pi}{2}\int_{-\infty}^{\infty}\frac{\cosh(\nu
t)}{\sqrt{2}\cosh\frac{t}{2}+1}\,dt=\frac{\pi}{2}\int_{-\infty}^{\infty}\frac{e^{\nu
t}}{\sqrt{2}\cosh\frac{t}{2}+1}\,dt.$
For a random variable $X$ following the _Perk’s distribution_ whose density is
$\frac{\sin\lambda}{2\lambda}\frac{dx}{\cosh x+\cos(\lambda)}$ with
$\lambda\in[0,\pi)$, it is known that
$\mathbbm{E}[e^{itX}]=\frac{\pi\sinh(\lambda t)}{\lambda\sinh(\pi t)}$ for
$t\in\mathbbm{R}$; see e.g. [Fis14, Section 2.1]. Taking
$\lambda=\frac{\pi}{4}$, we have $\int_{-\infty}^{\infty}\frac{e^{\nu
t}}{\sqrt{2}\cosh\frac{t}{2}+1}\,dt=\frac{\pi}{\cos(\pi\nu)\cos(\frac{1}{2}\pi\nu)}.$
This concludes the proof. ∎
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# Real-time Controllable Denoising for Image and Video
Zhaoyang Zhang1 Yitong Jiang111footnotemark: 1 Wenqi Shao1,3 Xiaogang Wang1
Ping Luo2,3 Kaimo Lin4 Jinwei Gu1,3
1The Chinese University of Hong Kong 2The Univesity of Hong Kong 3Shanghai AI
Laboratory 4SenseBrain
{zhaoyangzhang@link, ytjiang@link,weqish@link, xgwang<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>Equal contribution.
###### Abstract
Controllable image denoising aims to generate clean samples with human
perceptual priors and balance sharpness and smoothness. In traditional filter-
based denoising methods, this can be easily achieved by adjusting the
filtering strength. However, for NN (Neural Network)-based models, adjusting
the final denoising strength requires performing network inference each time,
making it almost impossible for real-time user interaction. In this paper, we
introduce Real-time Controllable Denoising (RCD), the first deep image and
video denoising pipeline that provides a fully controllable user interface to
edit arbitrary denoising levels in real-time with only one-time network
inference. Unlike existing controllable denoising methods that require
multiple denoisers and training stages, RCD replaces the last output layer
(which usually outputs a single noise map) of an existing CNN-based model with
a lightweight module that outputs multiple noise maps. We propose a novel
Noise Decorrelation process to enforce the orthogonality of the noise feature
maps, allowing arbitrary noise level control through noise map interpolation.
This process is network-free and does not require network inference. Our
experiments show that RCD can enable real-time editable image and video
denoising for various existing heavy-weight models without sacrificing their
original performance.
## 1 Introduction
Image and video denoising are fundamental problems in computational
photography and computer vision. With the development of deep neural networks
[26, 12, 50, 61], model-based denoising methods have achieved tremendous
success in generating clean images and videos with superior denoising scores
[57, 59, 4]. However, it should be noted that the improvement in
reconstruction accuracy (e.g., PSNR, SSIM) is not always accompanied by an
improvement in visual quality, which is known as the Perception-Distortion
trade-off [6]. In traditional denoising approaches, we can easily adjust the
denoising level by tuning related control parameters and deriving our
preferred visual results. However, for typical deep network methods, we can
only restore the degraded image or video to a fixed output with a
predetermined restoration level.
Figure 1: Real-time controllable denoising allows users further tuning the
restored results to achieve Perception-Distortion trade-off. A-B: tuning with
changing denoising intensity. C-E: tuning without changing denoising
intensity.
In recent years, several modulation methods have been proposed to generate
continuous restoration effects between two pre-defined denoising levels. These
methods can be categorized into two kinds: interpolation-based methods [52,
17, 24, 51], which use deep feature interpolation layers, and condition-
network-based methods, which import an extra condition network for denoising
control [25, 9, 39]. Essentially, both types of methods are designed based on
the observation that the outputs of the network change continuously with the
modulation of features/filters. This observation enables deep denoising
control, but it also introduces several limitations. First, there is a lack of
explainability, as the relationship between the control parameters (how to
modulate features) and the control operation (how the network outputs are
changed) is unclear [24]. This indicates that black-box operators (network
layers) must be used to encode them. Second, the use of control parameters as
network inputs requires entire network propagation each time control
parameters change, resulting in a lack of efficiency. Lastly, current
modulation methods often require an explicit degradation level during
training, which is hard to obtain for real-world samples. As a result, current
controllable denoising methods only focus on synthetic noise benchmarks.
Furthermore, both interpolation-based and condition-network-based methods have
their own drawbacks. Interpolation-based methods often require multiple
training stages, including pretraining two basic models (start level and end
level). On the other hand, condition-network-based methods are strenuous to
jointly optimize the base network and the condition network.
Figure 2: Comparison of pipelines between conventional controllable denoising
and our RCD. RCD achieves real-time noise control by manipulating editable
noises directly.
In this paper, we research on the problem: Can we achieve real-time
controllable denoising that abandons the auxiliary network and requires no
network forward propagation for changing restoration effects at test time?
Towards this goal, we propose Real-time Controllable Denoising method (RCD), a
lightweight pipeline for enabling rapid denoising control to achieve
Perception-Distortion Balance (See Fig. 1). Our RCD can be plugged into any
noise-generate-based restoration methods [11, 56, 47, 57] with just a few
additional calculations. Specifically, we replace the last layer of an
existing denoising network (which usually outputs a single noise map) with a
lightweight module that generates multiple noise maps with different noise
levels. We utilize a novel Noise Decorrelation process to enforce the
orthogonality of the noise distribution of these noise maps during training.
As a result, we can attain arbitrary denoising effects by simple linear
interpolation of these noise maps. Since this process does not require network
inference, it makes real-time user interaction possible even for heavy
denoising networks.
Fig. 2 illustrates the fundamental differences between our RCD approach and
conventional controllable denoising methods. In contrast to traditional
methods that rely on control networks, the RCD pipeline generates editable
noises of varying intensities/levels, providing explicit control by external
parameters and enabling network-free, real-time denoising editing. Real-time
editing capabilities offered by RCD create new opportunities for numerous
applications that were previously impossible using conventional techniques,
such as online video denoising editing, even during playback (e.g., mobile
phone camera video quality tuning for ISP tuning engineers), as well as
deploying controllable denoising on edge devices and embedded systems. Since
the editing stage of RCD only involves image interpolation, users can edit
their desired results on low-performance devices without the need for
GPUs/DSPs.
Moreover, unlike previous methods that only support changing noise levels, RCD
allows users to adjust denoising results at a specific noise level by
providing a new interface to modify the noise generation strategy. RCD is also
the first validated method for controllable denoising on real-world
benchmarks. It is noteworthy that existing controllable methods typically
require training data with fixed-level noise to establish their maximum and
minimum noise levels, which makes them unsuitable for most real-world
benchmarks comprising data with varying and unbalanced noise levels.
Our main contributions can be summarized as follows:
* •
We propose RCD, a controllable denoising pipeline that firstly supports real-
time denoising control ($>\textbf{2000}\times$ speedup compared to
conventional controllable methods) and larger control capacity (more than just
intensity) without multiple training stages [24] and auxiliary networks [51].
* •
RCD is the first method supporting controllable denoising on real-world
benchmarks.
* •
We propose a general Noise Decorrelation technique to estimate editable
noises.
* •
We achieve comparable or better results on widely-used real/synthetic image-
denoising and video-denoising datasets with minimal additional computational
cost.
## 2 Related Works
### 2.1 Denoising
Traditional image and video denoising methods are often based on prior
assumptions such as sparse image prior [3, 16, 20, 15], non-local similarity
[7, 18, 14, 13], and other similar techniques [22, 42, 54]. However, with the
recent development of deep learning networks, many learning-based methods have
been proposed and achieved state-of-the-art performance. Early works [8]
utilized multi-layer perceptron (MLP) to achieve comparable results with BM3D.
In recent years, there has been rapid progress on CNN-based denoising methods
[57, 59, 48, 21, 4, 10] and Transformer-based methods [56, 32, 61, 43], which
have started to dominate the image/video denoising task. However, the above-
mentioned works mainly focus on designing novel network architectures to
improve the denoising performance and usually generate a single output. Their
lack of ability to adjust the output denoising level based on user’s feedback
has greatly restricted their practical use in many real-world applications.
Moreover, although techniques like pruning [38, 62, 33] and quantization [46,
63] can accelerate such neural network-based methods, they are typically
heavy, which restricts their application to real-time denoising control.
Figure 3: Pipeline overview of proposed RCD framework. A: Backbone network for
generating multi-level noise maps. B: Noise Decorrelation module for editable
noises. C: AutoTune module for providing reference control parameters for
users.
### 2.2 Controllable denoising
Most conventional deep-learning methods for image/video denoising can only
generate a fixed result with a specific restoration level. Recently, some
controllable image/video denoising methods allow users to adjust the
restoration effect without retraining the network. DNI [52] and AdaFM [24]
used the observation that the learned filters of the models trained with
different restoration levels are similar in visual patterns. DNI interpolated
all the corresponding parameters between two related networks to derive smooth
and continuous restoration effects, while AdaFM adopted feature modulation
filters after each convolution layer. CFSNet [51] proposed an adaptive
learning strategy of using interpolation coefficients to couple the
intermediate features between the main branch and the tuning branch. Different
from these interpolation-base methods, some other methods [25, 9, 39] regarded
modulation as a conditional image restoration problem and adopted a joint
training strategy. CUGAN [9] proposed a GAN-based image restoration framework
to avoid the over-smooth problem, a common issue in PSNR-oriented methods.
However, all of the above controllable methods can only be trained with
synthetic degradations because they require explicit degradation levels during
training. When applied on real-world data, as shown in [23], methods that
trained for blind Additive White Gaussian Noise (AWGN) [57, 35] may be
overfitted and often suffer from dramatic performance drop. Besides the real-
world image issue, all these controllable methods utilize an auxiliary
conditional network and require one network inference for each different
target restoration level at test time, which makes them almost impossible for
real-time application.
## 3 Methods
### 3.1 Conventional Deep Denoising
Deep denoising methods trump traditional filter-based techniques by leveraging
the neural networks’ robust representation learning capability. Most current
denoising methods [11, 32, 47] reason about the relationship between clean and
noisy images by regressing noise maps with a neural generator. Specifically,
given a noisy image $\mathbf{I_{n}}$ and model
$\mathcal{M}:\mathbb{R}^{H\times W\times C}\to\mathbb{R}^{H\times W\times C}$,
we can derive the predicted clean image $\mathbf{I_{c}}$ by:
$\mathbf{I_{c}}=\mathbf{I_{n}}+\mathcal{M}(\mathbf{I_{n}})$, where model
$\mathcal{M}$ is updated by minimizing the distance between the denoising
result $\mathbf{I_{c}}$ and the ground truth $\mathbf{I_{gt}}$. As we can see,
this kind of approach generates a single fixed output result in a black-box
manner, making it almost impossible to adjust the denoise operation
explicitly.
### 3.2 Pipeline Overview
In this section, we present Real-time Controllable Denoising (RCD), a novel
deep learning-based pipeline for real-time controllable denoising. As
illustrated in Fig. 3, RCD essentially consists of three parts: (1) A backbone
network, _i.e_., $\mathcal{M}_{b}:\mathbb{R}^{H\times W\times
C}\to\mathbb{R}^{H\times W\times LC}$, generates multiple fixed-level noise
maps, where $L$ is the number of pre-defined noise levels (see (A) in Fig. 3).
(2) A Noise Decorrelation (ND) block that enforces the editability of the
generated noise maps (see (B) in Fig. 3). (3) An AutoTune module that gives a
default set of control parameters to generate the best denoising result.
Specifically, the backbone network will generate multiple fixed-level noise
maps, _i.e_., $\\{{\mathcal{N}}_{i}\\}_{i=1}^{L}$, for each noisy image input.
The noise maps are then fed into the proposed Noise Decorrelation (ND) block,
which makes noise maps orthogonal to each other. In this way, the decorrelated
noise maps $\\{\tilde{\mathcal{N}}_{i}\\}_{i=1}^{L}$ will be zero-correlated
and thus become linearly interpolable. At last, the AutoTune module will give
a set of suggested control parameters $\\{\bar{c}_{i}\\}_{i=1}^{L}$ to
generate the final denoising result as follows:
$\mathbf{I_{c}}=\mathbf{I_{n}}+\sum_{i=1}^{L}\bar{c}_{i}\tilde{\mathcal{N}}_{i},$
(1)
where $\sum_{i=1}^{L}\bar{c}_{i}=1$. Moreover, given the zero-correlated noise
maps, users can also generate arbitrary strength denoising results by
replacing $\\{\bar{c}_{i}\\}_{i=1}^{L}$ with their own customized control
parameters $\\{c_{i}\\}_{i=1}^{L}$.
### 3.3 Multi-level Noise Generation
Given a noisy input image $\mathbf{I_{n}}$, the backbone network aims to
generate multiple noise maps $\\{\mathcal{N}_{i}\\}_{i=1}^{L}$, corresponding
to a set of pre-defined noise levels $\\{l_{i}\\}_{i=1}^{L}$, _e.g_. noise
levels $\\{5,10,15,...,60\\}$. Hence, we have
$\sigma(\mathcal{N}_{i})=l_{i},\forall i=1,...,L,$ (2)
where $\sigma$ is the noise level operation that calculates the standard
deviation of pixels in each noise map. To obtain multi-level noise maps, we
replace the conventional last output layer of the denoising network with a
convolutional layer with an output channel size of $L\cdot C$. Moreover, the
level of the noise map is explicitly generated with the normalization
operation, as given by
$\mathcal{N}_{i}=l_{i}\frac{\mathcal{M}_{b}(\mathbf{I_{n}})^{(i)}}{\sigma(\mathcal{M}_{b}(\mathbf{I_{n}})^{(i)})},\forall
i=1,...,L.$ (3)
Here $\mathcal{M}_{b}(\mathbf{I_{n}})\in\mathbb{R}^{H\times W\times LC}$ is
network output, and
$\mathcal{M}_{b}(\mathbf{I_{n}})^{(i)}\in\mathbb{R}^{H\times W\times C}$ is
the i-th component separated from the channel dimension. The derived
$\mathcal{N}_{i}$ can be considered as the noise map estimated at the given
noise level $l_{i}$.
Different from prior controllable denoising methods with implicit
interpolation in the network, we propose to explicitly interpolate the noise
maps in Eqn. 3. Thanks to the separation of noise interpolation and network
inference, our RCD can achieve real-time user interaction.
However, the multi-level noise maps $\mathcal{N}_{i}$ directly obtained by
convolutional layers are usually highly correlated, which leads to the problem
of noise level collapse. In other words, the noise map representations in
different levels are redundant, implying that the number of noise maps at
different noise strengths that participate in the linear interpolation in Eqn.
1 is implicitly reduced. Without any constraint, our experiments show that the
single noise map at a certain noise level would dominate in the linear
interpolation for a variety of input noisy images. To address this issue, we
further introduce the Noise Decorrelation block to make representations of
these noise maps much more informative in the following section.
Figure 4: Demonstration of Noise Decorrelation’s influence on noise editing.
$|\Sigma|_{F}$ denotes norms of the covariance matrix for corresponding
learned noises and $\sigma$ is noise intensity.
### 3.4 Noise Decorrelation
The Noise Decorrelation (ND) block is designed to regularize the backbone
network to generate editable noise maps at varying levels. In particular, this
block is a parameter-free computational unit that enforces $\mathcal{N}_{i}$
to be approximately zero-correlated with each other:
$\mathbf{cov}(\mathcal{N}_{i},\mathcal{N}_{j})\approx 0,\forall
i,j\in\\{1,2,...,L\\}$, where $\mathbf{cov}(\cdot,\cdot)$ is covariance
operator. Inspired by the success of using the decorrelation technique in
network optimization and normalization, we adopt the whitening-based methods
[29, 30] for noise decorrelation here. The noise maps are decorrelated using
the inverse square root of their covariance matrix.
Specifically, for each predicted fixed-level noise map $\mathcal{N}_{i}$, it
will firstly be reshaped to $\mathcal{N}_{i}\in\mathbb{R}^{1\times M}$, where
$M=HWC$. By stacking the reshaped $\mathcal{N}_{i}$ over the first dimension,
we have $\mathbf{N}\in\mathbb{R}^{L\times M}$. We then calculate the noise
covariance matrix $\sigma$ by:
$\Sigma=\frac{1}{M-1}(\mathbf{N}-\bar{\mathbf{N}})(\mathbf{N}-\bar{\mathbf{N}})^{T}$
where $\bar{\mathbf{N}}$ is mean of $\mathbf{N}$ over channel $M$.
The Noise Decorrelation block needs to compute inverse square root
$\Sigma^{-\frac{1}{2}}\in\mathbb{R}^{L\times L}$, which can be done by eigen
decomposition or SVD. Since this kind of operation involves heavy computation
[29], we instead adopt the more efficient Newton’s Iteration to estimate
$\Sigma^{-\frac{1}{2}}$ as in [5, 27]. Giving a covariance matrix $\Sigma$,
Newton’s Iteration calculates $\Sigma^{-\frac{1}{2}}$ by following the
iterations below:
$\begin{array}[]{l}\Sigma_{0}=I,\\\
\Sigma_{k}=\frac{1}{2}(3\Sigma_{k-1}-(\Sigma_{k-1})^{3}\Sigma),k=1,2,..,T,\end{array}$
(4)
where $k$ is the iteration index and $T$ is the iteration number (in our
experiments $T$ = 3 or 4). $\Sigma_{k}$ is guaranteed to converge to
$\Sigma^{-\frac{1}{2}}$, if $\left\|I-\Sigma\right\|_{2}<1$ [5]. This
condition can be achieved by normalizing $\Sigma$ to
$\frac{\Sigma}{tr(\Sigma)}$, where $tr(.)$ is trace operator.
The derived $\Sigma^{-\frac{1}{2}}$ can be regarded as a whitening matrix
[44], which decorrelates the noise maps $\mathbf{N}$ in a differentiable
manner. The decorrelated noise maps $\tilde{\mathbf{N}\in\mathbb{R}^{H\times
W\times LC}}$ can be obtained by calculating:
$\tilde{\mathbf{N}}=\Sigma^{-\frac{1}{2}}\mathbf{N}$. We can then have our
editable fixed-level noises $\tilde{\mathcal{N}}_{i}\in\mathbb{R}^{H\times
W\times C}$ by reshaping $\tilde{\mathbf{N}}$ and splitting it into $L$ noise
maps. After Noise Decorrelation, we apply the same normalization as Eqn. 3 to
guarantee the noise strength of the decorrelated noises.
The zero-correlated noise maps $\tilde{\mathcal{N}}_{i}$ present several
excellent properties for controllable denoising. Firstly, the linearity of the
noise level’s square towards $\tilde{\mathcal{N}}_{i}$ is guaranteed. In other
words, given an arbitrary set of control parameters $\\{c_{i}\\}_{i=1}^{L}$,
we have
$\mathbf{Var}(\sum_{i=1}^{L}c_{i}\tilde{\mathcal{N}}_{i})=\sum_{i=1}^{L}c_{i}^{2}\mathbf{Var}(\tilde{\mathcal{N}}_{i}),$
(5)
where $\mathbf{Var}(\cdot)$ denotes variance operator. Apparently, Eqn. 5
holds when elements of $\\{\tilde{\mathcal{N}}_{i}\\}_{i=1,2,..,m}$ are
mutually zero-correlated. Eqn. 5 reveals the explicit relationship between the
control parameters and the target noise level, which allows us to directly
edit noises by interpolating $\mathcal{N}_{i}$ using $c_{i}$. Secondly, the
Noise Decorrelation block can be regarded as a regularization tool that forces
models to learn different noise formats for each level, which will increase
the representation capacity of the denoising network [30].
Fig. 4 demonstrates how the Noise Decorrelation block works. With ND block,
the covariance of learned noises is reduced to almost zero (without it,
$|\Sigma|_{F}$ can be 751 and unignorable), allowing us to derive determined
interpolated results with target noise intensity. In contrast, without the
Noise Decorrelation blocks, the output noise level can not be guaranteed.
### 3.5 AutoTune Module
Given the decorrelated noise maps from the Noise Decorrelation block, the
AutoTune module will predict a set of model-suggested control parameters,
_i.e_., $\\{\bar{c}_{i}\\}_{i=1}^{L}$, to generate the default denoising
result. Users can then use this set of parameters as a starting point to fine-
tune their final desired denoising strength. Our AutoTune module is extremely
lightweight, and is formulated as a single-layer module with temperature
softmax activation. Specifically, $\\{\bar{c}_{i}\\}_{i=1}^{L}$ can be
obtained by :
$\bar{c}_{i}=\frac{e^{\frac{\mathcal{A}(f)_{i}}{\tau}}}{\sum_{j=1}^{n}e^{\frac{\mathcal{A}(f)_{j}}{\tau}}},$
where $\mathcal{A}$ is the NN layer, $f$ is the input feature maps, and $\tau$
is temperature. In our experiments, $\tau$ is set to be $0.05$ for best
performance. Following the design ethos of efficiency and least coupling to
the backbone architecture, we directly choose the unnormalized model outputs
$\mathcal{M}(\mathbf{I_{n}})$ as $f$ (see (C) in Fig. 3).
Figure 5: Example of RCD denoising results by AutoTune and HumanTune on Set8.
AutoTune module provides reference control parameters, _i.e_.,
$\\{\bar{c}_{i}\\}_{i=1}^{L}$), to generate the denoising result, and it can
be further improved by fine-grained artificial tuning (HumanTune), _i.e_.,
$\\{c_{i}\\}$, without changing the noise intensity (both $\sigma=40$).
### 3.6 New Cardinality for Denoising Control.
Unlike existing methods that only modulate noise intensity, our RCD control
scheme allows users to further optimize the denoising result to a given noise
intensity by tuning $\\{c_{i}\\}$, as long as the weighted mean of $l_{i}$
towards $c_{i}$ remains the same. Eqn. 5 shows that when
$\sum_{i=1}^{L}c_{i}^{2}\mathbf{Var}(\tilde{\mathcal{N}}_{i})=\sum_{i=1}^{L}c_{i}^{2}l_{i}^{2}$
is fixed, the variance of the output noise
$\mathbf{Var}(\sum_{i=1}^{L}c_{i}\tilde{\mathcal{N}}_{i})$ would be also
fixed. By exploring $c_{i}$ under the condition of fixed
$\sum_{i=1}^{L}c_{i}^{2}l_{i}^{2}$, we can further optimize the denoising
results at a specific noise level by involving different components
($\mathcal{N}_{i}$) in the noise interpolation. As shown in Fig. 5, our
AutoTune module can generate high-quality results using just the reference
control parameters $\\{\bar{c_{i}}\\}$. Users can further improve the result
by artificially tuning $\\{c_{i}\\}$ around $\\{\bar{c_{i}}\\}$, even at the
same noise level.
Table 1: Gaussian single image denoising results (PSNR). RCD is evaluated with AutoTune results. “-”: not reported Method | Controllable | CBSD68 | Kodak24 | McMaster | Urban100
---|---|---|---|---|---
$\sigma$ = 15 $\sigma$ = 25 $\sigma$ = 50 | $\sigma$ = 15 $\sigma$ = 25 $\sigma$ = 50 | $\sigma$ = 15 $\sigma$ = 25 $\sigma$ = 50 | $\sigma$ = 15 $\sigma$ = 25 $\sigma$ = 50
IRCNN [58] | ✗ | 33.86 31.16 27.86 | 34.69 32.18 28.93 | 34.58 32.18 28.91 | 33.78 31.20 27.70
FFDNet [59] | ✗ | 33.87 31.21 27.96 | 34.63 32.13 28.98 | 34.66 32.35 29.18 | 33.83 31.40 28.05
DnCNN [57] | ✗ | 33.90 31.24 27.95 | 34.60 32.14 28.95 | 33.45 31.52 28.62 | 32.98 30.81 27.59
DSNet [41] | ✗ | 33.91 31.28 28.05 | 34.63 32.16 29.05 | 34.67 32.40 29.28 | -
CResMD [25] | ✓ | 33.97 - 28.06 | - | - | -
AdaFM-Net [24] | ✓ | 34.10 31.43 28.13 | - | - | -
NAFNet [11] | ✗ | 34.11 31.49 28.27 | 35.14 32.70 29.68 | 35.07 32.82 29.79 | 34.41 32.09 29.00
NAFNet-RCD (ours) | ✓ | 34.13 31.49 28.26 | 35.15 32.72 29.69 | 35.11 32.84 29.81 | 34.45 32.12 29.02
### 3.7 Optimization Targets
To guarantee the visual quality of arbitrarily edited noise
$\sum_{i=1}^{L}c_{i}\tilde{\mathcal{N}}_{i}$ for any given set of control
parameters $\\{c_{i}\\}_{i=1}^{L}$, we adopt a multi-level concurrent training
strategy by minimizing the difference between each level’s noise output and
the ground truth noise, _i.e_., $\mathcal{L}_{level}$, which is derived by:
$\mathcal{L}_{level}=\frac{1}{L}\sum^{L}_{i=1}\mathcal{L}(\mathbf{I}_{gt},\mathbf{I_{n}}+\tilde{\mathcal{N}}_{i})$
(6)
where $\mathbf{I}_{gt}$, $\mathbf{I_{n}}$ are ground truth clean image and the
input noisy image. $\mathcal{L}(.)$ can be any loss functions (e.g., L2 loss
or PSNR loss).
Spectacularly, $\mathbf{I_{n}}+\tilde{\mathcal{N}}_{i}$ can be regarded as
corner cases of RCD when we use one-hot control parameters as input. Joint
optimization of all the noise levels ensures that each element of
$\mathcal{N}_{i}$ can be trained as optimal noise estimation under the
condition of fixed noise level $l_{i}$.
Together with the AutoTune module optimization, our final cost function can be
written as:
$\mathcal{L}_{total}=\lambda\mathcal{L}_{level}+\mathcal{L}(\mathbf{I}_{gt},\mathbf{I_{n}}+\sum_{i=1}^{L}\bar{c}_{i}\tilde{\mathcal{N}}_{i}),$
(7)
where $\lambda$ is the loss weight ($\lambda=0.1$ in our experiments), and
$\mathcal{L}(\mathbf{I}_{gt},\mathbf{I_{n}}+\sum_{i=1}^{n}\bar{c}_{i}\tilde{\mathcal{N}}_{i})$
optimizes the denoising result derived by the model-suggested control
parameters.
## 4 Experiments
This section is organized as follows: First, we demonstrate the effectiveness
of our plug-in RCD with SOTA image denoising methods [11] in different scales
on synthetic noise datasets. Next, to evaluate the ability of blind denoising
on real-world data, we conduct experiments on popular real-world denoising
dataset SIDD [1]. Then, we apply our real-time controllable RCD pipeline on
video denoising applications. At last, we empirically discuss some design
details described in the previous sections.
Figure 6: Visual comparison of RCD and their baseline results on $\sigma=50$
denoising. GT: Ground truth. Base: Baseline model without RCD. AutoTune: RCD
results by applying control parameters from AutoTune module.
### 4.1 Gaussian Single Image Denoising
Experimental Setup. To fully demonstrate the effectiveness of the proposed
RCD, we choose the most recent SOTA method NAFNet [11] as our backbone.
Following [56], we first conduct denoising experiments on several widely-used
synthetic color image benchmarks (DIV2K [2], BSD400 [36], Flickr2K [55] and
WaterlooED [34]) with additive white Gaussian noise ($\sigma\in[0,60]$). The
training patch size is 128 × 128 and the batch size is 64. We train our model
with Adam [31] optimizer and learning rate $1e-3$ for total 60K iterations.
Consistent to [11], PSNR loss is adapted as the loss function. Both the
baseline model (NAFNet) and its RCD variants (NAFNet-RCD) are trained from
scratch. For settings of RCD, we initialize $L=12$ and
$\\{l_{i}\\}=[5,10,...,60]$ for synthetic denoising training.
Table 2: Running time comparison for RCD and other controllable methods during
test time. Full Pipeline compares full pipeline latency for the model to infer
1000 images, and Edit-only compares latency for editing one image with 1000
different control parameters.
Method | Multi-stage training | Full Pipeline | Edit-only
---|---|---|---
AdaFM-Net | required | 81.03s | 81.03s
CResMDNet | not required | 128.08 | 128.08s
NAFNet-RCD | not required | 64.84s | 0.04s
Table 3: Ablation of RCD on various backbone sizes. Method | CBSD68 | Kodak24 | McMaster | Urban100
---|---|---|---|---
$\sigma$ = 15 $\sigma$ = 25 $\sigma$ = 50 | $\sigma$ = 15 $\sigma$ = 25 $\sigma$ = 50 | $\sigma$ = 15 $\sigma$ = 25 $\sigma$ = 50 | $\sigma$ = 15 $\sigma$ = 25 $\sigma$ = 50
NAFNet-tiny | 33.58 30.91 27.62 | 34.33 31.84 28.63 | 33.85 31.61 28.55 | 32.96 30.37 26.92
NAFNet-RCD-tiny | 33.71 31.06 27.68 | 34.46 31.98 28.65 | 34.07 31.78 28.61 | 33.22 30.66 27.18
NAFNet-small | 33.84 31.18 27.91 | 34.68 32.18 29.01 | 34.68 32.18 29.01 | 33.61 31.10 27.68
NAFNet-RCD-small | 33.96 31.31 28.05 | 34.83 32.32 29.14 | 34.71 32.40 29.26 | 33.92 31.46 28.08
NAFNet | 34.11 31.49 28.27 | 35.14 32.70 29.68 | 35.07 32.82 29.79 | 34.41 32.09 29.00
NAFNet-RCD | 34.13 31.49 28.26 | 35.15 32.72 29.69 | 35.11 32.84 29.81 | 34.45 32.12 29.02
Complexity analysis. Extensive adjustments of controllable parameters are
often required to obtain one satisfying result for users. Therefore, editing
time is vital for controllable methods. This section compares the inference
and editing latency of our RCD and conventional controllable pipelines on GTX
1080Ti. As shown in Tab. 2, the proposed RCD not only outperforms other
conventional controllable pipelines on inference time, but more importantly,
can overwhelm those traditional controllable designs on editing time, which
can be more than 2000 times faster (as editing process of RCD is network-free,
without reliance on sub-networks). This comparison confirms that our RCD is
more than enough for real-time image editing.
Results Analysis. We evaluate our proposed method on widely used synthetic
noise datasets CBSD68 [37], Kodak24 [19], McMaster [60] and Urban100 [28] with
noise levels $\sigma(15)$, $\sigma(15)$ and $\sigma(50)$. RCD is evaluated
with denoising results using AutoTune outputs$\\{\bar{c}_{i}\\}$. As shown in
Tab. 1, NFANet-RCD achieves comparable performance to the baseline NFANet
consistently on multiple datasets, indicating that our plug-in RCD module
enables real-time controllable denoising for NAFNet without sacrificing its
original denoising performance. Please note that NAFNet-RCD can yield
comparable results to the backbone just by using the AutoTune outputs, and the
performance can be further improved by manually tuning the control parameters
( See Sec. 3.6.) We further show the qualitative performance of NAFNet-RCD in
Fig. 6. NAFNet-RCD can recover more details of some degraded images, which may
be benefited from RCD’s richer representation capacity by integrating multiple
noise maps.
Slimmer Model Variants. Towards the goal of evaluating the compatibility and
robustness of RCD, we conduct ablations by applying RCD to different-sized
backbones. Specifically, we shrink the width and block numbers of NAFNet,
denoting derived models as NAFNet-small ($\frac{1}{4}\times$) and NAFNet-tiny
($\frac{1}{16}\times$). Tab. 3 reports the results of RCD with those scaled
backbones. It can be observed that the RCD-variants can achieve comparable and
even slightly better denoising results compared to their baselines, which
further demonstrates RCD’s robustness and effectiveness for different-sized
backbones.
Table 4: Image denoising results on SIDD. Real noise: results on real-world SIDD test sets. Synthetic noise: results on SIDD test set with additive Gaussian noise ($\sigma=25$). Method | Real noise | Synthetic noise
---|---|---
PSNR | SSIM | PSNR | SSIM
NAFNet-tiny | 42.19 | 0.9796 | 38.46 | 0.9551
NAFNet-RCD-tiny | 41.86 | 0.9781 | 38.60 | 0.9558
NAFNet | 43.22 | 0.9818 | 38.85 | 0.9481
NAFNet-RCD | 42.91 | 0.9806 | 39.14 | 0.9580
### 4.2 Real Single Image Denoising
Experimental Setup (Real Image) Unlike existing controllable denoising
methods [24, 51] which focus on synthetic benchmarks, we are the first
solution that attempts to extend controllable denoising to real-world SIDD
datasets. SIDD consists of real noisy images captured by smartphones with
$\sigma\in[0,50]$. Instead of using full SIDD data, we choose subsets of SIDD
with $\sigma\in[0,12]$ (around 70% of the entire dataset) to train our RCD
model, which is initialized with $L=4$ and $\\{l_{i}\\}=[3,6,9,12]$. The main
reason is the lacking of high $\sigma$ data at given levels in SIDD because of
SIDD’s highly long-tailed noise level distribution. Specifically, most noisy
images in SIDD gather in $\sigma<12$ and the samples distribute sparsely when
$\sigma$ is large. Consistent to Sec. 4.1, we adopt NAFNet (SOTA methods for
SIDD challenge [11]) as our backbone at two scales ($1\times$,
$\frac{1}{16}\times$). Both NAFNet-RCD and the corresponding baselines are
trained on this subset with the same training settings as in [11].
Figure 7: Illustration of RCD control logics. Users can retouch the denoising
level by tuning the Intensity bar
($\sigma=\sqrt{\sum_{i=1}^{L}c_{i}^{2}l_{i}^{2}}$ ) and setup their perceptual
preference at fixed level by tuning Component bar (changing $\\{c_{i}\\}$
while keeping $\sigma$).
Results and Analysis We conduct blind denoising experiments on SIDD with
different RCD model scales to evaluate its adjustability to the real-world
dataset. As shown in Tab. 4 (left), our RCD (AutoTune results) can achieve
high-quality controllable real-world denoising in both model scales. However,
we note that enabling controllable denoising with RCD may still result in a
slight decrease in quantitative results (about 0.3dB), which may be a result
of unbalanced data for each level and short noise level interval
($|l_{i+1}-l_{i}|$, see more discussion in Sec. 4.4).
SIDD with synthetic noise. We extensively conduct synthetic denoising
experiments on SIDD to further show the compatibility of RCD on SIDD datasets.
Following Sec. 4.1, we add random Gaussian noise $\sigma\in[0,60]$ to SIDD
training data, and both methods are evaluated on $\sigma=50$ SIDD test
samples. As shown in Tab. 4 (right), RCD models slightly outperform their
baselines, demonstrating RCD’s compatibility for SIDD. Moreover, this result
can also indicate that RCD’s performance drop on SIDD real image may arise
from the noise distribution and RCD configurations, rather than RCD’s adaptive
capacity to SIDD data. See Appendix for more results and visualizations.
### 4.3 Video Denoising
Experiment Setup Following common practice [47, 45, 32], we train our models
on DAVIS training set and use DAVIS-test and Set-8 for benchmarking. Like in
[47], we add Gaussian noise with random standard deviation between 5-50 on the
DAVIS clean videos for training. The DAVIS set contains 30 color sequences of
resolution $854\times 480$, which will be randomly cropped into $128\times
128$ patches during training. Other training settings and hyperparameters are
kept the same as [47] for a fair comparison.
Choice of Basic model. We choose FastDVD [47] as our backbone model. Although
recent methods [49, 32] outperform FastDVD by 1-2 PSNR at most, they actually
introduce huge models and extra heavy operations like patch clustering [49]
and layer-wise frame-to-frame wrapping using optical flow [32] ( $>100\times$
slower than FastDVD).
Results and Analysis. Like [45], we evaluate our video denoising models with
the input length of one frame and five frames. We denote RCD models for video
denoising as “FastDVD-RC” and compare their quantitative AutoTune denoising
results to baseline FastDVD in Tab. 5. Consistent with preceding sections,
AutoTune results of FastDVD-RCD can demonstrate comparable performance to the
default FastDVD, which means our RCD can also achieve lossless real-time noise
editing in video scenarios. Unlike previous heavy controllable denoising
methods, our real-time RCD can even allow users to do online video denoising
editing without any latency.
Table 5: Video denoising results.
Test set | $\sigma$ | 1 frame | 5 frames
---|---|---|---
FastDVD | FastDVD-RCD | FastDVD | FastDVD-RCD
DAVIS | 20 | 34.17 | 34.21 | 35.69 | 35.65
30 | 32.45 | 32.69 | 34.06 | 34.04
40 | 31.39 | 31.60 | 32.80 | 32.78
50 | 30.26 | 30.57 | 31.83 | 31.85
Set 8 | 20 | 31.99 | 32.01 | 33.43 | 33.46
30 | 30.61 | 30.65 | 31.62 | 31.71
40 | 29.62 | 29.83 | 30.36 | 30.42
50 | 28.61 | 28.85 | 29.41 | 29.60
### 4.4 Discussions
Selection of Denoising Levels. Differing from conventional denoising methods,
RCD requires a group of predefined noise levels $\\{l_{i}\\}_{i=1}^{L}$. To
evaluate how the selection of $\\{l_{i}\\}_{i=1}^{L}$ affects RCD’s
performance, we conduct ablation studies on FastDVD-RCD by changing the number
of noise maps ($L$) (See Tab. 6.) All of the models are trained on noisy
images with $\sigma\in(0,60]$ and uniformly sampled noise levels that
$\\{l_{i}=\frac{60}{L}*i\\}_{i=1}^{L}$. We observe that larger $L$ means more
fine-grained control on denoising, but it may incur a performance drop. In
fact, when $n$ is large we find that $\mathcal{L}_{level}$ will also keep
large and be hard to optimize. Trading-off performance and control precision,
we empirically choose $L=12$ and noise level interval $|l_{i+1}-l_{i}|=5$ as
defaults.
Table 6: Ablations of FastDVD-RCD on different number of noise levels.
Reported scores are PSNR of AutoTune outputs and GT.
Test Set | $\sigma$ | $L=1$ | $L=2$ | $L=12$ | $L=30$ | $L=60$
---|---|---|---|---|---|---
Set8 | 20 | 31.87 | 31.42 | 32.01 | 31.39 | 31.07
30 | 30.51 | 30.09 | 30.65 | 30.12 | 29.75
40 | 29.60 | 29.31 | 29.83 | 29.33 | 29.01
50 | 28.62 | 28.29 | 28.85 | 28.22 | 28.05
Control Capacity. This section discusses the representation capacity of
$c_{i}$ as control parameters. Generally, $c_{i}$ controls the denoising
process on two aspects: intensity and components. Firstly, the noise levels of
RCD outputs are identical and can be derived by
$\sigma=\sqrt{\sum_{i=1}^{L}c_{i}^{2}l_{i}^{2}}$ (See Sec. 3.4), which allows
us to control the denoising intensity by changing $\\{c_{i}\\}$. Fig. 7
depicts visualizations of RCD-controlled denoising under different intensity
settings. Besides, as discussed in Sec. 3.6, RCD supports further optimization
of the denoising results at specific noise intensity by tuning $c_{i}$ by
involving different components of $\tilde{\mathcal{N}_{i}}$. (Please be
reminded that $\tilde{\mathcal{N}_{i}}$ is trained by $\mathcal{L}_{level}$,
denoting learned optimal denoising results at every fixed level $l_{i}$.)
## 5 Summary
We present RCD framework that enables real-time noise editing for controllable
denoising. Unlike existing continual-level denoising methods, RCD doesn’t
require multiple training stages and auxiliary networks. With the proposed
Noise Decorrelation module, RCD transforms the control of denoising into
white-box operations, with no requirement to feed control parameters to
networks at test time, which enables real-time editing even for heavy network
models. Extensive experiments on widely-used real/synthetic image and video
denoising datasets demonstrate the robustness and effectiveness of our RCD.
## 6 Acknowledgement
This paper is partially supported by the National Key R&D Program of China
No.2022ZD0161000, the General Research Fund of HK No.17200622, and Shanghai
Committee of Science and Technology (Grant No. 21DZ1100100).
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Figure 8: RCD real image denoising results on SIDD. GT: ground truth.
AutoTune: AutoTune results of RCD. Figure 9: Video denoising results. Base:
uncontrollable baseline.
## Appendix A More Visualization Results
This section shows more visual results to demonstrate the effectiveness of our
proposed RCD. Besides, we also provide demo video for showing features of RCD
(Please see in attached files of supplementary material).
Real Image Denoising. We visualize RCD denoising results on SIDD test set in
Fig.8. The left part shows the comparison of RCD (AutoTune) and real image
ground truth, and the right part gives RCD results by tuning the noise level.
As demonstrated, RCD can support controllable real image denoising and yield
high visual quality results.
Video denoising results. We further show the qualitative performance of
FastDVD-RCD in Fig.9, with comparison to baseline uncontrollable FastDVDnet.
Consistent with image denoising, FastDVD-RCD can recover more details of some
degraded images, which may be benefited from RCD’s richer representation
capacity by integrating multiple noise maps
Comparison of RCD and AdaFM on SIDD. In Fig.10 we show the comparison of RCD
and representative conventional controllable denoising method AdaFM. Compared
to RCD and GT, AdaFM results have more artifacts and remained noises.
Figure 10: Comparison of RCD and AdaFM on SIDD real image denoising.
## Appendix B Implementation Details
Choice of Basic Image Model. Considering the application of real-time image
controllable denoising, we need to select the base models with acceptable
running time and parameters. Besides, the base model is required to support
prior-free blind denoising to be applied on real-world data and should be able
to be trained in an end-to-end manner for level loss to be pluged-in readily.
Our models on single image denoising are based on the SOTA restoration model
NAFNet[11], which can be scaled flexibly from 1.1 GMACs to 65 GMACs. To
balance the running time and the performance, we adopt NAFNet with width 16/32
and number of blocks 8. We also conduct experiments on NAFNet base model with
width 32 and number of blocks 36 to verify the effectiveness of RCD on
relatively larger model.
Alteration of Basic Model Only two adjustments of the base model are required
to support our proposed editable denoising. First, we alter the output channel
number of the base model ending layer from output image channel number to
$L\times$ output image channel number, $L$ is the number of the predefined
noise level. For example, when training noise level from 0 to 60, the uniform
noise level gap between each noise map is 5 and there are 13 of them in total
([0,5,10,…55,60]). Then we apply noise decorrelation on those fixed-level
noise maps generated by our model and also feed them to the AutoTune module.
Second, for your AutoTune module, we add an additional CNN layer to predict a
feature-map. After the adaptive average pooling layer and temperatured softmax
activation, we attain a series of weights to fuse the 13 noise maps as model-
suggested guidance to the user.
Model Variants Details. We calculate the parameters of NAFNet base model and
our NAFNet-RCD model. Compared with NAFNet, NAFNet-RCD only has alterations on
two CNN layers, which is negligible for normal-size networks. Specifically,
these additional parameters account for 0.03%, 3.60%, and 9.17% of total
parameters for model size 1) NAFNet-RCD: width 32, number of blocks 36, 2)
NAFNet-RCD-small: width 32, number of blocks 8, 3) NAFNet-RCD-tiny: width 16,
number of blocks 8. Please be noted that the number of additional parameters
is only 7.7K even though it accounts for 9.17% of light model NAFNet-RCD-tiny.
## Appendix C Additional Experiments
Results on other real-world datasets. We further evaluate our RCD models on
the PolyU [53] and Nam [40] benchmarks. Both the RCD and baseline models are
trained on SIDD real-world data. Table 7 shows that on both benchmarks, the
RCD models can still perform controllable denoising without sacrificing much
performance, and on Nam, the RCD models even slightly outperform their
uncontrollable baselines.
Table 7: Image denoising results on PolyU and Nam. PolyU: results on real-
world PolyU test sets. Nam: results on real-world Nam test set.
Method | PolyU | Nam
---|---|---
PSNR | SSIM | PSNR | SSIM
NAFNet-tiny | 38.52 | 0.9827 | 38.93 | 0.9881
NAFNet-RCD-tiny | 38.36 | 0.9826 | 39.03 | 0.9881
NAFNet | 39.11 | 0.9837 | 39.54 | 0.9894
NAFNet-RCD | 39.07 | 0.9837 | 39.67 | 0.9896
Results on more architectures. We also test our RCD using the Restormer [56]
model. As shown in Table 8, Restormer-RCD slightly outperforms its baseline,
consistent with NAFNet.
Table 8: Restormer results on SIDD test set with additive Gaussian noise
($\sigma$ from 0 to 50 ).
Method | SIDD Synthetic noise
---|---
PSNR | SSIM
Restormer | 41.31 | 0.9763
Restormer-RCD | 41.79 | 0.9781
|
Connecting The Dots / Intelligent Trackers 2017
11institutetext: UC San Diego, La Jolla, CA, USA 92093 22institutetext:
Princeton University, Princeton, NJ, USA 08544 33institutetext: Cornell
University, Ithaca, NY, USA 14853 44institutetext: Fermilab, Batavia, IL, USA
60510-5011
# Parallelized Kalman-Filter-Based Reconstruction of Particle Tracks on Many-
Core Processors and GPUs
Giuseppe Cerati 44<EMAIL_ADDRESS>Peter Elmer 22<EMAIL_ADDRESS>Slava Krutelyov 11<EMAIL_ADDRESS>Steven Lantz 33
<EMAIL_ADDRESS>Matthieu Lefebvre 22<EMAIL_ADDRESS>Mario
Masciovecchio 11<EMAIL_ADDRESS>Kevin McDermott 33
<EMAIL_ADDRESS>Daniel Riley 33<EMAIL_ADDRESS>Matevž Tadel
11<EMAIL_ADDRESS>Peter Wittich 33<EMAIL_ADDRESS>Frank Würthwein
11<EMAIL_ADDRESS>Avi Yagil 11<EMAIL_ADDRESS>
###### Abstract
For over a decade now, physical and energy constraints have limited clock
speed improvements in commodity microprocessors. Instead, chipmakers have been
pushed into producing lower-power, multi-core processors such as Graphical
Processing Units (GPU), ARM CPUs, and Intel MICs. Broad-based efforts from
manufacturers and developers have been devoted to making these processors
user-friendly enough to perform general computations. However, extracting
performance from a larger number of cores, as well as specialized vector or
SIMD units, requires special care in algorithm design and code optimization.
One of the most computationally challenging problems in high-energy particle
experiments is finding and fitting the charged-particle tracks during event
reconstruction. This is expected to become by far the dominant problem at the
High-Luminosity Large Hadron Collider (HL-LHC), for example. Today the most
common track finding methods are those based on the Kalman filter. Experience
with Kalman techniques on real tracking detector systems has shown that they
are robust and provide high physics performance. This is why they are
currently in use at the LHC, both in the trigger and offline. Previously we
reported on the significant parallel speedups that resulted from our
investigations to adapt Kalman filters to track fitting and track building on
Intel Xeon and Xeon Phi. Here, we discuss our progresses toward the
understanding of these processors and the new developments to port the Kalman
filter to NVIDIA GPUs.
## 1 Introduction
The Large Hadron Collider (LHC) at CERN is the highest energy collider ever
constructed. It accelerates two counter-circulating proton beams and brings
them to collision in four locations around a 27 kilometer ring straddling the
border between Switzerland and France. By several measures it is the largest
man-made scientific device on the planet. The goal of the LHC is to probe the
basic building blocks of matter and their interactions. In 2012, the Higgs
boson was discovered by the CMS and ATLAS collaborations cmshiggs ; atlashiggs
. By measuring the energy and momentum of particles produced by the collision,
we can infer the existence of massive particles that were created and measure
their properties. This process is known as event reconstruction and consists
of integrating information from different detector components. Track
reconstruction, also known as tracking, is one step in event reconstruction
and determines the trajectories of charged particles (“tracks”) from a set of
positions of energy deposits within the various layers of our detectors
(“hits”). Tracking, in comparison to other aspects of event reconstruction, is
the most computationally time consuming, the most sensitive to increased
activity in the detector, and traditionally, the least amenable to
parallelized processing. The speed of online reconstruction has a direct
impact on how much data can be stored from the 40 MHz collisions rate, while
the speed on the offline reconstruction limits how much data can be processed
for physics analyses. This research is aimed at vastly speeding up tracking.
The large amount of time spent in tracking will become even more important in
the high-luminosity era of the LHC. The increase in event rate will lead to an
increase in detector occupancy (“pile-up”, PU), leading to an exponential gain
in the time required to perform track reconstruction, as can be seen in Figure
1 vertex . In the Figure, PU25 corresponds to the data taken during 2012, and
PU140 corresponds to the low end of estimates for the HL-LHC era. Clearly our
research on tracking performance will become increasingly important during
this era.
Figure 1: CPU time per event versus instantaneous luminosity, for both full
reconstruction and the dominant tracking portion. PU25 corresponds to the data
taken during 2012, and PU140 corresponds to the HL-LHC era. The time of the
reconstruction is dominated by track reconstruction vertex .
Accommodating this increasing need for computational power is complicated by
the change in computing architectures during the last decade. Around 2005, the
computing processor market reached a turning point: power density limitations
in chips ended the long-time trend of ever-increasing clock speeds, and our
applications no longer immediately run exponentially faster on subsequent
generations of processors. This is true even though the underlying transistor
count continues to increase per Moore’s law. Increases in processing speed
such as those required by the time increases in Figure 1 will no longer come
‘for free’ from faster processors. New, parallel processors instead are
aggregates of ‘small cores’ that in total still show large performance gains
from generation to generation, but their usage requires reworking of our
mostly serial software to exploit these gains. The processors in question
include ARM, Graphical Processing Units (GPU) and the Intel Xeon and Xeon Phi.
In what follows, we first update the reader about the algorithms we target and
the performance level we now obtain on Xeon and Xeon Phi. We then explain the
challenges and the steps we took in porting these algorithms to GPUs along
with some initial performance assessments.
## 2 Kalman Filter Tracking
The algorithm we are targeting for parallelized processing is a Kalman Filter
(KF) based algorithm Fruhwirth:1987fm . KF-based tracking algorithms are
widely used since they are fast and incorporate estimates of multiple
scattering in the detector material directly into the trajectory of the
particle. In addition, a combinatorial KF provides an elegant way to resolve
ambiguities when considering multiple hits to add to a track billoir ;
Mankel1997 . Other track finding algorithms, more naturally suited to
parallelization and coming from the image processing community, have been
explored by others. These include Hough Transforms and Cellular Automata,
among others (see, for instance, HALYO1 ). However, these are not the main
algorithms in use at the LHC today, whereas there is extensive understanding
on how KF algorithms perform. KF algorithms have proven to be robust and
perform well in the difficult experimental environment of the LHC vertex .
Rather than abandon the collective knowledge gained from KF algorithms, we
wish to extend this robust tool by porting KF tracking to parallel
architectures.
KF tracking proceeds in three main stages: seeding, building, and fitting.
Seeding provides the initial estimate of the track parameters based on a few
hits in a subset of layers. Realistic seeding is currently under development
and will not be reported here. Building then collects additional hits in other
detector layers to form a complete track, using the combinatorial KF as a
basis for deciding which hits to consider and keep. Track building is by far
the most time consuming step of tracking, as it requires branching to explore
potentially more than one track candidate per seed after finding compatible
hits on a given layer. After hits have been assigned to tracks in each layer,
a final fit with a KF and smoother can be performed over each track to provide
the best estimate of its parameters and to compute a measure of the track
quality in terms of a $\chi^{2}$.
To realize performance gains, we need to exploit two types of parallelism:
vectorization and parallelization. Vectorization aims to perform a single
instruction on multiple data at the same time by performing the same operation
across different data in lockstep. Parallelization aims to perform different
instructions at the same time on different data. In this work, parallelization
is done by distributing the workload between threads, which are concurrent
lightweight processes sharing resources (for instance, memory). The challenge
to vectorization is that KF tracking may branch to explore multiple candidates
per seed, interfering with the lockstep synchronization required for
vectorization performance. The challenge to parallelization is that hit
occupancy in a detector is not uniformly distributed on a per event basis,
creating the potential for workload imbalances across threads. For these and
other reasons, KF tracking cannot be ported in a straightforward way to run in
parallel on many-core processors. Past work by our group has shown progress in
porting sub-stages of KF tracking to support parallelism in simplified
detectors (see, e.g. our presentations at ACAT2014 acat2014 , CHEP2015
chep2015 , and NSS-MIC2015 nss2015 ). As the hit collection is completely
determined after track building, track fitting can repeatedly apply the KF
algorithm without branching, making this the ideal place to start in porting
KF tracking to Xeon and Xeon Phi, with our first results shown at ACAT2014
acat2014 . In more recent works (CHEP2016 chep2016 , CtD2016 ctd2016 ), we
discussed the improvements made to target a more realistic geometry. Profiling
and recent computational optimizations were also outlined.
### 2.1 Optimized Matrix Library Matriplex
The computational problem of KF-based tracking consists of a sequence of
matrix operations. Unlike Fruhwirth:1987fm , we parameterize the measurement
state and the track state in global coordinates with matrices of size
$N\times{}N=3\times{}3$ and $6\times{}6$, respectively. To allow maximum
flexibility for exploring SIMD operations involving many small-dimensional
matrices, and to decouple the specific matrix computations from the high level
algorithm, we have developed a matrix library, Matriplex, which relies on a
matrix-major memory layout (a layout where elements with the same index but
from different matrices are stored consecutively). A more complete description
of the Matriplex library can be found in the work we presented at ACAT2014
acat2014 . The adaptation of this data structure to GPUs will be discussed in
section 4.
## 3 Track Building on Xeon and Xeon Phi
Combinatorial KF track building shares the same core KF calculations as track
fitting, but has two major complications for utilizing vectorizable operations
compared to track fitting. The first such problem is the “nHits” problem: when
performing the KF operations, the hit set for a given track candidate in
building is undefined and one potentially has to choose between
$\mathcal{O}{(10^{4})}$ hits per layer. The second problem is the
“combinatorial” problem: when more than one hit on a given layer is compatible
with a given input candidate, branching occurs to explore all possible track
candidates up to some cutoff number of candidates per seed.
The key to reducing the nHits problem is to partition the tracks and hits
spatially in order to reduce the search space for a given track. We partition
space without reference to the detector structure, placing hits in each layer
into directionally projective bins. The bins provide a fast look-up of
compatible hits in a layer for a given track given the estimate from the KF.
With regard to the combinatorial problem, we first developed track building to
add only the best hit out of all compatible hits on a layer for each track
candidate. By definition, then, each seed only produces one track and does not
require copying of track candidates to explore multiple hits per layer, as in
the full combinatorial KF approach. The best hit is defined as the hit that
produced the lowest $\chi{}^{2}$ increment to the track candidate. The
vectorization and parallelization performance of this “best hit” approach were
presented at CHEP2015 chep2015 . After demonstrating the feasibility of track
building in the best hit approach, we then allowed for more than one track
candidate per seed per layer to be produced, as in the full combinatorial KF
approach and presented the results at CtD2016 ctd2016 .
### 3.1 Memory Management
With the full combinatorial approach in place, we performed extensive studies
of the performance of our software, in terms of both the physics performance
and the code performance. For the latter, we used the Intel VTune 2016 vtune
suite of tools to identify bottlenecks and understand the effects of our
optimization attempts. In particular, as can be expected, we determined that
memory management is of critical importance.
In the full combinatorial approach, if multiple compatible hits are found for
a track candidate, it becomes necessary to make copies of that track for each
compatible hit. To mitigate the impact from this serial work, we moved copying
outside of all vectorizable operations into what we term the “clone engine”.
The clone engine approach only copies the best N track candidates per seed
after reading and sorting a bookkeeping list of all compatible hits. This is
in contrast to a first attempt at the combinatorial KF which copied a
candidate each time a hit was deemed compatible, then sorting and keeping only
the best N candidates per seed after all the possible hits on a given layer
for all input candidates were explored. A more detailed discussion of this
work on memory management and impacts on performance was presented at NSS-
MIC2015 nss2015 and at CHEP 2016 chep2016
### 3.2 Results
We present here the latest vectorization and parallelization benchmarks in
track building since CHEP 2016. Results are given in average time per event.
For each event, building proceeds from $10,000$ seeds that are derived from
the first three hits of each simulated track in the event. We used a
simplified detector geometry consisting of equally spaced, fixed length
concentric cylinders. Tracks are simulated using a set of in-house simulation
routines replicating the detector resolution via Gaussian smearing of the hit
positions. Figure 2 contains two plots displaying the performance obtained on
a computer with two Sandy Bridge multi-core processors (E5-2620 $@$ 2.00GHz).
Plot 2(a) shows the vectorization performance of three different track
building approaches, as functions of the number of vector units enabled while
using a single thread. Plot 2(b) shows the parallelization performance of the
same track building approaches, as functions of the number of threads enabled
using Intel Thread Building Blocks (TBB) tbb_website . The Xeon machine we are
using has 12 physical cores which appear to be 24 logical cores to the OS due
to hyperthreading being enabled. The slight deviation from ideal scaling
before hyperthreading arises from load balancing issues between threads. The
large deviation from ideal scaling after enabling hyperthreading is due to
resource limitations: the two threads per core are contending for the same
instruction pipelines and data caches. Even so, nearly eightfold speedup is
seen for 21 threads.
(a) Impact of vector size on performance. Only one thread is enabled.
(b) Impact of the number of threads on performance. The full vector size (8
floats) is enabled.
Figure 2: Average time per event of $10,000$ tracks on Xeon (Sandy Bridge
E5-2620) processors. Both plots show results for three different track
building approaches. The blue curve is the “best hit” tracking approach. The
green and red curves use the full combinatorial approach in track building.
The red curve moves the copying of tracks outside of the vectorizable
operations.
(a) Impact of vector size on performance. Only one thread is enabled.
(b) Impact of the number of threads on performance. The full vector size (16
floating points) is enabled.
Figure 3: Average time per event of $10,000$ tracks on Xeon Phi (Knights
Corner 7120P) coprocessors. Both plots show results for three different track
building approaches. The blue curve is the “best hit” tracking approach. The
green and red curves use the full combinatorial approach in track building.
The red curve moves the copying of tracks outside of the vectorizable
operations.
The blue curve is the “best hit” approach described previously. Naturally,
this approach will have the lowest absolute time in comparison to the two
fully combinatorial approaches, the red and green curves. The green curve is
the original approach to combinatorial track finding where the copying of the
track candidates is inside the vectorizable KF operations. The red curve shows
the clone engine approach, which moves the serial work of copying outside of
the KF vectorizable operations. As expected, the clone engine approach has a
lower absolute time and higher speedup than the original. It is important to
note that while the best hit approach is the fastest, the physics performance
has inefficiencies in hit finding and track finding from not being fully able
to explore multiple track candidates per seed. Even in our simplified model,
this behavior is already apparent, and the best hit approach is expected to
become even more inefficient when using more realistic detector effects. For
reference, on this simplified problem, the best hit approach approximately has
as an efficiency of 93% and a fake rate of 3%, while the combinatorial
approaches have efficiencies over 99% and fake rates under 1%. (Candidates
matching fewer than 7 hits of a simulated track count as neither successes nor
fakes.)
Figure 3 shows the same plots as Figure 2, now with Xeon Phi (Knights Corner
7120P). Figure 3(a) shows the vectorization results with Xeon Phi which has
AVX-512 vector registers that are twice as wide as those in Xeon. With one
thread enabled, Xeon Phi sees the same 1.5$\times{}$ to 2$\times{}$ speedup as
Xeon. As seen on Xeon, the best hit approach still has the lowest absolute
timing. Figure 3(b) displays the parallelization performance, with Xeon Phi,
with the full vector width of 16 floats enabled using TBB. The Xeon Phi we are
using has 61 physical cores requiring 122 independent instruction streams for
full utilization, due to the fact that the Xeon Phi issues instructions for a
given thread every other clock cycle. Therefore, to keep a given physical core
busy every clock cycle, one has to schedule two threads per core alternating
in instruction execution. A form of hyperthreading is also present on Xeon
Phi, yielding a total of 244 hardware threads (logical cores). Overall, a
factor of about 30$\times{}$ speedup is observed.
## 4 Porting KF based algorithms to the GPU
The challenges to port the Kalman Filter to the GPU are similar in principle
to those on x86 multi-core platforms. The parallelization and vectorization
issues we encounter are only exacerbated by the fine grain parallelism
required by graphical processors. In particular, the large core count of GPUs
and their thread scheduling policies force a large number of threads to be
concurrently scheduled. It also implies that coping with branching is
critical, as threads of a warp (a group of 32 consecutive threads) execute the
same instruction at the same time.
We have followed the same incremental strategy to move onto GPU architectures
as the one we adopted for Xeon and Xeon Phi architectures. We first started by
studying and choosing a data structure adapted to operations on many small
matrices. We then ported our algorithms, starting from track fitting: its
simpler nature made it easier to understand the requirements of porting KF
based algorithms to GPUs. Armed with this knowledge, we made a first stab at
porting track building algorithms. The best hit approach —where only the
candidate yielding the best $\chi^{2}$ is considered at each layer, for each
seed —is the natural one to develop, as it introduces a limited number of new
problems. The main issue is to understand how to deal with numerous memory
indirections. Pursuing again the same stepwise improvements as those done for
x86 processors, we next looked at the combinatorial algorithm, where multiple
candidates per seed are considered at each layer. This introduces the
additional problem of branching.
### 4.1 Memory Layout
Matriplex has been found to be the most efficient memory layout for GPU, as it
naturally offers coalescent accesses. Coalescent accesses occur when, in the
case of 4-byte elements, 32 consecutive threads request 32 consecutive
elements from global memory. In conjunction with aligned memory accesses, it
has the desirable effect of allowing for a single memory transaction instead
of 32 separate ones. As a global memory (the large off-chip memory) request
takes between 200 and 400 clock cycles to complete, this leads to significant
gains. In general, poor performance can be expected unless memory accesses are
coalescent. A key difference with CPU Matriplex is the number of matrices
grouped within such a data structure. Instead of being tailored to the
dimension of a hardware vector unit, it contains as many matrices as possible
in order to increase the parallelism exposed to the device. Figure 4 shows
some results of a study we conducted to determine the best memory layout,
considering that a large part of the KF operations involves multiplying small
matrices. In all cases, shared memory is used to reduce the number of global
memory transactions, and this has a dramatic impact on the performance, as
much as two orders of magnitude. By using the Matriplex data structure, we are
able to outperform the finely tuned and batched cuBLAS approach. As a results,
all our subsequent work has been based on the GPU version of the Matriplex
data structure.
Figure 4: Performance comparison of different memory organizations for
multiplying $6\times 6$ matrices. It shows the time as a function of the
number of $6\times 6$ matrix multiplications. The orange and purple lines use
an array of matrices, while the green line uses the Matriplex data structure.
### 4.2 Track Fitting on the GPU
As mentioned, track fitting is suitable for developing a better understanding
of the requirements to obtain performance on an unexplored hardware
architecture, such as GPUs. Porting from C++ to CUDA is made tedious by the
lack of a library matching the C++ STL and usable from device code. The
existing options (for example NVIDIA Thrust nvidia_thrust ) are higher level
abstractions to be used from the host and have to be casted to raw pointers
when passed to device kernels. The process of optimizing the code so that it
runs fast enough is often orders of magnitude harder than the process of
rewriting to CUDA. In what follows, we choose the most straightforward way to
map computations to the GPU by using one GPU thread per track during fitting.
There is a number of places where optimization can occur. The most common
places are the kernels themselves (i.e. the CUDA routines) and the data
transfers between the host (CPU) and the device (GPU). We have observed that
optimizing kernels brings significant gains. The most beneficial kernel-level
optimizations were using read-only caches, merging kernels to limit the launch
overhead of many small kernels, and favoring registers over shared memory when
possible. These kernel optimizations resulted in a speedup of roughly 8 with
respect to a naive implementation. Reorganizing data, in particular to fill
Matriplex structures, is costly on the GPU because of the large number of
memory indirections it requires. For track fitting, we have chosen to let this
reorganization happen on the CPU, which is more suited to this task. By
launching kernels asynchronously inside streams, we are able to overlap these
reorganizations with actual computations. An additional lesson we learned is
the need to fill the GPU by fitting tracks of more than one event
concurrently. Indeed, one event is too small to completely fill the GPU by
itself. By using multiple CPU threads, each of them associated with a list of
events to process, significant improvements are obtained. Figure 5 shows the
impact of concurrent fitting. For small Matriplex sizes, the kernels perform
poorly due to a low compute intensity. For larger sizes, multiple CPU threads
are better able to fill the GPU, and performance improves by a factor of
roughly 3.
Figure 5: Concurrent fitting processes on the GPU. Different plots show
different number of CPU threads concurrently streaming events to the GPU. Time
is plotted as a function of the Matriplex size.
### 4.3 Track Building on the GPU
Track building is a more complex problem to deal with on multi- and many-core
architectures. We approach it with two different algorithms. The first one,
“best hit” considers a single candidate hit for each seed at each layer. The
second one is a combinatorial algorithm, keeping track of multiple candidates
for each seed at each layer, where candidates are ranked by the number of hits
found and by their $\chi^{2}$. Note that only the “clone engine” approach has
been attempted on GPUs: the other combinatorial algorithm involves too many
memory copies to be even considered.
Compared to track fitting, the “best hit” approach increases the complexity by
adding many memory indirections. For each track, hits within $\eta$ bins have
to be considered. This implies frequent reorganizations into Matriplex data
structures, making the overlapping of reorganizations and computations not
possible anymore. Reorganizing hits into Matriplex is consequently done on the
GPU. As in track fitting, the parallelization relies on having one GPU thread
per candidate. Modular and nested data structures can fragment data transfers,
even if they are asynchronous, and will in most cases have to be redesigned to
a shallower hierarchy. A data structure with multiple layers also forces the
inconvenience of having to use additional arrays to keep track of pointers to
GPU memory. As a side note, newer versions of CUDA offer unified memory as a
way to transparently access data that might be located either in the main RAM
or in the GPU global memory. The cost of such a convenience has been greatly
reduced over the past years, but the frequency of these transfers dictates a
more controlled approach, using the knowledge we have to pre-transfer data
asynchronously.
The combinatorial approach generates even more complications when aiming at
performant GPU computations. Having multiple candidates per seed at each layer
means that conditional branches are encountered frequently. Since warps always
schedule the same instruction at a given time, branches serialize the warp
execution when the condition differs for each thread of the warp. We address
this issue by defining a data structure to store information about multiple
candidates. While on x86 processors standard C++ vectors are used, on GPUs the
underlying choices are more limited and boil down to C-arrays. Our
implementation takes the form of a heap based approach, often used to
implement priority queues. Details are shown in Figure 6. A heap is a complete
binary tree and can be implemented as a 0-based array where an element at
index $i$ has children at indexes $2i+1$ and $2i+2$. This data structure uses
shared memory to avoid extraneous requests to global memory. At the beginning
of each layer, a heap is assigned to each track candidate to store the best
suited hits. Each heap is accessed by a single thread, and elements are
consecutive in the vertical direction to avoid bank conflicts. At the end of
each layer, new overall candidates for the best track need to be found by
sifting all the heaps. Figure 7 explains this sifting process. A number of
heaps are sorted and their best elements are push-popped into non-sorted
heaps. The process is repeated until a single heap remains. The last heap
contains information about the bests of all the candidates and is sorted in
order to proceed to the next layer. Initially, heaps are filled with guards,
allowing a different number of potential hits to be considerate per initial
track candidate. The number of potential hits is limited by the statically
defined heap size.
Figure 6: The diagram on the left represents the shared memory layout for
storing a seed’s candidates. Assuming a maximum of 4 candidates per seed, each
column represents a heap to store potential new hits for a track candidate.
Each different color represents a different initial track candidate of the
seed at a given layer. Numbers represent indexes of the heap’s elements.
Memory is consecutive in the vertical direction. The diagram on the right
shows an equivalent representation using binary trees. Figure 7: Reducing
candidates in shared memory for a single seed. At the end of each layer, the
new candidates computed from the starting 4 candidates are sifted to a new set
of 4 candidates. $\infty$ signs represent guards for the heap based algorithm.
Numbers are $\chi^{2}$ values chosen to be illustrative.
A large part of the development effort to port track building to GPUs has been
to bring the performance to a level comparable to the one observed on the
extensively tuned Xeon version. Results for the “best hit” approach to track
building are shown on Figure 8(a). Considering the computational time as well
as the data transfer, the GPU version is about 3 times faster than a single-
threaded execution on a Sandy Bridge (SNB) processor but about 4 times slower
than an execution using 24 hyperthreads. The performance comparison is even
more severe for the combinatorial version, as shown in Figure 8(b). It is a
consequence of the large number of synchronizations and branch predictions.
(a) Performance comparison for the “best hit” approach.
(b) Performance comparison for the combinatoric “clone engine” approach.
Figure 8: Track building performance comparison between the GPU version and
the optimized Xeon version. The GPU is a NVIDIA K40 and the Xeon a Sandy
Bridge E5-2620
Several future developments can improve the performance of the GPU code.
First, concurrent events should be adapted to track building because this will
help to fill the GPU’s stream multiprocessors more efficiently, as we have
seen for track fitting. Next, alternative parallelization patterns need to be
studied to completely understand what are the most important performance-
driving factors. For instance, using one GPU thread per seed may reduce the
amount of synchronization required between threads, but will make concurrent
event processing even more critical. Finally, data transfers need to be
drastically improved as they now account for almost half of the track building
wall time.
## 5 Conclusion and Outlook
We have made significant progress in parallelized and vectorized Kalman
Filter-based tracking on multi- and many-core processors. On Xeon and Xeon
Phi, improvements have been made to include a more realistic geometry. These
changes were not discussed here as they are not reflected on the GPU side,
yet. On all platforms many challenges remain to fully exploit the
computational power of these highly parallel architectures. For instance, on
the GPU, the bottlenecks we will have to address next are data transfers
between the host and the device, as well as branching and synchronization
within kernels. The project has produced promising results; however, much work
remains.
## 6 Acknowledgment
This work is supported by the U.S. National Science Foundation, under the
grants PHY-1520969, PHY-1521042, PHY-1520942 and PHY-1120138.
## References
* (1) CMS Collaboration, Physics Letters B716, 30 (2012)
* (2) ATLAS Collaboration, Physics Letters B716, 1 (2012)
* (3) G. Cerati, Proceedings of Science VERTEX2014, 037 (2015)
* (4) R. Fruehwirth, Nuclear Instrumentation Methods A 262, 444 (1987)
* (5) P. Billoir, Computer Physics Communications 225, 352 (1984)
* (6) R. Mankel, Nuclear Instruments and Methods in Physics Research A 395, 169 (1997)
* (7) V. Haylo, P. LeGresley, P. Lujan, V. Karpusenko, and A. Vladimirov, Journal of Instrumentation 9, P04005 (2014)
* (8) G. Cerati _et al._ , Journal of Physics: Conference Series 608, 012057 (2014)
* (9) G. Cerati _et al._ , Journal of Physics: Conference Series 664, 072008 (2015)
* (10) G. Cerati _et al._ , “Kalman-Filter-Based Particle Tracking on Parallel Architectures at Hadron Colliders” (2015), NSS-MIC
* (11) G. Cerati _et al._ , Journal of Physics: Conference Series (2016), https://arxiv.org/abs/1702.06359
* (12) G. Cerati _et al._ , EPJ Web Conf. 127, 00010 (2016)
* (13) _Intel VTune Amplifier_ , https://software.intel.com/en-us/intel-vtune-amplifier-xe
* (14) _Intel Thread Building Blocks_ , https://www.threadingbuildingblocks.org/
* (15) _NVIDIA Thrust_ , https://developer.nvidia.com/thrust
|
| Rapidly convergent coupled-cluster Monte Carlo using a Chebyshev projector
---|---
| Zijun Zhao,∗a‡ Maria-Andreea Filip,a§ and Alex J W Thoma
| The multi-reference coupled-cluster Monte Carlo (MR-CCMC) algorithm is a
determinant-based quantum Monte Carlo (QMC) algorithm that is conceptually
similar to Full Configuration Interaction QMC (FCIQMC). It has been shown to
offer a balanced treatment of both static and dynamic correlation while
retaining polynomial scaling, although scaling to large systems with
significant strong correlation remained impractical. In this paper, we
document recent algorithmic advances that enable rapid convergence and a more
black-box approach to the multi-reference problem. These include a
logarithmically scaling metric-tree based excitation acceptance algorithm to
search for determinants connected to the model space at the desired excitation
level and a symmetry-screening procedure for the reference space. We show
that, for moderately sized model spaces, the new search algorithm brings about
an approximately 8-fold acceleration of one MR-CCMC iteration, while the
symmetry screening procedure reduces the number of active model space
determinants at essentially no loss of accuracy. We also introduce a
stochastic implementation of an approximate wall projector, which is the
infinite imaginary time limit of the exponential projector, using a truncated
expansion of the wall function in Chebyshev polynomials. We show that this
wall-Chebyshev projector requires significantly fewer applications of the
Hamiltonian to achieve the same statistical convergence. We have applied these
methods to calculate the binding curve of the beryllium and carbon dimers with
large active spaces, obtaining binding curves that are close to FCIQMC
quality. Notably, the wall-Chebyshev projector presented here can be applied
to accelerate any projector-based QMC algorithm. This paper represents a
significant step in the algorithmic maturation of the MR-CCMC algorithm and
paves the way for large-scale calculations.
##
††footnotetext: a Yusuf Hamied Department of Chemistry, University of
Cambridge, Cambridge, UK. E-mail<EMAIL_ADDRESS>‡ Present
address: Department of Chemistry, Emory University, Atlanta, GA, USA.
††footnotetext: § Present address: Max Planck Institute for Solid State
Research, Stuttgart, Germany.
## 1 Introduction
Quantum Monte Carlo (QMC) methods have long provided a powerful alternative to
conventional electronic structure methods, by generating high accuracy results
at a fraction of the cost of standard approaches. The combination of
Variational Monte Carlo (VMC)1, 2 and Diffusion Monte Carlo (DMC)3, 4, 5 has
become a significant benchmarking approach in many areas of electronic
structure,6, 7, 8, 9, 10, but it is limited by the need to provide some
approximate nodal structure to avoid collapse onto bosonic solutions.
Fermionic Monte Carlo methods11, 12 have since been developed which act
directly in the anti-symmetrised Hilbert space of the electronic structure
problem, thereby removing the potential for bosonic solutions a priori.
First introduced in 2009 by Booth et al.11, full configuration interaction
quantum Monte Carlo (FCIQMC) can be variously described as a stochastic power
iteration algorithm or an iterative solution to the imaginary time
Schrödinger’s equation.
Here, we give a brief summary of theoretical underpinnings of FCIQMC by taking
the latter view. By applying a Wick rotation 13, $\tau\leftarrow it,\
\tau\in\mathbb{R}$, to the time-dependent Schrödinger’s equation
${i\dot{\Psi}=\hat{H}\Psi}$, one obtains the imaginary time Schrödinger’s
equation:
$\diffp{}{\tau}|\Psi(\tau)\rangle=-\hat{H}|\Psi(\tau)\rangle,\
\tau\in\mathbb{R}.$ (1)
It can be formally integrated to give
$|\Psi(\tau)\rangle=e^{-\tau(\hat{H}-S)}|\Psi(0)\rangle,$ (2)
with $S$ being the constant of integration, also known as the ‘shift’.
The reference wavefunction $|\Psi(0)\rangle$, commonly a Hartree–Fock (HF)
solution, can be expanded in the eigenbasis of the full Hamiltonian,
$\\{|\Psi_{i}^{\mathrm{FCI}}\rangle\\}$, leading to
$|\Psi(\tau)\rangle=\sum_{i}e^{-\tau(E_{i}-S)}c_{i}|\Psi_{i}^{\mathrm{FCI}}\rangle,$
(3)
with $\\{E_{i}\\}$ being the eigenspectrum of the full Hamiltonian and
$|\Psi(0)\rangle=c_{i}|\Psi_{i}^{\mathrm{FCI}}\rangle$. We can see that, if
$S=E_{0}$, in the limit of $\tau\rightarrow\infty$, we obtain the ground state
of the full Hamiltonian.
By discretising the projector in Equation 2 and further applying the first-
order Taylor series expansion, we obtain the ‘master equation’ of FCIQMC:
$|\Psi(\tau+\delta\tau)\rangle=[1-\delta\tau(\hat{H}-S)]|\Psi(\tau)\rangle.$
(4)
This equation can be projected onto the different determinants in the Hilbert
space to give
$\langle D_{\bm{\mathrm{i}}}|\Psi(\tau+\delta\tau)\rangle=\langle
D_{\bm{\mathrm{i}}}|\Psi(\tau)\rangle-\delta\tau\langle
D_{\bm{\mathrm{i}}}|\hat{H}-S|\Psi(\tau)\rangle,$ (5)
which gives an update equation for the corresponding FCI parameters
$c_{\bm{\mathrm{i}}}$, where
$|\Psi(\tau)\rangle=\sum_{\bm{\mathrm{i}}}c_{\bm{\mathrm{i}}}(\tau)|D_{\bm{\mathrm{i}}}\rangle$:
$c_{\bm{\mathrm{i}}}(\tau+\delta\tau)=c_{\bm{\mathrm{i}}}(\tau)-\delta\tau[(H_{\bm{\mathrm{ii}}}-S)c_{\bm{\mathrm{i}}}(\tau)+\sum_{\bm{\mathrm{j}}\neq\bm{\mathrm{i}}}H_{\bm{\mathrm{ij}}}c_{\bm{\mathrm{j}}}(\tau)],$
(6)
where $H_{\bm{\mathrm{ij}}}=\langle
D_{\bm{\mathrm{i}}}|\hat{H}|D_{\bm{\mathrm{j}}}\rangle$. This equation can be
viewed as describing the population dynamics of particles placed on the
different determinants and may be modelled by a stochastic process composed of
three steps:11
* •
Spawning: Given determinant $D_{\bm{\mathrm{i}}}$, generate new particles on
determinant $D_{\bm{\mathrm{j}}}$ with probability ${p\propto\delta\tau
H_{\bm{\mathrm{ij}}}c_{\bm{\mathrm{j}}}(\tau)}$.
* •
Death: Given determinant $D_{\bm{\mathrm{i}}}$, generate new particles on
determinant $D_{\bm{\mathrm{i}}}$ with probability
${p\propto\delta\tau(H_{\bm{\mathrm{ii}}}-S)c_{\bm{\mathrm{i}}}(\tau)}$
* •
Annihilation: For a given determinant, cancel out particles carrying opposite
signs.
The formulation of CCMC closely matches that of FCIQMC, with the difference
that instead of residing on determinants, walkers reside on excitors,
$\hat{a}_{\bm{\mathrm{n}}}$, defined as
$\hat{a}_{\bm{\mathrm{n}}}|D_{\bm{\mathrm{0}}}\rangle=\pm|D_{\bm{\mathrm{n}}}\rangle$.
Replacing the FCI wavefunction by the coupled cluster ansatz in Equation 4 and
left-multiplying by $\langle D_{\bm{\mathrm{i}}}|$ gives
$\begin{split}\langle
D_{\bm{\mathrm{i}}}|\Psi^{\mathrm{CC}}(\tau+\delta\tau)\rangle=&\langle
D_{\bm{\mathrm{i}}}|\Psi^{\mathrm{CC}}(\tau)\rangle\\\ -&\delta\tau\langle
D_{\bm{\mathrm{i}}}|(\hat{H}-S)|\Psi^{\mathrm{CC}}(\tau)\rangle\end{split}$
(7)
The coupled cluster ansatz parametrises the wavefunction with cluster
amplitudes in a non-linear fashion. The mapping of CI coefficients to cluster
amplitudes can be done by a simple projection, which reveals contributions
from multiple clusters. For example in a CCSD wavefunction (i.e.,
${\hat{T}}={\hat{T}}_{1}+{\hat{T}}_{2}$)
$\langle
D_{ij}^{ab}|e^{{\hat{T}}}D_{\bm{\mathrm{0}}}\rangle=t_{ij}^{ab}+t_{i}^{a}t_{j}^{b}-t_{i}^{b}t_{j}^{a}$
(8)
with the negative sign arising from the fact that
${\hat{a}_{b}^{\dagger}\hat{a}_{i}\hat{a}_{a}^{\dagger}\hat{a}_{j}=-\hat{a}_{a}^{\dagger}\hat{a}_{i}\hat{a}_{b}^{\dagger}\hat{a}_{j}}$,
due to the anti-commutation relations of the second-quantised creation and
annihilation operators 14. Terms like $t_{ij}^{ab}$ are known as non-composite
cluster amplitudes, and the rest as composite cluster amplitudes. Here we make
the approximation that composite clusters have much smaller contributions than
non-composite ones, their changes will be negligible per time step, and hence
we can cancel out the $\mathcal{O}(\hat{T}^{2})$ contributions on both sides
to write
$t_{\bm{\mathrm{i}}}(\tau+\delta\tau)\rangle=t_{\bm{\mathrm{i}}}(\tau)-\delta\tau\langle
D_{\bm{\mathrm{i}}}|(\hat{H}-S)|\Psi^{\mathrm{CC}}(\tau)\rangle$ (9)
Compared to FCIQMC, an additional step needs to be performed for each Monte
Carlo iteration: the sampling of the exponential ansatz. For $N_{\mathrm{ex}}$
total walkers, also called excips in CCMC, $\mathcal{O}(N_{\mathrm{ex}})$
clusters are formed randomly by combining present excitors according to
specific biasing rules 15.
Finally, the intermediate normalisation 16 of the wavefunction is redefined to
give the CCMC ansatz:
$|\Psi_{\text{CCMC}}\rangle=N_{0}e^{{\hat{T}}/N_{0}}|D_{\bm{\mathrm{0}}}\rangle,$
(10)
which introduces the reference population as a new independent variable,
solving the problem that Equation 7 does not lead to a viable update equation
for $\langle D_{\bm{\mathrm{i}}}|=\langle D_{\bm{\mathrm{0}}}|$.
A multi-reference formulation of the CCMC algorithm (MR-CCMC) 17 has been
implemented, retaining a single-reference formalism, in common with the so-
called SRMRCC methods in Ref. 18. The flexibility of the CCMC algorithm allows
this multireference approach to be implemented with minimal code changes to
the single-reference algorithm, bypassing what would be a challenging process
in deterministic methods. Essentially, for a coupled cluster truncation level
$P$, the algorithm allows any number of determinants to become a ‘secondary
reference’, stores excitors that are within $P$ excitations from any
references (instead of just the HF determinant), and allows clusters to form
that are within $P+2$ excitations from any reference. The set of references is
commonly known as the model or reference space. To summarise, the algorithmic
modifications relative to single-reference CCMC are:
* •
Store all the secondary references in some searchable data structure, and
additionally store the highest excitation level from the reference determinant
among the secondary references, $k_{\text{max}}$.
* •
Cluster expansion: allow clusters with excitation level of up to
$k_{\text{max}}+P+2$ to form, instead of $P+2$ in the single-reference case.
Discard those that are not $P+2$ excitations away from some reference
determinant.
* •
Spawning: for a randomly generated spawnee (i.e., $\langle D_{\bm{j}}|$),
check that it is within $P$ excitations of any secondary references.
* •
Cloning/death: allow death on excitors that are within $P$ excitations from
any secondary references.
While this MR-CCMC method can treat systems which conventional single-
reference CC struggles with, this comes at an increased computational cost.
Comparing contributor excitation levels to all references becomes expensive as
the number of references grows, particularly when the contributor turns out to
lie outside of the desired space. Therefore, non-trivial computational effort
is expended on attempts that will not contribute to the overall estimators and
propagation, while also making successful steps more expensive than their
single-reference equivalents. In the rest of this paper, we will first
introduce the wall-Chebyshev projector, and show that it can be applied to
(mr-)CCMC and FCIQMC to reduce the number of times the Hamiltonian needs to be
applied to reach statistical convergence, thereby reducing the computational
cost. We will then introduce a suite of further modifications to the MR-CCMC
algorithm that accelerates the handling of the model space. We apply the
resulting algorithm to several benchmark systems to show that it offers a way
for mr-CCMC to become a competitive alternative to FCIQMC even in the strong
correlation regime.
## 2 The wall-Chebyshev projector
### 2.1 Motivation and theory
In all projector-based QMC methods, such as FCIQMC and CCMC, the linear
projector (Equation 4) is used. The first-order Taylor expansion turns out to
be a very reasonable approximation, since we demonstrate in Appendix A that
there is no benefit whatsoever in going to higher orders of the Taylor
expansion of the exponential projector. However, this does not mean that one
cannot devise more efficient projectors. An example is a projector based on a
Chebyshev expansion of the wall function, which was first proposed in Ref. 19
in the context of a deterministic projector-based selected CI algorithm.
The wall function is given by
${\text{wall}}(x)=\begin{cases}\infty,\ x<0\\\ 1,\ x=0\\\ 0,\ x>0\end{cases},$
(11)
and is physically motivated as the infinite time limit of the exponential
projector:
$\lim_{\tau\rightarrow\infty}e^{-\tau(x-S)}=\text{wall}(x-S),$ (12)
which can map any trial wave function $|\Phi_{0}\rangle$ to the exact ground
state $|\Psi_{0}\rangle$, if $\langle\Phi_{0}|\Psi_{0}\rangle\neq 0$ and
$E_{0}\leq S<E_{1}$.
While a Taylor series expansion does not exist for the discontinuous wall
function, an expansion in Chebyshev polynomials, like a Fourier expansion, is
trivial. The Chebyshev polynomials of the first kind, defined as
$T_{n}(\cos(\theta))=\cos(n\theta)$, form an orthogonal basis (with metric
$(1-x^{2})^{-1/2}$) for functions defined over $x\in[-1,1]$:
$\int^{1}_{-1}T_{m}(x)T_{n}(x)(1-x^{2})^{-1/2}\dl
x=\delta_{mn}\pi/(2-\delta_{m0}).$ (13)
To facilitate the following discussion, we define the spectral range, $R$, of
a Hamiltonian as $R=E_{N-1}-E_{0}$, where $E_{i}$ is the $i$-th eigenvalue of
the full Hamiltonian and $N$ is the size of the Hilbert space. Furthermore,
our energy range $\epsilon\in[E_{0},E_{N-1}]$ requires the application of an
affine transformation to the Chebyshev polynomials
$\epsilon=E_{0}+\frac{R}{2}(x+1),\ x\in[-1,1].$ (14)
We show in Appendix B that the $m$-th order Chebyshev expansion of the wall
function is
$g^{\text{wall-
Ch}}_{m}(\epsilon)=\frac{1}{1+2m}\sum_{k=0}^{m}(2-\delta_{k0})T_{k}(-x).$ (15)
For illustration purposes, we plot several orders of Chebyshev expansion in
Figure 1, where we can also observe the monotonic divergence to $+\infty$ for
$\epsilon<E_{0}$. The other tail also diverges to $\pm\infty$ depending on the
parity of the order $m$.
In this instance, the nodes of Equation 15 are analytically known (see
derivation in Appendix B) as
$a_{\nu}=E_{0}+\frac{R}{2}\left(1-\cos\frac{\nu\pi}{m+1/2}\right).$ (16)
This allows us to decompose the $m$-th order projector into a product of $m$
linear projectors, each with their own weight that ensures $g^{\text{wall-
Ch}}_{m}(E_{0})=1$:
$g^{\text{wall-Ch}}_{m}(\epsilon)=\prod_{\nu=1}^{m}\frac{\epsilon-
a_{\nu}}{E_{0}-a_{\nu}}.$ (17)
A decomposition for a fifth order Chebyshev expansion of the wall function can
be seen in Figure 2.
Fig. 1: The Chebyshev expansions of the wall function in an arbitrary range of
$[-75,5]$, compared to the linear projector with the maximal time step of
$\delta\tau=0.025$, and its corresponding exponential projector. Fig. 2: The
fifth order Chebyshev expansion of the wall function, shown here to decompose
into a product of $5$ linear projectors, each with their own effective time
steps.
### 2.2 Application to FCIQMC and CCMC
In FCIQMC and CCMC, the lowest eigenvalue estimate is the shift, $S$, and the
upper spectral bound can be a constant, estimated from the Gershgorin circle
theorem 20 as
$\widetilde{E}_{N-1}=H_{N-1,N-1}+{{{\sum}^{\prime}_{{\bm{\mathrm{j}}}\in\\{\mathbf{S},\mathbf{D}\\}}}}H_{N-1,{\bm{\mathrm{j}}}},$
(18)
where the sum is over all determinants connected to the highest determinant
(singles and doubles), and the ‘′’ restricts it to ${\bm{\mathrm{j}}\neq
N-1}$.
The action of the wall-Chebyshev projector on
${|\Psi^{(n,0)}\rangle=[g^{\text{wall-Ch}}(\hat{H})]^{n}|\Phi\rangle}$ is
$|\Psi^{(n+1,0)}\rangle=g^{\text{wall-Ch}}(\hat{H})|\Psi^{(n,0)}\rangle,$ (19)
which gives the wavefunction after $n+1$ applications of the projector. We can
additionally define the ‘intermediate’ wavefunctions as
$|\Psi^{(n,\mu)}\rangle=\left[\prod_{\nu=0}^{\mu}\frac{\hat{H}-a_{\nu}}{S-a_{\nu}}\right]|\Psi^{(n,0)}\rangle.$
(20)
We are now ready to derive the update equations for FCIQMC and CCMC. We start
with the slightly more involved derivation for CCMC . Projecting these
intermediate wavefunctions onto determinants, we have
$\langle
D_{\bm{\mathrm{i}}}|\Psi^{(n,\nu+1)}\rangle=\frac{1}{S-a_{\nu}}\langle
D_{\bm{\mathrm{i}}}|{\hat{H}-a_{\nu}}|\Psi^{(n,\nu)}\rangle$ (21)
It is important now to distinguish between $\tilde{t}_{\bm{\mathrm{i}}}$, the
projection of a wavefunction onto determinant $D_{\bm{\mathrm{i}}}$, and the
corresponding excitor amplitude, $t_{\bm{\mathrm{i}}}$, with the former
including unconnected (‘composite’) contributions. At convergence,
$\displaystyle\begin{split}\tilde{t}_{\bm{\mathrm{i}}}=-\frac{1}{a_{\nu}-S}\left[\sum_{{\bm{\mathrm{j}}}\neq{\bm{\mathrm{i}}}}H_{\bm{\mathrm{ij}}}\tilde{t}_{\bm{\mathrm{j}}}+(H_{\bm{\mathrm{ii}}}-a_{\nu})\tilde{t}_{\bm{\mathrm{i}}}\right]&\\\
\tilde{t}_{\bm{\mathrm{i}}}-t_{\bm{\mathrm{i}}}+t_{\bm{\mathrm{i}}}=-\frac{1}{a_{\nu}-S}\left[\sum_{{\bm{\mathrm{j}}}\neq{\bm{\mathrm{i}}}}H_{\bm{\mathrm{ij}}}\tilde{t}_{\bm{\mathrm{j}}}+(H_{\bm{\mathrm{ii}}}-a_{\nu})\tilde{t}_{\bm{\mathrm{i}}}\right]&\\\
t_{\bm{\mathrm{i}}}=t_{\bm{\mathrm{i}}}-\frac{1}{a_{\nu}-S}\left[\sum_{\bm{\mathrm{j}}\neq\bm{\mathrm{i}}}H_{\bm{\mathrm{ij}}}\tilde{t}_{\bm{\mathrm{j}}}+(H_{\bm{\mathrm{ii}}}-S)\tilde{t}_{\bm{\mathrm{i}}}\right]&\end{split}$
(22)
We may now convert the last equation into an update step,
$t_{\bm{\mathrm{i}}}(\tau+\delta\tau)=t_{\bm{\mathrm{i}}}(\tau)-\frac{1}{a_{\nu}-S}\left[\sum_{\bm{\mathrm{j}}\neq\bm{\mathrm{i}}}H_{\bm{\mathrm{ij}}}\tilde{t}_{\bm{\mathrm{j}}}(\tau)+(H_{\bm{\mathrm{ii}}}-S)\tilde{t}_{\bm{\mathrm{i}}}(\tau)\right]$
(23)
Comparing with the original update equations, which are given by
$t_{\bm{\mathrm{i}}}(\tau+\delta\tau)=t_{\bm{\mathrm{i}}}(\tau)-\delta\tau\left[\sum_{\bm{\mathrm{j}}\neq\bm{\mathrm{i}}}H_{\bm{\mathrm{ij}}}\tilde{t}_{\bm{\mathrm{j}}}(\tau)+(H_{\bm{\mathrm{jj}}}-S)\tilde{t}_{\bm{\mathrm{i}}}(\tau)\right],$
(24)
we reach the conclusion that the necessary modifications are
1. 1.
Setting $\delta\tau=1$
2. 2.
Applying the $m$ constituent linear projectors in the $m$-th order wall-
Chebyshev projector. For linear projector ${\nu\in\\{1,\dots,m\\}}$, scale
Hamiltonian elements in spawning and death by $1/(a_{\nu}-S)$ (‘Chebyshev
weights’).
The same analysis can be performed on FCIQMC, without the complication of
composite amplitudes, to obtain a similar set of update equations:
$c_{\bm{\mathrm{i}}}=c_{\bm{\mathrm{i}}}-\frac{1}{a_{\nu}-S}\left[\sum_{\bm{\mathrm{j}}\neq\bm{\mathrm{i}}}H_{\bm{\mathrm{ij}}}c_{\bm{\mathrm{j}}}+(H_{\bm{\mathrm{ii}}}-S)c_{\bm{\mathrm{i}}}\right].$
(25)
In terms of implementation, the two sets of update equations are nearly
identical, and can share the same code in large parts.
Analysis of the asymptotic rate of convergence (see Appendix C) shows that the
theoretical speedup of an order $m$ wall-Chebyshev projector relative to the
linear projector with largest allowed $\delta\tau$ is $(m+1)/3$. 19 Due to
blooms, the largest $\delta\tau$ is never reached in the conventional
propagator, so real speedups are expected to be larger.
### 2.3 The shift update procedure
The original shift update equation for CCMC and FCIQMC is given by
$S^{(n+A)}=S^{(n)}-\frac{\zeta}{A\delta\tau}\ln\left(\frac{N_{\mathrm{w}}^{(n+A)}}{N_{\mathrm{w}}^{(n)}}\right),$
(26)
where the update is performed every $A$ time steps, $\zeta$ is the shift
damping parameter, and $N_{\mathrm{w}}$ is the total number of walkers.
Due to the time step $\delta\tau$ being set to unity, the shift update
procedure is expected to become rather unresponsive to the changes in particle
population. As a consequence, there can be vastly uncontained spawning,
unchecked by the lower-than-expected deaths, resulting in unmanageable
population growths. To remedy this, initially, a scaled update procedure was
experimented with, setting $A=1$:
$S^{(n+1)}=S^{(n)}-{\zeta}\sum_{\nu=1}^{\mu}[a_{\nu}^{(n)}-S^{(n)}]\ln\frac{N_{\mathrm{w}}^{(n,\nu)}}{N_{\mathrm{w}}^{(n,\nu-1)}},$
(27)
which seemed attractive as it reduces to Equation 21 in the first-order case
where the sum only contains one term or if all Chebyshev weights are the same.
However, this was not successful in reigning in the population growth. We
believe this is because the intermediate wavefunctions in Equation 20 are ill-
behaved due to being generated by an effective time step potentially larger
than $\tau_{\mathrm{max}}$. A series of population changes that start and end
at $N^{(n)}$ and $N^{(n+1)}$ respectively can produce very different values of
shift update in Equation 27, and the shift update produced is very sensitive
to the unreliable intermediate values. Hence the population information from
these intermediate wavefunctions should not be used.
Another procedure that was more successful was to decrease the damping (by
increasing $\zeta$) of the shift updates, causing the shift to be more
responsive to the changes in populations, which in turn helps stabilise the
population. We also found it helpful to use the improved shift update
procedure outlined in Ref. 21, where an additional term is added to the shift-
update procedure:
$S^{(n+A)}\leftarrow-\frac{\xi}{A\delta\tau}\ln\left(\frac{N_{\mathrm{w}}^{(n+A)}}{N_{t}}\right),$
(28)
where $\xi$ is the ‘forcing strength’, and $N_{t}$ is the target population.
This has the effect of additionally stabilising the population by ‘pinning’ it
to the pre-set target population. A further proposal from the same paper,
arising from an argument from a scalar model of population dynamics, is for
critical damping to be achieved by setting $\xi=\zeta^{2}/4$. This is also
found to be helpful. Altogether, these modifications result in greatly
improved population control and we were able to obtain dynamics that can be
used in a reblocking analysis, as shown in Figure 8, for example.
In practice, we have also found that with increasing order of Chebyshev
projector, a larger target population is usually needed, otherwise the
calculation may exhibit a sign-problem-like divergence. This may be attributed
to the larger effective time steps that the higher order projectors use and is
documented elsewhere, for example, see Fig. 2 in Ref. 22.
## 3 Accelerating the MR-CCMC algorithm
### 3.1 Efficient cluster acceptance algorithm
In the spawning step of the MR-CCMC algorithm, we check that a spawnee is
wihtin $P$ excitations of any secondary reference. The same check needs to be
performed in the death step. The original MR-CCMC algorithm performed a linear
scan through the list of secondary references, which is clearly a
$\mathcal{O}(n_{\mathrm{ref}})$ operation, where $n_{\mathrm{ref}}$ is the
number of secondary references. The subroutine that decides whether a spawn is
accepted is the second most frequently called subroutine in the program, after
the excitation generator. Therefore, a linear search in this step can quickly
become prohibitively expensive in a moderate to large model space
($n_{\mathrm{ref}}>1000$, for example). We note that traditional data
structures and search algorithms, such as a binary search on a sorted list of
secondary references, would not work here, as the definition of ‘distance’ in
this case, i.e., excitation-rank, is non-Euclidean. A search algorithm in a
general metric space is therefore needed.
The excitation-rank distance between two Slater determinants is equivalent to
half the Hamming distance between the bit strings representing these two
determinants, and the Hamming distance is a well-known example of a discrete
metric 23. A data structure, known as the BK tree 24, is particularly well
suited for efficient searches in discrete metric spaces. The tree, an example
of which is given in Figure 3, is constructed only once at the beginning of
the calculation. Subsequently, the search can be performed in
$\mathcal{O}(\log n_{\mathrm{ref}})$ time using a recursive tree traversal
algorithm. The tree construction and search algorithms are pictured in Figure
4.
Fig. 3: The BK tree can conduct efficient nearest neighbour searches in a
discrete metric space, like the excitation rank. In this figure a BK tree
built from $20$ arbitrary determinants is shown. The topology of the tree is
not unique, and is dependent on the order the nodes were added to the tree.
Fig. 4: Flowcharts for the building and searching of a BK-tree.
### 3.2 Compression of the model space
Whereas many classical multireference coupled cluster (MRCC) methods work with
complete active spaces (CAS), the MR-CCMC algorithm as presented here is
highly flexible as to the shape of the model space, and as such can be
considered a general model space (GMS) method 18. This enables us to consider
arbitrary subsets of the CAS as the model space, and fine-tune the balance
between cost and accuracy. One of us has devised a compression method in this
vein 25. Here we briefly summarise its main thrust.
Two of us observed that 17 that for some $(ne,no)$ active space, the results
of a MR-CCSD calculation using all of the determinants in this active space as
references (i.e., a CAS MR-CCSD calculation) is qualitatively very similar to
the results of a MR-CCSD…$m$ calculation, where $m=n/2$, using only the
‘bottom’ and ‘top’ determinants (i.e., the aufbau and anti-aufbau determinant
respectively) of the CAS as the references. We term the latter calculation as
‘2r-CCSD…$m$’. Using this observation, we aim to algorithmically generate only
those determinants that are in the Hilbert spaces of both the CAS MR-CCSD and
2r-CCSD…$m$ calculations, which should provide us with a compressed set of
reference determinants that captures the most important determinants in the
CAS. It was shown that this set of determinants can be generated by
enumerating determinants of up to $(m-2)$–fold excitations from the bottom and
top determinants.
## 4 Computational details
### 4.1 Basis sets and point group symmetries
In this work we will study the carbon and beryllium dimers. For these systems,
Dunning’s cc-pVXZ basis sets are used 26. The required electron integrals are
generated by the Psi4 27 and PySCF 28 packages. The electron integrals are
generated in the $D_{2h}$ point group symmetry and transformed into the basis
of $\hat{L}_{z}$ eigenfunctions based on the TransLz.f90 script provided in
the NECI package 29, which we re-wrote to interface with PySCF. Unless
otherwise noted, the core $1s$ electrons of C2 are frozen. The ‘heat bath’
excitation generator 30 is used whenever possible, otherwise the renormalised
excitation generator 31 is used.
The use of the $\hat{L}_{z}$ eigenfunctions helps not only further reduce the
size of the relevant symmetry sector, but also helps distinguish low-lying
states that would descend to the same irreducible representation in $D_{2h}$.
For C2, this would be the ${}^{1}\Sigma_{g}^{+}$ state and the
${}^{1}\Delta_{g}$ states, which both descend into the ${}^{1}A_{g}$ state in
$D_{2h}$. The two states approach and cross each other 32, 33, which would
prove challenging, if not impossible, to distinguish in $D_{2h}$.
When employing the wall-Chebyshev projector, the upper spectral range estimate
obtained from the Gershgorin theorem (Equation 18) is scaled up by $10\%$ by
default to guarantee an upper bound on the spectral width of the Hamiltonian.
Quantum Monte Carlo calculations are carried out using the HANDE-QMC
package34.
### 4.2 Symmetry screening
The full $(8e,8o)$ CAS model space of C2 contains $(^{8}C_{4})^{2}=4900$ model
space determinants that have $M_{s}=0$. However, like most classical quantum
chemistry programs, HANDE performs calculations strictly in the symmetry
sector requested by the user, which in our case is the totally symmetric
irreducible representation, unless otherwise specified. This brings into
question the effect, if any, of including secondary references outside the
symmetry sector under consideration. We first note that the secondary
references are only used to define the shape of the spawning/death space, and
are not required to be occupied. It is conceivable that there are
determinants/excitors in the correct symmetry sector that are included in the
spawning space only by virtue of being connected (but of course not coupled by
the Hamiltonian) to a wrong-symmetry secondary reference. To quantify the
effect of these wrong-symmetry secondary references, we implemented a simple
symmetry-screening switch in HANDE, which when turned on, only processes
secondary references of the currently considered symmetry.
## 5 Results and discussion
### 5.1 BK-tree
For a C2 system with a full $(8e,8o)$ CAS as the model space without symmetry
screening ($4900$ references that preserve the $M_{s}=0$ symmetry), the BK-
tree search is benchmarked against a naïve linear search, which loops over all
secondary references and terminates when one of the references is within $P$
excitations of the target determinant. The validity of the BK-tree search is
separately established by performing a normal calculation with either search
algorithm using the same random number generator seed, and asserting that the
results are the same. Benchmarking results are given in Table 1.
Table 1: Timing comparison between the BK tree and naive search algorithms, for C2 using a $(8e,8o)$ CAS as the model space for a multireference CCMCSD calculation. | Overall timing / s | Time per spawning attempt / $\mu$s
---|---|---
BK tree | $809.28$ | $12.761$
Linear | $5995.16$ | $94.533$
An apparent $8\times$ speedup is observed. Without performing a full profiling
study, the actual reduction in time cost of the acceptance search is expected
to be greater than $8\times$ as a complex series of operations is performed
per spawning attempt on top of the acceptance search.
### 5.2 Model space compression
For the $(8e,8o)$ CAS used for C2, the compression method discussed in Section
3.2 yields a total of $722$ determinants in the compressed model space. Here
we show the results for the C2/cc-pVDZ system at separations of $0.9$ to $1.5$
Å. The performance of the compression scheme is shown in Figure 5. Here we
have employed the default quasi-Newton acceleration35 implemented in HANDE. We
can see that despite the almost $7$-fold reduction in the model space (and
consequently a similar reduction in computational cost), the errors are within
chemical accuracy ($1.6$ m$E_{h}$).
Fig. 5: The correlation energy for C2/cc-pVDZ at $r_{\mathrm{C-C}}=0.9-1.5$Å
with the compressed set of $721$ secondary references, relative to the full
CAS model space. We observe that, despite a $7$-fold reduction in the size of
the model space, the reductions in the correlation energy captured are much
smaller, making this an attractive trade-off. The stochastic error bars are
too small to be seen, due to the use of the semi-stochastic algorithm 25.
### 5.3 Symmetry screening
Results with symmetry screening turned on for the same C2 system are shown in
Figure 6. We can see that, although statistically significant, including the
symmetry-forbidden secondary references only captures a negligible amount of
additional correlation energy. The full set of $4900$ references are reduced
to $635$ symmetry-allowed references, while the compressed set of $722$
references, whose results are not shown here, are reduced to $109$ references.
Both of these bring about a roughly $8$-fold reduction of the model space
size.
Fig. 6: The differences in energy between the full set of secondary
references, $E_{\mathrm{full}}$, and the set of symmetry-screened secondary
references, $E_{\mathrm{sym}}$, in units of m$E_{h}$. The very small error
bars are due to the semi-stochastic algorithm 25.
### 5.4 Binding curve of the carbon dimer
We studied the ${}^{1}\Sigma_{g}^{+}$ state of the carbon dimer in the cc-pVDZ
basis. The carbon dimer is a challenging test case for electronic structure
methods, and the challenge is two-fold: firstly, as mentioned in Section 4.1,
the ${}^{1}\Sigma_{g}^{+}$ state becomes very nearly degenerate with the
exceptionally low-lying $\Delta_{g}$ state at stretched bond lengths, and both
states descend to the $A_{g}$ state in the commonly used $D_{2h}$ point group
symmetry, rendering it very challenging to distinguish both states without the
use of the $L_{z}$ symmetry, with one paper resorting to tracking individual
CI coefficients 32; secondly, there is an abundance of avoided crossings, and
specifically, the first excited ${}^{1}\Sigma_{g}^{+}$ state participates in
an avoided crossing with the ground state at a bond length of around $1.6$ Å
36, resulting in a change in the most highly weighted diabatic state (i.e.,
determinant). This makes MR-CCMC calculations based on RHF orbitals exhibit
long-imaginary-time instabilities for stretched geometries, which preclude
obtaining accurate estimators. The binding curve presented in Figure 7 used
the full $(8e,8o)$ CAS as the model space for a MR-CCMCSD calculation, with
orbitals obtained using PySCF from an $(8e,8o)$ state-average CASSCF
calculation over the lowest three ${}^{1}A_{g}$ states (in the $D_{2h}$ point
group). The orbital coefficients are still tagged with their corresponding
$D_{\infty h}$ irreducible representations, enabling us to perform the $L_{z}$
transformation. We ensured that the $\pi_{u}$ manifold was included in the
reference determinant that generates the Hilbert space and secondary
references.
Fig. 7: The binding curve of the ${}^{1}\Sigma_{g}^{+}$ state of C2/cc-pVDZ in
the range of $0.9$-$2.8$ Å separation. MR-CCMC calculations are based on
CASSCF orbitals and use an $(8e,8o)$ CAS as a model space, with clusters
truncated at the double excitation level.The FCIQMC result is from Ref. 37,
and the DMRG results are from 32.
The non-parallelity errors (NPE), defined here as the difference between the
maximal and minimal deviation from the DMRG energies (which are interpolated
with a cubic spline using the scipy.interpolate.CubeSpline class) or the
FCIQMC energies, and a summary in shown in Table 2.
| NPE (m$E_{\mathrm{h}}$) | Max AD (m$E_{\mathrm{h}}$) | Min AD (m$E_{\mathrm{h}}$)
---|---|---|---
FCIQMC | 10.0 | 5.8 (2.8) | 0.7 (2.4)
DMRG | 10.4 | 7.1 (1.9) | 0.4 (2.5)
Table 2: The non-parallelity error, maximal, and minimal deviations of the
carbon dimer binding curve calculated with MR-CCMCSD using a $(8e,8o)$ CAS
model space, compared to FCIQMC and DMRG. The numbers in parenthesis indicate
the bond length (in angstrom) at which the maximal/minimal absolute deviations
occur.
### 5.5 Chebyshev Results
Figure 8 shows an example of the power of the Chebyshev propagation, applied
to an MR-CCMCSD calculation for C2. The shoulder height is reached after
around $50$ iterations, and the calculation is equilibriated essentially
instantaneously, which means all that is left to do is collecting statistics.
On an Intel(R) Xeon(R) E5-2650 v2 CPU, the calculation in the figure was run
for only $2$ hours on $6$ physical cores. Without the Chebyshev projector, the
same calculation takes around $24$ hours with $12$ physical cores to give the
same statistical error bar.
Fig. 8: A fifth order Chebyshev projector applied to MR-CCMCSD for C2/cc-pVDZ
at $1.2$ Å separation, with a full $(8e,8o)$ CAS as the model space, with a
target population of $2\times 10^{6}$ and a shift damping parameter of $0.5$,
as compared to a linear projector with $\delta\tau=0.001$, a target population
of $1\times 10^{6}$, and a shift damping parameter of $0.05$. Both
calculations are carried out with a two-stage harmonic forcing shift update
scheme. The inset shows that the projected energy only barely stabilises
around the true value at the end of the calculation using the linear
projector. This means that the reblocking algorithm will disregard essentially
all of the iterations shown here, due to the shift and the projected energy
not agreeing.
Part of the benefit of our Chebyshev propagator algorithm is the automatic
determination of the effective time steps, or ‘Chebyshev weights’, via the
Gershgorin circle theorem, which in the first-order expansion limit reduces to
an automatic way of choosing a good time step, instead of using trial and
error.
The following example shows Be2, a smaller, modestly multi-reference system,
treated in the cc-pVQZ basis set. Referring to Figure 9, compared with the
linear projector with a guessed $\delta\tau=0.002$, the second-order Chebyshev
expansion shows a clear speed-up in convergence. In fact, the Chebyshev
calculation took $66$ applications of the Hamiltonian to reach the target
population of $3\times 10^{6}$, whereas the linear projector took $3034$
applications to reach the same target population.
Fig. 9: The second order projector and the default linear projector at
$\delta\tau=0.002$ applied to MR-CCMCSD for the Be2/cc-pVQZ system at $2.5$ Å
separation, with a symmetry-screened $(4e,8o)$ (full $2s,2p$ valence) CAS as
the model space. Both calculations have a target population of $3\times
10^{6}$.
However, the second-order Chebyshev projector is expected to be as efficient
as the linear projector with the maximum allowed time step (see Section 2.2).
To provide a fairer comparison, we present in Figure 10 the first, second and
fourth order Chebyshev projector applied to the Be2/cc-pVQZ at $2.5$ Å system.
The first order Chebyshev projector is equivalent to a linear projector with
$\delta\tau=3/(E_{N-1}-E_{0})$, which is $2/3$ of $\delta\tau_{\text{max}}$,
but this maximal time step is commonly found to give rise to destabilising
population blooms, so the first order Chebyshev projector gives an estimate
for a practical time step to use.
Fig. 10: MR-CCMCSD for Be2/cc-pVQZ at $2.5$ Å separation with the $(4e,8o)$
CAS model space run with the first, second and fourth order Chebyshev
projectors. Only the time evolution of the shift parameters is shown for
clarity.
Here we can most clearly see the reduction in time needed to reach the
shoulder height. It is worth bearing in mind that the three calculations
require different target populations to stabilise, with the first and second
order projectors having target populations of $3\times 10^{6}$, and the fourth
order projector having a target population of $5\times 10^{6}$, which slightly
increases the number of iterations required to reach the target population.
An even clearer example of the relative time reduction is shown in Figure 11,
where we simulate the effect of the first order Chebyshev projector by
compressing the imaginary time evolution of the linear projector with
$\delta\tau=0.001$ in Figure 8 accordingly.
Fig. 11: C2/cc-pVDZ at $1.2$ Å separation, with a full $(8e,8o)$ CAS model
space, run with the first (simulated) and fifth order Chebyshev projectors.
The latter requires significantly fewer Hamiltonian applications to reach the
target population.
As a demonstration of the applicability of the Chebyshev projector in
different correlation regimes, we have computed the binding curve of the
Be2/cc-pVTZ system using a $(4e,8o)$ CAS as a model space for MR-CCMCSD, using
the fifth-order Chebyshev projector. CCSD(T) and semistochastic heat-bath
configuration interaction with second-order perturbation correction (SHCI-PT2)
38 results are also shown. The CCSD(T) results are from Psi4,27, and the SHCI-
PT2 results are generated using the Dice plug-in to the PySCF package28. For
SHCI-PT2, the full (8e, 60o) space was correlated, and hence it can be
considered a surrogate for FCI results. The variational threshold was set to
$\epsilon_{1}=8\times 10^{-5}$ $E_{\mathrm{h}}$, and the PT2 threshold was set
to $\epsilon_{2}=1\times 10^{-8}$ $E_{\mathrm{h}}$, using $N_{d}=200$
deterministic determinants, with 5 PT2 iterations. Figure 12 shows the binding
curves. The cc-pVTZ basis is known to severely overbind the beryllium dimer,39
compared to the experimental value of $934.9\pm 2.5$ cm-1.40, 41, 42
Notwithstanding, the MR-CCMC method using the Chebyshev projector was able to
provide consistently better description of the binding curve than the ‘gold-
standard’ CCSD(T). The MR-CCMC results are close to but not qualitatively the
same as FCI results near the equilibrium, where static correlation
dominates,43, and are near-identical to FCI results in the stretched region,
where dynamic correlation dominates, and a compact coupled-cluster
representation of the wavefunction is beneficial.
Fig. 12: The binding curve of the beryllium dimer using the cc-pVTZ basis,
computed using the CCSD(T), MR-CCMCSD with a $(4e,8o)$ CAS, and SHCI-PT2
methods.
## 6 Conclusion
We present here a series of algorithmic changes that can be used to accelerate
the MR-CCMC algorithm in particular and QMC algorithms in general.
Specific to the MR-CCMC algorithm, we have introduced a BK-tree-based search
algorithm to verify whether proposed clusters and spawns are within the
accepted manifold for a given reference space. This reduces the scaling of
this step from $\mathcal{O}(n_{\mathrm{ref}})$ to $\mathcal{O}(\log
n_{\mathrm{ref}})$, which translates to an $8\times$ speed-up for the
molecular systems studied. We have also designed a compression method for the
reference space, which preserves only what we expect to be the most
significant reference determinants. This decreases the size of the space by
close to an order of magnitude. Finally, we have shown that only including
references that belong to the same symmetry sector as the desired solution is
also effective as a means to reduce the size of the reference space, while
introducing only negligible additional error to the results.
We have also developed a new projector based on the Chebyshev polynomial
expansion of the wall function, which significantly accelerates the
convergence of QMC calculations. In an example calculation on the Be2
molecule, this reduced the number of Hamiltonian applications needed to reach
the target population by two orders of magnitude. The wall-Chebyshev projector
is generally applicable to different QMC algorithms, including FCIQMC and
(mr-)CCMC approaches, so we believe that, together with many recent
developments in increasing the apparent scarcity of the Hamiltonian and
optimising the shift behaviour,44, 45, 21it is a promising step in expanding
the range of applications for these methods.
## Author contributions
ZZ, MAF and AJWT all contributed to the conceptualization of the project. ZZ
developed the necessary software and analysed the data, under supervision from
MAF and AJWT. The manuscript was written by ZZ and MAF, with input from all
authors.
## Conflicts of interest
There are no conflicts to declare.
## Acknowledgements
ZZ was in part funded by the U.S. Department of Energy under grant DE-
SC0024532. MAF thanks Corpus Christi College, Cambridge and the Cambridge
Trust for a studentship, as well as Peterhouse for funding through a Research
Fellowship. This work used the ARCHER2 UK National Supercomputing Service
(https://www.archer2.ac.uk).
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## Appendix A Higher Taylor expansions of the exponential projector
In Ref. 19 it was proposed that there is no gain in going to higher order
Taylor expansions of the exponential projector, because all orders of
expansion have $\gamma=\tau$ (see Appendix C). The conclusion is correct, but
for a more subtle reason that we will now explain. The $m$-th order Taylor
series expansion of $g^{\text{exp}}$ is
$\sum_{k=0}^{m}\frac{1}{k!}(-\tau)^{k}(x-E_{0})^{k}.$ (29)
Eq. 6 in Ref. 19 requires that $g(E_{N-1})<1$, so, defining the spectral range
$R=E_{N-1}-E_{0}$, we have
$\left|\sum_{k=0}^{m}\frac{1}{k!}(-\tau)^{k}R^{k}\right|<1.$ (30)
The $m=1$ case leads to the familiar requirement in DMC, FCIQMC and CCMC that
$\tau_{\text{max}}<\frac{2}{E_{{N-1}}-E_{0}}$ (31)
and solving Equation 30 numerically shows that higher order expansions lead to
larger maximum allowed $\tau$. In fact, asymptotically, $\tau_{\text{max}}$
increases linearly with a gradient of $1/e\approx 0.368$ (see Figure 13). So,
naïvely we can expect the efficiency to increase linearly with $m$. To prove
the linearity, we note that $\tau_{\text{max}}R>1$, and we can approximate
$\sum_{k=0}^{m}|(-x)^{k}/{k!}|$ ($x\equiv\tau R$) with the leading order term
$|(-x)^{m}|/{m!}$, and so we are left with
$\displaystyle|(-x)^{m}|$ $\displaystyle<m!$ (32a) $\displaystyle m\ln x$
$\displaystyle\lessapprox m\ln m-m$ (32b) $\displaystyle x$
$\displaystyle<me^{-1},$ (32c)
where we used the Stirling’s formula in the second line.
Fig. 13: Calculated and fitted $\tau_{\text{max}}$ as a function of the order
of Taylor expansion of the exponential projector. There is indeed no gain
whatsoever in going to the second order Taylor expansion, but there is in
going to yet higher orders.
Therein lies the real reason for not using higher order expansions: a naive
implementation requires $m(m+1)/2$ applications of the Hamiltonian per
projection, and even a factorised implementation would require $m$
applications per projection, not to mention the lack of closed forms for the
roots of the $m$-th order expansion, due to the Abel-Ruffini theorem46, which
shows that no analytical solution can exist for $m\geq 5$. In any case, the
overall efficiency stays at best constant. Therefore, the conclusion that no
gains can be made is correct, although a more tortuous argument is needed.
## Appendix B Properties of the wall-Chebyshev projector
Assuming the entire spectral range is re-scaled such that $x\in[-1,1]$, where
$x=2(E-E_{0}))/R-1$, and $R=E_{N-1}-E_{0}$ is the spectral range or the
Hamiltonian, the Chebyshev expansion coefficients of the wall function is
$\begin{split}c_{k}=&\int_{-1}^{1}\text{wall}(x+1)T_{k}(x)(1-x^{2})^{-1/2}\dl
x\\\ \equiv&\int^{1}_{-1}\text{wall}(x+1)T_{k}(x)\delta(x+1)\dl x\\\
=&(2-\delta_{k0})T_{k}(-1)=(2-\delta_{k0})(-1)^{k},\end{split}$ (33)
where the second line uses the fact that the wall function is zero everywhere
but at the lower bound, so the weight function $(1-x^{2})^{-1/2}$ has the same
action as the delta function centred at $-1$, $\delta(x+1)$. The last equality
exploits a well-known identity of the Chebyshev polynomials 47. We can then
write the $m$-th order expansion as
$\begin{split}g^{\text{wall-
Ch}}_{m}(x)&=\frac{1}{1+2m}\sum_{k=0}^{m}(2-\delta_{k0})(-1)^{k}T_{k}(x)\\\
&=\frac{1}{1+2m}\sum_{k=0}^{m}(2-\delta_{k0})T_{k}(-x),\end{split}$ (34)
where the last equality exploits the fact that $T_{k}$ has the same parity as
$k$ 47, and we scale the sum such that $g^{\text{wall-Ch}}_{m}(-1)=1$.
To obtain an analytical expression for the zeroes of the wall-Chebyshev
projector, we use the trigonometric definition of the Chebyshev polynomials.
Inverting the sign of the argument in Equation 34, we have
$\begin{split}g^{\text{wall-
Ch}}_{m}(-\cos\theta)&\propto\sum_{k=0}^{m}(2-\delta_{k0})T_{k}(\cos\theta)\\\
&=1+2\left[\sum_{k=0}^{m}\cos(k\theta)\right]\\\
&=\frac{\sin\left[{\left(m+1/2\right)\theta}\right]}{\sin\left(\theta/2\right)},\end{split}$
(35)
where the last equality is the Dirichlet kernel 48. The zeroes of
$g^{\text{wall-Ch}}_{m}(x)$ are then transparently
$a_{\nu}=-\cos\left(\frac{\nu\pi}{m+1/2}\right),\ \nu=1,2,\dots,m,$ (36)
where the negative sign is from account for the sign inversion in Equation 35.
In an arbitrary spectral range other than $[-1,1]$, these zeroes are
$a_{\nu}=E_{0}+\frac{R}{2}\left(1-\cos\frac{\nu\pi}{m+1/2}\right).$ (37)
Knowing its zeroes, we can decompose $g^{\text{wall-Ch}}_{m}(x)$ into a
product of linear projectors:
$g^{\text{wall-Ch}}_{m}(x)=\prod_{\nu=1}^{m}\frac{x-a_{\nu}}{E_{0}-a_{\nu}},$
(38)
where the numerators ensure $g^{\text{wall-Ch}}_{m}(E_{0})=1$.
## Appendix C Convergence properties of generators
We summarise here some important properties of generators. The asymptotic rate
of convergence of a propagator is dominated by the slowest-decaying
eigencomponent:
$\mu=\lim_{n\rightarrow\infty}=\frac{||\Psi^{(n+1)}-\Psi_{0}||}{||\Psi^{(n)}-\Psi_{0}||}=\max_{i}\left|\frac{g(E_{i})}{g(E_{0})}\right|.$
(39)
Zhang and Evangelista 19 suggested that, in the common case that the first
excited state is the slowest-decaying component, and that the first excited
energy is small compared to the spectral range of $\hat{H}$, the above can be
approximated as
$\mu\approx\left|1+(E_{1}-E_{0})g^{\prime}(E_{0})\right|\equiv\left|1-\alpha\gamma\right|,$
(40)
where $\gamma=-g^{\prime}(E_{0})$ is the convergence factor for $g$. We now
derive the relation given in Ref. 49 that $\gamma$ is approximately the number
of times, $n$, that $g$ needs to be applied to achieve an error in the norm,
$\epsilon=||\Psi^{(n)}-\Psi_{0}||$ to the $N$-th decimal place:
$\displaystyle\epsilon=10^{-N}$ $\displaystyle\approx(1-\alpha\gamma)^{n}$
(41a) $\displaystyle 10^{-N}$ $\displaystyle\approx e^{-n\alpha\gamma}$ (41b)
$\displaystyle n$ $\displaystyle=\frac{N\ln 10}{\alpha\gamma}\equiv\kappa
N\cdot\frac{1}{\gamma},$ (41c)
where $\kappa=\ln 10/(E_{1}-E_{0})$ is the convergence prefactor, which is
inversely proportional to the first excited energy gap.
|
# EF-Calib: Spatiotemporal Calibration of
Event- and Frame-Based Cameras
Using Continuous-Time Trajectories
Shaoan Wang, Zhanhua Xin, Yaoqing Hu, Dongyue Li, Mingzhu Zhu, and Junzhi Yu
This work was supported in part by the National Natural Science Foundation of
China under Grant T2121002 and Grant 62233001, and in part by the Beijing
Natural Science Foundation under Grant 2022MQ05. _(Corresponding author:
Junzhi Yu.)_ Shaoan Wang, Zhanhua Xin, Yaoqing Hu, Dongyue Li, and Junzhi Yu
are with the State Key Laboratory for Turbulence and Complex Systems,
Department of Advanced Manufacturing and Robotics, College of Engineering,
Peking University, Beijing 100871, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>junzhi.yu@ia.ac.cn).Mingzhu Zhu is with the
Department of Mechanical Engineering, Fuzhou University, Fuzhou 350000, China
(e-mail: mzz@fzu.edu.cn).
###### Abstract
Event camera, a bio-inspired asynchronous triggered camera, offers promising
prospects for fusion with frame-based cameras owing to its low latency and
high dynamic range. However, calibrating stereo vision systems that
incorporate both event and frame-based cameras remains a significant
challenge. In this letter, we present EF-Calib, a spatiotemporal calibration
framework for event- and frame-based cameras using continuous-time
trajectories. A novel calibration pattern applicable to both camera types and
the corresponding event recognition algorithm is proposed. Leveraging the
asynchronous nature of events, a derivable piece-wise B-spline to represent
camera pose continuously is introduced, enabling calibration for intrinsic
parameters, extrinsic parameters, and time offset, with analytical Jacobians
provided. Various experiments are carried out to evaluate the calibration
performance of EF-Calib, including calibration experiments for intrinsic
parameters, extrinsic parameters, and time offset. Experimental results show
that EF-Calib achieves the most accurate intrinsic parameters compared to
current SOTA, the close accuracy of the extrinsic parameters compared to the
frame-based results, and accurate time offset estimation. EF-Calib provides a
convenient and accurate toolbox for calibrating the system that fuses events
and frames. The code of this paper will also be open-sourced at:
https://github.com/wsakobe/EF-Calib.
###### Index Terms:
Event camera, spatiotemporal calibration, continuous-time trajectory, time
offset estimation.
## I Introduction
Recent years, there has been a growing interest among researchers in a novel
bio-inspired camera called the event camera [1]. Abandoning the frame-
triggered concept of conventional cameras, each pixel of the event camera can
be considered as responding independently and asynchronously to changes in
illumination, resulting in an extremely low-latency and high-dynamic-range
response pattern. These advantages offer competitive prospects for event
cameras in areas such as robotics [2], autonomous driving [3], VR/AR [4], and
camera imaging [5].
Figure 1: Overview diagram of EF-Calib. (a) The novel calibration pattern
consists of the concentric circle and crosspoint. (b) The stereo vision system
consists of an event camera and a frame-based camera. (c) The calibration
process of EF-Calib.
However, due to the imaging principle of event cameras, they can only react to
changes in illumination, making it difficult to capture absolute amounts of
illumination and RGB values as frame-based cameras do. This limitation weakens
the ability of event cameras to perceive and understand the environment.
Therefore, many recent studies have attempted to fuse events with images in
order to fully utilize the unique advantages of both modalities, as
illustrated in Fig. 1. Some novel SLAM systems achieve more robust
localization under fast motion by fusing events and frames [2, 8, 7, 6]. In
addition, several studies are exploring how to fuse events and frames for
object detection in challenging environments [3, 9]. In recent years, some
event-centric datasets with multiple sensors, including frame-based cameras,
have also been widely proposed [12, 11, 10].
It is important to note that calibrating the intrinsic and extrinsic
parameters of each camera is an indispensable step in the context of multi-
camera fusion. Classical camera-to-camera calibration schemes typically
require time synchronization, followed by the acquisition of each camera’s
parameters through the synchronous acquisition of images of the calibration
board with different poses [13]. However, due to the asynchronous nature of
the event camera, it is difficult to combine multiple events into single
“frames” and time-synchronize them with an image. In addition, events are only
generated if there is relative motion, so the event camera cannot capture a
stationary calibration board. In summary, a new calibration framework must be
designed for the system including event- and frame-based cameras.
To address the aforementioned issues, this letter proposes a novel
spatiotemporal calibration framework for event- and frame-based cameras. To
the best of our knowledge, EF-Calib is the first calibration framework to
achieve joint calibration of event- and frame-based cameras without requiring
any time synchronization. The main contributions of this paper are as follows:
1. 1.
A novel spatiotemporal calibration framework for event- and frame-based
cameras is proposed. This framework can jointly obtain the intrinsic and
extrinsic parameters, as well as the time offset without requiring any
hardware synchronization.
2. 2.
Leveraging the asynchronous and low-latency properties of the event camera,
the framework introduces a continuous-time trajectory to optimize its motion
trajectory, facilitating arbitrary timestamp alignment with the frame-based
camera.
3. 3.
Extensive experiments are conducted in diverse scenarios to validate the
proposed calibration framework. The results indicate that the framework
achieves accuracy close to that of frame-based camera calibration methods and
consistently calibrates the time offset between the cameras.
The rest of the letter is organized as follows. Sec. II summarizes the related
works. Sec. III presents the preliminaries of event-based vision and
continuous-time trajectory. Sec. IV introduces the calibration framework. Sec.
V evaluates the calibration performance from different aspects. At last, Sec.
VI presents the conclusion of this letter.
## II Related Works
For geometric vision, camera calibration is particularly crucial as it serves
as the initial step in processing the input image signal, with the quality of
calibration often dictating the performance of subsequent tasks. Traditional
camera calibration has undergone significant evolution. The most prevalent
calibration method today involves capturing images of a calibration pattern
with a known size, such as a checkerboard, from various viewpoints to identify
corresponding feature points [13]. Subsequently, the intrinsic and distortion
parameters of each camera, as well as the extrinsic parameters between
cameras, are automatically calculated.
Nevertheless, applying this static and discrete calibration method to event
cameras, which are triggered by changes in illumination or relative motion,
presents challenges. Initially, many open-source event camera calibration
toolkits utilized a synchronized blinking LED calibration board with a known
size or a blinking checkerboard pattern generated by an LED screen [14, 15,
16]. This allowed the event camera to identify features similar to a
conventional camera and utilize traditional calibration methods. However,
these toolkits require complex device preparation and are unsuitable for
calibrating extrinsic parameters between event- and frame-based cameras.
In recent years, there has been increased focus on designing new calibration
frameworks to facilitate event camera calibration using existing calibration
boards. Muglikar et al. [17] utilize deep learning-based image reconstruction
networks, such as E2VID [18], to record events generated by moving the
calibration board and then apply the reconstructed images to classical
calibration methods. However, these methods heavily rely on the quality of
image reconstruction and face challenges in achieving time synchronization
with conventional cameras. Another approach involves directly utilizing events
generated during camera motion for camera calibration. Huang et al. [19]
proposed a calibration framework based on a circular calibration board and
employed B-splines to optimize the movement trajectory, which is the most
similar method to the one proposed in this letter. However, they directly use
clustered asynchronous events as features for optimization, compromising sub-
pixel accuracy and being highly sensitive to noise. Additionally, their focus
is solely on the intrinsic calibration of event cameras, without addressing
extrinsic calibration between event cameras and frame-based cameras. Salah et
al. [20] also utilize circular calibration boards and introduce eRWLS to fit
circular features with sub-pixel accuracy. However, they compress events over
a period into a fixed timestamp to obtain a reference “frame” making this
method challenging to synchronize with a conventional camera. Furthermore, it
does not account for the deformation of circular features at different viewing
angles, leading to reduced sub-pixel localization accuracy. The calibration
framework proposed in this letter continues this concept and provides an
improvement to address the problems of these methods.
## III Preliminaries
### III-A Event-Based Vision
Unlike conventional cameras, each pixel of the event camera is independently
triggered and responds to changes in the logarithmic illumination signal
$L(\mathbf{u}_{k},t_{k})$. An event $(\mathbf{u}_{k},t_{k},p_{k})$ is
triggered when the change in logarithmic illumination received by a pixel
$\mathbf{u}_{k}=(x_{k},y_{k})$ exceeds a threshold value $C$, i.e.
$\Delta L(\mathbf{u}_{k},t_{k})\doteq
L(\mathbf{u}_{k},t_{k})-L(\mathbf{u}_{k},t_{k}-\Delta t)=p_{k}C$ (1)
where $\Delta t$ is the time since the last triggered event by the same pixel,
$p_{k}\in\\{-\text{1},+\text{1}\\}$ is the polarity of the event.
### III-B Continuous-Time Trajectory Representation
Continuous-time trajectories are often represented utilizing a weighted
combination of the temporal basis functions [21], such as polynomial
functions, FFTs, and Bézier curves. In this letter, the uniform B-spline is
introduced as a representation of the continuous-time trajectory. B-splines
have the advantages of smoothness, local support, and analytic derivatives,
which are well-suited for representing the 6-DoF pose of the event camera [23,
22, 24]. Following the formulation of cumulative $k$th degree B-spline
$\mathcal{L}$, the event camera pose
$\mathbf{T}^{w}_{e}(\tau)\in\mathbb{SE}\text{3}$ at any time
$\tau\in[t_{i},t_{i+1})$, it can be represented by $N$ control points
$\mathbf{T}_{i}\in\mathbb{SE}\text{3},i\in[\text{0},\text{1},\ldots,N-\text{1}]$:
$\mathcal{L}:\mathbf{T}_{e}^{w}(\tau)=\mathbf{T}_{i}\cdot\prod_{j=\text{1}}^{k}\mathrm{Exp}\left(\tilde{\mathbf{B}}_{j}(\tau)\cdot\mathrm{Log}\left(\mathbf{T}_{i+j-1}^{-1}\mathbf{T}_{i+j}\right)\right)$
(2)
where $\tilde{\mathbf{B}}_{j}(\tau)$ is the cumulative basis function, which
is denoted by
$\tilde{\mathbf{B}}_{j}(\tau)=\tilde{\mathbf{M}}^{(k)}\mathbf{u}$ (3)
$\mathbf{u}=\begin{bmatrix}1&u&\cdots&u^{k}\end{bmatrix}^{T},u=\frac{\tau-
t_{i}}{t_{i+1}-t_{i}}$ (4)
where $\tilde{\mathbf{M}}^{(k)}\in\mathbb{R}^{(k+\text{1})\times(k+\text{1})}$
is the cumulative blending matrix of B-splines. Since the control points of
the B-splines are uniformly distributed on the time scale, the cumulative
blending matrix $\mathbf{M}^{(k)}$ is constant. In this letter, considering
the continuity and complexity, we use cubic B-splines to represent the camera
pose, i.e., $k=\text{3}$. The corresponding cumulative mixing matrix
$\tilde{\mathbf{M}}^{(3)}$ [25] is
$\tilde{\mathbf{M}}^{(\text{3})}=\dfrac{\text{1}}{\text{6}}\begin{bmatrix}\text{6}&\text{0}&\text{0}&\text{0}\\\
\text{5}&\text{3}&\text{-3}&\text{1}\\\
\text{1}&\text{3}&\text{3}&\text{-2}\\\
\text{0}&\text{0}&\text{0}&\text{1}\end{bmatrix}$ (5)
Commonly, to simplify the computation, some work decouples the rotation
$\mathbf{R}_{e}^{w}(\tau)\in\mathbb{SO}$3 and translation
$\mathbf{p}_{e}^{w}(\tau)\in\mathbb{R}^{\text{3}}$ into two independent cubic
B-splines, and the same process is carried out in this paper as well. Hence,
the continuous-time trajectory of the camera pose can be finally formulated as
$\mathbf{R}_{e}^{w}(\tau)=\mathbf{R}_{i}\cdot\prod_{j=\text{1}}^{3}\mathrm{Exp}\left(\tilde{\mathbf{B}}_{j}(\tau)\cdot\mathrm{Log}\left(\mathbf{R}_{i+j-1}^{-1}\mathbf{R}_{i+j}\right)\right)$
(6)
$\mathbf{p}_{e}^{w}(\tau)=\mathbf{p}_{i}+\sum_{j=\text{1}}^{3}\tilde{\mathbf{B}}_{j}(\tau)\cdot(\mathbf{p}_{i+j}-\mathbf{p}_{i+j-1})$
(7)
After decoupling the pose into two cubic B-splines, the corresponding analytic
derivatives [26] can also be derived
$\displaystyle\dot{\mathbf{R}_{e}^{w}}(\tau)$
$\displaystyle=\mathbf{R}_{e}^{w}(\tau)\cdot\left(\boldsymbol{\omega}^{(\text{3})}(\tau)\right)_{\wedge}$
(8)
$\displaystyle=\mathbf{R}_{i}\left(\dot{\mathbf{A}}_{\text{1}}\mathbf{A}_{\text{2}}\mathbf{A}_{\text{3}}+\mathbf{A}_{\text{1}}\dot{\mathbf{A}}_{\text{2}}\mathbf{A}_{\text{3}}+\mathbf{A}_{\text{1}}\mathbf{A}_{\text{2}}\dot{\mathbf{A}}_{\text{3}}\right)$
$\mathbf{v}_{e}^{w}(\tau)=\dot{\mathbf{p}}_{e}^{w}(\tau)=\mathbf{p}_{i}\cdot\sum_{j=\text{1}}^{\text{3}}\dot{\tilde{\mathbf{B}_{j}}}(\tau)\cdot(\mathbf{p}_{i+j}-\mathbf{p}_{i+j-1})$
(9)
where
$\mathbf{A}_{j}=\mathrm{Exp}\left(\tilde{\mathbf{B}}_{j}(\tau)\cdot\mathrm{Log}\left(\mathbf{R}_{i+j-1}^{-1}\mathbf{R}_{i+j}\right)\right)$
(10)
$\dot{\mathbf{A}}_{j}=\mathbf{A}_{j}\dot{\tilde{\mathbf{B}}}(\tau)_{j}\mathrm{Log}\left(\mathbf{R}_{i+j-1}^{-1}\mathbf{R}_{i+j}\right)$
(11)
$\dot{\tilde{\mathbf{B}_{j}}}(\tau)=\dfrac{1}{\Delta
t}\tilde{\mathbf{M}}^{(\text{3})}\begin{bmatrix}\text{0}\\\\[3.00003pt]
\text{1}\\\\[3.00003pt] \text{2}u\\\\[3.00003pt]
\text{3}u^{\text{2}}\end{bmatrix}$ (12)
Since this letter utilizes the uniform B-spline, $\Delta t$ is equal to the
time interval between any two consecutive knots, i.e. $\Delta
t=t_{i+1}-t_{i}$.
Figure 2: Flowchart of the proposed calibration framework
## IV Methodology
### IV-A Calibration Framework
For camera calibration, it is of utmost importance to accurately and robustly
identify features on the calibration pattern. However, the checkerboard
pattern [27], which is widely used, is difficult to apply to event cameras
because events disappear during parallel edge motion. Consequently,
calibration patterns for event cameras often employ circular features, but
these can produce blur in moving frames. To balance the characteristics of
event cameras and frame-based cameras, we designed a new calibration pattern
that combines isotropic circles and checkerboard crosspoints, as shown in Fig.
1(a). The center of each circle in this pattern coincides with the center of
the inner crosspoint. This hybrid pattern significantly enhances the
recognition efficiency and accuracy of the event camera while maintaining
compatibility with frame-based cameras.
Fig. 2 presents the flowchart of the proposed calibration framework. In this
letter, we divide the entire calibration process into two stages. The first
stage focuses on feature extraction and refinement of the calibration pattern.
The second stage is dedicated to optimizing the camera trajectory using
piecewise B-splines to achieve accurate calibration results. These two stages
will be elaborated in the following subsections.
### IV-B Event-Based Feature Recognizer
Unlike frame-based camera, event camera only output asynchronous events during
relative motion, posing a challenge for robust calibration pattern
recognition. To address this, we propose an event-based calibration pattern
feature recognizer, illustrated in Fig. 3. First, we accumulate events over a
short period of time $\Delta t$ according to their polarity to obtain
“accumulation frames” that resemble traditional images. The introduction of
this “accumulation frames” can help us to recognize the feature plate using
some classical image processing algorithms. The following subsections describe
the recognition algorithm based on “accumulation frames” in detail.
Figure 3: The pipeline of event-based feature recognizer.
#### IV-B1 Noise Suppression
Event cameras often generate a large number of events from the static
background when they are in motion. These noisy events can adversely affect
the recognition of calibration boards, significantly reducing the operation
speed and accuracy of the recognizer. Therefore, after obtaining the
“accumulation frames”, a noise suppression module is designed to filter out
most of the events that are not related to the calibration board.
For circular features, the triggered events typically consist of two
semicircular arcs connecting regions of opposite polarity. However, many
structures in the background have straight edges, making them more likely to
have connected regions that resemble straight lines. To leverage this
property, we introduce a fast and accurate connected component labeling (CCL)
algorithm called BBDT, proposed by Grana et al. [28]. This algorithm merges
neighboring events with the same polarity to obtain all the connectivity
regions. Then, the magnitudes of the two principal components of each
connected region are calculated using PCA. For background-triggered connected
regions, the magnitude of the second principal component
$\|\textbf{PC}_{\text{2}}\|$ should be much smaller than the magnitude of the
first principal component $\|\textbf{PC}_{\text{1}}\|$, so that a large number
of noisy regions can be suppressed by the principal component magnitude ratio
$\beta_{PC}$, as given that
$\beta_{PC}=\|\textbf{PC}_{\text{1}}\|/\|\textbf{PC}_{\text{2}}\|<T_{PC}$,
where $T_{PC}$ is a threshold for the $\beta_{PC}$, and any region with
$\beta_{PC}$ higher than $T_{PC}$ is suppressed and not involved in subsequent
operations.
#### IV-B2 Feature Extraction
Following noise suppression, we proceed to extract potential circular features
from the remaining regions. Specifically, we identify two candidate regions of
opposite polarity based on their distance. Subsequently, we fit the elliptic
equation using all pixels contained within these regions, exploiting the fact
that circular features adhere to the elliptic model under a projective
transform. Then the fitting error $e_{fit}$ is calculated, excluding candidate
regions with a fitting error exceeding the fitting threshold $T_{fit}$.
Furthermore, additional geometric constraints are needed to eliminate the
remaining false positive regions. Specifically, the two candidate regions
constituting the same ellipse should demonstrate similar PCA magnitudes;
regions with a notable discrepancy in PCA magnitude fail to meet this
criterion. Additionally, the contributions of the two candidate regions to the
circumference of the ellipse should be close. In other words, the angular
range $\theta_{r}$ of the two candidate regions with respect to the center of
the ellipse should be close to 180∘.
Regions that successfully meet these geometric constraints are recognized as
accurately representing the elliptical features within the calibration
pattern. Consequently, depending on the distribution of these elliptical
features, their relative positions on the calibration pattern can be decoded
to correspond with the results identified in the frame-based camera.
Figure 4: Schematic of the moving ellipse model. The set of events belonging
to the same elliptical feature can be considered as a three-dimensional
oblique elliptical cylinder.
#### IV-B3 Feature Refinement
Neglecting the timestamps of events during elliptical fitting inevitably
introduces errors, thereby affecting the camera calibration precision.
However, by regarding the timestamps $t$ as a third dimension alongside the
pixel coordinates $[x,y]^{T}$, events can be conceptualized as points within a
three-dimensional space. Each “accumulation frames”, corresponding to a brief
time interval, allows for the assumption of solely translational movement with
speed $[v_{x},v_{y}]^{T}$, parallel to the pixel plane, for each elliptical
feature at any given moment within this interval, as demonstrated in Fig. 4.
Consequently, the moving ellipse model $\mathcal{F}$ is described by the
following representation:
$\mathcal{F}:ax(t)^{2}+\beta x(t)y(t)+\gamma y(t)^{2}+\delta x(t)+\epsilon
y(t)+\zeta=\text{0}$ (13)
In matrix form,
$\begin{cases}\mathcal{F}:\mathbf{P}(t)^{T}\mathbf{Q}\mathbf{P}(t)=\text{0}\\\
\mathbf{P}(t)=\mathbf{P}(t_{0})-\mathbf{V}\cdot(t-t_{0})\\\
\mathbf{Q}=\begin{pmatrix}\alpha&\beta/\text{2}&\delta/\text{2}\\\
\beta/\text{2}&\gamma&\epsilon/\text{2}\\\
\delta/\text{2}&\epsilon/\text{2}&\zeta\end{pmatrix}\end{cases}$ (14)
where $\mathbf{P}(t)=[x(t),y(t),\text{1}]^{T}$,
$\mathbf{V}=[v_{x},v_{y}]^{T}$, $t\in[t_{0},t_{0}+\text{2}\delta_{t}]$, and
$t_{0}$ represents the starting time of current “accumulation frame”. Here the
cost function for model optimization is defined as
$\arg\min_{\mathcal{F},\mathbf{V}}\sum_{i\in\\{e\\}}\left(\|\mathbf{P}_{i}^{T}\mathbf{Q}\mathbf{P}_{i}\|^{1}\right)$
(15)
By substituting the events into the aforementioned model, we utilize the
Levenberg-Marquardt algorithm to solve the model, thereby refining the
elliptical features.
### IV-C Trajectory Optimization
Figure 5: Schematic of a piece-wise B-spline trajectory. The number of event
features inside the red box is insufficient; therefore, this segment of the
trajectory is omitted.
The refinement process converts the previously discrete elliptical features
into densely populated patterns within the corresponding time period of
associated events. This densely populated feature facilitates the optimization
of event camera poses within a continuous-time trajectory representation.
Nevertheless, maintaining continuous visibility of the entire calibration
plate during the calibration process poses a challenge. Incomplete calibration
patterns in a continuous event stream often result in unsuccessful
recognition. To mitigate this issue, the continuous-time trajectory is divided
into multiple segments based on the output from the recognizer. Consequently,
a piece-wise B-spline-based optimizer for event camera pose trajectories is
proposed.
First, based on the predefined knot interval $\Delta t$ and the results of the
recognizer, the features whose timestamps differ from the timestamps of other
features by more than $\Delta t$ are eliminated. In addition, segments
containing too few features are also excluded to ensure optimization accuracy,
and only the more desirable segments are preserved. Fig. 5 illustrates the
segmentation process of the trajectory. The entire calibration process
$\Lambda_{\mathcal{L}}$ is divided into a combination of $M$ segments of
trajectories $\mathcal{L}$:
$\Lambda_{\mathcal{L}}=\sum_{m}\\{\mathcal{L}_{m}\\}_{a_{m}}^{b_{m}},m=\text{1},\ldots,M$
(16)
where $a_{m}$ and $b_{m}$ are the starting and ending times corresponding to
the $m$th segment of B-splines, respectively. Each segment of B-splines is
optimized by only the features whose timestamps belong to its time period.
Figure 6: Schematic of sampling on a continuously moving ellipse model.
For $\mathcal{L}_{m}$, the corresponding state vector $\mathcal{X}_{e}$ is:
$\mathcal{X}_{e}=[\xi^{\text{1}}_{m},\ \xi^{\text{2}}_{m},\ \cdots,\
\xi^{N_{m}}_{m},\ K_{e},\ D_{e}]$ (17)
where $\xi_{m}^{i}$ is the control point of $\mathcal{L}_{m}$, $N_{m}$ is the
number of control points, and $K_{e}$ and $D_{e}$ are the intrinsic and
distortion parameters of the event camera. The corresponding visual residual
for the $i$th feature based on reprojection error is defined as:
$\displaystyle\mathbf{r}_{e}(\mathcal{X}_{e})=\sum_{k\in\mathcal{K}}\pi_{e}(\mathbf{R}_{w}^{e}(t_{i}+k\cdot\delta_{t_{i}})P_{i}^{w}+\mathbf{t}_{w}^{e}(t_{i}+k\cdot\delta_{t}))$
(18) $\displaystyle-(\begin{bmatrix}u_{i}^{e}\\\
v_{i}^{e}\end{bmatrix}+k\cdot\delta_{t}\mathbf{V}_{i})$
where $\mathbf{R}^{e}_{w}(\cdot)$ and $\mathbf{t}^{e}_{w}(\cdot)$ are derived
from the B-spline trajectory using equations (6) and (7), respectively. Since
the feature refinement yields a continuous moving ellipse model, the residuals
can be constructed by sampling the model at any time. Here, $\mathcal{K}$
denotes the partition of $\delta_{t_{i}}$, which defines the sampling interval
of the feature, as shown in Fig. 6. The function $\pi_{e}(\cdot)$ projects the
spatial point $P_{i}^{w}$ onto the “accumulation frame”.
The intrinsic and distortion parameters of the event camera, along with the
control points of the splines, are jointly optimized by minimizing the
following cost function:
$\arg\min_{\mathcal{X}_{e}}\left\\{\sum\rho(\|\mathbf{r}_{e}(\mathcal{X}_{e})\|^{2})\right\\}$
(19)
where $\rho(\cdot)$ is the Huber loss function.
### IV-D Spatialtemporal Calibration
The final step involves jointly optimizing the event camera and the frame-
based camera, utilizing the previously optimized trajectories, to determine
the extrinsic parameters and time offset. The corresponding state vector
$\mathcal{X}_{f}$ is:
$\mathcal{X}_{f}=[K_{f},\ D_{f},\ \mathbf{T}^{f}_{e},\ t_{d}]$ (20)
where $\mathbf{T}^{f}_{e}$ is the transformation matrix between the two
cameras and $t_{d}$ is the difference between the real timestamps of the two
cameras, i.e., the time offset.
Similarly, define the visual residuals based on reprojection errors in
spatiotemporal calibration as:
$\mathbf{r}_{f}(\mathcal{X}_{f})=\pi_{f}\left(\mathbf{R}^{f}_{e}\left(\mathbf{R}^{e}_{w}(t_{i}+t_{d})P^{w}_{i}+\mathbf{t}^{e}_{w}(t_{i}+t_{d})\right)+\mathbf{t}^{f}_{e}\right)-\begin{bmatrix}u^{f}_{i}\\\\[6.0pt]
v^{f}_{i}\end{bmatrix}$ (21)
where $\pi_{e}(\cdot)$ projects the spatial points onto the image plane of the
frame-based camera.
From (21), the Jacobian $J_{t_{d}}$ of $\mathbf{r}_{f}$ w.r.t $t_{d}$ can be
obtained by the chain rule:
$J_{t_{d}}=\frac{\partial\mathbf{r}_{f}}{\partial P_{i}^{f}}\frac{\partial
P_{i}^{f}}{\partial t_{d}}=\frac{\partial\mathbf{r}_{f}}{\partial
P_{i}^{e}}(\frac{\partial
P_{i}^{f}}{\partial\mathbf{R}_{w}^{e}}\frac{\partial\mathbf{R}_{w}^{e}}{\partial
t_{d}}+\frac{\partial
P_{i}^{f}}{\partial\mathbf{t}_{w}^{e}}\frac{\partial\mathbf{t}_{w}^{e}}{\partial
t_{d}})$ (22)
where
$P_{i}^{f}=\mathbf{R}^{f}_{e}\left(\mathbf{R}^{e}_{w}(t_{i}+t_{d})P^{w}_{i}+\mathbf{t}^{e}_{w}(t_{i}+t_{d})\right)+\mathbf{t}^{f}_{e}$.
Based on (8), (9), and (22), the structure of $\partial P_{i}^{f}/\partial
t_{d}$ can be derived straightforwardly:
$\frac{\partial
P_{i}^{f}}{\partial\mathbf{R}_{w}^{e}}\frac{\partial\mathbf{R}_{w}^{e}}{\partial
t_{d}}=\mathbf{R}^{f}_{e}\dot{\mathbf{R}}^{e}_{w}(t_{i}+t_{d})P^{w}_{i}+\mathbf{R}^{f}_{e}\mathbf{v}^{e}_{w}(t_{i}+t_{d})$
(23)
To jointly optimize the intrinsic and extrinsic parameters of the frame-based
camera, as well as the time offset, the following cost function is minimized:
$\arg\min_{\mathcal{X}_{f}}\left\\{\sum\rho(\|\mathbf{r}_{f}(\mathcal{X}_{f})\|^{2})\right\\}$
(24)
Notably, the optimization problems in (15), (19), and (24) are solved using
Google Ceres111https://ceres-solver.org/.
## V Experiments
In this section, several experiments are conducted to evaluate the performance
of EF-Calib, encompassing intrinsic calibration test, extrinsic calibration
test, and time offset calibration test. Additionally, an ablation study is
conducted to evaluate the contribution of several key modules within EF-Calib.
### V-A System Setup
A real-world stereo vision system was designed as Fig. 1(b) shows. It contains
an event camera and a frame-based camera. The two cameras are integrated by a
slide, on which the baseline and viewing angle can be arbitrarily changed to
test the calibration performance of EF-Calib in different situations
comprehensively. The event camera utilized in this letter is the Inivation
DAVIS 346, featuring a resolution of 346$\times$260 and a maximum temporal
resolution of 1 µs. Additionally, this type of camera can also generate
regular frames at a frequency of 30 Hz under standard illumination conditions.
This configuration can be readily employed to compare EF-Calib with a high-
quality, frame-based calibration pipeline, such as the OpenCV calibration
toolbox [29]. The frame-based camera employed is the HikVision MV-CE013-80UM
industrial camera with a global shutter and a resolution of 1280$\times$1024
pixels. Note that no hardware synchronization was utilized in the stereo
vision system. This deliberate choice was made to provide a more rigorous
evaluation of the calibration capability of EF-Calib under real-world
conditions.
### V-B Calibration Experiments
The calibration performance of a stereo vision system is usually affected by
the camera baseline and viewing angle. To fully evaluate the calibration
performance of EF-Calib, we conducted calibration experiments in three
settings and analyzed the corresponding calibration results separately. In the
first setting (Trial 1), the cameras are configured for a regular baseline. In
the second setting (Trial 2), the cameras are configured for a wide baseline.
In the third setup (Trial 3), the cameras are configured as a narrow baseline.
For each trial, the cameras are adjusted to obtain a reasonable viewing angle,
ensuring sufficient overlap of the camera field of view. For each trial,
images with event data are recorded simultaneously for sufficient time (about
40 s) to achieve converged calibration results.
#### V-B1 Intrinsic Calibration Test
We utilized only the event stream data from the above three trials and
completed the intrinsic calibration of the event camera respectively.
Previously, we completed the intrinsic calibration using OpenCV toolbox [29]
with the frame provided by DAVIS 346 and considered this calibration result as
the ground truth. In addition, EF-Calib is compared with two state-of-the-art
event camera intrinsic calibration methods [17, 20]. Note that the calibration
patterns used by the compared methods are the ones originally employed by
them: [17] utilizes a checkerboard pattern, while [20] employs an asymmetric
circular pattern.
Table I shows the intrinsic calibration results of each method. It can be seen
that the intrinsic parameter obtained by our method is closest to the ground
truth (GT) and the results corresponding to three trials are very stable. In
addition, Fig. 7 illustrates the plot of the intrinsic parameters over time.
It can be noticed that EF-Calib can get converged results in less than 20 s,
demonstrating the ease of use of our method.
Figure 7: Results of the intrinsic calibration test. TABLE I: Comparative Results of Intrinsic Calibration Test Methods | $f_{x}$ | $f_{y}$ | $c_{x}$ | $c_{y}$ | $k_{\text{1}}$ | $k_{\text{2}}$ | RPE
---|---|---|---|---|---|---|---
Frame-based (GT) | 413.84 | 413.80 | 157.42 | 132.25 | -0.38 | 0.31 | 0.13
E2Calib [17] | 417.56 | 417.24 | 159.86 | 132.35 | -0.36 | 0.09 | 0.41
E-Calib [20] | 404.66 | 403.99 | 159.81 | 132.69 | -0.37 | 0.32 | 0.33
EF-Calib (Trial 1) | 414.06 | 413.28 | 158.03 | 132.43 | -0.38 | 0.31 | 0.10
EF-Calib (Trial 2) | 414.79 | 413.85 | 158.74 | 131.90 | -0.38 | 0.34 | 0.12
EF-Calib (Trial 3) | 414.87 | 413.94 | 159.16 | 133.65 | -0.37 | 0.31 | 0.13
#### V-B2 Extrinsic Calibration Test
To evaluate the extrinsic calibration performance of the EF-Calib, we
calculated the errors corresponding to rotation and translation separately for
each trial, i.e.
$\displaystyle e_{t}$
$\displaystyle=\frac{1}{N}\sum_{i}^{N}\left\|\mathbf{t}^{e}_{w}(t_{i}+t_{d})-\mathbf{T}_{f}^{e}{\mathbf{t}^{f}_{w}}_{i}\right\|_{2}$
(25) $\displaystyle e_{r}$
$\displaystyle=\frac{1}{N}\sum_{i}^{N}\|\boldsymbol{\theta}(\mathbf{R}_{w}^{e}(t_{i}+t_{d}))-\boldsymbol{\theta}(\mathbf{R}_{f}^{e}{\mathbf{R}_{w}^{f}}_{i})\|_{2}$
where $N$ is the frame number and $\boldsymbol{\theta}(\cdot)$ represents the
Euler angle corresponding to the rotation matrix. Similarly to the intrinsic
calibration test, we also acquired 30 pairs of images containing a
checkerboard calibration board from the two cameras in different poses
simultaneously. These frames were calibrated using the OpenCV toolbox to
obtain the corresponding extrinsic parameters for both cameras. The computed
EF-Calib calibration errors were compared with the corresponding errors of the
frame-based extrinsic parameter calibration. From Table II, it can be seen
that EF-Calib can achieve the same level of error as the frame-based extrinsic
calibration, verifying its effectiveness in extrinsic calibration.
TABLE II: Comparative Results of Extrinsic Calibration Test Trial | Method | $e_{t}$ (mm) | $e_{r}$ (∘) | Frames
---|---|---|---|---
Trial 1 | Frame-based | 0.3499 | 0.0918 | 30
EF-Calib | 0.5336 | 0.1984 | 250
Trial 2 | Frame-based | 0.4199 | 0.1854 | 30
EF-Calib | 0.6572 | 0.3127 | 207
Trial 3 | Frame-based | 0.2541 | 0.0954 | 30
EF-Calib | 0.3638 | 0.2911 | 184
#### V-B3 Time Offset Calibration Test
Time offset estimation is crucial for multi-camera systems that lack hardware
triggering. In this test, we evaluate EF-Calib’s capability to calibrate time
offsets by manually adjusting the timestamp of each image frame. Specifically,
we compare the differences between the time offset calibration results
obtained with modified timestamps and the original results. This is achieved
by uniformly delaying or advancing the timestamps by a fixed time interval
$\Delta t_{d}$. Specifically, the timestamps were modified by $\pm$2.5 ms and
$\pm$5 ms for each trial, and the difference between the modified time delay
and the original time delay was calculated and compared to the delta. Fig. 8
illustrates the experimental results of the time offset calibration test, and
EF-Calib can accurately calibrate the time offset at different scenarios.
Figure 8: Results of the time offset calibration test. TABLE III: Ablation Study of EF-Calib Piece-wise trajectory | Temporal calibration | Feature refinement | Trial 1 | Trial 2 | Trial 3
---|---|---|---|---|---
$e_{t}$ | $e_{r}$ | $e_{t}$ | $e_{r}$ | $e_{t}$ | $e_{r}$
| | | 8.1633 | 2.8632 | 4.6642 | 2.6372 | 1.2161 | 1.4511
✔ | | | 8.0019 | 2.8187 | 3.8794 | 2.7859 | 0.4974 | 1.4322
✔ | ✔ | | 0.9141 | 5.6910 | 0.6322 | 0.3239 | 0.3707 | 0.3065
✔ | ✔ | ✔ | 0.5336 | 0.1984 | 0.6572 | 0.3127 | 0.3638 | 0.2911
### V-C Ablation Study
To thoroughly analyze and validate the performance and functionality of each
module within EF-Calib, we conducted an ablation study. Specifically, we
scrutinized three modules: piece-wise trajectories, temporal calibration, and
feature refinement. Table III demonstrates the impact of the introduction of
these three modules on the calibration error of the EF-Calib extrinsic
parameters. As can be seen from Table III, the introduction of all three
modules can greatly improve the extrinsic parameter calibration accuracy.
## VI Conclusion
In this letter, we propose a spatiotemporal calibration framework called EF-
Calib, aiming to achieve joint calibration of intrinsic parameters, extrinsic
parameters, and time offset for event and frame-based cameras. First, we
design a novel calibration pattern that accommodates the heterogeneous nature
of event and frame-based representations, incorporating both circles and
crosspoints to facilitate simultaneous recognition by both camera types. A
corresponding event-based recognition algorithm is developed to ensure robust
and accurate feature recovery using this pattern. Additionally, to manage the
asynchronous characteristics of the event stream, we introduce a piece-wise
B-spline to continuously represent the pose trajectory of the event camera.
Finally, we provide the analytic Jacobian of the error term and implement the
joint calibration of intrinsic, extrinsic, and time offset for both camera
types. Experimental results demonstrate that EF-Calib outperforms current
state-of-the-art methods in intrinsic parameter estimation while also
achieving high accuracy in extrinsic parameter and time offset estimation.
These results demonstrate the spatiotemporal calibration capabilities of EF-
Calib and lay a robust foundation for the fusion of event and frame.
In the future, we aim to explore markerless online calibration based on EF-
Calib. Additionally, we plan to utilize EF-Calib to create novel visual
perception frameworks that fuse events and frames.
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